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exists a. ((((exists ff_h_pfp_shift_bound_chosen. ff_h_pfp_shift_bound_chosen + S (a) = S ((S (i)) * c)) /\\ exists ff_q_pfp_shift_bound_chosen. b = ff_q_pfp_shift_bound_chosen * S ((S (i)) * c) + (a))) /\\ ((exists pfa_gap_shift_bound_value. pfa_gap_shift_bound_value + S (a) = (p))))", "specialize hc (i)", "apply hc", "exact ho_right", "cases ha", "cases ha_witness", "exists x", "split", "specialize hs_left (i)", "specialize hs_left (x)", "apply hs_left", "exact ho_right", "exact ha_witness_left", "exact ha_witness_right" ], "script_sha256": "6744eb2edc84481d8344376c06d87266d8b6846cc137aa135847a7ec37950db6", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_shift_candidate.py", "sha256": "325d3085482ee73a2c6ee90cd17e45cffe53273671edf89c40d88428335c9c4b" }, "statement": "forall p b c L d e. (~((p) = 1) /\\ forall pfa_factor_left_shift_bound_prime pfa_factor_right_shift_bound_prime. (p) = pfa_factor_left_shift_bound_prime * pfa_factor_right_shift_bound_prime -> pfa_factor_left_shift_bound_prime = 1 \\/ pfa_factor_right_shift_bound_prime = 1) -> (forall fom_index_pfp_shift_bound_source. (exists fom_gap_pfp_shift_bound_source_index_bound. fom_gap_pfp_shift_bound_source_index_bound + S (fom_index_pfp_shift_bound_source) = L) -> exists fom_value_pfp_shift_bound_source. ((((exists fom_beta_height_pfp_shift_bound_source_entry. fom_beta_height_pfp_shift_bound_source_entry + S (fom_value_pfp_shift_bound_source) = S ((S (fom_index_pfp_shift_bound_source)) * c)) /\\ exists fom_beta_quotient_pfp_shift_bound_source_entry. b = fom_beta_quotient_pfp_shift_bound_source_entry * S ((S (fom_index_pfp_shift_bound_source)) * c) + (fom_value_pfp_shift_bound_source))) /\\ (exists fom_gap_pfp_shift_bound_source_value_bound. fom_gap_pfp_shift_bound_source_value_bound + S (fom_value_pfp_shift_bound_source) = p))) -> (((forall mdr_i_pfp_shift_bound_relationprefix mdr_a_pfp_shift_bound_relationprefix. (exists mdr_gap_pfp_shift_bound_relationprefixb. mdr_gap_pfp_shift_bound_relationprefixb + S (mdr_i_pfp_shift_bound_relationprefix) = (L)) -> (((exists ff_h_mdr_pfp_shift_bound_relationprefixo. ff_h_mdr_pfp_shift_bound_relationprefixo + S (mdr_a_pfp_shift_bound_relationprefix) = S ((S (mdr_i_pfp_shift_bound_relationprefix)) * c)) /\\ exists ff_q_mdr_pfp_shift_bound_relationprefixo. b = ff_q_mdr_pfp_shift_bound_relationprefixo * S ((S (mdr_i_pfp_shift_bound_relationprefix)) * c) + (mdr_a_pfp_shift_bound_relationprefix))) -> (((exists ff_h_mdr_pfp_shift_bound_relationprefixn. ff_h_mdr_pfp_shift_bound_relationprefixn + S (mdr_a_pfp_shift_bound_relationprefix) = S ((S (mdr_i_pfp_shift_bound_relationprefix)) * e)) /\\ exists ff_q_mdr_pfp_shift_bound_relationprefixn. d = ff_q_mdr_pfp_shift_bound_relationprefixn * S ((S (mdr_i_pfp_shift_bound_relationprefix)) * e) + (mdr_a_pfp_shift_bound_relationprefix)))) /\\ ((((exists ff_h_pfp_shift_bound_relationlast. ff_h_pfp_shift_bound_relationlast + S (0) = S ((S (L)) * e)) /\\ exists ff_q_pfp_shift_bound_relationlast. d = ff_q_pfp_shift_bound_relationlast * S ((S (L)) * e) + (0)))))) -> (forall fom_index_pfp_shift_bound_result. (exists fom_gap_pfp_shift_bound_result_index_bound. fom_gap_pfp_shift_bound_result_index_bound + S (fom_index_pfp_shift_bound_result) = S L) -> exists fom_value_pfp_shift_bound_result. 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(((exists ff_h_pfp_shift_unique_source_value. ff_h_pfp_shift_unique_source_value + S (x) = S ((S (i)) * c)) /\\ exists ff_q_pfp_shift_unique_source_value. b = ff_q_pfp_shift_unique_source_value * S ((S (i)) * c) + (x)))", "specialize beta_at_exists (b)", "specialize beta_at_exists (c)", "specialize beta_at_exists (i)", "apply beta_at_exists", "cases hx", "have heq : a=x", "specialize beta_at_unique (d)", "specialize beta_at_unique (e)", "specialize beta_at_unique (i)", "specialize beta_at_unique (a)", "specialize beta_at_unique (x)", "apply beta_at_unique", "exact ha", "specialize hd_left (i)", "specialize hd_left (x)", "apply hd_left", "exact ho_right", "exact hx_witness", "rewrite heq", "rewrite heq", "specialize hf_left (i)", "specialize hf_left (x)", "apply hf_left", "exact ho_right", "exact hx_witness" ], "script_sha256": "3e54040442916d577258f6cb93224204e4754ba9dd062bcb9aa28dea757ce0b9", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_shift_candidate.py", "sha256": "325d3085482ee73a2c6ee90cd17e45cffe53273671edf89c40d88428335c9c4b" }, "statement": "forall b c L d e f g. (((forall mdr_i_pfp_shift_unique_firstprefix mdr_a_pfp_shift_unique_firstprefix. (exists mdr_gap_pfp_shift_unique_firstprefixb. mdr_gap_pfp_shift_unique_firstprefixb + S (mdr_i_pfp_shift_unique_firstprefix) = (L)) -> (((exists ff_h_mdr_pfp_shift_unique_firstprefixo. ff_h_mdr_pfp_shift_unique_firstprefixo + S (mdr_a_pfp_shift_unique_firstprefix) = S ((S (mdr_i_pfp_shift_unique_firstprefix)) * c)) /\\ exists ff_q_mdr_pfp_shift_unique_firstprefixo. b = ff_q_mdr_pfp_shift_unique_firstprefixo * S ((S (mdr_i_pfp_shift_unique_firstprefix)) * c) + (mdr_a_pfp_shift_unique_firstprefix))) -> (((exists ff_h_mdr_pfp_shift_unique_firstprefixn. ff_h_mdr_pfp_shift_unique_firstprefixn + S (mdr_a_pfp_shift_unique_firstprefix) = S ((S (mdr_i_pfp_shift_unique_firstprefix)) * e)) /\\ exists ff_q_mdr_pfp_shift_unique_firstprefixn. d = ff_q_mdr_pfp_shift_unique_firstprefixn * S ((S (mdr_i_pfp_shift_unique_firstprefix)) * e) + (mdr_a_pfp_shift_unique_firstprefix)))) /\\ ((((exists ff_h_pfp_shift_unique_firstlast. ff_h_pfp_shift_unique_firstlast + S (0) = S ((S (L)) * e)) /\\ exists ff_q_pfp_shift_unique_firstlast. d = ff_q_pfp_shift_unique_firstlast * S ((S (L)) * e) + (0)))))) -> (((forall mdr_i_pfp_shift_unique_secondprefix mdr_a_pfp_shift_unique_secondprefix. (exists mdr_gap_pfp_shift_unique_secondprefixb. mdr_gap_pfp_shift_unique_secondprefixb + S (mdr_i_pfp_shift_unique_secondprefix) = (L)) -> (((exists ff_h_mdr_pfp_shift_unique_secondprefixo. ff_h_mdr_pfp_shift_unique_secondprefixo + S (mdr_a_pfp_shift_unique_secondprefix) = S ((S (mdr_i_pfp_shift_unique_secondprefix)) * c)) /\\ exists ff_q_mdr_pfp_shift_unique_secondprefixo. b = ff_q_mdr_pfp_shift_unique_secondprefixo * S ((S (mdr_i_pfp_shift_unique_secondprefix)) * c) + (mdr_a_pfp_shift_unique_secondprefix))) -> (((exists ff_h_mdr_pfp_shift_unique_secondprefixn. ff_h_mdr_pfp_shift_unique_secondprefixn + S (mdr_a_pfp_shift_unique_secondprefix) = S ((S (mdr_i_pfp_shift_unique_secondprefix)) * g)) /\\ exists ff_q_mdr_pfp_shift_unique_secondprefixn. f = ff_q_mdr_pfp_shift_unique_secondprefixn * S ((S (mdr_i_pfp_shift_unique_secondprefix)) * g) + (mdr_a_pfp_shift_unique_secondprefix)))) /\\ ((((exists ff_h_pfp_shift_unique_secondlast. ff_h_pfp_shift_unique_secondlast + S (0) = S ((S (L)) * g)) /\\ exists ff_q_pfp_shift_unique_secondlast. f = ff_q_pfp_shift_unique_secondlast * S ((S (L)) * g) + (0)))))) -> (forall mdr_i_pfp_shift_unique_result mdr_a_pfp_shift_unique_result. 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"peano-lab/py/peano_lab/library/prime_field_polynomial_shift_candidate.py", "sha256": "325d3085482ee73a2c6ee90cd17e45cffe53273671edf89c40d88428335c9c4b" }, "statement": "forall ab ac L bb bc M BB BC i j t. (((forall mdr_i_pfp_shift_term_relationprefix mdr_a_pfp_shift_term_relationprefix. (exists mdr_gap_pfp_shift_term_relationprefixb. mdr_gap_pfp_shift_term_relationprefixb + S (mdr_i_pfp_shift_term_relationprefix) = (M)) -> (((exists ff_h_mdr_pfp_shift_term_relationprefixo. ff_h_mdr_pfp_shift_term_relationprefixo + S (mdr_a_pfp_shift_term_relationprefix) = S ((S (mdr_i_pfp_shift_term_relationprefix)) * bc)) /\\ exists ff_q_mdr_pfp_shift_term_relationprefixo. bb = ff_q_mdr_pfp_shift_term_relationprefixo * S ((S (mdr_i_pfp_shift_term_relationprefix)) * bc) + (mdr_a_pfp_shift_term_relationprefix))) -> (((exists ff_h_mdr_pfp_shift_term_relationprefixn. ff_h_mdr_pfp_shift_term_relationprefixn + S (mdr_a_pfp_shift_term_relationprefix) = S ((S (mdr_i_pfp_shift_term_relationprefix)) * BC)) /\\ exists ff_q_mdr_pfp_shift_term_relationprefixn. BB = ff_q_mdr_pfp_shift_term_relationprefixn * S ((S (mdr_i_pfp_shift_term_relationprefix)) * BC) + (mdr_a_pfp_shift_term_relationprefix)))) /\\ ((((exists ff_h_pfp_shift_term_relationlast. ff_h_pfp_shift_term_relationlast + S (0) = S ((S (M)) * BC)) /\\ exists ff_q_pfp_shift_term_relationlast. BB = ff_q_pfp_shift_term_relationlast * S ((S (M)) * BC) + (0)))))) -> ((((exists pfc_complement_shift_term_old pfc_left_shift_term_old pfc_right_shift_term_old. (((j)+pfc_complement_shift_term_old=(i)) /\\ ((((((exists pfa_gap_shift_term_oldleftinside. pfa_gap_shift_term_oldleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_shift_term_oldleftentry. ff_h_pfp_shift_term_oldleftentry + S (pfc_left_shift_term_old) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_shift_term_oldleftentry. ab = ff_q_pfp_shift_term_oldleftentry * S ((S (j)) * ac) + (pfc_left_shift_term_old)))))) \\/ (((exists pfc_gap_shift_term_oldleftoutside. pfc_gap_shift_term_oldleftoutside+(L)=(j)) /\\ (((pfc_left_shift_term_old)=0))))) /\\ ((((((exists pfa_gap_shift_term_oldrightinside. pfa_gap_shift_term_oldrightinside + S (pfc_complement_shift_term_old) = (M)) /\\ ((((exists ff_h_pfp_shift_term_oldrightentry. ff_h_pfp_shift_term_oldrightentry + S (pfc_right_shift_term_old) = S ((S (pfc_complement_shift_term_old)) * bc)) /\\ exists ff_q_pfp_shift_term_oldrightentry. bb = ff_q_pfp_shift_term_oldrightentry * S ((S (pfc_complement_shift_term_old)) * bc) + (pfc_right_shift_term_old)))))) \\/ (((exists pfc_gap_shift_term_oldrightoutside. pfc_gap_shift_term_oldrightoutside+(M)=(pfc_complement_shift_term_old)) /\\ (((pfc_right_shift_term_old)=0))))) /\\ (((t)=pfc_left_shift_term_old*pfc_right_shift_term_old)))))))) -> (exists pfc_complement_shift_term_new pfc_left_shift_term_new pfc_right_shift_term_new. (((j)+pfc_complement_shift_term_new=(i)) /\\ ((((((exists pfa_gap_shift_term_newleftinside. pfa_gap_shift_term_newleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_shift_term_newleftentry. ff_h_pfp_shift_term_newleftentry + S (pfc_left_shift_term_new) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_shift_term_newleftentry. ab = ff_q_pfp_shift_term_newleftentry * S ((S (j)) * ac) + (pfc_left_shift_term_new)))))) \\/ (((exists pfc_gap_shift_term_newleftoutside. pfc_gap_shift_term_newleftoutside+(L)=(j)) /\\ (((pfc_left_shift_term_new)=0))))) /\\ ((((((exists pfa_gap_shift_term_newrightinside. pfa_gap_shift_term_newrightinside + S (pfc_complement_shift_term_new) = (S M)) /\\ ((((exists ff_h_pfp_shift_term_newrightentry. ff_h_pfp_shift_term_newrightentry + S (pfc_right_shift_term_new) = S ((S (pfc_complement_shift_term_new)) * BC)) /\\ exists ff_q_pfp_shift_term_newrightentry. BB = ff_q_pfp_shift_term_newrightentry * S ((S (pfc_complement_shift_term_new)) * BC) + (pfc_right_shift_term_new)))))) \\/ (((exists pfc_gap_shift_term_newrightoutside. pfc_gap_shift_term_newrightoutside+(S M)=(pfc_complement_shift_term_new)) /\\ (((pfc_right_shift_term_new)=0))))) /\\ (((t)=pfc_left_shift_term_new*pfc_right_shift_term_new))))))))) /\\ (((exists pfc_complement_shift_term_new pfc_left_shift_term_new pfc_right_shift_term_new. (((j)+pfc_complement_shift_term_new=(i)) /\\ ((((((exists pfa_gap_shift_term_newleftinside. pfa_gap_shift_term_newleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_shift_term_newleftentry. ff_h_pfp_shift_term_newleftentry + S (pfc_left_shift_term_new) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_shift_term_newleftentry. ab = ff_q_pfp_shift_term_newleftentry * S ((S (j)) * ac) + (pfc_left_shift_term_new)))))) \\/ (((exists pfc_gap_shift_term_newleftoutside. pfc_gap_shift_term_newleftoutside+(L)=(j)) /\\ (((pfc_left_shift_term_new)=0))))) /\\ ((((((exists pfa_gap_shift_term_newrightinside. pfa_gap_shift_term_newrightinside + S (pfc_complement_shift_term_new) = (S M)) /\\ ((((exists ff_h_pfp_shift_term_newrightentry. ff_h_pfp_shift_term_newrightentry + S (pfc_right_shift_term_new) = S ((S (pfc_complement_shift_term_new)) * BC)) /\\ exists ff_q_pfp_shift_term_newrightentry. BB = ff_q_pfp_shift_term_newrightentry * S ((S (pfc_complement_shift_term_new)) * BC) + (pfc_right_shift_term_new)))))) \\/ (((exists pfc_gap_shift_term_newrightoutside. pfc_gap_shift_term_newrightoutside+(S M)=(pfc_complement_shift_term_new)) /\\ (((pfc_right_shift_term_new)=0))))) /\\ (((t)=pfc_left_shift_term_new*pfc_right_shift_term_new)))))))) -> (exists pfc_complement_shift_term_old pfc_left_shift_term_old pfc_right_shift_term_old. (((j)+pfc_complement_shift_term_old=(i)) /\\ ((((((exists pfa_gap_shift_term_oldleftinside. pfa_gap_shift_term_oldleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_shift_term_oldleftentry. ff_h_pfp_shift_term_oldleftentry + S (pfc_left_shift_term_old) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_shift_term_oldleftentry. ab = ff_q_pfp_shift_term_oldleftentry * S ((S (j)) * ac) + (pfc_left_shift_term_old)))))) \\/ (((exists pfc_gap_shift_term_oldleftoutside. pfc_gap_shift_term_oldleftoutside+(L)=(j)) /\\ (((pfc_left_shift_term_old)=0))))) /\\ ((((((exists pfa_gap_shift_term_oldrightinside. pfa_gap_shift_term_oldrightinside + S (pfc_complement_shift_term_old) = (M)) /\\ ((((exists ff_h_pfp_shift_term_oldrightentry. ff_h_pfp_shift_term_oldrightentry + S (pfc_right_shift_term_old) = S ((S (pfc_complement_shift_term_old)) * bc)) /\\ exists ff_q_pfp_shift_term_oldrightentry. bb = ff_q_pfp_shift_term_oldrightentry * S ((S (pfc_complement_shift_term_old)) * bc) + (pfc_right_shift_term_old)))))) \\/ (((exists pfc_gap_shift_term_oldrightoutside. pfc_gap_shift_term_oldrightoutside+(M)=(pfc_complement_shift_term_old)) /\\ (((pfc_right_shift_term_old)=0))))) /\\ (((t)=pfc_left_shift_term_old*pfc_right_shift_term_old))))))))))))", "statement_sha256": "eebfa364c6c0acbf6b3abf82c9804c30523f26edf41931d09e5736ffa7f1bcd6", "summary": "An actual trailing-zero shift of the right factor preserves exactly the same antidiagonal term witnesses in both directions.", "summary_sha256": "122a847f8301eedf20ab7d3820129bfdef5b937ce89034cd8f4d9553c030dafb" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "f3adab07e76247782092306cd325aeed4084c727a64412f43b8812cb4fe84fdf", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 177, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", 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hc_witness_witness_witness", "cases hc_witness_witness_witness_right", "exists x", "exists x1", "exists x2", "split", "intro j", "intro hj", "have ht : exists t. ((((exists ff_h_pfp_shift_coefficient_chosen_entry_left. ff_h_pfp_shift_coefficient_chosen_entry_left + S (t) = S ((S (j)) * x1)) /\\ exists ff_q_pfp_shift_coefficient_chosen_entry_left. x = ff_q_pfp_shift_coefficient_chosen_entry_left * S ((S (j)) * x1) + (t))) /\\ ((exists pfc_complement_shift_coefficient_chosen_term_left pfc_left_shift_coefficient_chosen_term_left pfc_right_shift_coefficient_chosen_term_left. (((j)+pfc_complement_shift_coefficient_chosen_term_left=(i)) /\\ ((((((exists pfa_gap_shift_coefficient_chosen_term_leftleftinside. pfa_gap_shift_coefficient_chosen_term_leftleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_shift_coefficient_chosen_term_leftleftentry. ff_h_pfp_shift_coefficient_chosen_term_leftleftentry + S (pfc_left_shift_coefficient_chosen_term_left) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_shift_coefficient_chosen_term_leftleftentry. ab = ff_q_pfp_shift_coefficient_chosen_term_leftleftentry * S ((S (j)) * ac) + (pfc_left_shift_coefficient_chosen_term_left)))))) \\/ (((exists pfc_gap_shift_coefficient_chosen_term_leftleftoutside. pfc_gap_shift_coefficient_chosen_term_leftleftoutside+(L)=(j)) /\\ (((pfc_left_shift_coefficient_chosen_term_left)=0))))) /\\ ((((((exists pfa_gap_shift_coefficient_chosen_term_leftrightinside. pfa_gap_shift_coefficient_chosen_term_leftrightinside + S (pfc_complement_shift_coefficient_chosen_term_left) = (M)) /\\ ((((exists ff_h_pfp_shift_coefficient_chosen_term_leftrightentry. ff_h_pfp_shift_coefficient_chosen_term_leftrightentry + S (pfc_right_shift_coefficient_chosen_term_left) = S ((S (pfc_complement_shift_coefficient_chosen_term_left)) * bc)) /\\ exists ff_q_pfp_shift_coefficient_chosen_term_leftrightentry. bb = ff_q_pfp_shift_coefficient_chosen_term_leftrightentry * S ((S (pfc_complement_shift_coefficient_chosen_term_left)) * bc) + (pfc_right_shift_coefficient_chosen_term_left)))))) \\/ (((exists pfc_gap_shift_coefficient_chosen_term_leftrightoutside. pfc_gap_shift_coefficient_chosen_term_leftrightoutside+(M)=(pfc_complement_shift_coefficient_chosen_term_left)) /\\ (((pfc_right_shift_coefficient_chosen_term_left)=0))))) /\\ (((t)=pfc_left_shift_coefficient_chosen_term_left*pfc_right_shift_coefficient_chosen_term_left))))))))))", "specialize hc_witness_witness_witness_left (j)", "apply hc_witness_witness_witness_left", "exact hj", "cases ht", "cases ht_witness", "exists x3", "split", "exact ht_witness_left", "have htiff : (((exists pfc_complement_shift_coefficient_old_term_left pfc_left_shift_coefficient_old_term_left pfc_right_shift_coefficient_old_term_left. (((j)+pfc_complement_shift_coefficient_old_term_left=(i)) /\\ ((((((exists pfa_gap_shift_coefficient_old_term_leftleftinside. pfa_gap_shift_coefficient_old_term_leftleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_shift_coefficient_old_term_leftleftentry. ff_h_pfp_shift_coefficient_old_term_leftleftentry + S (pfc_left_shift_coefficient_old_term_left) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_shift_coefficient_old_term_leftleftentry. ab = ff_q_pfp_shift_coefficient_old_term_leftleftentry * S ((S (j)) * ac) + (pfc_left_shift_coefficient_old_term_left)))))) \\/ (((exists pfc_gap_shift_coefficient_old_term_leftleftoutside. pfc_gap_shift_coefficient_old_term_leftleftoutside+(L)=(j)) /\\ (((pfc_left_shift_coefficient_old_term_left)=0))))) /\\ ((((((exists pfa_gap_shift_coefficient_old_term_leftrightinside. pfa_gap_shift_coefficient_old_term_leftrightinside + S (pfc_complement_shift_coefficient_old_term_left) = (M)) /\\ ((((exists ff_h_pfp_shift_coefficient_old_term_leftrightentry. ff_h_pfp_shift_coefficient_old_term_leftrightentry + S (pfc_right_shift_coefficient_old_term_left) = S ((S (pfc_complement_shift_coefficient_old_term_left)) * bc)) /\\ exists ff_q_pfp_shift_coefficient_old_term_leftrightentry. bb = ff_q_pfp_shift_coefficient_old_term_leftrightentry * S ((S (pfc_complement_shift_coefficient_old_term_left)) * bc) + (pfc_right_shift_coefficient_old_term_left)))))) \\/ (((exists pfc_gap_shift_coefficient_old_term_leftrightoutside. pfc_gap_shift_coefficient_old_term_leftrightoutside+(M)=(pfc_complement_shift_coefficient_old_term_left)) /\\ (((pfc_right_shift_coefficient_old_term_left)=0))))) /\\ (((x3)=pfc_left_shift_coefficient_old_term_left*pfc_right_shift_coefficient_old_term_left)))))))) -> (exists pfc_complement_shift_coefficient_new_term_left pfc_left_shift_coefficient_new_term_left pfc_right_shift_coefficient_new_term_left. (((j)+pfc_complement_shift_coefficient_new_term_left=(i)) /\\ ((((((exists pfa_gap_shift_coefficient_new_term_leftleftinside. pfa_gap_shift_coefficient_new_term_leftleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_shift_coefficient_new_term_leftleftentry. ff_h_pfp_shift_coefficient_new_term_leftleftentry + S (pfc_left_shift_coefficient_new_term_left) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_shift_coefficient_new_term_leftleftentry. ab = ff_q_pfp_shift_coefficient_new_term_leftleftentry * S ((S (j)) * ac) + (pfc_left_shift_coefficient_new_term_left)))))) \\/ (((exists pfc_gap_shift_coefficient_new_term_leftleftoutside. pfc_gap_shift_coefficient_new_term_leftleftoutside+(L)=(j)) /\\ (((pfc_left_shift_coefficient_new_term_left)=0))))) /\\ ((((((exists pfa_gap_shift_coefficient_new_term_leftrightinside. pfa_gap_shift_coefficient_new_term_leftrightinside + S (pfc_complement_shift_coefficient_new_term_left) = (S M)) /\\ ((((exists ff_h_pfp_shift_coefficient_new_term_leftrightentry. ff_h_pfp_shift_coefficient_new_term_leftrightentry + S (pfc_right_shift_coefficient_new_term_left) = S ((S (pfc_complement_shift_coefficient_new_term_left)) * BC)) /\\ exists ff_q_pfp_shift_coefficient_new_term_leftrightentry. BB = ff_q_pfp_shift_coefficient_new_term_leftrightentry * S ((S (pfc_complement_shift_coefficient_new_term_left)) * BC) + (pfc_right_shift_coefficient_new_term_left)))))) \\/ (((exists pfc_gap_shift_coefficient_new_term_leftrightoutside. pfc_gap_shift_coefficient_new_term_leftrightoutside+(S M)=(pfc_complement_shift_coefficient_new_term_left)) /\\ (((pfc_right_shift_coefficient_new_term_left)=0))))) /\\ (((x3)=pfc_left_shift_coefficient_new_term_left*pfc_right_shift_coefficient_new_term_left))))))))) /\\ (((exists pfc_complement_shift_coefficient_new_term_left pfc_left_shift_coefficient_new_term_left pfc_right_shift_coefficient_new_term_left. (((j)+pfc_complement_shift_coefficient_new_term_left=(i)) /\\ ((((((exists pfa_gap_shift_coefficient_new_term_leftleftinside. pfa_gap_shift_coefficient_new_term_leftleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_shift_coefficient_new_term_leftleftentry. ff_h_pfp_shift_coefficient_new_term_leftleftentry + S (pfc_left_shift_coefficient_new_term_left) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_shift_coefficient_new_term_leftleftentry. ab = ff_q_pfp_shift_coefficient_new_term_leftleftentry * S ((S (j)) * ac) + (pfc_left_shift_coefficient_new_term_left)))))) \\/ (((exists pfc_gap_shift_coefficient_new_term_leftleftoutside. pfc_gap_shift_coefficient_new_term_leftleftoutside+(L)=(j)) /\\ (((pfc_left_shift_coefficient_new_term_left)=0))))) /\\ ((((((exists pfa_gap_shift_coefficient_new_term_leftrightinside. pfa_gap_shift_coefficient_new_term_leftrightinside + S (pfc_complement_shift_coefficient_new_term_left) = (S M)) /\\ ((((exists ff_h_pfp_shift_coefficient_new_term_leftrightentry. ff_h_pfp_shift_coefficient_new_term_leftrightentry + S (pfc_right_shift_coefficient_new_term_left) = S ((S (pfc_complement_shift_coefficient_new_term_left)) * BC)) /\\ exists ff_q_pfp_shift_coefficient_new_term_leftrightentry. BB = ff_q_pfp_shift_coefficient_new_term_leftrightentry * S ((S (pfc_complement_shift_coefficient_new_term_left)) * BC) + (pfc_right_shift_coefficient_new_term_left)))))) \\/ (((exists pfc_gap_shift_coefficient_new_term_leftrightoutside. pfc_gap_shift_coefficient_new_term_leftrightoutside+(S M)=(pfc_complement_shift_coefficient_new_term_left)) /\\ (((pfc_right_shift_coefficient_new_term_left)=0))))) /\\ (((x3)=pfc_left_shift_coefficient_new_term_left*pfc_right_shift_coefficient_new_term_left)))))))) -> (exists pfc_complement_shift_coefficient_old_term_left pfc_left_shift_coefficient_old_term_left pfc_right_shift_coefficient_old_term_left. (((j)+pfc_complement_shift_coefficient_old_term_left=(i)) /\\ ((((((exists pfa_gap_shift_coefficient_old_term_leftleftinside. pfa_gap_shift_coefficient_old_term_leftleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_shift_coefficient_old_term_leftleftentry. ff_h_pfp_shift_coefficient_old_term_leftleftentry + S (pfc_left_shift_coefficient_old_term_left) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_shift_coefficient_old_term_leftleftentry. ab = ff_q_pfp_shift_coefficient_old_term_leftleftentry * S ((S (j)) * ac) + (pfc_left_shift_coefficient_old_term_left)))))) \\/ (((exists pfc_gap_shift_coefficient_old_term_leftleftoutside. pfc_gap_shift_coefficient_old_term_leftleftoutside+(L)=(j)) /\\ (((pfc_left_shift_coefficient_old_term_left)=0))))) /\\ ((((((exists pfa_gap_shift_coefficient_old_term_leftrightinside. pfa_gap_shift_coefficient_old_term_leftrightinside + S (pfc_complement_shift_coefficient_old_term_left) = (M)) /\\ ((((exists ff_h_pfp_shift_coefficient_old_term_leftrightentry. ff_h_pfp_shift_coefficient_old_term_leftrightentry + S (pfc_right_shift_coefficient_old_term_left) = S ((S (pfc_complement_shift_coefficient_old_term_left)) * bc)) /\\ exists ff_q_pfp_shift_coefficient_old_term_leftrightentry. bb = ff_q_pfp_shift_coefficient_old_term_leftrightentry * S ((S (pfc_complement_shift_coefficient_old_term_left)) * bc) + (pfc_right_shift_coefficient_old_term_left)))))) \\/ (((exists pfc_gap_shift_coefficient_old_term_leftrightoutside. pfc_gap_shift_coefficient_old_term_leftrightoutside+(M)=(pfc_complement_shift_coefficient_old_term_left)) /\\ (((pfc_right_shift_coefficient_old_term_left)=0))))) /\\ (((x3)=pfc_left_shift_coefficient_old_term_left*pfc_right_shift_coefficient_old_term_left)))))))))))", "specialize polynomial_diagonal_term_shift_right_iff (ab)", "specialize polynomial_diagonal_term_shift_right_iff (ac)", "specialize polynomial_diagonal_term_shift_right_iff (L)", "specialize polynomial_diagonal_term_shift_right_iff (bb)", "specialize polynomial_diagonal_term_shift_right_iff (bc)", "specialize polynomial_diagonal_term_shift_right_iff (M)", "specialize polynomial_diagonal_term_shift_right_iff (BB)", "specialize polynomial_diagonal_term_shift_right_iff (BC)", "specialize polynomial_diagonal_term_shift_right_iff (i)", "specialize polynomial_diagonal_term_shift_right_iff (j)", "specialize polynomial_diagonal_term_shift_right_iff (x3)", "apply polynomial_diagonal_term_shift_right_iff", "exact hs", "cases htiff", "apply htiff_left", "exact ht_witness_right", "split", "exact hc_witness_witness_witness_right_left", "exact hc_witness_witness_witness_right_right", "intro hc", "cases hc", "cases hc_witness", "cases hc_witness_witness", "cases hc_witness_witness_witness", "cases hc_witness_witness_witness_right", "exists x", "exists x1", "exists x2", "split", "intro j", "intro hj", "have ht : exists t. ((((exists ff_h_pfp_shift_coefficient_chosen_entry_right. ff_h_pfp_shift_coefficient_chosen_entry_right + S (t) = S ((S (j)) * x1)) /\\ exists ff_q_pfp_shift_coefficient_chosen_entry_right. x = ff_q_pfp_shift_coefficient_chosen_entry_right * S ((S (j)) * x1) + (t))) /\\ ((exists pfc_complement_shift_coefficient_chosen_term_right pfc_left_shift_coefficient_chosen_term_right pfc_right_shift_coefficient_chosen_term_right. (((j)+pfc_complement_shift_coefficient_chosen_term_right=(i)) /\\ ((((((exists pfa_gap_shift_coefficient_chosen_term_rightleftinside. pfa_gap_shift_coefficient_chosen_term_rightleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_shift_coefficient_chosen_term_rightleftentry. ff_h_pfp_shift_coefficient_chosen_term_rightleftentry + S (pfc_left_shift_coefficient_chosen_term_right) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_shift_coefficient_chosen_term_rightleftentry. ab = ff_q_pfp_shift_coefficient_chosen_term_rightleftentry * S ((S (j)) * ac) + (pfc_left_shift_coefficient_chosen_term_right)))))) \\/ (((exists pfc_gap_shift_coefficient_chosen_term_rightleftoutside. pfc_gap_shift_coefficient_chosen_term_rightleftoutside+(L)=(j)) /\\ (((pfc_left_shift_coefficient_chosen_term_right)=0))))) /\\ ((((((exists pfa_gap_shift_coefficient_chosen_term_rightrightinside. pfa_gap_shift_coefficient_chosen_term_rightrightinside + S (pfc_complement_shift_coefficient_chosen_term_right) = (S M)) /\\ ((((exists ff_h_pfp_shift_coefficient_chosen_term_rightrightentry. ff_h_pfp_shift_coefficient_chosen_term_rightrightentry + S (pfc_right_shift_coefficient_chosen_term_right) = S ((S (pfc_complement_shift_coefficient_chosen_term_right)) * BC)) /\\ exists ff_q_pfp_shift_coefficient_chosen_term_rightrightentry. BB = ff_q_pfp_shift_coefficient_chosen_term_rightrightentry * S ((S (pfc_complement_shift_coefficient_chosen_term_right)) * BC) + (pfc_right_shift_coefficient_chosen_term_right)))))) \\/ (((exists pfc_gap_shift_coefficient_chosen_term_rightrightoutside. pfc_gap_shift_coefficient_chosen_term_rightrightoutside+(S M)=(pfc_complement_shift_coefficient_chosen_term_right)) /\\ (((pfc_right_shift_coefficient_chosen_term_right)=0))))) /\\ (((t)=pfc_left_shift_coefficient_chosen_term_right*pfc_right_shift_coefficient_chosen_term_right))))))))))", "specialize hc_witness_witness_witness_left (j)", "apply hc_witness_witness_witness_left", "exact hj", "cases ht", "cases ht_witness", "exists x3", "split", "exact ht_witness_left", "have htiff : (((exists pfc_complement_shift_coefficient_old_term_right pfc_left_shift_coefficient_old_term_right pfc_right_shift_coefficient_old_term_right. (((j)+pfc_complement_shift_coefficient_old_term_right=(i)) /\\ ((((((exists pfa_gap_shift_coefficient_old_term_rightleftinside. pfa_gap_shift_coefficient_old_term_rightleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_shift_coefficient_old_term_rightleftentry. ff_h_pfp_shift_coefficient_old_term_rightleftentry + S (pfc_left_shift_coefficient_old_term_right) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_shift_coefficient_old_term_rightleftentry. ab = ff_q_pfp_shift_coefficient_old_term_rightleftentry * S ((S (j)) * ac) + (pfc_left_shift_coefficient_old_term_right)))))) \\/ (((exists pfc_gap_shift_coefficient_old_term_rightleftoutside. pfc_gap_shift_coefficient_old_term_rightleftoutside+(L)=(j)) /\\ (((pfc_left_shift_coefficient_old_term_right)=0))))) /\\ ((((((exists pfa_gap_shift_coefficient_old_term_rightrightinside. pfa_gap_shift_coefficient_old_term_rightrightinside + S (pfc_complement_shift_coefficient_old_term_right) = (M)) /\\ ((((exists ff_h_pfp_shift_coefficient_old_term_rightrightentry. ff_h_pfp_shift_coefficient_old_term_rightrightentry + S (pfc_right_shift_coefficient_old_term_right) = S ((S (pfc_complement_shift_coefficient_old_term_right)) * bc)) /\\ exists ff_q_pfp_shift_coefficient_old_term_rightrightentry. bb = ff_q_pfp_shift_coefficient_old_term_rightrightentry * S ((S (pfc_complement_shift_coefficient_old_term_right)) * bc) + (pfc_right_shift_coefficient_old_term_right)))))) \\/ (((exists pfc_gap_shift_coefficient_old_term_rightrightoutside. pfc_gap_shift_coefficient_old_term_rightrightoutside+(M)=(pfc_complement_shift_coefficient_old_term_right)) /\\ (((pfc_right_shift_coefficient_old_term_right)=0))))) /\\ (((x3)=pfc_left_shift_coefficient_old_term_right*pfc_right_shift_coefficient_old_term_right)))))))) -> (exists pfc_complement_shift_coefficient_new_term_right pfc_left_shift_coefficient_new_term_right pfc_right_shift_coefficient_new_term_right. (((j)+pfc_complement_shift_coefficient_new_term_right=(i)) /\\ ((((((exists pfa_gap_shift_coefficient_new_term_rightleftinside. pfa_gap_shift_coefficient_new_term_rightleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_shift_coefficient_new_term_rightleftentry. ff_h_pfp_shift_coefficient_new_term_rightleftentry + S (pfc_left_shift_coefficient_new_term_right) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_shift_coefficient_new_term_rightleftentry. ab = ff_q_pfp_shift_coefficient_new_term_rightleftentry * S ((S (j)) * ac) + (pfc_left_shift_coefficient_new_term_right)))))) \\/ (((exists pfc_gap_shift_coefficient_new_term_rightleftoutside. pfc_gap_shift_coefficient_new_term_rightleftoutside+(L)=(j)) /\\ (((pfc_left_shift_coefficient_new_term_right)=0))))) /\\ ((((((exists pfa_gap_shift_coefficient_new_term_rightrightinside. pfa_gap_shift_coefficient_new_term_rightrightinside + S (pfc_complement_shift_coefficient_new_term_right) = (S M)) /\\ ((((exists ff_h_pfp_shift_coefficient_new_term_rightrightentry. ff_h_pfp_shift_coefficient_new_term_rightrightentry + S (pfc_right_shift_coefficient_new_term_right) = S ((S (pfc_complement_shift_coefficient_new_term_right)) * BC)) /\\ exists ff_q_pfp_shift_coefficient_new_term_rightrightentry. BB = ff_q_pfp_shift_coefficient_new_term_rightrightentry * S ((S (pfc_complement_shift_coefficient_new_term_right)) * BC) + (pfc_right_shift_coefficient_new_term_right)))))) \\/ (((exists pfc_gap_shift_coefficient_new_term_rightrightoutside. pfc_gap_shift_coefficient_new_term_rightrightoutside+(S M)=(pfc_complement_shift_coefficient_new_term_right)) /\\ (((pfc_right_shift_coefficient_new_term_right)=0))))) /\\ (((x3)=pfc_left_shift_coefficient_new_term_right*pfc_right_shift_coefficient_new_term_right))))))))) /\\ (((exists pfc_complement_shift_coefficient_new_term_right pfc_left_shift_coefficient_new_term_right pfc_right_shift_coefficient_new_term_right. (((j)+pfc_complement_shift_coefficient_new_term_right=(i)) /\\ ((((((exists pfa_gap_shift_coefficient_new_term_rightleftinside. pfa_gap_shift_coefficient_new_term_rightleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_shift_coefficient_new_term_rightleftentry. ff_h_pfp_shift_coefficient_new_term_rightleftentry + S (pfc_left_shift_coefficient_new_term_right) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_shift_coefficient_new_term_rightleftentry. ab = ff_q_pfp_shift_coefficient_new_term_rightleftentry * S ((S (j)) * ac) + (pfc_left_shift_coefficient_new_term_right)))))) \\/ (((exists pfc_gap_shift_coefficient_new_term_rightleftoutside. pfc_gap_shift_coefficient_new_term_rightleftoutside+(L)=(j)) /\\ (((pfc_left_shift_coefficient_new_term_right)=0))))) /\\ ((((((exists pfa_gap_shift_coefficient_new_term_rightrightinside. pfa_gap_shift_coefficient_new_term_rightrightinside + S (pfc_complement_shift_coefficient_new_term_right) = (S M)) /\\ ((((exists ff_h_pfp_shift_coefficient_new_term_rightrightentry. ff_h_pfp_shift_coefficient_new_term_rightrightentry + S (pfc_right_shift_coefficient_new_term_right) = S ((S (pfc_complement_shift_coefficient_new_term_right)) * BC)) /\\ exists ff_q_pfp_shift_coefficient_new_term_rightrightentry. BB = ff_q_pfp_shift_coefficient_new_term_rightrightentry * S ((S (pfc_complement_shift_coefficient_new_term_right)) * BC) + (pfc_right_shift_coefficient_new_term_right)))))) \\/ (((exists pfc_gap_shift_coefficient_new_term_rightrightoutside. pfc_gap_shift_coefficient_new_term_rightrightoutside+(S M)=(pfc_complement_shift_coefficient_new_term_right)) /\\ (((pfc_right_shift_coefficient_new_term_right)=0))))) /\\ (((x3)=pfc_left_shift_coefficient_new_term_right*pfc_right_shift_coefficient_new_term_right)))))))) -> (exists pfc_complement_shift_coefficient_old_term_right pfc_left_shift_coefficient_old_term_right pfc_right_shift_coefficient_old_term_right. (((j)+pfc_complement_shift_coefficient_old_term_right=(i)) /\\ ((((((exists pfa_gap_shift_coefficient_old_term_rightleftinside. pfa_gap_shift_coefficient_old_term_rightleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_shift_coefficient_old_term_rightleftentry. ff_h_pfp_shift_coefficient_old_term_rightleftentry + S (pfc_left_shift_coefficient_old_term_right) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_shift_coefficient_old_term_rightleftentry. ab = ff_q_pfp_shift_coefficient_old_term_rightleftentry * S ((S (j)) * ac) + (pfc_left_shift_coefficient_old_term_right)))))) \\/ (((exists pfc_gap_shift_coefficient_old_term_rightleftoutside. pfc_gap_shift_coefficient_old_term_rightleftoutside+(L)=(j)) /\\ (((pfc_left_shift_coefficient_old_term_right)=0))))) /\\ ((((((exists pfa_gap_shift_coefficient_old_term_rightrightinside. pfa_gap_shift_coefficient_old_term_rightrightinside + S (pfc_complement_shift_coefficient_old_term_right) = (M)) /\\ ((((exists ff_h_pfp_shift_coefficient_old_term_rightrightentry. ff_h_pfp_shift_coefficient_old_term_rightrightentry + S (pfc_right_shift_coefficient_old_term_right) = S ((S (pfc_complement_shift_coefficient_old_term_right)) * bc)) /\\ exists ff_q_pfp_shift_coefficient_old_term_rightrightentry. bb = ff_q_pfp_shift_coefficient_old_term_rightrightentry * S ((S (pfc_complement_shift_coefficient_old_term_right)) * bc) + (pfc_right_shift_coefficient_old_term_right)))))) \\/ (((exists pfc_gap_shift_coefficient_old_term_rightrightoutside. pfc_gap_shift_coefficient_old_term_rightrightoutside+(M)=(pfc_complement_shift_coefficient_old_term_right)) /\\ (((pfc_right_shift_coefficient_old_term_right)=0))))) /\\ (((x3)=pfc_left_shift_coefficient_old_term_right*pfc_right_shift_coefficient_old_term_right)))))))))))", "specialize polynomial_diagonal_term_shift_right_iff (ab)", "specialize polynomial_diagonal_term_shift_right_iff (ac)", "specialize polynomial_diagonal_term_shift_right_iff (L)", "specialize polynomial_diagonal_term_shift_right_iff (bb)", "specialize polynomial_diagonal_term_shift_right_iff (bc)", "specialize polynomial_diagonal_term_shift_right_iff (M)", "specialize polynomial_diagonal_term_shift_right_iff (BB)", "specialize polynomial_diagonal_term_shift_right_iff (BC)", "specialize polynomial_diagonal_term_shift_right_iff (i)", "specialize polynomial_diagonal_term_shift_right_iff (j)", "specialize polynomial_diagonal_term_shift_right_iff (x3)", "apply polynomial_diagonal_term_shift_right_iff", "exact hs", "cases htiff", "apply htiff_right", "exact ht_witness_right", "split", "exact hc_witness_witness_witness_right_left", "exact hc_witness_witness_witness_right_right" ], "script_sha256": "82292575f9894e57566fbd259b7000ebd1b7b202918d9ab90cf466f3bf458c6f", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_shift_candidate.py", "sha256": "325d3085482ee73a2c6ee90cd17e45cffe53273671edf89c40d88428335c9c4b" }, "statement": "forall p ab ac L bb bc M BB BC i r. (((forall mdr_i_pfp_shift_coefficient_relationprefix mdr_a_pfp_shift_coefficient_relationprefix. (exists mdr_gap_pfp_shift_coefficient_relationprefixb. mdr_gap_pfp_shift_coefficient_relationprefixb + S (mdr_i_pfp_shift_coefficient_relationprefix) = (M)) -> (((exists ff_h_mdr_pfp_shift_coefficient_relationprefixo. ff_h_mdr_pfp_shift_coefficient_relationprefixo + S (mdr_a_pfp_shift_coefficient_relationprefix) = S ((S (mdr_i_pfp_shift_coefficient_relationprefix)) * bc)) /\\ exists ff_q_mdr_pfp_shift_coefficient_relationprefixo. bb = ff_q_mdr_pfp_shift_coefficient_relationprefixo * S ((S (mdr_i_pfp_shift_coefficient_relationprefix)) * bc) + (mdr_a_pfp_shift_coefficient_relationprefix))) -> (((exists ff_h_mdr_pfp_shift_coefficient_relationprefixn. ff_h_mdr_pfp_shift_coefficient_relationprefixn + S (mdr_a_pfp_shift_coefficient_relationprefix) = S ((S (mdr_i_pfp_shift_coefficient_relationprefix)) * BC)) /\\ exists ff_q_mdr_pfp_shift_coefficient_relationprefixn. BB = ff_q_mdr_pfp_shift_coefficient_relationprefixn * S ((S (mdr_i_pfp_shift_coefficient_relationprefix)) * BC) + (mdr_a_pfp_shift_coefficient_relationprefix)))) /\\ ((((exists ff_h_pfp_shift_coefficient_relationlast. ff_h_pfp_shift_coefficient_relationlast + S (0) = S ((S (M)) * BC)) /\\ exists ff_q_pfp_shift_coefficient_relationlast. BB = ff_q_pfp_shift_coefficient_relationlast * S ((S (M)) * BC) + (0)))))) -> ((((exists pfc_terms_code_shift_coefficient_old pfc_terms_scale_shift_coefficient_old pfc_natural_sum_shift_coefficient_old. ((forall pfc_index_shift_coefficient_olddiagonal. (exists pfa_gap_shift_coefficient_olddiagonalbound. pfa_gap_shift_coefficient_olddiagonalbound + S (pfc_index_shift_coefficient_olddiagonal) = (S (i))) -> exists pfc_value_shift_coefficient_olddiagonal. ((((exists ff_h_pfp_shift_coefficient_olddiagonalentry. ff_h_pfp_shift_coefficient_olddiagonalentry + S (pfc_value_shift_coefficient_olddiagonal) = S ((S (pfc_index_shift_coefficient_olddiagonal)) * pfc_terms_scale_shift_coefficient_old)) /\\ exists ff_q_pfp_shift_coefficient_olddiagonalentry. pfc_terms_code_shift_coefficient_old = ff_q_pfp_shift_coefficient_olddiagonalentry * S ((S (pfc_index_shift_coefficient_olddiagonal)) * pfc_terms_scale_shift_coefficient_old) + (pfc_value_shift_coefficient_olddiagonal))) /\\ ((exists pfc_complement_shift_coefficient_olddiagonalterm pfc_left_shift_coefficient_olddiagonalterm pfc_right_shift_coefficient_olddiagonalterm. (((pfc_index_shift_coefficient_olddiagonal)+pfc_complement_shift_coefficient_olddiagonalterm=(i)) /\\ ((((((exists pfa_gap_shift_coefficient_olddiagonaltermleftinside. pfa_gap_shift_coefficient_olddiagonaltermleftinside + S (pfc_index_shift_coefficient_olddiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_coefficient_olddiagonaltermleftentry. ff_h_pfp_shift_coefficient_olddiagonaltermleftentry + S (pfc_left_shift_coefficient_olddiagonalterm) = S ((S (pfc_index_shift_coefficient_olddiagonal)) * ac)) /\\ exists ff_q_pfp_shift_coefficient_olddiagonaltermleftentry. ab = ff_q_pfp_shift_coefficient_olddiagonaltermleftentry * S ((S (pfc_index_shift_coefficient_olddiagonal)) * ac) + (pfc_left_shift_coefficient_olddiagonalterm)))))) \\/ (((exists pfc_gap_shift_coefficient_olddiagonaltermleftoutside. pfc_gap_shift_coefficient_olddiagonaltermleftoutside+(L)=(pfc_index_shift_coefficient_olddiagonal)) /\\ (((pfc_left_shift_coefficient_olddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_coefficient_olddiagonaltermrightinside. pfa_gap_shift_coefficient_olddiagonaltermrightinside + S (pfc_complement_shift_coefficient_olddiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_shift_coefficient_olddiagonaltermrightentry. ff_h_pfp_shift_coefficient_olddiagonaltermrightentry + S (pfc_right_shift_coefficient_olddiagonalterm) = S ((S (pfc_complement_shift_coefficient_olddiagonalterm)) * bc)) /\\ exists ff_q_pfp_shift_coefficient_olddiagonaltermrightentry. bb = ff_q_pfp_shift_coefficient_olddiagonaltermrightentry * S ((S (pfc_complement_shift_coefficient_olddiagonalterm)) * bc) + (pfc_right_shift_coefficient_olddiagonalterm)))))) \\/ (((exists pfc_gap_shift_coefficient_olddiagonaltermrightoutside. pfc_gap_shift_coefficient_olddiagonaltermrightoutside+(M)=(pfc_complement_shift_coefficient_olddiagonalterm)) /\\ (((pfc_right_shift_coefficient_olddiagonalterm)=0))))) /\\ (((pfc_value_shift_coefficient_olddiagonal)=pfc_left_shift_coefficient_olddiagonalterm*pfc_right_shift_coefficient_olddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_coefficient_oldsum fs_v_pfc_shift_coefficient_oldsum. ((((exists fs_h_pfc_shift_coefficient_oldsum_body_start. fs_h_pfc_shift_coefficient_oldsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_coefficient_oldsum)) /\\ exists fs_q_pfc_shift_coefficient_oldsum_body_start. fs_u_pfc_shift_coefficient_oldsum = fs_q_pfc_shift_coefficient_oldsum_body_start * S ((S (0)) * fs_v_pfc_shift_coefficient_oldsum) + (0))) /\\ ((((exists fs_h_pfc_shift_coefficient_oldsum_body_terminal. fs_h_pfc_shift_coefficient_oldsum_body_terminal + S (pfc_natural_sum_shift_coefficient_old) = S ((S (S (i))) * fs_v_pfc_shift_coefficient_oldsum)) /\\ exists fs_q_pfc_shift_coefficient_oldsum_body_terminal. fs_u_pfc_shift_coefficient_oldsum = fs_q_pfc_shift_coefficient_oldsum_body_terminal * S ((S (S (i))) * fs_v_pfc_shift_coefficient_oldsum) + (pfc_natural_sum_shift_coefficient_old))) /\\ forall fs_i_pfc_shift_coefficient_oldsum_body_steps. (exists fs_lt_pfc_shift_coefficient_oldsum_body_steps_bound. fs_lt_pfc_shift_coefficient_oldsum_body_steps_bound + S fs_i_pfc_shift_coefficient_oldsum_body_steps = S (i)) -> exists fs_a_pfc_shift_coefficient_oldsum_body_steps fs_r_pfc_shift_coefficient_oldsum_body_steps fs_s_pfc_shift_coefficient_oldsum_body_steps. ((((exists fs_h_pfc_shift_coefficient_oldsum_body_steps_summand. fs_h_pfc_shift_coefficient_oldsum_body_steps_summand + S (fs_a_pfc_shift_coefficient_oldsum_body_steps) = S ((S (fs_i_pfc_shift_coefficient_oldsum_body_steps)) * pfc_terms_scale_shift_coefficient_old)) /\\ exists fs_q_pfc_shift_coefficient_oldsum_body_steps_summand. pfc_terms_code_shift_coefficient_old = fs_q_pfc_shift_coefficient_oldsum_body_steps_summand * S ((S (fs_i_pfc_shift_coefficient_oldsum_body_steps)) * pfc_terms_scale_shift_coefficient_old) + (fs_a_pfc_shift_coefficient_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_coefficient_oldsum_body_steps_partial. fs_h_pfc_shift_coefficient_oldsum_body_steps_partial + S (fs_r_pfc_shift_coefficient_oldsum_body_steps) = S ((S (fs_i_pfc_shift_coefficient_oldsum_body_steps)) * fs_v_pfc_shift_coefficient_oldsum)) /\\ exists fs_q_pfc_shift_coefficient_oldsum_body_steps_partial. fs_u_pfc_shift_coefficient_oldsum = fs_q_pfc_shift_coefficient_oldsum_body_steps_partial * S ((S (fs_i_pfc_shift_coefficient_oldsum_body_steps)) * fs_v_pfc_shift_coefficient_oldsum) + (fs_r_pfc_shift_coefficient_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_coefficient_oldsum_body_steps_successor. fs_h_pfc_shift_coefficient_oldsum_body_steps_successor + S (fs_s_pfc_shift_coefficient_oldsum_body_steps) = S ((S (S fs_i_pfc_shift_coefficient_oldsum_body_steps)) * fs_v_pfc_shift_coefficient_oldsum)) /\\ exists fs_q_pfc_shift_coefficient_oldsum_body_steps_successor. fs_u_pfc_shift_coefficient_oldsum = fs_q_pfc_shift_coefficient_oldsum_body_steps_successor * S ((S (S fs_i_pfc_shift_coefficient_oldsum_body_steps)) * fs_v_pfc_shift_coefficient_oldsum) + (fs_s_pfc_shift_coefficient_oldsum_body_steps))) /\\ fs_s_pfc_shift_coefficient_oldsum_body_steps = fs_r_pfc_shift_coefficient_oldsum_body_steps + fs_a_pfc_shift_coefficient_oldsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_coefficient_oldresiduebound. pfa_gap_shift_coefficient_oldresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_shift_coefficient_oldresiduecongruence pfa_offset_right_shift_coefficient_oldresiduecongruence. (pfc_natural_sum_shift_coefficient_old) + (p) * pfa_offset_left_shift_coefficient_oldresiduecongruence = (r) + (p) * pfa_offset_right_shift_coefficient_oldresiduecongruence))))))))) -> (exists pfc_terms_code_shift_coefficient_new pfc_terms_scale_shift_coefficient_new pfc_natural_sum_shift_coefficient_new. ((forall pfc_index_shift_coefficient_newdiagonal. (exists pfa_gap_shift_coefficient_newdiagonalbound. pfa_gap_shift_coefficient_newdiagonalbound + S (pfc_index_shift_coefficient_newdiagonal) = (S (i))) -> exists pfc_value_shift_coefficient_newdiagonal. ((((exists ff_h_pfp_shift_coefficient_newdiagonalentry. ff_h_pfp_shift_coefficient_newdiagonalentry + S (pfc_value_shift_coefficient_newdiagonal) = S ((S (pfc_index_shift_coefficient_newdiagonal)) * pfc_terms_scale_shift_coefficient_new)) /\\ exists ff_q_pfp_shift_coefficient_newdiagonalentry. pfc_terms_code_shift_coefficient_new = ff_q_pfp_shift_coefficient_newdiagonalentry * S ((S (pfc_index_shift_coefficient_newdiagonal)) * pfc_terms_scale_shift_coefficient_new) + (pfc_value_shift_coefficient_newdiagonal))) /\\ ((exists pfc_complement_shift_coefficient_newdiagonalterm pfc_left_shift_coefficient_newdiagonalterm pfc_right_shift_coefficient_newdiagonalterm. (((pfc_index_shift_coefficient_newdiagonal)+pfc_complement_shift_coefficient_newdiagonalterm=(i)) /\\ ((((((exists pfa_gap_shift_coefficient_newdiagonaltermleftinside. pfa_gap_shift_coefficient_newdiagonaltermleftinside + S (pfc_index_shift_coefficient_newdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_coefficient_newdiagonaltermleftentry. ff_h_pfp_shift_coefficient_newdiagonaltermleftentry + S (pfc_left_shift_coefficient_newdiagonalterm) = S ((S (pfc_index_shift_coefficient_newdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_coefficient_newdiagonaltermleftentry. ab = ff_q_pfp_shift_coefficient_newdiagonaltermleftentry * S ((S (pfc_index_shift_coefficient_newdiagonal)) * ac) + (pfc_left_shift_coefficient_newdiagonalterm)))))) \\/ (((exists pfc_gap_shift_coefficient_newdiagonaltermleftoutside. pfc_gap_shift_coefficient_newdiagonaltermleftoutside+(L)=(pfc_index_shift_coefficient_newdiagonal)) /\\ (((pfc_left_shift_coefficient_newdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_coefficient_newdiagonaltermrightinside. pfa_gap_shift_coefficient_newdiagonaltermrightinside + S (pfc_complement_shift_coefficient_newdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_shift_coefficient_newdiagonaltermrightentry. ff_h_pfp_shift_coefficient_newdiagonaltermrightentry + S (pfc_right_shift_coefficient_newdiagonalterm) = S ((S (pfc_complement_shift_coefficient_newdiagonalterm)) * BC)) /\\ exists ff_q_pfp_shift_coefficient_newdiagonaltermrightentry. BB = ff_q_pfp_shift_coefficient_newdiagonaltermrightentry * S ((S (pfc_complement_shift_coefficient_newdiagonalterm)) * BC) + (pfc_right_shift_coefficient_newdiagonalterm)))))) \\/ (((exists pfc_gap_shift_coefficient_newdiagonaltermrightoutside. pfc_gap_shift_coefficient_newdiagonaltermrightoutside+(S M)=(pfc_complement_shift_coefficient_newdiagonalterm)) /\\ (((pfc_right_shift_coefficient_newdiagonalterm)=0))))) /\\ (((pfc_value_shift_coefficient_newdiagonal)=pfc_left_shift_coefficient_newdiagonalterm*pfc_right_shift_coefficient_newdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_coefficient_newsum fs_v_pfc_shift_coefficient_newsum. ((((exists fs_h_pfc_shift_coefficient_newsum_body_start. fs_h_pfc_shift_coefficient_newsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_coefficient_newsum)) /\\ exists fs_q_pfc_shift_coefficient_newsum_body_start. fs_u_pfc_shift_coefficient_newsum = fs_q_pfc_shift_coefficient_newsum_body_start * S ((S (0)) * fs_v_pfc_shift_coefficient_newsum) + (0))) /\\ ((((exists fs_h_pfc_shift_coefficient_newsum_body_terminal. fs_h_pfc_shift_coefficient_newsum_body_terminal + S (pfc_natural_sum_shift_coefficient_new) = S ((S (S (i))) * fs_v_pfc_shift_coefficient_newsum)) /\\ exists fs_q_pfc_shift_coefficient_newsum_body_terminal. fs_u_pfc_shift_coefficient_newsum = fs_q_pfc_shift_coefficient_newsum_body_terminal * S ((S (S (i))) * fs_v_pfc_shift_coefficient_newsum) + (pfc_natural_sum_shift_coefficient_new))) /\\ forall fs_i_pfc_shift_coefficient_newsum_body_steps. (exists fs_lt_pfc_shift_coefficient_newsum_body_steps_bound. fs_lt_pfc_shift_coefficient_newsum_body_steps_bound + S fs_i_pfc_shift_coefficient_newsum_body_steps = S (i)) -> exists fs_a_pfc_shift_coefficient_newsum_body_steps fs_r_pfc_shift_coefficient_newsum_body_steps fs_s_pfc_shift_coefficient_newsum_body_steps. ((((exists fs_h_pfc_shift_coefficient_newsum_body_steps_summand. fs_h_pfc_shift_coefficient_newsum_body_steps_summand + S (fs_a_pfc_shift_coefficient_newsum_body_steps) = S ((S (fs_i_pfc_shift_coefficient_newsum_body_steps)) * pfc_terms_scale_shift_coefficient_new)) /\\ exists fs_q_pfc_shift_coefficient_newsum_body_steps_summand. pfc_terms_code_shift_coefficient_new = fs_q_pfc_shift_coefficient_newsum_body_steps_summand * S ((S (fs_i_pfc_shift_coefficient_newsum_body_steps)) * pfc_terms_scale_shift_coefficient_new) + (fs_a_pfc_shift_coefficient_newsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_coefficient_newsum_body_steps_partial. fs_h_pfc_shift_coefficient_newsum_body_steps_partial + S (fs_r_pfc_shift_coefficient_newsum_body_steps) = S ((S (fs_i_pfc_shift_coefficient_newsum_body_steps)) * fs_v_pfc_shift_coefficient_newsum)) /\\ exists fs_q_pfc_shift_coefficient_newsum_body_steps_partial. fs_u_pfc_shift_coefficient_newsum = fs_q_pfc_shift_coefficient_newsum_body_steps_partial * S ((S (fs_i_pfc_shift_coefficient_newsum_body_steps)) * fs_v_pfc_shift_coefficient_newsum) + (fs_r_pfc_shift_coefficient_newsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_coefficient_newsum_body_steps_successor. fs_h_pfc_shift_coefficient_newsum_body_steps_successor + S (fs_s_pfc_shift_coefficient_newsum_body_steps) = S ((S (S fs_i_pfc_shift_coefficient_newsum_body_steps)) * fs_v_pfc_shift_coefficient_newsum)) /\\ exists fs_q_pfc_shift_coefficient_newsum_body_steps_successor. fs_u_pfc_shift_coefficient_newsum = fs_q_pfc_shift_coefficient_newsum_body_steps_successor * S ((S (S fs_i_pfc_shift_coefficient_newsum_body_steps)) * fs_v_pfc_shift_coefficient_newsum) + (fs_s_pfc_shift_coefficient_newsum_body_steps))) /\\ fs_s_pfc_shift_coefficient_newsum_body_steps = fs_r_pfc_shift_coefficient_newsum_body_steps + fs_a_pfc_shift_coefficient_newsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_coefficient_newresiduebound. pfa_gap_shift_coefficient_newresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_shift_coefficient_newresiduecongruence pfa_offset_right_shift_coefficient_newresiduecongruence. (pfc_natural_sum_shift_coefficient_new) + (p) * pfa_offset_left_shift_coefficient_newresiduecongruence = (r) + (p) * pfa_offset_right_shift_coefficient_newresiduecongruence)))))))))) /\\ (((exists pfc_terms_code_shift_coefficient_new pfc_terms_scale_shift_coefficient_new pfc_natural_sum_shift_coefficient_new. ((forall pfc_index_shift_coefficient_newdiagonal. (exists pfa_gap_shift_coefficient_newdiagonalbound. pfa_gap_shift_coefficient_newdiagonalbound + S (pfc_index_shift_coefficient_newdiagonal) = (S (i))) -> exists pfc_value_shift_coefficient_newdiagonal. ((((exists ff_h_pfp_shift_coefficient_newdiagonalentry. ff_h_pfp_shift_coefficient_newdiagonalentry + S (pfc_value_shift_coefficient_newdiagonal) = S ((S (pfc_index_shift_coefficient_newdiagonal)) * pfc_terms_scale_shift_coefficient_new)) /\\ exists ff_q_pfp_shift_coefficient_newdiagonalentry. pfc_terms_code_shift_coefficient_new = ff_q_pfp_shift_coefficient_newdiagonalentry * S ((S (pfc_index_shift_coefficient_newdiagonal)) * pfc_terms_scale_shift_coefficient_new) + (pfc_value_shift_coefficient_newdiagonal))) /\\ ((exists pfc_complement_shift_coefficient_newdiagonalterm pfc_left_shift_coefficient_newdiagonalterm pfc_right_shift_coefficient_newdiagonalterm. (((pfc_index_shift_coefficient_newdiagonal)+pfc_complement_shift_coefficient_newdiagonalterm=(i)) /\\ ((((((exists pfa_gap_shift_coefficient_newdiagonaltermleftinside. pfa_gap_shift_coefficient_newdiagonaltermleftinside + S (pfc_index_shift_coefficient_newdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_coefficient_newdiagonaltermleftentry. ff_h_pfp_shift_coefficient_newdiagonaltermleftentry + S (pfc_left_shift_coefficient_newdiagonalterm) = S ((S (pfc_index_shift_coefficient_newdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_coefficient_newdiagonaltermleftentry. ab = ff_q_pfp_shift_coefficient_newdiagonaltermleftentry * S ((S (pfc_index_shift_coefficient_newdiagonal)) * ac) + (pfc_left_shift_coefficient_newdiagonalterm)))))) \\/ (((exists pfc_gap_shift_coefficient_newdiagonaltermleftoutside. pfc_gap_shift_coefficient_newdiagonaltermleftoutside+(L)=(pfc_index_shift_coefficient_newdiagonal)) /\\ (((pfc_left_shift_coefficient_newdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_coefficient_newdiagonaltermrightinside. pfa_gap_shift_coefficient_newdiagonaltermrightinside + S (pfc_complement_shift_coefficient_newdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_shift_coefficient_newdiagonaltermrightentry. ff_h_pfp_shift_coefficient_newdiagonaltermrightentry + S (pfc_right_shift_coefficient_newdiagonalterm) = S ((S (pfc_complement_shift_coefficient_newdiagonalterm)) * BC)) /\\ exists ff_q_pfp_shift_coefficient_newdiagonaltermrightentry. BB = ff_q_pfp_shift_coefficient_newdiagonaltermrightentry * S ((S (pfc_complement_shift_coefficient_newdiagonalterm)) * BC) + (pfc_right_shift_coefficient_newdiagonalterm)))))) \\/ (((exists pfc_gap_shift_coefficient_newdiagonaltermrightoutside. pfc_gap_shift_coefficient_newdiagonaltermrightoutside+(S M)=(pfc_complement_shift_coefficient_newdiagonalterm)) /\\ (((pfc_right_shift_coefficient_newdiagonalterm)=0))))) /\\ (((pfc_value_shift_coefficient_newdiagonal)=pfc_left_shift_coefficient_newdiagonalterm*pfc_right_shift_coefficient_newdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_coefficient_newsum fs_v_pfc_shift_coefficient_newsum. ((((exists fs_h_pfc_shift_coefficient_newsum_body_start. fs_h_pfc_shift_coefficient_newsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_coefficient_newsum)) /\\ exists fs_q_pfc_shift_coefficient_newsum_body_start. fs_u_pfc_shift_coefficient_newsum = fs_q_pfc_shift_coefficient_newsum_body_start * S ((S (0)) * fs_v_pfc_shift_coefficient_newsum) + (0))) /\\ ((((exists fs_h_pfc_shift_coefficient_newsum_body_terminal. fs_h_pfc_shift_coefficient_newsum_body_terminal + S (pfc_natural_sum_shift_coefficient_new) = S ((S (S (i))) * fs_v_pfc_shift_coefficient_newsum)) /\\ exists fs_q_pfc_shift_coefficient_newsum_body_terminal. fs_u_pfc_shift_coefficient_newsum = fs_q_pfc_shift_coefficient_newsum_body_terminal * S ((S (S (i))) * fs_v_pfc_shift_coefficient_newsum) + (pfc_natural_sum_shift_coefficient_new))) /\\ forall fs_i_pfc_shift_coefficient_newsum_body_steps. (exists fs_lt_pfc_shift_coefficient_newsum_body_steps_bound. fs_lt_pfc_shift_coefficient_newsum_body_steps_bound + S fs_i_pfc_shift_coefficient_newsum_body_steps = S (i)) -> exists fs_a_pfc_shift_coefficient_newsum_body_steps fs_r_pfc_shift_coefficient_newsum_body_steps fs_s_pfc_shift_coefficient_newsum_body_steps. ((((exists fs_h_pfc_shift_coefficient_newsum_body_steps_summand. fs_h_pfc_shift_coefficient_newsum_body_steps_summand + S (fs_a_pfc_shift_coefficient_newsum_body_steps) = S ((S (fs_i_pfc_shift_coefficient_newsum_body_steps)) * pfc_terms_scale_shift_coefficient_new)) /\\ exists fs_q_pfc_shift_coefficient_newsum_body_steps_summand. pfc_terms_code_shift_coefficient_new = fs_q_pfc_shift_coefficient_newsum_body_steps_summand * S ((S (fs_i_pfc_shift_coefficient_newsum_body_steps)) * pfc_terms_scale_shift_coefficient_new) + (fs_a_pfc_shift_coefficient_newsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_coefficient_newsum_body_steps_partial. fs_h_pfc_shift_coefficient_newsum_body_steps_partial + S (fs_r_pfc_shift_coefficient_newsum_body_steps) = S ((S (fs_i_pfc_shift_coefficient_newsum_body_steps)) * fs_v_pfc_shift_coefficient_newsum)) /\\ exists fs_q_pfc_shift_coefficient_newsum_body_steps_partial. fs_u_pfc_shift_coefficient_newsum = fs_q_pfc_shift_coefficient_newsum_body_steps_partial * S ((S (fs_i_pfc_shift_coefficient_newsum_body_steps)) * fs_v_pfc_shift_coefficient_newsum) + (fs_r_pfc_shift_coefficient_newsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_coefficient_newsum_body_steps_successor. fs_h_pfc_shift_coefficient_newsum_body_steps_successor + S (fs_s_pfc_shift_coefficient_newsum_body_steps) = S ((S (S fs_i_pfc_shift_coefficient_newsum_body_steps)) * fs_v_pfc_shift_coefficient_newsum)) /\\ exists fs_q_pfc_shift_coefficient_newsum_body_steps_successor. fs_u_pfc_shift_coefficient_newsum = fs_q_pfc_shift_coefficient_newsum_body_steps_successor * S ((S (S fs_i_pfc_shift_coefficient_newsum_body_steps)) * fs_v_pfc_shift_coefficient_newsum) + (fs_s_pfc_shift_coefficient_newsum_body_steps))) /\\ fs_s_pfc_shift_coefficient_newsum_body_steps = fs_r_pfc_shift_coefficient_newsum_body_steps + fs_a_pfc_shift_coefficient_newsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_coefficient_newresiduebound. pfa_gap_shift_coefficient_newresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_shift_coefficient_newresiduecongruence pfa_offset_right_shift_coefficient_newresiduecongruence. (pfc_natural_sum_shift_coefficient_new) + (p) * pfa_offset_left_shift_coefficient_newresiduecongruence = (r) + (p) * pfa_offset_right_shift_coefficient_newresiduecongruence))))))))) -> (exists pfc_terms_code_shift_coefficient_old pfc_terms_scale_shift_coefficient_old pfc_natural_sum_shift_coefficient_old. ((forall pfc_index_shift_coefficient_olddiagonal. (exists pfa_gap_shift_coefficient_olddiagonalbound. pfa_gap_shift_coefficient_olddiagonalbound + S (pfc_index_shift_coefficient_olddiagonal) = (S (i))) -> exists pfc_value_shift_coefficient_olddiagonal. ((((exists ff_h_pfp_shift_coefficient_olddiagonalentry. ff_h_pfp_shift_coefficient_olddiagonalentry + S (pfc_value_shift_coefficient_olddiagonal) = S ((S (pfc_index_shift_coefficient_olddiagonal)) * pfc_terms_scale_shift_coefficient_old)) /\\ exists ff_q_pfp_shift_coefficient_olddiagonalentry. pfc_terms_code_shift_coefficient_old = ff_q_pfp_shift_coefficient_olddiagonalentry * S ((S (pfc_index_shift_coefficient_olddiagonal)) * pfc_terms_scale_shift_coefficient_old) + (pfc_value_shift_coefficient_olddiagonal))) /\\ ((exists pfc_complement_shift_coefficient_olddiagonalterm pfc_left_shift_coefficient_olddiagonalterm pfc_right_shift_coefficient_olddiagonalterm. (((pfc_index_shift_coefficient_olddiagonal)+pfc_complement_shift_coefficient_olddiagonalterm=(i)) /\\ ((((((exists pfa_gap_shift_coefficient_olddiagonaltermleftinside. pfa_gap_shift_coefficient_olddiagonaltermleftinside + S (pfc_index_shift_coefficient_olddiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_coefficient_olddiagonaltermleftentry. ff_h_pfp_shift_coefficient_olddiagonaltermleftentry + S (pfc_left_shift_coefficient_olddiagonalterm) = S ((S (pfc_index_shift_coefficient_olddiagonal)) * ac)) /\\ exists ff_q_pfp_shift_coefficient_olddiagonaltermleftentry. ab = ff_q_pfp_shift_coefficient_olddiagonaltermleftentry * S ((S (pfc_index_shift_coefficient_olddiagonal)) * ac) + (pfc_left_shift_coefficient_olddiagonalterm)))))) \\/ (((exists pfc_gap_shift_coefficient_olddiagonaltermleftoutside. pfc_gap_shift_coefficient_olddiagonaltermleftoutside+(L)=(pfc_index_shift_coefficient_olddiagonal)) /\\ (((pfc_left_shift_coefficient_olddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_coefficient_olddiagonaltermrightinside. pfa_gap_shift_coefficient_olddiagonaltermrightinside + S (pfc_complement_shift_coefficient_olddiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_shift_coefficient_olddiagonaltermrightentry. ff_h_pfp_shift_coefficient_olddiagonaltermrightentry + S (pfc_right_shift_coefficient_olddiagonalterm) = S ((S (pfc_complement_shift_coefficient_olddiagonalterm)) * bc)) /\\ exists ff_q_pfp_shift_coefficient_olddiagonaltermrightentry. bb = ff_q_pfp_shift_coefficient_olddiagonaltermrightentry * S ((S (pfc_complement_shift_coefficient_olddiagonalterm)) * bc) + (pfc_right_shift_coefficient_olddiagonalterm)))))) \\/ (((exists pfc_gap_shift_coefficient_olddiagonaltermrightoutside. pfc_gap_shift_coefficient_olddiagonaltermrightoutside+(M)=(pfc_complement_shift_coefficient_olddiagonalterm)) /\\ (((pfc_right_shift_coefficient_olddiagonalterm)=0))))) /\\ (((pfc_value_shift_coefficient_olddiagonal)=pfc_left_shift_coefficient_olddiagonalterm*pfc_right_shift_coefficient_olddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_coefficient_oldsum fs_v_pfc_shift_coefficient_oldsum. ((((exists fs_h_pfc_shift_coefficient_oldsum_body_start. fs_h_pfc_shift_coefficient_oldsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_coefficient_oldsum)) /\\ exists fs_q_pfc_shift_coefficient_oldsum_body_start. fs_u_pfc_shift_coefficient_oldsum = fs_q_pfc_shift_coefficient_oldsum_body_start * S ((S (0)) * fs_v_pfc_shift_coefficient_oldsum) + (0))) /\\ ((((exists fs_h_pfc_shift_coefficient_oldsum_body_terminal. fs_h_pfc_shift_coefficient_oldsum_body_terminal + S (pfc_natural_sum_shift_coefficient_old) = S ((S (S (i))) * fs_v_pfc_shift_coefficient_oldsum)) /\\ exists fs_q_pfc_shift_coefficient_oldsum_body_terminal. fs_u_pfc_shift_coefficient_oldsum = fs_q_pfc_shift_coefficient_oldsum_body_terminal * S ((S (S (i))) * fs_v_pfc_shift_coefficient_oldsum) + (pfc_natural_sum_shift_coefficient_old))) /\\ forall fs_i_pfc_shift_coefficient_oldsum_body_steps. (exists fs_lt_pfc_shift_coefficient_oldsum_body_steps_bound. fs_lt_pfc_shift_coefficient_oldsum_body_steps_bound + S fs_i_pfc_shift_coefficient_oldsum_body_steps = S (i)) -> exists fs_a_pfc_shift_coefficient_oldsum_body_steps fs_r_pfc_shift_coefficient_oldsum_body_steps fs_s_pfc_shift_coefficient_oldsum_body_steps. ((((exists fs_h_pfc_shift_coefficient_oldsum_body_steps_summand. fs_h_pfc_shift_coefficient_oldsum_body_steps_summand + S (fs_a_pfc_shift_coefficient_oldsum_body_steps) = S ((S (fs_i_pfc_shift_coefficient_oldsum_body_steps)) * pfc_terms_scale_shift_coefficient_old)) /\\ exists fs_q_pfc_shift_coefficient_oldsum_body_steps_summand. pfc_terms_code_shift_coefficient_old = fs_q_pfc_shift_coefficient_oldsum_body_steps_summand * S ((S (fs_i_pfc_shift_coefficient_oldsum_body_steps)) * pfc_terms_scale_shift_coefficient_old) + (fs_a_pfc_shift_coefficient_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_coefficient_oldsum_body_steps_partial. fs_h_pfc_shift_coefficient_oldsum_body_steps_partial + S (fs_r_pfc_shift_coefficient_oldsum_body_steps) = S ((S (fs_i_pfc_shift_coefficient_oldsum_body_steps)) * fs_v_pfc_shift_coefficient_oldsum)) /\\ exists fs_q_pfc_shift_coefficient_oldsum_body_steps_partial. fs_u_pfc_shift_coefficient_oldsum = fs_q_pfc_shift_coefficient_oldsum_body_steps_partial * S ((S (fs_i_pfc_shift_coefficient_oldsum_body_steps)) * fs_v_pfc_shift_coefficient_oldsum) + (fs_r_pfc_shift_coefficient_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_coefficient_oldsum_body_steps_successor. fs_h_pfc_shift_coefficient_oldsum_body_steps_successor + S (fs_s_pfc_shift_coefficient_oldsum_body_steps) = S ((S (S fs_i_pfc_shift_coefficient_oldsum_body_steps)) * fs_v_pfc_shift_coefficient_oldsum)) /\\ exists fs_q_pfc_shift_coefficient_oldsum_body_steps_successor. fs_u_pfc_shift_coefficient_oldsum = fs_q_pfc_shift_coefficient_oldsum_body_steps_successor * S ((S (S fs_i_pfc_shift_coefficient_oldsum_body_steps)) * fs_v_pfc_shift_coefficient_oldsum) + (fs_s_pfc_shift_coefficient_oldsum_body_steps))) /\\ fs_s_pfc_shift_coefficient_oldsum_body_steps = fs_r_pfc_shift_coefficient_oldsum_body_steps + fs_a_pfc_shift_coefficient_oldsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_coefficient_oldresiduebound. pfa_gap_shift_coefficient_oldresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_shift_coefficient_oldresiduecongruence pfa_offset_right_shift_coefficient_oldresiduecongruence. 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"prime_field_polynomial_convolution_shift_right_nonempty", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro M", "intro cb", "intro cc", "intro N", "intro BB", "intro BC", "intro db", "intro dc", "intro K", "intro hp", "intro hL", "intro hM", "intro hs", "intro hc", "intro hd", "have hwhole : ((forall fom_index_pfp_shift_product_oldleft. (exists fom_gap_pfp_shift_product_oldleft_index_bound. fom_gap_pfp_shift_product_oldleft_index_bound + S (fom_index_pfp_shift_product_oldleft) = L) -> exists fom_value_pfp_shift_product_oldleft. ((((exists fom_beta_height_pfp_shift_product_oldleft_entry. fom_beta_height_pfp_shift_product_oldleft_entry + S (fom_value_pfp_shift_product_oldleft) = S ((S (fom_index_pfp_shift_product_oldleft)) * ac)) /\\ exists fom_beta_quotient_pfp_shift_product_oldleft_entry. ab = fom_beta_quotient_pfp_shift_product_oldleft_entry * S ((S (fom_index_pfp_shift_product_oldleft)) * ac) + (fom_value_pfp_shift_product_oldleft))) /\\ (exists fom_gap_pfp_shift_product_oldleft_value_bound. fom_gap_pfp_shift_product_oldleft_value_bound + S (fom_value_pfp_shift_product_oldleft) = p))) /\\ (((forall fom_index_pfp_shift_product_oldright. (exists fom_gap_pfp_shift_product_oldright_index_bound. fom_gap_pfp_shift_product_oldright_index_bound + S (fom_index_pfp_shift_product_oldright) = M) -> exists fom_value_pfp_shift_product_oldright. ((((exists fom_beta_height_pfp_shift_product_oldright_entry. fom_beta_height_pfp_shift_product_oldright_entry + S (fom_value_pfp_shift_product_oldright) = S ((S (fom_index_pfp_shift_product_oldright)) * bc)) /\\ exists fom_beta_quotient_pfp_shift_product_oldright_entry. bb = fom_beta_quotient_pfp_shift_product_oldright_entry * S ((S (fom_index_pfp_shift_product_oldright)) * bc) + (fom_value_pfp_shift_product_oldright))) /\\ (exists fom_gap_pfp_shift_product_oldright_value_bound. fom_gap_pfp_shift_product_oldright_value_bound + S (fom_value_pfp_shift_product_oldright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_shift_product_oldcoefficients. (exists pfa_gap_shift_product_oldcoefficientsbound. pfa_gap_shift_product_oldcoefficientsbound + S (pfc_index_shift_product_oldcoefficients) = (N)) -> exists pfc_value_shift_product_oldcoefficients. ((((exists ff_h_pfp_shift_product_oldcoefficientsentry. ff_h_pfp_shift_product_oldcoefficientsentry + S (pfc_value_shift_product_oldcoefficients) = S ((S (pfc_index_shift_product_oldcoefficients)) * cc)) /\\ exists ff_q_pfp_shift_product_oldcoefficientsentry. cb = ff_q_pfp_shift_product_oldcoefficientsentry * S ((S (pfc_index_shift_product_oldcoefficients)) * cc) + (pfc_value_shift_product_oldcoefficients))) /\\ ((exists pfc_terms_code_shift_product_oldcoefficientscoefficient pfc_terms_scale_shift_product_oldcoefficientscoefficient pfc_natural_sum_shift_product_oldcoefficientscoefficient. ((forall pfc_index_shift_product_oldcoefficientscoefficientdiagonal. (exists pfa_gap_shift_product_oldcoefficientscoefficientdiagonalbound. pfa_gap_shift_product_oldcoefficientscoefficientdiagonalbound + S (pfc_index_shift_product_oldcoefficientscoefficientdiagonal) = (S (pfc_index_shift_product_oldcoefficients))) -> exists pfc_value_shift_product_oldcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_shift_product_oldcoefficientscoefficientdiagonalentry. ff_h_pfp_shift_product_oldcoefficientscoefficientdiagonalentry + S (pfc_value_shift_product_oldcoefficientscoefficientdiagonal) = S ((S (pfc_index_shift_product_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_product_oldcoefficientscoefficient)) /\\ exists ff_q_pfp_shift_product_oldcoefficientscoefficientdiagonalentry. pfc_terms_code_shift_product_oldcoefficientscoefficient = ff_q_pfp_shift_product_oldcoefficientscoefficientdiagonalentry * S ((S (pfc_index_shift_product_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_product_oldcoefficientscoefficient) + (pfc_value_shift_product_oldcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_shift_product_oldcoefficientscoefficientdiagonalterm pfc_left_shift_product_oldcoefficientscoefficientdiagonalterm pfc_right_shift_product_oldcoefficientscoefficientdiagonalterm. (((pfc_index_shift_product_oldcoefficientscoefficientdiagonal)+pfc_complement_shift_product_oldcoefficientscoefficientdiagonalterm=(pfc_index_shift_product_oldcoefficients)) /\\ ((((((exists pfa_gap_shift_product_oldcoefficientscoefficientdiagonaltermleftinside. pfa_gap_shift_product_oldcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_shift_product_oldcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_oldcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_shift_product_oldcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_shift_product_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_shift_product_oldcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_oldcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_shift_product_oldcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_shift_product_oldcoefficientscoefficientdiagonal)) * ac) + (pfc_left_shift_product_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_oldcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_shift_product_oldcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_shift_product_oldcoefficientscoefficientdiagonal)) /\\ (((pfc_left_shift_product_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_oldcoefficientscoefficientdiagonaltermrightinside. pfa_gap_shift_product_oldcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_shift_product_oldcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_shift_product_oldcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_shift_product_oldcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_shift_product_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_shift_product_oldcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_shift_product_oldcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_shift_product_oldcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_shift_product_oldcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_shift_product_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_oldcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_shift_product_oldcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_shift_product_oldcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_shift_product_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_shift_product_oldcoefficientscoefficientdiagonal)=pfc_left_shift_product_oldcoefficientscoefficientdiagonalterm*pfc_right_shift_product_oldcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_oldcoefficientscoefficientsum fs_v_pfc_shift_product_oldcoefficientscoefficientsum. ((((exists fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_start. fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_start. fs_u_pfc_shift_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_terminal. fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_shift_product_oldcoefficientscoefficient) = S ((S (S (pfc_index_shift_product_oldcoefficients))) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_terminal. fs_u_pfc_shift_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_shift_product_oldcoefficients))) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum) + (pfc_natural_sum_shift_product_oldcoefficientscoefficient))) /\\ forall fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps = S (pfc_index_shift_product_oldcoefficients)) -> exists fs_a_pfc_shift_product_oldcoefficientscoefficientsum_body_steps fs_r_pfc_shift_product_oldcoefficientscoefficientsum_body_steps fs_s_pfc_shift_product_oldcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_shift_product_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_product_oldcoefficientscoefficient)) /\\ exists fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_shift_product_oldcoefficientscoefficient = fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_product_oldcoefficientscoefficient) + (fs_a_pfc_shift_product_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_shift_product_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_shift_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum) + (fs_r_pfc_shift_product_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_shift_product_oldcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_shift_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum) + (fs_s_pfc_shift_product_oldcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_shift_product_oldcoefficientscoefficientsum_body_steps = fs_r_pfc_shift_product_oldcoefficientscoefficientsum_body_steps + fs_a_pfc_shift_product_oldcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_oldcoefficientscoefficientresiduebound. pfa_gap_shift_product_oldcoefficientscoefficientresiduebound + S (pfc_value_shift_product_oldcoefficients) = (p)) /\\ ((exists pfa_offset_left_shift_product_oldcoefficientscoefficientresiduecongruence pfa_offset_right_shift_product_oldcoefficientscoefficientresiduecongruence. (pfc_natural_sum_shift_product_oldcoefficientscoefficient) + (p) * pfa_offset_left_shift_product_oldcoefficientscoefficientresiduecongruence = (pfc_value_shift_product_oldcoefficients) + (p) * pfa_offset_right_shift_product_oldcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hc", "cases hc", "cases hc_right", "cases hc_right_right", "cases hd", "cases hd_right", "cases hd_right_right", "have hk : K=S N", "specialize polynomial_product_length_shift_right_nonempty (L)", "specialize polynomial_product_length_shift_right_nonempty (M)", "specialize polynomial_product_length_shift_right_nonempty (N)", "specialize polynomial_product_length_shift_right_nonempty (K)", "apply polynomial_product_length_shift_right_nonempty", "exact hc_right_right_left", "exact hL", "exact hM", "exact hd_right_right_left", "split", "exact hk", "split", "intro i", "intro a", "intro hi", "intro ha", "have hca : exists pfc_terms_code_shift_product_source_coefficient pfc_terms_scale_shift_product_source_coefficient pfc_natural_sum_shift_product_source_coefficient. ((forall pfc_index_shift_product_source_coefficientdiagonal. (exists pfa_gap_shift_product_source_coefficientdiagonalbound. pfa_gap_shift_product_source_coefficientdiagonalbound + S (pfc_index_shift_product_source_coefficientdiagonal) = (S (i))) -> exists pfc_value_shift_product_source_coefficientdiagonal. ((((exists ff_h_pfp_shift_product_source_coefficientdiagonalentry. ff_h_pfp_shift_product_source_coefficientdiagonalentry + S (pfc_value_shift_product_source_coefficientdiagonal) = S ((S (pfc_index_shift_product_source_coefficientdiagonal)) * pfc_terms_scale_shift_product_source_coefficient)) /\\ exists ff_q_pfp_shift_product_source_coefficientdiagonalentry. pfc_terms_code_shift_product_source_coefficient = ff_q_pfp_shift_product_source_coefficientdiagonalentry * S ((S (pfc_index_shift_product_source_coefficientdiagonal)) * pfc_terms_scale_shift_product_source_coefficient) + (pfc_value_shift_product_source_coefficientdiagonal))) /\\ ((exists pfc_complement_shift_product_source_coefficientdiagonalterm pfc_left_shift_product_source_coefficientdiagonalterm pfc_right_shift_product_source_coefficientdiagonalterm. (((pfc_index_shift_product_source_coefficientdiagonal)+pfc_complement_shift_product_source_coefficientdiagonalterm=(i)) /\\ ((((((exists pfa_gap_shift_product_source_coefficientdiagonaltermleftinside. pfa_gap_shift_product_source_coefficientdiagonaltermleftinside + S (pfc_index_shift_product_source_coefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_source_coefficientdiagonaltermleftentry. ff_h_pfp_shift_product_source_coefficientdiagonaltermleftentry + S (pfc_left_shift_product_source_coefficientdiagonalterm) = S ((S (pfc_index_shift_product_source_coefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_source_coefficientdiagonaltermleftentry. ab = ff_q_pfp_shift_product_source_coefficientdiagonaltermleftentry * S ((S (pfc_index_shift_product_source_coefficientdiagonal)) * ac) + (pfc_left_shift_product_source_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_source_coefficientdiagonaltermleftoutside. pfc_gap_shift_product_source_coefficientdiagonaltermleftoutside+(L)=(pfc_index_shift_product_source_coefficientdiagonal)) /\\ (((pfc_left_shift_product_source_coefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_source_coefficientdiagonaltermrightinside. pfa_gap_shift_product_source_coefficientdiagonaltermrightinside + S (pfc_complement_shift_product_source_coefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_shift_product_source_coefficientdiagonaltermrightentry. ff_h_pfp_shift_product_source_coefficientdiagonaltermrightentry + S (pfc_right_shift_product_source_coefficientdiagonalterm) = S ((S (pfc_complement_shift_product_source_coefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_shift_product_source_coefficientdiagonaltermrightentry. bb = ff_q_pfp_shift_product_source_coefficientdiagonaltermrightentry * S ((S (pfc_complement_shift_product_source_coefficientdiagonalterm)) * bc) + (pfc_right_shift_product_source_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_source_coefficientdiagonaltermrightoutside. pfc_gap_shift_product_source_coefficientdiagonaltermrightoutside+(M)=(pfc_complement_shift_product_source_coefficientdiagonalterm)) /\\ (((pfc_right_shift_product_source_coefficientdiagonalterm)=0))))) /\\ (((pfc_value_shift_product_source_coefficientdiagonal)=pfc_left_shift_product_source_coefficientdiagonalterm*pfc_right_shift_product_source_coefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_source_coefficientsum fs_v_pfc_shift_product_source_coefficientsum. ((((exists fs_h_pfc_shift_product_source_coefficientsum_body_start. fs_h_pfc_shift_product_source_coefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_source_coefficientsum)) /\\ exists fs_q_pfc_shift_product_source_coefficientsum_body_start. fs_u_pfc_shift_product_source_coefficientsum = fs_q_pfc_shift_product_source_coefficientsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_source_coefficientsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_source_coefficientsum_body_terminal. fs_h_pfc_shift_product_source_coefficientsum_body_terminal + S (pfc_natural_sum_shift_product_source_coefficient) = S ((S (S (i))) * fs_v_pfc_shift_product_source_coefficientsum)) /\\ exists fs_q_pfc_shift_product_source_coefficientsum_body_terminal. fs_u_pfc_shift_product_source_coefficientsum = fs_q_pfc_shift_product_source_coefficientsum_body_terminal * S ((S (S (i))) * fs_v_pfc_shift_product_source_coefficientsum) + (pfc_natural_sum_shift_product_source_coefficient))) /\\ forall fs_i_pfc_shift_product_source_coefficientsum_body_steps. (exists fs_lt_pfc_shift_product_source_coefficientsum_body_steps_bound. fs_lt_pfc_shift_product_source_coefficientsum_body_steps_bound + S fs_i_pfc_shift_product_source_coefficientsum_body_steps = S (i)) -> exists fs_a_pfc_shift_product_source_coefficientsum_body_steps fs_r_pfc_shift_product_source_coefficientsum_body_steps fs_s_pfc_shift_product_source_coefficientsum_body_steps. ((((exists fs_h_pfc_shift_product_source_coefficientsum_body_steps_summand. fs_h_pfc_shift_product_source_coefficientsum_body_steps_summand + S (fs_a_pfc_shift_product_source_coefficientsum_body_steps) = S ((S (fs_i_pfc_shift_product_source_coefficientsum_body_steps)) * pfc_terms_scale_shift_product_source_coefficient)) /\\ exists fs_q_pfc_shift_product_source_coefficientsum_body_steps_summand. pfc_terms_code_shift_product_source_coefficient = fs_q_pfc_shift_product_source_coefficientsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_source_coefficientsum_body_steps)) * pfc_terms_scale_shift_product_source_coefficient) + (fs_a_pfc_shift_product_source_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_source_coefficientsum_body_steps_partial. fs_h_pfc_shift_product_source_coefficientsum_body_steps_partial + S (fs_r_pfc_shift_product_source_coefficientsum_body_steps) = S ((S (fs_i_pfc_shift_product_source_coefficientsum_body_steps)) * fs_v_pfc_shift_product_source_coefficientsum)) /\\ exists fs_q_pfc_shift_product_source_coefficientsum_body_steps_partial. fs_u_pfc_shift_product_source_coefficientsum = fs_q_pfc_shift_product_source_coefficientsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_source_coefficientsum_body_steps)) * fs_v_pfc_shift_product_source_coefficientsum) + (fs_r_pfc_shift_product_source_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_source_coefficientsum_body_steps_successor. fs_h_pfc_shift_product_source_coefficientsum_body_steps_successor + S (fs_s_pfc_shift_product_source_coefficientsum_body_steps) = S ((S (S fs_i_pfc_shift_product_source_coefficientsum_body_steps)) * fs_v_pfc_shift_product_source_coefficientsum)) /\\ exists fs_q_pfc_shift_product_source_coefficientsum_body_steps_successor. fs_u_pfc_shift_product_source_coefficientsum = fs_q_pfc_shift_product_source_coefficientsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_source_coefficientsum_body_steps)) * fs_v_pfc_shift_product_source_coefficientsum) + (fs_s_pfc_shift_product_source_coefficientsum_body_steps))) /\\ fs_s_pfc_shift_product_source_coefficientsum_body_steps = fs_r_pfc_shift_product_source_coefficientsum_body_steps + fs_a_pfc_shift_product_source_coefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_source_coefficientresiduebound. pfa_gap_shift_product_source_coefficientresiduebound + S (a) = (p)) /\\ ((exists pfa_offset_left_shift_product_source_coefficientresiduecongruence pfa_offset_right_shift_product_source_coefficientresiduecongruence. (pfc_natural_sum_shift_product_source_coefficient) + (p) * pfa_offset_left_shift_product_source_coefficientresiduecongruence = (a) + (p) * pfa_offset_right_shift_product_source_coefficientresiduecongruence))))))))", "specialize prime_field_convolution_prefix_entry (p)", "specialize prime_field_convolution_prefix_entry (ab)", "specialize prime_field_convolution_prefix_entry (ac)", "specialize prime_field_convolution_prefix_entry (L)", "specialize prime_field_convolution_prefix_entry (bb)", "specialize prime_field_convolution_prefix_entry (bc)", "specialize prime_field_convolution_prefix_entry (M)", "specialize prime_field_convolution_prefix_entry (cb)", "specialize prime_field_convolution_prefix_entry (cc)", "specialize prime_field_convolution_prefix_entry (N)", "specialize prime_field_convolution_prefix_entry (i)", "specialize prime_field_convolution_prefix_entry (a)", "apply prime_field_convolution_prefix_entry", "exact hc_right_right_right", "exact hi", "exact ha", "have hda : exists pfc_terms_code_shift_product_transported_coefficient pfc_terms_scale_shift_product_transported_coefficient pfc_natural_sum_shift_product_transported_coefficient. ((forall pfc_index_shift_product_transported_coefficientdiagonal. (exists pfa_gap_shift_product_transported_coefficientdiagonalbound. pfa_gap_shift_product_transported_coefficientdiagonalbound + S (pfc_index_shift_product_transported_coefficientdiagonal) = (S (i))) -> exists pfc_value_shift_product_transported_coefficientdiagonal. ((((exists ff_h_pfp_shift_product_transported_coefficientdiagonalentry. ff_h_pfp_shift_product_transported_coefficientdiagonalentry + S (pfc_value_shift_product_transported_coefficientdiagonal) = S ((S (pfc_index_shift_product_transported_coefficientdiagonal)) * pfc_terms_scale_shift_product_transported_coefficient)) /\\ exists ff_q_pfp_shift_product_transported_coefficientdiagonalentry. pfc_terms_code_shift_product_transported_coefficient = ff_q_pfp_shift_product_transported_coefficientdiagonalentry * S ((S (pfc_index_shift_product_transported_coefficientdiagonal)) * pfc_terms_scale_shift_product_transported_coefficient) + (pfc_value_shift_product_transported_coefficientdiagonal))) /\\ ((exists pfc_complement_shift_product_transported_coefficientdiagonalterm pfc_left_shift_product_transported_coefficientdiagonalterm pfc_right_shift_product_transported_coefficientdiagonalterm. (((pfc_index_shift_product_transported_coefficientdiagonal)+pfc_complement_shift_product_transported_coefficientdiagonalterm=(i)) /\\ ((((((exists pfa_gap_shift_product_transported_coefficientdiagonaltermleftinside. pfa_gap_shift_product_transported_coefficientdiagonaltermleftinside + S (pfc_index_shift_product_transported_coefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_transported_coefficientdiagonaltermleftentry. ff_h_pfp_shift_product_transported_coefficientdiagonaltermleftentry + S (pfc_left_shift_product_transported_coefficientdiagonalterm) = S ((S (pfc_index_shift_product_transported_coefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_transported_coefficientdiagonaltermleftentry. ab = ff_q_pfp_shift_product_transported_coefficientdiagonaltermleftentry * S ((S (pfc_index_shift_product_transported_coefficientdiagonal)) * ac) + (pfc_left_shift_product_transported_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_transported_coefficientdiagonaltermleftoutside. pfc_gap_shift_product_transported_coefficientdiagonaltermleftoutside+(L)=(pfc_index_shift_product_transported_coefficientdiagonal)) /\\ (((pfc_left_shift_product_transported_coefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_transported_coefficientdiagonaltermrightinside. pfa_gap_shift_product_transported_coefficientdiagonaltermrightinside + S (pfc_complement_shift_product_transported_coefficientdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_shift_product_transported_coefficientdiagonaltermrightentry. ff_h_pfp_shift_product_transported_coefficientdiagonaltermrightentry + S (pfc_right_shift_product_transported_coefficientdiagonalterm) = S ((S (pfc_complement_shift_product_transported_coefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_shift_product_transported_coefficientdiagonaltermrightentry. BB = ff_q_pfp_shift_product_transported_coefficientdiagonaltermrightentry * S ((S (pfc_complement_shift_product_transported_coefficientdiagonalterm)) * BC) + (pfc_right_shift_product_transported_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_transported_coefficientdiagonaltermrightoutside. pfc_gap_shift_product_transported_coefficientdiagonaltermrightoutside+(S M)=(pfc_complement_shift_product_transported_coefficientdiagonalterm)) /\\ (((pfc_right_shift_product_transported_coefficientdiagonalterm)=0))))) /\\ (((pfc_value_shift_product_transported_coefficientdiagonal)=pfc_left_shift_product_transported_coefficientdiagonalterm*pfc_right_shift_product_transported_coefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_transported_coefficientsum fs_v_pfc_shift_product_transported_coefficientsum. ((((exists fs_h_pfc_shift_product_transported_coefficientsum_body_start. fs_h_pfc_shift_product_transported_coefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_transported_coefficientsum)) /\\ exists fs_q_pfc_shift_product_transported_coefficientsum_body_start. fs_u_pfc_shift_product_transported_coefficientsum = fs_q_pfc_shift_product_transported_coefficientsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_transported_coefficientsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_transported_coefficientsum_body_terminal. fs_h_pfc_shift_product_transported_coefficientsum_body_terminal + S (pfc_natural_sum_shift_product_transported_coefficient) = S ((S (S (i))) * fs_v_pfc_shift_product_transported_coefficientsum)) /\\ exists fs_q_pfc_shift_product_transported_coefficientsum_body_terminal. fs_u_pfc_shift_product_transported_coefficientsum = fs_q_pfc_shift_product_transported_coefficientsum_body_terminal * S ((S (S (i))) * fs_v_pfc_shift_product_transported_coefficientsum) + (pfc_natural_sum_shift_product_transported_coefficient))) /\\ forall fs_i_pfc_shift_product_transported_coefficientsum_body_steps. (exists fs_lt_pfc_shift_product_transported_coefficientsum_body_steps_bound. fs_lt_pfc_shift_product_transported_coefficientsum_body_steps_bound + S fs_i_pfc_shift_product_transported_coefficientsum_body_steps = S (i)) -> exists fs_a_pfc_shift_product_transported_coefficientsum_body_steps fs_r_pfc_shift_product_transported_coefficientsum_body_steps fs_s_pfc_shift_product_transported_coefficientsum_body_steps. ((((exists fs_h_pfc_shift_product_transported_coefficientsum_body_steps_summand. fs_h_pfc_shift_product_transported_coefficientsum_body_steps_summand + S (fs_a_pfc_shift_product_transported_coefficientsum_body_steps) = S ((S (fs_i_pfc_shift_product_transported_coefficientsum_body_steps)) * pfc_terms_scale_shift_product_transported_coefficient)) /\\ exists fs_q_pfc_shift_product_transported_coefficientsum_body_steps_summand. pfc_terms_code_shift_product_transported_coefficient = fs_q_pfc_shift_product_transported_coefficientsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_transported_coefficientsum_body_steps)) * pfc_terms_scale_shift_product_transported_coefficient) + (fs_a_pfc_shift_product_transported_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_transported_coefficientsum_body_steps_partial. fs_h_pfc_shift_product_transported_coefficientsum_body_steps_partial + S (fs_r_pfc_shift_product_transported_coefficientsum_body_steps) = S ((S (fs_i_pfc_shift_product_transported_coefficientsum_body_steps)) * fs_v_pfc_shift_product_transported_coefficientsum)) /\\ exists fs_q_pfc_shift_product_transported_coefficientsum_body_steps_partial. fs_u_pfc_shift_product_transported_coefficientsum = fs_q_pfc_shift_product_transported_coefficientsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_transported_coefficientsum_body_steps)) * fs_v_pfc_shift_product_transported_coefficientsum) + (fs_r_pfc_shift_product_transported_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_transported_coefficientsum_body_steps_successor. fs_h_pfc_shift_product_transported_coefficientsum_body_steps_successor + S (fs_s_pfc_shift_product_transported_coefficientsum_body_steps) = S ((S (S fs_i_pfc_shift_product_transported_coefficientsum_body_steps)) * fs_v_pfc_shift_product_transported_coefficientsum)) /\\ exists fs_q_pfc_shift_product_transported_coefficientsum_body_steps_successor. fs_u_pfc_shift_product_transported_coefficientsum = fs_q_pfc_shift_product_transported_coefficientsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_transported_coefficientsum_body_steps)) * fs_v_pfc_shift_product_transported_coefficientsum) + (fs_s_pfc_shift_product_transported_coefficientsum_body_steps))) /\\ fs_s_pfc_shift_product_transported_coefficientsum_body_steps = fs_r_pfc_shift_product_transported_coefficientsum_body_steps + fs_a_pfc_shift_product_transported_coefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_transported_coefficientresiduebound. pfa_gap_shift_product_transported_coefficientresiduebound + S (a) = (p)) /\\ ((exists pfa_offset_left_shift_product_transported_coefficientresiduecongruence pfa_offset_right_shift_product_transported_coefficientresiduecongruence. (pfc_natural_sum_shift_product_transported_coefficient) + (p) * pfa_offset_left_shift_product_transported_coefficientresiduecongruence = (a) + (p) * pfa_offset_right_shift_product_transported_coefficientresiduecongruence))))))))", "have hiff : (((exists pfc_terms_code_shift_product_transport_old pfc_terms_scale_shift_product_transport_old pfc_natural_sum_shift_product_transport_old. ((forall pfc_index_shift_product_transport_olddiagonal. (exists pfa_gap_shift_product_transport_olddiagonalbound. pfa_gap_shift_product_transport_olddiagonalbound + S (pfc_index_shift_product_transport_olddiagonal) = (S (i))) -> exists pfc_value_shift_product_transport_olddiagonal. ((((exists ff_h_pfp_shift_product_transport_olddiagonalentry. ff_h_pfp_shift_product_transport_olddiagonalentry + S (pfc_value_shift_product_transport_olddiagonal) = S ((S (pfc_index_shift_product_transport_olddiagonal)) * pfc_terms_scale_shift_product_transport_old)) /\\ exists ff_q_pfp_shift_product_transport_olddiagonalentry. pfc_terms_code_shift_product_transport_old = ff_q_pfp_shift_product_transport_olddiagonalentry * S ((S (pfc_index_shift_product_transport_olddiagonal)) * pfc_terms_scale_shift_product_transport_old) + (pfc_value_shift_product_transport_olddiagonal))) /\\ ((exists pfc_complement_shift_product_transport_olddiagonalterm pfc_left_shift_product_transport_olddiagonalterm pfc_right_shift_product_transport_olddiagonalterm. (((pfc_index_shift_product_transport_olddiagonal)+pfc_complement_shift_product_transport_olddiagonalterm=(i)) /\\ ((((((exists pfa_gap_shift_product_transport_olddiagonaltermleftinside. pfa_gap_shift_product_transport_olddiagonaltermleftinside + S (pfc_index_shift_product_transport_olddiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_transport_olddiagonaltermleftentry. ff_h_pfp_shift_product_transport_olddiagonaltermleftentry + S (pfc_left_shift_product_transport_olddiagonalterm) = S ((S (pfc_index_shift_product_transport_olddiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_transport_olddiagonaltermleftentry. ab = ff_q_pfp_shift_product_transport_olddiagonaltermleftentry * S ((S (pfc_index_shift_product_transport_olddiagonal)) * ac) + (pfc_left_shift_product_transport_olddiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_transport_olddiagonaltermleftoutside. pfc_gap_shift_product_transport_olddiagonaltermleftoutside+(L)=(pfc_index_shift_product_transport_olddiagonal)) /\\ (((pfc_left_shift_product_transport_olddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_transport_olddiagonaltermrightinside. pfa_gap_shift_product_transport_olddiagonaltermrightinside + S (pfc_complement_shift_product_transport_olddiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_shift_product_transport_olddiagonaltermrightentry. ff_h_pfp_shift_product_transport_olddiagonaltermrightentry + S (pfc_right_shift_product_transport_olddiagonalterm) = S ((S (pfc_complement_shift_product_transport_olddiagonalterm)) * bc)) /\\ exists ff_q_pfp_shift_product_transport_olddiagonaltermrightentry. bb = ff_q_pfp_shift_product_transport_olddiagonaltermrightentry * S ((S (pfc_complement_shift_product_transport_olddiagonalterm)) * bc) + (pfc_right_shift_product_transport_olddiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_transport_olddiagonaltermrightoutside. pfc_gap_shift_product_transport_olddiagonaltermrightoutside+(M)=(pfc_complement_shift_product_transport_olddiagonalterm)) /\\ (((pfc_right_shift_product_transport_olddiagonalterm)=0))))) /\\ (((pfc_value_shift_product_transport_olddiagonal)=pfc_left_shift_product_transport_olddiagonalterm*pfc_right_shift_product_transport_olddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_transport_oldsum fs_v_pfc_shift_product_transport_oldsum. ((((exists fs_h_pfc_shift_product_transport_oldsum_body_start. fs_h_pfc_shift_product_transport_oldsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_transport_oldsum_body_start. fs_u_pfc_shift_product_transport_oldsum = fs_q_pfc_shift_product_transport_oldsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_transport_oldsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_transport_oldsum_body_terminal. fs_h_pfc_shift_product_transport_oldsum_body_terminal + S (pfc_natural_sum_shift_product_transport_old) = S ((S (S (i))) * fs_v_pfc_shift_product_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_transport_oldsum_body_terminal. fs_u_pfc_shift_product_transport_oldsum = fs_q_pfc_shift_product_transport_oldsum_body_terminal * S ((S (S (i))) * fs_v_pfc_shift_product_transport_oldsum) + (pfc_natural_sum_shift_product_transport_old))) /\\ forall fs_i_pfc_shift_product_transport_oldsum_body_steps. (exists fs_lt_pfc_shift_product_transport_oldsum_body_steps_bound. fs_lt_pfc_shift_product_transport_oldsum_body_steps_bound + S fs_i_pfc_shift_product_transport_oldsum_body_steps = S (i)) -> exists fs_a_pfc_shift_product_transport_oldsum_body_steps fs_r_pfc_shift_product_transport_oldsum_body_steps fs_s_pfc_shift_product_transport_oldsum_body_steps. ((((exists fs_h_pfc_shift_product_transport_oldsum_body_steps_summand. fs_h_pfc_shift_product_transport_oldsum_body_steps_summand + S (fs_a_pfc_shift_product_transport_oldsum_body_steps) = S ((S (fs_i_pfc_shift_product_transport_oldsum_body_steps)) * pfc_terms_scale_shift_product_transport_old)) /\\ exists fs_q_pfc_shift_product_transport_oldsum_body_steps_summand. pfc_terms_code_shift_product_transport_old = fs_q_pfc_shift_product_transport_oldsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_transport_oldsum_body_steps)) * pfc_terms_scale_shift_product_transport_old) + (fs_a_pfc_shift_product_transport_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_transport_oldsum_body_steps_partial. fs_h_pfc_shift_product_transport_oldsum_body_steps_partial + S (fs_r_pfc_shift_product_transport_oldsum_body_steps) = S ((S (fs_i_pfc_shift_product_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_transport_oldsum_body_steps_partial. fs_u_pfc_shift_product_transport_oldsum = fs_q_pfc_shift_product_transport_oldsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_transport_oldsum) + (fs_r_pfc_shift_product_transport_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_transport_oldsum_body_steps_successor. fs_h_pfc_shift_product_transport_oldsum_body_steps_successor + S (fs_s_pfc_shift_product_transport_oldsum_body_steps) = S ((S (S fs_i_pfc_shift_product_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_transport_oldsum_body_steps_successor. fs_u_pfc_shift_product_transport_oldsum = fs_q_pfc_shift_product_transport_oldsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_transport_oldsum) + (fs_s_pfc_shift_product_transport_oldsum_body_steps))) /\\ fs_s_pfc_shift_product_transport_oldsum_body_steps = fs_r_pfc_shift_product_transport_oldsum_body_steps + fs_a_pfc_shift_product_transport_oldsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_transport_oldresiduebound. pfa_gap_shift_product_transport_oldresiduebound + S (a) = (p)) /\\ ((exists pfa_offset_left_shift_product_transport_oldresiduecongruence pfa_offset_right_shift_product_transport_oldresiduecongruence. (pfc_natural_sum_shift_product_transport_old) + (p) * pfa_offset_left_shift_product_transport_oldresiduecongruence = (a) + (p) * pfa_offset_right_shift_product_transport_oldresiduecongruence))))))))) -> (exists pfc_terms_code_shift_product_transport_new pfc_terms_scale_shift_product_transport_new pfc_natural_sum_shift_product_transport_new. ((forall pfc_index_shift_product_transport_newdiagonal. (exists pfa_gap_shift_product_transport_newdiagonalbound. pfa_gap_shift_product_transport_newdiagonalbound + S (pfc_index_shift_product_transport_newdiagonal) = (S (i))) -> exists pfc_value_shift_product_transport_newdiagonal. ((((exists ff_h_pfp_shift_product_transport_newdiagonalentry. ff_h_pfp_shift_product_transport_newdiagonalentry + S (pfc_value_shift_product_transport_newdiagonal) = S ((S (pfc_index_shift_product_transport_newdiagonal)) * pfc_terms_scale_shift_product_transport_new)) /\\ exists ff_q_pfp_shift_product_transport_newdiagonalentry. pfc_terms_code_shift_product_transport_new = ff_q_pfp_shift_product_transport_newdiagonalentry * S ((S (pfc_index_shift_product_transport_newdiagonal)) * pfc_terms_scale_shift_product_transport_new) + (pfc_value_shift_product_transport_newdiagonal))) /\\ ((exists pfc_complement_shift_product_transport_newdiagonalterm pfc_left_shift_product_transport_newdiagonalterm pfc_right_shift_product_transport_newdiagonalterm. (((pfc_index_shift_product_transport_newdiagonal)+pfc_complement_shift_product_transport_newdiagonalterm=(i)) /\\ ((((((exists pfa_gap_shift_product_transport_newdiagonaltermleftinside. pfa_gap_shift_product_transport_newdiagonaltermleftinside + S (pfc_index_shift_product_transport_newdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_transport_newdiagonaltermleftentry. ff_h_pfp_shift_product_transport_newdiagonaltermleftentry + S (pfc_left_shift_product_transport_newdiagonalterm) = S ((S (pfc_index_shift_product_transport_newdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_transport_newdiagonaltermleftentry. ab = ff_q_pfp_shift_product_transport_newdiagonaltermleftentry * S ((S (pfc_index_shift_product_transport_newdiagonal)) * ac) + (pfc_left_shift_product_transport_newdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_transport_newdiagonaltermleftoutside. pfc_gap_shift_product_transport_newdiagonaltermleftoutside+(L)=(pfc_index_shift_product_transport_newdiagonal)) /\\ (((pfc_left_shift_product_transport_newdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_transport_newdiagonaltermrightinside. pfa_gap_shift_product_transport_newdiagonaltermrightinside + S (pfc_complement_shift_product_transport_newdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_shift_product_transport_newdiagonaltermrightentry. ff_h_pfp_shift_product_transport_newdiagonaltermrightentry + S (pfc_right_shift_product_transport_newdiagonalterm) = S ((S (pfc_complement_shift_product_transport_newdiagonalterm)) * BC)) /\\ exists ff_q_pfp_shift_product_transport_newdiagonaltermrightentry. BB = ff_q_pfp_shift_product_transport_newdiagonaltermrightentry * S ((S (pfc_complement_shift_product_transport_newdiagonalterm)) * BC) + (pfc_right_shift_product_transport_newdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_transport_newdiagonaltermrightoutside. pfc_gap_shift_product_transport_newdiagonaltermrightoutside+(S M)=(pfc_complement_shift_product_transport_newdiagonalterm)) /\\ (((pfc_right_shift_product_transport_newdiagonalterm)=0))))) /\\ (((pfc_value_shift_product_transport_newdiagonal)=pfc_left_shift_product_transport_newdiagonalterm*pfc_right_shift_product_transport_newdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_transport_newsum fs_v_pfc_shift_product_transport_newsum. ((((exists fs_h_pfc_shift_product_transport_newsum_body_start. fs_h_pfc_shift_product_transport_newsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_transport_newsum)) /\\ exists fs_q_pfc_shift_product_transport_newsum_body_start. fs_u_pfc_shift_product_transport_newsum = fs_q_pfc_shift_product_transport_newsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_transport_newsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_transport_newsum_body_terminal. fs_h_pfc_shift_product_transport_newsum_body_terminal + S (pfc_natural_sum_shift_product_transport_new) = S ((S (S (i))) * fs_v_pfc_shift_product_transport_newsum)) /\\ exists fs_q_pfc_shift_product_transport_newsum_body_terminal. fs_u_pfc_shift_product_transport_newsum = fs_q_pfc_shift_product_transport_newsum_body_terminal * S ((S (S (i))) * fs_v_pfc_shift_product_transport_newsum) + (pfc_natural_sum_shift_product_transport_new))) /\\ forall fs_i_pfc_shift_product_transport_newsum_body_steps. (exists fs_lt_pfc_shift_product_transport_newsum_body_steps_bound. fs_lt_pfc_shift_product_transport_newsum_body_steps_bound + S fs_i_pfc_shift_product_transport_newsum_body_steps = S (i)) -> exists fs_a_pfc_shift_product_transport_newsum_body_steps fs_r_pfc_shift_product_transport_newsum_body_steps fs_s_pfc_shift_product_transport_newsum_body_steps. ((((exists fs_h_pfc_shift_product_transport_newsum_body_steps_summand. fs_h_pfc_shift_product_transport_newsum_body_steps_summand + S (fs_a_pfc_shift_product_transport_newsum_body_steps) = S ((S (fs_i_pfc_shift_product_transport_newsum_body_steps)) * pfc_terms_scale_shift_product_transport_new)) /\\ exists fs_q_pfc_shift_product_transport_newsum_body_steps_summand. pfc_terms_code_shift_product_transport_new = fs_q_pfc_shift_product_transport_newsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_transport_newsum_body_steps)) * pfc_terms_scale_shift_product_transport_new) + (fs_a_pfc_shift_product_transport_newsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_transport_newsum_body_steps_partial. fs_h_pfc_shift_product_transport_newsum_body_steps_partial + S (fs_r_pfc_shift_product_transport_newsum_body_steps) = S ((S (fs_i_pfc_shift_product_transport_newsum_body_steps)) * fs_v_pfc_shift_product_transport_newsum)) /\\ exists fs_q_pfc_shift_product_transport_newsum_body_steps_partial. fs_u_pfc_shift_product_transport_newsum = fs_q_pfc_shift_product_transport_newsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_transport_newsum_body_steps)) * fs_v_pfc_shift_product_transport_newsum) + (fs_r_pfc_shift_product_transport_newsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_transport_newsum_body_steps_successor. fs_h_pfc_shift_product_transport_newsum_body_steps_successor + S (fs_s_pfc_shift_product_transport_newsum_body_steps) = S ((S (S fs_i_pfc_shift_product_transport_newsum_body_steps)) * fs_v_pfc_shift_product_transport_newsum)) /\\ exists fs_q_pfc_shift_product_transport_newsum_body_steps_successor. fs_u_pfc_shift_product_transport_newsum = fs_q_pfc_shift_product_transport_newsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_transport_newsum_body_steps)) * fs_v_pfc_shift_product_transport_newsum) + (fs_s_pfc_shift_product_transport_newsum_body_steps))) /\\ fs_s_pfc_shift_product_transport_newsum_body_steps = fs_r_pfc_shift_product_transport_newsum_body_steps + fs_a_pfc_shift_product_transport_newsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_transport_newresiduebound. pfa_gap_shift_product_transport_newresiduebound + S (a) = (p)) /\\ ((exists pfa_offset_left_shift_product_transport_newresiduecongruence pfa_offset_right_shift_product_transport_newresiduecongruence. (pfc_natural_sum_shift_product_transport_new) + (p) * pfa_offset_left_shift_product_transport_newresiduecongruence = (a) + (p) * pfa_offset_right_shift_product_transport_newresiduecongruence)))))))))) /\\ (((exists pfc_terms_code_shift_product_transport_new pfc_terms_scale_shift_product_transport_new pfc_natural_sum_shift_product_transport_new. ((forall pfc_index_shift_product_transport_newdiagonal. (exists pfa_gap_shift_product_transport_newdiagonalbound. pfa_gap_shift_product_transport_newdiagonalbound + S (pfc_index_shift_product_transport_newdiagonal) = (S (i))) -> exists pfc_value_shift_product_transport_newdiagonal. ((((exists ff_h_pfp_shift_product_transport_newdiagonalentry. ff_h_pfp_shift_product_transport_newdiagonalentry + S (pfc_value_shift_product_transport_newdiagonal) = S ((S (pfc_index_shift_product_transport_newdiagonal)) * pfc_terms_scale_shift_product_transport_new)) /\\ exists ff_q_pfp_shift_product_transport_newdiagonalentry. pfc_terms_code_shift_product_transport_new = ff_q_pfp_shift_product_transport_newdiagonalentry * S ((S (pfc_index_shift_product_transport_newdiagonal)) * pfc_terms_scale_shift_product_transport_new) + (pfc_value_shift_product_transport_newdiagonal))) /\\ ((exists pfc_complement_shift_product_transport_newdiagonalterm pfc_left_shift_product_transport_newdiagonalterm pfc_right_shift_product_transport_newdiagonalterm. (((pfc_index_shift_product_transport_newdiagonal)+pfc_complement_shift_product_transport_newdiagonalterm=(i)) /\\ ((((((exists pfa_gap_shift_product_transport_newdiagonaltermleftinside. pfa_gap_shift_product_transport_newdiagonaltermleftinside + S (pfc_index_shift_product_transport_newdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_transport_newdiagonaltermleftentry. ff_h_pfp_shift_product_transport_newdiagonaltermleftentry + S (pfc_left_shift_product_transport_newdiagonalterm) = S ((S (pfc_index_shift_product_transport_newdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_transport_newdiagonaltermleftentry. ab = ff_q_pfp_shift_product_transport_newdiagonaltermleftentry * S ((S (pfc_index_shift_product_transport_newdiagonal)) * ac) + (pfc_left_shift_product_transport_newdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_transport_newdiagonaltermleftoutside. pfc_gap_shift_product_transport_newdiagonaltermleftoutside+(L)=(pfc_index_shift_product_transport_newdiagonal)) /\\ (((pfc_left_shift_product_transport_newdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_transport_newdiagonaltermrightinside. pfa_gap_shift_product_transport_newdiagonaltermrightinside + S (pfc_complement_shift_product_transport_newdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_shift_product_transport_newdiagonaltermrightentry. ff_h_pfp_shift_product_transport_newdiagonaltermrightentry + S (pfc_right_shift_product_transport_newdiagonalterm) = S ((S (pfc_complement_shift_product_transport_newdiagonalterm)) * BC)) /\\ exists ff_q_pfp_shift_product_transport_newdiagonaltermrightentry. BB = ff_q_pfp_shift_product_transport_newdiagonaltermrightentry * S ((S (pfc_complement_shift_product_transport_newdiagonalterm)) * BC) + (pfc_right_shift_product_transport_newdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_transport_newdiagonaltermrightoutside. pfc_gap_shift_product_transport_newdiagonaltermrightoutside+(S M)=(pfc_complement_shift_product_transport_newdiagonalterm)) /\\ (((pfc_right_shift_product_transport_newdiagonalterm)=0))))) /\\ (((pfc_value_shift_product_transport_newdiagonal)=pfc_left_shift_product_transport_newdiagonalterm*pfc_right_shift_product_transport_newdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_transport_newsum fs_v_pfc_shift_product_transport_newsum. ((((exists fs_h_pfc_shift_product_transport_newsum_body_start. fs_h_pfc_shift_product_transport_newsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_transport_newsum)) /\\ exists fs_q_pfc_shift_product_transport_newsum_body_start. fs_u_pfc_shift_product_transport_newsum = fs_q_pfc_shift_product_transport_newsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_transport_newsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_transport_newsum_body_terminal. fs_h_pfc_shift_product_transport_newsum_body_terminal + S (pfc_natural_sum_shift_product_transport_new) = S ((S (S (i))) * fs_v_pfc_shift_product_transport_newsum)) /\\ exists fs_q_pfc_shift_product_transport_newsum_body_terminal. fs_u_pfc_shift_product_transport_newsum = fs_q_pfc_shift_product_transport_newsum_body_terminal * S ((S (S (i))) * fs_v_pfc_shift_product_transport_newsum) + (pfc_natural_sum_shift_product_transport_new))) /\\ forall fs_i_pfc_shift_product_transport_newsum_body_steps. (exists fs_lt_pfc_shift_product_transport_newsum_body_steps_bound. fs_lt_pfc_shift_product_transport_newsum_body_steps_bound + S fs_i_pfc_shift_product_transport_newsum_body_steps = S (i)) -> exists fs_a_pfc_shift_product_transport_newsum_body_steps fs_r_pfc_shift_product_transport_newsum_body_steps fs_s_pfc_shift_product_transport_newsum_body_steps. ((((exists fs_h_pfc_shift_product_transport_newsum_body_steps_summand. fs_h_pfc_shift_product_transport_newsum_body_steps_summand + S (fs_a_pfc_shift_product_transport_newsum_body_steps) = S ((S (fs_i_pfc_shift_product_transport_newsum_body_steps)) * pfc_terms_scale_shift_product_transport_new)) /\\ exists fs_q_pfc_shift_product_transport_newsum_body_steps_summand. pfc_terms_code_shift_product_transport_new = fs_q_pfc_shift_product_transport_newsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_transport_newsum_body_steps)) * pfc_terms_scale_shift_product_transport_new) + (fs_a_pfc_shift_product_transport_newsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_transport_newsum_body_steps_partial. fs_h_pfc_shift_product_transport_newsum_body_steps_partial + S (fs_r_pfc_shift_product_transport_newsum_body_steps) = S ((S (fs_i_pfc_shift_product_transport_newsum_body_steps)) * fs_v_pfc_shift_product_transport_newsum)) /\\ exists fs_q_pfc_shift_product_transport_newsum_body_steps_partial. fs_u_pfc_shift_product_transport_newsum = fs_q_pfc_shift_product_transport_newsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_transport_newsum_body_steps)) * fs_v_pfc_shift_product_transport_newsum) + (fs_r_pfc_shift_product_transport_newsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_transport_newsum_body_steps_successor. fs_h_pfc_shift_product_transport_newsum_body_steps_successor + S (fs_s_pfc_shift_product_transport_newsum_body_steps) = S ((S (S fs_i_pfc_shift_product_transport_newsum_body_steps)) * fs_v_pfc_shift_product_transport_newsum)) /\\ exists fs_q_pfc_shift_product_transport_newsum_body_steps_successor. fs_u_pfc_shift_product_transport_newsum = fs_q_pfc_shift_product_transport_newsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_transport_newsum_body_steps)) * fs_v_pfc_shift_product_transport_newsum) + (fs_s_pfc_shift_product_transport_newsum_body_steps))) /\\ fs_s_pfc_shift_product_transport_newsum_body_steps = fs_r_pfc_shift_product_transport_newsum_body_steps + fs_a_pfc_shift_product_transport_newsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_transport_newresiduebound. pfa_gap_shift_product_transport_newresiduebound + S (a) = (p)) /\\ ((exists pfa_offset_left_shift_product_transport_newresiduecongruence pfa_offset_right_shift_product_transport_newresiduecongruence. (pfc_natural_sum_shift_product_transport_new) + (p) * pfa_offset_left_shift_product_transport_newresiduecongruence = (a) + (p) * pfa_offset_right_shift_product_transport_newresiduecongruence))))))))) -> (exists pfc_terms_code_shift_product_transport_old pfc_terms_scale_shift_product_transport_old pfc_natural_sum_shift_product_transport_old. ((forall pfc_index_shift_product_transport_olddiagonal. (exists pfa_gap_shift_product_transport_olddiagonalbound. pfa_gap_shift_product_transport_olddiagonalbound + S (pfc_index_shift_product_transport_olddiagonal) = (S (i))) -> exists pfc_value_shift_product_transport_olddiagonal. ((((exists ff_h_pfp_shift_product_transport_olddiagonalentry. ff_h_pfp_shift_product_transport_olddiagonalentry + S (pfc_value_shift_product_transport_olddiagonal) = S ((S (pfc_index_shift_product_transport_olddiagonal)) * pfc_terms_scale_shift_product_transport_old)) /\\ exists ff_q_pfp_shift_product_transport_olddiagonalentry. pfc_terms_code_shift_product_transport_old = ff_q_pfp_shift_product_transport_olddiagonalentry * S ((S (pfc_index_shift_product_transport_olddiagonal)) * pfc_terms_scale_shift_product_transport_old) + (pfc_value_shift_product_transport_olddiagonal))) /\\ ((exists pfc_complement_shift_product_transport_olddiagonalterm pfc_left_shift_product_transport_olddiagonalterm pfc_right_shift_product_transport_olddiagonalterm. (((pfc_index_shift_product_transport_olddiagonal)+pfc_complement_shift_product_transport_olddiagonalterm=(i)) /\\ ((((((exists pfa_gap_shift_product_transport_olddiagonaltermleftinside. pfa_gap_shift_product_transport_olddiagonaltermleftinside + S (pfc_index_shift_product_transport_olddiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_transport_olddiagonaltermleftentry. ff_h_pfp_shift_product_transport_olddiagonaltermleftentry + S (pfc_left_shift_product_transport_olddiagonalterm) = S ((S (pfc_index_shift_product_transport_olddiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_transport_olddiagonaltermleftentry. ab = ff_q_pfp_shift_product_transport_olddiagonaltermleftentry * S ((S (pfc_index_shift_product_transport_olddiagonal)) * ac) + (pfc_left_shift_product_transport_olddiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_transport_olddiagonaltermleftoutside. pfc_gap_shift_product_transport_olddiagonaltermleftoutside+(L)=(pfc_index_shift_product_transport_olddiagonal)) /\\ (((pfc_left_shift_product_transport_olddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_transport_olddiagonaltermrightinside. pfa_gap_shift_product_transport_olddiagonaltermrightinside + S (pfc_complement_shift_product_transport_olddiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_shift_product_transport_olddiagonaltermrightentry. ff_h_pfp_shift_product_transport_olddiagonaltermrightentry + S (pfc_right_shift_product_transport_olddiagonalterm) = S ((S (pfc_complement_shift_product_transport_olddiagonalterm)) * bc)) /\\ exists ff_q_pfp_shift_product_transport_olddiagonaltermrightentry. bb = ff_q_pfp_shift_product_transport_olddiagonaltermrightentry * S ((S (pfc_complement_shift_product_transport_olddiagonalterm)) * bc) + (pfc_right_shift_product_transport_olddiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_transport_olddiagonaltermrightoutside. pfc_gap_shift_product_transport_olddiagonaltermrightoutside+(M)=(pfc_complement_shift_product_transport_olddiagonalterm)) /\\ (((pfc_right_shift_product_transport_olddiagonalterm)=0))))) /\\ (((pfc_value_shift_product_transport_olddiagonal)=pfc_left_shift_product_transport_olddiagonalterm*pfc_right_shift_product_transport_olddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_transport_oldsum fs_v_pfc_shift_product_transport_oldsum. ((((exists fs_h_pfc_shift_product_transport_oldsum_body_start. fs_h_pfc_shift_product_transport_oldsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_transport_oldsum_body_start. fs_u_pfc_shift_product_transport_oldsum = fs_q_pfc_shift_product_transport_oldsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_transport_oldsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_transport_oldsum_body_terminal. fs_h_pfc_shift_product_transport_oldsum_body_terminal + S (pfc_natural_sum_shift_product_transport_old) = S ((S (S (i))) * fs_v_pfc_shift_product_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_transport_oldsum_body_terminal. fs_u_pfc_shift_product_transport_oldsum = fs_q_pfc_shift_product_transport_oldsum_body_terminal * S ((S (S (i))) * fs_v_pfc_shift_product_transport_oldsum) + (pfc_natural_sum_shift_product_transport_old))) /\\ forall fs_i_pfc_shift_product_transport_oldsum_body_steps. (exists fs_lt_pfc_shift_product_transport_oldsum_body_steps_bound. fs_lt_pfc_shift_product_transport_oldsum_body_steps_bound + S fs_i_pfc_shift_product_transport_oldsum_body_steps = S (i)) -> exists fs_a_pfc_shift_product_transport_oldsum_body_steps fs_r_pfc_shift_product_transport_oldsum_body_steps fs_s_pfc_shift_product_transport_oldsum_body_steps. ((((exists fs_h_pfc_shift_product_transport_oldsum_body_steps_summand. fs_h_pfc_shift_product_transport_oldsum_body_steps_summand + S (fs_a_pfc_shift_product_transport_oldsum_body_steps) = S ((S (fs_i_pfc_shift_product_transport_oldsum_body_steps)) * pfc_terms_scale_shift_product_transport_old)) /\\ exists fs_q_pfc_shift_product_transport_oldsum_body_steps_summand. pfc_terms_code_shift_product_transport_old = fs_q_pfc_shift_product_transport_oldsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_transport_oldsum_body_steps)) * pfc_terms_scale_shift_product_transport_old) + (fs_a_pfc_shift_product_transport_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_transport_oldsum_body_steps_partial. fs_h_pfc_shift_product_transport_oldsum_body_steps_partial + S (fs_r_pfc_shift_product_transport_oldsum_body_steps) = S ((S (fs_i_pfc_shift_product_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_transport_oldsum_body_steps_partial. fs_u_pfc_shift_product_transport_oldsum = fs_q_pfc_shift_product_transport_oldsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_transport_oldsum) + (fs_r_pfc_shift_product_transport_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_transport_oldsum_body_steps_successor. fs_h_pfc_shift_product_transport_oldsum_body_steps_successor + S (fs_s_pfc_shift_product_transport_oldsum_body_steps) = S ((S (S fs_i_pfc_shift_product_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_transport_oldsum_body_steps_successor. fs_u_pfc_shift_product_transport_oldsum = fs_q_pfc_shift_product_transport_oldsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_transport_oldsum) + (fs_s_pfc_shift_product_transport_oldsum_body_steps))) /\\ fs_s_pfc_shift_product_transport_oldsum_body_steps = fs_r_pfc_shift_product_transport_oldsum_body_steps + fs_a_pfc_shift_product_transport_oldsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_transport_oldresiduebound. pfa_gap_shift_product_transport_oldresiduebound + S (a) = (p)) /\\ ((exists pfa_offset_left_shift_product_transport_oldresiduecongruence pfa_offset_right_shift_product_transport_oldresiduecongruence. (pfc_natural_sum_shift_product_transport_old) + (p) * pfa_offset_left_shift_product_transport_oldresiduecongruence = (a) + (p) * pfa_offset_right_shift_product_transport_oldresiduecongruence))))))))))))", "specialize prime_field_convolution_coefficient_shift_right_iff (p)", "specialize prime_field_convolution_coefficient_shift_right_iff (ab)", "specialize prime_field_convolution_coefficient_shift_right_iff (ac)", "specialize prime_field_convolution_coefficient_shift_right_iff (L)", "specialize prime_field_convolution_coefficient_shift_right_iff (bb)", "specialize prime_field_convolution_coefficient_shift_right_iff (bc)", "specialize prime_field_convolution_coefficient_shift_right_iff (M)", "specialize prime_field_convolution_coefficient_shift_right_iff (BB)", "specialize prime_field_convolution_coefficient_shift_right_iff (BC)", "specialize prime_field_convolution_coefficient_shift_right_iff (i)", "specialize prime_field_convolution_coefficient_shift_right_iff (a)", "apply prime_field_convolution_coefficient_shift_right_iff", "exact hs", "cases hiff", "apply hiff_left", "exact hca", "have hv : exists r. ((((exists ff_h_pfp_shift_product_chosen_entry. ff_h_pfp_shift_product_chosen_entry + S (r) = S ((S (i)) * dc)) /\\ exists ff_q_pfp_shift_product_chosen_entry. db = ff_q_pfp_shift_product_chosen_entry * S ((S (i)) * dc) + (r))) /\\ ((exists pfc_terms_code_shift_product_chosen_coefficient pfc_terms_scale_shift_product_chosen_coefficient pfc_natural_sum_shift_product_chosen_coefficient. ((forall pfc_index_shift_product_chosen_coefficientdiagonal. (exists pfa_gap_shift_product_chosen_coefficientdiagonalbound. pfa_gap_shift_product_chosen_coefficientdiagonalbound + S (pfc_index_shift_product_chosen_coefficientdiagonal) = (S (i))) -> exists pfc_value_shift_product_chosen_coefficientdiagonal. ((((exists ff_h_pfp_shift_product_chosen_coefficientdiagonalentry. ff_h_pfp_shift_product_chosen_coefficientdiagonalentry + S (pfc_value_shift_product_chosen_coefficientdiagonal) = S ((S (pfc_index_shift_product_chosen_coefficientdiagonal)) * pfc_terms_scale_shift_product_chosen_coefficient)) /\\ exists ff_q_pfp_shift_product_chosen_coefficientdiagonalentry. pfc_terms_code_shift_product_chosen_coefficient = ff_q_pfp_shift_product_chosen_coefficientdiagonalentry * S ((S (pfc_index_shift_product_chosen_coefficientdiagonal)) * pfc_terms_scale_shift_product_chosen_coefficient) + (pfc_value_shift_product_chosen_coefficientdiagonal))) /\\ ((exists pfc_complement_shift_product_chosen_coefficientdiagonalterm pfc_left_shift_product_chosen_coefficientdiagonalterm pfc_right_shift_product_chosen_coefficientdiagonalterm. (((pfc_index_shift_product_chosen_coefficientdiagonal)+pfc_complement_shift_product_chosen_coefficientdiagonalterm=(i)) /\\ ((((((exists pfa_gap_shift_product_chosen_coefficientdiagonaltermleftinside. pfa_gap_shift_product_chosen_coefficientdiagonaltermleftinside + S (pfc_index_shift_product_chosen_coefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_chosen_coefficientdiagonaltermleftentry. ff_h_pfp_shift_product_chosen_coefficientdiagonaltermleftentry + S (pfc_left_shift_product_chosen_coefficientdiagonalterm) = S ((S (pfc_index_shift_product_chosen_coefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_chosen_coefficientdiagonaltermleftentry. ab = ff_q_pfp_shift_product_chosen_coefficientdiagonaltermleftentry * S ((S (pfc_index_shift_product_chosen_coefficientdiagonal)) * ac) + (pfc_left_shift_product_chosen_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_chosen_coefficientdiagonaltermleftoutside. pfc_gap_shift_product_chosen_coefficientdiagonaltermleftoutside+(L)=(pfc_index_shift_product_chosen_coefficientdiagonal)) /\\ (((pfc_left_shift_product_chosen_coefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_chosen_coefficientdiagonaltermrightinside. pfa_gap_shift_product_chosen_coefficientdiagonaltermrightinside + S (pfc_complement_shift_product_chosen_coefficientdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_shift_product_chosen_coefficientdiagonaltermrightentry. ff_h_pfp_shift_product_chosen_coefficientdiagonaltermrightentry + S (pfc_right_shift_product_chosen_coefficientdiagonalterm) = S ((S (pfc_complement_shift_product_chosen_coefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_shift_product_chosen_coefficientdiagonaltermrightentry. BB = ff_q_pfp_shift_product_chosen_coefficientdiagonaltermrightentry * S ((S (pfc_complement_shift_product_chosen_coefficientdiagonalterm)) * BC) + (pfc_right_shift_product_chosen_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_chosen_coefficientdiagonaltermrightoutside. pfc_gap_shift_product_chosen_coefficientdiagonaltermrightoutside+(S M)=(pfc_complement_shift_product_chosen_coefficientdiagonalterm)) /\\ (((pfc_right_shift_product_chosen_coefficientdiagonalterm)=0))))) /\\ (((pfc_value_shift_product_chosen_coefficientdiagonal)=pfc_left_shift_product_chosen_coefficientdiagonalterm*pfc_right_shift_product_chosen_coefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_chosen_coefficientsum fs_v_pfc_shift_product_chosen_coefficientsum. ((((exists fs_h_pfc_shift_product_chosen_coefficientsum_body_start. fs_h_pfc_shift_product_chosen_coefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_chosen_coefficientsum)) /\\ exists fs_q_pfc_shift_product_chosen_coefficientsum_body_start. fs_u_pfc_shift_product_chosen_coefficientsum = fs_q_pfc_shift_product_chosen_coefficientsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_chosen_coefficientsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_chosen_coefficientsum_body_terminal. fs_h_pfc_shift_product_chosen_coefficientsum_body_terminal + S (pfc_natural_sum_shift_product_chosen_coefficient) = S ((S (S (i))) * fs_v_pfc_shift_product_chosen_coefficientsum)) /\\ exists fs_q_pfc_shift_product_chosen_coefficientsum_body_terminal. fs_u_pfc_shift_product_chosen_coefficientsum = fs_q_pfc_shift_product_chosen_coefficientsum_body_terminal * S ((S (S (i))) * fs_v_pfc_shift_product_chosen_coefficientsum) + (pfc_natural_sum_shift_product_chosen_coefficient))) /\\ forall fs_i_pfc_shift_product_chosen_coefficientsum_body_steps. (exists fs_lt_pfc_shift_product_chosen_coefficientsum_body_steps_bound. fs_lt_pfc_shift_product_chosen_coefficientsum_body_steps_bound + S fs_i_pfc_shift_product_chosen_coefficientsum_body_steps = S (i)) -> exists fs_a_pfc_shift_product_chosen_coefficientsum_body_steps fs_r_pfc_shift_product_chosen_coefficientsum_body_steps fs_s_pfc_shift_product_chosen_coefficientsum_body_steps. ((((exists fs_h_pfc_shift_product_chosen_coefficientsum_body_steps_summand. fs_h_pfc_shift_product_chosen_coefficientsum_body_steps_summand + S (fs_a_pfc_shift_product_chosen_coefficientsum_body_steps) = S ((S (fs_i_pfc_shift_product_chosen_coefficientsum_body_steps)) * pfc_terms_scale_shift_product_chosen_coefficient)) /\\ exists fs_q_pfc_shift_product_chosen_coefficientsum_body_steps_summand. pfc_terms_code_shift_product_chosen_coefficient = fs_q_pfc_shift_product_chosen_coefficientsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_chosen_coefficientsum_body_steps)) * pfc_terms_scale_shift_product_chosen_coefficient) + (fs_a_pfc_shift_product_chosen_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_chosen_coefficientsum_body_steps_partial. fs_h_pfc_shift_product_chosen_coefficientsum_body_steps_partial + S (fs_r_pfc_shift_product_chosen_coefficientsum_body_steps) = S ((S (fs_i_pfc_shift_product_chosen_coefficientsum_body_steps)) * fs_v_pfc_shift_product_chosen_coefficientsum)) /\\ exists fs_q_pfc_shift_product_chosen_coefficientsum_body_steps_partial. fs_u_pfc_shift_product_chosen_coefficientsum = fs_q_pfc_shift_product_chosen_coefficientsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_chosen_coefficientsum_body_steps)) * fs_v_pfc_shift_product_chosen_coefficientsum) + (fs_r_pfc_shift_product_chosen_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_chosen_coefficientsum_body_steps_successor. fs_h_pfc_shift_product_chosen_coefficientsum_body_steps_successor + S (fs_s_pfc_shift_product_chosen_coefficientsum_body_steps) = S ((S (S fs_i_pfc_shift_product_chosen_coefficientsum_body_steps)) * fs_v_pfc_shift_product_chosen_coefficientsum)) /\\ exists fs_q_pfc_shift_product_chosen_coefficientsum_body_steps_successor. fs_u_pfc_shift_product_chosen_coefficientsum = fs_q_pfc_shift_product_chosen_coefficientsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_chosen_coefficientsum_body_steps)) * fs_v_pfc_shift_product_chosen_coefficientsum) + (fs_s_pfc_shift_product_chosen_coefficientsum_body_steps))) /\\ fs_s_pfc_shift_product_chosen_coefficientsum_body_steps = fs_r_pfc_shift_product_chosen_coefficientsum_body_steps + fs_a_pfc_shift_product_chosen_coefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_chosen_coefficientresiduebound. pfa_gap_shift_product_chosen_coefficientresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_shift_product_chosen_coefficientresiduecongruence pfa_offset_right_shift_product_chosen_coefficientresiduecongruence. (pfc_natural_sum_shift_product_chosen_coefficient) + (p) * pfa_offset_left_shift_product_chosen_coefficientresiduecongruence = (r) + (p) * pfa_offset_right_shift_product_chosen_coefficientresiduecongruence)))))))))))", "specialize hd_right_right_right (i)", "apply hd_right_right_right", "rewrite hk", "specialize le_succ (S i)", "specialize le_succ (N)", "apply le_succ", "exact hi", "cases hv", "cases hv_witness", "have heq : a=x", "specialize prime_field_convolution_coefficient_functional (p)", "specialize prime_field_convolution_coefficient_functional (ab)", "specialize prime_field_convolution_coefficient_functional (ac)", "specialize prime_field_convolution_coefficient_functional (L)", "specialize prime_field_convolution_coefficient_functional (BB)", "specialize prime_field_convolution_coefficient_functional (BC)", "specialize prime_field_convolution_coefficient_functional (S M)", "specialize prime_field_convolution_coefficient_functional (i)", "specialize prime_field_convolution_coefficient_functional (a)", "specialize prime_field_convolution_coefficient_functional (x)", "apply prime_field_convolution_coefficient_functional", "exact hda", "exact hv_witness_right", "rewrite heq", "rewrite heq", "exact hv_witness_left", "have hv : exists r. ((((exists ff_h_pfp_shift_product_last_entry. ff_h_pfp_shift_product_last_entry + S (r) = S ((S (N)) * dc)) /\\ exists ff_q_pfp_shift_product_last_entry. db = ff_q_pfp_shift_product_last_entry * S ((S (N)) * dc) + (r))) /\\ ((exists pfc_terms_code_shift_product_last_coefficient pfc_terms_scale_shift_product_last_coefficient pfc_natural_sum_shift_product_last_coefficient. ((forall pfc_index_shift_product_last_coefficientdiagonal. (exists pfa_gap_shift_product_last_coefficientdiagonalbound. pfa_gap_shift_product_last_coefficientdiagonalbound + S (pfc_index_shift_product_last_coefficientdiagonal) = (S (N))) -> exists pfc_value_shift_product_last_coefficientdiagonal. ((((exists ff_h_pfp_shift_product_last_coefficientdiagonalentry. ff_h_pfp_shift_product_last_coefficientdiagonalentry + S (pfc_value_shift_product_last_coefficientdiagonal) = S ((S (pfc_index_shift_product_last_coefficientdiagonal)) * pfc_terms_scale_shift_product_last_coefficient)) /\\ exists ff_q_pfp_shift_product_last_coefficientdiagonalentry. pfc_terms_code_shift_product_last_coefficient = ff_q_pfp_shift_product_last_coefficientdiagonalentry * S ((S (pfc_index_shift_product_last_coefficientdiagonal)) * pfc_terms_scale_shift_product_last_coefficient) + (pfc_value_shift_product_last_coefficientdiagonal))) /\\ ((exists pfc_complement_shift_product_last_coefficientdiagonalterm pfc_left_shift_product_last_coefficientdiagonalterm pfc_right_shift_product_last_coefficientdiagonalterm. (((pfc_index_shift_product_last_coefficientdiagonal)+pfc_complement_shift_product_last_coefficientdiagonalterm=(N)) /\\ ((((((exists pfa_gap_shift_product_last_coefficientdiagonaltermleftinside. pfa_gap_shift_product_last_coefficientdiagonaltermleftinside + S (pfc_index_shift_product_last_coefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_last_coefficientdiagonaltermleftentry. ff_h_pfp_shift_product_last_coefficientdiagonaltermleftentry + S (pfc_left_shift_product_last_coefficientdiagonalterm) = S ((S (pfc_index_shift_product_last_coefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_last_coefficientdiagonaltermleftentry. ab = ff_q_pfp_shift_product_last_coefficientdiagonaltermleftentry * S ((S (pfc_index_shift_product_last_coefficientdiagonal)) * ac) + (pfc_left_shift_product_last_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_last_coefficientdiagonaltermleftoutside. pfc_gap_shift_product_last_coefficientdiagonaltermleftoutside+(L)=(pfc_index_shift_product_last_coefficientdiagonal)) /\\ (((pfc_left_shift_product_last_coefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_last_coefficientdiagonaltermrightinside. pfa_gap_shift_product_last_coefficientdiagonaltermrightinside + S (pfc_complement_shift_product_last_coefficientdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_shift_product_last_coefficientdiagonaltermrightentry. ff_h_pfp_shift_product_last_coefficientdiagonaltermrightentry + S (pfc_right_shift_product_last_coefficientdiagonalterm) = S ((S (pfc_complement_shift_product_last_coefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_shift_product_last_coefficientdiagonaltermrightentry. BB = ff_q_pfp_shift_product_last_coefficientdiagonaltermrightentry * S ((S (pfc_complement_shift_product_last_coefficientdiagonalterm)) * BC) + (pfc_right_shift_product_last_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_last_coefficientdiagonaltermrightoutside. pfc_gap_shift_product_last_coefficientdiagonaltermrightoutside+(S M)=(pfc_complement_shift_product_last_coefficientdiagonalterm)) /\\ (((pfc_right_shift_product_last_coefficientdiagonalterm)=0))))) /\\ (((pfc_value_shift_product_last_coefficientdiagonal)=pfc_left_shift_product_last_coefficientdiagonalterm*pfc_right_shift_product_last_coefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_last_coefficientsum fs_v_pfc_shift_product_last_coefficientsum. ((((exists fs_h_pfc_shift_product_last_coefficientsum_body_start. fs_h_pfc_shift_product_last_coefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_last_coefficientsum)) /\\ exists fs_q_pfc_shift_product_last_coefficientsum_body_start. fs_u_pfc_shift_product_last_coefficientsum = fs_q_pfc_shift_product_last_coefficientsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_last_coefficientsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_last_coefficientsum_body_terminal. fs_h_pfc_shift_product_last_coefficientsum_body_terminal + S (pfc_natural_sum_shift_product_last_coefficient) = S ((S (S (N))) * fs_v_pfc_shift_product_last_coefficientsum)) /\\ exists fs_q_pfc_shift_product_last_coefficientsum_body_terminal. fs_u_pfc_shift_product_last_coefficientsum = fs_q_pfc_shift_product_last_coefficientsum_body_terminal * S ((S (S (N))) * fs_v_pfc_shift_product_last_coefficientsum) + (pfc_natural_sum_shift_product_last_coefficient))) /\\ forall fs_i_pfc_shift_product_last_coefficientsum_body_steps. (exists fs_lt_pfc_shift_product_last_coefficientsum_body_steps_bound. fs_lt_pfc_shift_product_last_coefficientsum_body_steps_bound + S fs_i_pfc_shift_product_last_coefficientsum_body_steps = S (N)) -> exists fs_a_pfc_shift_product_last_coefficientsum_body_steps fs_r_pfc_shift_product_last_coefficientsum_body_steps fs_s_pfc_shift_product_last_coefficientsum_body_steps. ((((exists fs_h_pfc_shift_product_last_coefficientsum_body_steps_summand. fs_h_pfc_shift_product_last_coefficientsum_body_steps_summand + S (fs_a_pfc_shift_product_last_coefficientsum_body_steps) = S ((S (fs_i_pfc_shift_product_last_coefficientsum_body_steps)) * pfc_terms_scale_shift_product_last_coefficient)) /\\ exists fs_q_pfc_shift_product_last_coefficientsum_body_steps_summand. pfc_terms_code_shift_product_last_coefficient = fs_q_pfc_shift_product_last_coefficientsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_last_coefficientsum_body_steps)) * pfc_terms_scale_shift_product_last_coefficient) + (fs_a_pfc_shift_product_last_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_last_coefficientsum_body_steps_partial. fs_h_pfc_shift_product_last_coefficientsum_body_steps_partial + S (fs_r_pfc_shift_product_last_coefficientsum_body_steps) = S ((S (fs_i_pfc_shift_product_last_coefficientsum_body_steps)) * fs_v_pfc_shift_product_last_coefficientsum)) /\\ exists fs_q_pfc_shift_product_last_coefficientsum_body_steps_partial. fs_u_pfc_shift_product_last_coefficientsum = fs_q_pfc_shift_product_last_coefficientsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_last_coefficientsum_body_steps)) * fs_v_pfc_shift_product_last_coefficientsum) + (fs_r_pfc_shift_product_last_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_last_coefficientsum_body_steps_successor. fs_h_pfc_shift_product_last_coefficientsum_body_steps_successor + S (fs_s_pfc_shift_product_last_coefficientsum_body_steps) = S ((S (S fs_i_pfc_shift_product_last_coefficientsum_body_steps)) * fs_v_pfc_shift_product_last_coefficientsum)) /\\ exists fs_q_pfc_shift_product_last_coefficientsum_body_steps_successor. fs_u_pfc_shift_product_last_coefficientsum = fs_q_pfc_shift_product_last_coefficientsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_last_coefficientsum_body_steps)) * fs_v_pfc_shift_product_last_coefficientsum) + (fs_s_pfc_shift_product_last_coefficientsum_body_steps))) /\\ fs_s_pfc_shift_product_last_coefficientsum_body_steps = fs_r_pfc_shift_product_last_coefficientsum_body_steps + fs_a_pfc_shift_product_last_coefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_last_coefficientresiduebound. pfa_gap_shift_product_last_coefficientresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_shift_product_last_coefficientresiduecongruence pfa_offset_right_shift_product_last_coefficientresiduecongruence. (pfc_natural_sum_shift_product_last_coefficient) + (p) * pfa_offset_left_shift_product_last_coefficientresiduecongruence = (r) + (p) * pfa_offset_right_shift_product_last_coefficientresiduecongruence)))))))))))", "specialize hd_right_right_right (N)", "apply hd_right_right_right", "rewrite hk", "specialize le_refl (S N)", "apply le_refl", "cases hv", "cases hv_witness", "have hco : exists pfc_terms_code_shift_product_exterior_old pfc_terms_scale_shift_product_exterior_old pfc_natural_sum_shift_product_exterior_old. ((forall pfc_index_shift_product_exterior_olddiagonal. (exists pfa_gap_shift_product_exterior_olddiagonalbound. pfa_gap_shift_product_exterior_olddiagonalbound + S (pfc_index_shift_product_exterior_olddiagonal) = (S (N))) -> exists pfc_value_shift_product_exterior_olddiagonal. ((((exists ff_h_pfp_shift_product_exterior_olddiagonalentry. ff_h_pfp_shift_product_exterior_olddiagonalentry + S (pfc_value_shift_product_exterior_olddiagonal) = S ((S (pfc_index_shift_product_exterior_olddiagonal)) * pfc_terms_scale_shift_product_exterior_old)) /\\ exists ff_q_pfp_shift_product_exterior_olddiagonalentry. pfc_terms_code_shift_product_exterior_old = ff_q_pfp_shift_product_exterior_olddiagonalentry * S ((S (pfc_index_shift_product_exterior_olddiagonal)) * pfc_terms_scale_shift_product_exterior_old) + (pfc_value_shift_product_exterior_olddiagonal))) /\\ ((exists pfc_complement_shift_product_exterior_olddiagonalterm pfc_left_shift_product_exterior_olddiagonalterm pfc_right_shift_product_exterior_olddiagonalterm. (((pfc_index_shift_product_exterior_olddiagonal)+pfc_complement_shift_product_exterior_olddiagonalterm=(N)) /\\ ((((((exists pfa_gap_shift_product_exterior_olddiagonaltermleftinside. pfa_gap_shift_product_exterior_olddiagonaltermleftinside + S (pfc_index_shift_product_exterior_olddiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_exterior_olddiagonaltermleftentry. ff_h_pfp_shift_product_exterior_olddiagonaltermleftentry + S (pfc_left_shift_product_exterior_olddiagonalterm) = S ((S (pfc_index_shift_product_exterior_olddiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_exterior_olddiagonaltermleftentry. ab = ff_q_pfp_shift_product_exterior_olddiagonaltermleftentry * S ((S (pfc_index_shift_product_exterior_olddiagonal)) * ac) + (pfc_left_shift_product_exterior_olddiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_exterior_olddiagonaltermleftoutside. pfc_gap_shift_product_exterior_olddiagonaltermleftoutside+(L)=(pfc_index_shift_product_exterior_olddiagonal)) /\\ (((pfc_left_shift_product_exterior_olddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_exterior_olddiagonaltermrightinside. pfa_gap_shift_product_exterior_olddiagonaltermrightinside + S (pfc_complement_shift_product_exterior_olddiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_shift_product_exterior_olddiagonaltermrightentry. ff_h_pfp_shift_product_exterior_olddiagonaltermrightentry + S (pfc_right_shift_product_exterior_olddiagonalterm) = S ((S (pfc_complement_shift_product_exterior_olddiagonalterm)) * bc)) /\\ exists ff_q_pfp_shift_product_exterior_olddiagonaltermrightentry. bb = ff_q_pfp_shift_product_exterior_olddiagonaltermrightentry * S ((S (pfc_complement_shift_product_exterior_olddiagonalterm)) * bc) + (pfc_right_shift_product_exterior_olddiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_exterior_olddiagonaltermrightoutside. pfc_gap_shift_product_exterior_olddiagonaltermrightoutside+(M)=(pfc_complement_shift_product_exterior_olddiagonalterm)) /\\ (((pfc_right_shift_product_exterior_olddiagonalterm)=0))))) /\\ (((pfc_value_shift_product_exterior_olddiagonal)=pfc_left_shift_product_exterior_olddiagonalterm*pfc_right_shift_product_exterior_olddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_exterior_oldsum fs_v_pfc_shift_product_exterior_oldsum. ((((exists fs_h_pfc_shift_product_exterior_oldsum_body_start. fs_h_pfc_shift_product_exterior_oldsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_exterior_oldsum)) /\\ exists fs_q_pfc_shift_product_exterior_oldsum_body_start. fs_u_pfc_shift_product_exterior_oldsum = fs_q_pfc_shift_product_exterior_oldsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_exterior_oldsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_exterior_oldsum_body_terminal. fs_h_pfc_shift_product_exterior_oldsum_body_terminal + S (pfc_natural_sum_shift_product_exterior_old) = S ((S (S (N))) * fs_v_pfc_shift_product_exterior_oldsum)) /\\ exists fs_q_pfc_shift_product_exterior_oldsum_body_terminal. fs_u_pfc_shift_product_exterior_oldsum = fs_q_pfc_shift_product_exterior_oldsum_body_terminal * S ((S (S (N))) * fs_v_pfc_shift_product_exterior_oldsum) + (pfc_natural_sum_shift_product_exterior_old))) /\\ forall fs_i_pfc_shift_product_exterior_oldsum_body_steps. (exists fs_lt_pfc_shift_product_exterior_oldsum_body_steps_bound. fs_lt_pfc_shift_product_exterior_oldsum_body_steps_bound + S fs_i_pfc_shift_product_exterior_oldsum_body_steps = S (N)) -> exists fs_a_pfc_shift_product_exterior_oldsum_body_steps fs_r_pfc_shift_product_exterior_oldsum_body_steps fs_s_pfc_shift_product_exterior_oldsum_body_steps. ((((exists fs_h_pfc_shift_product_exterior_oldsum_body_steps_summand. fs_h_pfc_shift_product_exterior_oldsum_body_steps_summand + S (fs_a_pfc_shift_product_exterior_oldsum_body_steps) = S ((S (fs_i_pfc_shift_product_exterior_oldsum_body_steps)) * pfc_terms_scale_shift_product_exterior_old)) /\\ exists fs_q_pfc_shift_product_exterior_oldsum_body_steps_summand. pfc_terms_code_shift_product_exterior_old = fs_q_pfc_shift_product_exterior_oldsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_exterior_oldsum_body_steps)) * pfc_terms_scale_shift_product_exterior_old) + (fs_a_pfc_shift_product_exterior_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_exterior_oldsum_body_steps_partial. fs_h_pfc_shift_product_exterior_oldsum_body_steps_partial + S (fs_r_pfc_shift_product_exterior_oldsum_body_steps) = S ((S (fs_i_pfc_shift_product_exterior_oldsum_body_steps)) * fs_v_pfc_shift_product_exterior_oldsum)) /\\ exists fs_q_pfc_shift_product_exterior_oldsum_body_steps_partial. fs_u_pfc_shift_product_exterior_oldsum = fs_q_pfc_shift_product_exterior_oldsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_exterior_oldsum_body_steps)) * fs_v_pfc_shift_product_exterior_oldsum) + (fs_r_pfc_shift_product_exterior_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_exterior_oldsum_body_steps_successor. fs_h_pfc_shift_product_exterior_oldsum_body_steps_successor + S (fs_s_pfc_shift_product_exterior_oldsum_body_steps) = S ((S (S fs_i_pfc_shift_product_exterior_oldsum_body_steps)) * fs_v_pfc_shift_product_exterior_oldsum)) /\\ exists fs_q_pfc_shift_product_exterior_oldsum_body_steps_successor. fs_u_pfc_shift_product_exterior_oldsum = fs_q_pfc_shift_product_exterior_oldsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_exterior_oldsum_body_steps)) * fs_v_pfc_shift_product_exterior_oldsum) + (fs_s_pfc_shift_product_exterior_oldsum_body_steps))) /\\ fs_s_pfc_shift_product_exterior_oldsum_body_steps = fs_r_pfc_shift_product_exterior_oldsum_body_steps + fs_a_pfc_shift_product_exterior_oldsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_exterior_oldresiduebound. pfa_gap_shift_product_exterior_oldresiduebound + S (x) = (p)) /\\ ((exists pfa_offset_left_shift_product_exterior_oldresiduecongruence pfa_offset_right_shift_product_exterior_oldresiduecongruence. (pfc_natural_sum_shift_product_exterior_old) + (p) * pfa_offset_left_shift_product_exterior_oldresiduecongruence = (x) + (p) * pfa_offset_right_shift_product_exterior_oldresiduecongruence))))))))", "have hiff : (((exists pfc_terms_code_shift_product_last_transport_old pfc_terms_scale_shift_product_last_transport_old pfc_natural_sum_shift_product_last_transport_old. ((forall pfc_index_shift_product_last_transport_olddiagonal. (exists pfa_gap_shift_product_last_transport_olddiagonalbound. pfa_gap_shift_product_last_transport_olddiagonalbound + S (pfc_index_shift_product_last_transport_olddiagonal) = (S (N))) -> exists pfc_value_shift_product_last_transport_olddiagonal. ((((exists ff_h_pfp_shift_product_last_transport_olddiagonalentry. ff_h_pfp_shift_product_last_transport_olddiagonalentry + S (pfc_value_shift_product_last_transport_olddiagonal) = S ((S (pfc_index_shift_product_last_transport_olddiagonal)) * pfc_terms_scale_shift_product_last_transport_old)) /\\ exists ff_q_pfp_shift_product_last_transport_olddiagonalentry. pfc_terms_code_shift_product_last_transport_old = ff_q_pfp_shift_product_last_transport_olddiagonalentry * S ((S (pfc_index_shift_product_last_transport_olddiagonal)) * pfc_terms_scale_shift_product_last_transport_old) + (pfc_value_shift_product_last_transport_olddiagonal))) /\\ ((exists pfc_complement_shift_product_last_transport_olddiagonalterm pfc_left_shift_product_last_transport_olddiagonalterm pfc_right_shift_product_last_transport_olddiagonalterm. (((pfc_index_shift_product_last_transport_olddiagonal)+pfc_complement_shift_product_last_transport_olddiagonalterm=(N)) /\\ ((((((exists pfa_gap_shift_product_last_transport_olddiagonaltermleftinside. pfa_gap_shift_product_last_transport_olddiagonaltermleftinside + S (pfc_index_shift_product_last_transport_olddiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_last_transport_olddiagonaltermleftentry. ff_h_pfp_shift_product_last_transport_olddiagonaltermleftentry + S (pfc_left_shift_product_last_transport_olddiagonalterm) = S ((S (pfc_index_shift_product_last_transport_olddiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_last_transport_olddiagonaltermleftentry. ab = ff_q_pfp_shift_product_last_transport_olddiagonaltermleftentry * S ((S (pfc_index_shift_product_last_transport_olddiagonal)) * ac) + (pfc_left_shift_product_last_transport_olddiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_last_transport_olddiagonaltermleftoutside. pfc_gap_shift_product_last_transport_olddiagonaltermleftoutside+(L)=(pfc_index_shift_product_last_transport_olddiagonal)) /\\ (((pfc_left_shift_product_last_transport_olddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_last_transport_olddiagonaltermrightinside. pfa_gap_shift_product_last_transport_olddiagonaltermrightinside + S (pfc_complement_shift_product_last_transport_olddiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_shift_product_last_transport_olddiagonaltermrightentry. ff_h_pfp_shift_product_last_transport_olddiagonaltermrightentry + S (pfc_right_shift_product_last_transport_olddiagonalterm) = S ((S (pfc_complement_shift_product_last_transport_olddiagonalterm)) * bc)) /\\ exists ff_q_pfp_shift_product_last_transport_olddiagonaltermrightentry. bb = ff_q_pfp_shift_product_last_transport_olddiagonaltermrightentry * S ((S (pfc_complement_shift_product_last_transport_olddiagonalterm)) * bc) + (pfc_right_shift_product_last_transport_olddiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_last_transport_olddiagonaltermrightoutside. pfc_gap_shift_product_last_transport_olddiagonaltermrightoutside+(M)=(pfc_complement_shift_product_last_transport_olddiagonalterm)) /\\ (((pfc_right_shift_product_last_transport_olddiagonalterm)=0))))) /\\ (((pfc_value_shift_product_last_transport_olddiagonal)=pfc_left_shift_product_last_transport_olddiagonalterm*pfc_right_shift_product_last_transport_olddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_last_transport_oldsum fs_v_pfc_shift_product_last_transport_oldsum. ((((exists fs_h_pfc_shift_product_last_transport_oldsum_body_start. fs_h_pfc_shift_product_last_transport_oldsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_last_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_last_transport_oldsum_body_start. fs_u_pfc_shift_product_last_transport_oldsum = fs_q_pfc_shift_product_last_transport_oldsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_last_transport_oldsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_last_transport_oldsum_body_terminal. fs_h_pfc_shift_product_last_transport_oldsum_body_terminal + S (pfc_natural_sum_shift_product_last_transport_old) = S ((S (S (N))) * fs_v_pfc_shift_product_last_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_last_transport_oldsum_body_terminal. fs_u_pfc_shift_product_last_transport_oldsum = fs_q_pfc_shift_product_last_transport_oldsum_body_terminal * S ((S (S (N))) * fs_v_pfc_shift_product_last_transport_oldsum) + (pfc_natural_sum_shift_product_last_transport_old))) /\\ forall fs_i_pfc_shift_product_last_transport_oldsum_body_steps. (exists fs_lt_pfc_shift_product_last_transport_oldsum_body_steps_bound. fs_lt_pfc_shift_product_last_transport_oldsum_body_steps_bound + S fs_i_pfc_shift_product_last_transport_oldsum_body_steps = S (N)) -> exists fs_a_pfc_shift_product_last_transport_oldsum_body_steps fs_r_pfc_shift_product_last_transport_oldsum_body_steps fs_s_pfc_shift_product_last_transport_oldsum_body_steps. ((((exists fs_h_pfc_shift_product_last_transport_oldsum_body_steps_summand. fs_h_pfc_shift_product_last_transport_oldsum_body_steps_summand + S (fs_a_pfc_shift_product_last_transport_oldsum_body_steps) = S ((S (fs_i_pfc_shift_product_last_transport_oldsum_body_steps)) * pfc_terms_scale_shift_product_last_transport_old)) /\\ exists fs_q_pfc_shift_product_last_transport_oldsum_body_steps_summand. pfc_terms_code_shift_product_last_transport_old = fs_q_pfc_shift_product_last_transport_oldsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_last_transport_oldsum_body_steps)) * pfc_terms_scale_shift_product_last_transport_old) + (fs_a_pfc_shift_product_last_transport_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_last_transport_oldsum_body_steps_partial. fs_h_pfc_shift_product_last_transport_oldsum_body_steps_partial + S (fs_r_pfc_shift_product_last_transport_oldsum_body_steps) = S ((S (fs_i_pfc_shift_product_last_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_last_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_last_transport_oldsum_body_steps_partial. fs_u_pfc_shift_product_last_transport_oldsum = fs_q_pfc_shift_product_last_transport_oldsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_last_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_last_transport_oldsum) + (fs_r_pfc_shift_product_last_transport_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_last_transport_oldsum_body_steps_successor. fs_h_pfc_shift_product_last_transport_oldsum_body_steps_successor + S (fs_s_pfc_shift_product_last_transport_oldsum_body_steps) = S ((S (S fs_i_pfc_shift_product_last_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_last_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_last_transport_oldsum_body_steps_successor. fs_u_pfc_shift_product_last_transport_oldsum = fs_q_pfc_shift_product_last_transport_oldsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_last_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_last_transport_oldsum) + (fs_s_pfc_shift_product_last_transport_oldsum_body_steps))) /\\ fs_s_pfc_shift_product_last_transport_oldsum_body_steps = fs_r_pfc_shift_product_last_transport_oldsum_body_steps + fs_a_pfc_shift_product_last_transport_oldsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_last_transport_oldresiduebound. pfa_gap_shift_product_last_transport_oldresiduebound + S (x) = (p)) /\\ ((exists pfa_offset_left_shift_product_last_transport_oldresiduecongruence pfa_offset_right_shift_product_last_transport_oldresiduecongruence. (pfc_natural_sum_shift_product_last_transport_old) + (p) * pfa_offset_left_shift_product_last_transport_oldresiduecongruence = (x) + (p) * pfa_offset_right_shift_product_last_transport_oldresiduecongruence))))))))) -> (exists pfc_terms_code_shift_product_last_transport_new pfc_terms_scale_shift_product_last_transport_new pfc_natural_sum_shift_product_last_transport_new. ((forall pfc_index_shift_product_last_transport_newdiagonal. (exists pfa_gap_shift_product_last_transport_newdiagonalbound. pfa_gap_shift_product_last_transport_newdiagonalbound + S (pfc_index_shift_product_last_transport_newdiagonal) = (S (N))) -> exists pfc_value_shift_product_last_transport_newdiagonal. ((((exists ff_h_pfp_shift_product_last_transport_newdiagonalentry. ff_h_pfp_shift_product_last_transport_newdiagonalentry + S (pfc_value_shift_product_last_transport_newdiagonal) = S ((S (pfc_index_shift_product_last_transport_newdiagonal)) * pfc_terms_scale_shift_product_last_transport_new)) /\\ exists ff_q_pfp_shift_product_last_transport_newdiagonalentry. pfc_terms_code_shift_product_last_transport_new = ff_q_pfp_shift_product_last_transport_newdiagonalentry * S ((S (pfc_index_shift_product_last_transport_newdiagonal)) * pfc_terms_scale_shift_product_last_transport_new) + (pfc_value_shift_product_last_transport_newdiagonal))) /\\ ((exists pfc_complement_shift_product_last_transport_newdiagonalterm pfc_left_shift_product_last_transport_newdiagonalterm pfc_right_shift_product_last_transport_newdiagonalterm. (((pfc_index_shift_product_last_transport_newdiagonal)+pfc_complement_shift_product_last_transport_newdiagonalterm=(N)) /\\ ((((((exists pfa_gap_shift_product_last_transport_newdiagonaltermleftinside. pfa_gap_shift_product_last_transport_newdiagonaltermleftinside + S (pfc_index_shift_product_last_transport_newdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_last_transport_newdiagonaltermleftentry. ff_h_pfp_shift_product_last_transport_newdiagonaltermleftentry + S (pfc_left_shift_product_last_transport_newdiagonalterm) = S ((S (pfc_index_shift_product_last_transport_newdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_last_transport_newdiagonaltermleftentry. ab = ff_q_pfp_shift_product_last_transport_newdiagonaltermleftentry * S ((S (pfc_index_shift_product_last_transport_newdiagonal)) * ac) + (pfc_left_shift_product_last_transport_newdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_last_transport_newdiagonaltermleftoutside. pfc_gap_shift_product_last_transport_newdiagonaltermleftoutside+(L)=(pfc_index_shift_product_last_transport_newdiagonal)) /\\ (((pfc_left_shift_product_last_transport_newdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_last_transport_newdiagonaltermrightinside. pfa_gap_shift_product_last_transport_newdiagonaltermrightinside + S (pfc_complement_shift_product_last_transport_newdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_shift_product_last_transport_newdiagonaltermrightentry. ff_h_pfp_shift_product_last_transport_newdiagonaltermrightentry + S (pfc_right_shift_product_last_transport_newdiagonalterm) = S ((S (pfc_complement_shift_product_last_transport_newdiagonalterm)) * BC)) /\\ exists ff_q_pfp_shift_product_last_transport_newdiagonaltermrightentry. BB = ff_q_pfp_shift_product_last_transport_newdiagonaltermrightentry * S ((S (pfc_complement_shift_product_last_transport_newdiagonalterm)) * BC) + (pfc_right_shift_product_last_transport_newdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_last_transport_newdiagonaltermrightoutside. pfc_gap_shift_product_last_transport_newdiagonaltermrightoutside+(S M)=(pfc_complement_shift_product_last_transport_newdiagonalterm)) /\\ (((pfc_right_shift_product_last_transport_newdiagonalterm)=0))))) /\\ (((pfc_value_shift_product_last_transport_newdiagonal)=pfc_left_shift_product_last_transport_newdiagonalterm*pfc_right_shift_product_last_transport_newdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_last_transport_newsum fs_v_pfc_shift_product_last_transport_newsum. ((((exists fs_h_pfc_shift_product_last_transport_newsum_body_start. fs_h_pfc_shift_product_last_transport_newsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_last_transport_newsum)) /\\ exists fs_q_pfc_shift_product_last_transport_newsum_body_start. fs_u_pfc_shift_product_last_transport_newsum = fs_q_pfc_shift_product_last_transport_newsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_last_transport_newsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_last_transport_newsum_body_terminal. fs_h_pfc_shift_product_last_transport_newsum_body_terminal + S (pfc_natural_sum_shift_product_last_transport_new) = S ((S (S (N))) * fs_v_pfc_shift_product_last_transport_newsum)) /\\ exists fs_q_pfc_shift_product_last_transport_newsum_body_terminal. fs_u_pfc_shift_product_last_transport_newsum = fs_q_pfc_shift_product_last_transport_newsum_body_terminal * S ((S (S (N))) * fs_v_pfc_shift_product_last_transport_newsum) + (pfc_natural_sum_shift_product_last_transport_new))) /\\ forall fs_i_pfc_shift_product_last_transport_newsum_body_steps. (exists fs_lt_pfc_shift_product_last_transport_newsum_body_steps_bound. fs_lt_pfc_shift_product_last_transport_newsum_body_steps_bound + S fs_i_pfc_shift_product_last_transport_newsum_body_steps = S (N)) -> exists fs_a_pfc_shift_product_last_transport_newsum_body_steps fs_r_pfc_shift_product_last_transport_newsum_body_steps fs_s_pfc_shift_product_last_transport_newsum_body_steps. ((((exists fs_h_pfc_shift_product_last_transport_newsum_body_steps_summand. fs_h_pfc_shift_product_last_transport_newsum_body_steps_summand + S (fs_a_pfc_shift_product_last_transport_newsum_body_steps) = S ((S (fs_i_pfc_shift_product_last_transport_newsum_body_steps)) * pfc_terms_scale_shift_product_last_transport_new)) /\\ exists fs_q_pfc_shift_product_last_transport_newsum_body_steps_summand. pfc_terms_code_shift_product_last_transport_new = fs_q_pfc_shift_product_last_transport_newsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_last_transport_newsum_body_steps)) * pfc_terms_scale_shift_product_last_transport_new) + (fs_a_pfc_shift_product_last_transport_newsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_last_transport_newsum_body_steps_partial. fs_h_pfc_shift_product_last_transport_newsum_body_steps_partial + S (fs_r_pfc_shift_product_last_transport_newsum_body_steps) = S ((S (fs_i_pfc_shift_product_last_transport_newsum_body_steps)) * fs_v_pfc_shift_product_last_transport_newsum)) /\\ exists fs_q_pfc_shift_product_last_transport_newsum_body_steps_partial. fs_u_pfc_shift_product_last_transport_newsum = fs_q_pfc_shift_product_last_transport_newsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_last_transport_newsum_body_steps)) * fs_v_pfc_shift_product_last_transport_newsum) + (fs_r_pfc_shift_product_last_transport_newsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_last_transport_newsum_body_steps_successor. fs_h_pfc_shift_product_last_transport_newsum_body_steps_successor + S (fs_s_pfc_shift_product_last_transport_newsum_body_steps) = S ((S (S fs_i_pfc_shift_product_last_transport_newsum_body_steps)) * fs_v_pfc_shift_product_last_transport_newsum)) /\\ exists fs_q_pfc_shift_product_last_transport_newsum_body_steps_successor. fs_u_pfc_shift_product_last_transport_newsum = fs_q_pfc_shift_product_last_transport_newsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_last_transport_newsum_body_steps)) * fs_v_pfc_shift_product_last_transport_newsum) + (fs_s_pfc_shift_product_last_transport_newsum_body_steps))) /\\ fs_s_pfc_shift_product_last_transport_newsum_body_steps = fs_r_pfc_shift_product_last_transport_newsum_body_steps + fs_a_pfc_shift_product_last_transport_newsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_last_transport_newresiduebound. pfa_gap_shift_product_last_transport_newresiduebound + S (x) = (p)) /\\ ((exists pfa_offset_left_shift_product_last_transport_newresiduecongruence pfa_offset_right_shift_product_last_transport_newresiduecongruence. (pfc_natural_sum_shift_product_last_transport_new) + (p) * pfa_offset_left_shift_product_last_transport_newresiduecongruence = (x) + (p) * pfa_offset_right_shift_product_last_transport_newresiduecongruence)))))))))) /\\ (((exists pfc_terms_code_shift_product_last_transport_new pfc_terms_scale_shift_product_last_transport_new pfc_natural_sum_shift_product_last_transport_new. ((forall pfc_index_shift_product_last_transport_newdiagonal. (exists pfa_gap_shift_product_last_transport_newdiagonalbound. pfa_gap_shift_product_last_transport_newdiagonalbound + S (pfc_index_shift_product_last_transport_newdiagonal) = (S (N))) -> exists pfc_value_shift_product_last_transport_newdiagonal. ((((exists ff_h_pfp_shift_product_last_transport_newdiagonalentry. ff_h_pfp_shift_product_last_transport_newdiagonalentry + S (pfc_value_shift_product_last_transport_newdiagonal) = S ((S (pfc_index_shift_product_last_transport_newdiagonal)) * pfc_terms_scale_shift_product_last_transport_new)) /\\ exists ff_q_pfp_shift_product_last_transport_newdiagonalentry. pfc_terms_code_shift_product_last_transport_new = ff_q_pfp_shift_product_last_transport_newdiagonalentry * S ((S (pfc_index_shift_product_last_transport_newdiagonal)) * pfc_terms_scale_shift_product_last_transport_new) + (pfc_value_shift_product_last_transport_newdiagonal))) /\\ ((exists pfc_complement_shift_product_last_transport_newdiagonalterm pfc_left_shift_product_last_transport_newdiagonalterm pfc_right_shift_product_last_transport_newdiagonalterm. (((pfc_index_shift_product_last_transport_newdiagonal)+pfc_complement_shift_product_last_transport_newdiagonalterm=(N)) /\\ ((((((exists pfa_gap_shift_product_last_transport_newdiagonaltermleftinside. pfa_gap_shift_product_last_transport_newdiagonaltermleftinside + S (pfc_index_shift_product_last_transport_newdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_last_transport_newdiagonaltermleftentry. ff_h_pfp_shift_product_last_transport_newdiagonaltermleftentry + S (pfc_left_shift_product_last_transport_newdiagonalterm) = S ((S (pfc_index_shift_product_last_transport_newdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_last_transport_newdiagonaltermleftentry. ab = ff_q_pfp_shift_product_last_transport_newdiagonaltermleftentry * S ((S (pfc_index_shift_product_last_transport_newdiagonal)) * ac) + (pfc_left_shift_product_last_transport_newdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_last_transport_newdiagonaltermleftoutside. pfc_gap_shift_product_last_transport_newdiagonaltermleftoutside+(L)=(pfc_index_shift_product_last_transport_newdiagonal)) /\\ (((pfc_left_shift_product_last_transport_newdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_last_transport_newdiagonaltermrightinside. pfa_gap_shift_product_last_transport_newdiagonaltermrightinside + S (pfc_complement_shift_product_last_transport_newdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_shift_product_last_transport_newdiagonaltermrightentry. ff_h_pfp_shift_product_last_transport_newdiagonaltermrightentry + S (pfc_right_shift_product_last_transport_newdiagonalterm) = S ((S (pfc_complement_shift_product_last_transport_newdiagonalterm)) * BC)) /\\ exists ff_q_pfp_shift_product_last_transport_newdiagonaltermrightentry. BB = ff_q_pfp_shift_product_last_transport_newdiagonaltermrightentry * S ((S (pfc_complement_shift_product_last_transport_newdiagonalterm)) * BC) + (pfc_right_shift_product_last_transport_newdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_last_transport_newdiagonaltermrightoutside. pfc_gap_shift_product_last_transport_newdiagonaltermrightoutside+(S M)=(pfc_complement_shift_product_last_transport_newdiagonalterm)) /\\ (((pfc_right_shift_product_last_transport_newdiagonalterm)=0))))) /\\ (((pfc_value_shift_product_last_transport_newdiagonal)=pfc_left_shift_product_last_transport_newdiagonalterm*pfc_right_shift_product_last_transport_newdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_last_transport_newsum fs_v_pfc_shift_product_last_transport_newsum. ((((exists fs_h_pfc_shift_product_last_transport_newsum_body_start. fs_h_pfc_shift_product_last_transport_newsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_last_transport_newsum)) /\\ exists fs_q_pfc_shift_product_last_transport_newsum_body_start. fs_u_pfc_shift_product_last_transport_newsum = fs_q_pfc_shift_product_last_transport_newsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_last_transport_newsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_last_transport_newsum_body_terminal. fs_h_pfc_shift_product_last_transport_newsum_body_terminal + S (pfc_natural_sum_shift_product_last_transport_new) = S ((S (S (N))) * fs_v_pfc_shift_product_last_transport_newsum)) /\\ exists fs_q_pfc_shift_product_last_transport_newsum_body_terminal. fs_u_pfc_shift_product_last_transport_newsum = fs_q_pfc_shift_product_last_transport_newsum_body_terminal * S ((S (S (N))) * fs_v_pfc_shift_product_last_transport_newsum) + (pfc_natural_sum_shift_product_last_transport_new))) /\\ forall fs_i_pfc_shift_product_last_transport_newsum_body_steps. (exists fs_lt_pfc_shift_product_last_transport_newsum_body_steps_bound. fs_lt_pfc_shift_product_last_transport_newsum_body_steps_bound + S fs_i_pfc_shift_product_last_transport_newsum_body_steps = S (N)) -> exists fs_a_pfc_shift_product_last_transport_newsum_body_steps fs_r_pfc_shift_product_last_transport_newsum_body_steps fs_s_pfc_shift_product_last_transport_newsum_body_steps. ((((exists fs_h_pfc_shift_product_last_transport_newsum_body_steps_summand. fs_h_pfc_shift_product_last_transport_newsum_body_steps_summand + S (fs_a_pfc_shift_product_last_transport_newsum_body_steps) = S ((S (fs_i_pfc_shift_product_last_transport_newsum_body_steps)) * pfc_terms_scale_shift_product_last_transport_new)) /\\ exists fs_q_pfc_shift_product_last_transport_newsum_body_steps_summand. pfc_terms_code_shift_product_last_transport_new = fs_q_pfc_shift_product_last_transport_newsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_last_transport_newsum_body_steps)) * pfc_terms_scale_shift_product_last_transport_new) + (fs_a_pfc_shift_product_last_transport_newsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_last_transport_newsum_body_steps_partial. fs_h_pfc_shift_product_last_transport_newsum_body_steps_partial + S (fs_r_pfc_shift_product_last_transport_newsum_body_steps) = S ((S (fs_i_pfc_shift_product_last_transport_newsum_body_steps)) * fs_v_pfc_shift_product_last_transport_newsum)) /\\ exists fs_q_pfc_shift_product_last_transport_newsum_body_steps_partial. fs_u_pfc_shift_product_last_transport_newsum = fs_q_pfc_shift_product_last_transport_newsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_last_transport_newsum_body_steps)) * fs_v_pfc_shift_product_last_transport_newsum) + (fs_r_pfc_shift_product_last_transport_newsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_last_transport_newsum_body_steps_successor. fs_h_pfc_shift_product_last_transport_newsum_body_steps_successor + S (fs_s_pfc_shift_product_last_transport_newsum_body_steps) = S ((S (S fs_i_pfc_shift_product_last_transport_newsum_body_steps)) * fs_v_pfc_shift_product_last_transport_newsum)) /\\ exists fs_q_pfc_shift_product_last_transport_newsum_body_steps_successor. fs_u_pfc_shift_product_last_transport_newsum = fs_q_pfc_shift_product_last_transport_newsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_last_transport_newsum_body_steps)) * fs_v_pfc_shift_product_last_transport_newsum) + (fs_s_pfc_shift_product_last_transport_newsum_body_steps))) /\\ fs_s_pfc_shift_product_last_transport_newsum_body_steps = fs_r_pfc_shift_product_last_transport_newsum_body_steps + fs_a_pfc_shift_product_last_transport_newsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_last_transport_newresiduebound. pfa_gap_shift_product_last_transport_newresiduebound + S (x) = (p)) /\\ ((exists pfa_offset_left_shift_product_last_transport_newresiduecongruence pfa_offset_right_shift_product_last_transport_newresiduecongruence. (pfc_natural_sum_shift_product_last_transport_new) + (p) * pfa_offset_left_shift_product_last_transport_newresiduecongruence = (x) + (p) * pfa_offset_right_shift_product_last_transport_newresiduecongruence))))))))) -> (exists pfc_terms_code_shift_product_last_transport_old pfc_terms_scale_shift_product_last_transport_old pfc_natural_sum_shift_product_last_transport_old. ((forall pfc_index_shift_product_last_transport_olddiagonal. (exists pfa_gap_shift_product_last_transport_olddiagonalbound. pfa_gap_shift_product_last_transport_olddiagonalbound + S (pfc_index_shift_product_last_transport_olddiagonal) = (S (N))) -> exists pfc_value_shift_product_last_transport_olddiagonal. ((((exists ff_h_pfp_shift_product_last_transport_olddiagonalentry. ff_h_pfp_shift_product_last_transport_olddiagonalentry + S (pfc_value_shift_product_last_transport_olddiagonal) = S ((S (pfc_index_shift_product_last_transport_olddiagonal)) * pfc_terms_scale_shift_product_last_transport_old)) /\\ exists ff_q_pfp_shift_product_last_transport_olddiagonalentry. pfc_terms_code_shift_product_last_transport_old = ff_q_pfp_shift_product_last_transport_olddiagonalentry * S ((S (pfc_index_shift_product_last_transport_olddiagonal)) * pfc_terms_scale_shift_product_last_transport_old) + (pfc_value_shift_product_last_transport_olddiagonal))) /\\ ((exists pfc_complement_shift_product_last_transport_olddiagonalterm pfc_left_shift_product_last_transport_olddiagonalterm pfc_right_shift_product_last_transport_olddiagonalterm. (((pfc_index_shift_product_last_transport_olddiagonal)+pfc_complement_shift_product_last_transport_olddiagonalterm=(N)) /\\ ((((((exists pfa_gap_shift_product_last_transport_olddiagonaltermleftinside. pfa_gap_shift_product_last_transport_olddiagonaltermleftinside + S (pfc_index_shift_product_last_transport_olddiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_last_transport_olddiagonaltermleftentry. ff_h_pfp_shift_product_last_transport_olddiagonaltermleftentry + S (pfc_left_shift_product_last_transport_olddiagonalterm) = S ((S (pfc_index_shift_product_last_transport_olddiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_last_transport_olddiagonaltermleftentry. ab = ff_q_pfp_shift_product_last_transport_olddiagonaltermleftentry * S ((S (pfc_index_shift_product_last_transport_olddiagonal)) * ac) + (pfc_left_shift_product_last_transport_olddiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_last_transport_olddiagonaltermleftoutside. pfc_gap_shift_product_last_transport_olddiagonaltermleftoutside+(L)=(pfc_index_shift_product_last_transport_olddiagonal)) /\\ (((pfc_left_shift_product_last_transport_olddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_last_transport_olddiagonaltermrightinside. pfa_gap_shift_product_last_transport_olddiagonaltermrightinside + S (pfc_complement_shift_product_last_transport_olddiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_shift_product_last_transport_olddiagonaltermrightentry. ff_h_pfp_shift_product_last_transport_olddiagonaltermrightentry + S (pfc_right_shift_product_last_transport_olddiagonalterm) = S ((S (pfc_complement_shift_product_last_transport_olddiagonalterm)) * bc)) /\\ exists ff_q_pfp_shift_product_last_transport_olddiagonaltermrightentry. bb = ff_q_pfp_shift_product_last_transport_olddiagonaltermrightentry * S ((S (pfc_complement_shift_product_last_transport_olddiagonalterm)) * bc) + (pfc_right_shift_product_last_transport_olddiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_last_transport_olddiagonaltermrightoutside. pfc_gap_shift_product_last_transport_olddiagonaltermrightoutside+(M)=(pfc_complement_shift_product_last_transport_olddiagonalterm)) /\\ (((pfc_right_shift_product_last_transport_olddiagonalterm)=0))))) /\\ (((pfc_value_shift_product_last_transport_olddiagonal)=pfc_left_shift_product_last_transport_olddiagonalterm*pfc_right_shift_product_last_transport_olddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_last_transport_oldsum fs_v_pfc_shift_product_last_transport_oldsum. ((((exists fs_h_pfc_shift_product_last_transport_oldsum_body_start. fs_h_pfc_shift_product_last_transport_oldsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_last_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_last_transport_oldsum_body_start. fs_u_pfc_shift_product_last_transport_oldsum = fs_q_pfc_shift_product_last_transport_oldsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_last_transport_oldsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_last_transport_oldsum_body_terminal. fs_h_pfc_shift_product_last_transport_oldsum_body_terminal + S (pfc_natural_sum_shift_product_last_transport_old) = S ((S (S (N))) * fs_v_pfc_shift_product_last_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_last_transport_oldsum_body_terminal. fs_u_pfc_shift_product_last_transport_oldsum = fs_q_pfc_shift_product_last_transport_oldsum_body_terminal * S ((S (S (N))) * fs_v_pfc_shift_product_last_transport_oldsum) + (pfc_natural_sum_shift_product_last_transport_old))) /\\ forall fs_i_pfc_shift_product_last_transport_oldsum_body_steps. (exists fs_lt_pfc_shift_product_last_transport_oldsum_body_steps_bound. fs_lt_pfc_shift_product_last_transport_oldsum_body_steps_bound + S fs_i_pfc_shift_product_last_transport_oldsum_body_steps = S (N)) -> exists fs_a_pfc_shift_product_last_transport_oldsum_body_steps fs_r_pfc_shift_product_last_transport_oldsum_body_steps fs_s_pfc_shift_product_last_transport_oldsum_body_steps. ((((exists fs_h_pfc_shift_product_last_transport_oldsum_body_steps_summand. fs_h_pfc_shift_product_last_transport_oldsum_body_steps_summand + S (fs_a_pfc_shift_product_last_transport_oldsum_body_steps) = S ((S (fs_i_pfc_shift_product_last_transport_oldsum_body_steps)) * pfc_terms_scale_shift_product_last_transport_old)) /\\ exists fs_q_pfc_shift_product_last_transport_oldsum_body_steps_summand. pfc_terms_code_shift_product_last_transport_old = fs_q_pfc_shift_product_last_transport_oldsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_last_transport_oldsum_body_steps)) * pfc_terms_scale_shift_product_last_transport_old) + (fs_a_pfc_shift_product_last_transport_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_last_transport_oldsum_body_steps_partial. fs_h_pfc_shift_product_last_transport_oldsum_body_steps_partial + S (fs_r_pfc_shift_product_last_transport_oldsum_body_steps) = S ((S (fs_i_pfc_shift_product_last_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_last_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_last_transport_oldsum_body_steps_partial. fs_u_pfc_shift_product_last_transport_oldsum = fs_q_pfc_shift_product_last_transport_oldsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_last_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_last_transport_oldsum) + (fs_r_pfc_shift_product_last_transport_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_last_transport_oldsum_body_steps_successor. fs_h_pfc_shift_product_last_transport_oldsum_body_steps_successor + S (fs_s_pfc_shift_product_last_transport_oldsum_body_steps) = S ((S (S fs_i_pfc_shift_product_last_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_last_transport_oldsum)) /\\ exists fs_q_pfc_shift_product_last_transport_oldsum_body_steps_successor. fs_u_pfc_shift_product_last_transport_oldsum = fs_q_pfc_shift_product_last_transport_oldsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_last_transport_oldsum_body_steps)) * fs_v_pfc_shift_product_last_transport_oldsum) + (fs_s_pfc_shift_product_last_transport_oldsum_body_steps))) /\\ fs_s_pfc_shift_product_last_transport_oldsum_body_steps = fs_r_pfc_shift_product_last_transport_oldsum_body_steps + fs_a_pfc_shift_product_last_transport_oldsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_last_transport_oldresiduebound. pfa_gap_shift_product_last_transport_oldresiduebound + S (x) = (p)) /\\ ((exists pfa_offset_left_shift_product_last_transport_oldresiduecongruence pfa_offset_right_shift_product_last_transport_oldresiduecongruence. (pfc_natural_sum_shift_product_last_transport_old) + (p) * pfa_offset_left_shift_product_last_transport_oldresiduecongruence = (x) + (p) * pfa_offset_right_shift_product_last_transport_oldresiduecongruence))))))))))))", "specialize prime_field_convolution_coefficient_shift_right_iff (p)", "specialize prime_field_convolution_coefficient_shift_right_iff (ab)", "specialize prime_field_convolution_coefficient_shift_right_iff (ac)", "specialize prime_field_convolution_coefficient_shift_right_iff (L)", "specialize prime_field_convolution_coefficient_shift_right_iff (bb)", "specialize prime_field_convolution_coefficient_shift_right_iff (bc)", "specialize prime_field_convolution_coefficient_shift_right_iff (M)", "specialize prime_field_convolution_coefficient_shift_right_iff (BB)", "specialize prime_field_convolution_coefficient_shift_right_iff (BC)", "specialize prime_field_convolution_coefficient_shift_right_iff (N)", "specialize prime_field_convolution_coefficient_shift_right_iff (x)", "apply prime_field_convolution_coefficient_shift_right_iff", "exact hs", "cases hiff", "apply hiff_right", "exact hv_witness_right", "have hz : x=0", "specialize prime_field_polynomial_convolution_outside_zero (p)", "specialize prime_field_polynomial_convolution_outside_zero (ab)", "specialize prime_field_polynomial_convolution_outside_zero (ac)", "specialize prime_field_polynomial_convolution_outside_zero (L)", "specialize prime_field_polynomial_convolution_outside_zero (bb)", "specialize prime_field_polynomial_convolution_outside_zero (bc)", "specialize prime_field_polynomial_convolution_outside_zero (M)", "specialize prime_field_polynomial_convolution_outside_zero (cb)", "specialize prime_field_polynomial_convolution_outside_zero (cc)", "specialize prime_field_polynomial_convolution_outside_zero (N)", "specialize prime_field_polynomial_convolution_outside_zero (N)", "specialize prime_field_polynomial_convolution_outside_zero (x)", "apply prime_field_polynomial_convolution_outside_zero", "exact hp", "exact hwhole", "specialize le_refl (N)", "apply le_refl", "exact hco", "rewrite hz at hv_witness_left", "rewrite hz at hv_witness_left", "exact hv_witness_left" ], "script_sha256": "0539f647560b37423cb362737e98228354e74ff7ead5bcaf9894bdbadbcb3c61", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_shift_candidate.py", "sha256": "325d3085482ee73a2c6ee90cd17e45cffe53273671edf89c40d88428335c9c4b" }, "statement": "forall p ab ac L bb bc M cb cc N BB BC db dc K. (~(p=0)) -> (~(L=0)) -> (~(M=0)) -> (((forall mdr_i_pfp_shift_product_factorprefix mdr_a_pfp_shift_product_factorprefix. (exists mdr_gap_pfp_shift_product_factorprefixb. mdr_gap_pfp_shift_product_factorprefixb + S (mdr_i_pfp_shift_product_factorprefix) = (M)) -> (((exists ff_h_mdr_pfp_shift_product_factorprefixo. ff_h_mdr_pfp_shift_product_factorprefixo + S (mdr_a_pfp_shift_product_factorprefix) = S ((S (mdr_i_pfp_shift_product_factorprefix)) * bc)) /\\ exists ff_q_mdr_pfp_shift_product_factorprefixo. bb = ff_q_mdr_pfp_shift_product_factorprefixo * S ((S (mdr_i_pfp_shift_product_factorprefix)) * bc) + (mdr_a_pfp_shift_product_factorprefix))) -> (((exists ff_h_mdr_pfp_shift_product_factorprefixn. ff_h_mdr_pfp_shift_product_factorprefixn + S (mdr_a_pfp_shift_product_factorprefix) = S ((S (mdr_i_pfp_shift_product_factorprefix)) * BC)) /\\ exists ff_q_mdr_pfp_shift_product_factorprefixn. BB = ff_q_mdr_pfp_shift_product_factorprefixn * S ((S (mdr_i_pfp_shift_product_factorprefix)) * BC) + (mdr_a_pfp_shift_product_factorprefix)))) /\\ ((((exists ff_h_pfp_shift_product_factorlast. ff_h_pfp_shift_product_factorlast + S (0) = S ((S (M)) * BC)) /\\ exists ff_q_pfp_shift_product_factorlast. BB = ff_q_pfp_shift_product_factorlast * S ((S (M)) * BC) + (0)))))) -> (((forall fom_index_pfp_shift_product_oldleft. (exists fom_gap_pfp_shift_product_oldleft_index_bound. fom_gap_pfp_shift_product_oldleft_index_bound + S (fom_index_pfp_shift_product_oldleft) = L) -> exists fom_value_pfp_shift_product_oldleft. ((((exists fom_beta_height_pfp_shift_product_oldleft_entry. fom_beta_height_pfp_shift_product_oldleft_entry + S (fom_value_pfp_shift_product_oldleft) = S ((S (fom_index_pfp_shift_product_oldleft)) * ac)) /\\ exists fom_beta_quotient_pfp_shift_product_oldleft_entry. ab = fom_beta_quotient_pfp_shift_product_oldleft_entry * S ((S (fom_index_pfp_shift_product_oldleft)) * ac) + (fom_value_pfp_shift_product_oldleft))) /\\ (exists fom_gap_pfp_shift_product_oldleft_value_bound. fom_gap_pfp_shift_product_oldleft_value_bound + S (fom_value_pfp_shift_product_oldleft) = p))) /\\ (((forall fom_index_pfp_shift_product_oldright. (exists fom_gap_pfp_shift_product_oldright_index_bound. fom_gap_pfp_shift_product_oldright_index_bound + S (fom_index_pfp_shift_product_oldright) = M) -> exists fom_value_pfp_shift_product_oldright. ((((exists fom_beta_height_pfp_shift_product_oldright_entry. fom_beta_height_pfp_shift_product_oldright_entry + S (fom_value_pfp_shift_product_oldright) = S ((S (fom_index_pfp_shift_product_oldright)) * bc)) /\\ exists fom_beta_quotient_pfp_shift_product_oldright_entry. bb = fom_beta_quotient_pfp_shift_product_oldright_entry * S ((S (fom_index_pfp_shift_product_oldright)) * bc) + (fom_value_pfp_shift_product_oldright))) /\\ (exists fom_gap_pfp_shift_product_oldright_value_bound. fom_gap_pfp_shift_product_oldright_value_bound + S (fom_value_pfp_shift_product_oldright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_shift_product_oldcoefficients. (exists pfa_gap_shift_product_oldcoefficientsbound. pfa_gap_shift_product_oldcoefficientsbound + S (pfc_index_shift_product_oldcoefficients) = (N)) -> exists pfc_value_shift_product_oldcoefficients. ((((exists ff_h_pfp_shift_product_oldcoefficientsentry. ff_h_pfp_shift_product_oldcoefficientsentry + S (pfc_value_shift_product_oldcoefficients) = S ((S (pfc_index_shift_product_oldcoefficients)) * cc)) /\\ exists ff_q_pfp_shift_product_oldcoefficientsentry. cb = ff_q_pfp_shift_product_oldcoefficientsentry * S ((S (pfc_index_shift_product_oldcoefficients)) * cc) + (pfc_value_shift_product_oldcoefficients))) /\\ ((exists pfc_terms_code_shift_product_oldcoefficientscoefficient pfc_terms_scale_shift_product_oldcoefficientscoefficient pfc_natural_sum_shift_product_oldcoefficientscoefficient. ((forall pfc_index_shift_product_oldcoefficientscoefficientdiagonal. (exists pfa_gap_shift_product_oldcoefficientscoefficientdiagonalbound. pfa_gap_shift_product_oldcoefficientscoefficientdiagonalbound + S (pfc_index_shift_product_oldcoefficientscoefficientdiagonal) = (S (pfc_index_shift_product_oldcoefficients))) -> exists pfc_value_shift_product_oldcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_shift_product_oldcoefficientscoefficientdiagonalentry. ff_h_pfp_shift_product_oldcoefficientscoefficientdiagonalentry + S (pfc_value_shift_product_oldcoefficientscoefficientdiagonal) = S ((S (pfc_index_shift_product_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_product_oldcoefficientscoefficient)) /\\ exists ff_q_pfp_shift_product_oldcoefficientscoefficientdiagonalentry. pfc_terms_code_shift_product_oldcoefficientscoefficient = ff_q_pfp_shift_product_oldcoefficientscoefficientdiagonalentry * S ((S (pfc_index_shift_product_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_product_oldcoefficientscoefficient) + (pfc_value_shift_product_oldcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_shift_product_oldcoefficientscoefficientdiagonalterm pfc_left_shift_product_oldcoefficientscoefficientdiagonalterm pfc_right_shift_product_oldcoefficientscoefficientdiagonalterm. (((pfc_index_shift_product_oldcoefficientscoefficientdiagonal)+pfc_complement_shift_product_oldcoefficientscoefficientdiagonalterm=(pfc_index_shift_product_oldcoefficients)) /\\ ((((((exists pfa_gap_shift_product_oldcoefficientscoefficientdiagonaltermleftinside. pfa_gap_shift_product_oldcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_shift_product_oldcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_oldcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_shift_product_oldcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_shift_product_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_shift_product_oldcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_oldcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_shift_product_oldcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_shift_product_oldcoefficientscoefficientdiagonal)) * ac) + (pfc_left_shift_product_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_oldcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_shift_product_oldcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_shift_product_oldcoefficientscoefficientdiagonal)) /\\ (((pfc_left_shift_product_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_oldcoefficientscoefficientdiagonaltermrightinside. pfa_gap_shift_product_oldcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_shift_product_oldcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_shift_product_oldcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_shift_product_oldcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_shift_product_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_shift_product_oldcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_shift_product_oldcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_shift_product_oldcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_shift_product_oldcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_shift_product_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_oldcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_shift_product_oldcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_shift_product_oldcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_shift_product_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_shift_product_oldcoefficientscoefficientdiagonal)=pfc_left_shift_product_oldcoefficientscoefficientdiagonalterm*pfc_right_shift_product_oldcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_oldcoefficientscoefficientsum fs_v_pfc_shift_product_oldcoefficientscoefficientsum. ((((exists fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_start. fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_start. fs_u_pfc_shift_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_terminal. fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_shift_product_oldcoefficientscoefficient) = S ((S (S (pfc_index_shift_product_oldcoefficients))) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_terminal. fs_u_pfc_shift_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_shift_product_oldcoefficients))) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum) + (pfc_natural_sum_shift_product_oldcoefficientscoefficient))) /\\ forall fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps = S (pfc_index_shift_product_oldcoefficients)) -> exists fs_a_pfc_shift_product_oldcoefficientscoefficientsum_body_steps fs_r_pfc_shift_product_oldcoefficientscoefficientsum_body_steps fs_s_pfc_shift_product_oldcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_shift_product_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_product_oldcoefficientscoefficient)) /\\ exists fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_shift_product_oldcoefficientscoefficient = fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_product_oldcoefficientscoefficient) + (fs_a_pfc_shift_product_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_shift_product_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_shift_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum) + (fs_r_pfc_shift_product_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_shift_product_oldcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_shift_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_product_oldcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_product_oldcoefficientscoefficientsum) + (fs_s_pfc_shift_product_oldcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_shift_product_oldcoefficientscoefficientsum_body_steps = fs_r_pfc_shift_product_oldcoefficientscoefficientsum_body_steps + fs_a_pfc_shift_product_oldcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_oldcoefficientscoefficientresiduebound. pfa_gap_shift_product_oldcoefficientscoefficientresiduebound + S (pfc_value_shift_product_oldcoefficients) = (p)) /\\ ((exists pfa_offset_left_shift_product_oldcoefficientscoefficientresiduecongruence pfa_offset_right_shift_product_oldcoefficientscoefficientresiduecongruence. (pfc_natural_sum_shift_product_oldcoefficientscoefficient) + (p) * pfa_offset_left_shift_product_oldcoefficientscoefficientresiduecongruence = (pfc_value_shift_product_oldcoefficients) + (p) * pfa_offset_right_shift_product_oldcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_shift_product_newleft. (exists fom_gap_pfp_shift_product_newleft_index_bound. fom_gap_pfp_shift_product_newleft_index_bound + S (fom_index_pfp_shift_product_newleft) = L) -> exists fom_value_pfp_shift_product_newleft. ((((exists fom_beta_height_pfp_shift_product_newleft_entry. fom_beta_height_pfp_shift_product_newleft_entry + S (fom_value_pfp_shift_product_newleft) = S ((S (fom_index_pfp_shift_product_newleft)) * ac)) /\\ exists fom_beta_quotient_pfp_shift_product_newleft_entry. ab = fom_beta_quotient_pfp_shift_product_newleft_entry * S ((S (fom_index_pfp_shift_product_newleft)) * ac) + (fom_value_pfp_shift_product_newleft))) /\\ (exists fom_gap_pfp_shift_product_newleft_value_bound. fom_gap_pfp_shift_product_newleft_value_bound + S (fom_value_pfp_shift_product_newleft) = p))) /\\ (((forall fom_index_pfp_shift_product_newright. (exists fom_gap_pfp_shift_product_newright_index_bound. fom_gap_pfp_shift_product_newright_index_bound + S (fom_index_pfp_shift_product_newright) = S M) -> exists fom_value_pfp_shift_product_newright. ((((exists fom_beta_height_pfp_shift_product_newright_entry. fom_beta_height_pfp_shift_product_newright_entry + S (fom_value_pfp_shift_product_newright) = S ((S (fom_index_pfp_shift_product_newright)) * BC)) /\\ exists fom_beta_quotient_pfp_shift_product_newright_entry. BB = fom_beta_quotient_pfp_shift_product_newright_entry * S ((S (fom_index_pfp_shift_product_newright)) * BC) + (fom_value_pfp_shift_product_newright))) /\\ (exists fom_gap_pfp_shift_product_newright_value_bound. fom_gap_pfp_shift_product_newright_value_bound + S (fom_value_pfp_shift_product_newright) = p))) /\\ (((((((L)=0 \\/ (S M)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((S M)=0)) /\\ (((L)+(S M)=S (K)))))))) /\\ ((forall pfc_index_shift_product_newcoefficients. (exists pfa_gap_shift_product_newcoefficientsbound. pfa_gap_shift_product_newcoefficientsbound + S (pfc_index_shift_product_newcoefficients) = (K)) -> exists pfc_value_shift_product_newcoefficients. ((((exists ff_h_pfp_shift_product_newcoefficientsentry. ff_h_pfp_shift_product_newcoefficientsentry + S (pfc_value_shift_product_newcoefficients) = S ((S (pfc_index_shift_product_newcoefficients)) * dc)) /\\ exists ff_q_pfp_shift_product_newcoefficientsentry. db = ff_q_pfp_shift_product_newcoefficientsentry * S ((S (pfc_index_shift_product_newcoefficients)) * dc) + (pfc_value_shift_product_newcoefficients))) /\\ ((exists pfc_terms_code_shift_product_newcoefficientscoefficient pfc_terms_scale_shift_product_newcoefficientscoefficient pfc_natural_sum_shift_product_newcoefficientscoefficient. ((forall pfc_index_shift_product_newcoefficientscoefficientdiagonal. (exists pfa_gap_shift_product_newcoefficientscoefficientdiagonalbound. pfa_gap_shift_product_newcoefficientscoefficientdiagonalbound + S (pfc_index_shift_product_newcoefficientscoefficientdiagonal) = (S (pfc_index_shift_product_newcoefficients))) -> exists pfc_value_shift_product_newcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_shift_product_newcoefficientscoefficientdiagonalentry. ff_h_pfp_shift_product_newcoefficientscoefficientdiagonalentry + S (pfc_value_shift_product_newcoefficientscoefficientdiagonal) = S ((S (pfc_index_shift_product_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_product_newcoefficientscoefficient)) /\\ exists ff_q_pfp_shift_product_newcoefficientscoefficientdiagonalentry. pfc_terms_code_shift_product_newcoefficientscoefficient = ff_q_pfp_shift_product_newcoefficientscoefficientdiagonalentry * S ((S (pfc_index_shift_product_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_product_newcoefficientscoefficient) + (pfc_value_shift_product_newcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_shift_product_newcoefficientscoefficientdiagonalterm pfc_left_shift_product_newcoefficientscoefficientdiagonalterm pfc_right_shift_product_newcoefficientscoefficientdiagonalterm. (((pfc_index_shift_product_newcoefficientscoefficientdiagonal)+pfc_complement_shift_product_newcoefficientscoefficientdiagonalterm=(pfc_index_shift_product_newcoefficients)) /\\ ((((((exists pfa_gap_shift_product_newcoefficientscoefficientdiagonaltermleftinside. pfa_gap_shift_product_newcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_shift_product_newcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_product_newcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_shift_product_newcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_shift_product_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_shift_product_newcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_product_newcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_shift_product_newcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_shift_product_newcoefficientscoefficientdiagonal)) * ac) + (pfc_left_shift_product_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_newcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_shift_product_newcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_shift_product_newcoefficientscoefficientdiagonal)) /\\ (((pfc_left_shift_product_newcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_product_newcoefficientscoefficientdiagonaltermrightinside. pfa_gap_shift_product_newcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_shift_product_newcoefficientscoefficientdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_shift_product_newcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_shift_product_newcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_shift_product_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_shift_product_newcoefficientscoefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_shift_product_newcoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_shift_product_newcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_shift_product_newcoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_shift_product_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_product_newcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_shift_product_newcoefficientscoefficientdiagonaltermrightoutside+(S M)=(pfc_complement_shift_product_newcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_shift_product_newcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_shift_product_newcoefficientscoefficientdiagonal)=pfc_left_shift_product_newcoefficientscoefficientdiagonalterm*pfc_right_shift_product_newcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_product_newcoefficientscoefficientsum fs_v_pfc_shift_product_newcoefficientscoefficientsum. ((((exists fs_h_pfc_shift_product_newcoefficientscoefficientsum_body_start. fs_h_pfc_shift_product_newcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_product_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_product_newcoefficientscoefficientsum_body_start. fs_u_pfc_shift_product_newcoefficientscoefficientsum = fs_q_pfc_shift_product_newcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_shift_product_newcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_shift_product_newcoefficientscoefficientsum_body_terminal. fs_h_pfc_shift_product_newcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_shift_product_newcoefficientscoefficient) = S ((S (S (pfc_index_shift_product_newcoefficients))) * fs_v_pfc_shift_product_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_product_newcoefficientscoefficientsum_body_terminal. fs_u_pfc_shift_product_newcoefficientscoefficientsum = fs_q_pfc_shift_product_newcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_shift_product_newcoefficients))) * fs_v_pfc_shift_product_newcoefficientscoefficientsum) + (pfc_natural_sum_shift_product_newcoefficientscoefficient))) /\\ forall fs_i_pfc_shift_product_newcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_shift_product_newcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_shift_product_newcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_shift_product_newcoefficientscoefficientsum_body_steps = S (pfc_index_shift_product_newcoefficients)) -> exists fs_a_pfc_shift_product_newcoefficientscoefficientsum_body_steps fs_r_pfc_shift_product_newcoefficientscoefficientsum_body_steps fs_s_pfc_shift_product_newcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_shift_product_newcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_shift_product_newcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_shift_product_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_product_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_product_newcoefficientscoefficient)) /\\ exists fs_q_pfc_shift_product_newcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_shift_product_newcoefficientscoefficient = fs_q_pfc_shift_product_newcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_shift_product_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_product_newcoefficientscoefficient) + (fs_a_pfc_shift_product_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_newcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_shift_product_newcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_shift_product_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_product_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_product_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_product_newcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_shift_product_newcoefficientscoefficientsum = fs_q_pfc_shift_product_newcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_shift_product_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_product_newcoefficientscoefficientsum) + (fs_r_pfc_shift_product_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_product_newcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_shift_product_newcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_shift_product_newcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_shift_product_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_product_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_product_newcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_shift_product_newcoefficientscoefficientsum = fs_q_pfc_shift_product_newcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_shift_product_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_product_newcoefficientscoefficientsum) + (fs_s_pfc_shift_product_newcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_shift_product_newcoefficientscoefficientsum_body_steps = fs_r_pfc_shift_product_newcoefficientscoefficientsum_body_steps + fs_a_pfc_shift_product_newcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_product_newcoefficientscoefficientresiduebound. pfa_gap_shift_product_newcoefficientscoefficientresiduebound + S (pfc_value_shift_product_newcoefficients) = (p)) /\\ ((exists pfa_offset_left_shift_product_newcoefficientscoefficientresiduecongruence pfa_offset_right_shift_product_newcoefficientscoefficientresiduecongruence. (pfc_natural_sum_shift_product_newcoefficientscoefficient) + (p) * pfa_offset_left_shift_product_newcoefficientscoefficientresiduecongruence = (pfc_value_shift_product_newcoefficients) + (p) * pfa_offset_right_shift_product_newcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((K=S N) /\\ ((((forall mdr_i_pfp_shift_product_resultprefix mdr_a_pfp_shift_product_resultprefix. (exists mdr_gap_pfp_shift_product_resultprefixb. mdr_gap_pfp_shift_product_resultprefixb + S (mdr_i_pfp_shift_product_resultprefix) = (N)) -> (((exists ff_h_mdr_pfp_shift_product_resultprefixo. ff_h_mdr_pfp_shift_product_resultprefixo + S (mdr_a_pfp_shift_product_resultprefix) = S ((S (mdr_i_pfp_shift_product_resultprefix)) * cc)) /\\ exists ff_q_mdr_pfp_shift_product_resultprefixo. cb = ff_q_mdr_pfp_shift_product_resultprefixo * S ((S (mdr_i_pfp_shift_product_resultprefix)) * cc) + (mdr_a_pfp_shift_product_resultprefix))) -> (((exists ff_h_mdr_pfp_shift_product_resultprefixn. ff_h_mdr_pfp_shift_product_resultprefixn + S (mdr_a_pfp_shift_product_resultprefix) = S ((S (mdr_i_pfp_shift_product_resultprefix)) * dc)) /\\ exists ff_q_mdr_pfp_shift_product_resultprefixn. db = ff_q_mdr_pfp_shift_product_resultprefixn * S ((S (mdr_i_pfp_shift_product_resultprefix)) * dc) + (mdr_a_pfp_shift_product_resultprefix)))) /\\ ((((exists ff_h_pfp_shift_product_resultlast. ff_h_pfp_shift_product_resultlast + S (0) = S ((S (N)) * dc)) /\\ exists ff_q_pfp_shift_product_resultlast. db = ff_q_pfp_shift_product_resultlast * S ((S (N)) * dc) + (0)))))))))", "statement_sha256": "6f60c7f6c17e34de78a145b9a6cb532ca29ba7a0a3b13d3c7b4abc78973bbe00", "summary": "For actual nonempty factors, the shifted product is exactly a trailing-zero extension of the original decoded product, at its proved successor length.", "summary_sha256": "794b4349cea9f6ff934a79e701c92737be5feaa0d587b11683e5d4691bbbd5d1" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "58f467040533a74a75978a5737894f6dfc238202da1c656e725db2ac1df05c2d", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 197, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": 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"intro hs", "intro hc", "intro hd", "intro hempty", "cases hempty", "have hz : forall pfp_repeat_index_shift_empty_source_left. (exists pfa_gap_shift_empty_source_leftindex. pfa_gap_shift_empty_source_leftindex + S (pfp_repeat_index_shift_empty_source_left) = (L)) -> (((exists ff_h_pfp_shift_empty_source_leftentry. ff_h_pfp_shift_empty_source_leftentry + S (0) = S ((S (pfp_repeat_index_shift_empty_source_left)) * ac)) /\\ exists ff_q_pfp_shift_empty_source_leftentry. ab = ff_q_pfp_shift_empty_source_leftentry * S ((S (pfp_repeat_index_shift_empty_source_left)) * ac) + (0)))", "intro i", "intro hi", "rewrite hempty_left at hi", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hi", "specialize zero_le (i)", "apply zero_le", "split", "specialize prime_field_polynomial_convolution_zero_left (p)", "specialize prime_field_polynomial_convolution_zero_left (ab)", "specialize prime_field_polynomial_convolution_zero_left (ac)", "specialize prime_field_polynomial_convolution_zero_left (L)", "specialize prime_field_polynomial_convolution_zero_left (bb)", "specialize prime_field_polynomial_convolution_zero_left (bc)", "specialize prime_field_polynomial_convolution_zero_left (M)", "specialize prime_field_polynomial_convolution_zero_left (cb)", "specialize prime_field_polynomial_convolution_zero_left (cc)", "specialize prime_field_polynomial_convolution_zero_left (N)", "apply prime_field_polynomial_convolution_zero_left", "exact hp", "exact hz", "exact hc", "specialize prime_field_polynomial_convolution_zero_left (p)", "specialize prime_field_polynomial_convolution_zero_left (ab)", "specialize prime_field_polynomial_convolution_zero_left (ac)", "specialize prime_field_polynomial_convolution_zero_left (L)", "specialize prime_field_polynomial_convolution_zero_left (BB)", "specialize prime_field_polynomial_convolution_zero_left (BC)", "specialize prime_field_polynomial_convolution_zero_left (S M)", "specialize prime_field_polynomial_convolution_zero_left (db)", "specialize prime_field_polynomial_convolution_zero_left (dc)", "specialize prime_field_polynomial_convolution_zero_left (K)", "apply prime_field_polynomial_convolution_zero_left", "exact hp", "exact hz", "exact hd", "have hz : forall pfp_repeat_index_shift_empty_source_right. (exists pfa_gap_shift_empty_source_rightindex. pfa_gap_shift_empty_source_rightindex + S (pfp_repeat_index_shift_empty_source_right) = (M)) -> (((exists ff_h_pfp_shift_empty_source_rightentry. ff_h_pfp_shift_empty_source_rightentry + S (0) = S ((S (pfp_repeat_index_shift_empty_source_right)) * bc)) /\\ exists ff_q_pfp_shift_empty_source_rightentry. bb = ff_q_pfp_shift_empty_source_rightentry * S ((S (pfp_repeat_index_shift_empty_source_right)) * bc) + (0)))", "intro i", "intro hi", "rewrite hempty_right at hi", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hi", "specialize zero_le (i)", "apply zero_le", "split", "specialize prime_field_polynomial_convolution_zero_right (p)", "specialize prime_field_polynomial_convolution_zero_right (ab)", "specialize prime_field_polynomial_convolution_zero_right (ac)", "specialize prime_field_polynomial_convolution_zero_right (L)", "specialize prime_field_polynomial_convolution_zero_right (bb)", "specialize prime_field_polynomial_convolution_zero_right (bc)", "specialize prime_field_polynomial_convolution_zero_right (M)", "specialize prime_field_polynomial_convolution_zero_right (cb)", "specialize prime_field_polynomial_convolution_zero_right (cc)", "specialize prime_field_polynomial_convolution_zero_right (N)", "apply prime_field_polynomial_convolution_zero_right", "exact hp", "exact hz", "exact hc", "specialize prime_field_polynomial_convolution_zero_right (p)", "specialize prime_field_polynomial_convolution_zero_right (ab)", "specialize prime_field_polynomial_convolution_zero_right (ac)", "specialize prime_field_polynomial_convolution_zero_right (L)", "specialize prime_field_polynomial_convolution_zero_right (BB)", "specialize prime_field_polynomial_convolution_zero_right (BC)", "specialize prime_field_polynomial_convolution_zero_right (S M)", "specialize prime_field_polynomial_convolution_zero_right (db)", "specialize prime_field_polynomial_convolution_zero_right (dc)", "specialize prime_field_polynomial_convolution_zero_right (K)", "apply prime_field_polynomial_convolution_zero_right", "exact hp", "specialize prime_field_polynomial_shift_zero_prefix (bb)", "specialize prime_field_polynomial_shift_zero_prefix (bc)", "specialize prime_field_polynomial_shift_zero_prefix (M)", "specialize prime_field_polynomial_shift_zero_prefix (BB)", "specialize prime_field_polynomial_shift_zero_prefix (BC)", "apply prime_field_polynomial_shift_zero_prefix", "exact hz", "exact hs", "exact hd" ], "script_sha256": "53f2875d66196cec5cffb76a96129e284466b6bf778a8f81bb817953804a5236", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_shift_candidate.py", "sha256": "325d3085482ee73a2c6ee90cd17e45cffe53273671edf89c40d88428335c9c4b" }, "statement": "forall p ab ac L bb bc M cb cc N BB BC db dc K. (~(p=0)) -> (((forall mdr_i_pfp_shift_empty_factorprefix mdr_a_pfp_shift_empty_factorprefix. (exists mdr_gap_pfp_shift_empty_factorprefixb. mdr_gap_pfp_shift_empty_factorprefixb + S (mdr_i_pfp_shift_empty_factorprefix) = (M)) -> (((exists ff_h_mdr_pfp_shift_empty_factorprefixo. ff_h_mdr_pfp_shift_empty_factorprefixo + S (mdr_a_pfp_shift_empty_factorprefix) = S ((S (mdr_i_pfp_shift_empty_factorprefix)) * bc)) /\\ exists ff_q_mdr_pfp_shift_empty_factorprefixo. bb = ff_q_mdr_pfp_shift_empty_factorprefixo * S ((S (mdr_i_pfp_shift_empty_factorprefix)) * bc) + (mdr_a_pfp_shift_empty_factorprefix))) -> (((exists ff_h_mdr_pfp_shift_empty_factorprefixn. ff_h_mdr_pfp_shift_empty_factorprefixn + S (mdr_a_pfp_shift_empty_factorprefix) = S ((S (mdr_i_pfp_shift_empty_factorprefix)) * BC)) /\\ exists ff_q_mdr_pfp_shift_empty_factorprefixn. BB = ff_q_mdr_pfp_shift_empty_factorprefixn * S ((S (mdr_i_pfp_shift_empty_factorprefix)) * BC) + (mdr_a_pfp_shift_empty_factorprefix)))) /\\ ((((exists ff_h_pfp_shift_empty_factorlast. ff_h_pfp_shift_empty_factorlast + S (0) = S ((S (M)) * BC)) /\\ exists ff_q_pfp_shift_empty_factorlast. BB = ff_q_pfp_shift_empty_factorlast * S ((S (M)) * BC) + (0)))))) -> (((forall fom_index_pfp_shift_empty_oldleft. (exists fom_gap_pfp_shift_empty_oldleft_index_bound. fom_gap_pfp_shift_empty_oldleft_index_bound + S (fom_index_pfp_shift_empty_oldleft) = L) -> exists fom_value_pfp_shift_empty_oldleft. ((((exists fom_beta_height_pfp_shift_empty_oldleft_entry. fom_beta_height_pfp_shift_empty_oldleft_entry + S (fom_value_pfp_shift_empty_oldleft) = S ((S (fom_index_pfp_shift_empty_oldleft)) * ac)) /\\ exists fom_beta_quotient_pfp_shift_empty_oldleft_entry. ab = fom_beta_quotient_pfp_shift_empty_oldleft_entry * S ((S (fom_index_pfp_shift_empty_oldleft)) * ac) + (fom_value_pfp_shift_empty_oldleft))) /\\ (exists fom_gap_pfp_shift_empty_oldleft_value_bound. fom_gap_pfp_shift_empty_oldleft_value_bound + S (fom_value_pfp_shift_empty_oldleft) = p))) /\\ (((forall fom_index_pfp_shift_empty_oldright. (exists fom_gap_pfp_shift_empty_oldright_index_bound. fom_gap_pfp_shift_empty_oldright_index_bound + S (fom_index_pfp_shift_empty_oldright) = M) -> exists fom_value_pfp_shift_empty_oldright. ((((exists fom_beta_height_pfp_shift_empty_oldright_entry. fom_beta_height_pfp_shift_empty_oldright_entry + S (fom_value_pfp_shift_empty_oldright) = S ((S (fom_index_pfp_shift_empty_oldright)) * bc)) /\\ exists fom_beta_quotient_pfp_shift_empty_oldright_entry. bb = fom_beta_quotient_pfp_shift_empty_oldright_entry * S ((S (fom_index_pfp_shift_empty_oldright)) * bc) + (fom_value_pfp_shift_empty_oldright))) /\\ (exists fom_gap_pfp_shift_empty_oldright_value_bound. fom_gap_pfp_shift_empty_oldright_value_bound + S (fom_value_pfp_shift_empty_oldright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_shift_empty_oldcoefficients. (exists pfa_gap_shift_empty_oldcoefficientsbound. pfa_gap_shift_empty_oldcoefficientsbound + S (pfc_index_shift_empty_oldcoefficients) = (N)) -> exists pfc_value_shift_empty_oldcoefficients. ((((exists ff_h_pfp_shift_empty_oldcoefficientsentry. ff_h_pfp_shift_empty_oldcoefficientsentry + S (pfc_value_shift_empty_oldcoefficients) = S ((S (pfc_index_shift_empty_oldcoefficients)) * cc)) /\\ exists ff_q_pfp_shift_empty_oldcoefficientsentry. cb = ff_q_pfp_shift_empty_oldcoefficientsentry * S ((S (pfc_index_shift_empty_oldcoefficients)) * cc) + (pfc_value_shift_empty_oldcoefficients))) /\\ ((exists pfc_terms_code_shift_empty_oldcoefficientscoefficient pfc_terms_scale_shift_empty_oldcoefficientscoefficient pfc_natural_sum_shift_empty_oldcoefficientscoefficient. ((forall pfc_index_shift_empty_oldcoefficientscoefficientdiagonal. (exists pfa_gap_shift_empty_oldcoefficientscoefficientdiagonalbound. pfa_gap_shift_empty_oldcoefficientscoefficientdiagonalbound + S (pfc_index_shift_empty_oldcoefficientscoefficientdiagonal) = (S (pfc_index_shift_empty_oldcoefficients))) -> exists pfc_value_shift_empty_oldcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_shift_empty_oldcoefficientscoefficientdiagonalentry. ff_h_pfp_shift_empty_oldcoefficientscoefficientdiagonalentry + S (pfc_value_shift_empty_oldcoefficientscoefficientdiagonal) = S ((S (pfc_index_shift_empty_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_empty_oldcoefficientscoefficient)) /\\ exists ff_q_pfp_shift_empty_oldcoefficientscoefficientdiagonalentry. pfc_terms_code_shift_empty_oldcoefficientscoefficient = ff_q_pfp_shift_empty_oldcoefficientscoefficientdiagonalentry * S ((S (pfc_index_shift_empty_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_empty_oldcoefficientscoefficient) + (pfc_value_shift_empty_oldcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_shift_empty_oldcoefficientscoefficientdiagonalterm pfc_left_shift_empty_oldcoefficientscoefficientdiagonalterm pfc_right_shift_empty_oldcoefficientscoefficientdiagonalterm. (((pfc_index_shift_empty_oldcoefficientscoefficientdiagonal)+pfc_complement_shift_empty_oldcoefficientscoefficientdiagonalterm=(pfc_index_shift_empty_oldcoefficients)) /\\ ((((((exists pfa_gap_shift_empty_oldcoefficientscoefficientdiagonaltermleftinside. pfa_gap_shift_empty_oldcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_shift_empty_oldcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_empty_oldcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_shift_empty_oldcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_shift_empty_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_shift_empty_oldcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_empty_oldcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_shift_empty_oldcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_shift_empty_oldcoefficientscoefficientdiagonal)) * ac) + (pfc_left_shift_empty_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_empty_oldcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_shift_empty_oldcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_shift_empty_oldcoefficientscoefficientdiagonal)) /\\ (((pfc_left_shift_empty_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_empty_oldcoefficientscoefficientdiagonaltermrightinside. pfa_gap_shift_empty_oldcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_shift_empty_oldcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_shift_empty_oldcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_shift_empty_oldcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_shift_empty_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_shift_empty_oldcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_shift_empty_oldcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_shift_empty_oldcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_shift_empty_oldcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_shift_empty_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_empty_oldcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_shift_empty_oldcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_shift_empty_oldcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_shift_empty_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_shift_empty_oldcoefficientscoefficientdiagonal)=pfc_left_shift_empty_oldcoefficientscoefficientdiagonalterm*pfc_right_shift_empty_oldcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_empty_oldcoefficientscoefficientsum fs_v_pfc_shift_empty_oldcoefficientscoefficientsum. ((((exists fs_h_pfc_shift_empty_oldcoefficientscoefficientsum_body_start. fs_h_pfc_shift_empty_oldcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_empty_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_empty_oldcoefficientscoefficientsum_body_start. fs_u_pfc_shift_empty_oldcoefficientscoefficientsum = fs_q_pfc_shift_empty_oldcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_shift_empty_oldcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_shift_empty_oldcoefficientscoefficientsum_body_terminal. fs_h_pfc_shift_empty_oldcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_shift_empty_oldcoefficientscoefficient) = S ((S (S (pfc_index_shift_empty_oldcoefficients))) * fs_v_pfc_shift_empty_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_empty_oldcoefficientscoefficientsum_body_terminal. fs_u_pfc_shift_empty_oldcoefficientscoefficientsum = fs_q_pfc_shift_empty_oldcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_shift_empty_oldcoefficients))) * fs_v_pfc_shift_empty_oldcoefficientscoefficientsum) + (pfc_natural_sum_shift_empty_oldcoefficientscoefficient))) /\\ forall fs_i_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps = S (pfc_index_shift_empty_oldcoefficients)) -> exists fs_a_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps fs_r_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps fs_s_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_empty_oldcoefficientscoefficient)) /\\ exists fs_q_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_shift_empty_oldcoefficientscoefficient = fs_q_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_empty_oldcoefficientscoefficient) + (fs_a_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_empty_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_shift_empty_oldcoefficientscoefficientsum = fs_q_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_empty_oldcoefficientscoefficientsum) + (fs_r_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_empty_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_shift_empty_oldcoefficientscoefficientsum = fs_q_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_empty_oldcoefficientscoefficientsum) + (fs_s_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps = fs_r_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps + fs_a_pfc_shift_empty_oldcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_empty_oldcoefficientscoefficientresiduebound. pfa_gap_shift_empty_oldcoefficientscoefficientresiduebound + S (pfc_value_shift_empty_oldcoefficients) = (p)) /\\ ((exists pfa_offset_left_shift_empty_oldcoefficientscoefficientresiduecongruence pfa_offset_right_shift_empty_oldcoefficientscoefficientresiduecongruence. (pfc_natural_sum_shift_empty_oldcoefficientscoefficient) + (p) * pfa_offset_left_shift_empty_oldcoefficientscoefficientresiduecongruence = (pfc_value_shift_empty_oldcoefficients) + (p) * pfa_offset_right_shift_empty_oldcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_shift_empty_newleft. (exists fom_gap_pfp_shift_empty_newleft_index_bound. fom_gap_pfp_shift_empty_newleft_index_bound + S (fom_index_pfp_shift_empty_newleft) = L) -> exists fom_value_pfp_shift_empty_newleft. ((((exists fom_beta_height_pfp_shift_empty_newleft_entry. fom_beta_height_pfp_shift_empty_newleft_entry + S (fom_value_pfp_shift_empty_newleft) = S ((S (fom_index_pfp_shift_empty_newleft)) * ac)) /\\ exists fom_beta_quotient_pfp_shift_empty_newleft_entry. ab = fom_beta_quotient_pfp_shift_empty_newleft_entry * S ((S (fom_index_pfp_shift_empty_newleft)) * ac) + (fom_value_pfp_shift_empty_newleft))) /\\ (exists fom_gap_pfp_shift_empty_newleft_value_bound. fom_gap_pfp_shift_empty_newleft_value_bound + S (fom_value_pfp_shift_empty_newleft) = p))) /\\ (((forall fom_index_pfp_shift_empty_newright. (exists fom_gap_pfp_shift_empty_newright_index_bound. fom_gap_pfp_shift_empty_newright_index_bound + S (fom_index_pfp_shift_empty_newright) = S M) -> exists fom_value_pfp_shift_empty_newright. ((((exists fom_beta_height_pfp_shift_empty_newright_entry. fom_beta_height_pfp_shift_empty_newright_entry + S (fom_value_pfp_shift_empty_newright) = S ((S (fom_index_pfp_shift_empty_newright)) * BC)) /\\ exists fom_beta_quotient_pfp_shift_empty_newright_entry. BB = fom_beta_quotient_pfp_shift_empty_newright_entry * S ((S (fom_index_pfp_shift_empty_newright)) * BC) + (fom_value_pfp_shift_empty_newright))) /\\ (exists fom_gap_pfp_shift_empty_newright_value_bound. fom_gap_pfp_shift_empty_newright_value_bound + S (fom_value_pfp_shift_empty_newright) = p))) /\\ (((((((L)=0 \\/ (S M)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((S M)=0)) /\\ (((L)+(S M)=S (K)))))))) /\\ ((forall pfc_index_shift_empty_newcoefficients. (exists pfa_gap_shift_empty_newcoefficientsbound. pfa_gap_shift_empty_newcoefficientsbound + S (pfc_index_shift_empty_newcoefficients) = (K)) -> exists pfc_value_shift_empty_newcoefficients. ((((exists ff_h_pfp_shift_empty_newcoefficientsentry. ff_h_pfp_shift_empty_newcoefficientsentry + S (pfc_value_shift_empty_newcoefficients) = S ((S (pfc_index_shift_empty_newcoefficients)) * dc)) /\\ exists ff_q_pfp_shift_empty_newcoefficientsentry. db = ff_q_pfp_shift_empty_newcoefficientsentry * S ((S (pfc_index_shift_empty_newcoefficients)) * dc) + (pfc_value_shift_empty_newcoefficients))) /\\ ((exists pfc_terms_code_shift_empty_newcoefficientscoefficient pfc_terms_scale_shift_empty_newcoefficientscoefficient pfc_natural_sum_shift_empty_newcoefficientscoefficient. ((forall pfc_index_shift_empty_newcoefficientscoefficientdiagonal. (exists pfa_gap_shift_empty_newcoefficientscoefficientdiagonalbound. pfa_gap_shift_empty_newcoefficientscoefficientdiagonalbound + S (pfc_index_shift_empty_newcoefficientscoefficientdiagonal) = (S (pfc_index_shift_empty_newcoefficients))) -> exists pfc_value_shift_empty_newcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_shift_empty_newcoefficientscoefficientdiagonalentry. ff_h_pfp_shift_empty_newcoefficientscoefficientdiagonalentry + S (pfc_value_shift_empty_newcoefficientscoefficientdiagonal) = S ((S (pfc_index_shift_empty_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_empty_newcoefficientscoefficient)) /\\ exists ff_q_pfp_shift_empty_newcoefficientscoefficientdiagonalentry. pfc_terms_code_shift_empty_newcoefficientscoefficient = ff_q_pfp_shift_empty_newcoefficientscoefficientdiagonalentry * S ((S (pfc_index_shift_empty_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_empty_newcoefficientscoefficient) + (pfc_value_shift_empty_newcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_shift_empty_newcoefficientscoefficientdiagonalterm pfc_left_shift_empty_newcoefficientscoefficientdiagonalterm pfc_right_shift_empty_newcoefficientscoefficientdiagonalterm. (((pfc_index_shift_empty_newcoefficientscoefficientdiagonal)+pfc_complement_shift_empty_newcoefficientscoefficientdiagonalterm=(pfc_index_shift_empty_newcoefficients)) /\\ ((((((exists pfa_gap_shift_empty_newcoefficientscoefficientdiagonaltermleftinside. pfa_gap_shift_empty_newcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_shift_empty_newcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_empty_newcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_shift_empty_newcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_shift_empty_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_shift_empty_newcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_empty_newcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_shift_empty_newcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_shift_empty_newcoefficientscoefficientdiagonal)) * ac) + (pfc_left_shift_empty_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_empty_newcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_shift_empty_newcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_shift_empty_newcoefficientscoefficientdiagonal)) /\\ (((pfc_left_shift_empty_newcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_empty_newcoefficientscoefficientdiagonaltermrightinside. pfa_gap_shift_empty_newcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_shift_empty_newcoefficientscoefficientdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_shift_empty_newcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_shift_empty_newcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_shift_empty_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_shift_empty_newcoefficientscoefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_shift_empty_newcoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_shift_empty_newcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_shift_empty_newcoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_shift_empty_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_empty_newcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_shift_empty_newcoefficientscoefficientdiagonaltermrightoutside+(S M)=(pfc_complement_shift_empty_newcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_shift_empty_newcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_shift_empty_newcoefficientscoefficientdiagonal)=pfc_left_shift_empty_newcoefficientscoefficientdiagonalterm*pfc_right_shift_empty_newcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_empty_newcoefficientscoefficientsum fs_v_pfc_shift_empty_newcoefficientscoefficientsum. ((((exists fs_h_pfc_shift_empty_newcoefficientscoefficientsum_body_start. fs_h_pfc_shift_empty_newcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_empty_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_empty_newcoefficientscoefficientsum_body_start. fs_u_pfc_shift_empty_newcoefficientscoefficientsum = fs_q_pfc_shift_empty_newcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_shift_empty_newcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_shift_empty_newcoefficientscoefficientsum_body_terminal. fs_h_pfc_shift_empty_newcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_shift_empty_newcoefficientscoefficient) = S ((S (S (pfc_index_shift_empty_newcoefficients))) * fs_v_pfc_shift_empty_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_empty_newcoefficientscoefficientsum_body_terminal. fs_u_pfc_shift_empty_newcoefficientscoefficientsum = fs_q_pfc_shift_empty_newcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_shift_empty_newcoefficients))) * fs_v_pfc_shift_empty_newcoefficientscoefficientsum) + (pfc_natural_sum_shift_empty_newcoefficientscoefficient))) /\\ forall fs_i_pfc_shift_empty_newcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_shift_empty_newcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_shift_empty_newcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_shift_empty_newcoefficientscoefficientsum_body_steps = S (pfc_index_shift_empty_newcoefficients)) -> exists fs_a_pfc_shift_empty_newcoefficientscoefficientsum_body_steps fs_r_pfc_shift_empty_newcoefficientscoefficientsum_body_steps fs_s_pfc_shift_empty_newcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_shift_empty_newcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_shift_empty_newcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_shift_empty_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_empty_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_empty_newcoefficientscoefficient)) /\\ exists fs_q_pfc_shift_empty_newcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_shift_empty_newcoefficientscoefficient = fs_q_pfc_shift_empty_newcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_shift_empty_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_empty_newcoefficientscoefficient) + (fs_a_pfc_shift_empty_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_empty_newcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_shift_empty_newcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_shift_empty_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_empty_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_empty_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_empty_newcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_shift_empty_newcoefficientscoefficientsum = fs_q_pfc_shift_empty_newcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_shift_empty_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_empty_newcoefficientscoefficientsum) + (fs_r_pfc_shift_empty_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_empty_newcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_shift_empty_newcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_shift_empty_newcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_shift_empty_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_empty_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_empty_newcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_shift_empty_newcoefficientscoefficientsum = fs_q_pfc_shift_empty_newcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_shift_empty_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_empty_newcoefficientscoefficientsum) + (fs_s_pfc_shift_empty_newcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_shift_empty_newcoefficientscoefficientsum_body_steps = fs_r_pfc_shift_empty_newcoefficientscoefficientsum_body_steps + fs_a_pfc_shift_empty_newcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_empty_newcoefficientscoefficientresiduebound. pfa_gap_shift_empty_newcoefficientscoefficientresiduebound + S (pfc_value_shift_empty_newcoefficients) = (p)) /\\ ((exists pfa_offset_left_shift_empty_newcoefficientscoefficientresiduecongruence pfa_offset_right_shift_empty_newcoefficientscoefficientresiduecongruence. (pfc_natural_sum_shift_empty_newcoefficientscoefficient) + (p) * pfa_offset_left_shift_empty_newcoefficientscoefficientresiduecongruence = (pfc_value_shift_empty_newcoefficients) + (p) * pfa_offset_right_shift_empty_newcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (L=0 \\/ M=0) -> (((forall pfp_repeat_index_shift_empty_old_zero. (exists pfa_gap_shift_empty_old_zeroindex. pfa_gap_shift_empty_old_zeroindex + S (pfp_repeat_index_shift_empty_old_zero) = (N)) -> (((exists ff_h_pfp_shift_empty_old_zeroentry. ff_h_pfp_shift_empty_old_zeroentry + S (0) = S ((S (pfp_repeat_index_shift_empty_old_zero)) * cc)) /\\ exists ff_q_pfp_shift_empty_old_zeroentry. cb = ff_q_pfp_shift_empty_old_zeroentry * S ((S (pfp_repeat_index_shift_empty_old_zero)) * cc) + (0)))) /\\ ((forall pfp_repeat_index_shift_empty_new_zero. 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(exists pfa_gap_shift_equivalent_empty_left_old_zeroindex. pfa_gap_shift_equivalent_empty_left_old_zeroindex + S (pfp_repeat_index_shift_equivalent_empty_left_old_zero) = (N)) -> (((exists ff_h_pfp_shift_equivalent_empty_left_old_zeroentry. ff_h_pfp_shift_equivalent_empty_left_old_zeroentry + S (0) = S ((S (pfp_repeat_index_shift_equivalent_empty_left_old_zero)) * cc)) /\\ exists ff_q_pfp_shift_equivalent_empty_left_old_zeroentry. cb = ff_q_pfp_shift_equivalent_empty_left_old_zeroentry * S ((S (pfp_repeat_index_shift_equivalent_empty_left_old_zero)) * cc) + (0)))) /\\ ((forall pfp_repeat_index_shift_equivalent_empty_left_new_zero. (exists pfa_gap_shift_equivalent_empty_left_new_zeroindex. pfa_gap_shift_equivalent_empty_left_new_zeroindex + S (pfp_repeat_index_shift_equivalent_empty_left_new_zero) = (K)) -> (((exists ff_h_pfp_shift_equivalent_empty_left_new_zeroentry. ff_h_pfp_shift_equivalent_empty_left_new_zeroentry + S (0) = S ((S (pfp_repeat_index_shift_equivalent_empty_left_new_zero)) * dc)) /\\ exists ff_q_pfp_shift_equivalent_empty_left_new_zeroentry. db = ff_q_pfp_shift_equivalent_empty_left_new_zeroentry * S ((S (pfp_repeat_index_shift_equivalent_empty_left_new_zero)) * dc) + (0))))))", "specialize prime_field_polynomial_convolution_shift_right_empty (p)", "specialize prime_field_polynomial_convolution_shift_right_empty (ab)", "specialize prime_field_polynomial_convolution_shift_right_empty (ac)", "specialize prime_field_polynomial_convolution_shift_right_empty (L)", "specialize prime_field_polynomial_convolution_shift_right_empty (bb)", "specialize prime_field_polynomial_convolution_shift_right_empty (bc)", "specialize prime_field_polynomial_convolution_shift_right_empty (M)", "specialize prime_field_polynomial_convolution_shift_right_empty (cb)", "specialize prime_field_polynomial_convolution_shift_right_empty (cc)", "specialize prime_field_polynomial_convolution_shift_right_empty (N)", "specialize prime_field_polynomial_convolution_shift_right_empty (BB)", "specialize prime_field_polynomial_convolution_shift_right_empty (BC)", "specialize prime_field_polynomial_convolution_shift_right_empty (db)", "specialize prime_field_polynomial_convolution_shift_right_empty (dc)", "specialize prime_field_polynomial_convolution_shift_right_empty (K)", "apply prime_field_polynomial_convolution_shift_right_empty", "exact hp", "exact hs", "exact hc", "exact hd", "left", "exact hL_left", "cases hz", "have hezero : forall pfp_repeat_index_shift_equivalent_empty_left_shifted_zero. 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(exists pfa_gap_shift_equivalent_empty_right_old_zeroindex. pfa_gap_shift_equivalent_empty_right_old_zeroindex + S (pfp_repeat_index_shift_equivalent_empty_right_old_zero) = (N)) -> (((exists ff_h_pfp_shift_equivalent_empty_right_old_zeroentry. ff_h_pfp_shift_equivalent_empty_right_old_zeroentry + S (0) = S ((S (pfp_repeat_index_shift_equivalent_empty_right_old_zero)) * cc)) /\\ exists ff_q_pfp_shift_equivalent_empty_right_old_zeroentry. cb = ff_q_pfp_shift_equivalent_empty_right_old_zeroentry * S ((S (pfp_repeat_index_shift_equivalent_empty_right_old_zero)) * cc) + (0)))) /\\ ((forall pfp_repeat_index_shift_equivalent_empty_right_new_zero. (exists pfa_gap_shift_equivalent_empty_right_new_zeroindex. pfa_gap_shift_equivalent_empty_right_new_zeroindex + S (pfp_repeat_index_shift_equivalent_empty_right_new_zero) = (K)) -> (((exists ff_h_pfp_shift_equivalent_empty_right_new_zeroentry. ff_h_pfp_shift_equivalent_empty_right_new_zeroentry + S (0) = S ((S (pfp_repeat_index_shift_equivalent_empty_right_new_zero)) * dc)) /\\ exists ff_q_pfp_shift_equivalent_empty_right_new_zeroentry. db = ff_q_pfp_shift_equivalent_empty_right_new_zeroentry * S ((S (pfp_repeat_index_shift_equivalent_empty_right_new_zero)) * dc) + (0))))))", "specialize prime_field_polynomial_convolution_shift_right_empty (p)", "specialize prime_field_polynomial_convolution_shift_right_empty (ab)", "specialize prime_field_polynomial_convolution_shift_right_empty (ac)", "specialize prime_field_polynomial_convolution_shift_right_empty (L)", "specialize prime_field_polynomial_convolution_shift_right_empty (bb)", "specialize prime_field_polynomial_convolution_shift_right_empty (bc)", "specialize prime_field_polynomial_convolution_shift_right_empty (M)", "specialize prime_field_polynomial_convolution_shift_right_empty (cb)", "specialize prime_field_polynomial_convolution_shift_right_empty (cc)", "specialize prime_field_polynomial_convolution_shift_right_empty (N)", "specialize prime_field_polynomial_convolution_shift_right_empty (BB)", "specialize prime_field_polynomial_convolution_shift_right_empty (BC)", "specialize prime_field_polynomial_convolution_shift_right_empty (db)", "specialize prime_field_polynomial_convolution_shift_right_empty (dc)", "specialize prime_field_polynomial_convolution_shift_right_empty (K)", "apply prime_field_polynomial_convolution_shift_right_empty", "exact hp", "exact hs", "exact hc", "exact hd", "right", "exact hM_left", "cases hz", "have hezero : forall pfp_repeat_index_shift_equivalent_empty_right_shifted_zero. (exists pfa_gap_shift_equivalent_empty_right_shifted_zeroindex. pfa_gap_shift_equivalent_empty_right_shifted_zeroindex + S (pfp_repeat_index_shift_equivalent_empty_right_shifted_zero) = (S N)) -> (((exists ff_h_pfp_shift_equivalent_empty_right_shifted_zeroentry. ff_h_pfp_shift_equivalent_empty_right_shifted_zeroentry + S (0) = S ((S (pfp_repeat_index_shift_equivalent_empty_right_shifted_zero)) * ec)) /\\ exists ff_q_pfp_shift_equivalent_empty_right_shifted_zeroentry. eb = ff_q_pfp_shift_equivalent_empty_right_shifted_zeroentry * S ((S (pfp_repeat_index_shift_equivalent_empty_right_shifted_zero)) * ec) + (0)))", "specialize prime_field_polynomial_shift_zero_prefix (cb)", "specialize prime_field_polynomial_shift_zero_prefix (cc)", "specialize prime_field_polynomial_shift_zero_prefix (N)", "specialize prime_field_polynomial_shift_zero_prefix (eb)", "specialize prime_field_polynomial_shift_zero_prefix (ec)", "apply prime_field_polynomial_shift_zero_prefix", "exact hz_left", "exact he", "specialize prime_field_polynomial_equivalent_transitive (db)", "specialize prime_field_polynomial_equivalent_transitive (dc)", "specialize prime_field_polynomial_equivalent_transitive (K)", "specialize prime_field_polynomial_equivalent_transitive (0)", "specialize prime_field_polynomial_equivalent_transitive (0)", "specialize prime_field_polynomial_equivalent_transitive (0)", "specialize prime_field_polynomial_equivalent_transitive (eb)", "specialize prime_field_polynomial_equivalent_transitive (ec)", "specialize prime_field_polynomial_equivalent_transitive (S N)", "apply prime_field_polynomial_equivalent_transitive", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (db)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (dc)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (K)", "apply prime_field_polynomial_zero_prefix_equivalent_empty", "exact hz_right", "specialize prime_field_polynomial_equivalent_symmetric (eb)", "specialize prime_field_polynomial_equivalent_symmetric (ec)", "specialize prime_field_polynomial_equivalent_symmetric (S N)", "specialize prime_field_polynomial_equivalent_symmetric (0)", "specialize prime_field_polynomial_equivalent_symmetric (0)", "specialize prime_field_polynomial_equivalent_symmetric (0)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (eb)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (ec)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (S N)", "apply prime_field_polynomial_zero_prefix_equivalent_empty", "exact hezero", "have hdata : ((K=S N) /\\ ((((forall mdr_i_pfp_shift_equivalent_nonempty_dataprefix mdr_a_pfp_shift_equivalent_nonempty_dataprefix. (exists mdr_gap_pfp_shift_equivalent_nonempty_dataprefixb. mdr_gap_pfp_shift_equivalent_nonempty_dataprefixb + S (mdr_i_pfp_shift_equivalent_nonempty_dataprefix) = (N)) -> (((exists ff_h_mdr_pfp_shift_equivalent_nonempty_dataprefixo. ff_h_mdr_pfp_shift_equivalent_nonempty_dataprefixo + S (mdr_a_pfp_shift_equivalent_nonempty_dataprefix) = S ((S (mdr_i_pfp_shift_equivalent_nonempty_dataprefix)) * cc)) /\\ exists ff_q_mdr_pfp_shift_equivalent_nonempty_dataprefixo. cb = ff_q_mdr_pfp_shift_equivalent_nonempty_dataprefixo * S ((S (mdr_i_pfp_shift_equivalent_nonempty_dataprefix)) * cc) + (mdr_a_pfp_shift_equivalent_nonempty_dataprefix))) -> (((exists ff_h_mdr_pfp_shift_equivalent_nonempty_dataprefixn. ff_h_mdr_pfp_shift_equivalent_nonempty_dataprefixn + S (mdr_a_pfp_shift_equivalent_nonempty_dataprefix) = S ((S (mdr_i_pfp_shift_equivalent_nonempty_dataprefix)) * dc)) /\\ exists ff_q_mdr_pfp_shift_equivalent_nonempty_dataprefixn. db = ff_q_mdr_pfp_shift_equivalent_nonempty_dataprefixn * S ((S (mdr_i_pfp_shift_equivalent_nonempty_dataprefix)) * dc) + (mdr_a_pfp_shift_equivalent_nonempty_dataprefix)))) /\\ ((((exists ff_h_pfp_shift_equivalent_nonempty_datalast. ff_h_pfp_shift_equivalent_nonempty_datalast + S (0) = S ((S (N)) * dc)) /\\ exists ff_q_pfp_shift_equivalent_nonempty_datalast. db = ff_q_pfp_shift_equivalent_nonempty_datalast * S ((S (N)) * dc) + (0))))))))", "specialize prime_field_polynomial_convolution_shift_right_nonempty (p)", "specialize prime_field_polynomial_convolution_shift_right_nonempty (ab)", "specialize prime_field_polynomial_convolution_shift_right_nonempty (ac)", "specialize prime_field_polynomial_convolution_shift_right_nonempty (L)", "specialize prime_field_polynomial_convolution_shift_right_nonempty (bb)", "specialize prime_field_polynomial_convolution_shift_right_nonempty (bc)", "specialize prime_field_polynomial_convolution_shift_right_nonempty (M)", "specialize prime_field_polynomial_convolution_shift_right_nonempty (cb)", "specialize prime_field_polynomial_convolution_shift_right_nonempty (cc)", "specialize prime_field_polynomial_convolution_shift_right_nonempty (N)", "specialize prime_field_polynomial_convolution_shift_right_nonempty (BB)", "specialize prime_field_polynomial_convolution_shift_right_nonempty (BC)", "specialize prime_field_polynomial_convolution_shift_right_nonempty (db)", "specialize prime_field_polynomial_convolution_shift_right_nonempty (dc)", "specialize prime_field_polynomial_convolution_shift_right_nonempty (K)", "apply prime_field_polynomial_convolution_shift_right_nonempty", "exact hp", "exact hL_right", "exact hM_right", "exact hs", "exact hc", "exact hd", "cases hdata", "rewrite hdata_left", "rewrite hdata_left", "specialize prime_field_polynomial_equal_implies_equivalent (db)", "specialize prime_field_polynomial_equal_implies_equivalent (dc)", "specialize prime_field_polynomial_equal_implies_equivalent (eb)", "specialize prime_field_polynomial_equal_implies_equivalent (ec)", "specialize prime_field_polynomial_equal_implies_equivalent (S N)", "apply prime_field_polynomial_equal_implies_equivalent", "specialize prime_field_polynomial_shift_functional (cb)", "specialize prime_field_polynomial_shift_functional (cc)", "specialize prime_field_polynomial_shift_functional (N)", "specialize prime_field_polynomial_shift_functional (db)", "specialize prime_field_polynomial_shift_functional (dc)", "specialize prime_field_polynomial_shift_functional (eb)", "specialize prime_field_polynomial_shift_functional (ec)", "apply prime_field_polynomial_shift_functional", "exact hdata_right", "exact he" ], "script_sha256": "037cf599baf62c68a00a4ae7c769e5c516a147e44abf36355a5b011e19a33e0d", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_shift_candidate.py", "sha256": "325d3085482ee73a2c6ee90cd17e45cffe53273671edf89c40d88428335c9c4b" }, "statement": "forall p ab ac L bb bc M cb cc N BB BC db dc K eb ec. (~(p=0)) -> (((forall mdr_i_pfp_shift_equivalent_factorprefix mdr_a_pfp_shift_equivalent_factorprefix. (exists mdr_gap_pfp_shift_equivalent_factorprefixb. mdr_gap_pfp_shift_equivalent_factorprefixb + S (mdr_i_pfp_shift_equivalent_factorprefix) = (M)) -> (((exists ff_h_mdr_pfp_shift_equivalent_factorprefixo. ff_h_mdr_pfp_shift_equivalent_factorprefixo + S (mdr_a_pfp_shift_equivalent_factorprefix) = S ((S (mdr_i_pfp_shift_equivalent_factorprefix)) * bc)) /\\ exists ff_q_mdr_pfp_shift_equivalent_factorprefixo. bb = ff_q_mdr_pfp_shift_equivalent_factorprefixo * S ((S (mdr_i_pfp_shift_equivalent_factorprefix)) * bc) + (mdr_a_pfp_shift_equivalent_factorprefix))) -> (((exists ff_h_mdr_pfp_shift_equivalent_factorprefixn. ff_h_mdr_pfp_shift_equivalent_factorprefixn + S (mdr_a_pfp_shift_equivalent_factorprefix) = S ((S (mdr_i_pfp_shift_equivalent_factorprefix)) * BC)) /\\ exists ff_q_mdr_pfp_shift_equivalent_factorprefixn. BB = ff_q_mdr_pfp_shift_equivalent_factorprefixn * S ((S (mdr_i_pfp_shift_equivalent_factorprefix)) * BC) + (mdr_a_pfp_shift_equivalent_factorprefix)))) /\\ ((((exists ff_h_pfp_shift_equivalent_factorlast. ff_h_pfp_shift_equivalent_factorlast + S (0) = S ((S (M)) * BC)) /\\ exists ff_q_pfp_shift_equivalent_factorlast. BB = ff_q_pfp_shift_equivalent_factorlast * S ((S (M)) * BC) + (0)))))) -> (((forall fom_index_pfp_shift_equivalent_oldleft. (exists fom_gap_pfp_shift_equivalent_oldleft_index_bound. fom_gap_pfp_shift_equivalent_oldleft_index_bound + S (fom_index_pfp_shift_equivalent_oldleft) = L) -> exists fom_value_pfp_shift_equivalent_oldleft. ((((exists fom_beta_height_pfp_shift_equivalent_oldleft_entry. fom_beta_height_pfp_shift_equivalent_oldleft_entry + S (fom_value_pfp_shift_equivalent_oldleft) = S ((S (fom_index_pfp_shift_equivalent_oldleft)) * ac)) /\\ exists fom_beta_quotient_pfp_shift_equivalent_oldleft_entry. ab = fom_beta_quotient_pfp_shift_equivalent_oldleft_entry * S ((S (fom_index_pfp_shift_equivalent_oldleft)) * ac) + (fom_value_pfp_shift_equivalent_oldleft))) /\\ (exists fom_gap_pfp_shift_equivalent_oldleft_value_bound. fom_gap_pfp_shift_equivalent_oldleft_value_bound + S (fom_value_pfp_shift_equivalent_oldleft) = p))) /\\ (((forall fom_index_pfp_shift_equivalent_oldright. (exists fom_gap_pfp_shift_equivalent_oldright_index_bound. fom_gap_pfp_shift_equivalent_oldright_index_bound + S (fom_index_pfp_shift_equivalent_oldright) = M) -> exists fom_value_pfp_shift_equivalent_oldright. ((((exists fom_beta_height_pfp_shift_equivalent_oldright_entry. fom_beta_height_pfp_shift_equivalent_oldright_entry + S (fom_value_pfp_shift_equivalent_oldright) = S ((S (fom_index_pfp_shift_equivalent_oldright)) * bc)) /\\ exists fom_beta_quotient_pfp_shift_equivalent_oldright_entry. bb = fom_beta_quotient_pfp_shift_equivalent_oldright_entry * S ((S (fom_index_pfp_shift_equivalent_oldright)) * bc) + (fom_value_pfp_shift_equivalent_oldright))) /\\ (exists fom_gap_pfp_shift_equivalent_oldright_value_bound. fom_gap_pfp_shift_equivalent_oldright_value_bound + S (fom_value_pfp_shift_equivalent_oldright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_shift_equivalent_oldcoefficients. (exists pfa_gap_shift_equivalent_oldcoefficientsbound. pfa_gap_shift_equivalent_oldcoefficientsbound + S (pfc_index_shift_equivalent_oldcoefficients) = (N)) -> exists pfc_value_shift_equivalent_oldcoefficients. ((((exists ff_h_pfp_shift_equivalent_oldcoefficientsentry. ff_h_pfp_shift_equivalent_oldcoefficientsentry + S (pfc_value_shift_equivalent_oldcoefficients) = S ((S (pfc_index_shift_equivalent_oldcoefficients)) * cc)) /\\ exists ff_q_pfp_shift_equivalent_oldcoefficientsentry. cb = ff_q_pfp_shift_equivalent_oldcoefficientsentry * S ((S (pfc_index_shift_equivalent_oldcoefficients)) * cc) + (pfc_value_shift_equivalent_oldcoefficients))) /\\ ((exists pfc_terms_code_shift_equivalent_oldcoefficientscoefficient pfc_terms_scale_shift_equivalent_oldcoefficientscoefficient pfc_natural_sum_shift_equivalent_oldcoefficientscoefficient. ((forall pfc_index_shift_equivalent_oldcoefficientscoefficientdiagonal. (exists pfa_gap_shift_equivalent_oldcoefficientscoefficientdiagonalbound. pfa_gap_shift_equivalent_oldcoefficientscoefficientdiagonalbound + S (pfc_index_shift_equivalent_oldcoefficientscoefficientdiagonal) = (S (pfc_index_shift_equivalent_oldcoefficients))) -> exists pfc_value_shift_equivalent_oldcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_shift_equivalent_oldcoefficientscoefficientdiagonalentry. ff_h_pfp_shift_equivalent_oldcoefficientscoefficientdiagonalentry + S (pfc_value_shift_equivalent_oldcoefficientscoefficientdiagonal) = S ((S (pfc_index_shift_equivalent_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_equivalent_oldcoefficientscoefficient)) /\\ exists ff_q_pfp_shift_equivalent_oldcoefficientscoefficientdiagonalentry. pfc_terms_code_shift_equivalent_oldcoefficientscoefficient = ff_q_pfp_shift_equivalent_oldcoefficientscoefficientdiagonalentry * S ((S (pfc_index_shift_equivalent_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_equivalent_oldcoefficientscoefficient) + (pfc_value_shift_equivalent_oldcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_shift_equivalent_oldcoefficientscoefficientdiagonalterm pfc_left_shift_equivalent_oldcoefficientscoefficientdiagonalterm pfc_right_shift_equivalent_oldcoefficientscoefficientdiagonalterm. (((pfc_index_shift_equivalent_oldcoefficientscoefficientdiagonal)+pfc_complement_shift_equivalent_oldcoefficientscoefficientdiagonalterm=(pfc_index_shift_equivalent_oldcoefficients)) /\\ ((((((exists pfa_gap_shift_equivalent_oldcoefficientscoefficientdiagonaltermleftinside. pfa_gap_shift_equivalent_oldcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_shift_equivalent_oldcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_equivalent_oldcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_shift_equivalent_oldcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_shift_equivalent_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_shift_equivalent_oldcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_equivalent_oldcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_shift_equivalent_oldcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_shift_equivalent_oldcoefficientscoefficientdiagonal)) * ac) + (pfc_left_shift_equivalent_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_equivalent_oldcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_shift_equivalent_oldcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_shift_equivalent_oldcoefficientscoefficientdiagonal)) /\\ (((pfc_left_shift_equivalent_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_equivalent_oldcoefficientscoefficientdiagonaltermrightinside. pfa_gap_shift_equivalent_oldcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_shift_equivalent_oldcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_shift_equivalent_oldcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_shift_equivalent_oldcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_shift_equivalent_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_shift_equivalent_oldcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_shift_equivalent_oldcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_shift_equivalent_oldcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_shift_equivalent_oldcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_shift_equivalent_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_equivalent_oldcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_shift_equivalent_oldcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_shift_equivalent_oldcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_shift_equivalent_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_shift_equivalent_oldcoefficientscoefficientdiagonal)=pfc_left_shift_equivalent_oldcoefficientscoefficientdiagonalterm*pfc_right_shift_equivalent_oldcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_equivalent_oldcoefficientscoefficientsum fs_v_pfc_shift_equivalent_oldcoefficientscoefficientsum. ((((exists fs_h_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_start. fs_h_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_equivalent_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_start. fs_u_pfc_shift_equivalent_oldcoefficientscoefficientsum = fs_q_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_shift_equivalent_oldcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_terminal. fs_h_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_shift_equivalent_oldcoefficientscoefficient) = S ((S (S (pfc_index_shift_equivalent_oldcoefficients))) * fs_v_pfc_shift_equivalent_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_terminal. fs_u_pfc_shift_equivalent_oldcoefficientscoefficientsum = fs_q_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_shift_equivalent_oldcoefficients))) * fs_v_pfc_shift_equivalent_oldcoefficientscoefficientsum) + (pfc_natural_sum_shift_equivalent_oldcoefficientscoefficient))) /\\ forall fs_i_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps = S (pfc_index_shift_equivalent_oldcoefficients)) -> exists fs_a_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps fs_r_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps fs_s_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_equivalent_oldcoefficientscoefficient)) /\\ exists fs_q_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_shift_equivalent_oldcoefficientscoefficient = fs_q_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_equivalent_oldcoefficientscoefficient) + (fs_a_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_equivalent_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_shift_equivalent_oldcoefficientscoefficientsum = fs_q_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_equivalent_oldcoefficientscoefficientsum) + (fs_r_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_equivalent_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_shift_equivalent_oldcoefficientscoefficientsum = fs_q_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_equivalent_oldcoefficientscoefficientsum) + (fs_s_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps = fs_r_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps + fs_a_pfc_shift_equivalent_oldcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_equivalent_oldcoefficientscoefficientresiduebound. pfa_gap_shift_equivalent_oldcoefficientscoefficientresiduebound + S (pfc_value_shift_equivalent_oldcoefficients) = (p)) /\\ ((exists pfa_offset_left_shift_equivalent_oldcoefficientscoefficientresiduecongruence pfa_offset_right_shift_equivalent_oldcoefficientscoefficientresiduecongruence. (pfc_natural_sum_shift_equivalent_oldcoefficientscoefficient) + (p) * pfa_offset_left_shift_equivalent_oldcoefficientscoefficientresiduecongruence = (pfc_value_shift_equivalent_oldcoefficients) + (p) * pfa_offset_right_shift_equivalent_oldcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_shift_equivalent_newleft. (exists fom_gap_pfp_shift_equivalent_newleft_index_bound. fom_gap_pfp_shift_equivalent_newleft_index_bound + S (fom_index_pfp_shift_equivalent_newleft) = L) -> exists fom_value_pfp_shift_equivalent_newleft. ((((exists fom_beta_height_pfp_shift_equivalent_newleft_entry. fom_beta_height_pfp_shift_equivalent_newleft_entry + S (fom_value_pfp_shift_equivalent_newleft) = S ((S (fom_index_pfp_shift_equivalent_newleft)) * ac)) /\\ exists fom_beta_quotient_pfp_shift_equivalent_newleft_entry. ab = fom_beta_quotient_pfp_shift_equivalent_newleft_entry * S ((S (fom_index_pfp_shift_equivalent_newleft)) * ac) + (fom_value_pfp_shift_equivalent_newleft))) /\\ (exists fom_gap_pfp_shift_equivalent_newleft_value_bound. fom_gap_pfp_shift_equivalent_newleft_value_bound + S (fom_value_pfp_shift_equivalent_newleft) = p))) /\\ (((forall fom_index_pfp_shift_equivalent_newright. (exists fom_gap_pfp_shift_equivalent_newright_index_bound. fom_gap_pfp_shift_equivalent_newright_index_bound + S (fom_index_pfp_shift_equivalent_newright) = S M) -> exists fom_value_pfp_shift_equivalent_newright. ((((exists fom_beta_height_pfp_shift_equivalent_newright_entry. fom_beta_height_pfp_shift_equivalent_newright_entry + S (fom_value_pfp_shift_equivalent_newright) = S ((S (fom_index_pfp_shift_equivalent_newright)) * BC)) /\\ exists fom_beta_quotient_pfp_shift_equivalent_newright_entry. BB = fom_beta_quotient_pfp_shift_equivalent_newright_entry * S ((S (fom_index_pfp_shift_equivalent_newright)) * BC) + (fom_value_pfp_shift_equivalent_newright))) /\\ (exists fom_gap_pfp_shift_equivalent_newright_value_bound. fom_gap_pfp_shift_equivalent_newright_value_bound + S (fom_value_pfp_shift_equivalent_newright) = p))) /\\ (((((((L)=0 \\/ (S M)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((S M)=0)) /\\ (((L)+(S M)=S (K)))))))) /\\ ((forall pfc_index_shift_equivalent_newcoefficients. (exists pfa_gap_shift_equivalent_newcoefficientsbound. pfa_gap_shift_equivalent_newcoefficientsbound + S (pfc_index_shift_equivalent_newcoefficients) = (K)) -> exists pfc_value_shift_equivalent_newcoefficients. ((((exists ff_h_pfp_shift_equivalent_newcoefficientsentry. ff_h_pfp_shift_equivalent_newcoefficientsentry + S (pfc_value_shift_equivalent_newcoefficients) = S ((S (pfc_index_shift_equivalent_newcoefficients)) * dc)) /\\ exists ff_q_pfp_shift_equivalent_newcoefficientsentry. db = ff_q_pfp_shift_equivalent_newcoefficientsentry * S ((S (pfc_index_shift_equivalent_newcoefficients)) * dc) + (pfc_value_shift_equivalent_newcoefficients))) /\\ ((exists pfc_terms_code_shift_equivalent_newcoefficientscoefficient pfc_terms_scale_shift_equivalent_newcoefficientscoefficient pfc_natural_sum_shift_equivalent_newcoefficientscoefficient. ((forall pfc_index_shift_equivalent_newcoefficientscoefficientdiagonal. (exists pfa_gap_shift_equivalent_newcoefficientscoefficientdiagonalbound. pfa_gap_shift_equivalent_newcoefficientscoefficientdiagonalbound + S (pfc_index_shift_equivalent_newcoefficientscoefficientdiagonal) = (S (pfc_index_shift_equivalent_newcoefficients))) -> exists pfc_value_shift_equivalent_newcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_shift_equivalent_newcoefficientscoefficientdiagonalentry. ff_h_pfp_shift_equivalent_newcoefficientscoefficientdiagonalentry + S (pfc_value_shift_equivalent_newcoefficientscoefficientdiagonal) = S ((S (pfc_index_shift_equivalent_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_equivalent_newcoefficientscoefficient)) /\\ exists ff_q_pfp_shift_equivalent_newcoefficientscoefficientdiagonalentry. pfc_terms_code_shift_equivalent_newcoefficientscoefficient = ff_q_pfp_shift_equivalent_newcoefficientscoefficientdiagonalentry * S ((S (pfc_index_shift_equivalent_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_equivalent_newcoefficientscoefficient) + (pfc_value_shift_equivalent_newcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_shift_equivalent_newcoefficientscoefficientdiagonalterm pfc_left_shift_equivalent_newcoefficientscoefficientdiagonalterm pfc_right_shift_equivalent_newcoefficientscoefficientdiagonalterm. (((pfc_index_shift_equivalent_newcoefficientscoefficientdiagonal)+pfc_complement_shift_equivalent_newcoefficientscoefficientdiagonalterm=(pfc_index_shift_equivalent_newcoefficients)) /\\ ((((((exists pfa_gap_shift_equivalent_newcoefficientscoefficientdiagonaltermleftinside. pfa_gap_shift_equivalent_newcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_shift_equivalent_newcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_equivalent_newcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_shift_equivalent_newcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_shift_equivalent_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_shift_equivalent_newcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_equivalent_newcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_shift_equivalent_newcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_shift_equivalent_newcoefficientscoefficientdiagonal)) * ac) + (pfc_left_shift_equivalent_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_equivalent_newcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_shift_equivalent_newcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_shift_equivalent_newcoefficientscoefficientdiagonal)) /\\ (((pfc_left_shift_equivalent_newcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_equivalent_newcoefficientscoefficientdiagonaltermrightinside. pfa_gap_shift_equivalent_newcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_shift_equivalent_newcoefficientscoefficientdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_shift_equivalent_newcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_shift_equivalent_newcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_shift_equivalent_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_shift_equivalent_newcoefficientscoefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_shift_equivalent_newcoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_shift_equivalent_newcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_shift_equivalent_newcoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_shift_equivalent_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_equivalent_newcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_shift_equivalent_newcoefficientscoefficientdiagonaltermrightoutside+(S M)=(pfc_complement_shift_equivalent_newcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_shift_equivalent_newcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_shift_equivalent_newcoefficientscoefficientdiagonal)=pfc_left_shift_equivalent_newcoefficientscoefficientdiagonalterm*pfc_right_shift_equivalent_newcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_equivalent_newcoefficientscoefficientsum fs_v_pfc_shift_equivalent_newcoefficientscoefficientsum. ((((exists fs_h_pfc_shift_equivalent_newcoefficientscoefficientsum_body_start. fs_h_pfc_shift_equivalent_newcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_equivalent_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_equivalent_newcoefficientscoefficientsum_body_start. fs_u_pfc_shift_equivalent_newcoefficientscoefficientsum = fs_q_pfc_shift_equivalent_newcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_shift_equivalent_newcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_shift_equivalent_newcoefficientscoefficientsum_body_terminal. fs_h_pfc_shift_equivalent_newcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_shift_equivalent_newcoefficientscoefficient) = S ((S (S (pfc_index_shift_equivalent_newcoefficients))) * fs_v_pfc_shift_equivalent_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_equivalent_newcoefficientscoefficientsum_body_terminal. fs_u_pfc_shift_equivalent_newcoefficientscoefficientsum = fs_q_pfc_shift_equivalent_newcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_shift_equivalent_newcoefficients))) * fs_v_pfc_shift_equivalent_newcoefficientscoefficientsum) + (pfc_natural_sum_shift_equivalent_newcoefficientscoefficient))) /\\ forall fs_i_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps = S (pfc_index_shift_equivalent_newcoefficients)) -> exists fs_a_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps fs_r_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps fs_s_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_equivalent_newcoefficientscoefficient)) /\\ exists fs_q_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_shift_equivalent_newcoefficientscoefficient = fs_q_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_equivalent_newcoefficientscoefficient) + (fs_a_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_equivalent_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_shift_equivalent_newcoefficientscoefficientsum = fs_q_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_equivalent_newcoefficientscoefficientsum) + (fs_r_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_equivalent_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_shift_equivalent_newcoefficientscoefficientsum = fs_q_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_equivalent_newcoefficientscoefficientsum) + (fs_s_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps = fs_r_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps + fs_a_pfc_shift_equivalent_newcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_equivalent_newcoefficientscoefficientresiduebound. pfa_gap_shift_equivalent_newcoefficientscoefficientresiduebound + S (pfc_value_shift_equivalent_newcoefficients) = (p)) /\\ ((exists pfa_offset_left_shift_equivalent_newcoefficientscoefficientresiduecongruence pfa_offset_right_shift_equivalent_newcoefficientscoefficientresiduecongruence. (pfc_natural_sum_shift_equivalent_newcoefficientscoefficient) + (p) * pfa_offset_left_shift_equivalent_newcoefficientscoefficientresiduecongruence = (pfc_value_shift_equivalent_newcoefficients) + (p) * pfa_offset_right_shift_equivalent_newcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall mdr_i_pfp_shift_equivalent_comparisonprefix mdr_a_pfp_shift_equivalent_comparisonprefix. (exists mdr_gap_pfp_shift_equivalent_comparisonprefixb. mdr_gap_pfp_shift_equivalent_comparisonprefixb + S (mdr_i_pfp_shift_equivalent_comparisonprefix) = (N)) -> (((exists ff_h_mdr_pfp_shift_equivalent_comparisonprefixo. ff_h_mdr_pfp_shift_equivalent_comparisonprefixo + S (mdr_a_pfp_shift_equivalent_comparisonprefix) = S ((S (mdr_i_pfp_shift_equivalent_comparisonprefix)) * cc)) /\\ exists ff_q_mdr_pfp_shift_equivalent_comparisonprefixo. cb = ff_q_mdr_pfp_shift_equivalent_comparisonprefixo * S ((S (mdr_i_pfp_shift_equivalent_comparisonprefix)) * cc) + (mdr_a_pfp_shift_equivalent_comparisonprefix))) -> (((exists ff_h_mdr_pfp_shift_equivalent_comparisonprefixn. ff_h_mdr_pfp_shift_equivalent_comparisonprefixn + S (mdr_a_pfp_shift_equivalent_comparisonprefix) = S ((S (mdr_i_pfp_shift_equivalent_comparisonprefix)) * ec)) /\\ exists ff_q_mdr_pfp_shift_equivalent_comparisonprefixn. eb = ff_q_mdr_pfp_shift_equivalent_comparisonprefixn * S ((S (mdr_i_pfp_shift_equivalent_comparisonprefix)) * ec) + (mdr_a_pfp_shift_equivalent_comparisonprefix)))) /\\ ((((exists ff_h_pfp_shift_equivalent_comparisonlast. ff_h_pfp_shift_equivalent_comparisonlast + S (0) = S ((S (N)) * ec)) /\\ exists ff_q_pfp_shift_equivalent_comparisonlast. eb = ff_q_pfp_shift_equivalent_comparisonlast * S ((S (N)) * ec) + (0)))))) -> (forall pfrep_power_shift_equivalent_result pfrep_left_shift_equivalent_result pfrep_right_shift_equivalent_result. ((exists pfrep_position_shift_equivalent_resultfirst. ((pfrep_position_shift_equivalent_resultfirst+S (pfrep_power_shift_equivalent_result)=(K)) /\\ ((((exists ff_h_pfp_shift_equivalent_resultfirstentry. ff_h_pfp_shift_equivalent_resultfirstentry + S (pfrep_left_shift_equivalent_result) = S ((S (pfrep_position_shift_equivalent_resultfirst)) * dc)) /\\ exists ff_q_pfp_shift_equivalent_resultfirstentry. db = ff_q_pfp_shift_equivalent_resultfirstentry * S ((S (pfrep_position_shift_equivalent_resultfirst)) * dc) + (pfrep_left_shift_equivalent_result)))))) \\/ (((exists pfrep_gap_shift_equivalent_resultfirstoutside. pfrep_gap_shift_equivalent_resultfirstoutside+(K)=(pfrep_power_shift_equivalent_result)) /\\ (((pfrep_left_shift_equivalent_result)=0))))) -> ((exists pfrep_position_shift_equivalent_resultsecond. ((pfrep_position_shift_equivalent_resultsecond+S (pfrep_power_shift_equivalent_result)=(S N)) /\\ ((((exists ff_h_pfp_shift_equivalent_resultsecondentry. ff_h_pfp_shift_equivalent_resultsecondentry + S (pfrep_right_shift_equivalent_result) = S ((S (pfrep_position_shift_equivalent_resultsecond)) * ec)) /\\ exists ff_q_pfp_shift_equivalent_resultsecondentry. eb = ff_q_pfp_shift_equivalent_resultsecondentry * S ((S (pfrep_position_shift_equivalent_resultsecond)) * ec) + (pfrep_right_shift_equivalent_result)))))) \\/ (((exists pfrep_gap_shift_equivalent_resultsecondoutside. pfrep_gap_shift_equivalent_resultsecondoutside+(S N)=(pfrep_power_shift_equivalent_result)) /\\ (((pfrep_right_shift_equivalent_result)=0))))) -> pfrep_left_shift_equivalent_result=pfrep_right_shift_equivalent_result)", "statement_sha256": "56aeba7667a7fc9ee6253ce009cc56e950d24249a3e1fbd4efb25f3bef7558b0", "summary": "At every nonzero modulus, the actual product with a shifted right factor is formally 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(exists fom_gap_pfp_shift_exists_product_oldleft_index_bound. fom_gap_pfp_shift_exists_product_oldleft_index_bound + S (fom_index_pfp_shift_exists_product_oldleft) = L) -> exists fom_value_pfp_shift_exists_product_oldleft. ((((exists fom_beta_height_pfp_shift_exists_product_oldleft_entry. fom_beta_height_pfp_shift_exists_product_oldleft_entry + S (fom_value_pfp_shift_exists_product_oldleft) = S ((S (fom_index_pfp_shift_exists_product_oldleft)) * ac)) /\\ exists fom_beta_quotient_pfp_shift_exists_product_oldleft_entry. ab = fom_beta_quotient_pfp_shift_exists_product_oldleft_entry * S ((S (fom_index_pfp_shift_exists_product_oldleft)) * ac) + (fom_value_pfp_shift_exists_product_oldleft))) /\\ (exists fom_gap_pfp_shift_exists_product_oldleft_value_bound. fom_gap_pfp_shift_exists_product_oldleft_value_bound + S (fom_value_pfp_shift_exists_product_oldleft) = p))) /\\ (((forall fom_index_pfp_shift_exists_product_oldright. (exists fom_gap_pfp_shift_exists_product_oldright_index_bound. fom_gap_pfp_shift_exists_product_oldright_index_bound + S (fom_index_pfp_shift_exists_product_oldright) = M) -> exists fom_value_pfp_shift_exists_product_oldright. ((((exists fom_beta_height_pfp_shift_exists_product_oldright_entry. fom_beta_height_pfp_shift_exists_product_oldright_entry + S (fom_value_pfp_shift_exists_product_oldright) = S ((S (fom_index_pfp_shift_exists_product_oldright)) * bc)) /\\ exists fom_beta_quotient_pfp_shift_exists_product_oldright_entry. bb = fom_beta_quotient_pfp_shift_exists_product_oldright_entry * S ((S (fom_index_pfp_shift_exists_product_oldright)) * bc) + (fom_value_pfp_shift_exists_product_oldright))) /\\ (exists fom_gap_pfp_shift_exists_product_oldright_value_bound. fom_gap_pfp_shift_exists_product_oldright_value_bound + S (fom_value_pfp_shift_exists_product_oldright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_shift_exists_product_oldcoefficients. (exists pfa_gap_shift_exists_product_oldcoefficientsbound. pfa_gap_shift_exists_product_oldcoefficientsbound + S (pfc_index_shift_exists_product_oldcoefficients) = (N)) -> exists pfc_value_shift_exists_product_oldcoefficients. ((((exists ff_h_pfp_shift_exists_product_oldcoefficientsentry. ff_h_pfp_shift_exists_product_oldcoefficientsentry + S (pfc_value_shift_exists_product_oldcoefficients) = S ((S (pfc_index_shift_exists_product_oldcoefficients)) * cc)) /\\ exists ff_q_pfp_shift_exists_product_oldcoefficientsentry. cb = ff_q_pfp_shift_exists_product_oldcoefficientsentry * S ((S (pfc_index_shift_exists_product_oldcoefficients)) * cc) + (pfc_value_shift_exists_product_oldcoefficients))) /\\ ((exists pfc_terms_code_shift_exists_product_oldcoefficientscoefficient pfc_terms_scale_shift_exists_product_oldcoefficientscoefficient pfc_natural_sum_shift_exists_product_oldcoefficientscoefficient. ((forall pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal. (exists pfa_gap_shift_exists_product_oldcoefficientscoefficientdiagonalbound. pfa_gap_shift_exists_product_oldcoefficientscoefficientdiagonalbound + S (pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal) = (S (pfc_index_shift_exists_product_oldcoefficients))) -> exists pfc_value_shift_exists_product_oldcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_shift_exists_product_oldcoefficientscoefficientdiagonalentry. ff_h_pfp_shift_exists_product_oldcoefficientscoefficientdiagonalentry + S (pfc_value_shift_exists_product_oldcoefficientscoefficientdiagonal) = S ((S (pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_exists_product_oldcoefficientscoefficient)) /\\ exists ff_q_pfp_shift_exists_product_oldcoefficientscoefficientdiagonalentry. pfc_terms_code_shift_exists_product_oldcoefficientscoefficient = ff_q_pfp_shift_exists_product_oldcoefficientscoefficientdiagonalentry * S ((S (pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_exists_product_oldcoefficientscoefficient) + (pfc_value_shift_exists_product_oldcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_shift_exists_product_oldcoefficientscoefficientdiagonalterm pfc_left_shift_exists_product_oldcoefficientscoefficientdiagonalterm pfc_right_shift_exists_product_oldcoefficientscoefficientdiagonalterm. (((pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal)+pfc_complement_shift_exists_product_oldcoefficientscoefficientdiagonalterm=(pfc_index_shift_exists_product_oldcoefficients)) /\\ ((((((exists pfa_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftinside. pfa_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_shift_exists_product_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal)) * ac) + (pfc_left_shift_exists_product_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal)) /\\ (((pfc_left_shift_exists_product_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightinside. pfa_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_shift_exists_product_oldcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_shift_exists_product_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_shift_exists_product_oldcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_shift_exists_product_oldcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_shift_exists_product_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_shift_exists_product_oldcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_shift_exists_product_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_shift_exists_product_oldcoefficientscoefficientdiagonal)=pfc_left_shift_exists_product_oldcoefficientscoefficientdiagonalterm*pfc_right_shift_exists_product_oldcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_exists_product_oldcoefficientscoefficientsum fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum. ((((exists fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_start. fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_start. fs_u_pfc_shift_exists_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_terminal. fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_shift_exists_product_oldcoefficientscoefficient) = S ((S (S (pfc_index_shift_exists_product_oldcoefficients))) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_terminal. fs_u_pfc_shift_exists_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_shift_exists_product_oldcoefficients))) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum) + (pfc_natural_sum_shift_exists_product_oldcoefficientscoefficient))) /\\ forall fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps = S (pfc_index_shift_exists_product_oldcoefficients)) -> exists fs_a_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps fs_r_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps fs_s_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_exists_product_oldcoefficientscoefficient)) /\\ exists fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_shift_exists_product_oldcoefficientscoefficient = fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_exists_product_oldcoefficientscoefficient) + (fs_a_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_shift_exists_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum) + (fs_r_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_shift_exists_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum) + (fs_s_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps = fs_r_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps + fs_a_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_exists_product_oldcoefficientscoefficientresiduebound. pfa_gap_shift_exists_product_oldcoefficientscoefficientresiduebound + S (pfc_value_shift_exists_product_oldcoefficients) = (p)) /\\ ((exists pfa_offset_left_shift_exists_product_oldcoefficientscoefficientresiduecongruence pfa_offset_right_shift_exists_product_oldcoefficientscoefficientresiduecongruence. (pfc_natural_sum_shift_exists_product_oldcoefficientscoefficient) + (p) * pfa_offset_left_shift_exists_product_oldcoefficientscoefficientresiduecongruence = (pfc_value_shift_exists_product_oldcoefficients) + (p) * pfa_offset_right_shift_exists_product_oldcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hc", "cases hcopy", "cases hcopy_right", "cases hcopy_right_right", "have hp : ~(p=0)", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hprime", "exact hz", "have hlength : exists K. (((((L)=0 \\/ (S M)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((S M)=0)) /\\ (((L)+(S M)=S (K))))))))", "specialize polynomial_product_length_exists (L)", "specialize polynomial_product_length_exists (S M)", "apply polynomial_product_length_exists", "cases hlength", "have hv : exists d e. ((forall fom_index_pfp_shift_exists_product_chosenleft. (exists fom_gap_pfp_shift_exists_product_chosenleft_index_bound. fom_gap_pfp_shift_exists_product_chosenleft_index_bound + S (fom_index_pfp_shift_exists_product_chosenleft) = L) -> exists fom_value_pfp_shift_exists_product_chosenleft. ((((exists fom_beta_height_pfp_shift_exists_product_chosenleft_entry. fom_beta_height_pfp_shift_exists_product_chosenleft_entry + S (fom_value_pfp_shift_exists_product_chosenleft) = S ((S (fom_index_pfp_shift_exists_product_chosenleft)) * ac)) /\\ exists fom_beta_quotient_pfp_shift_exists_product_chosenleft_entry. ab = fom_beta_quotient_pfp_shift_exists_product_chosenleft_entry * S ((S (fom_index_pfp_shift_exists_product_chosenleft)) * ac) + (fom_value_pfp_shift_exists_product_chosenleft))) /\\ (exists fom_gap_pfp_shift_exists_product_chosenleft_value_bound. fom_gap_pfp_shift_exists_product_chosenleft_value_bound + S (fom_value_pfp_shift_exists_product_chosenleft) = p))) /\\ (((forall fom_index_pfp_shift_exists_product_chosenright. (exists fom_gap_pfp_shift_exists_product_chosenright_index_bound. fom_gap_pfp_shift_exists_product_chosenright_index_bound + S (fom_index_pfp_shift_exists_product_chosenright) = S M) -> exists fom_value_pfp_shift_exists_product_chosenright. ((((exists fom_beta_height_pfp_shift_exists_product_chosenright_entry. fom_beta_height_pfp_shift_exists_product_chosenright_entry + S (fom_value_pfp_shift_exists_product_chosenright) = S ((S (fom_index_pfp_shift_exists_product_chosenright)) * BC)) /\\ exists fom_beta_quotient_pfp_shift_exists_product_chosenright_entry. BB = fom_beta_quotient_pfp_shift_exists_product_chosenright_entry * S ((S (fom_index_pfp_shift_exists_product_chosenright)) * BC) + (fom_value_pfp_shift_exists_product_chosenright))) /\\ (exists fom_gap_pfp_shift_exists_product_chosenright_value_bound. fom_gap_pfp_shift_exists_product_chosenright_value_bound + S (fom_value_pfp_shift_exists_product_chosenright) = p))) /\\ (((((((L)=0 \\/ (S M)=0) /\\ (((x)=0)))) \\/ (((~((L)=0)) /\\ (((~((S M)=0)) /\\ (((L)+(S M)=S (x)))))))) /\\ ((forall pfc_index_shift_exists_product_chosencoefficients. (exists pfa_gap_shift_exists_product_chosencoefficientsbound. pfa_gap_shift_exists_product_chosencoefficientsbound + S (pfc_index_shift_exists_product_chosencoefficients) = (x)) -> exists pfc_value_shift_exists_product_chosencoefficients. ((((exists ff_h_pfp_shift_exists_product_chosencoefficientsentry. ff_h_pfp_shift_exists_product_chosencoefficientsentry + S (pfc_value_shift_exists_product_chosencoefficients) = S ((S (pfc_index_shift_exists_product_chosencoefficients)) * e)) /\\ exists ff_q_pfp_shift_exists_product_chosencoefficientsentry. d = ff_q_pfp_shift_exists_product_chosencoefficientsentry * S ((S (pfc_index_shift_exists_product_chosencoefficients)) * e) + (pfc_value_shift_exists_product_chosencoefficients))) /\\ ((exists pfc_terms_code_shift_exists_product_chosencoefficientscoefficient pfc_terms_scale_shift_exists_product_chosencoefficientscoefficient pfc_natural_sum_shift_exists_product_chosencoefficientscoefficient. ((forall pfc_index_shift_exists_product_chosencoefficientscoefficientdiagonal. (exists pfa_gap_shift_exists_product_chosencoefficientscoefficientdiagonalbound. pfa_gap_shift_exists_product_chosencoefficientscoefficientdiagonalbound + S (pfc_index_shift_exists_product_chosencoefficientscoefficientdiagonal) = (S (pfc_index_shift_exists_product_chosencoefficients))) -> exists pfc_value_shift_exists_product_chosencoefficientscoefficientdiagonal. ((((exists ff_h_pfp_shift_exists_product_chosencoefficientscoefficientdiagonalentry. ff_h_pfp_shift_exists_product_chosencoefficientscoefficientdiagonalentry + S (pfc_value_shift_exists_product_chosencoefficientscoefficientdiagonal) = S ((S (pfc_index_shift_exists_product_chosencoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_exists_product_chosencoefficientscoefficient)) /\\ exists ff_q_pfp_shift_exists_product_chosencoefficientscoefficientdiagonalentry. pfc_terms_code_shift_exists_product_chosencoefficientscoefficient = ff_q_pfp_shift_exists_product_chosencoefficientscoefficientdiagonalentry * S ((S (pfc_index_shift_exists_product_chosencoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_exists_product_chosencoefficientscoefficient) + (pfc_value_shift_exists_product_chosencoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_shift_exists_product_chosencoefficientscoefficientdiagonalterm pfc_left_shift_exists_product_chosencoefficientscoefficientdiagonalterm pfc_right_shift_exists_product_chosencoefficientscoefficientdiagonalterm. (((pfc_index_shift_exists_product_chosencoefficientscoefficientdiagonal)+pfc_complement_shift_exists_product_chosencoefficientscoefficientdiagonalterm=(pfc_index_shift_exists_product_chosencoefficients)) /\\ ((((((exists pfa_gap_shift_exists_product_chosencoefficientscoefficientdiagonaltermleftinside. pfa_gap_shift_exists_product_chosencoefficientscoefficientdiagonaltermleftinside + S (pfc_index_shift_exists_product_chosencoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_exists_product_chosencoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_shift_exists_product_chosencoefficientscoefficientdiagonaltermleftentry + S (pfc_left_shift_exists_product_chosencoefficientscoefficientdiagonalterm) = S ((S (pfc_index_shift_exists_product_chosencoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_exists_product_chosencoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_shift_exists_product_chosencoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_shift_exists_product_chosencoefficientscoefficientdiagonal)) * ac) + (pfc_left_shift_exists_product_chosencoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_exists_product_chosencoefficientscoefficientdiagonaltermleftoutside. pfc_gap_shift_exists_product_chosencoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_shift_exists_product_chosencoefficientscoefficientdiagonal)) /\\ (((pfc_left_shift_exists_product_chosencoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_exists_product_chosencoefficientscoefficientdiagonaltermrightinside. pfa_gap_shift_exists_product_chosencoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_shift_exists_product_chosencoefficientscoefficientdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_shift_exists_product_chosencoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_shift_exists_product_chosencoefficientscoefficientdiagonaltermrightentry + S (pfc_right_shift_exists_product_chosencoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_shift_exists_product_chosencoefficientscoefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_shift_exists_product_chosencoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_shift_exists_product_chosencoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_shift_exists_product_chosencoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_shift_exists_product_chosencoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_exists_product_chosencoefficientscoefficientdiagonaltermrightoutside. pfc_gap_shift_exists_product_chosencoefficientscoefficientdiagonaltermrightoutside+(S M)=(pfc_complement_shift_exists_product_chosencoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_shift_exists_product_chosencoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_shift_exists_product_chosencoefficientscoefficientdiagonal)=pfc_left_shift_exists_product_chosencoefficientscoefficientdiagonalterm*pfc_right_shift_exists_product_chosencoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_exists_product_chosencoefficientscoefficientsum fs_v_pfc_shift_exists_product_chosencoefficientscoefficientsum. ((((exists fs_h_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_start. fs_h_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_exists_product_chosencoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_start. fs_u_pfc_shift_exists_product_chosencoefficientscoefficientsum = fs_q_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_shift_exists_product_chosencoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_terminal. fs_h_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_shift_exists_product_chosencoefficientscoefficient) = S ((S (S (pfc_index_shift_exists_product_chosencoefficients))) * fs_v_pfc_shift_exists_product_chosencoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_terminal. fs_u_pfc_shift_exists_product_chosencoefficientscoefficientsum = fs_q_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_shift_exists_product_chosencoefficients))) * fs_v_pfc_shift_exists_product_chosencoefficientscoefficientsum) + (pfc_natural_sum_shift_exists_product_chosencoefficientscoefficient))) /\\ forall fs_i_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps = S (pfc_index_shift_exists_product_chosencoefficients)) -> exists fs_a_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps fs_r_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps fs_s_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps_summand. fs_h_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_exists_product_chosencoefficientscoefficient)) /\\ exists fs_q_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps_summand. pfc_terms_code_shift_exists_product_chosencoefficientscoefficient = fs_q_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_exists_product_chosencoefficientscoefficient) + (fs_a_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps_partial. fs_h_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_chosencoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps_partial. fs_u_pfc_shift_exists_product_chosencoefficientscoefficientsum = fs_q_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_chosencoefficientscoefficientsum) + (fs_r_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps_successor. fs_h_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_chosencoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps_successor. fs_u_pfc_shift_exists_product_chosencoefficientscoefficientsum = fs_q_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_chosencoefficientscoefficientsum) + (fs_s_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps = fs_r_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps + fs_a_pfc_shift_exists_product_chosencoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_exists_product_chosencoefficientscoefficientresiduebound. pfa_gap_shift_exists_product_chosencoefficientscoefficientresiduebound + S (pfc_value_shift_exists_product_chosencoefficients) = (p)) /\\ ((exists pfa_offset_left_shift_exists_product_chosencoefficientscoefficientresiduecongruence pfa_offset_right_shift_exists_product_chosencoefficientscoefficientresiduecongruence. (pfc_natural_sum_shift_exists_product_chosencoefficientscoefficient) + (p) * pfa_offset_left_shift_exists_product_chosencoefficientscoefficientresiduecongruence = (pfc_value_shift_exists_product_chosencoefficients) + (p) * pfa_offset_right_shift_exists_product_chosencoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (ab)", "specialize prime_field_polynomial_convolution_at_length_exists (ac)", "specialize prime_field_polynomial_convolution_at_length_exists (L)", "specialize prime_field_polynomial_convolution_at_length_exists (BB)", "specialize prime_field_polynomial_convolution_at_length_exists (BC)", "specialize prime_field_polynomial_convolution_at_length_exists (S M)", "specialize prime_field_polynomial_convolution_at_length_exists (x)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp", "exact hcopy_left", "specialize prime_field_polynomial_shift_bounded (p)", "specialize prime_field_polynomial_shift_bounded (bb)", "specialize prime_field_polynomial_shift_bounded (bc)", "specialize prime_field_polynomial_shift_bounded (M)", "specialize prime_field_polynomial_shift_bounded (BB)", "specialize prime_field_polynomial_shift_bounded (BC)", "apply prime_field_polynomial_shift_bounded", "exact hprime", "exact hcopy_right_left", "exact hs", "exact hlength_witness", "cases hv", "cases hv_witness", "have he : exists e f. (((forall mdr_i_pfp_shift_exists_product_actual_comparisonprefix mdr_a_pfp_shift_exists_product_actual_comparisonprefix. (exists mdr_gap_pfp_shift_exists_product_actual_comparisonprefixb. mdr_gap_pfp_shift_exists_product_actual_comparisonprefixb + S (mdr_i_pfp_shift_exists_product_actual_comparisonprefix) = (N)) -> (((exists ff_h_mdr_pfp_shift_exists_product_actual_comparisonprefixo. ff_h_mdr_pfp_shift_exists_product_actual_comparisonprefixo + S (mdr_a_pfp_shift_exists_product_actual_comparisonprefix) = S ((S (mdr_i_pfp_shift_exists_product_actual_comparisonprefix)) * cc)) /\\ exists ff_q_mdr_pfp_shift_exists_product_actual_comparisonprefixo. cb = ff_q_mdr_pfp_shift_exists_product_actual_comparisonprefixo * S ((S (mdr_i_pfp_shift_exists_product_actual_comparisonprefix)) * cc) + (mdr_a_pfp_shift_exists_product_actual_comparisonprefix))) -> (((exists ff_h_mdr_pfp_shift_exists_product_actual_comparisonprefixn. ff_h_mdr_pfp_shift_exists_product_actual_comparisonprefixn + S (mdr_a_pfp_shift_exists_product_actual_comparisonprefix) = S ((S (mdr_i_pfp_shift_exists_product_actual_comparisonprefix)) * f)) /\\ exists ff_q_mdr_pfp_shift_exists_product_actual_comparisonprefixn. e = ff_q_mdr_pfp_shift_exists_product_actual_comparisonprefixn * S ((S (mdr_i_pfp_shift_exists_product_actual_comparisonprefix)) * f) + (mdr_a_pfp_shift_exists_product_actual_comparisonprefix)))) /\\ ((((exists ff_h_pfp_shift_exists_product_actual_comparisonlast. ff_h_pfp_shift_exists_product_actual_comparisonlast + S (0) = S ((S (N)) * f)) /\\ exists ff_q_pfp_shift_exists_product_actual_comparisonlast. e = ff_q_pfp_shift_exists_product_actual_comparisonlast * S ((S (N)) * f) + (0))))))", "specialize prime_field_polynomial_shift_exists (cb)", "specialize prime_field_polynomial_shift_exists (cc)", "specialize prime_field_polynomial_shift_exists (N)", "apply prime_field_polynomial_shift_exists", "cases he", "cases he_witness", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "split", "exact hv_witness_witness", "split", "exact he_witness_witness", "specialize prime_field_polynomial_convolution_shift_right_equivalent (p)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (ab)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (ac)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (L)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (bb)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (bc)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (M)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (cb)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (cc)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (N)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (BB)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (BC)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (x1)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (x2)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (x)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (x3)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (x4)", "apply prime_field_polynomial_convolution_shift_right_equivalent", "exact hp", "exact hs", "exact hc", "exact hv_witness_witness", "exact he_witness_witness" ], "script_sha256": "353895f21f9ddfe8e754af4ad2904fbb406ffb6b07e9f5f7d3c53dbfbd52a39e", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_shift_candidate.py", "sha256": "325d3085482ee73a2c6ee90cd17e45cffe53273671edf89c40d88428335c9c4b" }, "statement": "forall p ab ac L bb bc M cb cc N BB BC. (~((p) = 1) /\\ forall pfa_factor_left_shift_exists_product_prime pfa_factor_right_shift_exists_product_prime. (p) = pfa_factor_left_shift_exists_product_prime * pfa_factor_right_shift_exists_product_prime -> pfa_factor_left_shift_exists_product_prime = 1 \\/ pfa_factor_right_shift_exists_product_prime = 1) -> (((forall mdr_i_pfp_shift_exists_product_factorprefix mdr_a_pfp_shift_exists_product_factorprefix. (exists mdr_gap_pfp_shift_exists_product_factorprefixb. mdr_gap_pfp_shift_exists_product_factorprefixb + S (mdr_i_pfp_shift_exists_product_factorprefix) = (M)) -> (((exists ff_h_mdr_pfp_shift_exists_product_factorprefixo. ff_h_mdr_pfp_shift_exists_product_factorprefixo + S (mdr_a_pfp_shift_exists_product_factorprefix) = S ((S (mdr_i_pfp_shift_exists_product_factorprefix)) * bc)) /\\ exists ff_q_mdr_pfp_shift_exists_product_factorprefixo. bb = ff_q_mdr_pfp_shift_exists_product_factorprefixo * S ((S (mdr_i_pfp_shift_exists_product_factorprefix)) * bc) + (mdr_a_pfp_shift_exists_product_factorprefix))) -> (((exists ff_h_mdr_pfp_shift_exists_product_factorprefixn. ff_h_mdr_pfp_shift_exists_product_factorprefixn + S (mdr_a_pfp_shift_exists_product_factorprefix) = S ((S (mdr_i_pfp_shift_exists_product_factorprefix)) * BC)) /\\ exists ff_q_mdr_pfp_shift_exists_product_factorprefixn. BB = ff_q_mdr_pfp_shift_exists_product_factorprefixn * S ((S (mdr_i_pfp_shift_exists_product_factorprefix)) * BC) + (mdr_a_pfp_shift_exists_product_factorprefix)))) /\\ ((((exists ff_h_pfp_shift_exists_product_factorlast. ff_h_pfp_shift_exists_product_factorlast + S (0) = S ((S (M)) * BC)) /\\ exists ff_q_pfp_shift_exists_product_factorlast. BB = ff_q_pfp_shift_exists_product_factorlast * S ((S (M)) * BC) + (0)))))) -> (((forall fom_index_pfp_shift_exists_product_oldleft. (exists fom_gap_pfp_shift_exists_product_oldleft_index_bound. fom_gap_pfp_shift_exists_product_oldleft_index_bound + S (fom_index_pfp_shift_exists_product_oldleft) = L) -> exists fom_value_pfp_shift_exists_product_oldleft. ((((exists fom_beta_height_pfp_shift_exists_product_oldleft_entry. fom_beta_height_pfp_shift_exists_product_oldleft_entry + S (fom_value_pfp_shift_exists_product_oldleft) = S ((S (fom_index_pfp_shift_exists_product_oldleft)) * ac)) /\\ exists fom_beta_quotient_pfp_shift_exists_product_oldleft_entry. ab = fom_beta_quotient_pfp_shift_exists_product_oldleft_entry * S ((S (fom_index_pfp_shift_exists_product_oldleft)) * ac) + (fom_value_pfp_shift_exists_product_oldleft))) /\\ (exists fom_gap_pfp_shift_exists_product_oldleft_value_bound. fom_gap_pfp_shift_exists_product_oldleft_value_bound + S (fom_value_pfp_shift_exists_product_oldleft) = p))) /\\ (((forall fom_index_pfp_shift_exists_product_oldright. (exists fom_gap_pfp_shift_exists_product_oldright_index_bound. fom_gap_pfp_shift_exists_product_oldright_index_bound + S (fom_index_pfp_shift_exists_product_oldright) = M) -> exists fom_value_pfp_shift_exists_product_oldright. ((((exists fom_beta_height_pfp_shift_exists_product_oldright_entry. fom_beta_height_pfp_shift_exists_product_oldright_entry + S (fom_value_pfp_shift_exists_product_oldright) = S ((S (fom_index_pfp_shift_exists_product_oldright)) * bc)) /\\ exists fom_beta_quotient_pfp_shift_exists_product_oldright_entry. bb = fom_beta_quotient_pfp_shift_exists_product_oldright_entry * S ((S (fom_index_pfp_shift_exists_product_oldright)) * bc) + (fom_value_pfp_shift_exists_product_oldright))) /\\ (exists fom_gap_pfp_shift_exists_product_oldright_value_bound. fom_gap_pfp_shift_exists_product_oldright_value_bound + S (fom_value_pfp_shift_exists_product_oldright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_shift_exists_product_oldcoefficients. (exists pfa_gap_shift_exists_product_oldcoefficientsbound. pfa_gap_shift_exists_product_oldcoefficientsbound + S (pfc_index_shift_exists_product_oldcoefficients) = (N)) -> exists pfc_value_shift_exists_product_oldcoefficients. ((((exists ff_h_pfp_shift_exists_product_oldcoefficientsentry. ff_h_pfp_shift_exists_product_oldcoefficientsentry + S (pfc_value_shift_exists_product_oldcoefficients) = S ((S (pfc_index_shift_exists_product_oldcoefficients)) * cc)) /\\ exists ff_q_pfp_shift_exists_product_oldcoefficientsentry. cb = ff_q_pfp_shift_exists_product_oldcoefficientsentry * S ((S (pfc_index_shift_exists_product_oldcoefficients)) * cc) + (pfc_value_shift_exists_product_oldcoefficients))) /\\ ((exists pfc_terms_code_shift_exists_product_oldcoefficientscoefficient pfc_terms_scale_shift_exists_product_oldcoefficientscoefficient pfc_natural_sum_shift_exists_product_oldcoefficientscoefficient. ((forall pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal. (exists pfa_gap_shift_exists_product_oldcoefficientscoefficientdiagonalbound. pfa_gap_shift_exists_product_oldcoefficientscoefficientdiagonalbound + S (pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal) = (S (pfc_index_shift_exists_product_oldcoefficients))) -> exists pfc_value_shift_exists_product_oldcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_shift_exists_product_oldcoefficientscoefficientdiagonalentry. ff_h_pfp_shift_exists_product_oldcoefficientscoefficientdiagonalentry + S (pfc_value_shift_exists_product_oldcoefficientscoefficientdiagonal) = S ((S (pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_exists_product_oldcoefficientscoefficient)) /\\ exists ff_q_pfp_shift_exists_product_oldcoefficientscoefficientdiagonalentry. pfc_terms_code_shift_exists_product_oldcoefficientscoefficient = ff_q_pfp_shift_exists_product_oldcoefficientscoefficientdiagonalentry * S ((S (pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_exists_product_oldcoefficientscoefficient) + (pfc_value_shift_exists_product_oldcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_shift_exists_product_oldcoefficientscoefficientdiagonalterm pfc_left_shift_exists_product_oldcoefficientscoefficientdiagonalterm pfc_right_shift_exists_product_oldcoefficientscoefficientdiagonalterm. (((pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal)+pfc_complement_shift_exists_product_oldcoefficientscoefficientdiagonalterm=(pfc_index_shift_exists_product_oldcoefficients)) /\\ ((((((exists pfa_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftinside. pfa_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_shift_exists_product_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal)) * ac) + (pfc_left_shift_exists_product_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_shift_exists_product_oldcoefficientscoefficientdiagonal)) /\\ (((pfc_left_shift_exists_product_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightinside. pfa_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_shift_exists_product_oldcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_shift_exists_product_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_shift_exists_product_oldcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_shift_exists_product_oldcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_shift_exists_product_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_shift_exists_product_oldcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_shift_exists_product_oldcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_shift_exists_product_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_shift_exists_product_oldcoefficientscoefficientdiagonal)=pfc_left_shift_exists_product_oldcoefficientscoefficientdiagonalterm*pfc_right_shift_exists_product_oldcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_exists_product_oldcoefficientscoefficientsum fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum. ((((exists fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_start. fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_start. fs_u_pfc_shift_exists_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_terminal. fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_shift_exists_product_oldcoefficientscoefficient) = S ((S (S (pfc_index_shift_exists_product_oldcoefficients))) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_terminal. fs_u_pfc_shift_exists_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_shift_exists_product_oldcoefficients))) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum) + (pfc_natural_sum_shift_exists_product_oldcoefficientscoefficient))) /\\ forall fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps = S (pfc_index_shift_exists_product_oldcoefficients)) -> exists fs_a_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps fs_r_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps fs_s_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_exists_product_oldcoefficientscoefficient)) /\\ exists fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_shift_exists_product_oldcoefficientscoefficient = fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_exists_product_oldcoefficientscoefficient) + (fs_a_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_shift_exists_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum) + (fs_r_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_shift_exists_product_oldcoefficientscoefficientsum = fs_q_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_oldcoefficientscoefficientsum) + (fs_s_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps = fs_r_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps + fs_a_pfc_shift_exists_product_oldcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_exists_product_oldcoefficientscoefficientresiduebound. pfa_gap_shift_exists_product_oldcoefficientscoefficientresiduebound + S (pfc_value_shift_exists_product_oldcoefficients) = (p)) /\\ ((exists pfa_offset_left_shift_exists_product_oldcoefficientscoefficientresiduecongruence pfa_offset_right_shift_exists_product_oldcoefficientscoefficientresiduecongruence. (pfc_natural_sum_shift_exists_product_oldcoefficientscoefficient) + (p) * pfa_offset_left_shift_exists_product_oldcoefficientscoefficientresiduecongruence = (pfc_value_shift_exists_product_oldcoefficients) + (p) * pfa_offset_right_shift_exists_product_oldcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (exists K db dc eb ec. ((((forall fom_index_pfp_shift_exists_product_newleft. (exists fom_gap_pfp_shift_exists_product_newleft_index_bound. fom_gap_pfp_shift_exists_product_newleft_index_bound + S (fom_index_pfp_shift_exists_product_newleft) = L) -> exists fom_value_pfp_shift_exists_product_newleft. ((((exists fom_beta_height_pfp_shift_exists_product_newleft_entry. fom_beta_height_pfp_shift_exists_product_newleft_entry + S (fom_value_pfp_shift_exists_product_newleft) = S ((S (fom_index_pfp_shift_exists_product_newleft)) * ac)) /\\ exists fom_beta_quotient_pfp_shift_exists_product_newleft_entry. ab = fom_beta_quotient_pfp_shift_exists_product_newleft_entry * S ((S (fom_index_pfp_shift_exists_product_newleft)) * ac) + (fom_value_pfp_shift_exists_product_newleft))) /\\ (exists fom_gap_pfp_shift_exists_product_newleft_value_bound. fom_gap_pfp_shift_exists_product_newleft_value_bound + S (fom_value_pfp_shift_exists_product_newleft) = p))) /\\ (((forall fom_index_pfp_shift_exists_product_newright. (exists fom_gap_pfp_shift_exists_product_newright_index_bound. fom_gap_pfp_shift_exists_product_newright_index_bound + S (fom_index_pfp_shift_exists_product_newright) = S M) -> exists fom_value_pfp_shift_exists_product_newright. ((((exists fom_beta_height_pfp_shift_exists_product_newright_entry. fom_beta_height_pfp_shift_exists_product_newright_entry + S (fom_value_pfp_shift_exists_product_newright) = S ((S (fom_index_pfp_shift_exists_product_newright)) * BC)) /\\ exists fom_beta_quotient_pfp_shift_exists_product_newright_entry. BB = fom_beta_quotient_pfp_shift_exists_product_newright_entry * S ((S (fom_index_pfp_shift_exists_product_newright)) * BC) + (fom_value_pfp_shift_exists_product_newright))) /\\ (exists fom_gap_pfp_shift_exists_product_newright_value_bound. fom_gap_pfp_shift_exists_product_newright_value_bound + S (fom_value_pfp_shift_exists_product_newright) = p))) /\\ (((((((L)=0 \\/ (S M)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((S M)=0)) /\\ (((L)+(S M)=S (K)))))))) /\\ ((forall pfc_index_shift_exists_product_newcoefficients. (exists pfa_gap_shift_exists_product_newcoefficientsbound. pfa_gap_shift_exists_product_newcoefficientsbound + S (pfc_index_shift_exists_product_newcoefficients) = (K)) -> exists pfc_value_shift_exists_product_newcoefficients. ((((exists ff_h_pfp_shift_exists_product_newcoefficientsentry. ff_h_pfp_shift_exists_product_newcoefficientsentry + S (pfc_value_shift_exists_product_newcoefficients) = S ((S (pfc_index_shift_exists_product_newcoefficients)) * dc)) /\\ exists ff_q_pfp_shift_exists_product_newcoefficientsentry. db = ff_q_pfp_shift_exists_product_newcoefficientsentry * S ((S (pfc_index_shift_exists_product_newcoefficients)) * dc) + (pfc_value_shift_exists_product_newcoefficients))) /\\ ((exists pfc_terms_code_shift_exists_product_newcoefficientscoefficient pfc_terms_scale_shift_exists_product_newcoefficientscoefficient pfc_natural_sum_shift_exists_product_newcoefficientscoefficient. ((forall pfc_index_shift_exists_product_newcoefficientscoefficientdiagonal. (exists pfa_gap_shift_exists_product_newcoefficientscoefficientdiagonalbound. pfa_gap_shift_exists_product_newcoefficientscoefficientdiagonalbound + S (pfc_index_shift_exists_product_newcoefficientscoefficientdiagonal) = (S (pfc_index_shift_exists_product_newcoefficients))) -> exists pfc_value_shift_exists_product_newcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_shift_exists_product_newcoefficientscoefficientdiagonalentry. ff_h_pfp_shift_exists_product_newcoefficientscoefficientdiagonalentry + S (pfc_value_shift_exists_product_newcoefficientscoefficientdiagonal) = S ((S (pfc_index_shift_exists_product_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_exists_product_newcoefficientscoefficient)) /\\ exists ff_q_pfp_shift_exists_product_newcoefficientscoefficientdiagonalentry. pfc_terms_code_shift_exists_product_newcoefficientscoefficient = ff_q_pfp_shift_exists_product_newcoefficientscoefficientdiagonalentry * S ((S (pfc_index_shift_exists_product_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_shift_exists_product_newcoefficientscoefficient) + (pfc_value_shift_exists_product_newcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_shift_exists_product_newcoefficientscoefficientdiagonalterm pfc_left_shift_exists_product_newcoefficientscoefficientdiagonalterm pfc_right_shift_exists_product_newcoefficientscoefficientdiagonalterm. (((pfc_index_shift_exists_product_newcoefficientscoefficientdiagonal)+pfc_complement_shift_exists_product_newcoefficientscoefficientdiagonalterm=(pfc_index_shift_exists_product_newcoefficients)) /\\ ((((((exists pfa_gap_shift_exists_product_newcoefficientscoefficientdiagonaltermleftinside. pfa_gap_shift_exists_product_newcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_shift_exists_product_newcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_shift_exists_product_newcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_shift_exists_product_newcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_shift_exists_product_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_shift_exists_product_newcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_shift_exists_product_newcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_shift_exists_product_newcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_shift_exists_product_newcoefficientscoefficientdiagonal)) * ac) + (pfc_left_shift_exists_product_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_exists_product_newcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_shift_exists_product_newcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_shift_exists_product_newcoefficientscoefficientdiagonal)) /\\ (((pfc_left_shift_exists_product_newcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_shift_exists_product_newcoefficientscoefficientdiagonaltermrightinside. pfa_gap_shift_exists_product_newcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_shift_exists_product_newcoefficientscoefficientdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_shift_exists_product_newcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_shift_exists_product_newcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_shift_exists_product_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_shift_exists_product_newcoefficientscoefficientdiagonalterm)) * BC)) /\\ exists ff_q_pfp_shift_exists_product_newcoefficientscoefficientdiagonaltermrightentry. BB = ff_q_pfp_shift_exists_product_newcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_shift_exists_product_newcoefficientscoefficientdiagonalterm)) * BC) + (pfc_right_shift_exists_product_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_shift_exists_product_newcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_shift_exists_product_newcoefficientscoefficientdiagonaltermrightoutside+(S M)=(pfc_complement_shift_exists_product_newcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_shift_exists_product_newcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_shift_exists_product_newcoefficientscoefficientdiagonal)=pfc_left_shift_exists_product_newcoefficientscoefficientdiagonalterm*pfc_right_shift_exists_product_newcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_shift_exists_product_newcoefficientscoefficientsum fs_v_pfc_shift_exists_product_newcoefficientscoefficientsum. ((((exists fs_h_pfc_shift_exists_product_newcoefficientscoefficientsum_body_start. fs_h_pfc_shift_exists_product_newcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_shift_exists_product_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_newcoefficientscoefficientsum_body_start. fs_u_pfc_shift_exists_product_newcoefficientscoefficientsum = fs_q_pfc_shift_exists_product_newcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_shift_exists_product_newcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_shift_exists_product_newcoefficientscoefficientsum_body_terminal. fs_h_pfc_shift_exists_product_newcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_shift_exists_product_newcoefficientscoefficient) = S ((S (S (pfc_index_shift_exists_product_newcoefficients))) * fs_v_pfc_shift_exists_product_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_newcoefficientscoefficientsum_body_terminal. fs_u_pfc_shift_exists_product_newcoefficientscoefficientsum = fs_q_pfc_shift_exists_product_newcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_shift_exists_product_newcoefficients))) * fs_v_pfc_shift_exists_product_newcoefficientscoefficientsum) + (pfc_natural_sum_shift_exists_product_newcoefficientscoefficient))) /\\ forall fs_i_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps = S (pfc_index_shift_exists_product_newcoefficients)) -> exists fs_a_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps fs_r_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps fs_s_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_exists_product_newcoefficientscoefficient)) /\\ exists fs_q_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_shift_exists_product_newcoefficientscoefficient = fs_q_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_shift_exists_product_newcoefficientscoefficient) + (fs_a_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_shift_exists_product_newcoefficientscoefficientsum = fs_q_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_newcoefficientscoefficientsum) + (fs_r_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_shift_exists_product_newcoefficientscoefficientsum = fs_q_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_shift_exists_product_newcoefficientscoefficientsum) + (fs_s_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps = fs_r_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps + fs_a_pfc_shift_exists_product_newcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_shift_exists_product_newcoefficientscoefficientresiduebound. pfa_gap_shift_exists_product_newcoefficientscoefficientresiduebound + S (pfc_value_shift_exists_product_newcoefficients) = (p)) /\\ ((exists pfa_offset_left_shift_exists_product_newcoefficientscoefficientresiduecongruence pfa_offset_right_shift_exists_product_newcoefficientscoefficientresiduecongruence. (pfc_natural_sum_shift_exists_product_newcoefficientscoefficient) + (p) * pfa_offset_left_shift_exists_product_newcoefficientscoefficientresiduecongruence = (pfc_value_shift_exists_product_newcoefficients) + (p) * pfa_offset_right_shift_exists_product_newcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall mdr_i_pfp_shift_exists_product_shiftprefix mdr_a_pfp_shift_exists_product_shiftprefix. (exists mdr_gap_pfp_shift_exists_product_shiftprefixb. mdr_gap_pfp_shift_exists_product_shiftprefixb + S (mdr_i_pfp_shift_exists_product_shiftprefix) = (N)) -> (((exists ff_h_mdr_pfp_shift_exists_product_shiftprefixo. ff_h_mdr_pfp_shift_exists_product_shiftprefixo + S (mdr_a_pfp_shift_exists_product_shiftprefix) = S ((S (mdr_i_pfp_shift_exists_product_shiftprefix)) * cc)) /\\ exists ff_q_mdr_pfp_shift_exists_product_shiftprefixo. cb = ff_q_mdr_pfp_shift_exists_product_shiftprefixo * S ((S (mdr_i_pfp_shift_exists_product_shiftprefix)) * cc) + (mdr_a_pfp_shift_exists_product_shiftprefix))) -> (((exists ff_h_mdr_pfp_shift_exists_product_shiftprefixn. ff_h_mdr_pfp_shift_exists_product_shiftprefixn + S (mdr_a_pfp_shift_exists_product_shiftprefix) = S ((S (mdr_i_pfp_shift_exists_product_shiftprefix)) * ec)) /\\ exists ff_q_mdr_pfp_shift_exists_product_shiftprefixn. eb = ff_q_mdr_pfp_shift_exists_product_shiftprefixn * S ((S (mdr_i_pfp_shift_exists_product_shiftprefix)) * ec) + (mdr_a_pfp_shift_exists_product_shiftprefix)))) /\\ ((((exists ff_h_pfp_shift_exists_product_shiftlast. ff_h_pfp_shift_exists_product_shiftlast + S (0) = S ((S (N)) * ec)) /\\ exists ff_q_pfp_shift_exists_product_shiftlast. eb = ff_q_pfp_shift_exists_product_shiftlast * S ((S (N)) * ec) + (0)))))) /\\ ((forall pfrep_power_shift_exists_product_equivalence pfrep_left_shift_exists_product_equivalence pfrep_right_shift_exists_product_equivalence. ((exists pfrep_position_shift_exists_product_equivalencefirst. ((pfrep_position_shift_exists_product_equivalencefirst+S (pfrep_power_shift_exists_product_equivalence)=(K)) /\\ ((((exists ff_h_pfp_shift_exists_product_equivalencefirstentry. ff_h_pfp_shift_exists_product_equivalencefirstentry + S (pfrep_left_shift_exists_product_equivalence) = S ((S (pfrep_position_shift_exists_product_equivalencefirst)) * dc)) /\\ exists ff_q_pfp_shift_exists_product_equivalencefirstentry. db = ff_q_pfp_shift_exists_product_equivalencefirstentry * S ((S (pfrep_position_shift_exists_product_equivalencefirst)) * dc) + (pfrep_left_shift_exists_product_equivalence)))))) \\/ (((exists pfrep_gap_shift_exists_product_equivalencefirstoutside. pfrep_gap_shift_exists_product_equivalencefirstoutside+(K)=(pfrep_power_shift_exists_product_equivalence)) /\\ (((pfrep_left_shift_exists_product_equivalence)=0))))) -> ((exists pfrep_position_shift_exists_product_equivalencesecond. ((pfrep_position_shift_exists_product_equivalencesecond+S (pfrep_power_shift_exists_product_equivalence)=(S N)) /\\ ((((exists ff_h_pfp_shift_exists_product_equivalencesecondentry. ff_h_pfp_shift_exists_product_equivalencesecondentry + S (pfrep_right_shift_exists_product_equivalence) = S ((S (pfrep_position_shift_exists_product_equivalencesecond)) * ec)) /\\ exists ff_q_pfp_shift_exists_product_equivalencesecondentry. eb = ff_q_pfp_shift_exists_product_equivalencesecondentry * S ((S (pfrep_position_shift_exists_product_equivalencesecond)) * ec) + (pfrep_right_shift_exists_product_equivalence)))))) \\/ (((exists pfrep_gap_shift_exists_product_equivalencesecondoutside. pfrep_gap_shift_exists_product_equivalencesecondoutside+(S N)=(pfrep_power_shift_exists_product_equivalence)) /\\ (((pfrep_right_shift_exists_product_equivalence)=0))))) -> pfrep_left_shift_exists_product_equivalence=pfrep_right_shift_exists_product_equivalence))))))", "statement_sha256": 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(((forall mdr_i_pfp_shift_power_zero_relationprefix mdr_a_pfp_shift_power_zero_relationprefix. (exists mdr_gap_pfp_shift_power_zero_relationprefixb. mdr_gap_pfp_shift_power_zero_relationprefixb + S (mdr_i_pfp_shift_power_zero_relationprefix) = (L)) -> (((exists ff_h_mdr_pfp_shift_power_zero_relationprefixo. ff_h_mdr_pfp_shift_power_zero_relationprefixo + S (mdr_a_pfp_shift_power_zero_relationprefix) = S ((S (mdr_i_pfp_shift_power_zero_relationprefix)) * c)) /\\ exists ff_q_mdr_pfp_shift_power_zero_relationprefixo. b = ff_q_mdr_pfp_shift_power_zero_relationprefixo * S ((S (mdr_i_pfp_shift_power_zero_relationprefix)) * c) + (mdr_a_pfp_shift_power_zero_relationprefix))) -> (((exists ff_h_mdr_pfp_shift_power_zero_relationprefixn. ff_h_mdr_pfp_shift_power_zero_relationprefixn + S (mdr_a_pfp_shift_power_zero_relationprefix) = S ((S (mdr_i_pfp_shift_power_zero_relationprefix)) * e)) /\\ exists ff_q_mdr_pfp_shift_power_zero_relationprefixn. d = ff_q_mdr_pfp_shift_power_zero_relationprefixn * S ((S (mdr_i_pfp_shift_power_zero_relationprefix)) * e) + (mdr_a_pfp_shift_power_zero_relationprefix)))) /\\ ((((exists ff_h_pfp_shift_power_zero_relationlast. ff_h_pfp_shift_power_zero_relationlast + S (0) = S ((S (L)) * e)) /\\ exists ff_q_pfp_shift_power_zero_relationlast. d = ff_q_pfp_shift_power_zero_relationlast * S ((S (L)) * e) + (0)))))) -> ((exists pfrep_position_shift_power_zero_result. 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(((forall mdr_i_pfp_shift_power_successor_relationprefix mdr_a_pfp_shift_power_successor_relationprefix. (exists mdr_gap_pfp_shift_power_successor_relationprefixb. mdr_gap_pfp_shift_power_successor_relationprefixb + S (mdr_i_pfp_shift_power_successor_relationprefix) = (L)) -> (((exists ff_h_mdr_pfp_shift_power_successor_relationprefixo. ff_h_mdr_pfp_shift_power_successor_relationprefixo + S (mdr_a_pfp_shift_power_successor_relationprefix) = S ((S (mdr_i_pfp_shift_power_successor_relationprefix)) * c)) /\\ exists ff_q_mdr_pfp_shift_power_successor_relationprefixo. b = ff_q_mdr_pfp_shift_power_successor_relationprefixo * S ((S (mdr_i_pfp_shift_power_successor_relationprefix)) * c) + (mdr_a_pfp_shift_power_successor_relationprefix))) -> (((exists ff_h_mdr_pfp_shift_power_successor_relationprefixn. ff_h_mdr_pfp_shift_power_successor_relationprefixn + S (mdr_a_pfp_shift_power_successor_relationprefix) = S ((S (mdr_i_pfp_shift_power_successor_relationprefix)) * e)) /\\ exists ff_q_mdr_pfp_shift_power_successor_relationprefixn. d = ff_q_mdr_pfp_shift_power_successor_relationprefixn * S ((S (mdr_i_pfp_shift_power_successor_relationprefix)) * e) + (mdr_a_pfp_shift_power_successor_relationprefix)))) /\\ ((((exists ff_h_pfp_shift_power_successor_relationlast. ff_h_pfp_shift_power_successor_relationlast + S (0) = S ((S (L)) * e)) /\\ exists ff_q_pfp_shift_power_successor_relationlast. d = ff_q_pfp_shift_power_successor_relationlast * S ((S (L)) * e) + (0)))))) -> ((exists pfrep_position_shift_power_successor_source. ((pfrep_position_shift_power_successor_source+S (k)=(L)) /\\ ((((exists ff_h_pfp_shift_power_successor_sourceentry. ff_h_pfp_shift_power_successor_sourceentry + S (a) = S ((S (pfrep_position_shift_power_successor_source)) * c)) /\\ exists ff_q_pfp_shift_power_successor_sourceentry. b = ff_q_pfp_shift_power_successor_sourceentry * S ((S (pfrep_position_shift_power_successor_source)) * c) + (a)))))) \\/ (((exists pfrep_gap_shift_power_successor_sourceoutside. pfrep_gap_shift_power_successor_sourceoutside+(L)=(k)) /\\ (((a)=0))))) -> ((exists pfrep_position_shift_power_successor_result. 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((((exists ff_h_pfp_scalar_sum_old_last. ff_h_pfp_scalar_sum_old_last + S (a) = S ((S (L)) * ac)) /\\ exists ff_q_pfp_scalar_sum_old_last. ab = ff_q_pfp_scalar_sum_old_last * S ((S (L)) * ac) + (a))) /\\ (((exists fs_u_pfc_scalar_sum_old_prefix fs_v_pfc_scalar_sum_old_prefix. ((((exists fs_h_pfc_scalar_sum_old_prefix_body_start. fs_h_pfc_scalar_sum_old_prefix_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_sum_old_prefix)) /\\ exists fs_q_pfc_scalar_sum_old_prefix_body_start. fs_u_pfc_scalar_sum_old_prefix = fs_q_pfc_scalar_sum_old_prefix_body_start * S ((S (0)) * fs_v_pfc_scalar_sum_old_prefix) + (0))) /\\ ((((exists fs_h_pfc_scalar_sum_old_prefix_body_terminal. fs_h_pfc_scalar_sum_old_prefix_body_terminal + S (n) = S ((S (L)) * fs_v_pfc_scalar_sum_old_prefix)) /\\ exists fs_q_pfc_scalar_sum_old_prefix_body_terminal. fs_u_pfc_scalar_sum_old_prefix = fs_q_pfc_scalar_sum_old_prefix_body_terminal * S ((S (L)) * fs_v_pfc_scalar_sum_old_prefix) + (n))) /\\ forall fs_i_pfc_scalar_sum_old_prefix_body_steps. (exists fs_lt_pfc_scalar_sum_old_prefix_body_steps_bound. fs_lt_pfc_scalar_sum_old_prefix_body_steps_bound + S fs_i_pfc_scalar_sum_old_prefix_body_steps = L) -> exists fs_a_pfc_scalar_sum_old_prefix_body_steps fs_r_pfc_scalar_sum_old_prefix_body_steps fs_s_pfc_scalar_sum_old_prefix_body_steps. ((((exists fs_h_pfc_scalar_sum_old_prefix_body_steps_summand. fs_h_pfc_scalar_sum_old_prefix_body_steps_summand + S (fs_a_pfc_scalar_sum_old_prefix_body_steps) = S ((S (fs_i_pfc_scalar_sum_old_prefix_body_steps)) * ac)) /\\ exists fs_q_pfc_scalar_sum_old_prefix_body_steps_summand. ab = fs_q_pfc_scalar_sum_old_prefix_body_steps_summand * S ((S (fs_i_pfc_scalar_sum_old_prefix_body_steps)) * ac) + (fs_a_pfc_scalar_sum_old_prefix_body_steps))) /\\ ((((exists fs_h_pfc_scalar_sum_old_prefix_body_steps_partial. fs_h_pfc_scalar_sum_old_prefix_body_steps_partial + S (fs_r_pfc_scalar_sum_old_prefix_body_steps) = S ((S (fs_i_pfc_scalar_sum_old_prefix_body_steps)) * fs_v_pfc_scalar_sum_old_prefix)) /\\ exists fs_q_pfc_scalar_sum_old_prefix_body_steps_partial. fs_u_pfc_scalar_sum_old_prefix = fs_q_pfc_scalar_sum_old_prefix_body_steps_partial * S ((S (fs_i_pfc_scalar_sum_old_prefix_body_steps)) * fs_v_pfc_scalar_sum_old_prefix) + (fs_r_pfc_scalar_sum_old_prefix_body_steps))) /\\ ((((exists fs_h_pfc_scalar_sum_old_prefix_body_steps_successor. fs_h_pfc_scalar_sum_old_prefix_body_steps_successor + S (fs_s_pfc_scalar_sum_old_prefix_body_steps) = S ((S (S fs_i_pfc_scalar_sum_old_prefix_body_steps)) * fs_v_pfc_scalar_sum_old_prefix)) /\\ exists fs_q_pfc_scalar_sum_old_prefix_body_steps_successor. fs_u_pfc_scalar_sum_old_prefix = fs_q_pfc_scalar_sum_old_prefix_body_steps_successor * S ((S (S fs_i_pfc_scalar_sum_old_prefix_body_steps)) * fs_v_pfc_scalar_sum_old_prefix) + (fs_s_pfc_scalar_sum_old_prefix_body_steps))) /\\ fs_s_pfc_scalar_sum_old_prefix_body_steps = fs_r_pfc_scalar_sum_old_prefix_body_steps + fs_a_pfc_scalar_sum_old_prefix_body_steps)))))) /\\ ((u=n+a)))))", "specialize beta_sum_succ_decompose (ab)", "specialize beta_sum_succ_decompose (ac)", "specialize beta_sum_succ_decompose (L)", "specialize beta_sum_succ_decompose (u)", "apply beta_sum_succ_decompose", "exact hu", "cases hfirst", "cases hfirst_witness", "cases hfirst_witness_witness", "cases hfirst_witness_witness_right", "have hsecond : exists a n. ((((exists ff_h_pfp_scalar_sum_new_last. ff_h_pfp_scalar_sum_new_last + S (a) = S ((S (L)) * bc)) /\\ exists ff_q_pfp_scalar_sum_new_last. bb = ff_q_pfp_scalar_sum_new_last * S ((S (L)) * bc) + (a))) /\\ (((exists fs_u_pfc_scalar_sum_new_prefix fs_v_pfc_scalar_sum_new_prefix. ((((exists fs_h_pfc_scalar_sum_new_prefix_body_start. fs_h_pfc_scalar_sum_new_prefix_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_sum_new_prefix)) /\\ exists fs_q_pfc_scalar_sum_new_prefix_body_start. fs_u_pfc_scalar_sum_new_prefix = fs_q_pfc_scalar_sum_new_prefix_body_start * S ((S (0)) * fs_v_pfc_scalar_sum_new_prefix) + (0))) /\\ ((((exists fs_h_pfc_scalar_sum_new_prefix_body_terminal. fs_h_pfc_scalar_sum_new_prefix_body_terminal + S (n) = S ((S (L)) * fs_v_pfc_scalar_sum_new_prefix)) /\\ exists fs_q_pfc_scalar_sum_new_prefix_body_terminal. fs_u_pfc_scalar_sum_new_prefix = fs_q_pfc_scalar_sum_new_prefix_body_terminal * S ((S (L)) * fs_v_pfc_scalar_sum_new_prefix) + (n))) /\\ forall fs_i_pfc_scalar_sum_new_prefix_body_steps. (exists fs_lt_pfc_scalar_sum_new_prefix_body_steps_bound. fs_lt_pfc_scalar_sum_new_prefix_body_steps_bound + S fs_i_pfc_scalar_sum_new_prefix_body_steps = L) -> exists fs_a_pfc_scalar_sum_new_prefix_body_steps fs_r_pfc_scalar_sum_new_prefix_body_steps fs_s_pfc_scalar_sum_new_prefix_body_steps. ((((exists fs_h_pfc_scalar_sum_new_prefix_body_steps_summand. fs_h_pfc_scalar_sum_new_prefix_body_steps_summand + S (fs_a_pfc_scalar_sum_new_prefix_body_steps) = S ((S (fs_i_pfc_scalar_sum_new_prefix_body_steps)) * bc)) /\\ exists fs_q_pfc_scalar_sum_new_prefix_body_steps_summand. bb = fs_q_pfc_scalar_sum_new_prefix_body_steps_summand * S ((S (fs_i_pfc_scalar_sum_new_prefix_body_steps)) * bc) + (fs_a_pfc_scalar_sum_new_prefix_body_steps))) /\\ ((((exists fs_h_pfc_scalar_sum_new_prefix_body_steps_partial. fs_h_pfc_scalar_sum_new_prefix_body_steps_partial + S (fs_r_pfc_scalar_sum_new_prefix_body_steps) = S ((S (fs_i_pfc_scalar_sum_new_prefix_body_steps)) * fs_v_pfc_scalar_sum_new_prefix)) /\\ exists fs_q_pfc_scalar_sum_new_prefix_body_steps_partial. fs_u_pfc_scalar_sum_new_prefix = fs_q_pfc_scalar_sum_new_prefix_body_steps_partial * S ((S (fs_i_pfc_scalar_sum_new_prefix_body_steps)) * fs_v_pfc_scalar_sum_new_prefix) + (fs_r_pfc_scalar_sum_new_prefix_body_steps))) /\\ ((((exists fs_h_pfc_scalar_sum_new_prefix_body_steps_successor. fs_h_pfc_scalar_sum_new_prefix_body_steps_successor + S (fs_s_pfc_scalar_sum_new_prefix_body_steps) = S ((S (S fs_i_pfc_scalar_sum_new_prefix_body_steps)) * fs_v_pfc_scalar_sum_new_prefix)) /\\ exists fs_q_pfc_scalar_sum_new_prefix_body_steps_successor. fs_u_pfc_scalar_sum_new_prefix = fs_q_pfc_scalar_sum_new_prefix_body_steps_successor * S ((S (S fs_i_pfc_scalar_sum_new_prefix_body_steps)) * fs_v_pfc_scalar_sum_new_prefix) + (fs_s_pfc_scalar_sum_new_prefix_body_steps))) /\\ fs_s_pfc_scalar_sum_new_prefix_body_steps = fs_r_pfc_scalar_sum_new_prefix_body_steps + fs_a_pfc_scalar_sum_new_prefix_body_steps)))))) /\\ ((v=n+a)))))", "specialize beta_sum_succ_decompose (bb)", "specialize beta_sum_succ_decompose (bc)", "specialize beta_sum_succ_decompose (L)", "specialize beta_sum_succ_decompose (v)", "apply beta_sum_succ_decompose", "exact hv", "cases hsecond", "cases hsecond_witness", "cases hsecond_witness_witness", "cases hsecond_witness_witness_right", "have hprefix : exists pfa_offset_left_scalar_sum_prefix_congruence pfa_offset_right_scalar_sum_prefix_congruence. (k*x1) + (p) * pfa_offset_left_scalar_sum_prefix_congruence = (x3) + (p) * pfa_offset_right_scalar_sum_prefix_congruence", "specialize IH (x1)", "specialize IH (x3)", "apply IH", "exact hfirst_witness_witness_right_left", "exact hsecond_witness_witness_right_left", "intro i", "intro a", "intro b", "intro hi", "intro ha", "intro hb", "specialize hw (i)", "specialize hw (a)", "specialize hw (b)", "apply hw", "specialize le_succ (S i)", "specialize le_succ (L)", "apply le_succ", "exact hi", "exact ha", "exact hb", "have hlast : exists pfa_offset_left_scalar_sum_last_congruence pfa_offset_right_scalar_sum_last_congruence. (k*x) + (p) * pfa_offset_left_scalar_sum_last_congruence = (x2) + (p) * pfa_offset_right_scalar_sum_last_congruence", "specialize hw (L)", "specialize hw (x)", "specialize hw (x2)", "apply hw", "specialize le_refl (S L)", "apply le_refl", "exact hfirst_witness_witness_left", "exact hsecond_witness_witness_left", "have hcombined : exists pfa_offset_left_scalar_sum_combined pfa_offset_right_scalar_sum_combined. (k*x1+k*x) + (p) * pfa_offset_left_scalar_sum_combined = (x3+x2) + (p) * pfa_offset_right_scalar_sum_combined", "specialize mod_eq_add (p)", "specialize mod_eq_add (k*x1)", "specialize mod_eq_add (x3)", "specialize mod_eq_add (k*x)", "specialize mod_eq_add (x2)", "apply mod_eq_add", "exact hprefix", "exact hlast", "rewrite hfirst_witness_witness_right_right", "rewrite hsecond_witness_witness_right_right", "have hdistribute : k*(x1+x)=k*x1+k*x", "apply mul_add", "rewrite hdistribute", "exact hcombined" ], "script_sha256": "6830408c899c68c775b784406c65c2853ab68c1f253d6446846beca179296c32", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_scalar_convolution_candidate.py", "sha256": "e84f1c77c6c03fa5f08635aeede53591625d1c2bfcdfb64fbd379c33878aee0e" }, "statement": "forall p k ab ac bb bc L u v. (exists fs_u_pfc_scalar_sum_source fs_v_pfc_scalar_sum_source. ((((exists fs_h_pfc_scalar_sum_source_body_start. fs_h_pfc_scalar_sum_source_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_sum_source)) /\\ exists fs_q_pfc_scalar_sum_source_body_start. fs_u_pfc_scalar_sum_source = fs_q_pfc_scalar_sum_source_body_start * S ((S (0)) * fs_v_pfc_scalar_sum_source) + (0))) /\\ ((((exists fs_h_pfc_scalar_sum_source_body_terminal. fs_h_pfc_scalar_sum_source_body_terminal + S (u) = S ((S (L)) * fs_v_pfc_scalar_sum_source)) /\\ exists fs_q_pfc_scalar_sum_source_body_terminal. fs_u_pfc_scalar_sum_source = fs_q_pfc_scalar_sum_source_body_terminal * S ((S (L)) * fs_v_pfc_scalar_sum_source) + (u))) /\\ forall fs_i_pfc_scalar_sum_source_body_steps. (exists fs_lt_pfc_scalar_sum_source_body_steps_bound. fs_lt_pfc_scalar_sum_source_body_steps_bound + S fs_i_pfc_scalar_sum_source_body_steps = L) -> exists fs_a_pfc_scalar_sum_source_body_steps fs_r_pfc_scalar_sum_source_body_steps fs_s_pfc_scalar_sum_source_body_steps. ((((exists fs_h_pfc_scalar_sum_source_body_steps_summand. fs_h_pfc_scalar_sum_source_body_steps_summand + S (fs_a_pfc_scalar_sum_source_body_steps) = S ((S (fs_i_pfc_scalar_sum_source_body_steps)) * ac)) /\\ exists fs_q_pfc_scalar_sum_source_body_steps_summand. ab = fs_q_pfc_scalar_sum_source_body_steps_summand * S ((S (fs_i_pfc_scalar_sum_source_body_steps)) * ac) + (fs_a_pfc_scalar_sum_source_body_steps))) /\\ ((((exists fs_h_pfc_scalar_sum_source_body_steps_partial. fs_h_pfc_scalar_sum_source_body_steps_partial + S (fs_r_pfc_scalar_sum_source_body_steps) = S ((S (fs_i_pfc_scalar_sum_source_body_steps)) * fs_v_pfc_scalar_sum_source)) /\\ exists fs_q_pfc_scalar_sum_source_body_steps_partial. fs_u_pfc_scalar_sum_source = fs_q_pfc_scalar_sum_source_body_steps_partial * S ((S (fs_i_pfc_scalar_sum_source_body_steps)) * fs_v_pfc_scalar_sum_source) + (fs_r_pfc_scalar_sum_source_body_steps))) /\\ ((((exists fs_h_pfc_scalar_sum_source_body_steps_successor. fs_h_pfc_scalar_sum_source_body_steps_successor + S (fs_s_pfc_scalar_sum_source_body_steps) = S ((S (S fs_i_pfc_scalar_sum_source_body_steps)) * fs_v_pfc_scalar_sum_source)) /\\ exists fs_q_pfc_scalar_sum_source_body_steps_successor. fs_u_pfc_scalar_sum_source = fs_q_pfc_scalar_sum_source_body_steps_successor * S ((S (S fs_i_pfc_scalar_sum_source_body_steps)) * fs_v_pfc_scalar_sum_source) + (fs_s_pfc_scalar_sum_source_body_steps))) /\\ fs_s_pfc_scalar_sum_source_body_steps = fs_r_pfc_scalar_sum_source_body_steps + fs_a_pfc_scalar_sum_source_body_steps)))))) -> (exists fs_u_pfc_scalar_sum_target fs_v_pfc_scalar_sum_target. ((((exists fs_h_pfc_scalar_sum_target_body_start. fs_h_pfc_scalar_sum_target_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_sum_target)) /\\ exists fs_q_pfc_scalar_sum_target_body_start. fs_u_pfc_scalar_sum_target = fs_q_pfc_scalar_sum_target_body_start * S ((S (0)) * fs_v_pfc_scalar_sum_target) + (0))) /\\ ((((exists fs_h_pfc_scalar_sum_target_body_terminal. fs_h_pfc_scalar_sum_target_body_terminal + S (v) = S ((S (L)) * fs_v_pfc_scalar_sum_target)) /\\ exists fs_q_pfc_scalar_sum_target_body_terminal. fs_u_pfc_scalar_sum_target = fs_q_pfc_scalar_sum_target_body_terminal * S ((S (L)) * fs_v_pfc_scalar_sum_target) + (v))) /\\ forall fs_i_pfc_scalar_sum_target_body_steps. (exists fs_lt_pfc_scalar_sum_target_body_steps_bound. fs_lt_pfc_scalar_sum_target_body_steps_bound + S fs_i_pfc_scalar_sum_target_body_steps = L) -> exists fs_a_pfc_scalar_sum_target_body_steps fs_r_pfc_scalar_sum_target_body_steps fs_s_pfc_scalar_sum_target_body_steps. ((((exists fs_h_pfc_scalar_sum_target_body_steps_summand. fs_h_pfc_scalar_sum_target_body_steps_summand + S (fs_a_pfc_scalar_sum_target_body_steps) = S ((S (fs_i_pfc_scalar_sum_target_body_steps)) * bc)) /\\ exists fs_q_pfc_scalar_sum_target_body_steps_summand. bb = fs_q_pfc_scalar_sum_target_body_steps_summand * S ((S (fs_i_pfc_scalar_sum_target_body_steps)) * bc) + (fs_a_pfc_scalar_sum_target_body_steps))) /\\ ((((exists fs_h_pfc_scalar_sum_target_body_steps_partial. fs_h_pfc_scalar_sum_target_body_steps_partial + S (fs_r_pfc_scalar_sum_target_body_steps) = S ((S (fs_i_pfc_scalar_sum_target_body_steps)) * fs_v_pfc_scalar_sum_target)) /\\ exists fs_q_pfc_scalar_sum_target_body_steps_partial. fs_u_pfc_scalar_sum_target = fs_q_pfc_scalar_sum_target_body_steps_partial * S ((S (fs_i_pfc_scalar_sum_target_body_steps)) * fs_v_pfc_scalar_sum_target) + (fs_r_pfc_scalar_sum_target_body_steps))) /\\ ((((exists fs_h_pfc_scalar_sum_target_body_steps_successor. fs_h_pfc_scalar_sum_target_body_steps_successor + S (fs_s_pfc_scalar_sum_target_body_steps) = S ((S (S fs_i_pfc_scalar_sum_target_body_steps)) * fs_v_pfc_scalar_sum_target)) /\\ exists fs_q_pfc_scalar_sum_target_body_steps_successor. fs_u_pfc_scalar_sum_target = fs_q_pfc_scalar_sum_target_body_steps_successor * S ((S (S fs_i_pfc_scalar_sum_target_body_steps)) * fs_v_pfc_scalar_sum_target) + (fs_s_pfc_scalar_sum_target_body_steps))) /\\ fs_s_pfc_scalar_sum_target_body_steps = fs_r_pfc_scalar_sum_target_body_steps + fs_a_pfc_scalar_sum_target_body_steps)))))) -> (forall pfscalar_index_scalar_sum_points pfscalar_source_scalar_sum_points pfscalar_target_scalar_sum_points. (exists pfa_gap_scalar_sum_pointsindex. pfa_gap_scalar_sum_pointsindex + S (pfscalar_index_scalar_sum_points) = (L)) -> (((exists ff_h_pfp_scalar_sum_pointssource. ff_h_pfp_scalar_sum_pointssource + S (pfscalar_source_scalar_sum_points) = S ((S (pfscalar_index_scalar_sum_points)) * ac)) /\\ exists ff_q_pfp_scalar_sum_pointssource. ab = ff_q_pfp_scalar_sum_pointssource * S ((S (pfscalar_index_scalar_sum_points)) * ac) + (pfscalar_source_scalar_sum_points))) -> (((exists ff_h_pfp_scalar_sum_pointstarget. ff_h_pfp_scalar_sum_pointstarget + S (pfscalar_target_scalar_sum_points) = S ((S (pfscalar_index_scalar_sum_points)) * bc)) /\\ exists ff_q_pfp_scalar_sum_pointstarget. bb = ff_q_pfp_scalar_sum_pointstarget * S ((S (pfscalar_index_scalar_sum_points)) * bc) + (pfscalar_target_scalar_sum_points))) -> (exists pfa_offset_left_scalar_sum_pointscongruence pfa_offset_right_scalar_sum_pointscongruence. 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"alpha_v34_frontier_executable_audit", "path": "peano-lab/py/tests/test_campaign_research_v34_closure.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "kind": "alpha_v34_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-field-polynomial-gcd-bezout-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "kind": "alpha_v34_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=222]" }, { "document_sha256": "81943a545e7b9f667ad758c24de14bb47280e8cf3fbb3ca714efec55fb096408", "kind": "alpha_v34_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "0997866499ff8ce721589dcb20d457d91f237653d569c3c46f7aa6476b6b5224", "membership": "alpha_only", "name": "polynomial_zero_extended_scale_congruent", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro k", "intro bb", "intro bc", "intro sb", "intro sc", "intro M", "intro i", "intro b", "intro s", "intro hs", "intro hb", "intro hr", "cases hb", "cases hb_left", "have htarget : ((exists ff_h_pfp_scalar_pad_inside_target. ff_h_pfp_scalar_pad_inside_target + S (s) = S ((S (i)) * sc)) /\\ exists ff_q_pfp_scalar_pad_inside_target. sb = ff_q_pfp_scalar_pad_inside_target * S ((S (i)) * sc) + (s))", "specialize polynomial_zero_extended_entry_inside (sb)", "specialize polynomial_zero_extended_entry_inside (sc)", "specialize polynomial_zero_extended_entry_inside (M)", "specialize polynomial_zero_extended_entry_inside (i)", "specialize polynomial_zero_extended_entry_inside (s)", "apply polynomial_zero_extended_entry_inside", "exact hb_left_left", "exact hr", "have hm : ((exists pfa_gap_scalar_pad_inside_productleft. pfa_gap_scalar_pad_inside_productleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_pad_inside_productright. pfa_gap_scalar_pad_inside_productright + S (b) = (p)) /\\ ((((exists pfa_gap_scalar_pad_inside_productresultbound. pfa_gap_scalar_pad_inside_productresultbound + S (s) = (p)) /\\ ((exists pfa_offset_left_scalar_pad_inside_productresultcongruence pfa_offset_right_scalar_pad_inside_productresultcongruence. ((k) * (b)) + (p) * pfa_offset_left_scalar_pad_inside_productresultcongruence = (s) + (p) * pfa_offset_right_scalar_pad_inside_productresultcongruence))))))))", "specialize prime_field_polynomial_scale_entry (p)", "specialize prime_field_polynomial_scale_entry (k)", "specialize prime_field_polynomial_scale_entry (bb)", "specialize prime_field_polynomial_scale_entry (bc)", "specialize prime_field_polynomial_scale_entry (sb)", "specialize prime_field_polynomial_scale_entry (sc)", "specialize prime_field_polynomial_scale_entry (M)", "specialize prime_field_polynomial_scale_entry (i)", "specialize prime_field_polynomial_scale_entry (b)", "specialize prime_field_polynomial_scale_entry (s)", "apply prime_field_polynomial_scale_entry", "exact hs", "exact hb_left_left", "exact hb_left_right", "exact htarget", "cases hm", "cases hm_right", "cases hm_right_right", "exact hm_right_right_right", "cases hb_right", "have hz : s=0", "specialize polynomial_zero_extended_entry_functional (sb)", "specialize polynomial_zero_extended_entry_functional (sc)", "specialize polynomial_zero_extended_entry_functional (M)", "specialize polynomial_zero_extended_entry_functional (i)", "specialize polynomial_zero_extended_entry_functional (s)", "specialize polynomial_zero_extended_entry_functional (0)", "apply polynomial_zero_extended_entry_functional", "exact hr", "right", "split", "exact hb_right_left", "refl", "rewrite hb_right_right", "rewrite hz", "exists 0", "exists 0", "simp" ], "script_sha256": "c7c9f54ff59e4c62ee32bf68ff60be019e9d4585955575cee88198fc3cad577e", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_scalar_convolution_candidate.py", "sha256": "e84f1c77c6c03fa5f08635aeede53591625d1c2bfcdfb64fbd379c33878aee0e" }, "statement": "forall p k bb bc sb sc M i b s. (((exists pfa_gap_scalar_pad_operationscalar. pfa_gap_scalar_pad_operationscalar + S (k) = (p)) /\\ ((forall pfp_index_scalar_pad_operation. (exists pfa_gap_scalar_pad_operationindex. pfa_gap_scalar_pad_operationindex + S (pfp_index_scalar_pad_operation) = (M)) -> exists pfp_source_scalar_pad_operation pfp_value_scalar_pad_operation. ((((exists ff_h_pfp_scalar_pad_operationsource. ff_h_pfp_scalar_pad_operationsource + S (pfp_source_scalar_pad_operation) = S ((S (pfp_index_scalar_pad_operation)) * bc)) /\\ exists ff_q_pfp_scalar_pad_operationsource. bb = ff_q_pfp_scalar_pad_operationsource * S ((S (pfp_index_scalar_pad_operation)) * bc) + (pfp_source_scalar_pad_operation))) /\\ (((((exists ff_h_pfp_scalar_pad_operationtarget. ff_h_pfp_scalar_pad_operationtarget + S (pfp_value_scalar_pad_operation) = S ((S (pfp_index_scalar_pad_operation)) * sc)) /\\ exists ff_q_pfp_scalar_pad_operationtarget. sb = ff_q_pfp_scalar_pad_operationtarget * S ((S (pfp_index_scalar_pad_operation)) * sc) + (pfp_value_scalar_pad_operation))) /\\ ((((exists pfa_gap_scalar_pad_operationoperationleft. pfa_gap_scalar_pad_operationoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_pad_operationoperationright. pfa_gap_scalar_pad_operationoperationright + S (pfp_source_scalar_pad_operation) = (p)) /\\ ((((exists pfa_gap_scalar_pad_operationoperationresultbound. pfa_gap_scalar_pad_operationoperationresultbound + S (pfp_value_scalar_pad_operation) = (p)) /\\ ((exists pfa_offset_left_scalar_pad_operationoperationresultcongruence pfa_offset_right_scalar_pad_operationoperationresultcongruence. ((k) * (pfp_source_scalar_pad_operation)) + (p) * pfa_offset_left_scalar_pad_operationoperationresultcongruence = (pfp_value_scalar_pad_operation) + (p) * pfa_offset_right_scalar_pad_operationoperationresultcongruence))))))))))))))))) -> ((((exists pfa_gap_scalar_pad_sourceinside. pfa_gap_scalar_pad_sourceinside + S (i) = (M)) /\\ ((((exists ff_h_pfp_scalar_pad_sourceentry. ff_h_pfp_scalar_pad_sourceentry + S (b) = S ((S (i)) * bc)) /\\ exists ff_q_pfp_scalar_pad_sourceentry. bb = ff_q_pfp_scalar_pad_sourceentry * S ((S (i)) * bc) + (b)))))) \\/ (((exists pfc_gap_scalar_pad_sourceoutside. pfc_gap_scalar_pad_sourceoutside+(M)=(i)) /\\ (((b)=0))))) -> ((((exists pfa_gap_scalar_pad_targetinside. pfa_gap_scalar_pad_targetinside + S (i) = (M)) /\\ ((((exists ff_h_pfp_scalar_pad_targetentry. ff_h_pfp_scalar_pad_targetentry + S (s) = S ((S (i)) * sc)) /\\ exists ff_q_pfp_scalar_pad_targetentry. sb = ff_q_pfp_scalar_pad_targetentry * S ((S (i)) * sc) + (s)))))) \\/ (((exists pfc_gap_scalar_pad_targetoutside. pfc_gap_scalar_pad_targetoutside+(M)=(i)) /\\ (((s)=0))))) -> (exists pfa_offset_left_scalar_pad_result pfa_offset_right_scalar_pad_result. (k*b) + (p) * pfa_offset_left_scalar_pad_result = (s) + (p) * pfa_offset_right_scalar_pad_result)", "statement_sha256": "0ea8b4433b1e99cbd033476e4b1384c25293f1b1cc8d0999e8e3e2b30fea44e6", "summary": "Actual coefficient scaling extends by genuine exterior zeros to a scalar congruence at every array index, with no condition on raw entries after the prefixes.", "summary_sha256": "8d501e68f2309187d41df1febcce2bf25521cc07997452884814f10cf4877ceb" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "8fae76e15f019120cc8acc460dd6f594c1f1c1198dcbd0f38144ffeef60e311c", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 223, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "e84f1c77c6c03fa5f08635aeede53591625d1c2bfcdfb64fbd379c33878aee0e", "test_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85" }, "body_checked": true, "body_receipt": { "command_count": 81, "dependency_count": 6, "dne_command_count": 0, "name": "polynomial_diagonal_term_right_scale_congruent", "proof_depth": 47, "proof_edges": 159, "proof_nodes": 160, "proof_objects": 160, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "add_left_cancel", "polynomial_zero_extended_entry_functional", "polynomial_zero_extended_scale_congruent", "mod_eq_mul_left", "mul_assoc", "mul_comm" ], "dependencies_sha256": "d65621e0a509ae505486b64de83ba852005564b21a997a83d2df8e25367ea93b", "empty_context_closure": { "body_proof_depth": 47, "body_proof_nodes": 160, "bundle_campaign": "polynomial-gcd-bezout", "bundle_dependency_edge_count": 1578, "bundle_node_count": 493, "bundle_node_id": 223, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "bundle_root_id": 492, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "8e1a0db8e0acf723d5f525f43092d1aecbec2921153ed8667f136b8ae561c25a", "status": "checked" }, "enrollment_index": 4109, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "e84f1c77c6c03fa5f08635aeede53591625d1c2bfcdfb64fbd379c33878aee0e", "kind": "alpha_v34_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_scalar_convolution_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85", "kind": "alpha_v34_frontier_executable_audit", "path": "peano-lab/py/tests/test_campaign_research_v34_closure.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "kind": "alpha_v34_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-field-polynomial-gcd-bezout-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "kind": "alpha_v34_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=223]" }, { "document_sha256": "81943a545e7b9f667ad758c24de14bb47280e8cf3fbb3ca714efec55fb096408", "kind": "alpha_v34_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "b3732c361d5e8f5121bea88ecc455fb11cc257642c21eadd483f1481032f50f1", "membership": "alpha_only", "name": "polynomial_diagonal_term_right_scale_congruent", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro k", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro M", "intro sb", "intro sc", "intro i", "intro j", "intro t", "intro r", "intro hs", "intro ht", "intro hr", "cases ht", "cases ht_witness", "cases ht_witness_witness", "cases ht_witness_witness_witness", "cases ht_witness_witness_witness_right", "cases ht_witness_witness_witness_right_right", "cases hr", "cases hr_witness", "cases hr_witness_witness", "cases hr_witness_witness_witness", "cases hr_witness_witness_witness_right", "cases hr_witness_witness_witness_right_right", "have hindex : x=x3", "specialize add_left_cancel (j)", "specialize add_left_cancel (x)", "specialize add_left_cancel (x3)", "apply add_left_cancel", "trans i", "exact ht_witness_witness_witness_left", "symm", "exact hr_witness_witness_witness_left", "rewrite hindex at ht_witness_witness_witness_right_right_left", "rewrite hindex at ht_witness_witness_witness_right_right_left", "rewrite hindex at ht_witness_witness_witness_right_right_left", "rewrite hindex at ht_witness_witness_witness_right_right_left", "have hleft : x1=x4", "specialize polynomial_zero_extended_entry_functional (ab)", "specialize polynomial_zero_extended_entry_functional (ac)", "specialize polynomial_zero_extended_entry_functional (L)", "specialize polynomial_zero_extended_entry_functional (j)", "specialize polynomial_zero_extended_entry_functional (x1)", "specialize polynomial_zero_extended_entry_functional (x4)", "apply polynomial_zero_extended_entry_functional", "exact ht_witness_witness_witness_right_left", "exact hr_witness_witness_witness_right_left", "have hscale : exists pfa_offset_left_scalar_term_actual_right_scale pfa_offset_right_scalar_term_actual_right_scale. (k*x2) + (p) * pfa_offset_left_scalar_term_actual_right_scale = (x5) + (p) * pfa_offset_right_scalar_term_actual_right_scale", "specialize polynomial_zero_extended_scale_congruent (p)", "specialize polynomial_zero_extended_scale_congruent (k)", "specialize polynomial_zero_extended_scale_congruent (bb)", "specialize polynomial_zero_extended_scale_congruent (bc)", "specialize polynomial_zero_extended_scale_congruent (sb)", "specialize polynomial_zero_extended_scale_congruent (sc)", "specialize polynomial_zero_extended_scale_congruent (M)", "specialize polynomial_zero_extended_scale_congruent (x3)", "specialize polynomial_zero_extended_scale_congruent (x2)", "specialize polynomial_zero_extended_scale_congruent (x5)", "apply polynomial_zero_extended_scale_congruent", "exact hs", "exact ht_witness_witness_witness_right_right_left", "exact hr_witness_witness_witness_right_right_left", "have hm : exists pfa_offset_left_scalar_term_product_congruence pfa_offset_right_scalar_term_product_congruence. (x4*(k*x2)) + (p) * pfa_offset_left_scalar_term_product_congruence = (x4*x5) + (p) * pfa_offset_right_scalar_term_product_congruence", "specialize mod_eq_mul_left (p)", "specialize mod_eq_mul_left (k*x2)", "specialize mod_eq_mul_left (x5)", "specialize mod_eq_mul_left (x4)", "apply mod_eq_mul_left", "exact hscale", "have hshuffle : k*(x1*x2)=x4*(k*x2)", "rewrite hleft", "simp [mul_assoc,mul_comm]", "rewrite ht_witness_witness_witness_right_right_right", "rewrite hr_witness_witness_witness_right_right_right", "rewrite hshuffle", "exact hm" ], "script_sha256": "fc65c1ae468401182944f6e88947961b19d3217b9028c3aa331a584a705fbb96", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_scalar_convolution_candidate.py", "sha256": "e84f1c77c6c03fa5f08635aeede53591625d1c2bfcdfb64fbd379c33878aee0e" }, "statement": "forall p k ab ac L bb bc M sb sc i j t r. (((exists pfa_gap_scalar_term_operationscalar. pfa_gap_scalar_term_operationscalar + S (k) = (p)) /\\ ((forall pfp_index_scalar_term_operation. (exists pfa_gap_scalar_term_operationindex. pfa_gap_scalar_term_operationindex + S (pfp_index_scalar_term_operation) = (M)) -> exists pfp_source_scalar_term_operation pfp_value_scalar_term_operation. ((((exists ff_h_pfp_scalar_term_operationsource. ff_h_pfp_scalar_term_operationsource + S (pfp_source_scalar_term_operation) = S ((S (pfp_index_scalar_term_operation)) * bc)) /\\ exists ff_q_pfp_scalar_term_operationsource. bb = ff_q_pfp_scalar_term_operationsource * S ((S (pfp_index_scalar_term_operation)) * bc) + (pfp_source_scalar_term_operation))) /\\ (((((exists ff_h_pfp_scalar_term_operationtarget. ff_h_pfp_scalar_term_operationtarget + S (pfp_value_scalar_term_operation) = S ((S (pfp_index_scalar_term_operation)) * sc)) /\\ exists ff_q_pfp_scalar_term_operationtarget. sb = ff_q_pfp_scalar_term_operationtarget * S ((S (pfp_index_scalar_term_operation)) * sc) + (pfp_value_scalar_term_operation))) /\\ ((((exists pfa_gap_scalar_term_operationoperationleft. pfa_gap_scalar_term_operationoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_term_operationoperationright. pfa_gap_scalar_term_operationoperationright + S (pfp_source_scalar_term_operation) = (p)) /\\ ((((exists pfa_gap_scalar_term_operationoperationresultbound. pfa_gap_scalar_term_operationoperationresultbound + S (pfp_value_scalar_term_operation) = (p)) /\\ ((exists pfa_offset_left_scalar_term_operationoperationresultcongruence pfa_offset_right_scalar_term_operationoperationresultcongruence. ((k) * (pfp_source_scalar_term_operation)) + (p) * pfa_offset_left_scalar_term_operationoperationresultcongruence = (pfp_value_scalar_term_operation) + (p) * pfa_offset_right_scalar_term_operationoperationresultcongruence))))))))))))))))) -> (exists pfc_complement_scalar_term_original pfc_left_scalar_term_original pfc_right_scalar_term_original. (((j)+pfc_complement_scalar_term_original=(i)) /\\ ((((((exists pfa_gap_scalar_term_originalleftinside. pfa_gap_scalar_term_originalleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_scalar_term_originalleftentry. ff_h_pfp_scalar_term_originalleftentry + S (pfc_left_scalar_term_original) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_scalar_term_originalleftentry. ab = ff_q_pfp_scalar_term_originalleftentry * S ((S (j)) * ac) + (pfc_left_scalar_term_original)))))) \\/ (((exists pfc_gap_scalar_term_originalleftoutside. pfc_gap_scalar_term_originalleftoutside+(L)=(j)) /\\ (((pfc_left_scalar_term_original)=0))))) /\\ ((((((exists pfa_gap_scalar_term_originalrightinside. pfa_gap_scalar_term_originalrightinside + S (pfc_complement_scalar_term_original) = (M)) /\\ ((((exists ff_h_pfp_scalar_term_originalrightentry. ff_h_pfp_scalar_term_originalrightentry + S (pfc_right_scalar_term_original) = S ((S (pfc_complement_scalar_term_original)) * bc)) /\\ exists ff_q_pfp_scalar_term_originalrightentry. bb = ff_q_pfp_scalar_term_originalrightentry * S ((S (pfc_complement_scalar_term_original)) * bc) + (pfc_right_scalar_term_original)))))) \\/ (((exists pfc_gap_scalar_term_originalrightoutside. pfc_gap_scalar_term_originalrightoutside+(M)=(pfc_complement_scalar_term_original)) /\\ (((pfc_right_scalar_term_original)=0))))) /\\ (((t)=pfc_left_scalar_term_original*pfc_right_scalar_term_original)))))))) -> (exists pfc_complement_scalar_term_scaled pfc_left_scalar_term_scaled pfc_right_scalar_term_scaled. (((j)+pfc_complement_scalar_term_scaled=(i)) /\\ ((((((exists pfa_gap_scalar_term_scaledleftinside. pfa_gap_scalar_term_scaledleftinside + S (j) = (L)) /\\ ((((exists ff_h_pfp_scalar_term_scaledleftentry. ff_h_pfp_scalar_term_scaledleftentry + S (pfc_left_scalar_term_scaled) = S ((S (j)) * ac)) /\\ exists ff_q_pfp_scalar_term_scaledleftentry. ab = ff_q_pfp_scalar_term_scaledleftentry * S ((S (j)) * ac) + (pfc_left_scalar_term_scaled)))))) \\/ (((exists pfc_gap_scalar_term_scaledleftoutside. pfc_gap_scalar_term_scaledleftoutside+(L)=(j)) /\\ (((pfc_left_scalar_term_scaled)=0))))) /\\ ((((((exists pfa_gap_scalar_term_scaledrightinside. pfa_gap_scalar_term_scaledrightinside + S (pfc_complement_scalar_term_scaled) = (M)) /\\ ((((exists ff_h_pfp_scalar_term_scaledrightentry. ff_h_pfp_scalar_term_scaledrightentry + S (pfc_right_scalar_term_scaled) = S ((S (pfc_complement_scalar_term_scaled)) * sc)) /\\ exists ff_q_pfp_scalar_term_scaledrightentry. sb = ff_q_pfp_scalar_term_scaledrightentry * S ((S (pfc_complement_scalar_term_scaled)) * sc) + (pfc_right_scalar_term_scaled)))))) \\/ (((exists pfc_gap_scalar_term_scaledrightoutside. pfc_gap_scalar_term_scaledrightoutside+(M)=(pfc_complement_scalar_term_scaled)) /\\ (((pfc_right_scalar_term_scaled)=0))))) /\\ (((r)=pfc_left_scalar_term_scaled*pfc_right_scalar_term_scaled)))))))) -> (exists pfa_offset_left_scalar_term_result pfa_offset_right_scalar_term_result. (k*t) + (p) * pfa_offset_left_scalar_term_result = (r) + (p) * pfa_offset_right_scalar_term_result)", "statement_sha256": "8e1a0db8e0acf723d5f525f43092d1aecbec2921153ed8667f136b8ae561c25a", "summary": "The uniquely identified complementary index and unchanged left coefficient turn actual right-input scaling into pointwise antidiagonal scalar congruence.", "summary_sha256": "1f67036c345fff545e10a13fea06722df2e40413a995232d6f9dd6e18e141341" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "dad90387f2d0adc05424c1b61e1d95b1f313024719175836d8b6c9a4e21810e1", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 224, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": 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(((exists pfa_gap_scalar_diagonal_operationscalar. pfa_gap_scalar_diagonal_operationscalar + S (k) = (p)) /\\ ((forall pfp_index_scalar_diagonal_operation. (exists pfa_gap_scalar_diagonal_operationindex. pfa_gap_scalar_diagonal_operationindex + S (pfp_index_scalar_diagonal_operation) = (M)) -> exists pfp_source_scalar_diagonal_operation pfp_value_scalar_diagonal_operation. ((((exists ff_h_pfp_scalar_diagonal_operationsource. ff_h_pfp_scalar_diagonal_operationsource + S (pfp_source_scalar_diagonal_operation) = S ((S (pfp_index_scalar_diagonal_operation)) * bc)) /\\ exists ff_q_pfp_scalar_diagonal_operationsource. bb = ff_q_pfp_scalar_diagonal_operationsource * S ((S (pfp_index_scalar_diagonal_operation)) * bc) + (pfp_source_scalar_diagonal_operation))) /\\ (((((exists ff_h_pfp_scalar_diagonal_operationtarget. ff_h_pfp_scalar_diagonal_operationtarget + S (pfp_value_scalar_diagonal_operation) = S ((S (pfp_index_scalar_diagonal_operation)) * sc)) /\\ exists ff_q_pfp_scalar_diagonal_operationtarget. sb = ff_q_pfp_scalar_diagonal_operationtarget * S ((S (pfp_index_scalar_diagonal_operation)) * sc) + (pfp_value_scalar_diagonal_operation))) /\\ ((((exists pfa_gap_scalar_diagonal_operationoperationleft. pfa_gap_scalar_diagonal_operationoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_diagonal_operationoperationright. pfa_gap_scalar_diagonal_operationoperationright + S (pfp_source_scalar_diagonal_operation) = (p)) /\\ ((((exists pfa_gap_scalar_diagonal_operationoperationresultbound. pfa_gap_scalar_diagonal_operationoperationresultbound + S (pfp_value_scalar_diagonal_operation) = (p)) /\\ ((exists pfa_offset_left_scalar_diagonal_operationoperationresultcongruence pfa_offset_right_scalar_diagonal_operationoperationresultcongruence. ((k) * (pfp_source_scalar_diagonal_operation)) + (p) * pfa_offset_left_scalar_diagonal_operationoperationresultcongruence = (pfp_value_scalar_diagonal_operation) + (p) * pfa_offset_right_scalar_diagonal_operationoperationresultcongruence))))))))))))))))) -> (forall pfc_index_scalar_diagonal_old. (exists pfa_gap_scalar_diagonal_oldbound. pfa_gap_scalar_diagonal_oldbound + S (pfc_index_scalar_diagonal_old) = (N)) -> exists pfc_value_scalar_diagonal_old. ((((exists ff_h_pfp_scalar_diagonal_oldentry. ff_h_pfp_scalar_diagonal_oldentry + S (pfc_value_scalar_diagonal_old) = S ((S (pfc_index_scalar_diagonal_old)) * dc)) /\\ exists ff_q_pfp_scalar_diagonal_oldentry. db = ff_q_pfp_scalar_diagonal_oldentry * S ((S (pfc_index_scalar_diagonal_old)) * dc) + (pfc_value_scalar_diagonal_old))) /\\ ((exists pfc_complement_scalar_diagonal_oldterm pfc_left_scalar_diagonal_oldterm pfc_right_scalar_diagonal_oldterm. (((pfc_index_scalar_diagonal_old)+pfc_complement_scalar_diagonal_oldterm=(i)) /\\ ((((((exists pfa_gap_scalar_diagonal_oldtermleftinside. pfa_gap_scalar_diagonal_oldtermleftinside + S (pfc_index_scalar_diagonal_old) = (L)) /\\ ((((exists ff_h_pfp_scalar_diagonal_oldtermleftentry. ff_h_pfp_scalar_diagonal_oldtermleftentry + S (pfc_left_scalar_diagonal_oldterm) = S ((S (pfc_index_scalar_diagonal_old)) * ac)) /\\ exists ff_q_pfp_scalar_diagonal_oldtermleftentry. ab = ff_q_pfp_scalar_diagonal_oldtermleftentry * S ((S (pfc_index_scalar_diagonal_old)) * ac) + (pfc_left_scalar_diagonal_oldterm)))))) \\/ (((exists pfc_gap_scalar_diagonal_oldtermleftoutside. pfc_gap_scalar_diagonal_oldtermleftoutside+(L)=(pfc_index_scalar_diagonal_old)) /\\ (((pfc_left_scalar_diagonal_oldterm)=0))))) /\\ ((((((exists pfa_gap_scalar_diagonal_oldtermrightinside. pfa_gap_scalar_diagonal_oldtermrightinside + S (pfc_complement_scalar_diagonal_oldterm) = (M)) /\\ ((((exists ff_h_pfp_scalar_diagonal_oldtermrightentry. ff_h_pfp_scalar_diagonal_oldtermrightentry + S (pfc_right_scalar_diagonal_oldterm) = S ((S (pfc_complement_scalar_diagonal_oldterm)) * bc)) /\\ exists ff_q_pfp_scalar_diagonal_oldtermrightentry. bb = ff_q_pfp_scalar_diagonal_oldtermrightentry * S ((S (pfc_complement_scalar_diagonal_oldterm)) * bc) + (pfc_right_scalar_diagonal_oldterm)))))) \\/ (((exists pfc_gap_scalar_diagonal_oldtermrightoutside. pfc_gap_scalar_diagonal_oldtermrightoutside+(M)=(pfc_complement_scalar_diagonal_oldterm)) /\\ (((pfc_right_scalar_diagonal_oldterm)=0))))) /\\ (((pfc_value_scalar_diagonal_old)=pfc_left_scalar_diagonal_oldterm*pfc_right_scalar_diagonal_oldterm))))))))))) -> (exists fs_u_pfc_scalar_diagonal_old_sum fs_v_pfc_scalar_diagonal_old_sum. ((((exists fs_h_pfc_scalar_diagonal_old_sum_body_start. fs_h_pfc_scalar_diagonal_old_sum_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_diagonal_old_sum)) /\\ exists fs_q_pfc_scalar_diagonal_old_sum_body_start. fs_u_pfc_scalar_diagonal_old_sum = fs_q_pfc_scalar_diagonal_old_sum_body_start * S ((S (0)) * fs_v_pfc_scalar_diagonal_old_sum) + (0))) /\\ ((((exists fs_h_pfc_scalar_diagonal_old_sum_body_terminal. fs_h_pfc_scalar_diagonal_old_sum_body_terminal + S (u) = S ((S (N)) * fs_v_pfc_scalar_diagonal_old_sum)) /\\ exists fs_q_pfc_scalar_diagonal_old_sum_body_terminal. fs_u_pfc_scalar_diagonal_old_sum = fs_q_pfc_scalar_diagonal_old_sum_body_terminal * S ((S (N)) * fs_v_pfc_scalar_diagonal_old_sum) + (u))) /\\ forall fs_i_pfc_scalar_diagonal_old_sum_body_steps. (exists fs_lt_pfc_scalar_diagonal_old_sum_body_steps_bound. fs_lt_pfc_scalar_diagonal_old_sum_body_steps_bound + S fs_i_pfc_scalar_diagonal_old_sum_body_steps = N) -> exists fs_a_pfc_scalar_diagonal_old_sum_body_steps fs_r_pfc_scalar_diagonal_old_sum_body_steps fs_s_pfc_scalar_diagonal_old_sum_body_steps. ((((exists fs_h_pfc_scalar_diagonal_old_sum_body_steps_summand. fs_h_pfc_scalar_diagonal_old_sum_body_steps_summand + S (fs_a_pfc_scalar_diagonal_old_sum_body_steps) = S ((S (fs_i_pfc_scalar_diagonal_old_sum_body_steps)) * dc)) /\\ exists fs_q_pfc_scalar_diagonal_old_sum_body_steps_summand. db = fs_q_pfc_scalar_diagonal_old_sum_body_steps_summand * S ((S (fs_i_pfc_scalar_diagonal_old_sum_body_steps)) * dc) + (fs_a_pfc_scalar_diagonal_old_sum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_diagonal_old_sum_body_steps_partial. fs_h_pfc_scalar_diagonal_old_sum_body_steps_partial + S (fs_r_pfc_scalar_diagonal_old_sum_body_steps) = S ((S (fs_i_pfc_scalar_diagonal_old_sum_body_steps)) * fs_v_pfc_scalar_diagonal_old_sum)) /\\ exists fs_q_pfc_scalar_diagonal_old_sum_body_steps_partial. fs_u_pfc_scalar_diagonal_old_sum = fs_q_pfc_scalar_diagonal_old_sum_body_steps_partial * S ((S (fs_i_pfc_scalar_diagonal_old_sum_body_steps)) * fs_v_pfc_scalar_diagonal_old_sum) + (fs_r_pfc_scalar_diagonal_old_sum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_diagonal_old_sum_body_steps_successor. fs_h_pfc_scalar_diagonal_old_sum_body_steps_successor + S (fs_s_pfc_scalar_diagonal_old_sum_body_steps) = S ((S (S fs_i_pfc_scalar_diagonal_old_sum_body_steps)) * fs_v_pfc_scalar_diagonal_old_sum)) /\\ exists fs_q_pfc_scalar_diagonal_old_sum_body_steps_successor. fs_u_pfc_scalar_diagonal_old_sum = fs_q_pfc_scalar_diagonal_old_sum_body_steps_successor * S ((S (S fs_i_pfc_scalar_diagonal_old_sum_body_steps)) * fs_v_pfc_scalar_diagonal_old_sum) + (fs_s_pfc_scalar_diagonal_old_sum_body_steps))) /\\ fs_s_pfc_scalar_diagonal_old_sum_body_steps = fs_r_pfc_scalar_diagonal_old_sum_body_steps + fs_a_pfc_scalar_diagonal_old_sum_body_steps)))))) -> (forall pfc_index_scalar_diagonal_new. (exists pfa_gap_scalar_diagonal_newbound. pfa_gap_scalar_diagonal_newbound + S (pfc_index_scalar_diagonal_new) = (N)) -> exists pfc_value_scalar_diagonal_new. ((((exists ff_h_pfp_scalar_diagonal_newentry. ff_h_pfp_scalar_diagonal_newentry + S (pfc_value_scalar_diagonal_new) = S ((S (pfc_index_scalar_diagonal_new)) * ec)) /\\ exists ff_q_pfp_scalar_diagonal_newentry. eb = ff_q_pfp_scalar_diagonal_newentry * S ((S (pfc_index_scalar_diagonal_new)) * ec) + (pfc_value_scalar_diagonal_new))) /\\ ((exists pfc_complement_scalar_diagonal_newterm pfc_left_scalar_diagonal_newterm pfc_right_scalar_diagonal_newterm. (((pfc_index_scalar_diagonal_new)+pfc_complement_scalar_diagonal_newterm=(i)) /\\ ((((((exists pfa_gap_scalar_diagonal_newtermleftinside. pfa_gap_scalar_diagonal_newtermleftinside + S (pfc_index_scalar_diagonal_new) = (L)) /\\ ((((exists ff_h_pfp_scalar_diagonal_newtermleftentry. ff_h_pfp_scalar_diagonal_newtermleftentry + S (pfc_left_scalar_diagonal_newterm) = S ((S (pfc_index_scalar_diagonal_new)) * ac)) /\\ exists ff_q_pfp_scalar_diagonal_newtermleftentry. ab = ff_q_pfp_scalar_diagonal_newtermleftentry * S ((S (pfc_index_scalar_diagonal_new)) * ac) + (pfc_left_scalar_diagonal_newterm)))))) \\/ (((exists pfc_gap_scalar_diagonal_newtermleftoutside. pfc_gap_scalar_diagonal_newtermleftoutside+(L)=(pfc_index_scalar_diagonal_new)) /\\ (((pfc_left_scalar_diagonal_newterm)=0))))) /\\ ((((((exists pfa_gap_scalar_diagonal_newtermrightinside. pfa_gap_scalar_diagonal_newtermrightinside + S (pfc_complement_scalar_diagonal_newterm) = (M)) /\\ ((((exists ff_h_pfp_scalar_diagonal_newtermrightentry. ff_h_pfp_scalar_diagonal_newtermrightentry + S (pfc_right_scalar_diagonal_newterm) = S ((S (pfc_complement_scalar_diagonal_newterm)) * sc)) /\\ exists ff_q_pfp_scalar_diagonal_newtermrightentry. sb = ff_q_pfp_scalar_diagonal_newtermrightentry * S ((S (pfc_complement_scalar_diagonal_newterm)) * sc) + (pfc_right_scalar_diagonal_newterm)))))) \\/ (((exists pfc_gap_scalar_diagonal_newtermrightoutside. pfc_gap_scalar_diagonal_newtermrightoutside+(M)=(pfc_complement_scalar_diagonal_newterm)) /\\ (((pfc_right_scalar_diagonal_newterm)=0))))) /\\ (((pfc_value_scalar_diagonal_new)=pfc_left_scalar_diagonal_newterm*pfc_right_scalar_diagonal_newterm))))))))))) -> (exists fs_u_pfc_scalar_diagonal_new_sum fs_v_pfc_scalar_diagonal_new_sum. ((((exists fs_h_pfc_scalar_diagonal_new_sum_body_start. fs_h_pfc_scalar_diagonal_new_sum_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_diagonal_new_sum)) /\\ exists fs_q_pfc_scalar_diagonal_new_sum_body_start. fs_u_pfc_scalar_diagonal_new_sum = fs_q_pfc_scalar_diagonal_new_sum_body_start * S ((S (0)) * fs_v_pfc_scalar_diagonal_new_sum) + (0))) /\\ ((((exists fs_h_pfc_scalar_diagonal_new_sum_body_terminal. fs_h_pfc_scalar_diagonal_new_sum_body_terminal + S (v) = S ((S (N)) * fs_v_pfc_scalar_diagonal_new_sum)) /\\ exists fs_q_pfc_scalar_diagonal_new_sum_body_terminal. fs_u_pfc_scalar_diagonal_new_sum = fs_q_pfc_scalar_diagonal_new_sum_body_terminal * S ((S (N)) * fs_v_pfc_scalar_diagonal_new_sum) + (v))) /\\ forall fs_i_pfc_scalar_diagonal_new_sum_body_steps. (exists fs_lt_pfc_scalar_diagonal_new_sum_body_steps_bound. fs_lt_pfc_scalar_diagonal_new_sum_body_steps_bound + S fs_i_pfc_scalar_diagonal_new_sum_body_steps = N) -> exists fs_a_pfc_scalar_diagonal_new_sum_body_steps fs_r_pfc_scalar_diagonal_new_sum_body_steps fs_s_pfc_scalar_diagonal_new_sum_body_steps. ((((exists fs_h_pfc_scalar_diagonal_new_sum_body_steps_summand. fs_h_pfc_scalar_diagonal_new_sum_body_steps_summand + S (fs_a_pfc_scalar_diagonal_new_sum_body_steps) = S ((S (fs_i_pfc_scalar_diagonal_new_sum_body_steps)) * ec)) /\\ exists fs_q_pfc_scalar_diagonal_new_sum_body_steps_summand. eb = fs_q_pfc_scalar_diagonal_new_sum_body_steps_summand * S ((S (fs_i_pfc_scalar_diagonal_new_sum_body_steps)) * ec) + (fs_a_pfc_scalar_diagonal_new_sum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_diagonal_new_sum_body_steps_partial. fs_h_pfc_scalar_diagonal_new_sum_body_steps_partial + S (fs_r_pfc_scalar_diagonal_new_sum_body_steps) = S ((S (fs_i_pfc_scalar_diagonal_new_sum_body_steps)) * fs_v_pfc_scalar_diagonal_new_sum)) /\\ exists fs_q_pfc_scalar_diagonal_new_sum_body_steps_partial. fs_u_pfc_scalar_diagonal_new_sum = fs_q_pfc_scalar_diagonal_new_sum_body_steps_partial * S ((S (fs_i_pfc_scalar_diagonal_new_sum_body_steps)) * fs_v_pfc_scalar_diagonal_new_sum) + (fs_r_pfc_scalar_diagonal_new_sum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_diagonal_new_sum_body_steps_successor. fs_h_pfc_scalar_diagonal_new_sum_body_steps_successor + S (fs_s_pfc_scalar_diagonal_new_sum_body_steps) = S ((S (S fs_i_pfc_scalar_diagonal_new_sum_body_steps)) * fs_v_pfc_scalar_diagonal_new_sum)) /\\ exists fs_q_pfc_scalar_diagonal_new_sum_body_steps_successor. fs_u_pfc_scalar_diagonal_new_sum = fs_q_pfc_scalar_diagonal_new_sum_body_steps_successor * S ((S (S fs_i_pfc_scalar_diagonal_new_sum_body_steps)) * fs_v_pfc_scalar_diagonal_new_sum) + (fs_s_pfc_scalar_diagonal_new_sum_body_steps))) /\\ fs_s_pfc_scalar_diagonal_new_sum_body_steps = fs_r_pfc_scalar_diagonal_new_sum_body_steps + fs_a_pfc_scalar_diagonal_new_sum_body_steps)))))) -> (exists pfa_offset_left_scalar_diagonal_result pfa_offset_right_scalar_diagonal_result. (k*u) + (p) * pfa_offset_left_scalar_diagonal_result = (v) + (p) * pfa_offset_right_scalar_diagonal_result)", "statement_sha256": "d626f52fd37a69ed75ab3f857caec8dec4a7824cc1833345b9ef3a7550332137", "summary": "Two actual antidiagonal tables and their actual natural sums satisfy scalar congruence; no supplied output identity or Fubini oracle is used.", "summary_sha256": "3df6efcc1c21dd79d63c55433c6f3ab016ff0b4e74ca4304379532d891e18b21" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "4bf438fd994f87921efcf7e96b5d5b7611eba370127abc866fd04d953016ed8c", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 225, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "e84f1c77c6c03fa5f08635aeede53591625d1c2bfcdfb64fbd379c33878aee0e", "test_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85" }, "body_checked": true, "body_receipt": { "command_count": 84, "dependency_count": 4, "dne_command_count": 0, "name": "prime_field_convolution_coefficient_right_scale", "proof_depth": 56, "proof_edges": 112, "proof_nodes": 113, "proof_objects": 113, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "polynomial_diagonal_sum_right_scale_congruent", "mod_eq_trans", "mod_eq_symm", "mod_eq_mul_left" ], "dependencies_sha256": "818ef11356ee9ca8dd6854ea2fd3564fb230cbd666c810230de25b3c84694097", "empty_context_closure": { "body_proof_depth": 56, "body_proof_nodes": 113, "bundle_campaign": "polynomial-gcd-bezout", "bundle_dependency_edge_count": 1578, "bundle_node_count": 493, "bundle_node_id": 225, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "bundle_root_id": 492, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "773621c979a21853a6166127334031545c60a0df4cbbfe9d2a4c3eb88dc92e70", "status": "checked" }, "enrollment_index": 4111, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "e84f1c77c6c03fa5f08635aeede53591625d1c2bfcdfb64fbd379c33878aee0e", "kind": "alpha_v34_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_scalar_convolution_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85", "kind": "alpha_v34_frontier_executable_audit", "path": "peano-lab/py/tests/test_campaign_research_v34_closure.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "kind": "alpha_v34_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-field-polynomial-gcd-bezout-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "kind": "alpha_v34_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=225]" }, { "document_sha256": "81943a545e7b9f667ad758c24de14bb47280e8cf3fbb3ca714efec55fb096408", "kind": "alpha_v34_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "c451e101a0acb014051f904b6b71d162409191c02d50b77595d5e7d16ac1bec1", "membership": "alpha_only", "name": "prime_field_convolution_coefficient_right_scale", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro k", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro M", "intro sb", "intro sc", "intro i", "intro c", "intro r", "intro hs", "intro hc", "intro hr", "cases hc", "cases hc_witness", "cases hc_witness_witness", "cases hc_witness_witness_witness", "cases hc_witness_witness_witness_right", "cases hr", "cases hr_witness", "cases hr_witness_witness", "cases hr_witness_witness_witness", "cases hr_witness_witness_witness_right", "have hsum : exists pfa_offset_left_scalar_coefficient_actual_sum pfa_offset_right_scalar_coefficient_actual_sum. (k*x2) + (p) * pfa_offset_left_scalar_coefficient_actual_sum = (x5) + (p) * pfa_offset_right_scalar_coefficient_actual_sum", "specialize polynomial_diagonal_sum_right_scale_congruent (p)", "specialize polynomial_diagonal_sum_right_scale_congruent (k)", "specialize polynomial_diagonal_sum_right_scale_congruent (ab)", "specialize polynomial_diagonal_sum_right_scale_congruent (ac)", "specialize polynomial_diagonal_sum_right_scale_congruent (L)", "specialize polynomial_diagonal_sum_right_scale_congruent (bb)", "specialize polynomial_diagonal_sum_right_scale_congruent (bc)", "specialize polynomial_diagonal_sum_right_scale_congruent (M)", "specialize polynomial_diagonal_sum_right_scale_congruent (sb)", "specialize polynomial_diagonal_sum_right_scale_congruent (sc)", "specialize polynomial_diagonal_sum_right_scale_congruent (i)", "specialize polynomial_diagonal_sum_right_scale_congruent (x)", "specialize polynomial_diagonal_sum_right_scale_congruent (x1)", "specialize polynomial_diagonal_sum_right_scale_congruent (x3)", "specialize polynomial_diagonal_sum_right_scale_congruent (x4)", "specialize polynomial_diagonal_sum_right_scale_congruent (S i)", "specialize polynomial_diagonal_sum_right_scale_congruent (x2)", "specialize polynomial_diagonal_sum_right_scale_congruent (x5)", "apply polynomial_diagonal_sum_right_scale_congruent", "exact hs", "exact hc_witness_witness_witness_left", "exact hc_witness_witness_witness_right_left", "exact hr_witness_witness_witness_left", "exact hr_witness_witness_witness_right_left", "cases hs", "cases hc_witness_witness_witness_right_right", "cases hr_witness_witness_witness_right_right", "have htail : exists pfa_offset_left_scalar_coefficient_tail pfa_offset_right_scalar_coefficient_tail. (k*x2) + (p) * pfa_offset_left_scalar_coefficient_tail = (r) + (p) * pfa_offset_right_scalar_coefficient_tail", "specialize mod_eq_trans (p)", "specialize mod_eq_trans (k*x2)", "specialize mod_eq_trans (x5)", "specialize mod_eq_trans (r)", "apply mod_eq_trans", "exact hsum", "exact hr_witness_witness_witness_right_right_right", "split", "exact hs_left", "split", "exact hc_witness_witness_witness_right_right_left", "split", "exact hr_witness_witness_witness_right_right_left", "specialize mod_eq_trans (p)", "specialize mod_eq_trans (k*c)", "specialize mod_eq_trans (k*x2)", "specialize mod_eq_trans (r)", "apply mod_eq_trans", "specialize mod_eq_symm (p)", "specialize mod_eq_symm (k*x2)", "specialize mod_eq_symm (k*c)", "apply mod_eq_symm", "specialize mod_eq_mul_left (p)", "specialize mod_eq_mul_left (x2)", "specialize mod_eq_mul_left (c)", "specialize mod_eq_mul_left (k)", "apply mod_eq_mul_left", "exact hc_witness_witness_witness_right_right_right", "exact htail" ], "script_sha256": "d699db95813a981019175b87a6cebd4fcc8986a34f32d1002dd915093e3166c7", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_scalar_convolution_candidate.py", "sha256": "e84f1c77c6c03fa5f08635aeede53591625d1c2bfcdfb64fbd379c33878aee0e" }, "statement": "forall p k ab ac L bb bc M sb sc i c r. (((exists pfa_gap_scalar_coefficient_operationscalar. pfa_gap_scalar_coefficient_operationscalar + S (k) = (p)) /\\ ((forall pfp_index_scalar_coefficient_operation. (exists pfa_gap_scalar_coefficient_operationindex. pfa_gap_scalar_coefficient_operationindex + S (pfp_index_scalar_coefficient_operation) = (M)) -> exists pfp_source_scalar_coefficient_operation pfp_value_scalar_coefficient_operation. ((((exists ff_h_pfp_scalar_coefficient_operationsource. ff_h_pfp_scalar_coefficient_operationsource + S (pfp_source_scalar_coefficient_operation) = S ((S (pfp_index_scalar_coefficient_operation)) * bc)) /\\ exists ff_q_pfp_scalar_coefficient_operationsource. bb = ff_q_pfp_scalar_coefficient_operationsource * S ((S (pfp_index_scalar_coefficient_operation)) * bc) + (pfp_source_scalar_coefficient_operation))) /\\ (((((exists ff_h_pfp_scalar_coefficient_operationtarget. ff_h_pfp_scalar_coefficient_operationtarget + S (pfp_value_scalar_coefficient_operation) = S ((S (pfp_index_scalar_coefficient_operation)) * sc)) /\\ exists ff_q_pfp_scalar_coefficient_operationtarget. sb = ff_q_pfp_scalar_coefficient_operationtarget * S ((S (pfp_index_scalar_coefficient_operation)) * sc) + (pfp_value_scalar_coefficient_operation))) /\\ ((((exists pfa_gap_scalar_coefficient_operationoperationleft. pfa_gap_scalar_coefficient_operationoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_coefficient_operationoperationright. pfa_gap_scalar_coefficient_operationoperationright + S (pfp_source_scalar_coefficient_operation) = (p)) /\\ ((((exists pfa_gap_scalar_coefficient_operationoperationresultbound. pfa_gap_scalar_coefficient_operationoperationresultbound + S (pfp_value_scalar_coefficient_operation) = (p)) /\\ ((exists pfa_offset_left_scalar_coefficient_operationoperationresultcongruence pfa_offset_right_scalar_coefficient_operationoperationresultcongruence. ((k) * (pfp_source_scalar_coefficient_operation)) + (p) * pfa_offset_left_scalar_coefficient_operationoperationresultcongruence = (pfp_value_scalar_coefficient_operation) + (p) * pfa_offset_right_scalar_coefficient_operationoperationresultcongruence))))))))))))))))) -> (exists pfc_terms_code_scalar_coefficient_original pfc_terms_scale_scalar_coefficient_original pfc_natural_sum_scalar_coefficient_original. ((forall pfc_index_scalar_coefficient_originaldiagonal. (exists pfa_gap_scalar_coefficient_originaldiagonalbound. pfa_gap_scalar_coefficient_originaldiagonalbound + S (pfc_index_scalar_coefficient_originaldiagonal) = (S (i))) -> exists pfc_value_scalar_coefficient_originaldiagonal. ((((exists ff_h_pfp_scalar_coefficient_originaldiagonalentry. ff_h_pfp_scalar_coefficient_originaldiagonalentry + S (pfc_value_scalar_coefficient_originaldiagonal) = S ((S (pfc_index_scalar_coefficient_originaldiagonal)) * pfc_terms_scale_scalar_coefficient_original)) /\\ exists ff_q_pfp_scalar_coefficient_originaldiagonalentry. pfc_terms_code_scalar_coefficient_original = ff_q_pfp_scalar_coefficient_originaldiagonalentry * S ((S (pfc_index_scalar_coefficient_originaldiagonal)) * pfc_terms_scale_scalar_coefficient_original) + (pfc_value_scalar_coefficient_originaldiagonal))) /\\ ((exists pfc_complement_scalar_coefficient_originaldiagonalterm pfc_left_scalar_coefficient_originaldiagonalterm pfc_right_scalar_coefficient_originaldiagonalterm. (((pfc_index_scalar_coefficient_originaldiagonal)+pfc_complement_scalar_coefficient_originaldiagonalterm=(i)) /\\ ((((((exists pfa_gap_scalar_coefficient_originaldiagonaltermleftinside. pfa_gap_scalar_coefficient_originaldiagonaltermleftinside + S (pfc_index_scalar_coefficient_originaldiagonal) = (L)) /\\ ((((exists ff_h_pfp_scalar_coefficient_originaldiagonaltermleftentry. ff_h_pfp_scalar_coefficient_originaldiagonaltermleftentry + S (pfc_left_scalar_coefficient_originaldiagonalterm) = S ((S (pfc_index_scalar_coefficient_originaldiagonal)) * ac)) /\\ exists ff_q_pfp_scalar_coefficient_originaldiagonaltermleftentry. ab = ff_q_pfp_scalar_coefficient_originaldiagonaltermleftentry * S ((S (pfc_index_scalar_coefficient_originaldiagonal)) * ac) + (pfc_left_scalar_coefficient_originaldiagonalterm)))))) \\/ (((exists pfc_gap_scalar_coefficient_originaldiagonaltermleftoutside. pfc_gap_scalar_coefficient_originaldiagonaltermleftoutside+(L)=(pfc_index_scalar_coefficient_originaldiagonal)) /\\ (((pfc_left_scalar_coefficient_originaldiagonalterm)=0))))) /\\ ((((((exists pfa_gap_scalar_coefficient_originaldiagonaltermrightinside. pfa_gap_scalar_coefficient_originaldiagonaltermrightinside + S (pfc_complement_scalar_coefficient_originaldiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_scalar_coefficient_originaldiagonaltermrightentry. ff_h_pfp_scalar_coefficient_originaldiagonaltermrightentry + S (pfc_right_scalar_coefficient_originaldiagonalterm) = S ((S (pfc_complement_scalar_coefficient_originaldiagonalterm)) * bc)) /\\ exists ff_q_pfp_scalar_coefficient_originaldiagonaltermrightentry. bb = ff_q_pfp_scalar_coefficient_originaldiagonaltermrightentry * S ((S (pfc_complement_scalar_coefficient_originaldiagonalterm)) * bc) + (pfc_right_scalar_coefficient_originaldiagonalterm)))))) \\/ (((exists pfc_gap_scalar_coefficient_originaldiagonaltermrightoutside. pfc_gap_scalar_coefficient_originaldiagonaltermrightoutside+(M)=(pfc_complement_scalar_coefficient_originaldiagonalterm)) /\\ (((pfc_right_scalar_coefficient_originaldiagonalterm)=0))))) /\\ (((pfc_value_scalar_coefficient_originaldiagonal)=pfc_left_scalar_coefficient_originaldiagonalterm*pfc_right_scalar_coefficient_originaldiagonalterm))))))))))) /\\ (((exists fs_u_pfc_scalar_coefficient_originalsum fs_v_pfc_scalar_coefficient_originalsum. ((((exists fs_h_pfc_scalar_coefficient_originalsum_body_start. fs_h_pfc_scalar_coefficient_originalsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_coefficient_originalsum)) /\\ exists fs_q_pfc_scalar_coefficient_originalsum_body_start. fs_u_pfc_scalar_coefficient_originalsum = fs_q_pfc_scalar_coefficient_originalsum_body_start * S ((S (0)) * fs_v_pfc_scalar_coefficient_originalsum) + (0))) /\\ ((((exists fs_h_pfc_scalar_coefficient_originalsum_body_terminal. fs_h_pfc_scalar_coefficient_originalsum_body_terminal + S (pfc_natural_sum_scalar_coefficient_original) = S ((S (S (i))) * fs_v_pfc_scalar_coefficient_originalsum)) /\\ exists fs_q_pfc_scalar_coefficient_originalsum_body_terminal. fs_u_pfc_scalar_coefficient_originalsum = fs_q_pfc_scalar_coefficient_originalsum_body_terminal * S ((S (S (i))) * fs_v_pfc_scalar_coefficient_originalsum) + (pfc_natural_sum_scalar_coefficient_original))) /\\ forall fs_i_pfc_scalar_coefficient_originalsum_body_steps. (exists fs_lt_pfc_scalar_coefficient_originalsum_body_steps_bound. fs_lt_pfc_scalar_coefficient_originalsum_body_steps_bound + S fs_i_pfc_scalar_coefficient_originalsum_body_steps = S (i)) -> exists fs_a_pfc_scalar_coefficient_originalsum_body_steps fs_r_pfc_scalar_coefficient_originalsum_body_steps fs_s_pfc_scalar_coefficient_originalsum_body_steps. ((((exists fs_h_pfc_scalar_coefficient_originalsum_body_steps_summand. fs_h_pfc_scalar_coefficient_originalsum_body_steps_summand + S (fs_a_pfc_scalar_coefficient_originalsum_body_steps) = S ((S (fs_i_pfc_scalar_coefficient_originalsum_body_steps)) * pfc_terms_scale_scalar_coefficient_original)) /\\ exists fs_q_pfc_scalar_coefficient_originalsum_body_steps_summand. pfc_terms_code_scalar_coefficient_original = fs_q_pfc_scalar_coefficient_originalsum_body_steps_summand * S ((S (fs_i_pfc_scalar_coefficient_originalsum_body_steps)) * pfc_terms_scale_scalar_coefficient_original) + (fs_a_pfc_scalar_coefficient_originalsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_coefficient_originalsum_body_steps_partial. fs_h_pfc_scalar_coefficient_originalsum_body_steps_partial + S (fs_r_pfc_scalar_coefficient_originalsum_body_steps) = S ((S (fs_i_pfc_scalar_coefficient_originalsum_body_steps)) * fs_v_pfc_scalar_coefficient_originalsum)) /\\ exists fs_q_pfc_scalar_coefficient_originalsum_body_steps_partial. fs_u_pfc_scalar_coefficient_originalsum = fs_q_pfc_scalar_coefficient_originalsum_body_steps_partial * S ((S (fs_i_pfc_scalar_coefficient_originalsum_body_steps)) * fs_v_pfc_scalar_coefficient_originalsum) + (fs_r_pfc_scalar_coefficient_originalsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_coefficient_originalsum_body_steps_successor. fs_h_pfc_scalar_coefficient_originalsum_body_steps_successor + S (fs_s_pfc_scalar_coefficient_originalsum_body_steps) = S ((S (S fs_i_pfc_scalar_coefficient_originalsum_body_steps)) * fs_v_pfc_scalar_coefficient_originalsum)) /\\ exists fs_q_pfc_scalar_coefficient_originalsum_body_steps_successor. fs_u_pfc_scalar_coefficient_originalsum = fs_q_pfc_scalar_coefficient_originalsum_body_steps_successor * S ((S (S fs_i_pfc_scalar_coefficient_originalsum_body_steps)) * fs_v_pfc_scalar_coefficient_originalsum) + (fs_s_pfc_scalar_coefficient_originalsum_body_steps))) /\\ fs_s_pfc_scalar_coefficient_originalsum_body_steps = fs_r_pfc_scalar_coefficient_originalsum_body_steps + fs_a_pfc_scalar_coefficient_originalsum_body_steps)))))) /\\ ((((exists pfa_gap_scalar_coefficient_originalresiduebound. pfa_gap_scalar_coefficient_originalresiduebound + S (c) = (p)) /\\ ((exists pfa_offset_left_scalar_coefficient_originalresiduecongruence pfa_offset_right_scalar_coefficient_originalresiduecongruence. (pfc_natural_sum_scalar_coefficient_original) + (p) * pfa_offset_left_scalar_coefficient_originalresiduecongruence = (c) + (p) * pfa_offset_right_scalar_coefficient_originalresiduecongruence))))))))) -> (exists pfc_terms_code_scalar_coefficient_scaled pfc_terms_scale_scalar_coefficient_scaled pfc_natural_sum_scalar_coefficient_scaled. ((forall pfc_index_scalar_coefficient_scaleddiagonal. (exists pfa_gap_scalar_coefficient_scaleddiagonalbound. pfa_gap_scalar_coefficient_scaleddiagonalbound + S (pfc_index_scalar_coefficient_scaleddiagonal) = (S (i))) -> exists pfc_value_scalar_coefficient_scaleddiagonal. ((((exists ff_h_pfp_scalar_coefficient_scaleddiagonalentry. ff_h_pfp_scalar_coefficient_scaleddiagonalentry + S (pfc_value_scalar_coefficient_scaleddiagonal) = S ((S (pfc_index_scalar_coefficient_scaleddiagonal)) * pfc_terms_scale_scalar_coefficient_scaled)) /\\ exists ff_q_pfp_scalar_coefficient_scaleddiagonalentry. pfc_terms_code_scalar_coefficient_scaled = ff_q_pfp_scalar_coefficient_scaleddiagonalentry * S ((S (pfc_index_scalar_coefficient_scaleddiagonal)) * pfc_terms_scale_scalar_coefficient_scaled) + (pfc_value_scalar_coefficient_scaleddiagonal))) /\\ ((exists pfc_complement_scalar_coefficient_scaleddiagonalterm pfc_left_scalar_coefficient_scaleddiagonalterm pfc_right_scalar_coefficient_scaleddiagonalterm. (((pfc_index_scalar_coefficient_scaleddiagonal)+pfc_complement_scalar_coefficient_scaleddiagonalterm=(i)) /\\ ((((((exists pfa_gap_scalar_coefficient_scaleddiagonaltermleftinside. pfa_gap_scalar_coefficient_scaleddiagonaltermleftinside + S (pfc_index_scalar_coefficient_scaleddiagonal) = (L)) /\\ ((((exists ff_h_pfp_scalar_coefficient_scaleddiagonaltermleftentry. ff_h_pfp_scalar_coefficient_scaleddiagonaltermleftentry + S (pfc_left_scalar_coefficient_scaleddiagonalterm) = S ((S (pfc_index_scalar_coefficient_scaleddiagonal)) * ac)) /\\ exists ff_q_pfp_scalar_coefficient_scaleddiagonaltermleftentry. ab = ff_q_pfp_scalar_coefficient_scaleddiagonaltermleftentry * S ((S (pfc_index_scalar_coefficient_scaleddiagonal)) * ac) + (pfc_left_scalar_coefficient_scaleddiagonalterm)))))) \\/ (((exists pfc_gap_scalar_coefficient_scaleddiagonaltermleftoutside. pfc_gap_scalar_coefficient_scaleddiagonaltermleftoutside+(L)=(pfc_index_scalar_coefficient_scaleddiagonal)) /\\ (((pfc_left_scalar_coefficient_scaleddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_scalar_coefficient_scaleddiagonaltermrightinside. pfa_gap_scalar_coefficient_scaleddiagonaltermrightinside + S (pfc_complement_scalar_coefficient_scaleddiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_scalar_coefficient_scaleddiagonaltermrightentry. ff_h_pfp_scalar_coefficient_scaleddiagonaltermrightentry + S (pfc_right_scalar_coefficient_scaleddiagonalterm) = S ((S (pfc_complement_scalar_coefficient_scaleddiagonalterm)) * sc)) /\\ exists ff_q_pfp_scalar_coefficient_scaleddiagonaltermrightentry. sb = ff_q_pfp_scalar_coefficient_scaleddiagonaltermrightentry * S ((S (pfc_complement_scalar_coefficient_scaleddiagonalterm)) * sc) + (pfc_right_scalar_coefficient_scaleddiagonalterm)))))) \\/ (((exists pfc_gap_scalar_coefficient_scaleddiagonaltermrightoutside. pfc_gap_scalar_coefficient_scaleddiagonaltermrightoutside+(M)=(pfc_complement_scalar_coefficient_scaleddiagonalterm)) /\\ (((pfc_right_scalar_coefficient_scaleddiagonalterm)=0))))) /\\ (((pfc_value_scalar_coefficient_scaleddiagonal)=pfc_left_scalar_coefficient_scaleddiagonalterm*pfc_right_scalar_coefficient_scaleddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_scalar_coefficient_scaledsum fs_v_pfc_scalar_coefficient_scaledsum. ((((exists fs_h_pfc_scalar_coefficient_scaledsum_body_start. fs_h_pfc_scalar_coefficient_scaledsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_coefficient_scaledsum)) /\\ exists fs_q_pfc_scalar_coefficient_scaledsum_body_start. fs_u_pfc_scalar_coefficient_scaledsum = fs_q_pfc_scalar_coefficient_scaledsum_body_start * S ((S (0)) * fs_v_pfc_scalar_coefficient_scaledsum) + (0))) /\\ ((((exists fs_h_pfc_scalar_coefficient_scaledsum_body_terminal. fs_h_pfc_scalar_coefficient_scaledsum_body_terminal + S (pfc_natural_sum_scalar_coefficient_scaled) = S ((S (S (i))) * fs_v_pfc_scalar_coefficient_scaledsum)) /\\ exists fs_q_pfc_scalar_coefficient_scaledsum_body_terminal. fs_u_pfc_scalar_coefficient_scaledsum = fs_q_pfc_scalar_coefficient_scaledsum_body_terminal * S ((S (S (i))) * fs_v_pfc_scalar_coefficient_scaledsum) + (pfc_natural_sum_scalar_coefficient_scaled))) /\\ forall fs_i_pfc_scalar_coefficient_scaledsum_body_steps. (exists fs_lt_pfc_scalar_coefficient_scaledsum_body_steps_bound. fs_lt_pfc_scalar_coefficient_scaledsum_body_steps_bound + S fs_i_pfc_scalar_coefficient_scaledsum_body_steps = S (i)) -> exists fs_a_pfc_scalar_coefficient_scaledsum_body_steps fs_r_pfc_scalar_coefficient_scaledsum_body_steps fs_s_pfc_scalar_coefficient_scaledsum_body_steps. ((((exists fs_h_pfc_scalar_coefficient_scaledsum_body_steps_summand. fs_h_pfc_scalar_coefficient_scaledsum_body_steps_summand + S (fs_a_pfc_scalar_coefficient_scaledsum_body_steps) = S ((S (fs_i_pfc_scalar_coefficient_scaledsum_body_steps)) * pfc_terms_scale_scalar_coefficient_scaled)) /\\ exists fs_q_pfc_scalar_coefficient_scaledsum_body_steps_summand. pfc_terms_code_scalar_coefficient_scaled = fs_q_pfc_scalar_coefficient_scaledsum_body_steps_summand * S ((S (fs_i_pfc_scalar_coefficient_scaledsum_body_steps)) * pfc_terms_scale_scalar_coefficient_scaled) + (fs_a_pfc_scalar_coefficient_scaledsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_coefficient_scaledsum_body_steps_partial. fs_h_pfc_scalar_coefficient_scaledsum_body_steps_partial + S (fs_r_pfc_scalar_coefficient_scaledsum_body_steps) = S ((S (fs_i_pfc_scalar_coefficient_scaledsum_body_steps)) * fs_v_pfc_scalar_coefficient_scaledsum)) /\\ exists fs_q_pfc_scalar_coefficient_scaledsum_body_steps_partial. fs_u_pfc_scalar_coefficient_scaledsum = fs_q_pfc_scalar_coefficient_scaledsum_body_steps_partial * S ((S (fs_i_pfc_scalar_coefficient_scaledsum_body_steps)) * fs_v_pfc_scalar_coefficient_scaledsum) + (fs_r_pfc_scalar_coefficient_scaledsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_coefficient_scaledsum_body_steps_successor. fs_h_pfc_scalar_coefficient_scaledsum_body_steps_successor + S (fs_s_pfc_scalar_coefficient_scaledsum_body_steps) = S ((S (S fs_i_pfc_scalar_coefficient_scaledsum_body_steps)) * fs_v_pfc_scalar_coefficient_scaledsum)) /\\ exists fs_q_pfc_scalar_coefficient_scaledsum_body_steps_successor. fs_u_pfc_scalar_coefficient_scaledsum = fs_q_pfc_scalar_coefficient_scaledsum_body_steps_successor * S ((S (S fs_i_pfc_scalar_coefficient_scaledsum_body_steps)) * fs_v_pfc_scalar_coefficient_scaledsum) + (fs_s_pfc_scalar_coefficient_scaledsum_body_steps))) /\\ fs_s_pfc_scalar_coefficient_scaledsum_body_steps = fs_r_pfc_scalar_coefficient_scaledsum_body_steps + fs_a_pfc_scalar_coefficient_scaledsum_body_steps)))))) /\\ ((((exists pfa_gap_scalar_coefficient_scaledresiduebound. pfa_gap_scalar_coefficient_scaledresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_scalar_coefficient_scaledresiduecongruence pfa_offset_right_scalar_coefficient_scaledresiduecongruence. (pfc_natural_sum_scalar_coefficient_scaled) + (p) * pfa_offset_left_scalar_coefficient_scaledresiduecongruence = (r) + (p) * pfa_offset_right_scalar_coefficient_scaledresiduecongruence))))))))) -> (((exists pfa_gap_scalar_coefficient_resultleft. pfa_gap_scalar_coefficient_resultleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_coefficient_resultright. pfa_gap_scalar_coefficient_resultright + S (c) = (p)) /\\ ((((exists pfa_gap_scalar_coefficient_resultresultbound. pfa_gap_scalar_coefficient_resultresultbound + S (r) = (p)) /\\ ((exists pfa_offset_left_scalar_coefficient_resultresultcongruence pfa_offset_right_scalar_coefficient_resultresultcongruence. 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"cases hc_right_right", "cases hd", "cases hd_right", "cases hd_right_right", "have hk : K=N", "specialize polynomial_product_length_functional (L)", "specialize polynomial_product_length_functional (M)", "specialize polynomial_product_length_functional (K)", "specialize polynomial_product_length_functional (N)", "apply polynomial_product_length_functional", "exact hd_right_right_left", "exact hc_right_right_left", "split", "exact hk", "have hcopy : ((exists pfa_gap_scalar_product_scale_copyscalar. pfa_gap_scalar_product_scale_copyscalar + S (k) = (p)) /\\ ((forall pfp_index_scalar_product_scale_copy. (exists pfa_gap_scalar_product_scale_copyindex. pfa_gap_scalar_product_scale_copyindex + S (pfp_index_scalar_product_scale_copy) = (M)) -> exists pfp_source_scalar_product_scale_copy pfp_value_scalar_product_scale_copy. ((((exists ff_h_pfp_scalar_product_scale_copysource. ff_h_pfp_scalar_product_scale_copysource + S (pfp_source_scalar_product_scale_copy) = S ((S (pfp_index_scalar_product_scale_copy)) * bc)) /\\ exists ff_q_pfp_scalar_product_scale_copysource. bb = ff_q_pfp_scalar_product_scale_copysource * S ((S (pfp_index_scalar_product_scale_copy)) * bc) + (pfp_source_scalar_product_scale_copy))) /\\ (((((exists ff_h_pfp_scalar_product_scale_copytarget. ff_h_pfp_scalar_product_scale_copytarget + S (pfp_value_scalar_product_scale_copy) = S ((S (pfp_index_scalar_product_scale_copy)) * sc)) /\\ exists ff_q_pfp_scalar_product_scale_copytarget. sb = ff_q_pfp_scalar_product_scale_copytarget * S ((S (pfp_index_scalar_product_scale_copy)) * sc) + (pfp_value_scalar_product_scale_copy))) /\\ ((((exists pfa_gap_scalar_product_scale_copyoperationleft. pfa_gap_scalar_product_scale_copyoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_product_scale_copyoperationright. pfa_gap_scalar_product_scale_copyoperationright + S (pfp_source_scalar_product_scale_copy) = (p)) /\\ ((((exists pfa_gap_scalar_product_scale_copyoperationresultbound. pfa_gap_scalar_product_scale_copyoperationresultbound + S (pfp_value_scalar_product_scale_copy) = (p)) /\\ ((exists pfa_offset_left_scalar_product_scale_copyoperationresultcongruence pfa_offset_right_scalar_product_scale_copyoperationresultcongruence. ((k) * (pfp_source_scalar_product_scale_copy)) + (p) * pfa_offset_left_scalar_product_scale_copyoperationresultcongruence = (pfp_value_scalar_product_scale_copy) + (p) * pfa_offset_right_scalar_product_scale_copyoperationresultcongruence))))))))))))))))", "exact hs", "cases hcopy", "split", "exact hcopy_left", "intro i", "intro hi", "have ha : exists a. ((((exists ff_h_pfp_scalar_product_original_entry. ff_h_pfp_scalar_product_original_entry + S (a) = S ((S (i)) * cc)) /\\ exists ff_q_pfp_scalar_product_original_entry. cb = ff_q_pfp_scalar_product_original_entry * S ((S (i)) * cc) + (a))) /\\ ((exists pfc_terms_code_scalar_product_original_coefficient pfc_terms_scale_scalar_product_original_coefficient pfc_natural_sum_scalar_product_original_coefficient. ((forall pfc_index_scalar_product_original_coefficientdiagonal. (exists pfa_gap_scalar_product_original_coefficientdiagonalbound. pfa_gap_scalar_product_original_coefficientdiagonalbound + S (pfc_index_scalar_product_original_coefficientdiagonal) = (S (i))) -> exists pfc_value_scalar_product_original_coefficientdiagonal. ((((exists ff_h_pfp_scalar_product_original_coefficientdiagonalentry. ff_h_pfp_scalar_product_original_coefficientdiagonalentry + S (pfc_value_scalar_product_original_coefficientdiagonal) = S ((S (pfc_index_scalar_product_original_coefficientdiagonal)) * pfc_terms_scale_scalar_product_original_coefficient)) /\\ exists ff_q_pfp_scalar_product_original_coefficientdiagonalentry. pfc_terms_code_scalar_product_original_coefficient = ff_q_pfp_scalar_product_original_coefficientdiagonalentry * S ((S (pfc_index_scalar_product_original_coefficientdiagonal)) * pfc_terms_scale_scalar_product_original_coefficient) + (pfc_value_scalar_product_original_coefficientdiagonal))) /\\ ((exists pfc_complement_scalar_product_original_coefficientdiagonalterm pfc_left_scalar_product_original_coefficientdiagonalterm pfc_right_scalar_product_original_coefficientdiagonalterm. (((pfc_index_scalar_product_original_coefficientdiagonal)+pfc_complement_scalar_product_original_coefficientdiagonalterm=(i)) /\\ ((((((exists pfa_gap_scalar_product_original_coefficientdiagonaltermleftinside. pfa_gap_scalar_product_original_coefficientdiagonaltermleftinside + S (pfc_index_scalar_product_original_coefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_scalar_product_original_coefficientdiagonaltermleftentry. ff_h_pfp_scalar_product_original_coefficientdiagonaltermleftentry + S (pfc_left_scalar_product_original_coefficientdiagonalterm) = S ((S (pfc_index_scalar_product_original_coefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_scalar_product_original_coefficientdiagonaltermleftentry. ab = ff_q_pfp_scalar_product_original_coefficientdiagonaltermleftentry * S ((S (pfc_index_scalar_product_original_coefficientdiagonal)) * ac) + (pfc_left_scalar_product_original_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_product_original_coefficientdiagonaltermleftoutside. pfc_gap_scalar_product_original_coefficientdiagonaltermleftoutside+(L)=(pfc_index_scalar_product_original_coefficientdiagonal)) /\\ (((pfc_left_scalar_product_original_coefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_scalar_product_original_coefficientdiagonaltermrightinside. pfa_gap_scalar_product_original_coefficientdiagonaltermrightinside + S (pfc_complement_scalar_product_original_coefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_scalar_product_original_coefficientdiagonaltermrightentry. ff_h_pfp_scalar_product_original_coefficientdiagonaltermrightentry + S (pfc_right_scalar_product_original_coefficientdiagonalterm) = S ((S (pfc_complement_scalar_product_original_coefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_scalar_product_original_coefficientdiagonaltermrightentry. bb = ff_q_pfp_scalar_product_original_coefficientdiagonaltermrightentry * S ((S (pfc_complement_scalar_product_original_coefficientdiagonalterm)) * bc) + (pfc_right_scalar_product_original_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_product_original_coefficientdiagonaltermrightoutside. pfc_gap_scalar_product_original_coefficientdiagonaltermrightoutside+(M)=(pfc_complement_scalar_product_original_coefficientdiagonalterm)) /\\ (((pfc_right_scalar_product_original_coefficientdiagonalterm)=0))))) /\\ (((pfc_value_scalar_product_original_coefficientdiagonal)=pfc_left_scalar_product_original_coefficientdiagonalterm*pfc_right_scalar_product_original_coefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_scalar_product_original_coefficientsum fs_v_pfc_scalar_product_original_coefficientsum. ((((exists fs_h_pfc_scalar_product_original_coefficientsum_body_start. fs_h_pfc_scalar_product_original_coefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_product_original_coefficientsum)) /\\ exists fs_q_pfc_scalar_product_original_coefficientsum_body_start. fs_u_pfc_scalar_product_original_coefficientsum = fs_q_pfc_scalar_product_original_coefficientsum_body_start * S ((S (0)) * fs_v_pfc_scalar_product_original_coefficientsum) + (0))) /\\ ((((exists fs_h_pfc_scalar_product_original_coefficientsum_body_terminal. fs_h_pfc_scalar_product_original_coefficientsum_body_terminal + S (pfc_natural_sum_scalar_product_original_coefficient) = S ((S (S (i))) * fs_v_pfc_scalar_product_original_coefficientsum)) /\\ exists fs_q_pfc_scalar_product_original_coefficientsum_body_terminal. fs_u_pfc_scalar_product_original_coefficientsum = fs_q_pfc_scalar_product_original_coefficientsum_body_terminal * S ((S (S (i))) * fs_v_pfc_scalar_product_original_coefficientsum) + (pfc_natural_sum_scalar_product_original_coefficient))) /\\ forall fs_i_pfc_scalar_product_original_coefficientsum_body_steps. (exists fs_lt_pfc_scalar_product_original_coefficientsum_body_steps_bound. fs_lt_pfc_scalar_product_original_coefficientsum_body_steps_bound + S fs_i_pfc_scalar_product_original_coefficientsum_body_steps = S (i)) -> exists fs_a_pfc_scalar_product_original_coefficientsum_body_steps fs_r_pfc_scalar_product_original_coefficientsum_body_steps fs_s_pfc_scalar_product_original_coefficientsum_body_steps. ((((exists fs_h_pfc_scalar_product_original_coefficientsum_body_steps_summand. fs_h_pfc_scalar_product_original_coefficientsum_body_steps_summand + S (fs_a_pfc_scalar_product_original_coefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_product_original_coefficientsum_body_steps)) * pfc_terms_scale_scalar_product_original_coefficient)) /\\ exists fs_q_pfc_scalar_product_original_coefficientsum_body_steps_summand. pfc_terms_code_scalar_product_original_coefficient = fs_q_pfc_scalar_product_original_coefficientsum_body_steps_summand * S ((S (fs_i_pfc_scalar_product_original_coefficientsum_body_steps)) * pfc_terms_scale_scalar_product_original_coefficient) + (fs_a_pfc_scalar_product_original_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_product_original_coefficientsum_body_steps_partial. fs_h_pfc_scalar_product_original_coefficientsum_body_steps_partial + S (fs_r_pfc_scalar_product_original_coefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_product_original_coefficientsum_body_steps)) * fs_v_pfc_scalar_product_original_coefficientsum)) /\\ exists fs_q_pfc_scalar_product_original_coefficientsum_body_steps_partial. fs_u_pfc_scalar_product_original_coefficientsum = fs_q_pfc_scalar_product_original_coefficientsum_body_steps_partial * S ((S (fs_i_pfc_scalar_product_original_coefficientsum_body_steps)) * fs_v_pfc_scalar_product_original_coefficientsum) + (fs_r_pfc_scalar_product_original_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_product_original_coefficientsum_body_steps_successor. fs_h_pfc_scalar_product_original_coefficientsum_body_steps_successor + S (fs_s_pfc_scalar_product_original_coefficientsum_body_steps) = S ((S (S fs_i_pfc_scalar_product_original_coefficientsum_body_steps)) * fs_v_pfc_scalar_product_original_coefficientsum)) /\\ exists fs_q_pfc_scalar_product_original_coefficientsum_body_steps_successor. fs_u_pfc_scalar_product_original_coefficientsum = fs_q_pfc_scalar_product_original_coefficientsum_body_steps_successor * S ((S (S fs_i_pfc_scalar_product_original_coefficientsum_body_steps)) * fs_v_pfc_scalar_product_original_coefficientsum) + (fs_s_pfc_scalar_product_original_coefficientsum_body_steps))) /\\ fs_s_pfc_scalar_product_original_coefficientsum_body_steps = fs_r_pfc_scalar_product_original_coefficientsum_body_steps + fs_a_pfc_scalar_product_original_coefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_scalar_product_original_coefficientresiduebound. pfa_gap_scalar_product_original_coefficientresiduebound + S (a) = (p)) /\\ ((exists pfa_offset_left_scalar_product_original_coefficientresiduecongruence pfa_offset_right_scalar_product_original_coefficientresiduecongruence. (pfc_natural_sum_scalar_product_original_coefficient) + (p) * pfa_offset_left_scalar_product_original_coefficientresiduecongruence = (a) + (p) * pfa_offset_right_scalar_product_original_coefficientresiduecongruence)))))))))))", "specialize hc_right_right_right (i)", "apply hc_right_right_right", "exact hi", "cases ha", "cases ha_witness", "have hb : exists a. ((((exists ff_h_pfp_scalar_product_scaled_entry. ff_h_pfp_scalar_product_scaled_entry + S (a) = S ((S (i)) * dc)) /\\ exists ff_q_pfp_scalar_product_scaled_entry. db = ff_q_pfp_scalar_product_scaled_entry * S ((S (i)) * dc) + (a))) /\\ ((exists pfc_terms_code_scalar_product_scaled_coefficient pfc_terms_scale_scalar_product_scaled_coefficient pfc_natural_sum_scalar_product_scaled_coefficient. ((forall pfc_index_scalar_product_scaled_coefficientdiagonal. (exists pfa_gap_scalar_product_scaled_coefficientdiagonalbound. pfa_gap_scalar_product_scaled_coefficientdiagonalbound + S (pfc_index_scalar_product_scaled_coefficientdiagonal) = (S (i))) -> exists pfc_value_scalar_product_scaled_coefficientdiagonal. ((((exists ff_h_pfp_scalar_product_scaled_coefficientdiagonalentry. ff_h_pfp_scalar_product_scaled_coefficientdiagonalentry + S (pfc_value_scalar_product_scaled_coefficientdiagonal) = S ((S (pfc_index_scalar_product_scaled_coefficientdiagonal)) * pfc_terms_scale_scalar_product_scaled_coefficient)) /\\ exists ff_q_pfp_scalar_product_scaled_coefficientdiagonalentry. pfc_terms_code_scalar_product_scaled_coefficient = ff_q_pfp_scalar_product_scaled_coefficientdiagonalentry * S ((S (pfc_index_scalar_product_scaled_coefficientdiagonal)) * pfc_terms_scale_scalar_product_scaled_coefficient) + (pfc_value_scalar_product_scaled_coefficientdiagonal))) /\\ ((exists pfc_complement_scalar_product_scaled_coefficientdiagonalterm pfc_left_scalar_product_scaled_coefficientdiagonalterm pfc_right_scalar_product_scaled_coefficientdiagonalterm. (((pfc_index_scalar_product_scaled_coefficientdiagonal)+pfc_complement_scalar_product_scaled_coefficientdiagonalterm=(i)) /\\ ((((((exists pfa_gap_scalar_product_scaled_coefficientdiagonaltermleftinside. pfa_gap_scalar_product_scaled_coefficientdiagonaltermleftinside + S (pfc_index_scalar_product_scaled_coefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_scalar_product_scaled_coefficientdiagonaltermleftentry. ff_h_pfp_scalar_product_scaled_coefficientdiagonaltermleftentry + S (pfc_left_scalar_product_scaled_coefficientdiagonalterm) = S ((S (pfc_index_scalar_product_scaled_coefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_scalar_product_scaled_coefficientdiagonaltermleftentry. ab = ff_q_pfp_scalar_product_scaled_coefficientdiagonaltermleftentry * S ((S (pfc_index_scalar_product_scaled_coefficientdiagonal)) * ac) + (pfc_left_scalar_product_scaled_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_product_scaled_coefficientdiagonaltermleftoutside. pfc_gap_scalar_product_scaled_coefficientdiagonaltermleftoutside+(L)=(pfc_index_scalar_product_scaled_coefficientdiagonal)) /\\ (((pfc_left_scalar_product_scaled_coefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_scalar_product_scaled_coefficientdiagonaltermrightinside. pfa_gap_scalar_product_scaled_coefficientdiagonaltermrightinside + S (pfc_complement_scalar_product_scaled_coefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_scalar_product_scaled_coefficientdiagonaltermrightentry. ff_h_pfp_scalar_product_scaled_coefficientdiagonaltermrightentry + S (pfc_right_scalar_product_scaled_coefficientdiagonalterm) = S ((S (pfc_complement_scalar_product_scaled_coefficientdiagonalterm)) * sc)) /\\ exists ff_q_pfp_scalar_product_scaled_coefficientdiagonaltermrightentry. sb = ff_q_pfp_scalar_product_scaled_coefficientdiagonaltermrightentry * S ((S (pfc_complement_scalar_product_scaled_coefficientdiagonalterm)) * sc) + (pfc_right_scalar_product_scaled_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_product_scaled_coefficientdiagonaltermrightoutside. pfc_gap_scalar_product_scaled_coefficientdiagonaltermrightoutside+(M)=(pfc_complement_scalar_product_scaled_coefficientdiagonalterm)) /\\ (((pfc_right_scalar_product_scaled_coefficientdiagonalterm)=0))))) /\\ (((pfc_value_scalar_product_scaled_coefficientdiagonal)=pfc_left_scalar_product_scaled_coefficientdiagonalterm*pfc_right_scalar_product_scaled_coefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_scalar_product_scaled_coefficientsum fs_v_pfc_scalar_product_scaled_coefficientsum. ((((exists fs_h_pfc_scalar_product_scaled_coefficientsum_body_start. fs_h_pfc_scalar_product_scaled_coefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_product_scaled_coefficientsum)) /\\ exists fs_q_pfc_scalar_product_scaled_coefficientsum_body_start. fs_u_pfc_scalar_product_scaled_coefficientsum = fs_q_pfc_scalar_product_scaled_coefficientsum_body_start * S ((S (0)) * fs_v_pfc_scalar_product_scaled_coefficientsum) + (0))) /\\ ((((exists fs_h_pfc_scalar_product_scaled_coefficientsum_body_terminal. fs_h_pfc_scalar_product_scaled_coefficientsum_body_terminal + S (pfc_natural_sum_scalar_product_scaled_coefficient) = S ((S (S (i))) * fs_v_pfc_scalar_product_scaled_coefficientsum)) /\\ exists fs_q_pfc_scalar_product_scaled_coefficientsum_body_terminal. fs_u_pfc_scalar_product_scaled_coefficientsum = fs_q_pfc_scalar_product_scaled_coefficientsum_body_terminal * S ((S (S (i))) * fs_v_pfc_scalar_product_scaled_coefficientsum) + (pfc_natural_sum_scalar_product_scaled_coefficient))) /\\ forall fs_i_pfc_scalar_product_scaled_coefficientsum_body_steps. (exists fs_lt_pfc_scalar_product_scaled_coefficientsum_body_steps_bound. fs_lt_pfc_scalar_product_scaled_coefficientsum_body_steps_bound + S fs_i_pfc_scalar_product_scaled_coefficientsum_body_steps = S (i)) -> exists fs_a_pfc_scalar_product_scaled_coefficientsum_body_steps fs_r_pfc_scalar_product_scaled_coefficientsum_body_steps fs_s_pfc_scalar_product_scaled_coefficientsum_body_steps. ((((exists fs_h_pfc_scalar_product_scaled_coefficientsum_body_steps_summand. fs_h_pfc_scalar_product_scaled_coefficientsum_body_steps_summand + S (fs_a_pfc_scalar_product_scaled_coefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_product_scaled_coefficientsum_body_steps)) * pfc_terms_scale_scalar_product_scaled_coefficient)) /\\ exists fs_q_pfc_scalar_product_scaled_coefficientsum_body_steps_summand. pfc_terms_code_scalar_product_scaled_coefficient = fs_q_pfc_scalar_product_scaled_coefficientsum_body_steps_summand * S ((S (fs_i_pfc_scalar_product_scaled_coefficientsum_body_steps)) * pfc_terms_scale_scalar_product_scaled_coefficient) + (fs_a_pfc_scalar_product_scaled_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_product_scaled_coefficientsum_body_steps_partial. fs_h_pfc_scalar_product_scaled_coefficientsum_body_steps_partial + S (fs_r_pfc_scalar_product_scaled_coefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_product_scaled_coefficientsum_body_steps)) * fs_v_pfc_scalar_product_scaled_coefficientsum)) /\\ exists fs_q_pfc_scalar_product_scaled_coefficientsum_body_steps_partial. fs_u_pfc_scalar_product_scaled_coefficientsum = fs_q_pfc_scalar_product_scaled_coefficientsum_body_steps_partial * S ((S (fs_i_pfc_scalar_product_scaled_coefficientsum_body_steps)) * fs_v_pfc_scalar_product_scaled_coefficientsum) + (fs_r_pfc_scalar_product_scaled_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_product_scaled_coefficientsum_body_steps_successor. fs_h_pfc_scalar_product_scaled_coefficientsum_body_steps_successor + S (fs_s_pfc_scalar_product_scaled_coefficientsum_body_steps) = S ((S (S fs_i_pfc_scalar_product_scaled_coefficientsum_body_steps)) * fs_v_pfc_scalar_product_scaled_coefficientsum)) /\\ exists fs_q_pfc_scalar_product_scaled_coefficientsum_body_steps_successor. fs_u_pfc_scalar_product_scaled_coefficientsum = fs_q_pfc_scalar_product_scaled_coefficientsum_body_steps_successor * S ((S (S fs_i_pfc_scalar_product_scaled_coefficientsum_body_steps)) * fs_v_pfc_scalar_product_scaled_coefficientsum) + (fs_s_pfc_scalar_product_scaled_coefficientsum_body_steps))) /\\ fs_s_pfc_scalar_product_scaled_coefficientsum_body_steps = fs_r_pfc_scalar_product_scaled_coefficientsum_body_steps + fs_a_pfc_scalar_product_scaled_coefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_scalar_product_scaled_coefficientresiduebound. pfa_gap_scalar_product_scaled_coefficientresiduebound + S (a) = (p)) /\\ ((exists pfa_offset_left_scalar_product_scaled_coefficientresiduecongruence pfa_offset_right_scalar_product_scaled_coefficientresiduecongruence. (pfc_natural_sum_scalar_product_scaled_coefficient) + (p) * pfa_offset_left_scalar_product_scaled_coefficientresiduecongruence = (a) + (p) * pfa_offset_right_scalar_product_scaled_coefficientresiduecongruence)))))))))))", "specialize hd_right_right_right (i)", "apply hd_right_right_right", "rewrite hk", "exact hi", "cases hb", "cases hb_witness", "exists x", "exists x1", "split", "exact ha_witness_left", "split", "exact hb_witness_left", "specialize prime_field_convolution_coefficient_right_scale (p)", "specialize prime_field_convolution_coefficient_right_scale (k)", "specialize prime_field_convolution_coefficient_right_scale (ab)", "specialize prime_field_convolution_coefficient_right_scale (ac)", "specialize prime_field_convolution_coefficient_right_scale (L)", "specialize prime_field_convolution_coefficient_right_scale (bb)", "specialize prime_field_convolution_coefficient_right_scale (bc)", "specialize prime_field_convolution_coefficient_right_scale (M)", "specialize prime_field_convolution_coefficient_right_scale (sb)", "specialize prime_field_convolution_coefficient_right_scale (sc)", "specialize prime_field_convolution_coefficient_right_scale (i)", "specialize prime_field_convolution_coefficient_right_scale (x)", "specialize prime_field_convolution_coefficient_right_scale (x1)", "apply prime_field_convolution_coefficient_right_scale", "exact hs", "exact ha_witness_right", "exact hb_witness_right" ], "script_sha256": "e6d5d780f8515c37eced142d806b3608ee37abcb0f53c4578d57f48cba8e84f2", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_scalar_convolution_candidate.py", "sha256": "e84f1c77c6c03fa5f08635aeede53591625d1c2bfcdfb64fbd379c33878aee0e" }, "statement": "forall p k ab ac L bb bc M sb sc cb cc N db dc K. (((exists pfa_gap_scalar_product_inputscalar. pfa_gap_scalar_product_inputscalar + S (k) = (p)) /\\ ((forall pfp_index_scalar_product_input. (exists pfa_gap_scalar_product_inputindex. pfa_gap_scalar_product_inputindex + S (pfp_index_scalar_product_input) = (M)) -> exists pfp_source_scalar_product_input pfp_value_scalar_product_input. ((((exists ff_h_pfp_scalar_product_inputsource. ff_h_pfp_scalar_product_inputsource + S (pfp_source_scalar_product_input) = S ((S (pfp_index_scalar_product_input)) * bc)) /\\ exists ff_q_pfp_scalar_product_inputsource. bb = ff_q_pfp_scalar_product_inputsource * S ((S (pfp_index_scalar_product_input)) * bc) + (pfp_source_scalar_product_input))) /\\ (((((exists ff_h_pfp_scalar_product_inputtarget. ff_h_pfp_scalar_product_inputtarget + S (pfp_value_scalar_product_input) = S ((S (pfp_index_scalar_product_input)) * sc)) /\\ exists ff_q_pfp_scalar_product_inputtarget. sb = ff_q_pfp_scalar_product_inputtarget * S ((S (pfp_index_scalar_product_input)) * sc) + (pfp_value_scalar_product_input))) /\\ ((((exists pfa_gap_scalar_product_inputoperationleft. pfa_gap_scalar_product_inputoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_product_inputoperationright. pfa_gap_scalar_product_inputoperationright + S (pfp_source_scalar_product_input) = (p)) /\\ ((((exists pfa_gap_scalar_product_inputoperationresultbound. pfa_gap_scalar_product_inputoperationresultbound + S (pfp_value_scalar_product_input) = (p)) /\\ ((exists pfa_offset_left_scalar_product_inputoperationresultcongruence pfa_offset_right_scalar_product_inputoperationresultcongruence. ((k) * (pfp_source_scalar_product_input)) + (p) * pfa_offset_left_scalar_product_inputoperationresultcongruence = (pfp_value_scalar_product_input) + (p) * pfa_offset_right_scalar_product_inputoperationresultcongruence))))))))))))))))) -> (((forall fom_index_pfp_scalar_product_oldleft. (exists fom_gap_pfp_scalar_product_oldleft_index_bound. fom_gap_pfp_scalar_product_oldleft_index_bound + S (fom_index_pfp_scalar_product_oldleft) = L) -> exists fom_value_pfp_scalar_product_oldleft. ((((exists fom_beta_height_pfp_scalar_product_oldleft_entry. fom_beta_height_pfp_scalar_product_oldleft_entry + S (fom_value_pfp_scalar_product_oldleft) = S ((S (fom_index_pfp_scalar_product_oldleft)) * ac)) /\\ exists fom_beta_quotient_pfp_scalar_product_oldleft_entry. ab = fom_beta_quotient_pfp_scalar_product_oldleft_entry * S ((S (fom_index_pfp_scalar_product_oldleft)) * ac) + (fom_value_pfp_scalar_product_oldleft))) /\\ (exists fom_gap_pfp_scalar_product_oldleft_value_bound. fom_gap_pfp_scalar_product_oldleft_value_bound + S (fom_value_pfp_scalar_product_oldleft) = p))) /\\ (((forall fom_index_pfp_scalar_product_oldright. (exists fom_gap_pfp_scalar_product_oldright_index_bound. fom_gap_pfp_scalar_product_oldright_index_bound + S (fom_index_pfp_scalar_product_oldright) = M) -> exists fom_value_pfp_scalar_product_oldright. ((((exists fom_beta_height_pfp_scalar_product_oldright_entry. fom_beta_height_pfp_scalar_product_oldright_entry + S (fom_value_pfp_scalar_product_oldright) = S ((S (fom_index_pfp_scalar_product_oldright)) * bc)) /\\ exists fom_beta_quotient_pfp_scalar_product_oldright_entry. bb = fom_beta_quotient_pfp_scalar_product_oldright_entry * S ((S (fom_index_pfp_scalar_product_oldright)) * bc) + (fom_value_pfp_scalar_product_oldright))) /\\ (exists fom_gap_pfp_scalar_product_oldright_value_bound. fom_gap_pfp_scalar_product_oldright_value_bound + S (fom_value_pfp_scalar_product_oldright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_scalar_product_oldcoefficients. (exists pfa_gap_scalar_product_oldcoefficientsbound. pfa_gap_scalar_product_oldcoefficientsbound + S (pfc_index_scalar_product_oldcoefficients) = (N)) -> exists pfc_value_scalar_product_oldcoefficients. ((((exists ff_h_pfp_scalar_product_oldcoefficientsentry. ff_h_pfp_scalar_product_oldcoefficientsentry + S (pfc_value_scalar_product_oldcoefficients) = S ((S (pfc_index_scalar_product_oldcoefficients)) * cc)) /\\ exists ff_q_pfp_scalar_product_oldcoefficientsentry. cb = ff_q_pfp_scalar_product_oldcoefficientsentry * S ((S (pfc_index_scalar_product_oldcoefficients)) * cc) + (pfc_value_scalar_product_oldcoefficients))) /\\ ((exists pfc_terms_code_scalar_product_oldcoefficientscoefficient pfc_terms_scale_scalar_product_oldcoefficientscoefficient pfc_natural_sum_scalar_product_oldcoefficientscoefficient. ((forall pfc_index_scalar_product_oldcoefficientscoefficientdiagonal. (exists pfa_gap_scalar_product_oldcoefficientscoefficientdiagonalbound. pfa_gap_scalar_product_oldcoefficientscoefficientdiagonalbound + S (pfc_index_scalar_product_oldcoefficientscoefficientdiagonal) = (S (pfc_index_scalar_product_oldcoefficients))) -> exists pfc_value_scalar_product_oldcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_scalar_product_oldcoefficientscoefficientdiagonalentry. ff_h_pfp_scalar_product_oldcoefficientscoefficientdiagonalentry + S (pfc_value_scalar_product_oldcoefficientscoefficientdiagonal) = S ((S (pfc_index_scalar_product_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_product_oldcoefficientscoefficient)) /\\ exists ff_q_pfp_scalar_product_oldcoefficientscoefficientdiagonalentry. pfc_terms_code_scalar_product_oldcoefficientscoefficient = ff_q_pfp_scalar_product_oldcoefficientscoefficientdiagonalentry * S ((S (pfc_index_scalar_product_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_product_oldcoefficientscoefficient) + (pfc_value_scalar_product_oldcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_scalar_product_oldcoefficientscoefficientdiagonalterm pfc_left_scalar_product_oldcoefficientscoefficientdiagonalterm pfc_right_scalar_product_oldcoefficientscoefficientdiagonalterm. (((pfc_index_scalar_product_oldcoefficientscoefficientdiagonal)+pfc_complement_scalar_product_oldcoefficientscoefficientdiagonalterm=(pfc_index_scalar_product_oldcoefficients)) /\\ ((((((exists pfa_gap_scalar_product_oldcoefficientscoefficientdiagonaltermleftinside. pfa_gap_scalar_product_oldcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_scalar_product_oldcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_scalar_product_oldcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_scalar_product_oldcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_scalar_product_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_scalar_product_oldcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_scalar_product_oldcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_scalar_product_oldcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_scalar_product_oldcoefficientscoefficientdiagonal)) * ac) + (pfc_left_scalar_product_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_product_oldcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_scalar_product_oldcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_scalar_product_oldcoefficientscoefficientdiagonal)) /\\ (((pfc_left_scalar_product_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_scalar_product_oldcoefficientscoefficientdiagonaltermrightinside. pfa_gap_scalar_product_oldcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_scalar_product_oldcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_scalar_product_oldcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_scalar_product_oldcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_scalar_product_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_scalar_product_oldcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_scalar_product_oldcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_scalar_product_oldcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_scalar_product_oldcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_scalar_product_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_product_oldcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_scalar_product_oldcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_scalar_product_oldcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_scalar_product_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_scalar_product_oldcoefficientscoefficientdiagonal)=pfc_left_scalar_product_oldcoefficientscoefficientdiagonalterm*pfc_right_scalar_product_oldcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_scalar_product_oldcoefficientscoefficientsum fs_v_pfc_scalar_product_oldcoefficientscoefficientsum. ((((exists fs_h_pfc_scalar_product_oldcoefficientscoefficientsum_body_start. fs_h_pfc_scalar_product_oldcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_product_oldcoefficientscoefficientsum_body_start. fs_u_pfc_scalar_product_oldcoefficientscoefficientsum = fs_q_pfc_scalar_product_oldcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_scalar_product_oldcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_scalar_product_oldcoefficientscoefficientsum_body_terminal. fs_h_pfc_scalar_product_oldcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_scalar_product_oldcoefficientscoefficient) = S ((S (S (pfc_index_scalar_product_oldcoefficients))) * fs_v_pfc_scalar_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_product_oldcoefficientscoefficientsum_body_terminal. fs_u_pfc_scalar_product_oldcoefficientscoefficientsum = fs_q_pfc_scalar_product_oldcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_scalar_product_oldcoefficients))) * fs_v_pfc_scalar_product_oldcoefficientscoefficientsum) + (pfc_natural_sum_scalar_product_oldcoefficientscoefficient))) /\\ forall fs_i_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps = S (pfc_index_scalar_product_oldcoefficients)) -> exists fs_a_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps fs_r_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps fs_s_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_product_oldcoefficientscoefficient)) /\\ exists fs_q_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_scalar_product_oldcoefficientscoefficient = fs_q_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_product_oldcoefficientscoefficient) + (fs_a_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_scalar_product_oldcoefficientscoefficientsum = fs_q_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_product_oldcoefficientscoefficientsum) + (fs_r_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_product_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_scalar_product_oldcoefficientscoefficientsum = fs_q_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_product_oldcoefficientscoefficientsum) + (fs_s_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps = fs_r_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps + fs_a_pfc_scalar_product_oldcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_scalar_product_oldcoefficientscoefficientresiduebound. pfa_gap_scalar_product_oldcoefficientscoefficientresiduebound + S (pfc_value_scalar_product_oldcoefficients) = (p)) /\\ ((exists pfa_offset_left_scalar_product_oldcoefficientscoefficientresiduecongruence pfa_offset_right_scalar_product_oldcoefficientscoefficientresiduecongruence. (pfc_natural_sum_scalar_product_oldcoefficientscoefficient) + (p) * pfa_offset_left_scalar_product_oldcoefficientscoefficientresiduecongruence = (pfc_value_scalar_product_oldcoefficients) + (p) * pfa_offset_right_scalar_product_oldcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_scalar_product_scaledleft. (exists fom_gap_pfp_scalar_product_scaledleft_index_bound. fom_gap_pfp_scalar_product_scaledleft_index_bound + S (fom_index_pfp_scalar_product_scaledleft) = L) -> exists fom_value_pfp_scalar_product_scaledleft. ((((exists fom_beta_height_pfp_scalar_product_scaledleft_entry. fom_beta_height_pfp_scalar_product_scaledleft_entry + S (fom_value_pfp_scalar_product_scaledleft) = S ((S (fom_index_pfp_scalar_product_scaledleft)) * ac)) /\\ exists fom_beta_quotient_pfp_scalar_product_scaledleft_entry. ab = fom_beta_quotient_pfp_scalar_product_scaledleft_entry * S ((S (fom_index_pfp_scalar_product_scaledleft)) * ac) + (fom_value_pfp_scalar_product_scaledleft))) /\\ (exists fom_gap_pfp_scalar_product_scaledleft_value_bound. fom_gap_pfp_scalar_product_scaledleft_value_bound + S (fom_value_pfp_scalar_product_scaledleft) = p))) /\\ (((forall fom_index_pfp_scalar_product_scaledright. (exists fom_gap_pfp_scalar_product_scaledright_index_bound. fom_gap_pfp_scalar_product_scaledright_index_bound + S (fom_index_pfp_scalar_product_scaledright) = M) -> exists fom_value_pfp_scalar_product_scaledright. ((((exists fom_beta_height_pfp_scalar_product_scaledright_entry. fom_beta_height_pfp_scalar_product_scaledright_entry + S (fom_value_pfp_scalar_product_scaledright) = S ((S (fom_index_pfp_scalar_product_scaledright)) * sc)) /\\ exists fom_beta_quotient_pfp_scalar_product_scaledright_entry. sb = fom_beta_quotient_pfp_scalar_product_scaledright_entry * S ((S (fom_index_pfp_scalar_product_scaledright)) * sc) + (fom_value_pfp_scalar_product_scaledright))) /\\ (exists fom_gap_pfp_scalar_product_scaledright_value_bound. fom_gap_pfp_scalar_product_scaledright_value_bound + S (fom_value_pfp_scalar_product_scaledright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (K)))))))) /\\ ((forall pfc_index_scalar_product_scaledcoefficients. (exists pfa_gap_scalar_product_scaledcoefficientsbound. pfa_gap_scalar_product_scaledcoefficientsbound + S (pfc_index_scalar_product_scaledcoefficients) = (K)) -> exists pfc_value_scalar_product_scaledcoefficients. ((((exists ff_h_pfp_scalar_product_scaledcoefficientsentry. ff_h_pfp_scalar_product_scaledcoefficientsentry + S (pfc_value_scalar_product_scaledcoefficients) = S ((S (pfc_index_scalar_product_scaledcoefficients)) * dc)) /\\ exists ff_q_pfp_scalar_product_scaledcoefficientsentry. db = ff_q_pfp_scalar_product_scaledcoefficientsentry * S ((S (pfc_index_scalar_product_scaledcoefficients)) * dc) + (pfc_value_scalar_product_scaledcoefficients))) /\\ ((exists pfc_terms_code_scalar_product_scaledcoefficientscoefficient pfc_terms_scale_scalar_product_scaledcoefficientscoefficient pfc_natural_sum_scalar_product_scaledcoefficientscoefficient. ((forall pfc_index_scalar_product_scaledcoefficientscoefficientdiagonal. (exists pfa_gap_scalar_product_scaledcoefficientscoefficientdiagonalbound. pfa_gap_scalar_product_scaledcoefficientscoefficientdiagonalbound + S (pfc_index_scalar_product_scaledcoefficientscoefficientdiagonal) = (S (pfc_index_scalar_product_scaledcoefficients))) -> exists pfc_value_scalar_product_scaledcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_scalar_product_scaledcoefficientscoefficientdiagonalentry. ff_h_pfp_scalar_product_scaledcoefficientscoefficientdiagonalentry + S (pfc_value_scalar_product_scaledcoefficientscoefficientdiagonal) = S ((S (pfc_index_scalar_product_scaledcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_product_scaledcoefficientscoefficient)) /\\ exists ff_q_pfp_scalar_product_scaledcoefficientscoefficientdiagonalentry. pfc_terms_code_scalar_product_scaledcoefficientscoefficient = ff_q_pfp_scalar_product_scaledcoefficientscoefficientdiagonalentry * S ((S (pfc_index_scalar_product_scaledcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_product_scaledcoefficientscoefficient) + (pfc_value_scalar_product_scaledcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_scalar_product_scaledcoefficientscoefficientdiagonalterm pfc_left_scalar_product_scaledcoefficientscoefficientdiagonalterm pfc_right_scalar_product_scaledcoefficientscoefficientdiagonalterm. (((pfc_index_scalar_product_scaledcoefficientscoefficientdiagonal)+pfc_complement_scalar_product_scaledcoefficientscoefficientdiagonalterm=(pfc_index_scalar_product_scaledcoefficients)) /\\ ((((((exists pfa_gap_scalar_product_scaledcoefficientscoefficientdiagonaltermleftinside. pfa_gap_scalar_product_scaledcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_scalar_product_scaledcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_scalar_product_scaledcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_scalar_product_scaledcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_scalar_product_scaledcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_scalar_product_scaledcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_scalar_product_scaledcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_scalar_product_scaledcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_scalar_product_scaledcoefficientscoefficientdiagonal)) * ac) + (pfc_left_scalar_product_scaledcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_product_scaledcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_scalar_product_scaledcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_scalar_product_scaledcoefficientscoefficientdiagonal)) /\\ (((pfc_left_scalar_product_scaledcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_scalar_product_scaledcoefficientscoefficientdiagonaltermrightinside. pfa_gap_scalar_product_scaledcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_scalar_product_scaledcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_scalar_product_scaledcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_scalar_product_scaledcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_scalar_product_scaledcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_scalar_product_scaledcoefficientscoefficientdiagonalterm)) * sc)) /\\ exists ff_q_pfp_scalar_product_scaledcoefficientscoefficientdiagonaltermrightentry. sb = ff_q_pfp_scalar_product_scaledcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_scalar_product_scaledcoefficientscoefficientdiagonalterm)) * sc) + (pfc_right_scalar_product_scaledcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_product_scaledcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_scalar_product_scaledcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_scalar_product_scaledcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_scalar_product_scaledcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_scalar_product_scaledcoefficientscoefficientdiagonal)=pfc_left_scalar_product_scaledcoefficientscoefficientdiagonalterm*pfc_right_scalar_product_scaledcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_scalar_product_scaledcoefficientscoefficientsum fs_v_pfc_scalar_product_scaledcoefficientscoefficientsum. ((((exists fs_h_pfc_scalar_product_scaledcoefficientscoefficientsum_body_start. fs_h_pfc_scalar_product_scaledcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_product_scaledcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_product_scaledcoefficientscoefficientsum_body_start. fs_u_pfc_scalar_product_scaledcoefficientscoefficientsum = fs_q_pfc_scalar_product_scaledcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_scalar_product_scaledcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_scalar_product_scaledcoefficientscoefficientsum_body_terminal. fs_h_pfc_scalar_product_scaledcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_scalar_product_scaledcoefficientscoefficient) = S ((S (S (pfc_index_scalar_product_scaledcoefficients))) * fs_v_pfc_scalar_product_scaledcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_product_scaledcoefficientscoefficientsum_body_terminal. fs_u_pfc_scalar_product_scaledcoefficientscoefficientsum = fs_q_pfc_scalar_product_scaledcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_scalar_product_scaledcoefficients))) * fs_v_pfc_scalar_product_scaledcoefficientscoefficientsum) + (pfc_natural_sum_scalar_product_scaledcoefficientscoefficient))) /\\ forall fs_i_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps = S (pfc_index_scalar_product_scaledcoefficients)) -> exists fs_a_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps fs_r_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps fs_s_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_product_scaledcoefficientscoefficient)) /\\ exists fs_q_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_scalar_product_scaledcoefficientscoefficient = fs_q_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_product_scaledcoefficientscoefficient) + (fs_a_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_product_scaledcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_scalar_product_scaledcoefficientscoefficientsum = fs_q_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_product_scaledcoefficientscoefficientsum) + (fs_r_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_product_scaledcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_scalar_product_scaledcoefficientscoefficientsum = fs_q_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_product_scaledcoefficientscoefficientsum) + (fs_s_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps = fs_r_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps + fs_a_pfc_scalar_product_scaledcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_scalar_product_scaledcoefficientscoefficientresiduebound. pfa_gap_scalar_product_scaledcoefficientscoefficientresiduebound + S (pfc_value_scalar_product_scaledcoefficients) = (p)) /\\ ((exists pfa_offset_left_scalar_product_scaledcoefficientscoefficientresiduecongruence pfa_offset_right_scalar_product_scaledcoefficientscoefficientresiduecongruence. (pfc_natural_sum_scalar_product_scaledcoefficientscoefficient) + (p) * pfa_offset_left_scalar_product_scaledcoefficientscoefficientresiduecongruence = (pfc_value_scalar_product_scaledcoefficients) + (p) * pfa_offset_right_scalar_product_scaledcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((K=N) /\\ ((((exists pfa_gap_scalar_product_resultscalar. pfa_gap_scalar_product_resultscalar + S (k) = (p)) /\\ ((forall pfp_index_scalar_product_result. (exists pfa_gap_scalar_product_resultindex. pfa_gap_scalar_product_resultindex + S (pfp_index_scalar_product_result) = (N)) -> exists pfp_source_scalar_product_result pfp_value_scalar_product_result. ((((exists ff_h_pfp_scalar_product_resultsource. ff_h_pfp_scalar_product_resultsource + S (pfp_source_scalar_product_result) = S ((S (pfp_index_scalar_product_result)) * cc)) /\\ exists ff_q_pfp_scalar_product_resultsource. cb = ff_q_pfp_scalar_product_resultsource * S ((S (pfp_index_scalar_product_result)) * cc) + (pfp_source_scalar_product_result))) /\\ (((((exists ff_h_pfp_scalar_product_resulttarget. ff_h_pfp_scalar_product_resulttarget + S (pfp_value_scalar_product_result) = S ((S (pfp_index_scalar_product_result)) * dc)) /\\ exists ff_q_pfp_scalar_product_resulttarget. db = ff_q_pfp_scalar_product_resulttarget * S ((S (pfp_index_scalar_product_result)) * dc) + (pfp_value_scalar_product_result))) /\\ ((((exists pfa_gap_scalar_product_resultoperationleft. pfa_gap_scalar_product_resultoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_product_resultoperationright. pfa_gap_scalar_product_resultoperationright + S (pfp_source_scalar_product_result) = (p)) /\\ ((((exists pfa_gap_scalar_product_resultoperationresultbound. pfa_gap_scalar_product_resultoperationresultbound + S (pfp_value_scalar_product_result) = (p)) /\\ ((exists pfa_offset_left_scalar_product_resultoperationresultcongruence pfa_offset_right_scalar_product_resultoperationresultcongruence. 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"intro cb", "intro cc", "intro N", "intro db", "intro dc", "intro K", "intro eb", "intro ec", "intro hs", "intro hc", "intro hd", "intro he", "have hdata : ((K=N) /\\ ((((exists pfa_gap_scalar_comparison_datascalar. pfa_gap_scalar_comparison_datascalar + S (k) = (p)) /\\ ((forall pfp_index_scalar_comparison_data. (exists pfa_gap_scalar_comparison_dataindex. pfa_gap_scalar_comparison_dataindex + S (pfp_index_scalar_comparison_data) = (N)) -> exists pfp_source_scalar_comparison_data pfp_value_scalar_comparison_data. ((((exists ff_h_pfp_scalar_comparison_datasource. ff_h_pfp_scalar_comparison_datasource + S (pfp_source_scalar_comparison_data) = S ((S (pfp_index_scalar_comparison_data)) * cc)) /\\ exists ff_q_pfp_scalar_comparison_datasource. cb = ff_q_pfp_scalar_comparison_datasource * S ((S (pfp_index_scalar_comparison_data)) * cc) + (pfp_source_scalar_comparison_data))) /\\ (((((exists ff_h_pfp_scalar_comparison_datatarget. ff_h_pfp_scalar_comparison_datatarget + S (pfp_value_scalar_comparison_data) = S ((S (pfp_index_scalar_comparison_data)) * dc)) /\\ exists ff_q_pfp_scalar_comparison_datatarget. db = ff_q_pfp_scalar_comparison_datatarget * S ((S (pfp_index_scalar_comparison_data)) * dc) + (pfp_value_scalar_comparison_data))) /\\ ((((exists pfa_gap_scalar_comparison_dataoperationleft. pfa_gap_scalar_comparison_dataoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_comparison_dataoperationright. pfa_gap_scalar_comparison_dataoperationright + S (pfp_source_scalar_comparison_data) = (p)) /\\ ((((exists pfa_gap_scalar_comparison_dataoperationresultbound. pfa_gap_scalar_comparison_dataoperationresultbound + S (pfp_value_scalar_comparison_data) = (p)) /\\ ((exists pfa_offset_left_scalar_comparison_dataoperationresultcongruence pfa_offset_right_scalar_comparison_dataoperationresultcongruence. ((k) * (pfp_source_scalar_comparison_data)) + (p) * pfa_offset_left_scalar_comparison_dataoperationresultcongruence = (pfp_value_scalar_comparison_data) + (p) * pfa_offset_right_scalar_comparison_dataoperationresultcongruence)))))))))))))))))))", "specialize prime_field_polynomial_convolution_right_scale (p)", "specialize prime_field_polynomial_convolution_right_scale (k)", "specialize prime_field_polynomial_convolution_right_scale (ab)", "specialize prime_field_polynomial_convolution_right_scale (ac)", "specialize prime_field_polynomial_convolution_right_scale (L)", "specialize prime_field_polynomial_convolution_right_scale (bb)", "specialize prime_field_polynomial_convolution_right_scale (bc)", "specialize prime_field_polynomial_convolution_right_scale (M)", "specialize prime_field_polynomial_convolution_right_scale (sb)", "specialize prime_field_polynomial_convolution_right_scale (sc)", "specialize prime_field_polynomial_convolution_right_scale (cb)", "specialize prime_field_polynomial_convolution_right_scale (cc)", "specialize prime_field_polynomial_convolution_right_scale (N)", "specialize prime_field_polynomial_convolution_right_scale (db)", "specialize prime_field_polynomial_convolution_right_scale (dc)", "specialize prime_field_polynomial_convolution_right_scale (K)", "apply prime_field_polynomial_convolution_right_scale", "exact hs", "exact hc", "exact hd", "cases hdata", "split", "exact hdata_left", "specialize prime_field_polynomial_scale_functional (p)", "specialize prime_field_polynomial_scale_functional (k)", "specialize prime_field_polynomial_scale_functional (cb)", "specialize prime_field_polynomial_scale_functional (cc)", "specialize prime_field_polynomial_scale_functional (db)", "specialize prime_field_polynomial_scale_functional (dc)", "specialize prime_field_polynomial_scale_functional (eb)", "specialize prime_field_polynomial_scale_functional (ec)", "specialize prime_field_polynomial_scale_functional (N)", "apply prime_field_polynomial_scale_functional", "exact hdata_right", "exact he" ], "script_sha256": "e1b9e282d6bfe56d30739fbfdbab003226ce150b61a7f662861f1f8fd4e0ff13", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_scalar_convolution_candidate.py", "sha256": "e84f1c77c6c03fa5f08635aeede53591625d1c2bfcdfb64fbd379c33878aee0e" }, "statement": "forall p k ab ac L bb bc M sb sc cb cc N db dc K eb ec. (((exists pfa_gap_scalar_comparison_inputscalar. pfa_gap_scalar_comparison_inputscalar + S (k) = (p)) /\\ ((forall pfp_index_scalar_comparison_input. (exists pfa_gap_scalar_comparison_inputindex. pfa_gap_scalar_comparison_inputindex + S (pfp_index_scalar_comparison_input) = (M)) -> exists pfp_source_scalar_comparison_input pfp_value_scalar_comparison_input. ((((exists ff_h_pfp_scalar_comparison_inputsource. ff_h_pfp_scalar_comparison_inputsource + S (pfp_source_scalar_comparison_input) = S ((S (pfp_index_scalar_comparison_input)) * bc)) /\\ exists ff_q_pfp_scalar_comparison_inputsource. bb = ff_q_pfp_scalar_comparison_inputsource * S ((S (pfp_index_scalar_comparison_input)) * bc) + (pfp_source_scalar_comparison_input))) /\\ (((((exists ff_h_pfp_scalar_comparison_inputtarget. ff_h_pfp_scalar_comparison_inputtarget + S (pfp_value_scalar_comparison_input) = S ((S (pfp_index_scalar_comparison_input)) * sc)) /\\ exists ff_q_pfp_scalar_comparison_inputtarget. sb = ff_q_pfp_scalar_comparison_inputtarget * S ((S (pfp_index_scalar_comparison_input)) * sc) + (pfp_value_scalar_comparison_input))) /\\ ((((exists pfa_gap_scalar_comparison_inputoperationleft. pfa_gap_scalar_comparison_inputoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_comparison_inputoperationright. pfa_gap_scalar_comparison_inputoperationright + S (pfp_source_scalar_comparison_input) = (p)) /\\ ((((exists pfa_gap_scalar_comparison_inputoperationresultbound. pfa_gap_scalar_comparison_inputoperationresultbound + S (pfp_value_scalar_comparison_input) = (p)) /\\ ((exists pfa_offset_left_scalar_comparison_inputoperationresultcongruence pfa_offset_right_scalar_comparison_inputoperationresultcongruence. ((k) * (pfp_source_scalar_comparison_input)) + (p) * pfa_offset_left_scalar_comparison_inputoperationresultcongruence = (pfp_value_scalar_comparison_input) + (p) * pfa_offset_right_scalar_comparison_inputoperationresultcongruence))))))))))))))))) -> (((forall fom_index_pfp_scalar_comparison_oldleft. (exists fom_gap_pfp_scalar_comparison_oldleft_index_bound. fom_gap_pfp_scalar_comparison_oldleft_index_bound + S (fom_index_pfp_scalar_comparison_oldleft) = L) -> exists fom_value_pfp_scalar_comparison_oldleft. ((((exists fom_beta_height_pfp_scalar_comparison_oldleft_entry. fom_beta_height_pfp_scalar_comparison_oldleft_entry + S (fom_value_pfp_scalar_comparison_oldleft) = S ((S (fom_index_pfp_scalar_comparison_oldleft)) * ac)) /\\ exists fom_beta_quotient_pfp_scalar_comparison_oldleft_entry. ab = fom_beta_quotient_pfp_scalar_comparison_oldleft_entry * S ((S (fom_index_pfp_scalar_comparison_oldleft)) * ac) + (fom_value_pfp_scalar_comparison_oldleft))) /\\ (exists fom_gap_pfp_scalar_comparison_oldleft_value_bound. fom_gap_pfp_scalar_comparison_oldleft_value_bound + S (fom_value_pfp_scalar_comparison_oldleft) = p))) /\\ (((forall fom_index_pfp_scalar_comparison_oldright. (exists fom_gap_pfp_scalar_comparison_oldright_index_bound. fom_gap_pfp_scalar_comparison_oldright_index_bound + S (fom_index_pfp_scalar_comparison_oldright) = M) -> exists fom_value_pfp_scalar_comparison_oldright. ((((exists fom_beta_height_pfp_scalar_comparison_oldright_entry. fom_beta_height_pfp_scalar_comparison_oldright_entry + S (fom_value_pfp_scalar_comparison_oldright) = S ((S (fom_index_pfp_scalar_comparison_oldright)) * bc)) /\\ exists fom_beta_quotient_pfp_scalar_comparison_oldright_entry. bb = fom_beta_quotient_pfp_scalar_comparison_oldright_entry * S ((S (fom_index_pfp_scalar_comparison_oldright)) * bc) + (fom_value_pfp_scalar_comparison_oldright))) /\\ (exists fom_gap_pfp_scalar_comparison_oldright_value_bound. fom_gap_pfp_scalar_comparison_oldright_value_bound + S (fom_value_pfp_scalar_comparison_oldright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_scalar_comparison_oldcoefficients. (exists pfa_gap_scalar_comparison_oldcoefficientsbound. pfa_gap_scalar_comparison_oldcoefficientsbound + S (pfc_index_scalar_comparison_oldcoefficients) = (N)) -> exists pfc_value_scalar_comparison_oldcoefficients. ((((exists ff_h_pfp_scalar_comparison_oldcoefficientsentry. ff_h_pfp_scalar_comparison_oldcoefficientsentry + S (pfc_value_scalar_comparison_oldcoefficients) = S ((S (pfc_index_scalar_comparison_oldcoefficients)) * cc)) /\\ exists ff_q_pfp_scalar_comparison_oldcoefficientsentry. cb = ff_q_pfp_scalar_comparison_oldcoefficientsentry * S ((S (pfc_index_scalar_comparison_oldcoefficients)) * cc) + (pfc_value_scalar_comparison_oldcoefficients))) /\\ ((exists pfc_terms_code_scalar_comparison_oldcoefficientscoefficient pfc_terms_scale_scalar_comparison_oldcoefficientscoefficient pfc_natural_sum_scalar_comparison_oldcoefficientscoefficient. ((forall pfc_index_scalar_comparison_oldcoefficientscoefficientdiagonal. (exists pfa_gap_scalar_comparison_oldcoefficientscoefficientdiagonalbound. pfa_gap_scalar_comparison_oldcoefficientscoefficientdiagonalbound + S (pfc_index_scalar_comparison_oldcoefficientscoefficientdiagonal) = (S (pfc_index_scalar_comparison_oldcoefficients))) -> exists pfc_value_scalar_comparison_oldcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_scalar_comparison_oldcoefficientscoefficientdiagonalentry. ff_h_pfp_scalar_comparison_oldcoefficientscoefficientdiagonalentry + S (pfc_value_scalar_comparison_oldcoefficientscoefficientdiagonal) = S ((S (pfc_index_scalar_comparison_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_comparison_oldcoefficientscoefficient)) /\\ exists ff_q_pfp_scalar_comparison_oldcoefficientscoefficientdiagonalentry. pfc_terms_code_scalar_comparison_oldcoefficientscoefficient = ff_q_pfp_scalar_comparison_oldcoefficientscoefficientdiagonalentry * S ((S (pfc_index_scalar_comparison_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_comparison_oldcoefficientscoefficient) + (pfc_value_scalar_comparison_oldcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_scalar_comparison_oldcoefficientscoefficientdiagonalterm pfc_left_scalar_comparison_oldcoefficientscoefficientdiagonalterm pfc_right_scalar_comparison_oldcoefficientscoefficientdiagonalterm. (((pfc_index_scalar_comparison_oldcoefficientscoefficientdiagonal)+pfc_complement_scalar_comparison_oldcoefficientscoefficientdiagonalterm=(pfc_index_scalar_comparison_oldcoefficients)) /\\ ((((((exists pfa_gap_scalar_comparison_oldcoefficientscoefficientdiagonaltermleftinside. pfa_gap_scalar_comparison_oldcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_scalar_comparison_oldcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_scalar_comparison_oldcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_scalar_comparison_oldcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_scalar_comparison_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_scalar_comparison_oldcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_scalar_comparison_oldcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_scalar_comparison_oldcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_scalar_comparison_oldcoefficientscoefficientdiagonal)) * ac) + (pfc_left_scalar_comparison_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_comparison_oldcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_scalar_comparison_oldcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_scalar_comparison_oldcoefficientscoefficientdiagonal)) /\\ (((pfc_left_scalar_comparison_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_scalar_comparison_oldcoefficientscoefficientdiagonaltermrightinside. pfa_gap_scalar_comparison_oldcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_scalar_comparison_oldcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_scalar_comparison_oldcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_scalar_comparison_oldcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_scalar_comparison_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_scalar_comparison_oldcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_scalar_comparison_oldcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_scalar_comparison_oldcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_scalar_comparison_oldcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_scalar_comparison_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_comparison_oldcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_scalar_comparison_oldcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_scalar_comparison_oldcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_scalar_comparison_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_scalar_comparison_oldcoefficientscoefficientdiagonal)=pfc_left_scalar_comparison_oldcoefficientscoefficientdiagonalterm*pfc_right_scalar_comparison_oldcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_scalar_comparison_oldcoefficientscoefficientsum fs_v_pfc_scalar_comparison_oldcoefficientscoefficientsum. ((((exists fs_h_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_start. fs_h_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_comparison_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_start. fs_u_pfc_scalar_comparison_oldcoefficientscoefficientsum = fs_q_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_scalar_comparison_oldcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_terminal. fs_h_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_scalar_comparison_oldcoefficientscoefficient) = S ((S (S (pfc_index_scalar_comparison_oldcoefficients))) * fs_v_pfc_scalar_comparison_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_terminal. fs_u_pfc_scalar_comparison_oldcoefficientscoefficientsum = fs_q_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_scalar_comparison_oldcoefficients))) * fs_v_pfc_scalar_comparison_oldcoefficientscoefficientsum) + (pfc_natural_sum_scalar_comparison_oldcoefficientscoefficient))) /\\ forall fs_i_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps = S (pfc_index_scalar_comparison_oldcoefficients)) -> exists fs_a_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps fs_r_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps fs_s_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_comparison_oldcoefficientscoefficient)) /\\ exists fs_q_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_scalar_comparison_oldcoefficientscoefficient = fs_q_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_comparison_oldcoefficientscoefficient) + (fs_a_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_comparison_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_scalar_comparison_oldcoefficientscoefficientsum = fs_q_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_comparison_oldcoefficientscoefficientsum) + (fs_r_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_comparison_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_scalar_comparison_oldcoefficientscoefficientsum = fs_q_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_comparison_oldcoefficientscoefficientsum) + (fs_s_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps = fs_r_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps + fs_a_pfc_scalar_comparison_oldcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_scalar_comparison_oldcoefficientscoefficientresiduebound. pfa_gap_scalar_comparison_oldcoefficientscoefficientresiduebound + S (pfc_value_scalar_comparison_oldcoefficients) = (p)) /\\ ((exists pfa_offset_left_scalar_comparison_oldcoefficientscoefficientresiduecongruence pfa_offset_right_scalar_comparison_oldcoefficientscoefficientresiduecongruence. (pfc_natural_sum_scalar_comparison_oldcoefficientscoefficient) + (p) * pfa_offset_left_scalar_comparison_oldcoefficientscoefficientresiduecongruence = (pfc_value_scalar_comparison_oldcoefficients) + (p) * pfa_offset_right_scalar_comparison_oldcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_scalar_comparison_scaledleft. (exists fom_gap_pfp_scalar_comparison_scaledleft_index_bound. fom_gap_pfp_scalar_comparison_scaledleft_index_bound + S (fom_index_pfp_scalar_comparison_scaledleft) = L) -> exists fom_value_pfp_scalar_comparison_scaledleft. ((((exists fom_beta_height_pfp_scalar_comparison_scaledleft_entry. fom_beta_height_pfp_scalar_comparison_scaledleft_entry + S (fom_value_pfp_scalar_comparison_scaledleft) = S ((S (fom_index_pfp_scalar_comparison_scaledleft)) * ac)) /\\ exists fom_beta_quotient_pfp_scalar_comparison_scaledleft_entry. ab = fom_beta_quotient_pfp_scalar_comparison_scaledleft_entry * S ((S (fom_index_pfp_scalar_comparison_scaledleft)) * ac) + (fom_value_pfp_scalar_comparison_scaledleft))) /\\ (exists fom_gap_pfp_scalar_comparison_scaledleft_value_bound. fom_gap_pfp_scalar_comparison_scaledleft_value_bound + S (fom_value_pfp_scalar_comparison_scaledleft) = p))) /\\ (((forall fom_index_pfp_scalar_comparison_scaledright. (exists fom_gap_pfp_scalar_comparison_scaledright_index_bound. fom_gap_pfp_scalar_comparison_scaledright_index_bound + S (fom_index_pfp_scalar_comparison_scaledright) = M) -> exists fom_value_pfp_scalar_comparison_scaledright. ((((exists fom_beta_height_pfp_scalar_comparison_scaledright_entry. fom_beta_height_pfp_scalar_comparison_scaledright_entry + S (fom_value_pfp_scalar_comparison_scaledright) = S ((S (fom_index_pfp_scalar_comparison_scaledright)) * sc)) /\\ exists fom_beta_quotient_pfp_scalar_comparison_scaledright_entry. sb = fom_beta_quotient_pfp_scalar_comparison_scaledright_entry * S ((S (fom_index_pfp_scalar_comparison_scaledright)) * sc) + (fom_value_pfp_scalar_comparison_scaledright))) /\\ (exists fom_gap_pfp_scalar_comparison_scaledright_value_bound. fom_gap_pfp_scalar_comparison_scaledright_value_bound + S (fom_value_pfp_scalar_comparison_scaledright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (K)))))))) /\\ ((forall pfc_index_scalar_comparison_scaledcoefficients. (exists pfa_gap_scalar_comparison_scaledcoefficientsbound. pfa_gap_scalar_comparison_scaledcoefficientsbound + S (pfc_index_scalar_comparison_scaledcoefficients) = (K)) -> exists pfc_value_scalar_comparison_scaledcoefficients. ((((exists ff_h_pfp_scalar_comparison_scaledcoefficientsentry. ff_h_pfp_scalar_comparison_scaledcoefficientsentry + S (pfc_value_scalar_comparison_scaledcoefficients) = S ((S (pfc_index_scalar_comparison_scaledcoefficients)) * dc)) /\\ exists ff_q_pfp_scalar_comparison_scaledcoefficientsentry. db = ff_q_pfp_scalar_comparison_scaledcoefficientsentry * S ((S (pfc_index_scalar_comparison_scaledcoefficients)) * dc) + (pfc_value_scalar_comparison_scaledcoefficients))) /\\ ((exists pfc_terms_code_scalar_comparison_scaledcoefficientscoefficient pfc_terms_scale_scalar_comparison_scaledcoefficientscoefficient pfc_natural_sum_scalar_comparison_scaledcoefficientscoefficient. ((forall pfc_index_scalar_comparison_scaledcoefficientscoefficientdiagonal. (exists pfa_gap_scalar_comparison_scaledcoefficientscoefficientdiagonalbound. pfa_gap_scalar_comparison_scaledcoefficientscoefficientdiagonalbound + S (pfc_index_scalar_comparison_scaledcoefficientscoefficientdiagonal) = (S (pfc_index_scalar_comparison_scaledcoefficients))) -> exists pfc_value_scalar_comparison_scaledcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_scalar_comparison_scaledcoefficientscoefficientdiagonalentry. ff_h_pfp_scalar_comparison_scaledcoefficientscoefficientdiagonalentry + S (pfc_value_scalar_comparison_scaledcoefficientscoefficientdiagonal) = S ((S (pfc_index_scalar_comparison_scaledcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_comparison_scaledcoefficientscoefficient)) /\\ exists ff_q_pfp_scalar_comparison_scaledcoefficientscoefficientdiagonalentry. pfc_terms_code_scalar_comparison_scaledcoefficientscoefficient = ff_q_pfp_scalar_comparison_scaledcoefficientscoefficientdiagonalentry * S ((S (pfc_index_scalar_comparison_scaledcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_comparison_scaledcoefficientscoefficient) + (pfc_value_scalar_comparison_scaledcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_scalar_comparison_scaledcoefficientscoefficientdiagonalterm pfc_left_scalar_comparison_scaledcoefficientscoefficientdiagonalterm pfc_right_scalar_comparison_scaledcoefficientscoefficientdiagonalterm. (((pfc_index_scalar_comparison_scaledcoefficientscoefficientdiagonal)+pfc_complement_scalar_comparison_scaledcoefficientscoefficientdiagonalterm=(pfc_index_scalar_comparison_scaledcoefficients)) /\\ ((((((exists pfa_gap_scalar_comparison_scaledcoefficientscoefficientdiagonaltermleftinside. pfa_gap_scalar_comparison_scaledcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_scalar_comparison_scaledcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_scalar_comparison_scaledcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_scalar_comparison_scaledcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_scalar_comparison_scaledcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_scalar_comparison_scaledcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_scalar_comparison_scaledcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_scalar_comparison_scaledcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_scalar_comparison_scaledcoefficientscoefficientdiagonal)) * ac) + (pfc_left_scalar_comparison_scaledcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_comparison_scaledcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_scalar_comparison_scaledcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_scalar_comparison_scaledcoefficientscoefficientdiagonal)) /\\ (((pfc_left_scalar_comparison_scaledcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_scalar_comparison_scaledcoefficientscoefficientdiagonaltermrightinside. pfa_gap_scalar_comparison_scaledcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_scalar_comparison_scaledcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_scalar_comparison_scaledcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_scalar_comparison_scaledcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_scalar_comparison_scaledcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_scalar_comparison_scaledcoefficientscoefficientdiagonalterm)) * sc)) /\\ exists ff_q_pfp_scalar_comparison_scaledcoefficientscoefficientdiagonaltermrightentry. sb = ff_q_pfp_scalar_comparison_scaledcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_scalar_comparison_scaledcoefficientscoefficientdiagonalterm)) * sc) + (pfc_right_scalar_comparison_scaledcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_comparison_scaledcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_scalar_comparison_scaledcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_scalar_comparison_scaledcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_scalar_comparison_scaledcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_scalar_comparison_scaledcoefficientscoefficientdiagonal)=pfc_left_scalar_comparison_scaledcoefficientscoefficientdiagonalterm*pfc_right_scalar_comparison_scaledcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_scalar_comparison_scaledcoefficientscoefficientsum fs_v_pfc_scalar_comparison_scaledcoefficientscoefficientsum. ((((exists fs_h_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_start. fs_h_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_comparison_scaledcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_start. fs_u_pfc_scalar_comparison_scaledcoefficientscoefficientsum = fs_q_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_scalar_comparison_scaledcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_terminal. fs_h_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_scalar_comparison_scaledcoefficientscoefficient) = S ((S (S (pfc_index_scalar_comparison_scaledcoefficients))) * fs_v_pfc_scalar_comparison_scaledcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_terminal. fs_u_pfc_scalar_comparison_scaledcoefficientscoefficientsum = fs_q_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_scalar_comparison_scaledcoefficients))) * fs_v_pfc_scalar_comparison_scaledcoefficientscoefficientsum) + (pfc_natural_sum_scalar_comparison_scaledcoefficientscoefficient))) /\\ forall fs_i_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps = S (pfc_index_scalar_comparison_scaledcoefficients)) -> exists fs_a_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps fs_r_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps fs_s_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_comparison_scaledcoefficientscoefficient)) /\\ exists fs_q_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_scalar_comparison_scaledcoefficientscoefficient = fs_q_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_comparison_scaledcoefficientscoefficient) + (fs_a_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_comparison_scaledcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_scalar_comparison_scaledcoefficientscoefficientsum = fs_q_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_comparison_scaledcoefficientscoefficientsum) + (fs_r_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_comparison_scaledcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_scalar_comparison_scaledcoefficientscoefficientsum = fs_q_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_comparison_scaledcoefficientscoefficientsum) + (fs_s_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps = fs_r_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps + fs_a_pfc_scalar_comparison_scaledcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_scalar_comparison_scaledcoefficientscoefficientresiduebound. pfa_gap_scalar_comparison_scaledcoefficientscoefficientresiduebound + S (pfc_value_scalar_comparison_scaledcoefficients) = (p)) /\\ ((exists pfa_offset_left_scalar_comparison_scaledcoefficientscoefficientresiduecongruence pfa_offset_right_scalar_comparison_scaledcoefficientscoefficientresiduecongruence. (pfc_natural_sum_scalar_comparison_scaledcoefficientscoefficient) + (p) * pfa_offset_left_scalar_comparison_scaledcoefficientscoefficientresiduecongruence = (pfc_value_scalar_comparison_scaledcoefficients) + (p) * pfa_offset_right_scalar_comparison_scaledcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((exists pfa_gap_scalar_comparison_outputscalar. pfa_gap_scalar_comparison_outputscalar + S (k) = (p)) /\\ ((forall pfp_index_scalar_comparison_output. (exists pfa_gap_scalar_comparison_outputindex. pfa_gap_scalar_comparison_outputindex + S (pfp_index_scalar_comparison_output) = (N)) -> exists pfp_source_scalar_comparison_output pfp_value_scalar_comparison_output. ((((exists ff_h_pfp_scalar_comparison_outputsource. ff_h_pfp_scalar_comparison_outputsource + S (pfp_source_scalar_comparison_output) = S ((S (pfp_index_scalar_comparison_output)) * cc)) /\\ exists ff_q_pfp_scalar_comparison_outputsource. cb = ff_q_pfp_scalar_comparison_outputsource * S ((S (pfp_index_scalar_comparison_output)) * cc) + (pfp_source_scalar_comparison_output))) /\\ (((((exists ff_h_pfp_scalar_comparison_outputtarget. ff_h_pfp_scalar_comparison_outputtarget + S (pfp_value_scalar_comparison_output) = S ((S (pfp_index_scalar_comparison_output)) * ec)) /\\ exists ff_q_pfp_scalar_comparison_outputtarget. eb = ff_q_pfp_scalar_comparison_outputtarget * S ((S (pfp_index_scalar_comparison_output)) * ec) + (pfp_value_scalar_comparison_output))) /\\ ((((exists pfa_gap_scalar_comparison_outputoperationleft. pfa_gap_scalar_comparison_outputoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_comparison_outputoperationright. pfa_gap_scalar_comparison_outputoperationright + S (pfp_source_scalar_comparison_output) = (p)) /\\ ((((exists pfa_gap_scalar_comparison_outputoperationresultbound. pfa_gap_scalar_comparison_outputoperationresultbound + S (pfp_value_scalar_comparison_output) = (p)) /\\ ((exists pfa_offset_left_scalar_comparison_outputoperationresultcongruence pfa_offset_right_scalar_comparison_outputoperationresultcongruence. ((k) * (pfp_source_scalar_comparison_output)) + (p) * pfa_offset_left_scalar_comparison_outputoperationresultcongruence = (pfp_value_scalar_comparison_output) + (p) * pfa_offset_right_scalar_comparison_outputoperationresultcongruence))))))))))))))))) -> (((K=N) /\\ ((forall mdr_i_pfp_scalar_comparison_result mdr_a_pfp_scalar_comparison_result. (exists mdr_gap_pfp_scalar_comparison_resultb. mdr_gap_pfp_scalar_comparison_resultb + S (mdr_i_pfp_scalar_comparison_result) = (N)) -> (((exists ff_h_mdr_pfp_scalar_comparison_resulto. ff_h_mdr_pfp_scalar_comparison_resulto + S (mdr_a_pfp_scalar_comparison_result) = S ((S (mdr_i_pfp_scalar_comparison_result)) * dc)) /\\ exists ff_q_mdr_pfp_scalar_comparison_resulto. db = ff_q_mdr_pfp_scalar_comparison_resulto * S ((S (mdr_i_pfp_scalar_comparison_result)) * dc) + (mdr_a_pfp_scalar_comparison_result))) -> (((exists ff_h_mdr_pfp_scalar_comparison_resultn. ff_h_mdr_pfp_scalar_comparison_resultn + S (mdr_a_pfp_scalar_comparison_result) = S ((S (mdr_i_pfp_scalar_comparison_result)) * ec)) /\\ exists ff_q_mdr_pfp_scalar_comparison_resultn. eb = ff_q_mdr_pfp_scalar_comparison_resultn * S ((S (mdr_i_pfp_scalar_comparison_result)) * ec) + (mdr_a_pfp_scalar_comparison_result)))))))", "statement_sha256": "42cae7e1cc12bbe6b7b33d8060e1c66b7b46555d983ade711fd09d5545cb5e6c", "summary": "Every actual product A*(k B) agrees coefficientwise with every actual scalar result k*(A*B), at the same proper length; 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(exists fom_gap_pfp_scalar_exists_originalleft_index_bound. fom_gap_pfp_scalar_exists_originalleft_index_bound + S (fom_index_pfp_scalar_exists_originalleft) = L) -> exists fom_value_pfp_scalar_exists_originalleft. ((((exists fom_beta_height_pfp_scalar_exists_originalleft_entry. fom_beta_height_pfp_scalar_exists_originalleft_entry + S (fom_value_pfp_scalar_exists_originalleft) = S ((S (fom_index_pfp_scalar_exists_originalleft)) * ac)) /\\ exists fom_beta_quotient_pfp_scalar_exists_originalleft_entry. ab = fom_beta_quotient_pfp_scalar_exists_originalleft_entry * S ((S (fom_index_pfp_scalar_exists_originalleft)) * ac) + (fom_value_pfp_scalar_exists_originalleft))) /\\ (exists fom_gap_pfp_scalar_exists_originalleft_value_bound. fom_gap_pfp_scalar_exists_originalleft_value_bound + S (fom_value_pfp_scalar_exists_originalleft) = p))) /\\ (((forall fom_index_pfp_scalar_exists_originalright. (exists fom_gap_pfp_scalar_exists_originalright_index_bound. fom_gap_pfp_scalar_exists_originalright_index_bound + S (fom_index_pfp_scalar_exists_originalright) = M) -> exists fom_value_pfp_scalar_exists_originalright. ((((exists fom_beta_height_pfp_scalar_exists_originalright_entry. fom_beta_height_pfp_scalar_exists_originalright_entry + S (fom_value_pfp_scalar_exists_originalright) = S ((S (fom_index_pfp_scalar_exists_originalright)) * bc)) /\\ exists fom_beta_quotient_pfp_scalar_exists_originalright_entry. bb = fom_beta_quotient_pfp_scalar_exists_originalright_entry * S ((S (fom_index_pfp_scalar_exists_originalright)) * bc) + (fom_value_pfp_scalar_exists_originalright))) /\\ (exists fom_gap_pfp_scalar_exists_originalright_value_bound. fom_gap_pfp_scalar_exists_originalright_value_bound + S (fom_value_pfp_scalar_exists_originalright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_scalar_exists_originalcoefficients. (exists pfa_gap_scalar_exists_originalcoefficientsbound. pfa_gap_scalar_exists_originalcoefficientsbound + S (pfc_index_scalar_exists_originalcoefficients) = (N)) -> exists pfc_value_scalar_exists_originalcoefficients. ((((exists ff_h_pfp_scalar_exists_originalcoefficientsentry. ff_h_pfp_scalar_exists_originalcoefficientsentry + S (pfc_value_scalar_exists_originalcoefficients) = S ((S (pfc_index_scalar_exists_originalcoefficients)) * cc)) /\\ exists ff_q_pfp_scalar_exists_originalcoefficientsentry. cb = ff_q_pfp_scalar_exists_originalcoefficientsentry * S ((S (pfc_index_scalar_exists_originalcoefficients)) * cc) + (pfc_value_scalar_exists_originalcoefficients))) /\\ ((exists pfc_terms_code_scalar_exists_originalcoefficientscoefficient pfc_terms_scale_scalar_exists_originalcoefficientscoefficient pfc_natural_sum_scalar_exists_originalcoefficientscoefficient. ((forall pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal. (exists pfa_gap_scalar_exists_originalcoefficientscoefficientdiagonalbound. pfa_gap_scalar_exists_originalcoefficientscoefficientdiagonalbound + S (pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal) = (S (pfc_index_scalar_exists_originalcoefficients))) -> exists pfc_value_scalar_exists_originalcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_scalar_exists_originalcoefficientscoefficientdiagonalentry. ff_h_pfp_scalar_exists_originalcoefficientscoefficientdiagonalentry + S (pfc_value_scalar_exists_originalcoefficientscoefficientdiagonal) = S ((S (pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_exists_originalcoefficientscoefficient)) /\\ exists ff_q_pfp_scalar_exists_originalcoefficientscoefficientdiagonalentry. pfc_terms_code_scalar_exists_originalcoefficientscoefficient = ff_q_pfp_scalar_exists_originalcoefficientscoefficientdiagonalentry * S ((S (pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_exists_originalcoefficientscoefficient) + (pfc_value_scalar_exists_originalcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_scalar_exists_originalcoefficientscoefficientdiagonalterm pfc_left_scalar_exists_originalcoefficientscoefficientdiagonalterm pfc_right_scalar_exists_originalcoefficientscoefficientdiagonalterm. (((pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal)+pfc_complement_scalar_exists_originalcoefficientscoefficientdiagonalterm=(pfc_index_scalar_exists_originalcoefficients)) /\\ ((((((exists pfa_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermleftinside. pfa_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_scalar_exists_originalcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal)) * ac) + (pfc_left_scalar_exists_originalcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal)) /\\ (((pfc_left_scalar_exists_originalcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermrightinside. pfa_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_scalar_exists_originalcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_scalar_exists_originalcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_scalar_exists_originalcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_scalar_exists_originalcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_scalar_exists_originalcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_scalar_exists_originalcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_scalar_exists_originalcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_scalar_exists_originalcoefficientscoefficientdiagonal)=pfc_left_scalar_exists_originalcoefficientscoefficientdiagonalterm*pfc_right_scalar_exists_originalcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_scalar_exists_originalcoefficientscoefficientsum fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum. ((((exists fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_start. fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_start. fs_u_pfc_scalar_exists_originalcoefficientscoefficientsum = fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_terminal. fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_scalar_exists_originalcoefficientscoefficient) = S ((S (S (pfc_index_scalar_exists_originalcoefficients))) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_terminal. fs_u_pfc_scalar_exists_originalcoefficientscoefficientsum = fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_scalar_exists_originalcoefficients))) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum) + (pfc_natural_sum_scalar_exists_originalcoefficientscoefficient))) /\\ forall fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps = S (pfc_index_scalar_exists_originalcoefficients)) -> exists fs_a_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps fs_r_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps fs_s_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_exists_originalcoefficientscoefficient)) /\\ exists fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_scalar_exists_originalcoefficientscoefficient = fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_exists_originalcoefficientscoefficient) + (fs_a_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_scalar_exists_originalcoefficientscoefficientsum = fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum) + (fs_r_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_scalar_exists_originalcoefficientscoefficientsum = fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum) + (fs_s_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps = fs_r_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps + fs_a_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_scalar_exists_originalcoefficientscoefficientresiduebound. pfa_gap_scalar_exists_originalcoefficientscoefficientresiduebound + S (pfc_value_scalar_exists_originalcoefficients) = (p)) /\\ ((exists pfa_offset_left_scalar_exists_originalcoefficientscoefficientresiduecongruence pfa_offset_right_scalar_exists_originalcoefficientscoefficientresiduecongruence. (pfc_natural_sum_scalar_exists_originalcoefficientscoefficient) + (p) * pfa_offset_left_scalar_exists_originalcoefficientscoefficientresiduecongruence = (pfc_value_scalar_exists_originalcoefficients) + (p) * pfa_offset_right_scalar_exists_originalcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hc", "cases hcopy", "cases hcopy_right", "cases hcopy_right_right", "have hs : exists sb sc. ((exists pfa_gap_scalar_exists_inputscalar. pfa_gap_scalar_exists_inputscalar + S (k) = (p)) /\\ ((forall pfp_index_scalar_exists_input. (exists pfa_gap_scalar_exists_inputindex. pfa_gap_scalar_exists_inputindex + S (pfp_index_scalar_exists_input) = (M)) -> exists pfp_source_scalar_exists_input pfp_value_scalar_exists_input. ((((exists ff_h_pfp_scalar_exists_inputsource. ff_h_pfp_scalar_exists_inputsource + S (pfp_source_scalar_exists_input) = S ((S (pfp_index_scalar_exists_input)) * bc)) /\\ exists ff_q_pfp_scalar_exists_inputsource. bb = ff_q_pfp_scalar_exists_inputsource * S ((S (pfp_index_scalar_exists_input)) * bc) + (pfp_source_scalar_exists_input))) /\\ (((((exists ff_h_pfp_scalar_exists_inputtarget. ff_h_pfp_scalar_exists_inputtarget + S (pfp_value_scalar_exists_input) = S ((S (pfp_index_scalar_exists_input)) * sc)) /\\ exists ff_q_pfp_scalar_exists_inputtarget. sb = ff_q_pfp_scalar_exists_inputtarget * S ((S (pfp_index_scalar_exists_input)) * sc) + (pfp_value_scalar_exists_input))) /\\ ((((exists pfa_gap_scalar_exists_inputoperationleft. pfa_gap_scalar_exists_inputoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_exists_inputoperationright. pfa_gap_scalar_exists_inputoperationright + S (pfp_source_scalar_exists_input) = (p)) /\\ ((((exists pfa_gap_scalar_exists_inputoperationresultbound. pfa_gap_scalar_exists_inputoperationresultbound + S (pfp_value_scalar_exists_input) = (p)) /\\ ((exists pfa_offset_left_scalar_exists_inputoperationresultcongruence pfa_offset_right_scalar_exists_inputoperationresultcongruence. ((k) * (pfp_source_scalar_exists_input)) + (p) * pfa_offset_left_scalar_exists_inputoperationresultcongruence = (pfp_value_scalar_exists_input) + (p) * pfa_offset_right_scalar_exists_inputoperationresultcongruence))))))))))))))))", "specialize prime_field_polynomial_scale_exists (p)", "specialize prime_field_polynomial_scale_exists (k)", "specialize prime_field_polynomial_scale_exists (bb)", "specialize prime_field_polynomial_scale_exists (bc)", "specialize prime_field_polynomial_scale_exists (M)", "apply prime_field_polynomial_scale_exists", "exact hp", "exact hk", "exact hcopy_right_left", "cases hs", "cases hs_witness", "have hb : ((forall fom_index_pfp_scalar_exists_old_bound. (exists fom_gap_pfp_scalar_exists_old_bound_index_bound. fom_gap_pfp_scalar_exists_old_bound_index_bound + S (fom_index_pfp_scalar_exists_old_bound) = M) -> exists fom_value_pfp_scalar_exists_old_bound. ((((exists fom_beta_height_pfp_scalar_exists_old_bound_entry. fom_beta_height_pfp_scalar_exists_old_bound_entry + S (fom_value_pfp_scalar_exists_old_bound) = S ((S (fom_index_pfp_scalar_exists_old_bound)) * bc)) /\\ exists fom_beta_quotient_pfp_scalar_exists_old_bound_entry. bb = fom_beta_quotient_pfp_scalar_exists_old_bound_entry * S ((S (fom_index_pfp_scalar_exists_old_bound)) * bc) + (fom_value_pfp_scalar_exists_old_bound))) /\\ (exists fom_gap_pfp_scalar_exists_old_bound_value_bound. fom_gap_pfp_scalar_exists_old_bound_value_bound + S (fom_value_pfp_scalar_exists_old_bound) = p))) /\\ ((forall fom_index_pfp_scalar_exists_scaled_bound. (exists fom_gap_pfp_scalar_exists_scaled_bound_index_bound. fom_gap_pfp_scalar_exists_scaled_bound_index_bound + S (fom_index_pfp_scalar_exists_scaled_bound) = M) -> exists fom_value_pfp_scalar_exists_scaled_bound. ((((exists fom_beta_height_pfp_scalar_exists_scaled_bound_entry. fom_beta_height_pfp_scalar_exists_scaled_bound_entry + S (fom_value_pfp_scalar_exists_scaled_bound) = S ((S (fom_index_pfp_scalar_exists_scaled_bound)) * x1)) /\\ exists fom_beta_quotient_pfp_scalar_exists_scaled_bound_entry. x = fom_beta_quotient_pfp_scalar_exists_scaled_bound_entry * S ((S (fom_index_pfp_scalar_exists_scaled_bound)) * x1) + (fom_value_pfp_scalar_exists_scaled_bound))) /\\ (exists fom_gap_pfp_scalar_exists_scaled_bound_value_bound. fom_gap_pfp_scalar_exists_scaled_bound_value_bound + S (fom_value_pfp_scalar_exists_scaled_bound) = p)))))", "specialize prime_field_polynomial_scale_bounded (p)", "specialize prime_field_polynomial_scale_bounded (k)", "specialize prime_field_polynomial_scale_bounded (bb)", "specialize prime_field_polynomial_scale_bounded (bc)", "specialize prime_field_polynomial_scale_bounded (x)", "specialize prime_field_polynomial_scale_bounded (x1)", "specialize prime_field_polynomial_scale_bounded (M)", "apply prime_field_polynomial_scale_bounded", "exact hs_witness_witness", "cases hb", "have hd : exists d e. ((forall fom_index_pfp_scalar_exists_chosen_productleft. (exists fom_gap_pfp_scalar_exists_chosen_productleft_index_bound. fom_gap_pfp_scalar_exists_chosen_productleft_index_bound + S (fom_index_pfp_scalar_exists_chosen_productleft) = L) -> exists fom_value_pfp_scalar_exists_chosen_productleft. ((((exists fom_beta_height_pfp_scalar_exists_chosen_productleft_entry. fom_beta_height_pfp_scalar_exists_chosen_productleft_entry + S (fom_value_pfp_scalar_exists_chosen_productleft) = S ((S (fom_index_pfp_scalar_exists_chosen_productleft)) * ac)) /\\ exists fom_beta_quotient_pfp_scalar_exists_chosen_productleft_entry. ab = fom_beta_quotient_pfp_scalar_exists_chosen_productleft_entry * S ((S (fom_index_pfp_scalar_exists_chosen_productleft)) * ac) + (fom_value_pfp_scalar_exists_chosen_productleft))) /\\ (exists fom_gap_pfp_scalar_exists_chosen_productleft_value_bound. fom_gap_pfp_scalar_exists_chosen_productleft_value_bound + S (fom_value_pfp_scalar_exists_chosen_productleft) = p))) /\\ (((forall fom_index_pfp_scalar_exists_chosen_productright. (exists fom_gap_pfp_scalar_exists_chosen_productright_index_bound. fom_gap_pfp_scalar_exists_chosen_productright_index_bound + S (fom_index_pfp_scalar_exists_chosen_productright) = M) -> exists fom_value_pfp_scalar_exists_chosen_productright. ((((exists fom_beta_height_pfp_scalar_exists_chosen_productright_entry. fom_beta_height_pfp_scalar_exists_chosen_productright_entry + S (fom_value_pfp_scalar_exists_chosen_productright) = S ((S (fom_index_pfp_scalar_exists_chosen_productright)) * x1)) /\\ exists fom_beta_quotient_pfp_scalar_exists_chosen_productright_entry. x = fom_beta_quotient_pfp_scalar_exists_chosen_productright_entry * S ((S (fom_index_pfp_scalar_exists_chosen_productright)) * x1) + (fom_value_pfp_scalar_exists_chosen_productright))) /\\ (exists fom_gap_pfp_scalar_exists_chosen_productright_value_bound. fom_gap_pfp_scalar_exists_chosen_productright_value_bound + S (fom_value_pfp_scalar_exists_chosen_productright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_scalar_exists_chosen_productcoefficients. (exists pfa_gap_scalar_exists_chosen_productcoefficientsbound. pfa_gap_scalar_exists_chosen_productcoefficientsbound + S (pfc_index_scalar_exists_chosen_productcoefficients) = (N)) -> exists pfc_value_scalar_exists_chosen_productcoefficients. ((((exists ff_h_pfp_scalar_exists_chosen_productcoefficientsentry. ff_h_pfp_scalar_exists_chosen_productcoefficientsentry + S (pfc_value_scalar_exists_chosen_productcoefficients) = S ((S (pfc_index_scalar_exists_chosen_productcoefficients)) * e)) /\\ exists ff_q_pfp_scalar_exists_chosen_productcoefficientsentry. d = ff_q_pfp_scalar_exists_chosen_productcoefficientsentry * S ((S (pfc_index_scalar_exists_chosen_productcoefficients)) * e) + (pfc_value_scalar_exists_chosen_productcoefficients))) /\\ ((exists pfc_terms_code_scalar_exists_chosen_productcoefficientscoefficient pfc_terms_scale_scalar_exists_chosen_productcoefficientscoefficient pfc_natural_sum_scalar_exists_chosen_productcoefficientscoefficient. ((forall pfc_index_scalar_exists_chosen_productcoefficientscoefficientdiagonal. (exists pfa_gap_scalar_exists_chosen_productcoefficientscoefficientdiagonalbound. pfa_gap_scalar_exists_chosen_productcoefficientscoefficientdiagonalbound + S (pfc_index_scalar_exists_chosen_productcoefficientscoefficientdiagonal) = (S (pfc_index_scalar_exists_chosen_productcoefficients))) -> exists pfc_value_scalar_exists_chosen_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_scalar_exists_chosen_productcoefficientscoefficientdiagonalentry. ff_h_pfp_scalar_exists_chosen_productcoefficientscoefficientdiagonalentry + S (pfc_value_scalar_exists_chosen_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_scalar_exists_chosen_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_exists_chosen_productcoefficientscoefficient)) /\\ exists ff_q_pfp_scalar_exists_chosen_productcoefficientscoefficientdiagonalentry. pfc_terms_code_scalar_exists_chosen_productcoefficientscoefficient = ff_q_pfp_scalar_exists_chosen_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_scalar_exists_chosen_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_exists_chosen_productcoefficientscoefficient) + (pfc_value_scalar_exists_chosen_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm pfc_left_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm pfc_right_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm. (((pfc_index_scalar_exists_chosen_productcoefficientscoefficientdiagonal)+pfc_complement_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm=(pfc_index_scalar_exists_chosen_productcoefficients)) /\\ ((((((exists pfa_gap_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_scalar_exists_chosen_productcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_scalar_exists_chosen_productcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_scalar_exists_chosen_productcoefficientscoefficientdiagonal)) * ac) + (pfc_left_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_scalar_exists_chosen_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermrightentry. x = ff_q_pfp_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm)) * x1) + (pfc_right_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_scalar_exists_chosen_productcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_scalar_exists_chosen_productcoefficientscoefficientdiagonal)=pfc_left_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm*pfc_right_scalar_exists_chosen_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_scalar_exists_chosen_productcoefficientscoefficientsum fs_v_pfc_scalar_exists_chosen_productcoefficientscoefficientsum. ((((exists fs_h_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_start. fs_h_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_exists_chosen_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_start. fs_u_pfc_scalar_exists_chosen_productcoefficientscoefficientsum = fs_q_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_scalar_exists_chosen_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_scalar_exists_chosen_productcoefficientscoefficient) = S ((S (S (pfc_index_scalar_exists_chosen_productcoefficients))) * fs_v_pfc_scalar_exists_chosen_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_scalar_exists_chosen_productcoefficientscoefficientsum = fs_q_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_scalar_exists_chosen_productcoefficients))) * fs_v_pfc_scalar_exists_chosen_productcoefficientscoefficientsum) + (pfc_natural_sum_scalar_exists_chosen_productcoefficientscoefficient))) /\\ forall fs_i_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps = S (pfc_index_scalar_exists_chosen_productcoefficients)) -> exists fs_a_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps fs_r_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps fs_s_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_exists_chosen_productcoefficientscoefficient)) /\\ exists fs_q_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_scalar_exists_chosen_productcoefficientscoefficient = fs_q_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_exists_chosen_productcoefficientscoefficient) + (fs_a_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_chosen_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_scalar_exists_chosen_productcoefficientscoefficientsum = fs_q_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_chosen_productcoefficientscoefficientsum) + (fs_r_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_chosen_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_scalar_exists_chosen_productcoefficientscoefficientsum = fs_q_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_chosen_productcoefficientscoefficientsum) + (fs_s_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps = fs_r_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps + fs_a_pfc_scalar_exists_chosen_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_scalar_exists_chosen_productcoefficientscoefficientresiduebound. pfa_gap_scalar_exists_chosen_productcoefficientscoefficientresiduebound + S (pfc_value_scalar_exists_chosen_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_scalar_exists_chosen_productcoefficientscoefficientresiduecongruence pfa_offset_right_scalar_exists_chosen_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_scalar_exists_chosen_productcoefficientscoefficient) + (p) * pfa_offset_left_scalar_exists_chosen_productcoefficientscoefficientresiduecongruence = (pfc_value_scalar_exists_chosen_productcoefficients) + (p) * pfa_offset_right_scalar_exists_chosen_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (ab)", "specialize prime_field_polynomial_convolution_at_length_exists (ac)", "specialize prime_field_polynomial_convolution_at_length_exists (L)", "specialize prime_field_polynomial_convolution_at_length_exists (x)", "specialize prime_field_polynomial_convolution_at_length_exists (x1)", "specialize prime_field_polynomial_convolution_at_length_exists (M)", "specialize prime_field_polynomial_convolution_at_length_exists (N)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp", "exact hcopy_left", "exact hb_right", "exact hcopy_right_right_left", "cases hd", "cases hd_witness", "have he : exists eb ec. ((exists pfa_gap_scalar_exists_outputscalar. pfa_gap_scalar_exists_outputscalar + S (k) = (p)) /\\ ((forall pfp_index_scalar_exists_output. (exists pfa_gap_scalar_exists_outputindex. pfa_gap_scalar_exists_outputindex + S (pfp_index_scalar_exists_output) = (N)) -> exists pfp_source_scalar_exists_output pfp_value_scalar_exists_output. ((((exists ff_h_pfp_scalar_exists_outputsource. ff_h_pfp_scalar_exists_outputsource + S (pfp_source_scalar_exists_output) = S ((S (pfp_index_scalar_exists_output)) * cc)) /\\ exists ff_q_pfp_scalar_exists_outputsource. cb = ff_q_pfp_scalar_exists_outputsource * S ((S (pfp_index_scalar_exists_output)) * cc) + (pfp_source_scalar_exists_output))) /\\ (((((exists ff_h_pfp_scalar_exists_outputtarget. ff_h_pfp_scalar_exists_outputtarget + S (pfp_value_scalar_exists_output) = S ((S (pfp_index_scalar_exists_output)) * ec)) /\\ exists ff_q_pfp_scalar_exists_outputtarget. eb = ff_q_pfp_scalar_exists_outputtarget * S ((S (pfp_index_scalar_exists_output)) * ec) + (pfp_value_scalar_exists_output))) /\\ ((((exists pfa_gap_scalar_exists_outputoperationleft. pfa_gap_scalar_exists_outputoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_exists_outputoperationright. pfa_gap_scalar_exists_outputoperationright + S (pfp_source_scalar_exists_output) = (p)) /\\ ((((exists pfa_gap_scalar_exists_outputoperationresultbound. pfa_gap_scalar_exists_outputoperationresultbound + S (pfp_value_scalar_exists_output) = (p)) /\\ ((exists pfa_offset_left_scalar_exists_outputoperationresultcongruence pfa_offset_right_scalar_exists_outputoperationresultcongruence. ((k) * (pfp_source_scalar_exists_output)) + (p) * pfa_offset_left_scalar_exists_outputoperationresultcongruence = (pfp_value_scalar_exists_output) + (p) * pfa_offset_right_scalar_exists_outputoperationresultcongruence))))))))))))))))", "specialize prime_field_polynomial_scale_exists (p)", "specialize prime_field_polynomial_scale_exists (k)", "specialize prime_field_polynomial_scale_exists (cb)", "specialize prime_field_polynomial_scale_exists (cc)", "specialize prime_field_polynomial_scale_exists (N)", "apply prime_field_polynomial_scale_exists", "exact hp", "exact hk", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (ab)", "specialize prime_field_polynomial_convolution_bounded (ac)", "specialize prime_field_polynomial_convolution_bounded (L)", "specialize prime_field_polynomial_convolution_bounded (bb)", "specialize prime_field_polynomial_convolution_bounded (bc)", "specialize prime_field_polynomial_convolution_bounded (M)", "specialize prime_field_polynomial_convolution_bounded (cb)", "specialize prime_field_polynomial_convolution_bounded (cc)", "specialize prime_field_polynomial_convolution_bounded (N)", "apply prime_field_polynomial_convolution_bounded", "exact hc", "cases he", "cases he_witness", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "split", "exact hs_witness_witness", "split", "exact hd_witness_witness", "split", "exact he_witness_witness", "have hdata : ((N=N) /\\ ((forall mdr_i_pfp_scalar_exists_comparison mdr_a_pfp_scalar_exists_comparison. (exists mdr_gap_pfp_scalar_exists_comparisonb. mdr_gap_pfp_scalar_exists_comparisonb + S (mdr_i_pfp_scalar_exists_comparison) = (N)) -> (((exists ff_h_mdr_pfp_scalar_exists_comparisono. ff_h_mdr_pfp_scalar_exists_comparisono + S (mdr_a_pfp_scalar_exists_comparison) = S ((S (mdr_i_pfp_scalar_exists_comparison)) * x3)) /\\ exists ff_q_mdr_pfp_scalar_exists_comparisono. x2 = ff_q_mdr_pfp_scalar_exists_comparisono * S ((S (mdr_i_pfp_scalar_exists_comparison)) * x3) + (mdr_a_pfp_scalar_exists_comparison))) -> (((exists ff_h_mdr_pfp_scalar_exists_comparisonn. ff_h_mdr_pfp_scalar_exists_comparisonn + S (mdr_a_pfp_scalar_exists_comparison) = S ((S (mdr_i_pfp_scalar_exists_comparison)) * x5)) /\\ exists ff_q_mdr_pfp_scalar_exists_comparisonn. x4 = ff_q_mdr_pfp_scalar_exists_comparisonn * S ((S (mdr_i_pfp_scalar_exists_comparison)) * x5) + (mdr_a_pfp_scalar_exists_comparison))))))", "specialize prime_field_polynomial_convolution_right_scale_equal (p)", "specialize prime_field_polynomial_convolution_right_scale_equal (k)", "specialize prime_field_polynomial_convolution_right_scale_equal (ab)", "specialize prime_field_polynomial_convolution_right_scale_equal (ac)", "specialize prime_field_polynomial_convolution_right_scale_equal (L)", "specialize prime_field_polynomial_convolution_right_scale_equal (bb)", "specialize prime_field_polynomial_convolution_right_scale_equal (bc)", "specialize prime_field_polynomial_convolution_right_scale_equal (M)", "specialize prime_field_polynomial_convolution_right_scale_equal (x)", "specialize prime_field_polynomial_convolution_right_scale_equal (x1)", "specialize prime_field_polynomial_convolution_right_scale_equal (cb)", "specialize prime_field_polynomial_convolution_right_scale_equal (cc)", "specialize prime_field_polynomial_convolution_right_scale_equal (N)", "specialize prime_field_polynomial_convolution_right_scale_equal (x2)", "specialize prime_field_polynomial_convolution_right_scale_equal (x3)", "specialize prime_field_polynomial_convolution_right_scale_equal (N)", "specialize prime_field_polynomial_convolution_right_scale_equal (x4)", "specialize prime_field_polynomial_convolution_right_scale_equal (x5)", "apply prime_field_polynomial_convolution_right_scale_equal", "exact hs_witness_witness", "exact hc", "exact hd_witness_witness", "exact he_witness_witness", "cases hdata", "exact hdata_right" ], "script_sha256": "e953743fb1f6cdc49efa7a3e0c95ef2535832f88553722633c26644f1299c26a", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_scalar_convolution_candidate.py", "sha256": "e84f1c77c6c03fa5f08635aeede53591625d1c2bfcdfb64fbd379c33878aee0e" }, "statement": "forall p k ab ac L bb bc M cb cc N. (~(p=0)) -> (exists pfa_gap_scalar_exists_scalar. pfa_gap_scalar_exists_scalar + S (k) = (p)) -> (((forall fom_index_pfp_scalar_exists_originalleft. (exists fom_gap_pfp_scalar_exists_originalleft_index_bound. fom_gap_pfp_scalar_exists_originalleft_index_bound + S (fom_index_pfp_scalar_exists_originalleft) = L) -> exists fom_value_pfp_scalar_exists_originalleft. ((((exists fom_beta_height_pfp_scalar_exists_originalleft_entry. fom_beta_height_pfp_scalar_exists_originalleft_entry + S (fom_value_pfp_scalar_exists_originalleft) = S ((S (fom_index_pfp_scalar_exists_originalleft)) * ac)) /\\ exists fom_beta_quotient_pfp_scalar_exists_originalleft_entry. ab = fom_beta_quotient_pfp_scalar_exists_originalleft_entry * S ((S (fom_index_pfp_scalar_exists_originalleft)) * ac) + (fom_value_pfp_scalar_exists_originalleft))) /\\ (exists fom_gap_pfp_scalar_exists_originalleft_value_bound. fom_gap_pfp_scalar_exists_originalleft_value_bound + S (fom_value_pfp_scalar_exists_originalleft) = p))) /\\ (((forall fom_index_pfp_scalar_exists_originalright. (exists fom_gap_pfp_scalar_exists_originalright_index_bound. fom_gap_pfp_scalar_exists_originalright_index_bound + S (fom_index_pfp_scalar_exists_originalright) = M) -> exists fom_value_pfp_scalar_exists_originalright. ((((exists fom_beta_height_pfp_scalar_exists_originalright_entry. fom_beta_height_pfp_scalar_exists_originalright_entry + S (fom_value_pfp_scalar_exists_originalright) = S ((S (fom_index_pfp_scalar_exists_originalright)) * bc)) /\\ exists fom_beta_quotient_pfp_scalar_exists_originalright_entry. bb = fom_beta_quotient_pfp_scalar_exists_originalright_entry * S ((S (fom_index_pfp_scalar_exists_originalright)) * bc) + (fom_value_pfp_scalar_exists_originalright))) /\\ (exists fom_gap_pfp_scalar_exists_originalright_value_bound. fom_gap_pfp_scalar_exists_originalright_value_bound + S (fom_value_pfp_scalar_exists_originalright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_scalar_exists_originalcoefficients. (exists pfa_gap_scalar_exists_originalcoefficientsbound. pfa_gap_scalar_exists_originalcoefficientsbound + S (pfc_index_scalar_exists_originalcoefficients) = (N)) -> exists pfc_value_scalar_exists_originalcoefficients. ((((exists ff_h_pfp_scalar_exists_originalcoefficientsentry. ff_h_pfp_scalar_exists_originalcoefficientsentry + S (pfc_value_scalar_exists_originalcoefficients) = S ((S (pfc_index_scalar_exists_originalcoefficients)) * cc)) /\\ exists ff_q_pfp_scalar_exists_originalcoefficientsentry. cb = ff_q_pfp_scalar_exists_originalcoefficientsentry * S ((S (pfc_index_scalar_exists_originalcoefficients)) * cc) + (pfc_value_scalar_exists_originalcoefficients))) /\\ ((exists pfc_terms_code_scalar_exists_originalcoefficientscoefficient pfc_terms_scale_scalar_exists_originalcoefficientscoefficient pfc_natural_sum_scalar_exists_originalcoefficientscoefficient. ((forall pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal. (exists pfa_gap_scalar_exists_originalcoefficientscoefficientdiagonalbound. pfa_gap_scalar_exists_originalcoefficientscoefficientdiagonalbound + S (pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal) = (S (pfc_index_scalar_exists_originalcoefficients))) -> exists pfc_value_scalar_exists_originalcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_scalar_exists_originalcoefficientscoefficientdiagonalentry. ff_h_pfp_scalar_exists_originalcoefficientscoefficientdiagonalentry + S (pfc_value_scalar_exists_originalcoefficientscoefficientdiagonal) = S ((S (pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_exists_originalcoefficientscoefficient)) /\\ exists ff_q_pfp_scalar_exists_originalcoefficientscoefficientdiagonalentry. pfc_terms_code_scalar_exists_originalcoefficientscoefficient = ff_q_pfp_scalar_exists_originalcoefficientscoefficientdiagonalentry * S ((S (pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_exists_originalcoefficientscoefficient) + (pfc_value_scalar_exists_originalcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_scalar_exists_originalcoefficientscoefficientdiagonalterm pfc_left_scalar_exists_originalcoefficientscoefficientdiagonalterm pfc_right_scalar_exists_originalcoefficientscoefficientdiagonalterm. (((pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal)+pfc_complement_scalar_exists_originalcoefficientscoefficientdiagonalterm=(pfc_index_scalar_exists_originalcoefficients)) /\\ ((((((exists pfa_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermleftinside. pfa_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_scalar_exists_originalcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal)) * ac) + (pfc_left_scalar_exists_originalcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_scalar_exists_originalcoefficientscoefficientdiagonal)) /\\ (((pfc_left_scalar_exists_originalcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermrightinside. pfa_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_scalar_exists_originalcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_scalar_exists_originalcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_scalar_exists_originalcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_scalar_exists_originalcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_scalar_exists_originalcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_scalar_exists_originalcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_scalar_exists_originalcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_scalar_exists_originalcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_scalar_exists_originalcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_scalar_exists_originalcoefficientscoefficientdiagonal)=pfc_left_scalar_exists_originalcoefficientscoefficientdiagonalterm*pfc_right_scalar_exists_originalcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_scalar_exists_originalcoefficientscoefficientsum fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum. ((((exists fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_start. fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_start. fs_u_pfc_scalar_exists_originalcoefficientscoefficientsum = fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_terminal. fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_scalar_exists_originalcoefficientscoefficient) = S ((S (S (pfc_index_scalar_exists_originalcoefficients))) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_terminal. fs_u_pfc_scalar_exists_originalcoefficientscoefficientsum = fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_scalar_exists_originalcoefficients))) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum) + (pfc_natural_sum_scalar_exists_originalcoefficientscoefficient))) /\\ forall fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps = S (pfc_index_scalar_exists_originalcoefficients)) -> exists fs_a_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps fs_r_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps fs_s_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_exists_originalcoefficientscoefficient)) /\\ exists fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_scalar_exists_originalcoefficientscoefficient = fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_exists_originalcoefficientscoefficient) + (fs_a_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_scalar_exists_originalcoefficientscoefficientsum = fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum) + (fs_r_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_scalar_exists_originalcoefficientscoefficientsum = fs_q_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_originalcoefficientscoefficientsum) + (fs_s_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps = fs_r_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps + fs_a_pfc_scalar_exists_originalcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_scalar_exists_originalcoefficientscoefficientresiduebound. pfa_gap_scalar_exists_originalcoefficientscoefficientresiduebound + S (pfc_value_scalar_exists_originalcoefficients) = (p)) /\\ ((exists pfa_offset_left_scalar_exists_originalcoefficientscoefficientresiduecongruence pfa_offset_right_scalar_exists_originalcoefficientscoefficientresiduecongruence. (pfc_natural_sum_scalar_exists_originalcoefficientscoefficient) + (p) * pfa_offset_left_scalar_exists_originalcoefficientscoefficientresiduecongruence = (pfc_value_scalar_exists_originalcoefficients) + (p) * pfa_offset_right_scalar_exists_originalcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (exists sb sc db dc eb ec. ((((exists pfa_gap_scalar_exists_inputscalar. pfa_gap_scalar_exists_inputscalar + S (k) = (p)) /\\ ((forall pfp_index_scalar_exists_input. (exists pfa_gap_scalar_exists_inputindex. pfa_gap_scalar_exists_inputindex + S (pfp_index_scalar_exists_input) = (M)) -> exists pfp_source_scalar_exists_input pfp_value_scalar_exists_input. ((((exists ff_h_pfp_scalar_exists_inputsource. ff_h_pfp_scalar_exists_inputsource + S (pfp_source_scalar_exists_input) = S ((S (pfp_index_scalar_exists_input)) * bc)) /\\ exists ff_q_pfp_scalar_exists_inputsource. bb = ff_q_pfp_scalar_exists_inputsource * S ((S (pfp_index_scalar_exists_input)) * bc) + (pfp_source_scalar_exists_input))) /\\ (((((exists ff_h_pfp_scalar_exists_inputtarget. ff_h_pfp_scalar_exists_inputtarget + S (pfp_value_scalar_exists_input) = S ((S (pfp_index_scalar_exists_input)) * sc)) /\\ exists ff_q_pfp_scalar_exists_inputtarget. sb = ff_q_pfp_scalar_exists_inputtarget * S ((S (pfp_index_scalar_exists_input)) * sc) + (pfp_value_scalar_exists_input))) /\\ ((((exists pfa_gap_scalar_exists_inputoperationleft. pfa_gap_scalar_exists_inputoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_exists_inputoperationright. pfa_gap_scalar_exists_inputoperationright + S (pfp_source_scalar_exists_input) = (p)) /\\ ((((exists pfa_gap_scalar_exists_inputoperationresultbound. pfa_gap_scalar_exists_inputoperationresultbound + S (pfp_value_scalar_exists_input) = (p)) /\\ ((exists pfa_offset_left_scalar_exists_inputoperationresultcongruence pfa_offset_right_scalar_exists_inputoperationresultcongruence. ((k) * (pfp_source_scalar_exists_input)) + (p) * pfa_offset_left_scalar_exists_inputoperationresultcongruence = (pfp_value_scalar_exists_input) + (p) * pfa_offset_right_scalar_exists_inputoperationresultcongruence))))))))))))))))) /\\ (((((forall fom_index_pfp_scalar_exists_newleft. (exists fom_gap_pfp_scalar_exists_newleft_index_bound. fom_gap_pfp_scalar_exists_newleft_index_bound + S (fom_index_pfp_scalar_exists_newleft) = L) -> exists fom_value_pfp_scalar_exists_newleft. ((((exists fom_beta_height_pfp_scalar_exists_newleft_entry. fom_beta_height_pfp_scalar_exists_newleft_entry + S (fom_value_pfp_scalar_exists_newleft) = S ((S (fom_index_pfp_scalar_exists_newleft)) * ac)) /\\ exists fom_beta_quotient_pfp_scalar_exists_newleft_entry. ab = fom_beta_quotient_pfp_scalar_exists_newleft_entry * S ((S (fom_index_pfp_scalar_exists_newleft)) * ac) + (fom_value_pfp_scalar_exists_newleft))) /\\ (exists fom_gap_pfp_scalar_exists_newleft_value_bound. fom_gap_pfp_scalar_exists_newleft_value_bound + S (fom_value_pfp_scalar_exists_newleft) = p))) /\\ (((forall fom_index_pfp_scalar_exists_newright. (exists fom_gap_pfp_scalar_exists_newright_index_bound. fom_gap_pfp_scalar_exists_newright_index_bound + S (fom_index_pfp_scalar_exists_newright) = M) -> exists fom_value_pfp_scalar_exists_newright. ((((exists fom_beta_height_pfp_scalar_exists_newright_entry. fom_beta_height_pfp_scalar_exists_newright_entry + S (fom_value_pfp_scalar_exists_newright) = S ((S (fom_index_pfp_scalar_exists_newright)) * sc)) /\\ exists fom_beta_quotient_pfp_scalar_exists_newright_entry. sb = fom_beta_quotient_pfp_scalar_exists_newright_entry * S ((S (fom_index_pfp_scalar_exists_newright)) * sc) + (fom_value_pfp_scalar_exists_newright))) /\\ (exists fom_gap_pfp_scalar_exists_newright_value_bound. fom_gap_pfp_scalar_exists_newright_value_bound + S (fom_value_pfp_scalar_exists_newright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_scalar_exists_newcoefficients. (exists pfa_gap_scalar_exists_newcoefficientsbound. pfa_gap_scalar_exists_newcoefficientsbound + S (pfc_index_scalar_exists_newcoefficients) = (N)) -> exists pfc_value_scalar_exists_newcoefficients. ((((exists ff_h_pfp_scalar_exists_newcoefficientsentry. ff_h_pfp_scalar_exists_newcoefficientsentry + S (pfc_value_scalar_exists_newcoefficients) = S ((S (pfc_index_scalar_exists_newcoefficients)) * dc)) /\\ exists ff_q_pfp_scalar_exists_newcoefficientsentry. db = ff_q_pfp_scalar_exists_newcoefficientsentry * S ((S (pfc_index_scalar_exists_newcoefficients)) * dc) + (pfc_value_scalar_exists_newcoefficients))) /\\ ((exists pfc_terms_code_scalar_exists_newcoefficientscoefficient pfc_terms_scale_scalar_exists_newcoefficientscoefficient pfc_natural_sum_scalar_exists_newcoefficientscoefficient. ((forall pfc_index_scalar_exists_newcoefficientscoefficientdiagonal. (exists pfa_gap_scalar_exists_newcoefficientscoefficientdiagonalbound. pfa_gap_scalar_exists_newcoefficientscoefficientdiagonalbound + S (pfc_index_scalar_exists_newcoefficientscoefficientdiagonal) = (S (pfc_index_scalar_exists_newcoefficients))) -> exists pfc_value_scalar_exists_newcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_scalar_exists_newcoefficientscoefficientdiagonalentry. ff_h_pfp_scalar_exists_newcoefficientscoefficientdiagonalentry + S (pfc_value_scalar_exists_newcoefficientscoefficientdiagonal) = S ((S (pfc_index_scalar_exists_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_exists_newcoefficientscoefficient)) /\\ exists ff_q_pfp_scalar_exists_newcoefficientscoefficientdiagonalentry. pfc_terms_code_scalar_exists_newcoefficientscoefficient = ff_q_pfp_scalar_exists_newcoefficientscoefficientdiagonalentry * S ((S (pfc_index_scalar_exists_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_exists_newcoefficientscoefficient) + (pfc_value_scalar_exists_newcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_scalar_exists_newcoefficientscoefficientdiagonalterm pfc_left_scalar_exists_newcoefficientscoefficientdiagonalterm pfc_right_scalar_exists_newcoefficientscoefficientdiagonalterm. (((pfc_index_scalar_exists_newcoefficientscoefficientdiagonal)+pfc_complement_scalar_exists_newcoefficientscoefficientdiagonalterm=(pfc_index_scalar_exists_newcoefficients)) /\\ ((((((exists pfa_gap_scalar_exists_newcoefficientscoefficientdiagonaltermleftinside. pfa_gap_scalar_exists_newcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_scalar_exists_newcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_scalar_exists_newcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_scalar_exists_newcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_scalar_exists_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_scalar_exists_newcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_scalar_exists_newcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_scalar_exists_newcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_scalar_exists_newcoefficientscoefficientdiagonal)) * ac) + (pfc_left_scalar_exists_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_exists_newcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_scalar_exists_newcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_scalar_exists_newcoefficientscoefficientdiagonal)) /\\ (((pfc_left_scalar_exists_newcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_scalar_exists_newcoefficientscoefficientdiagonaltermrightinside. pfa_gap_scalar_exists_newcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_scalar_exists_newcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_scalar_exists_newcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_scalar_exists_newcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_scalar_exists_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_scalar_exists_newcoefficientscoefficientdiagonalterm)) * sc)) /\\ exists ff_q_pfp_scalar_exists_newcoefficientscoefficientdiagonaltermrightentry. sb = ff_q_pfp_scalar_exists_newcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_scalar_exists_newcoefficientscoefficientdiagonalterm)) * sc) + (pfc_right_scalar_exists_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_exists_newcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_scalar_exists_newcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_scalar_exists_newcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_scalar_exists_newcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_scalar_exists_newcoefficientscoefficientdiagonal)=pfc_left_scalar_exists_newcoefficientscoefficientdiagonalterm*pfc_right_scalar_exists_newcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_scalar_exists_newcoefficientscoefficientsum fs_v_pfc_scalar_exists_newcoefficientscoefficientsum. ((((exists fs_h_pfc_scalar_exists_newcoefficientscoefficientsum_body_start. fs_h_pfc_scalar_exists_newcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_exists_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_newcoefficientscoefficientsum_body_start. fs_u_pfc_scalar_exists_newcoefficientscoefficientsum = fs_q_pfc_scalar_exists_newcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_scalar_exists_newcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_scalar_exists_newcoefficientscoefficientsum_body_terminal. fs_h_pfc_scalar_exists_newcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_scalar_exists_newcoefficientscoefficient) = S ((S (S (pfc_index_scalar_exists_newcoefficients))) * fs_v_pfc_scalar_exists_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_newcoefficientscoefficientsum_body_terminal. fs_u_pfc_scalar_exists_newcoefficientscoefficientsum = fs_q_pfc_scalar_exists_newcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_scalar_exists_newcoefficients))) * fs_v_pfc_scalar_exists_newcoefficientscoefficientsum) + (pfc_natural_sum_scalar_exists_newcoefficientscoefficient))) /\\ forall fs_i_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps = S (pfc_index_scalar_exists_newcoefficients)) -> exists fs_a_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps fs_r_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps fs_s_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_exists_newcoefficientscoefficient)) /\\ exists fs_q_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_scalar_exists_newcoefficientscoefficient = fs_q_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_exists_newcoefficientscoefficient) + (fs_a_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_scalar_exists_newcoefficientscoefficientsum = fs_q_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_newcoefficientscoefficientsum) + (fs_r_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_scalar_exists_newcoefficientscoefficientsum = fs_q_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_exists_newcoefficientscoefficientsum) + (fs_s_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps = fs_r_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps + fs_a_pfc_scalar_exists_newcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_scalar_exists_newcoefficientscoefficientresiduebound. pfa_gap_scalar_exists_newcoefficientscoefficientresiduebound + S (pfc_value_scalar_exists_newcoefficients) = (p)) /\\ ((exists pfa_offset_left_scalar_exists_newcoefficientscoefficientresiduecongruence pfa_offset_right_scalar_exists_newcoefficientscoefficientresiduecongruence. (pfc_natural_sum_scalar_exists_newcoefficientscoefficient) + (p) * pfa_offset_left_scalar_exists_newcoefficientscoefficientresiduecongruence = (pfc_value_scalar_exists_newcoefficients) + (p) * pfa_offset_right_scalar_exists_newcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((exists pfa_gap_scalar_exists_outputscalar. pfa_gap_scalar_exists_outputscalar + S (k) = (p)) /\\ ((forall pfp_index_scalar_exists_output. (exists pfa_gap_scalar_exists_outputindex. pfa_gap_scalar_exists_outputindex + S (pfp_index_scalar_exists_output) = (N)) -> exists pfp_source_scalar_exists_output pfp_value_scalar_exists_output. ((((exists ff_h_pfp_scalar_exists_outputsource. ff_h_pfp_scalar_exists_outputsource + S (pfp_source_scalar_exists_output) = S ((S (pfp_index_scalar_exists_output)) * cc)) /\\ exists ff_q_pfp_scalar_exists_outputsource. cb = ff_q_pfp_scalar_exists_outputsource * S ((S (pfp_index_scalar_exists_output)) * cc) + (pfp_source_scalar_exists_output))) /\\ (((((exists ff_h_pfp_scalar_exists_outputtarget. ff_h_pfp_scalar_exists_outputtarget + S (pfp_value_scalar_exists_output) = S ((S (pfp_index_scalar_exists_output)) * ec)) /\\ exists ff_q_pfp_scalar_exists_outputtarget. eb = ff_q_pfp_scalar_exists_outputtarget * S ((S (pfp_index_scalar_exists_output)) * ec) + (pfp_value_scalar_exists_output))) /\\ ((((exists pfa_gap_scalar_exists_outputoperationleft. pfa_gap_scalar_exists_outputoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_scalar_exists_outputoperationright. pfa_gap_scalar_exists_outputoperationright + S (pfp_source_scalar_exists_output) = (p)) /\\ ((((exists pfa_gap_scalar_exists_outputoperationresultbound. pfa_gap_scalar_exists_outputoperationresultbound + S (pfp_value_scalar_exists_output) = (p)) /\\ ((exists pfa_offset_left_scalar_exists_outputoperationresultcongruence pfa_offset_right_scalar_exists_outputoperationresultcongruence. ((k) * (pfp_source_scalar_exists_output)) + (p) * pfa_offset_left_scalar_exists_outputoperationresultcongruence = (pfp_value_scalar_exists_output) + (p) * pfa_offset_right_scalar_exists_outputoperationresultcongruence))))))))))))))))) /\\ ((forall mdr_i_pfp_scalar_exists_equal mdr_a_pfp_scalar_exists_equal. 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((((exists ff_h_pfp_scalar_zero_chosen_source. ff_h_pfp_scalar_zero_chosen_source + S (a) = S ((S (i)) * bc)) /\\ exists ff_q_pfp_scalar_zero_chosen_source. bb = ff_q_pfp_scalar_zero_chosen_source * S ((S (i)) * bc) + (a))) /\\ (((((exists ff_h_pfp_scalar_zero_chosen_target. ff_h_pfp_scalar_zero_chosen_target + S (r) = S ((S (i)) * sc)) /\\ exists ff_q_pfp_scalar_zero_chosen_target. sb = ff_q_pfp_scalar_zero_chosen_target * S ((S (i)) * sc) + (r))) /\\ ((((exists pfa_gap_scalar_zero_chosen_productleft. pfa_gap_scalar_zero_chosen_productleft + S (0) = (p)) /\\ (((exists pfa_gap_scalar_zero_chosen_productright. pfa_gap_scalar_zero_chosen_productright + S (a) = (p)) /\\ ((((exists pfa_gap_scalar_zero_chosen_productresultbound. pfa_gap_scalar_zero_chosen_productresultbound + S (r) = (p)) /\\ ((exists pfa_offset_left_scalar_zero_chosen_productresultcongruence pfa_offset_right_scalar_zero_chosen_productresultcongruence. ((0) * (a)) + (p) * pfa_offset_left_scalar_zero_chosen_productresultcongruence = (r) + (p) * pfa_offset_right_scalar_zero_chosen_productresultcongruence)))))))))))))", "specialize hs_right (i)", "apply hs_right", "exact hi", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_right", "have hm : ((exists pfa_gap_scalar_zero_actual_productleft. pfa_gap_scalar_zero_actual_productleft + S (0) = (p)) /\\ (((exists pfa_gap_scalar_zero_actual_productright. pfa_gap_scalar_zero_actual_productright + S (x) = (p)) /\\ ((((exists pfa_gap_scalar_zero_actual_productresultbound. pfa_gap_scalar_zero_actual_productresultbound + S (x1) = (p)) /\\ ((exists pfa_offset_left_scalar_zero_actual_productresultcongruence pfa_offset_right_scalar_zero_actual_productresultcongruence. ((0) * (x)) + (p) * pfa_offset_left_scalar_zero_actual_productresultcongruence = (x1) + (p) * pfa_offset_right_scalar_zero_actual_productresultcongruence))))))))", "exact hv_witness_witness_right_right", "cases hm", "cases hm_right", "have hzero : 0*x=0", "apply mul_zero_left", "rewrite hzero at hm_right_right", "have heq : x1=0", "specialize prime_field_residue_bounded_value (p)", "specialize prime_field_residue_bounded_value (0)", "specialize prime_field_residue_bounded_value (x1)", "apply prime_field_residue_bounded_value", "exact hs_left", "exact hm_right_right", "rewrite heq at hv_witness_witness_right_left", "rewrite heq at hv_witness_witness_right_left", "exact hv_witness_witness_right_left" ], "script_sha256": "7c80eb40df4828bd209890cf02d441d4ab3d28dbbe758281e3012806b0c6c1d7", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_scalar_convolution_candidate.py", "sha256": "e84f1c77c6c03fa5f08635aeede53591625d1c2bfcdfb64fbd379c33878aee0e" }, "statement": "forall p bb bc sb sc M. (((exists pfa_gap_scalar_zero_actualscalar. pfa_gap_scalar_zero_actualscalar + S (0) = (p)) /\\ ((forall pfp_index_scalar_zero_actual. (exists pfa_gap_scalar_zero_actualindex. pfa_gap_scalar_zero_actualindex + S (pfp_index_scalar_zero_actual) = (M)) -> exists pfp_source_scalar_zero_actual pfp_value_scalar_zero_actual. ((((exists ff_h_pfp_scalar_zero_actualsource. ff_h_pfp_scalar_zero_actualsource + S (pfp_source_scalar_zero_actual) = S ((S (pfp_index_scalar_zero_actual)) * bc)) /\\ exists ff_q_pfp_scalar_zero_actualsource. bb = ff_q_pfp_scalar_zero_actualsource * S ((S (pfp_index_scalar_zero_actual)) * bc) + (pfp_source_scalar_zero_actual))) /\\ (((((exists ff_h_pfp_scalar_zero_actualtarget. ff_h_pfp_scalar_zero_actualtarget + S (pfp_value_scalar_zero_actual) = S ((S (pfp_index_scalar_zero_actual)) * sc)) /\\ exists ff_q_pfp_scalar_zero_actualtarget. sb = ff_q_pfp_scalar_zero_actualtarget * S ((S (pfp_index_scalar_zero_actual)) * sc) + (pfp_value_scalar_zero_actual))) /\\ ((((exists pfa_gap_scalar_zero_actualoperationleft. pfa_gap_scalar_zero_actualoperationleft + S (0) = (p)) /\\ (((exists pfa_gap_scalar_zero_actualoperationright. pfa_gap_scalar_zero_actualoperationright + S (pfp_source_scalar_zero_actual) = (p)) /\\ ((((exists pfa_gap_scalar_zero_actualoperationresultbound. pfa_gap_scalar_zero_actualoperationresultbound + S (pfp_value_scalar_zero_actual) = (p)) /\\ ((exists pfa_offset_left_scalar_zero_actualoperationresultcongruence pfa_offset_right_scalar_zero_actualoperationresultcongruence. ((0) * (pfp_source_scalar_zero_actual)) + (p) * pfa_offset_left_scalar_zero_actualoperationresultcongruence = (pfp_value_scalar_zero_actual) + (p) * pfa_offset_right_scalar_zero_actualoperationresultcongruence))))))))))))))))) -> (forall pfp_repeat_index_scalar_zero_result. (exists pfa_gap_scalar_zero_resultindex. pfa_gap_scalar_zero_resultindex + S (pfp_repeat_index_scalar_zero_result) = (M)) -> (((exists ff_h_pfp_scalar_zero_resultentry. ff_h_pfp_scalar_zero_resultentry + S (0) = S ((S (pfp_repeat_index_scalar_zero_result)) * sc)) /\\ exists ff_q_pfp_scalar_zero_resultentry. sb = ff_q_pfp_scalar_zero_resultentry * S ((S (pfp_repeat_index_scalar_zero_result)) * sc) + (0))))", "statement_sha256": "204eb32da7c0fab12c7d9d38094bff073e68792d2cac8ea095431a88859c3d33", "summary": "Every actual scalar-zero output is an actually all-zero prefix, without primality and without dropping the scalar bound on an empty input.", "summary_sha256": "828215220132a407e96e617adb297e4eeca84ced672d7c3fdaa1964d9be6ba25" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "11bdf85e8b21c4e814747cb8b5f619492e8b0d131ed9c20e7ffc3a9dc3c85924", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 242, "bundle_sha256": 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"prime_field_polynomial_convolution_right_scale_zero", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro M", "intro sb", "intro sc", "intro db", "intro dc", "intro N", "intro hp", "intro hs", "intro hd", "specialize prime_field_polynomial_convolution_zero_right (p)", "specialize prime_field_polynomial_convolution_zero_right (ab)", "specialize prime_field_polynomial_convolution_zero_right (ac)", "specialize prime_field_polynomial_convolution_zero_right (L)", "specialize prime_field_polynomial_convolution_zero_right (sb)", "specialize prime_field_polynomial_convolution_zero_right (sc)", "specialize prime_field_polynomial_convolution_zero_right (M)", "specialize prime_field_polynomial_convolution_zero_right (db)", "specialize prime_field_polynomial_convolution_zero_right (dc)", "specialize prime_field_polynomial_convolution_zero_right (N)", "apply prime_field_polynomial_convolution_zero_right", "exact hp", "specialize prime_field_polynomial_scale_zero_value (p)", "specialize prime_field_polynomial_scale_zero_value (bb)", "specialize prime_field_polynomial_scale_zero_value (bc)", "specialize prime_field_polynomial_scale_zero_value (sb)", "specialize prime_field_polynomial_scale_zero_value (sc)", "specialize prime_field_polynomial_scale_zero_value (M)", "apply prime_field_polynomial_scale_zero_value", "exact hs", "exact hd" ], "script_sha256": "eef69720264d25673287f21e1a6a3f4d8fddbfb5d2526978830d69f162d8097f", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_scalar_convolution_candidate.py", "sha256": "e84f1c77c6c03fa5f08635aeede53591625d1c2bfcdfb64fbd379c33878aee0e" }, "statement": "forall p ab ac L bb bc M sb sc db dc N. (~(p=0)) -> (((exists pfa_gap_scalar_zero_product_inputscalar. pfa_gap_scalar_zero_product_inputscalar + S (0) = (p)) /\\ ((forall pfp_index_scalar_zero_product_input. (exists pfa_gap_scalar_zero_product_inputindex. pfa_gap_scalar_zero_product_inputindex + S (pfp_index_scalar_zero_product_input) = (M)) -> exists pfp_source_scalar_zero_product_input pfp_value_scalar_zero_product_input. ((((exists ff_h_pfp_scalar_zero_product_inputsource. ff_h_pfp_scalar_zero_product_inputsource + S (pfp_source_scalar_zero_product_input) = S ((S (pfp_index_scalar_zero_product_input)) * bc)) /\\ exists ff_q_pfp_scalar_zero_product_inputsource. bb = ff_q_pfp_scalar_zero_product_inputsource * S ((S (pfp_index_scalar_zero_product_input)) * bc) + (pfp_source_scalar_zero_product_input))) /\\ (((((exists ff_h_pfp_scalar_zero_product_inputtarget. ff_h_pfp_scalar_zero_product_inputtarget + S (pfp_value_scalar_zero_product_input) = S ((S (pfp_index_scalar_zero_product_input)) * sc)) /\\ exists ff_q_pfp_scalar_zero_product_inputtarget. sb = ff_q_pfp_scalar_zero_product_inputtarget * S ((S (pfp_index_scalar_zero_product_input)) * sc) + (pfp_value_scalar_zero_product_input))) /\\ ((((exists pfa_gap_scalar_zero_product_inputoperationleft. pfa_gap_scalar_zero_product_inputoperationleft + S (0) = (p)) /\\ (((exists pfa_gap_scalar_zero_product_inputoperationright. pfa_gap_scalar_zero_product_inputoperationright + S (pfp_source_scalar_zero_product_input) = (p)) /\\ ((((exists pfa_gap_scalar_zero_product_inputoperationresultbound. pfa_gap_scalar_zero_product_inputoperationresultbound + S (pfp_value_scalar_zero_product_input) = (p)) /\\ ((exists pfa_offset_left_scalar_zero_product_inputoperationresultcongruence pfa_offset_right_scalar_zero_product_inputoperationresultcongruence. ((0) * (pfp_source_scalar_zero_product_input)) + (p) * pfa_offset_left_scalar_zero_product_inputoperationresultcongruence = (pfp_value_scalar_zero_product_input) + (p) * pfa_offset_right_scalar_zero_product_inputoperationresultcongruence))))))))))))))))) -> (((forall fom_index_pfp_scalar_zero_product_actualleft. (exists fom_gap_pfp_scalar_zero_product_actualleft_index_bound. fom_gap_pfp_scalar_zero_product_actualleft_index_bound + S (fom_index_pfp_scalar_zero_product_actualleft) = L) -> exists fom_value_pfp_scalar_zero_product_actualleft. ((((exists fom_beta_height_pfp_scalar_zero_product_actualleft_entry. fom_beta_height_pfp_scalar_zero_product_actualleft_entry + S (fom_value_pfp_scalar_zero_product_actualleft) = S ((S (fom_index_pfp_scalar_zero_product_actualleft)) * ac)) /\\ exists fom_beta_quotient_pfp_scalar_zero_product_actualleft_entry. ab = fom_beta_quotient_pfp_scalar_zero_product_actualleft_entry * S ((S (fom_index_pfp_scalar_zero_product_actualleft)) * ac) + (fom_value_pfp_scalar_zero_product_actualleft))) /\\ (exists fom_gap_pfp_scalar_zero_product_actualleft_value_bound. fom_gap_pfp_scalar_zero_product_actualleft_value_bound + S (fom_value_pfp_scalar_zero_product_actualleft) = p))) /\\ (((forall fom_index_pfp_scalar_zero_product_actualright. (exists fom_gap_pfp_scalar_zero_product_actualright_index_bound. fom_gap_pfp_scalar_zero_product_actualright_index_bound + S (fom_index_pfp_scalar_zero_product_actualright) = M) -> exists fom_value_pfp_scalar_zero_product_actualright. ((((exists fom_beta_height_pfp_scalar_zero_product_actualright_entry. fom_beta_height_pfp_scalar_zero_product_actualright_entry + S (fom_value_pfp_scalar_zero_product_actualright) = S ((S (fom_index_pfp_scalar_zero_product_actualright)) * sc)) /\\ exists fom_beta_quotient_pfp_scalar_zero_product_actualright_entry. sb = fom_beta_quotient_pfp_scalar_zero_product_actualright_entry * S ((S (fom_index_pfp_scalar_zero_product_actualright)) * sc) + (fom_value_pfp_scalar_zero_product_actualright))) /\\ (exists fom_gap_pfp_scalar_zero_product_actualright_value_bound. fom_gap_pfp_scalar_zero_product_actualright_value_bound + S (fom_value_pfp_scalar_zero_product_actualright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_scalar_zero_product_actualcoefficients. (exists pfa_gap_scalar_zero_product_actualcoefficientsbound. pfa_gap_scalar_zero_product_actualcoefficientsbound + S (pfc_index_scalar_zero_product_actualcoefficients) = (N)) -> exists pfc_value_scalar_zero_product_actualcoefficients. ((((exists ff_h_pfp_scalar_zero_product_actualcoefficientsentry. ff_h_pfp_scalar_zero_product_actualcoefficientsentry + S (pfc_value_scalar_zero_product_actualcoefficients) = S ((S (pfc_index_scalar_zero_product_actualcoefficients)) * dc)) /\\ exists ff_q_pfp_scalar_zero_product_actualcoefficientsentry. db = ff_q_pfp_scalar_zero_product_actualcoefficientsentry * S ((S (pfc_index_scalar_zero_product_actualcoefficients)) * dc) + (pfc_value_scalar_zero_product_actualcoefficients))) /\\ ((exists pfc_terms_code_scalar_zero_product_actualcoefficientscoefficient pfc_terms_scale_scalar_zero_product_actualcoefficientscoefficient pfc_natural_sum_scalar_zero_product_actualcoefficientscoefficient. ((forall pfc_index_scalar_zero_product_actualcoefficientscoefficientdiagonal. (exists pfa_gap_scalar_zero_product_actualcoefficientscoefficientdiagonalbound. pfa_gap_scalar_zero_product_actualcoefficientscoefficientdiagonalbound + S (pfc_index_scalar_zero_product_actualcoefficientscoefficientdiagonal) = (S (pfc_index_scalar_zero_product_actualcoefficients))) -> exists pfc_value_scalar_zero_product_actualcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_scalar_zero_product_actualcoefficientscoefficientdiagonalentry. ff_h_pfp_scalar_zero_product_actualcoefficientscoefficientdiagonalentry + S (pfc_value_scalar_zero_product_actualcoefficientscoefficientdiagonal) = S ((S (pfc_index_scalar_zero_product_actualcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_zero_product_actualcoefficientscoefficient)) /\\ exists ff_q_pfp_scalar_zero_product_actualcoefficientscoefficientdiagonalentry. pfc_terms_code_scalar_zero_product_actualcoefficientscoefficient = ff_q_pfp_scalar_zero_product_actualcoefficientscoefficientdiagonalentry * S ((S (pfc_index_scalar_zero_product_actualcoefficientscoefficientdiagonal)) * pfc_terms_scale_scalar_zero_product_actualcoefficientscoefficient) + (pfc_value_scalar_zero_product_actualcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_scalar_zero_product_actualcoefficientscoefficientdiagonalterm pfc_left_scalar_zero_product_actualcoefficientscoefficientdiagonalterm pfc_right_scalar_zero_product_actualcoefficientscoefficientdiagonalterm. (((pfc_index_scalar_zero_product_actualcoefficientscoefficientdiagonal)+pfc_complement_scalar_zero_product_actualcoefficientscoefficientdiagonalterm=(pfc_index_scalar_zero_product_actualcoefficients)) /\\ ((((((exists pfa_gap_scalar_zero_product_actualcoefficientscoefficientdiagonaltermleftinside. pfa_gap_scalar_zero_product_actualcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_scalar_zero_product_actualcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_scalar_zero_product_actualcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_scalar_zero_product_actualcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_scalar_zero_product_actualcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_scalar_zero_product_actualcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_scalar_zero_product_actualcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_scalar_zero_product_actualcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_scalar_zero_product_actualcoefficientscoefficientdiagonal)) * ac) + (pfc_left_scalar_zero_product_actualcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_zero_product_actualcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_scalar_zero_product_actualcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_scalar_zero_product_actualcoefficientscoefficientdiagonal)) /\\ (((pfc_left_scalar_zero_product_actualcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_scalar_zero_product_actualcoefficientscoefficientdiagonaltermrightinside. pfa_gap_scalar_zero_product_actualcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_scalar_zero_product_actualcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_scalar_zero_product_actualcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_scalar_zero_product_actualcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_scalar_zero_product_actualcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_scalar_zero_product_actualcoefficientscoefficientdiagonalterm)) * sc)) /\\ exists ff_q_pfp_scalar_zero_product_actualcoefficientscoefficientdiagonaltermrightentry. sb = ff_q_pfp_scalar_zero_product_actualcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_scalar_zero_product_actualcoefficientscoefficientdiagonalterm)) * sc) + (pfc_right_scalar_zero_product_actualcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_scalar_zero_product_actualcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_scalar_zero_product_actualcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_scalar_zero_product_actualcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_scalar_zero_product_actualcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_scalar_zero_product_actualcoefficientscoefficientdiagonal)=pfc_left_scalar_zero_product_actualcoefficientscoefficientdiagonalterm*pfc_right_scalar_zero_product_actualcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_scalar_zero_product_actualcoefficientscoefficientsum fs_v_pfc_scalar_zero_product_actualcoefficientscoefficientsum. ((((exists fs_h_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_start. fs_h_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_scalar_zero_product_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_start. fs_u_pfc_scalar_zero_product_actualcoefficientscoefficientsum = fs_q_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_scalar_zero_product_actualcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_terminal. fs_h_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_scalar_zero_product_actualcoefficientscoefficient) = S ((S (S (pfc_index_scalar_zero_product_actualcoefficients))) * fs_v_pfc_scalar_zero_product_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_terminal. fs_u_pfc_scalar_zero_product_actualcoefficientscoefficientsum = fs_q_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_scalar_zero_product_actualcoefficients))) * fs_v_pfc_scalar_zero_product_actualcoefficientscoefficientsum) + (pfc_natural_sum_scalar_zero_product_actualcoefficientscoefficient))) /\\ forall fs_i_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps = S (pfc_index_scalar_zero_product_actualcoefficients)) -> exists fs_a_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps fs_r_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps fs_s_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_zero_product_actualcoefficientscoefficient)) /\\ exists fs_q_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_scalar_zero_product_actualcoefficientscoefficient = fs_q_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_scalar_zero_product_actualcoefficientscoefficient) + (fs_a_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_zero_product_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_scalar_zero_product_actualcoefficientscoefficientsum = fs_q_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_zero_product_actualcoefficientscoefficientsum) + (fs_r_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_zero_product_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_scalar_zero_product_actualcoefficientscoefficientsum = fs_q_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_scalar_zero_product_actualcoefficientscoefficientsum) + (fs_s_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps = fs_r_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps + fs_a_pfc_scalar_zero_product_actualcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_scalar_zero_product_actualcoefficientscoefficientresiduebound. pfa_gap_scalar_zero_product_actualcoefficientscoefficientresiduebound + S (pfc_value_scalar_zero_product_actualcoefficients) = (p)) /\\ ((exists pfa_offset_left_scalar_zero_product_actualcoefficientscoefficientresiduecongruence pfa_offset_right_scalar_zero_product_actualcoefficientscoefficientresiduecongruence. (pfc_natural_sum_scalar_zero_product_actualcoefficientscoefficient) + (p) * pfa_offset_left_scalar_zero_product_actualcoefficientscoefficientresiduecongruence = (pfc_value_scalar_zero_product_actualcoefficients) + (p) * pfa_offset_right_scalar_zero_product_actualcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfp_repeat_index_scalar_zero_product_result. 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((((exists ff_h_pfp_append_sum_old_value. ff_h_pfp_append_sum_old_value + S (a) = S ((S (i)) * bc)) /\\ exists ff_q_pfp_append_sum_old_value. bb = ff_q_pfp_append_sum_old_value * S ((S (i)) * bc) + (a))) /\\ ((exists pfa_gap_append_sum_old_bound. pfa_gap_append_sum_old_bound + S (a) = (p))))", "specialize hb (i)", "apply hb", "exact hcase_right", "cases ha", "cases ha_witness", "exists x", "exists 0", "exists x", "split", "specialize hs_left (i)", "specialize hs_left (x)", "apply hs_left", "exact hcase_right", "exact ha_witness_left", "split", "specialize ht_left (i)", "apply ht_left", "exact hcase_right", "split", "specialize he (i)", "specialize he (x)", "apply he", "exact hcase_right", "exact ha_witness_left", "specialize prime_field_add_zero_right (p)", "specialize prime_field_add_zero_right (x)", "apply prime_field_add_zero_right", "exact hp", "exact ha_witness_right" ], "script_sha256": "a9625320f15222e943db612ed984b75dbbf7f0482c9e04e3988bb3cae1330482", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_append_candidate.py", "sha256": "271845bfffc7e513fdb0bd0c3666dcccace8436d4d3a0f4db64b67bcd4b87042" }, "statement": "forall p bb bc M c db dc sb sc kb kc tb tc. (~((p) = 1) /\\ forall pfa_factor_left_append_sum_prime pfa_factor_right_append_sum_prime. (p) = pfa_factor_left_append_sum_prime * pfa_factor_right_append_sum_prime -> pfa_factor_left_append_sum_prime = 1 \\/ pfa_factor_right_append_sum_prime = 1) -> (forall fom_index_pfp_append_sum_old_coefficients. (exists fom_gap_pfp_append_sum_old_coefficients_index_bound. fom_gap_pfp_append_sum_old_coefficients_index_bound + S (fom_index_pfp_append_sum_old_coefficients) = M) -> exists fom_value_pfp_append_sum_old_coefficients. ((((exists fom_beta_height_pfp_append_sum_old_coefficients_entry. fom_beta_height_pfp_append_sum_old_coefficients_entry + S (fom_value_pfp_append_sum_old_coefficients) = S ((S (fom_index_pfp_append_sum_old_coefficients)) * bc)) /\\ exists fom_beta_quotient_pfp_append_sum_old_coefficients_entry. bb = fom_beta_quotient_pfp_append_sum_old_coefficients_entry * S ((S (fom_index_pfp_append_sum_old_coefficients)) * bc) + (fom_value_pfp_append_sum_old_coefficients))) /\\ (exists fom_gap_pfp_append_sum_old_coefficients_value_bound. fom_gap_pfp_append_sum_old_coefficients_value_bound + S (fom_value_pfp_append_sum_old_coefficients) = p))) -> (exists pfa_gap_append_sum_scalar. pfa_gap_append_sum_scalar + S (c) = (p)) -> (forall mdr_i_pfp_append_sum_preserve mdr_a_pfp_append_sum_preserve. (exists mdr_gap_pfp_append_sum_preserveb. mdr_gap_pfp_append_sum_preserveb + S (mdr_i_pfp_append_sum_preserve) = (M)) -> (((exists ff_h_mdr_pfp_append_sum_preserveo. ff_h_mdr_pfp_append_sum_preserveo + S (mdr_a_pfp_append_sum_preserve) = S ((S (mdr_i_pfp_append_sum_preserve)) * bc)) /\\ exists ff_q_mdr_pfp_append_sum_preserveo. bb = ff_q_mdr_pfp_append_sum_preserveo * S ((S (mdr_i_pfp_append_sum_preserve)) * bc) + (mdr_a_pfp_append_sum_preserve))) -> (((exists ff_h_mdr_pfp_append_sum_preserven. ff_h_mdr_pfp_append_sum_preserven + S (mdr_a_pfp_append_sum_preserve) = S ((S (mdr_i_pfp_append_sum_preserve)) * dc)) /\\ exists ff_q_mdr_pfp_append_sum_preserven. db = ff_q_mdr_pfp_append_sum_preserven * S ((S (mdr_i_pfp_append_sum_preserve)) * dc) + (mdr_a_pfp_append_sum_preserve)))) -> (((exists ff_h_pfp_append_sum_actual_last. ff_h_pfp_append_sum_actual_last + S (c) = S ((S (M)) * dc)) /\\ exists ff_q_pfp_append_sum_actual_last. db = ff_q_pfp_append_sum_actual_last * S ((S (M)) * dc) + (c))) -> (((forall mdr_i_pfp_append_sum_shiftprefix mdr_a_pfp_append_sum_shiftprefix. (exists mdr_gap_pfp_append_sum_shiftprefixb. mdr_gap_pfp_append_sum_shiftprefixb + S (mdr_i_pfp_append_sum_shiftprefix) = (M)) -> (((exists ff_h_mdr_pfp_append_sum_shiftprefixo. ff_h_mdr_pfp_append_sum_shiftprefixo + S (mdr_a_pfp_append_sum_shiftprefix) = S ((S (mdr_i_pfp_append_sum_shiftprefix)) * bc)) /\\ exists ff_q_mdr_pfp_append_sum_shiftprefixo. bb = ff_q_mdr_pfp_append_sum_shiftprefixo * S ((S (mdr_i_pfp_append_sum_shiftprefix)) * bc) + (mdr_a_pfp_append_sum_shiftprefix))) -> (((exists ff_h_mdr_pfp_append_sum_shiftprefixn. ff_h_mdr_pfp_append_sum_shiftprefixn + S (mdr_a_pfp_append_sum_shiftprefix) = S ((S (mdr_i_pfp_append_sum_shiftprefix)) * sc)) /\\ exists ff_q_mdr_pfp_append_sum_shiftprefixn. sb = ff_q_mdr_pfp_append_sum_shiftprefixn * S ((S (mdr_i_pfp_append_sum_shiftprefix)) * sc) + (mdr_a_pfp_append_sum_shiftprefix)))) /\\ ((((exists ff_h_pfp_append_sum_shiftlast. ff_h_pfp_append_sum_shiftlast + S (0) = S ((S (M)) * sc)) /\\ exists ff_q_pfp_append_sum_shiftlast. sb = ff_q_pfp_append_sum_shiftlast * S ((S (M)) * sc) + (0)))))) -> (((exists ff_h_pfp_append_sum_singleton. ff_h_pfp_append_sum_singleton + S (c) = S ((S (0)) * kc)) /\\ exists ff_q_pfp_append_sum_singleton. kb = ff_q_pfp_append_sum_singleton * S ((S (0)) * kc) + (c))) -> (((forall pfp_repeat_index_append_sum_left_padzeros. (exists pfa_gap_append_sum_left_padzerosindex. pfa_gap_append_sum_left_padzerosindex + S (pfp_repeat_index_append_sum_left_padzeros) = (M)) -> (((exists ff_h_pfp_append_sum_left_padzerosentry. ff_h_pfp_append_sum_left_padzerosentry + S (0) = S ((S (pfp_repeat_index_append_sum_left_padzeros)) * tc)) /\\ exists ff_q_pfp_append_sum_left_padzerosentry. tb = ff_q_pfp_append_sum_left_padzerosentry * S ((S (pfp_repeat_index_append_sum_left_padzeros)) * tc) + (0)))) /\\ ((forall pfrep_index_append_sum_left_pad pfrep_value_append_sum_left_pad. (exists pfa_gap_append_sum_left_padbound. pfa_gap_append_sum_left_padbound + S (pfrep_index_append_sum_left_pad) = (1)) -> (((exists ff_h_pfp_append_sum_left_padinput. ff_h_pfp_append_sum_left_padinput + S (pfrep_value_append_sum_left_pad) = S ((S (pfrep_index_append_sum_left_pad)) * kc)) /\\ exists ff_q_pfp_append_sum_left_padinput. kb = ff_q_pfp_append_sum_left_padinput * S ((S (pfrep_index_append_sum_left_pad)) * kc) + (pfrep_value_append_sum_left_pad))) -> (((exists ff_h_pfp_append_sum_left_padoutput. ff_h_pfp_append_sum_left_padoutput + S (pfrep_value_append_sum_left_pad) = S ((S ((M)+pfrep_index_append_sum_left_pad)) * tc)) /\\ exists ff_q_pfp_append_sum_left_padoutput. tb = ff_q_pfp_append_sum_left_padoutput * S ((S ((M)+pfrep_index_append_sum_left_pad)) * tc) + (pfrep_value_append_sum_left_pad))))))) -> (forall pfp_index_append_sum_result. (exists pfa_gap_append_sum_resultindex. pfa_gap_append_sum_resultindex + S (pfp_index_append_sum_result) = (S M)) -> exists pfp_left_append_sum_result pfp_right_append_sum_result pfp_value_append_sum_result. ((((exists ff_h_pfp_append_sum_resultleft. ff_h_pfp_append_sum_resultleft + S (pfp_left_append_sum_result) = S ((S (pfp_index_append_sum_result)) * sc)) /\\ exists ff_q_pfp_append_sum_resultleft. sb = ff_q_pfp_append_sum_resultleft * S ((S (pfp_index_append_sum_result)) * sc) + (pfp_left_append_sum_result))) /\\ (((((exists ff_h_pfp_append_sum_resultright. ff_h_pfp_append_sum_resultright + S (pfp_right_append_sum_result) = S ((S (pfp_index_append_sum_result)) * tc)) /\\ exists ff_q_pfp_append_sum_resultright. tb = ff_q_pfp_append_sum_resultright * S ((S (pfp_index_append_sum_result)) * tc) + (pfp_right_append_sum_result))) /\\ (((((exists ff_h_pfp_append_sum_resulttarget. ff_h_pfp_append_sum_resulttarget + S (pfp_value_append_sum_result) = S ((S (pfp_index_append_sum_result)) * dc)) /\\ exists ff_q_pfp_append_sum_resulttarget. db = ff_q_pfp_append_sum_resulttarget * S ((S (pfp_index_append_sum_result)) * dc) + (pfp_value_append_sum_result))) /\\ ((((exists pfa_gap_append_sum_resultoperationleft. pfa_gap_append_sum_resultoperationleft + S (pfp_left_append_sum_result) = (p)) /\\ (((exists pfa_gap_append_sum_resultoperationright. pfa_gap_append_sum_resultoperationright + S (pfp_right_append_sum_result) = (p)) /\\ ((((exists pfa_gap_append_sum_resultoperationresultbound. pfa_gap_append_sum_resultoperationresultbound + S (pfp_value_append_sum_result) = (p)) /\\ ((exists pfa_offset_left_append_sum_resultoperationresultcongruence pfa_offset_right_append_sum_resultoperationresultcongruence. 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"script": [ "intro p", "intro bb", "intro bc", "intro M", "intro c", "intro db", "intro dc", "intro hp", "intro hb", "intro hc", "intro he", "intro hlast", "have hs : exists sb sc. ((forall mdr_i_pfp_append_decomposition_shiftprefix mdr_a_pfp_append_decomposition_shiftprefix. (exists mdr_gap_pfp_append_decomposition_shiftprefixb. mdr_gap_pfp_append_decomposition_shiftprefixb + S (mdr_i_pfp_append_decomposition_shiftprefix) = (M)) -> (((exists ff_h_mdr_pfp_append_decomposition_shiftprefixo. ff_h_mdr_pfp_append_decomposition_shiftprefixo + S (mdr_a_pfp_append_decomposition_shiftprefix) = S ((S (mdr_i_pfp_append_decomposition_shiftprefix)) * bc)) /\\ exists ff_q_mdr_pfp_append_decomposition_shiftprefixo. bb = ff_q_mdr_pfp_append_decomposition_shiftprefixo * S ((S (mdr_i_pfp_append_decomposition_shiftprefix)) * bc) + (mdr_a_pfp_append_decomposition_shiftprefix))) -> (((exists ff_h_mdr_pfp_append_decomposition_shiftprefixn. ff_h_mdr_pfp_append_decomposition_shiftprefixn + S (mdr_a_pfp_append_decomposition_shiftprefix) = S ((S (mdr_i_pfp_append_decomposition_shiftprefix)) * sc)) /\\ exists ff_q_mdr_pfp_append_decomposition_shiftprefixn. sb = ff_q_mdr_pfp_append_decomposition_shiftprefixn * S ((S (mdr_i_pfp_append_decomposition_shiftprefix)) * sc) + (mdr_a_pfp_append_decomposition_shiftprefix)))) /\\ ((((exists ff_h_pfp_append_decomposition_shiftlast. ff_h_pfp_append_decomposition_shiftlast + S (0) = S ((S (M)) * sc)) /\\ exists ff_q_pfp_append_decomposition_shiftlast. sb = ff_q_pfp_append_decomposition_shiftlast * S ((S (M)) * sc) + (0)))))", "specialize prime_field_polynomial_shift_exists (bb)", "specialize prime_field_polynomial_shift_exists (bc)", "specialize prime_field_polynomial_shift_exists (M)", "apply prime_field_polynomial_shift_exists", "cases hs", "cases hs_witness", "have hk : exists kb kc. forall pfp_repeat_index_append_decomposition_repeat. (exists pfa_gap_append_decomposition_repeatindex. pfa_gap_append_decomposition_repeatindex + S (pfp_repeat_index_append_decomposition_repeat) = (1)) -> (((exists ff_h_pfp_append_decomposition_repeatentry. ff_h_pfp_append_decomposition_repeatentry + S (c) = S ((S (pfp_repeat_index_append_decomposition_repeat)) * kc)) /\\ exists ff_q_pfp_append_decomposition_repeatentry. kb = ff_q_pfp_append_decomposition_repeatentry * S ((S (pfp_repeat_index_append_decomposition_repeat)) * kc) + (c)))", "specialize beta_repeat_exists (c)", "specialize beta_repeat_exists (1)", "apply beta_repeat_exists", "cases hk", "cases hk_witness", "have hconstant : ((exists ff_h_pfp_append_decomposition_chosen_constant. ff_h_pfp_append_decomposition_chosen_constant + S (c) = S ((S (0)) * x3)) /\\ exists ff_q_pfp_append_decomposition_chosen_constant. x2 = ff_q_pfp_append_decomposition_chosen_constant * S ((S (0)) * x3) + (c))", "specialize hk_witness_witness (0)", "apply hk_witness_witness", "exists 0", "simp", "have hbounded : forall fom_index_pfp_append_decomposition_chosen_bound. (exists fom_gap_pfp_append_decomposition_chosen_bound_index_bound. fom_gap_pfp_append_decomposition_chosen_bound_index_bound + S (fom_index_pfp_append_decomposition_chosen_bound) = 1) -> exists fom_value_pfp_append_decomposition_chosen_bound. ((((exists fom_beta_height_pfp_append_decomposition_chosen_bound_entry. fom_beta_height_pfp_append_decomposition_chosen_bound_entry + S (fom_value_pfp_append_decomposition_chosen_bound) = S ((S (fom_index_pfp_append_decomposition_chosen_bound)) * x3)) /\\ exists fom_beta_quotient_pfp_append_decomposition_chosen_bound_entry. x2 = fom_beta_quotient_pfp_append_decomposition_chosen_bound_entry * S ((S (fom_index_pfp_append_decomposition_chosen_bound)) * x3) + (fom_value_pfp_append_decomposition_chosen_bound))) /\\ (exists fom_gap_pfp_append_decomposition_chosen_bound_value_bound. fom_gap_pfp_append_decomposition_chosen_bound_value_bound + S (fom_value_pfp_append_decomposition_chosen_bound) = p))", "intro i", "intro hi", "exists c", "split", "specialize hk_witness_witness (i)", "apply hk_witness_witness", "exact hi", "exact hc", "have ht : exists tb tc. ((forall pfp_repeat_index_append_decomposition_chosen_padzeros. (exists pfa_gap_append_decomposition_chosen_padzerosindex. pfa_gap_append_decomposition_chosen_padzerosindex + S (pfp_repeat_index_append_decomposition_chosen_padzeros) = (M)) -> (((exists ff_h_pfp_append_decomposition_chosen_padzerosentry. ff_h_pfp_append_decomposition_chosen_padzerosentry + S (0) = S ((S (pfp_repeat_index_append_decomposition_chosen_padzeros)) * tc)) /\\ exists ff_q_pfp_append_decomposition_chosen_padzerosentry. tb = ff_q_pfp_append_decomposition_chosen_padzerosentry * S ((S (pfp_repeat_index_append_decomposition_chosen_padzeros)) * tc) + (0)))) /\\ ((forall pfrep_index_append_decomposition_chosen_pad pfrep_value_append_decomposition_chosen_pad. (exists pfa_gap_append_decomposition_chosen_padbound. pfa_gap_append_decomposition_chosen_padbound + S (pfrep_index_append_decomposition_chosen_pad) = (1)) -> (((exists ff_h_pfp_append_decomposition_chosen_padinput. ff_h_pfp_append_decomposition_chosen_padinput + S (pfrep_value_append_decomposition_chosen_pad) = S ((S (pfrep_index_append_decomposition_chosen_pad)) * x3)) /\\ exists ff_q_pfp_append_decomposition_chosen_padinput. x2 = ff_q_pfp_append_decomposition_chosen_padinput * S ((S (pfrep_index_append_decomposition_chosen_pad)) * x3) + (pfrep_value_append_decomposition_chosen_pad))) -> (((exists ff_h_pfp_append_decomposition_chosen_padoutput. ff_h_pfp_append_decomposition_chosen_padoutput + S (pfrep_value_append_decomposition_chosen_pad) = S ((S ((M)+pfrep_index_append_decomposition_chosen_pad)) * tc)) /\\ exists ff_q_pfp_append_decomposition_chosen_padoutput. tb = ff_q_pfp_append_decomposition_chosen_padoutput * S ((S ((M)+pfrep_index_append_decomposition_chosen_pad)) * tc) + (pfrep_value_append_decomposition_chosen_pad))))))", "specialize prime_field_polynomial_left_pad_exists (x2)", "specialize prime_field_polynomial_left_pad_exists (x3)", "specialize prime_field_polynomial_left_pad_exists (M)", "specialize prime_field_polynomial_left_pad_exists (1)", "apply prime_field_polynomial_left_pad_exists", "cases ht", "cases ht_witness", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "split", "exact hs_witness_witness", "split", "exact hbounded", "split", "exact hconstant", "split", "exact ht_witness_witness", "specialize prime_field_polynomial_append_shift_constant_add (p)", "specialize prime_field_polynomial_append_shift_constant_add (bb)", "specialize prime_field_polynomial_append_shift_constant_add (bc)", "specialize prime_field_polynomial_append_shift_constant_add (M)", "specialize prime_field_polynomial_append_shift_constant_add (c)", "specialize prime_field_polynomial_append_shift_constant_add (db)", "specialize prime_field_polynomial_append_shift_constant_add (dc)", "specialize prime_field_polynomial_append_shift_constant_add (x)", "specialize prime_field_polynomial_append_shift_constant_add (x1)", "specialize prime_field_polynomial_append_shift_constant_add (x2)", "specialize prime_field_polynomial_append_shift_constant_add (x3)", "specialize prime_field_polynomial_append_shift_constant_add (x4)", "specialize prime_field_polynomial_append_shift_constant_add (x5)", "apply prime_field_polynomial_append_shift_constant_add", "exact hp", "exact hb", "exact hc", "exact he", "exact hlast", "exact hs_witness_witness", "exact hconstant", "exact ht_witness_witness" ], "script_sha256": "e612fcf410cd4967225407aac75b7509cf11d35f3316779a7a94d9c5ee79f426", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_append_candidate.py", "sha256": "271845bfffc7e513fdb0bd0c3666dcccace8436d4d3a0f4db64b67bcd4b87042" }, "statement": "forall p bb bc M c db dc. (~((p) = 1) /\\ forall pfa_factor_left_append_decomposition_prime pfa_factor_right_append_decomposition_prime. (p) = pfa_factor_left_append_decomposition_prime * pfa_factor_right_append_decomposition_prime -> pfa_factor_left_append_decomposition_prime = 1 \\/ pfa_factor_right_append_decomposition_prime = 1) -> (forall fom_index_pfp_append_decomposition_old_coefficients. (exists fom_gap_pfp_append_decomposition_old_coefficients_index_bound. fom_gap_pfp_append_decomposition_old_coefficients_index_bound + S (fom_index_pfp_append_decomposition_old_coefficients) = M) -> exists fom_value_pfp_append_decomposition_old_coefficients. ((((exists fom_beta_height_pfp_append_decomposition_old_coefficients_entry. fom_beta_height_pfp_append_decomposition_old_coefficients_entry + S (fom_value_pfp_append_decomposition_old_coefficients) = S ((S (fom_index_pfp_append_decomposition_old_coefficients)) * bc)) /\\ exists fom_beta_quotient_pfp_append_decomposition_old_coefficients_entry. bb = fom_beta_quotient_pfp_append_decomposition_old_coefficients_entry * S ((S (fom_index_pfp_append_decomposition_old_coefficients)) * bc) + (fom_value_pfp_append_decomposition_old_coefficients))) /\\ (exists fom_gap_pfp_append_decomposition_old_coefficients_value_bound. fom_gap_pfp_append_decomposition_old_coefficients_value_bound + S (fom_value_pfp_append_decomposition_old_coefficients) = p))) -> (exists pfa_gap_append_decomposition_scalar. pfa_gap_append_decomposition_scalar + S (c) = (p)) -> (forall mdr_i_pfp_append_decomposition_preserve mdr_a_pfp_append_decomposition_preserve. (exists mdr_gap_pfp_append_decomposition_preserveb. mdr_gap_pfp_append_decomposition_preserveb + S (mdr_i_pfp_append_decomposition_preserve) = (M)) -> (((exists ff_h_mdr_pfp_append_decomposition_preserveo. ff_h_mdr_pfp_append_decomposition_preserveo + S (mdr_a_pfp_append_decomposition_preserve) = S ((S (mdr_i_pfp_append_decomposition_preserve)) * bc)) /\\ exists ff_q_mdr_pfp_append_decomposition_preserveo. bb = ff_q_mdr_pfp_append_decomposition_preserveo * S ((S (mdr_i_pfp_append_decomposition_preserve)) * bc) + (mdr_a_pfp_append_decomposition_preserve))) -> (((exists ff_h_mdr_pfp_append_decomposition_preserven. ff_h_mdr_pfp_append_decomposition_preserven + S (mdr_a_pfp_append_decomposition_preserve) = S ((S (mdr_i_pfp_append_decomposition_preserve)) * dc)) /\\ exists ff_q_mdr_pfp_append_decomposition_preserven. db = ff_q_mdr_pfp_append_decomposition_preserven * S ((S (mdr_i_pfp_append_decomposition_preserve)) * dc) + (mdr_a_pfp_append_decomposition_preserve)))) -> (((exists ff_h_pfp_append_decomposition_actual_last. ff_h_pfp_append_decomposition_actual_last + S (c) = S ((S (M)) * dc)) /\\ exists ff_q_pfp_append_decomposition_actual_last. db = ff_q_pfp_append_decomposition_actual_last * S ((S (M)) * dc) + (c))) -> (exists sb sc kb kc tb tc. ((((forall mdr_i_pfp_append_decomposition_shiftprefix mdr_a_pfp_append_decomposition_shiftprefix. (exists mdr_gap_pfp_append_decomposition_shiftprefixb. mdr_gap_pfp_append_decomposition_shiftprefixb + S (mdr_i_pfp_append_decomposition_shiftprefix) = (M)) -> (((exists ff_h_mdr_pfp_append_decomposition_shiftprefixo. ff_h_mdr_pfp_append_decomposition_shiftprefixo + S (mdr_a_pfp_append_decomposition_shiftprefix) = S ((S (mdr_i_pfp_append_decomposition_shiftprefix)) * bc)) /\\ exists ff_q_mdr_pfp_append_decomposition_shiftprefixo. bb = ff_q_mdr_pfp_append_decomposition_shiftprefixo * S ((S (mdr_i_pfp_append_decomposition_shiftprefix)) * bc) + (mdr_a_pfp_append_decomposition_shiftprefix))) -> (((exists ff_h_mdr_pfp_append_decomposition_shiftprefixn. ff_h_mdr_pfp_append_decomposition_shiftprefixn + S (mdr_a_pfp_append_decomposition_shiftprefix) = S ((S (mdr_i_pfp_append_decomposition_shiftprefix)) * sc)) /\\ exists ff_q_mdr_pfp_append_decomposition_shiftprefixn. sb = ff_q_mdr_pfp_append_decomposition_shiftprefixn * S ((S (mdr_i_pfp_append_decomposition_shiftprefix)) * sc) + (mdr_a_pfp_append_decomposition_shiftprefix)))) /\\ ((((exists ff_h_pfp_append_decomposition_shiftlast. ff_h_pfp_append_decomposition_shiftlast + S (0) = S ((S (M)) * sc)) /\\ exists ff_q_pfp_append_decomposition_shiftlast. sb = ff_q_pfp_append_decomposition_shiftlast * S ((S (M)) * sc) + (0)))))) /\\ (((forall fom_index_pfp_append_decomposition_constant_bound. (exists fom_gap_pfp_append_decomposition_constant_bound_index_bound. fom_gap_pfp_append_decomposition_constant_bound_index_bound + S (fom_index_pfp_append_decomposition_constant_bound) = 1) -> exists fom_value_pfp_append_decomposition_constant_bound. ((((exists fom_beta_height_pfp_append_decomposition_constant_bound_entry. fom_beta_height_pfp_append_decomposition_constant_bound_entry + S (fom_value_pfp_append_decomposition_constant_bound) = S ((S (fom_index_pfp_append_decomposition_constant_bound)) * kc)) /\\ exists fom_beta_quotient_pfp_append_decomposition_constant_bound_entry. kb = fom_beta_quotient_pfp_append_decomposition_constant_bound_entry * S ((S (fom_index_pfp_append_decomposition_constant_bound)) * kc) + (fom_value_pfp_append_decomposition_constant_bound))) /\\ (exists fom_gap_pfp_append_decomposition_constant_bound_value_bound. fom_gap_pfp_append_decomposition_constant_bound_value_bound + S (fom_value_pfp_append_decomposition_constant_bound) = p))) /\\ (((((exists ff_h_pfp_append_decomposition_constant. ff_h_pfp_append_decomposition_constant + S (c) = S ((S (0)) * kc)) /\\ exists ff_q_pfp_append_decomposition_constant. kb = ff_q_pfp_append_decomposition_constant * S ((S (0)) * kc) + (c))) /\\ (((((forall pfp_repeat_index_append_decomposition_padzeros. (exists pfa_gap_append_decomposition_padzerosindex. pfa_gap_append_decomposition_padzerosindex + S (pfp_repeat_index_append_decomposition_padzeros) = (M)) -> (((exists ff_h_pfp_append_decomposition_padzerosentry. ff_h_pfp_append_decomposition_padzerosentry + S (0) = S ((S (pfp_repeat_index_append_decomposition_padzeros)) * tc)) /\\ exists ff_q_pfp_append_decomposition_padzerosentry. tb = ff_q_pfp_append_decomposition_padzerosentry * S ((S (pfp_repeat_index_append_decomposition_padzeros)) * tc) + (0)))) /\\ ((forall pfrep_index_append_decomposition_pad pfrep_value_append_decomposition_pad. (exists pfa_gap_append_decomposition_padbound. pfa_gap_append_decomposition_padbound + S (pfrep_index_append_decomposition_pad) = (1)) -> (((exists ff_h_pfp_append_decomposition_padinput. ff_h_pfp_append_decomposition_padinput + S (pfrep_value_append_decomposition_pad) = S ((S (pfrep_index_append_decomposition_pad)) * kc)) /\\ exists ff_q_pfp_append_decomposition_padinput. kb = ff_q_pfp_append_decomposition_padinput * S ((S (pfrep_index_append_decomposition_pad)) * kc) + (pfrep_value_append_decomposition_pad))) -> (((exists ff_h_pfp_append_decomposition_padoutput. ff_h_pfp_append_decomposition_padoutput + S (pfrep_value_append_decomposition_pad) = S ((S ((M)+pfrep_index_append_decomposition_pad)) * tc)) /\\ exists ff_q_pfp_append_decomposition_padoutput. tb = ff_q_pfp_append_decomposition_padoutput * S ((S ((M)+pfrep_index_append_decomposition_pad)) * tc) + (pfrep_value_append_decomposition_pad))))))) /\\ ((forall pfp_index_append_decomposition_add. (exists pfa_gap_append_decomposition_addindex. pfa_gap_append_decomposition_addindex + S (pfp_index_append_decomposition_add) = (S M)) -> exists pfp_left_append_decomposition_add pfp_right_append_decomposition_add pfp_value_append_decomposition_add. ((((exists ff_h_pfp_append_decomposition_addleft. ff_h_pfp_append_decomposition_addleft + S (pfp_left_append_decomposition_add) = S ((S (pfp_index_append_decomposition_add)) * sc)) /\\ exists ff_q_pfp_append_decomposition_addleft. sb = ff_q_pfp_append_decomposition_addleft * S ((S (pfp_index_append_decomposition_add)) * sc) + (pfp_left_append_decomposition_add))) /\\ (((((exists ff_h_pfp_append_decomposition_addright. ff_h_pfp_append_decomposition_addright + S (pfp_right_append_decomposition_add) = S ((S (pfp_index_append_decomposition_add)) * tc)) /\\ exists ff_q_pfp_append_decomposition_addright. tb = ff_q_pfp_append_decomposition_addright * S ((S (pfp_index_append_decomposition_add)) * tc) + (pfp_right_append_decomposition_add))) /\\ (((((exists ff_h_pfp_append_decomposition_addtarget. ff_h_pfp_append_decomposition_addtarget + S (pfp_value_append_decomposition_add) = S ((S (pfp_index_append_decomposition_add)) * dc)) /\\ exists ff_q_pfp_append_decomposition_addtarget. db = ff_q_pfp_append_decomposition_addtarget * S ((S (pfp_index_append_decomposition_add)) * dc) + (pfp_value_append_decomposition_add))) /\\ ((((exists pfa_gap_append_decomposition_addoperationleft. pfa_gap_append_decomposition_addoperationleft + S (pfp_left_append_decomposition_add) = (p)) /\\ (((exists pfa_gap_append_decomposition_addoperationright. pfa_gap_append_decomposition_addoperationright + S (pfp_right_append_decomposition_add) = (p)) /\\ ((((exists pfa_gap_append_decomposition_addoperationresultbound. pfa_gap_append_decomposition_addoperationresultbound + S (pfp_value_append_decomposition_add) = (p)) /\\ ((exists pfa_offset_left_append_decomposition_addoperationresultcongruence pfa_offset_right_append_decomposition_addoperationresultcongruence. 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p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro M", "intro c", "intro db", "intro dc", "intro kb", "intro kc", "intro tb", "intro tc", "intro i", "intro u", "intro v", "intro w", "intro hp", "intro hb", "intro hc", "intro he", "intro hlast", "intro hk", "intro ht", "intro hu", "intro hv", "intro hw", "have hs : exists sb sc. ((forall mdr_i_pfp_append_coefficient_chosen_shiftprefix mdr_a_pfp_append_coefficient_chosen_shiftprefix. (exists mdr_gap_pfp_append_coefficient_chosen_shiftprefixb. mdr_gap_pfp_append_coefficient_chosen_shiftprefixb + S (mdr_i_pfp_append_coefficient_chosen_shiftprefix) = (M)) -> (((exists ff_h_mdr_pfp_append_coefficient_chosen_shiftprefixo. ff_h_mdr_pfp_append_coefficient_chosen_shiftprefixo + S (mdr_a_pfp_append_coefficient_chosen_shiftprefix) = S ((S (mdr_i_pfp_append_coefficient_chosen_shiftprefix)) * bc)) /\\ exists ff_q_mdr_pfp_append_coefficient_chosen_shiftprefixo. bb = ff_q_mdr_pfp_append_coefficient_chosen_shiftprefixo * S ((S (mdr_i_pfp_append_coefficient_chosen_shiftprefix)) * bc) + (mdr_a_pfp_append_coefficient_chosen_shiftprefix))) -> (((exists ff_h_mdr_pfp_append_coefficient_chosen_shiftprefixn. ff_h_mdr_pfp_append_coefficient_chosen_shiftprefixn + S (mdr_a_pfp_append_coefficient_chosen_shiftprefix) = S ((S (mdr_i_pfp_append_coefficient_chosen_shiftprefix)) * sc)) /\\ exists ff_q_mdr_pfp_append_coefficient_chosen_shiftprefixn. sb = ff_q_mdr_pfp_append_coefficient_chosen_shiftprefixn * S ((S (mdr_i_pfp_append_coefficient_chosen_shiftprefix)) * sc) + (mdr_a_pfp_append_coefficient_chosen_shiftprefix)))) /\\ ((((exists ff_h_pfp_append_coefficient_chosen_shiftlast. ff_h_pfp_append_coefficient_chosen_shiftlast + S (0) = S ((S (M)) * sc)) /\\ exists ff_q_pfp_append_coefficient_chosen_shiftlast. sb = ff_q_pfp_append_coefficient_chosen_shiftlast * S ((S (M)) * sc) + (0)))))", "specialize prime_field_polynomial_shift_exists (bb)", "specialize prime_field_polynomial_shift_exists (bc)", "specialize prime_field_polynomial_shift_exists (M)", "apply prime_field_polynomial_shift_exists", "cases hs", "cases hs_witness", "have hshifted : exists pfc_terms_code_append_coefficient_shifted pfc_terms_scale_append_coefficient_shifted pfc_natural_sum_append_coefficient_shifted. ((forall pfc_index_append_coefficient_shifteddiagonal. (exists pfa_gap_append_coefficient_shifteddiagonalbound. pfa_gap_append_coefficient_shifteddiagonalbound + S (pfc_index_append_coefficient_shifteddiagonal) = (S (i))) -> exists pfc_value_append_coefficient_shifteddiagonal. ((((exists ff_h_pfp_append_coefficient_shifteddiagonalentry. ff_h_pfp_append_coefficient_shifteddiagonalentry + S (pfc_value_append_coefficient_shifteddiagonal) = S ((S (pfc_index_append_coefficient_shifteddiagonal)) * pfc_terms_scale_append_coefficient_shifted)) /\\ exists ff_q_pfp_append_coefficient_shifteddiagonalentry. pfc_terms_code_append_coefficient_shifted = ff_q_pfp_append_coefficient_shifteddiagonalentry * S ((S (pfc_index_append_coefficient_shifteddiagonal)) * pfc_terms_scale_append_coefficient_shifted) + (pfc_value_append_coefficient_shifteddiagonal))) /\\ ((exists pfc_complement_append_coefficient_shifteddiagonalterm pfc_left_append_coefficient_shifteddiagonalterm pfc_right_append_coefficient_shifteddiagonalterm. (((pfc_index_append_coefficient_shifteddiagonal)+pfc_complement_append_coefficient_shifteddiagonalterm=(i)) /\\ ((((((exists pfa_gap_append_coefficient_shifteddiagonaltermleftinside. pfa_gap_append_coefficient_shifteddiagonaltermleftinside + S (pfc_index_append_coefficient_shifteddiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_coefficient_shifteddiagonaltermleftentry. ff_h_pfp_append_coefficient_shifteddiagonaltermleftentry + S (pfc_left_append_coefficient_shifteddiagonalterm) = S ((S (pfc_index_append_coefficient_shifteddiagonal)) * ac)) /\\ exists ff_q_pfp_append_coefficient_shifteddiagonaltermleftentry. ab = ff_q_pfp_append_coefficient_shifteddiagonaltermleftentry * S ((S (pfc_index_append_coefficient_shifteddiagonal)) * ac) + (pfc_left_append_coefficient_shifteddiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_shifteddiagonaltermleftoutside. pfc_gap_append_coefficient_shifteddiagonaltermleftoutside+(L)=(pfc_index_append_coefficient_shifteddiagonal)) /\\ (((pfc_left_append_coefficient_shifteddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_coefficient_shifteddiagonaltermrightinside. pfa_gap_append_coefficient_shifteddiagonaltermrightinside + S (pfc_complement_append_coefficient_shifteddiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_append_coefficient_shifteddiagonaltermrightentry. ff_h_pfp_append_coefficient_shifteddiagonaltermrightentry + S (pfc_right_append_coefficient_shifteddiagonalterm) = S ((S (pfc_complement_append_coefficient_shifteddiagonalterm)) * x1)) /\\ exists ff_q_pfp_append_coefficient_shifteddiagonaltermrightentry. x = ff_q_pfp_append_coefficient_shifteddiagonaltermrightentry * S ((S (pfc_complement_append_coefficient_shifteddiagonalterm)) * x1) + (pfc_right_append_coefficient_shifteddiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_shifteddiagonaltermrightoutside. pfc_gap_append_coefficient_shifteddiagonaltermrightoutside+(S M)=(pfc_complement_append_coefficient_shifteddiagonalterm)) /\\ (((pfc_right_append_coefficient_shifteddiagonalterm)=0))))) /\\ (((pfc_value_append_coefficient_shifteddiagonal)=pfc_left_append_coefficient_shifteddiagonalterm*pfc_right_append_coefficient_shifteddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_coefficient_shiftedsum fs_v_pfc_append_coefficient_shiftedsum. ((((exists fs_h_pfc_append_coefficient_shiftedsum_body_start. fs_h_pfc_append_coefficient_shiftedsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_coefficient_shiftedsum)) /\\ exists fs_q_pfc_append_coefficient_shiftedsum_body_start. fs_u_pfc_append_coefficient_shiftedsum = fs_q_pfc_append_coefficient_shiftedsum_body_start * S ((S (0)) * fs_v_pfc_append_coefficient_shiftedsum) + (0))) /\\ ((((exists fs_h_pfc_append_coefficient_shiftedsum_body_terminal. fs_h_pfc_append_coefficient_shiftedsum_body_terminal + S (pfc_natural_sum_append_coefficient_shifted) = S ((S (S (i))) * fs_v_pfc_append_coefficient_shiftedsum)) /\\ exists fs_q_pfc_append_coefficient_shiftedsum_body_terminal. fs_u_pfc_append_coefficient_shiftedsum = fs_q_pfc_append_coefficient_shiftedsum_body_terminal * S ((S (S (i))) * fs_v_pfc_append_coefficient_shiftedsum) + (pfc_natural_sum_append_coefficient_shifted))) /\\ forall fs_i_pfc_append_coefficient_shiftedsum_body_steps. (exists fs_lt_pfc_append_coefficient_shiftedsum_body_steps_bound. fs_lt_pfc_append_coefficient_shiftedsum_body_steps_bound + S fs_i_pfc_append_coefficient_shiftedsum_body_steps = S (i)) -> exists fs_a_pfc_append_coefficient_shiftedsum_body_steps fs_r_pfc_append_coefficient_shiftedsum_body_steps fs_s_pfc_append_coefficient_shiftedsum_body_steps. ((((exists fs_h_pfc_append_coefficient_shiftedsum_body_steps_summand. fs_h_pfc_append_coefficient_shiftedsum_body_steps_summand + S (fs_a_pfc_append_coefficient_shiftedsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * pfc_terms_scale_append_coefficient_shifted)) /\\ exists fs_q_pfc_append_coefficient_shiftedsum_body_steps_summand. pfc_terms_code_append_coefficient_shifted = fs_q_pfc_append_coefficient_shiftedsum_body_steps_summand * S ((S (fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * pfc_terms_scale_append_coefficient_shifted) + (fs_a_pfc_append_coefficient_shiftedsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_shiftedsum_body_steps_partial. fs_h_pfc_append_coefficient_shiftedsum_body_steps_partial + S (fs_r_pfc_append_coefficient_shiftedsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * fs_v_pfc_append_coefficient_shiftedsum)) /\\ exists fs_q_pfc_append_coefficient_shiftedsum_body_steps_partial. fs_u_pfc_append_coefficient_shiftedsum = fs_q_pfc_append_coefficient_shiftedsum_body_steps_partial * S ((S (fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * fs_v_pfc_append_coefficient_shiftedsum) + (fs_r_pfc_append_coefficient_shiftedsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_shiftedsum_body_steps_successor. fs_h_pfc_append_coefficient_shiftedsum_body_steps_successor + S (fs_s_pfc_append_coefficient_shiftedsum_body_steps) = S ((S (S fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * fs_v_pfc_append_coefficient_shiftedsum)) /\\ exists fs_q_pfc_append_coefficient_shiftedsum_body_steps_successor. fs_u_pfc_append_coefficient_shiftedsum = fs_q_pfc_append_coefficient_shiftedsum_body_steps_successor * S ((S (S fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * fs_v_pfc_append_coefficient_shiftedsum) + (fs_s_pfc_append_coefficient_shiftedsum_body_steps))) /\\ fs_s_pfc_append_coefficient_shiftedsum_body_steps = fs_r_pfc_append_coefficient_shiftedsum_body_steps + fs_a_pfc_append_coefficient_shiftedsum_body_steps)))))) /\\ ((((exists pfa_gap_append_coefficient_shiftedresiduebound. pfa_gap_append_coefficient_shiftedresiduebound + S (u) = (p)) /\\ ((exists pfa_offset_left_append_coefficient_shiftedresiduecongruence pfa_offset_right_append_coefficient_shiftedresiduecongruence. (pfc_natural_sum_append_coefficient_shifted) + (p) * pfa_offset_left_append_coefficient_shiftedresiduecongruence = (u) + (p) * pfa_offset_right_append_coefficient_shiftedresiduecongruence))))))))", "have hboth : (((exists pfc_terms_code_append_coefficient_old pfc_terms_scale_append_coefficient_old pfc_natural_sum_append_coefficient_old. ((forall pfc_index_append_coefficient_olddiagonal. (exists pfa_gap_append_coefficient_olddiagonalbound. pfa_gap_append_coefficient_olddiagonalbound + S (pfc_index_append_coefficient_olddiagonal) = (S (i))) -> exists pfc_value_append_coefficient_olddiagonal. ((((exists ff_h_pfp_append_coefficient_olddiagonalentry. ff_h_pfp_append_coefficient_olddiagonalentry + S (pfc_value_append_coefficient_olddiagonal) = S ((S (pfc_index_append_coefficient_olddiagonal)) * pfc_terms_scale_append_coefficient_old)) /\\ exists ff_q_pfp_append_coefficient_olddiagonalentry. pfc_terms_code_append_coefficient_old = ff_q_pfp_append_coefficient_olddiagonalentry * S ((S (pfc_index_append_coefficient_olddiagonal)) * pfc_terms_scale_append_coefficient_old) + (pfc_value_append_coefficient_olddiagonal))) /\\ ((exists pfc_complement_append_coefficient_olddiagonalterm pfc_left_append_coefficient_olddiagonalterm pfc_right_append_coefficient_olddiagonalterm. (((pfc_index_append_coefficient_olddiagonal)+pfc_complement_append_coefficient_olddiagonalterm=(i)) /\\ ((((((exists pfa_gap_append_coefficient_olddiagonaltermleftinside. pfa_gap_append_coefficient_olddiagonaltermleftinside + S (pfc_index_append_coefficient_olddiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_coefficient_olddiagonaltermleftentry. ff_h_pfp_append_coefficient_olddiagonaltermleftentry + S (pfc_left_append_coefficient_olddiagonalterm) = S ((S (pfc_index_append_coefficient_olddiagonal)) * ac)) /\\ exists ff_q_pfp_append_coefficient_olddiagonaltermleftentry. ab = ff_q_pfp_append_coefficient_olddiagonaltermleftentry * S ((S (pfc_index_append_coefficient_olddiagonal)) * ac) + (pfc_left_append_coefficient_olddiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_olddiagonaltermleftoutside. pfc_gap_append_coefficient_olddiagonaltermleftoutside+(L)=(pfc_index_append_coefficient_olddiagonal)) /\\ (((pfc_left_append_coefficient_olddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_coefficient_olddiagonaltermrightinside. pfa_gap_append_coefficient_olddiagonaltermrightinside + S (pfc_complement_append_coefficient_olddiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_append_coefficient_olddiagonaltermrightentry. ff_h_pfp_append_coefficient_olddiagonaltermrightentry + S (pfc_right_append_coefficient_olddiagonalterm) = S ((S (pfc_complement_append_coefficient_olddiagonalterm)) * bc)) /\\ exists ff_q_pfp_append_coefficient_olddiagonaltermrightentry. bb = ff_q_pfp_append_coefficient_olddiagonaltermrightentry * S ((S (pfc_complement_append_coefficient_olddiagonalterm)) * bc) + (pfc_right_append_coefficient_olddiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_olddiagonaltermrightoutside. pfc_gap_append_coefficient_olddiagonaltermrightoutside+(M)=(pfc_complement_append_coefficient_olddiagonalterm)) /\\ (((pfc_right_append_coefficient_olddiagonalterm)=0))))) /\\ (((pfc_value_append_coefficient_olddiagonal)=pfc_left_append_coefficient_olddiagonalterm*pfc_right_append_coefficient_olddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_coefficient_oldsum fs_v_pfc_append_coefficient_oldsum. ((((exists fs_h_pfc_append_coefficient_oldsum_body_start. fs_h_pfc_append_coefficient_oldsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_coefficient_oldsum)) /\\ exists fs_q_pfc_append_coefficient_oldsum_body_start. fs_u_pfc_append_coefficient_oldsum = fs_q_pfc_append_coefficient_oldsum_body_start * S ((S (0)) * fs_v_pfc_append_coefficient_oldsum) + (0))) /\\ ((((exists fs_h_pfc_append_coefficient_oldsum_body_terminal. fs_h_pfc_append_coefficient_oldsum_body_terminal + S (pfc_natural_sum_append_coefficient_old) = S ((S (S (i))) * fs_v_pfc_append_coefficient_oldsum)) /\\ exists fs_q_pfc_append_coefficient_oldsum_body_terminal. fs_u_pfc_append_coefficient_oldsum = fs_q_pfc_append_coefficient_oldsum_body_terminal * S ((S (S (i))) * fs_v_pfc_append_coefficient_oldsum) + (pfc_natural_sum_append_coefficient_old))) /\\ forall fs_i_pfc_append_coefficient_oldsum_body_steps. (exists fs_lt_pfc_append_coefficient_oldsum_body_steps_bound. fs_lt_pfc_append_coefficient_oldsum_body_steps_bound + S fs_i_pfc_append_coefficient_oldsum_body_steps = S (i)) -> exists fs_a_pfc_append_coefficient_oldsum_body_steps fs_r_pfc_append_coefficient_oldsum_body_steps fs_s_pfc_append_coefficient_oldsum_body_steps. ((((exists fs_h_pfc_append_coefficient_oldsum_body_steps_summand. fs_h_pfc_append_coefficient_oldsum_body_steps_summand + S (fs_a_pfc_append_coefficient_oldsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_oldsum_body_steps)) * pfc_terms_scale_append_coefficient_old)) /\\ exists fs_q_pfc_append_coefficient_oldsum_body_steps_summand. pfc_terms_code_append_coefficient_old = fs_q_pfc_append_coefficient_oldsum_body_steps_summand * S ((S (fs_i_pfc_append_coefficient_oldsum_body_steps)) * pfc_terms_scale_append_coefficient_old) + (fs_a_pfc_append_coefficient_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_oldsum_body_steps_partial. fs_h_pfc_append_coefficient_oldsum_body_steps_partial + S (fs_r_pfc_append_coefficient_oldsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_oldsum_body_steps)) * fs_v_pfc_append_coefficient_oldsum)) /\\ exists fs_q_pfc_append_coefficient_oldsum_body_steps_partial. fs_u_pfc_append_coefficient_oldsum = fs_q_pfc_append_coefficient_oldsum_body_steps_partial * S ((S (fs_i_pfc_append_coefficient_oldsum_body_steps)) * fs_v_pfc_append_coefficient_oldsum) + (fs_r_pfc_append_coefficient_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_oldsum_body_steps_successor. fs_h_pfc_append_coefficient_oldsum_body_steps_successor + S (fs_s_pfc_append_coefficient_oldsum_body_steps) = S ((S (S fs_i_pfc_append_coefficient_oldsum_body_steps)) * fs_v_pfc_append_coefficient_oldsum)) /\\ exists fs_q_pfc_append_coefficient_oldsum_body_steps_successor. fs_u_pfc_append_coefficient_oldsum = fs_q_pfc_append_coefficient_oldsum_body_steps_successor * S ((S (S fs_i_pfc_append_coefficient_oldsum_body_steps)) * fs_v_pfc_append_coefficient_oldsum) + (fs_s_pfc_append_coefficient_oldsum_body_steps))) /\\ fs_s_pfc_append_coefficient_oldsum_body_steps = fs_r_pfc_append_coefficient_oldsum_body_steps + fs_a_pfc_append_coefficient_oldsum_body_steps)))))) /\\ ((((exists pfa_gap_append_coefficient_oldresiduebound. pfa_gap_append_coefficient_oldresiduebound + S (u) = (p)) /\\ ((exists pfa_offset_left_append_coefficient_oldresiduecongruence pfa_offset_right_append_coefficient_oldresiduecongruence. (pfc_natural_sum_append_coefficient_old) + (p) * pfa_offset_left_append_coefficient_oldresiduecongruence = (u) + (p) * pfa_offset_right_append_coefficient_oldresiduecongruence))))))))) -> (exists pfc_terms_code_append_coefficient_shifted pfc_terms_scale_append_coefficient_shifted pfc_natural_sum_append_coefficient_shifted. ((forall pfc_index_append_coefficient_shifteddiagonal. (exists pfa_gap_append_coefficient_shifteddiagonalbound. pfa_gap_append_coefficient_shifteddiagonalbound + S (pfc_index_append_coefficient_shifteddiagonal) = (S (i))) -> exists pfc_value_append_coefficient_shifteddiagonal. ((((exists ff_h_pfp_append_coefficient_shifteddiagonalentry. ff_h_pfp_append_coefficient_shifteddiagonalentry + S (pfc_value_append_coefficient_shifteddiagonal) = S ((S (pfc_index_append_coefficient_shifteddiagonal)) * pfc_terms_scale_append_coefficient_shifted)) /\\ exists ff_q_pfp_append_coefficient_shifteddiagonalentry. pfc_terms_code_append_coefficient_shifted = ff_q_pfp_append_coefficient_shifteddiagonalentry * S ((S (pfc_index_append_coefficient_shifteddiagonal)) * pfc_terms_scale_append_coefficient_shifted) + (pfc_value_append_coefficient_shifteddiagonal))) /\\ ((exists pfc_complement_append_coefficient_shifteddiagonalterm pfc_left_append_coefficient_shifteddiagonalterm pfc_right_append_coefficient_shifteddiagonalterm. (((pfc_index_append_coefficient_shifteddiagonal)+pfc_complement_append_coefficient_shifteddiagonalterm=(i)) /\\ ((((((exists pfa_gap_append_coefficient_shifteddiagonaltermleftinside. pfa_gap_append_coefficient_shifteddiagonaltermleftinside + S (pfc_index_append_coefficient_shifteddiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_coefficient_shifteddiagonaltermleftentry. ff_h_pfp_append_coefficient_shifteddiagonaltermleftentry + S (pfc_left_append_coefficient_shifteddiagonalterm) = S ((S (pfc_index_append_coefficient_shifteddiagonal)) * ac)) /\\ exists ff_q_pfp_append_coefficient_shifteddiagonaltermleftentry. ab = ff_q_pfp_append_coefficient_shifteddiagonaltermleftentry * S ((S (pfc_index_append_coefficient_shifteddiagonal)) * ac) + (pfc_left_append_coefficient_shifteddiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_shifteddiagonaltermleftoutside. pfc_gap_append_coefficient_shifteddiagonaltermleftoutside+(L)=(pfc_index_append_coefficient_shifteddiagonal)) /\\ (((pfc_left_append_coefficient_shifteddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_coefficient_shifteddiagonaltermrightinside. pfa_gap_append_coefficient_shifteddiagonaltermrightinside + S (pfc_complement_append_coefficient_shifteddiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_append_coefficient_shifteddiagonaltermrightentry. ff_h_pfp_append_coefficient_shifteddiagonaltermrightentry + S (pfc_right_append_coefficient_shifteddiagonalterm) = S ((S (pfc_complement_append_coefficient_shifteddiagonalterm)) * x1)) /\\ exists ff_q_pfp_append_coefficient_shifteddiagonaltermrightentry. x = ff_q_pfp_append_coefficient_shifteddiagonaltermrightentry * S ((S (pfc_complement_append_coefficient_shifteddiagonalterm)) * x1) + (pfc_right_append_coefficient_shifteddiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_shifteddiagonaltermrightoutside. pfc_gap_append_coefficient_shifteddiagonaltermrightoutside+(S M)=(pfc_complement_append_coefficient_shifteddiagonalterm)) /\\ (((pfc_right_append_coefficient_shifteddiagonalterm)=0))))) /\\ (((pfc_value_append_coefficient_shifteddiagonal)=pfc_left_append_coefficient_shifteddiagonalterm*pfc_right_append_coefficient_shifteddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_coefficient_shiftedsum fs_v_pfc_append_coefficient_shiftedsum. ((((exists fs_h_pfc_append_coefficient_shiftedsum_body_start. fs_h_pfc_append_coefficient_shiftedsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_coefficient_shiftedsum)) /\\ exists fs_q_pfc_append_coefficient_shiftedsum_body_start. fs_u_pfc_append_coefficient_shiftedsum = fs_q_pfc_append_coefficient_shiftedsum_body_start * S ((S (0)) * fs_v_pfc_append_coefficient_shiftedsum) + (0))) /\\ ((((exists fs_h_pfc_append_coefficient_shiftedsum_body_terminal. fs_h_pfc_append_coefficient_shiftedsum_body_terminal + S (pfc_natural_sum_append_coefficient_shifted) = S ((S (S (i))) * fs_v_pfc_append_coefficient_shiftedsum)) /\\ exists fs_q_pfc_append_coefficient_shiftedsum_body_terminal. fs_u_pfc_append_coefficient_shiftedsum = fs_q_pfc_append_coefficient_shiftedsum_body_terminal * S ((S (S (i))) * fs_v_pfc_append_coefficient_shiftedsum) + (pfc_natural_sum_append_coefficient_shifted))) /\\ forall fs_i_pfc_append_coefficient_shiftedsum_body_steps. (exists fs_lt_pfc_append_coefficient_shiftedsum_body_steps_bound. fs_lt_pfc_append_coefficient_shiftedsum_body_steps_bound + S fs_i_pfc_append_coefficient_shiftedsum_body_steps = S (i)) -> exists fs_a_pfc_append_coefficient_shiftedsum_body_steps fs_r_pfc_append_coefficient_shiftedsum_body_steps fs_s_pfc_append_coefficient_shiftedsum_body_steps. ((((exists fs_h_pfc_append_coefficient_shiftedsum_body_steps_summand. fs_h_pfc_append_coefficient_shiftedsum_body_steps_summand + S (fs_a_pfc_append_coefficient_shiftedsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * pfc_terms_scale_append_coefficient_shifted)) /\\ exists fs_q_pfc_append_coefficient_shiftedsum_body_steps_summand. pfc_terms_code_append_coefficient_shifted = fs_q_pfc_append_coefficient_shiftedsum_body_steps_summand * S ((S (fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * pfc_terms_scale_append_coefficient_shifted) + (fs_a_pfc_append_coefficient_shiftedsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_shiftedsum_body_steps_partial. fs_h_pfc_append_coefficient_shiftedsum_body_steps_partial + S (fs_r_pfc_append_coefficient_shiftedsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * fs_v_pfc_append_coefficient_shiftedsum)) /\\ exists fs_q_pfc_append_coefficient_shiftedsum_body_steps_partial. fs_u_pfc_append_coefficient_shiftedsum = fs_q_pfc_append_coefficient_shiftedsum_body_steps_partial * S ((S (fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * fs_v_pfc_append_coefficient_shiftedsum) + (fs_r_pfc_append_coefficient_shiftedsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_shiftedsum_body_steps_successor. fs_h_pfc_append_coefficient_shiftedsum_body_steps_successor + S (fs_s_pfc_append_coefficient_shiftedsum_body_steps) = S ((S (S fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * fs_v_pfc_append_coefficient_shiftedsum)) /\\ exists fs_q_pfc_append_coefficient_shiftedsum_body_steps_successor. fs_u_pfc_append_coefficient_shiftedsum = fs_q_pfc_append_coefficient_shiftedsum_body_steps_successor * S ((S (S fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * fs_v_pfc_append_coefficient_shiftedsum) + (fs_s_pfc_append_coefficient_shiftedsum_body_steps))) /\\ fs_s_pfc_append_coefficient_shiftedsum_body_steps = fs_r_pfc_append_coefficient_shiftedsum_body_steps + fs_a_pfc_append_coefficient_shiftedsum_body_steps)))))) /\\ ((((exists pfa_gap_append_coefficient_shiftedresiduebound. pfa_gap_append_coefficient_shiftedresiduebound + S (u) = (p)) /\\ ((exists pfa_offset_left_append_coefficient_shiftedresiduecongruence pfa_offset_right_append_coefficient_shiftedresiduecongruence. (pfc_natural_sum_append_coefficient_shifted) + (p) * pfa_offset_left_append_coefficient_shiftedresiduecongruence = (u) + (p) * pfa_offset_right_append_coefficient_shiftedresiduecongruence)))))))))) /\\ (((exists pfc_terms_code_append_coefficient_shifted pfc_terms_scale_append_coefficient_shifted pfc_natural_sum_append_coefficient_shifted. ((forall pfc_index_append_coefficient_shifteddiagonal. (exists pfa_gap_append_coefficient_shifteddiagonalbound. pfa_gap_append_coefficient_shifteddiagonalbound + S (pfc_index_append_coefficient_shifteddiagonal) = (S (i))) -> exists pfc_value_append_coefficient_shifteddiagonal. ((((exists ff_h_pfp_append_coefficient_shifteddiagonalentry. ff_h_pfp_append_coefficient_shifteddiagonalentry + S (pfc_value_append_coefficient_shifteddiagonal) = S ((S (pfc_index_append_coefficient_shifteddiagonal)) * pfc_terms_scale_append_coefficient_shifted)) /\\ exists ff_q_pfp_append_coefficient_shifteddiagonalentry. pfc_terms_code_append_coefficient_shifted = ff_q_pfp_append_coefficient_shifteddiagonalentry * S ((S (pfc_index_append_coefficient_shifteddiagonal)) * pfc_terms_scale_append_coefficient_shifted) + (pfc_value_append_coefficient_shifteddiagonal))) /\\ ((exists pfc_complement_append_coefficient_shifteddiagonalterm pfc_left_append_coefficient_shifteddiagonalterm pfc_right_append_coefficient_shifteddiagonalterm. (((pfc_index_append_coefficient_shifteddiagonal)+pfc_complement_append_coefficient_shifteddiagonalterm=(i)) /\\ ((((((exists pfa_gap_append_coefficient_shifteddiagonaltermleftinside. pfa_gap_append_coefficient_shifteddiagonaltermleftinside + S (pfc_index_append_coefficient_shifteddiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_coefficient_shifteddiagonaltermleftentry. ff_h_pfp_append_coefficient_shifteddiagonaltermleftentry + S (pfc_left_append_coefficient_shifteddiagonalterm) = S ((S (pfc_index_append_coefficient_shifteddiagonal)) * ac)) /\\ exists ff_q_pfp_append_coefficient_shifteddiagonaltermleftentry. ab = ff_q_pfp_append_coefficient_shifteddiagonaltermleftentry * S ((S (pfc_index_append_coefficient_shifteddiagonal)) * ac) + (pfc_left_append_coefficient_shifteddiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_shifteddiagonaltermleftoutside. pfc_gap_append_coefficient_shifteddiagonaltermleftoutside+(L)=(pfc_index_append_coefficient_shifteddiagonal)) /\\ (((pfc_left_append_coefficient_shifteddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_coefficient_shifteddiagonaltermrightinside. pfa_gap_append_coefficient_shifteddiagonaltermrightinside + S (pfc_complement_append_coefficient_shifteddiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_append_coefficient_shifteddiagonaltermrightentry. ff_h_pfp_append_coefficient_shifteddiagonaltermrightentry + S (pfc_right_append_coefficient_shifteddiagonalterm) = S ((S (pfc_complement_append_coefficient_shifteddiagonalterm)) * x1)) /\\ exists ff_q_pfp_append_coefficient_shifteddiagonaltermrightentry. x = ff_q_pfp_append_coefficient_shifteddiagonaltermrightentry * S ((S (pfc_complement_append_coefficient_shifteddiagonalterm)) * x1) + (pfc_right_append_coefficient_shifteddiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_shifteddiagonaltermrightoutside. pfc_gap_append_coefficient_shifteddiagonaltermrightoutside+(S M)=(pfc_complement_append_coefficient_shifteddiagonalterm)) /\\ (((pfc_right_append_coefficient_shifteddiagonalterm)=0))))) /\\ (((pfc_value_append_coefficient_shifteddiagonal)=pfc_left_append_coefficient_shifteddiagonalterm*pfc_right_append_coefficient_shifteddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_coefficient_shiftedsum fs_v_pfc_append_coefficient_shiftedsum. ((((exists fs_h_pfc_append_coefficient_shiftedsum_body_start. fs_h_pfc_append_coefficient_shiftedsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_coefficient_shiftedsum)) /\\ exists fs_q_pfc_append_coefficient_shiftedsum_body_start. fs_u_pfc_append_coefficient_shiftedsum = fs_q_pfc_append_coefficient_shiftedsum_body_start * S ((S (0)) * fs_v_pfc_append_coefficient_shiftedsum) + (0))) /\\ ((((exists fs_h_pfc_append_coefficient_shiftedsum_body_terminal. fs_h_pfc_append_coefficient_shiftedsum_body_terminal + S (pfc_natural_sum_append_coefficient_shifted) = S ((S (S (i))) * fs_v_pfc_append_coefficient_shiftedsum)) /\\ exists fs_q_pfc_append_coefficient_shiftedsum_body_terminal. fs_u_pfc_append_coefficient_shiftedsum = fs_q_pfc_append_coefficient_shiftedsum_body_terminal * S ((S (S (i))) * fs_v_pfc_append_coefficient_shiftedsum) + (pfc_natural_sum_append_coefficient_shifted))) /\\ forall fs_i_pfc_append_coefficient_shiftedsum_body_steps. (exists fs_lt_pfc_append_coefficient_shiftedsum_body_steps_bound. fs_lt_pfc_append_coefficient_shiftedsum_body_steps_bound + S fs_i_pfc_append_coefficient_shiftedsum_body_steps = S (i)) -> exists fs_a_pfc_append_coefficient_shiftedsum_body_steps fs_r_pfc_append_coefficient_shiftedsum_body_steps fs_s_pfc_append_coefficient_shiftedsum_body_steps. ((((exists fs_h_pfc_append_coefficient_shiftedsum_body_steps_summand. fs_h_pfc_append_coefficient_shiftedsum_body_steps_summand + S (fs_a_pfc_append_coefficient_shiftedsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * pfc_terms_scale_append_coefficient_shifted)) /\\ exists fs_q_pfc_append_coefficient_shiftedsum_body_steps_summand. pfc_terms_code_append_coefficient_shifted = fs_q_pfc_append_coefficient_shiftedsum_body_steps_summand * S ((S (fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * pfc_terms_scale_append_coefficient_shifted) + (fs_a_pfc_append_coefficient_shiftedsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_shiftedsum_body_steps_partial. fs_h_pfc_append_coefficient_shiftedsum_body_steps_partial + S (fs_r_pfc_append_coefficient_shiftedsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * fs_v_pfc_append_coefficient_shiftedsum)) /\\ exists fs_q_pfc_append_coefficient_shiftedsum_body_steps_partial. fs_u_pfc_append_coefficient_shiftedsum = fs_q_pfc_append_coefficient_shiftedsum_body_steps_partial * S ((S (fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * fs_v_pfc_append_coefficient_shiftedsum) + (fs_r_pfc_append_coefficient_shiftedsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_shiftedsum_body_steps_successor. fs_h_pfc_append_coefficient_shiftedsum_body_steps_successor + S (fs_s_pfc_append_coefficient_shiftedsum_body_steps) = S ((S (S fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * fs_v_pfc_append_coefficient_shiftedsum)) /\\ exists fs_q_pfc_append_coefficient_shiftedsum_body_steps_successor. fs_u_pfc_append_coefficient_shiftedsum = fs_q_pfc_append_coefficient_shiftedsum_body_steps_successor * S ((S (S fs_i_pfc_append_coefficient_shiftedsum_body_steps)) * fs_v_pfc_append_coefficient_shiftedsum) + (fs_s_pfc_append_coefficient_shiftedsum_body_steps))) /\\ fs_s_pfc_append_coefficient_shiftedsum_body_steps = fs_r_pfc_append_coefficient_shiftedsum_body_steps + fs_a_pfc_append_coefficient_shiftedsum_body_steps)))))) /\\ ((((exists pfa_gap_append_coefficient_shiftedresiduebound. pfa_gap_append_coefficient_shiftedresiduebound + S (u) = (p)) /\\ ((exists pfa_offset_left_append_coefficient_shiftedresiduecongruence pfa_offset_right_append_coefficient_shiftedresiduecongruence. (pfc_natural_sum_append_coefficient_shifted) + (p) * pfa_offset_left_append_coefficient_shiftedresiduecongruence = (u) + (p) * pfa_offset_right_append_coefficient_shiftedresiduecongruence))))))))) -> (exists pfc_terms_code_append_coefficient_old pfc_terms_scale_append_coefficient_old pfc_natural_sum_append_coefficient_old. ((forall pfc_index_append_coefficient_olddiagonal. (exists pfa_gap_append_coefficient_olddiagonalbound. pfa_gap_append_coefficient_olddiagonalbound + S (pfc_index_append_coefficient_olddiagonal) = (S (i))) -> exists pfc_value_append_coefficient_olddiagonal. ((((exists ff_h_pfp_append_coefficient_olddiagonalentry. ff_h_pfp_append_coefficient_olddiagonalentry + S (pfc_value_append_coefficient_olddiagonal) = S ((S (pfc_index_append_coefficient_olddiagonal)) * pfc_terms_scale_append_coefficient_old)) /\\ exists ff_q_pfp_append_coefficient_olddiagonalentry. pfc_terms_code_append_coefficient_old = ff_q_pfp_append_coefficient_olddiagonalentry * S ((S (pfc_index_append_coefficient_olddiagonal)) * pfc_terms_scale_append_coefficient_old) + (pfc_value_append_coefficient_olddiagonal))) /\\ ((exists pfc_complement_append_coefficient_olddiagonalterm pfc_left_append_coefficient_olddiagonalterm pfc_right_append_coefficient_olddiagonalterm. (((pfc_index_append_coefficient_olddiagonal)+pfc_complement_append_coefficient_olddiagonalterm=(i)) /\\ ((((((exists pfa_gap_append_coefficient_olddiagonaltermleftinside. pfa_gap_append_coefficient_olddiagonaltermleftinside + S (pfc_index_append_coefficient_olddiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_coefficient_olddiagonaltermleftentry. ff_h_pfp_append_coefficient_olddiagonaltermleftentry + S (pfc_left_append_coefficient_olddiagonalterm) = S ((S (pfc_index_append_coefficient_olddiagonal)) * ac)) /\\ exists ff_q_pfp_append_coefficient_olddiagonaltermleftentry. ab = ff_q_pfp_append_coefficient_olddiagonaltermleftentry * S ((S (pfc_index_append_coefficient_olddiagonal)) * ac) + (pfc_left_append_coefficient_olddiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_olddiagonaltermleftoutside. pfc_gap_append_coefficient_olddiagonaltermleftoutside+(L)=(pfc_index_append_coefficient_olddiagonal)) /\\ (((pfc_left_append_coefficient_olddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_coefficient_olddiagonaltermrightinside. pfa_gap_append_coefficient_olddiagonaltermrightinside + S (pfc_complement_append_coefficient_olddiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_append_coefficient_olddiagonaltermrightentry. ff_h_pfp_append_coefficient_olddiagonaltermrightentry + S (pfc_right_append_coefficient_olddiagonalterm) = S ((S (pfc_complement_append_coefficient_olddiagonalterm)) * bc)) /\\ exists ff_q_pfp_append_coefficient_olddiagonaltermrightentry. bb = ff_q_pfp_append_coefficient_olddiagonaltermrightentry * S ((S (pfc_complement_append_coefficient_olddiagonalterm)) * bc) + (pfc_right_append_coefficient_olddiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_olddiagonaltermrightoutside. pfc_gap_append_coefficient_olddiagonaltermrightoutside+(M)=(pfc_complement_append_coefficient_olddiagonalterm)) /\\ (((pfc_right_append_coefficient_olddiagonalterm)=0))))) /\\ (((pfc_value_append_coefficient_olddiagonal)=pfc_left_append_coefficient_olddiagonalterm*pfc_right_append_coefficient_olddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_coefficient_oldsum fs_v_pfc_append_coefficient_oldsum. ((((exists fs_h_pfc_append_coefficient_oldsum_body_start. fs_h_pfc_append_coefficient_oldsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_coefficient_oldsum)) /\\ exists fs_q_pfc_append_coefficient_oldsum_body_start. fs_u_pfc_append_coefficient_oldsum = fs_q_pfc_append_coefficient_oldsum_body_start * S ((S (0)) * fs_v_pfc_append_coefficient_oldsum) + (0))) /\\ ((((exists fs_h_pfc_append_coefficient_oldsum_body_terminal. fs_h_pfc_append_coefficient_oldsum_body_terminal + S (pfc_natural_sum_append_coefficient_old) = S ((S (S (i))) * fs_v_pfc_append_coefficient_oldsum)) /\\ exists fs_q_pfc_append_coefficient_oldsum_body_terminal. fs_u_pfc_append_coefficient_oldsum = fs_q_pfc_append_coefficient_oldsum_body_terminal * S ((S (S (i))) * fs_v_pfc_append_coefficient_oldsum) + (pfc_natural_sum_append_coefficient_old))) /\\ forall fs_i_pfc_append_coefficient_oldsum_body_steps. (exists fs_lt_pfc_append_coefficient_oldsum_body_steps_bound. fs_lt_pfc_append_coefficient_oldsum_body_steps_bound + S fs_i_pfc_append_coefficient_oldsum_body_steps = S (i)) -> exists fs_a_pfc_append_coefficient_oldsum_body_steps fs_r_pfc_append_coefficient_oldsum_body_steps fs_s_pfc_append_coefficient_oldsum_body_steps. ((((exists fs_h_pfc_append_coefficient_oldsum_body_steps_summand. fs_h_pfc_append_coefficient_oldsum_body_steps_summand + S (fs_a_pfc_append_coefficient_oldsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_oldsum_body_steps)) * pfc_terms_scale_append_coefficient_old)) /\\ exists fs_q_pfc_append_coefficient_oldsum_body_steps_summand. pfc_terms_code_append_coefficient_old = fs_q_pfc_append_coefficient_oldsum_body_steps_summand * S ((S (fs_i_pfc_append_coefficient_oldsum_body_steps)) * pfc_terms_scale_append_coefficient_old) + (fs_a_pfc_append_coefficient_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_oldsum_body_steps_partial. fs_h_pfc_append_coefficient_oldsum_body_steps_partial + S (fs_r_pfc_append_coefficient_oldsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_oldsum_body_steps)) * fs_v_pfc_append_coefficient_oldsum)) /\\ exists fs_q_pfc_append_coefficient_oldsum_body_steps_partial. fs_u_pfc_append_coefficient_oldsum = fs_q_pfc_append_coefficient_oldsum_body_steps_partial * S ((S (fs_i_pfc_append_coefficient_oldsum_body_steps)) * fs_v_pfc_append_coefficient_oldsum) + (fs_r_pfc_append_coefficient_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_oldsum_body_steps_successor. fs_h_pfc_append_coefficient_oldsum_body_steps_successor + S (fs_s_pfc_append_coefficient_oldsum_body_steps) = S ((S (S fs_i_pfc_append_coefficient_oldsum_body_steps)) * fs_v_pfc_append_coefficient_oldsum)) /\\ exists fs_q_pfc_append_coefficient_oldsum_body_steps_successor. fs_u_pfc_append_coefficient_oldsum = fs_q_pfc_append_coefficient_oldsum_body_steps_successor * S ((S (S fs_i_pfc_append_coefficient_oldsum_body_steps)) * fs_v_pfc_append_coefficient_oldsum) + (fs_s_pfc_append_coefficient_oldsum_body_steps))) /\\ fs_s_pfc_append_coefficient_oldsum_body_steps = fs_r_pfc_append_coefficient_oldsum_body_steps + fs_a_pfc_append_coefficient_oldsum_body_steps)))))) /\\ ((((exists pfa_gap_append_coefficient_oldresiduebound. pfa_gap_append_coefficient_oldresiduebound + S (u) = (p)) /\\ ((exists pfa_offset_left_append_coefficient_oldresiduecongruence pfa_offset_right_append_coefficient_oldresiduecongruence. (pfc_natural_sum_append_coefficient_old) + (p) * pfa_offset_left_append_coefficient_oldresiduecongruence = (u) + (p) * pfa_offset_right_append_coefficient_oldresiduecongruence))))))))))))", "specialize prime_field_convolution_coefficient_shift_right_iff (p)", "specialize prime_field_convolution_coefficient_shift_right_iff (ab)", "specialize prime_field_convolution_coefficient_shift_right_iff (ac)", "specialize prime_field_convolution_coefficient_shift_right_iff (L)", "specialize prime_field_convolution_coefficient_shift_right_iff (bb)", "specialize prime_field_convolution_coefficient_shift_right_iff (bc)", "specialize prime_field_convolution_coefficient_shift_right_iff (M)", "specialize prime_field_convolution_coefficient_shift_right_iff (x)", "specialize prime_field_convolution_coefficient_shift_right_iff (x1)", "specialize prime_field_convolution_coefficient_shift_right_iff (i)", "specialize prime_field_convolution_coefficient_shift_right_iff (u)", "apply prime_field_convolution_coefficient_shift_right_iff", "exact hs_witness_witness", "cases hboth", "apply hboth_left", "exact hu", "specialize prime_field_convolution_coefficient_left_add (p)", "specialize prime_field_convolution_coefficient_left_add (x)", "specialize prime_field_convolution_coefficient_left_add (x1)", "specialize prime_field_convolution_coefficient_left_add (tb)", "specialize prime_field_convolution_coefficient_left_add (tc)", "specialize prime_field_convolution_coefficient_left_add (db)", "specialize prime_field_convolution_coefficient_left_add (dc)", "specialize prime_field_convolution_coefficient_left_add (S M)", "specialize prime_field_convolution_coefficient_left_add (ab)", "specialize prime_field_convolution_coefficient_left_add (ac)", "specialize prime_field_convolution_coefficient_left_add (L)", "specialize prime_field_convolution_coefficient_left_add (i)", "specialize prime_field_convolution_coefficient_left_add (u)", "specialize prime_field_convolution_coefficient_left_add (v)", "specialize prime_field_convolution_coefficient_left_add (w)", "apply prime_field_convolution_coefficient_left_add", "specialize prime_field_polynomial_append_shift_constant_add (p)", "specialize prime_field_polynomial_append_shift_constant_add (bb)", "specialize prime_field_polynomial_append_shift_constant_add (bc)", "specialize prime_field_polynomial_append_shift_constant_add (M)", "specialize prime_field_polynomial_append_shift_constant_add (c)", "specialize prime_field_polynomial_append_shift_constant_add (db)", "specialize prime_field_polynomial_append_shift_constant_add (dc)", "specialize prime_field_polynomial_append_shift_constant_add (x)", "specialize prime_field_polynomial_append_shift_constant_add (x1)", "specialize prime_field_polynomial_append_shift_constant_add (kb)", "specialize prime_field_polynomial_append_shift_constant_add (kc)", "specialize prime_field_polynomial_append_shift_constant_add (tb)", "specialize prime_field_polynomial_append_shift_constant_add (tc)", "apply prime_field_polynomial_append_shift_constant_add", "exact hp", "exact hb", "exact hc", "exact he", "exact hlast", "exact hs_witness_witness", "exact hk", "exact ht", "exact hshifted", "exact hv", "exact hw" ], "script_sha256": "9af42fa4af9481b31202233a0c3da8da083ed1441a352a5322dd150e44a7a67b", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_append_candidate.py", "sha256": "271845bfffc7e513fdb0bd0c3666dcccace8436d4d3a0f4db64b67bcd4b87042" }, "statement": "forall p ab ac L bb bc M c db dc kb kc tb tc i u v w. (~((p) = 1) /\\ forall pfa_factor_left_append_coefficient_prime pfa_factor_right_append_coefficient_prime. (p) = pfa_factor_left_append_coefficient_prime * pfa_factor_right_append_coefficient_prime -> pfa_factor_left_append_coefficient_prime = 1 \\/ pfa_factor_right_append_coefficient_prime = 1) -> (forall fom_index_pfp_append_coefficient_old_coefficients. (exists fom_gap_pfp_append_coefficient_old_coefficients_index_bound. fom_gap_pfp_append_coefficient_old_coefficients_index_bound + S (fom_index_pfp_append_coefficient_old_coefficients) = M) -> exists fom_value_pfp_append_coefficient_old_coefficients. ((((exists fom_beta_height_pfp_append_coefficient_old_coefficients_entry. fom_beta_height_pfp_append_coefficient_old_coefficients_entry + S (fom_value_pfp_append_coefficient_old_coefficients) = S ((S (fom_index_pfp_append_coefficient_old_coefficients)) * bc)) /\\ exists fom_beta_quotient_pfp_append_coefficient_old_coefficients_entry. bb = fom_beta_quotient_pfp_append_coefficient_old_coefficients_entry * S ((S (fom_index_pfp_append_coefficient_old_coefficients)) * bc) + (fom_value_pfp_append_coefficient_old_coefficients))) /\\ (exists fom_gap_pfp_append_coefficient_old_coefficients_value_bound. fom_gap_pfp_append_coefficient_old_coefficients_value_bound + S (fom_value_pfp_append_coefficient_old_coefficients) = p))) -> (exists pfa_gap_append_coefficient_scalar. pfa_gap_append_coefficient_scalar + S (c) = (p)) -> (forall mdr_i_pfp_append_coefficient_preserve mdr_a_pfp_append_coefficient_preserve. (exists mdr_gap_pfp_append_coefficient_preserveb. mdr_gap_pfp_append_coefficient_preserveb + S (mdr_i_pfp_append_coefficient_preserve) = (M)) -> (((exists ff_h_mdr_pfp_append_coefficient_preserveo. ff_h_mdr_pfp_append_coefficient_preserveo + S (mdr_a_pfp_append_coefficient_preserve) = S ((S (mdr_i_pfp_append_coefficient_preserve)) * bc)) /\\ exists ff_q_mdr_pfp_append_coefficient_preserveo. bb = ff_q_mdr_pfp_append_coefficient_preserveo * S ((S (mdr_i_pfp_append_coefficient_preserve)) * bc) + (mdr_a_pfp_append_coefficient_preserve))) -> (((exists ff_h_mdr_pfp_append_coefficient_preserven. ff_h_mdr_pfp_append_coefficient_preserven + S (mdr_a_pfp_append_coefficient_preserve) = S ((S (mdr_i_pfp_append_coefficient_preserve)) * dc)) /\\ exists ff_q_mdr_pfp_append_coefficient_preserven. db = ff_q_mdr_pfp_append_coefficient_preserven * S ((S (mdr_i_pfp_append_coefficient_preserve)) * dc) + (mdr_a_pfp_append_coefficient_preserve)))) -> (((exists ff_h_pfp_append_coefficient_actual_last. ff_h_pfp_append_coefficient_actual_last + S (c) = S ((S (M)) * dc)) /\\ exists ff_q_pfp_append_coefficient_actual_last. db = ff_q_pfp_append_coefficient_actual_last * S ((S (M)) * dc) + (c))) -> (((exists ff_h_pfp_append_coefficient_singleton. ff_h_pfp_append_coefficient_singleton + S (c) = S ((S (0)) * kc)) /\\ exists ff_q_pfp_append_coefficient_singleton. kb = ff_q_pfp_append_coefficient_singleton * S ((S (0)) * kc) + (c))) -> (((forall pfp_repeat_index_append_coefficient_padzeros. (exists pfa_gap_append_coefficient_padzerosindex. pfa_gap_append_coefficient_padzerosindex + S (pfp_repeat_index_append_coefficient_padzeros) = (M)) -> (((exists ff_h_pfp_append_coefficient_padzerosentry. ff_h_pfp_append_coefficient_padzerosentry + S (0) = S ((S (pfp_repeat_index_append_coefficient_padzeros)) * tc)) /\\ exists ff_q_pfp_append_coefficient_padzerosentry. tb = ff_q_pfp_append_coefficient_padzerosentry * S ((S (pfp_repeat_index_append_coefficient_padzeros)) * tc) + (0)))) /\\ ((forall pfrep_index_append_coefficient_pad pfrep_value_append_coefficient_pad. (exists pfa_gap_append_coefficient_padbound. pfa_gap_append_coefficient_padbound + S (pfrep_index_append_coefficient_pad) = (1)) -> (((exists ff_h_pfp_append_coefficient_padinput. ff_h_pfp_append_coefficient_padinput + S (pfrep_value_append_coefficient_pad) = S ((S (pfrep_index_append_coefficient_pad)) * kc)) /\\ exists ff_q_pfp_append_coefficient_padinput. kb = ff_q_pfp_append_coefficient_padinput * S ((S (pfrep_index_append_coefficient_pad)) * kc) + (pfrep_value_append_coefficient_pad))) -> (((exists ff_h_pfp_append_coefficient_padoutput. ff_h_pfp_append_coefficient_padoutput + S (pfrep_value_append_coefficient_pad) = S ((S ((M)+pfrep_index_append_coefficient_pad)) * tc)) /\\ exists ff_q_pfp_append_coefficient_padoutput. tb = ff_q_pfp_append_coefficient_padoutput * S ((S ((M)+pfrep_index_append_coefficient_pad)) * tc) + (pfrep_value_append_coefficient_pad))))))) -> (exists pfc_terms_code_append_coefficient_old pfc_terms_scale_append_coefficient_old pfc_natural_sum_append_coefficient_old. ((forall pfc_index_append_coefficient_olddiagonal. (exists pfa_gap_append_coefficient_olddiagonalbound. pfa_gap_append_coefficient_olddiagonalbound + S (pfc_index_append_coefficient_olddiagonal) = (S (i))) -> exists pfc_value_append_coefficient_olddiagonal. ((((exists ff_h_pfp_append_coefficient_olddiagonalentry. ff_h_pfp_append_coefficient_olddiagonalentry + S (pfc_value_append_coefficient_olddiagonal) = S ((S (pfc_index_append_coefficient_olddiagonal)) * pfc_terms_scale_append_coefficient_old)) /\\ exists ff_q_pfp_append_coefficient_olddiagonalentry. pfc_terms_code_append_coefficient_old = ff_q_pfp_append_coefficient_olddiagonalentry * S ((S (pfc_index_append_coefficient_olddiagonal)) * pfc_terms_scale_append_coefficient_old) + (pfc_value_append_coefficient_olddiagonal))) /\\ ((exists pfc_complement_append_coefficient_olddiagonalterm pfc_left_append_coefficient_olddiagonalterm pfc_right_append_coefficient_olddiagonalterm. (((pfc_index_append_coefficient_olddiagonal)+pfc_complement_append_coefficient_olddiagonalterm=(i)) /\\ ((((((exists pfa_gap_append_coefficient_olddiagonaltermleftinside. pfa_gap_append_coefficient_olddiagonaltermleftinside + S (pfc_index_append_coefficient_olddiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_coefficient_olddiagonaltermleftentry. ff_h_pfp_append_coefficient_olddiagonaltermleftentry + S (pfc_left_append_coefficient_olddiagonalterm) = S ((S (pfc_index_append_coefficient_olddiagonal)) * ac)) /\\ exists ff_q_pfp_append_coefficient_olddiagonaltermleftentry. ab = ff_q_pfp_append_coefficient_olddiagonaltermleftentry * S ((S (pfc_index_append_coefficient_olddiagonal)) * ac) + (pfc_left_append_coefficient_olddiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_olddiagonaltermleftoutside. pfc_gap_append_coefficient_olddiagonaltermleftoutside+(L)=(pfc_index_append_coefficient_olddiagonal)) /\\ (((pfc_left_append_coefficient_olddiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_coefficient_olddiagonaltermrightinside. pfa_gap_append_coefficient_olddiagonaltermrightinside + S (pfc_complement_append_coefficient_olddiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_append_coefficient_olddiagonaltermrightentry. ff_h_pfp_append_coefficient_olddiagonaltermrightentry + S (pfc_right_append_coefficient_olddiagonalterm) = S ((S (pfc_complement_append_coefficient_olddiagonalterm)) * bc)) /\\ exists ff_q_pfp_append_coefficient_olddiagonaltermrightentry. bb = ff_q_pfp_append_coefficient_olddiagonaltermrightentry * S ((S (pfc_complement_append_coefficient_olddiagonalterm)) * bc) + (pfc_right_append_coefficient_olddiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_olddiagonaltermrightoutside. pfc_gap_append_coefficient_olddiagonaltermrightoutside+(M)=(pfc_complement_append_coefficient_olddiagonalterm)) /\\ (((pfc_right_append_coefficient_olddiagonalterm)=0))))) /\\ (((pfc_value_append_coefficient_olddiagonal)=pfc_left_append_coefficient_olddiagonalterm*pfc_right_append_coefficient_olddiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_coefficient_oldsum fs_v_pfc_append_coefficient_oldsum. ((((exists fs_h_pfc_append_coefficient_oldsum_body_start. fs_h_pfc_append_coefficient_oldsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_coefficient_oldsum)) /\\ exists fs_q_pfc_append_coefficient_oldsum_body_start. fs_u_pfc_append_coefficient_oldsum = fs_q_pfc_append_coefficient_oldsum_body_start * S ((S (0)) * fs_v_pfc_append_coefficient_oldsum) + (0))) /\\ ((((exists fs_h_pfc_append_coefficient_oldsum_body_terminal. fs_h_pfc_append_coefficient_oldsum_body_terminal + S (pfc_natural_sum_append_coefficient_old) = S ((S (S (i))) * fs_v_pfc_append_coefficient_oldsum)) /\\ exists fs_q_pfc_append_coefficient_oldsum_body_terminal. fs_u_pfc_append_coefficient_oldsum = fs_q_pfc_append_coefficient_oldsum_body_terminal * S ((S (S (i))) * fs_v_pfc_append_coefficient_oldsum) + (pfc_natural_sum_append_coefficient_old))) /\\ forall fs_i_pfc_append_coefficient_oldsum_body_steps. (exists fs_lt_pfc_append_coefficient_oldsum_body_steps_bound. fs_lt_pfc_append_coefficient_oldsum_body_steps_bound + S fs_i_pfc_append_coefficient_oldsum_body_steps = S (i)) -> exists fs_a_pfc_append_coefficient_oldsum_body_steps fs_r_pfc_append_coefficient_oldsum_body_steps fs_s_pfc_append_coefficient_oldsum_body_steps. ((((exists fs_h_pfc_append_coefficient_oldsum_body_steps_summand. fs_h_pfc_append_coefficient_oldsum_body_steps_summand + S (fs_a_pfc_append_coefficient_oldsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_oldsum_body_steps)) * pfc_terms_scale_append_coefficient_old)) /\\ exists fs_q_pfc_append_coefficient_oldsum_body_steps_summand. pfc_terms_code_append_coefficient_old = fs_q_pfc_append_coefficient_oldsum_body_steps_summand * S ((S (fs_i_pfc_append_coefficient_oldsum_body_steps)) * pfc_terms_scale_append_coefficient_old) + (fs_a_pfc_append_coefficient_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_oldsum_body_steps_partial. fs_h_pfc_append_coefficient_oldsum_body_steps_partial + S (fs_r_pfc_append_coefficient_oldsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_oldsum_body_steps)) * fs_v_pfc_append_coefficient_oldsum)) /\\ exists fs_q_pfc_append_coefficient_oldsum_body_steps_partial. fs_u_pfc_append_coefficient_oldsum = fs_q_pfc_append_coefficient_oldsum_body_steps_partial * S ((S (fs_i_pfc_append_coefficient_oldsum_body_steps)) * fs_v_pfc_append_coefficient_oldsum) + (fs_r_pfc_append_coefficient_oldsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_oldsum_body_steps_successor. fs_h_pfc_append_coefficient_oldsum_body_steps_successor + S (fs_s_pfc_append_coefficient_oldsum_body_steps) = S ((S (S fs_i_pfc_append_coefficient_oldsum_body_steps)) * fs_v_pfc_append_coefficient_oldsum)) /\\ exists fs_q_pfc_append_coefficient_oldsum_body_steps_successor. fs_u_pfc_append_coefficient_oldsum = fs_q_pfc_append_coefficient_oldsum_body_steps_successor * S ((S (S fs_i_pfc_append_coefficient_oldsum_body_steps)) * fs_v_pfc_append_coefficient_oldsum) + (fs_s_pfc_append_coefficient_oldsum_body_steps))) /\\ fs_s_pfc_append_coefficient_oldsum_body_steps = fs_r_pfc_append_coefficient_oldsum_body_steps + fs_a_pfc_append_coefficient_oldsum_body_steps)))))) /\\ ((((exists pfa_gap_append_coefficient_oldresiduebound. pfa_gap_append_coefficient_oldresiduebound + S (u) = (p)) /\\ ((exists pfa_offset_left_append_coefficient_oldresiduecongruence pfa_offset_right_append_coefficient_oldresiduecongruence. (pfc_natural_sum_append_coefficient_old) + (p) * pfa_offset_left_append_coefficient_oldresiduecongruence = (u) + (p) * pfa_offset_right_append_coefficient_oldresiduecongruence))))))))) -> (exists pfc_terms_code_append_coefficient_constant pfc_terms_scale_append_coefficient_constant pfc_natural_sum_append_coefficient_constant. ((forall pfc_index_append_coefficient_constantdiagonal. (exists pfa_gap_append_coefficient_constantdiagonalbound. pfa_gap_append_coefficient_constantdiagonalbound + S (pfc_index_append_coefficient_constantdiagonal) = (S (i))) -> exists pfc_value_append_coefficient_constantdiagonal. ((((exists ff_h_pfp_append_coefficient_constantdiagonalentry. ff_h_pfp_append_coefficient_constantdiagonalentry + S (pfc_value_append_coefficient_constantdiagonal) = S ((S (pfc_index_append_coefficient_constantdiagonal)) * pfc_terms_scale_append_coefficient_constant)) /\\ exists ff_q_pfp_append_coefficient_constantdiagonalentry. pfc_terms_code_append_coefficient_constant = ff_q_pfp_append_coefficient_constantdiagonalentry * S ((S (pfc_index_append_coefficient_constantdiagonal)) * pfc_terms_scale_append_coefficient_constant) + (pfc_value_append_coefficient_constantdiagonal))) /\\ ((exists pfc_complement_append_coefficient_constantdiagonalterm pfc_left_append_coefficient_constantdiagonalterm pfc_right_append_coefficient_constantdiagonalterm. (((pfc_index_append_coefficient_constantdiagonal)+pfc_complement_append_coefficient_constantdiagonalterm=(i)) /\\ ((((((exists pfa_gap_append_coefficient_constantdiagonaltermleftinside. pfa_gap_append_coefficient_constantdiagonaltermleftinside + S (pfc_index_append_coefficient_constantdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_coefficient_constantdiagonaltermleftentry. ff_h_pfp_append_coefficient_constantdiagonaltermleftentry + S (pfc_left_append_coefficient_constantdiagonalterm) = S ((S (pfc_index_append_coefficient_constantdiagonal)) * ac)) /\\ exists ff_q_pfp_append_coefficient_constantdiagonaltermleftentry. ab = ff_q_pfp_append_coefficient_constantdiagonaltermleftentry * S ((S (pfc_index_append_coefficient_constantdiagonal)) * ac) + (pfc_left_append_coefficient_constantdiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_constantdiagonaltermleftoutside. pfc_gap_append_coefficient_constantdiagonaltermleftoutside+(L)=(pfc_index_append_coefficient_constantdiagonal)) /\\ (((pfc_left_append_coefficient_constantdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_coefficient_constantdiagonaltermrightinside. pfa_gap_append_coefficient_constantdiagonaltermrightinside + S (pfc_complement_append_coefficient_constantdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_append_coefficient_constantdiagonaltermrightentry. ff_h_pfp_append_coefficient_constantdiagonaltermrightentry + S (pfc_right_append_coefficient_constantdiagonalterm) = S ((S (pfc_complement_append_coefficient_constantdiagonalterm)) * tc)) /\\ exists ff_q_pfp_append_coefficient_constantdiagonaltermrightentry. tb = ff_q_pfp_append_coefficient_constantdiagonaltermrightentry * S ((S (pfc_complement_append_coefficient_constantdiagonalterm)) * tc) + (pfc_right_append_coefficient_constantdiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_constantdiagonaltermrightoutside. pfc_gap_append_coefficient_constantdiagonaltermrightoutside+(S M)=(pfc_complement_append_coefficient_constantdiagonalterm)) /\\ (((pfc_right_append_coefficient_constantdiagonalterm)=0))))) /\\ (((pfc_value_append_coefficient_constantdiagonal)=pfc_left_append_coefficient_constantdiagonalterm*pfc_right_append_coefficient_constantdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_coefficient_constantsum fs_v_pfc_append_coefficient_constantsum. ((((exists fs_h_pfc_append_coefficient_constantsum_body_start. fs_h_pfc_append_coefficient_constantsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_coefficient_constantsum)) /\\ exists fs_q_pfc_append_coefficient_constantsum_body_start. fs_u_pfc_append_coefficient_constantsum = fs_q_pfc_append_coefficient_constantsum_body_start * S ((S (0)) * fs_v_pfc_append_coefficient_constantsum) + (0))) /\\ ((((exists fs_h_pfc_append_coefficient_constantsum_body_terminal. fs_h_pfc_append_coefficient_constantsum_body_terminal + S (pfc_natural_sum_append_coefficient_constant) = S ((S (S (i))) * fs_v_pfc_append_coefficient_constantsum)) /\\ exists fs_q_pfc_append_coefficient_constantsum_body_terminal. fs_u_pfc_append_coefficient_constantsum = fs_q_pfc_append_coefficient_constantsum_body_terminal * S ((S (S (i))) * fs_v_pfc_append_coefficient_constantsum) + (pfc_natural_sum_append_coefficient_constant))) /\\ forall fs_i_pfc_append_coefficient_constantsum_body_steps. (exists fs_lt_pfc_append_coefficient_constantsum_body_steps_bound. fs_lt_pfc_append_coefficient_constantsum_body_steps_bound + S fs_i_pfc_append_coefficient_constantsum_body_steps = S (i)) -> exists fs_a_pfc_append_coefficient_constantsum_body_steps fs_r_pfc_append_coefficient_constantsum_body_steps fs_s_pfc_append_coefficient_constantsum_body_steps. ((((exists fs_h_pfc_append_coefficient_constantsum_body_steps_summand. fs_h_pfc_append_coefficient_constantsum_body_steps_summand + S (fs_a_pfc_append_coefficient_constantsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_constantsum_body_steps)) * pfc_terms_scale_append_coefficient_constant)) /\\ exists fs_q_pfc_append_coefficient_constantsum_body_steps_summand. pfc_terms_code_append_coefficient_constant = fs_q_pfc_append_coefficient_constantsum_body_steps_summand * S ((S (fs_i_pfc_append_coefficient_constantsum_body_steps)) * pfc_terms_scale_append_coefficient_constant) + (fs_a_pfc_append_coefficient_constantsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_constantsum_body_steps_partial. fs_h_pfc_append_coefficient_constantsum_body_steps_partial + S (fs_r_pfc_append_coefficient_constantsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_constantsum_body_steps)) * fs_v_pfc_append_coefficient_constantsum)) /\\ exists fs_q_pfc_append_coefficient_constantsum_body_steps_partial. fs_u_pfc_append_coefficient_constantsum = fs_q_pfc_append_coefficient_constantsum_body_steps_partial * S ((S (fs_i_pfc_append_coefficient_constantsum_body_steps)) * fs_v_pfc_append_coefficient_constantsum) + (fs_r_pfc_append_coefficient_constantsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_constantsum_body_steps_successor. fs_h_pfc_append_coefficient_constantsum_body_steps_successor + S (fs_s_pfc_append_coefficient_constantsum_body_steps) = S ((S (S fs_i_pfc_append_coefficient_constantsum_body_steps)) * fs_v_pfc_append_coefficient_constantsum)) /\\ exists fs_q_pfc_append_coefficient_constantsum_body_steps_successor. fs_u_pfc_append_coefficient_constantsum = fs_q_pfc_append_coefficient_constantsum_body_steps_successor * S ((S (S fs_i_pfc_append_coefficient_constantsum_body_steps)) * fs_v_pfc_append_coefficient_constantsum) + (fs_s_pfc_append_coefficient_constantsum_body_steps))) /\\ fs_s_pfc_append_coefficient_constantsum_body_steps = fs_r_pfc_append_coefficient_constantsum_body_steps + fs_a_pfc_append_coefficient_constantsum_body_steps)))))) /\\ ((((exists pfa_gap_append_coefficient_constantresiduebound. pfa_gap_append_coefficient_constantresiduebound + S (v) = (p)) /\\ ((exists pfa_offset_left_append_coefficient_constantresiduecongruence pfa_offset_right_append_coefficient_constantresiduecongruence. (pfc_natural_sum_append_coefficient_constant) + (p) * pfa_offset_left_append_coefficient_constantresiduecongruence = (v) + (p) * pfa_offset_right_append_coefficient_constantresiduecongruence))))))))) -> (exists pfc_terms_code_append_coefficient_new pfc_terms_scale_append_coefficient_new pfc_natural_sum_append_coefficient_new. ((forall pfc_index_append_coefficient_newdiagonal. (exists pfa_gap_append_coefficient_newdiagonalbound. pfa_gap_append_coefficient_newdiagonalbound + S (pfc_index_append_coefficient_newdiagonal) = (S (i))) -> exists pfc_value_append_coefficient_newdiagonal. ((((exists ff_h_pfp_append_coefficient_newdiagonalentry. ff_h_pfp_append_coefficient_newdiagonalentry + S (pfc_value_append_coefficient_newdiagonal) = S ((S (pfc_index_append_coefficient_newdiagonal)) * pfc_terms_scale_append_coefficient_new)) /\\ exists ff_q_pfp_append_coefficient_newdiagonalentry. pfc_terms_code_append_coefficient_new = ff_q_pfp_append_coefficient_newdiagonalentry * S ((S (pfc_index_append_coefficient_newdiagonal)) * pfc_terms_scale_append_coefficient_new) + (pfc_value_append_coefficient_newdiagonal))) /\\ ((exists pfc_complement_append_coefficient_newdiagonalterm pfc_left_append_coefficient_newdiagonalterm pfc_right_append_coefficient_newdiagonalterm. (((pfc_index_append_coefficient_newdiagonal)+pfc_complement_append_coefficient_newdiagonalterm=(i)) /\\ ((((((exists pfa_gap_append_coefficient_newdiagonaltermleftinside. pfa_gap_append_coefficient_newdiagonaltermleftinside + S (pfc_index_append_coefficient_newdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_coefficient_newdiagonaltermleftentry. ff_h_pfp_append_coefficient_newdiagonaltermleftentry + S (pfc_left_append_coefficient_newdiagonalterm) = S ((S (pfc_index_append_coefficient_newdiagonal)) * ac)) /\\ exists ff_q_pfp_append_coefficient_newdiagonaltermleftentry. ab = ff_q_pfp_append_coefficient_newdiagonaltermleftentry * S ((S (pfc_index_append_coefficient_newdiagonal)) * ac) + (pfc_left_append_coefficient_newdiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_newdiagonaltermleftoutside. pfc_gap_append_coefficient_newdiagonaltermleftoutside+(L)=(pfc_index_append_coefficient_newdiagonal)) /\\ (((pfc_left_append_coefficient_newdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_coefficient_newdiagonaltermrightinside. pfa_gap_append_coefficient_newdiagonaltermrightinside + S (pfc_complement_append_coefficient_newdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_append_coefficient_newdiagonaltermrightentry. ff_h_pfp_append_coefficient_newdiagonaltermrightentry + S (pfc_right_append_coefficient_newdiagonalterm) = S ((S (pfc_complement_append_coefficient_newdiagonalterm)) * dc)) /\\ exists ff_q_pfp_append_coefficient_newdiagonaltermrightentry. db = ff_q_pfp_append_coefficient_newdiagonaltermrightentry * S ((S (pfc_complement_append_coefficient_newdiagonalterm)) * dc) + (pfc_right_append_coefficient_newdiagonalterm)))))) \\/ (((exists pfc_gap_append_coefficient_newdiagonaltermrightoutside. pfc_gap_append_coefficient_newdiagonaltermrightoutside+(S M)=(pfc_complement_append_coefficient_newdiagonalterm)) /\\ (((pfc_right_append_coefficient_newdiagonalterm)=0))))) /\\ (((pfc_value_append_coefficient_newdiagonal)=pfc_left_append_coefficient_newdiagonalterm*pfc_right_append_coefficient_newdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_coefficient_newsum fs_v_pfc_append_coefficient_newsum. ((((exists fs_h_pfc_append_coefficient_newsum_body_start. fs_h_pfc_append_coefficient_newsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_coefficient_newsum)) /\\ exists fs_q_pfc_append_coefficient_newsum_body_start. fs_u_pfc_append_coefficient_newsum = fs_q_pfc_append_coefficient_newsum_body_start * S ((S (0)) * fs_v_pfc_append_coefficient_newsum) + (0))) /\\ ((((exists fs_h_pfc_append_coefficient_newsum_body_terminal. fs_h_pfc_append_coefficient_newsum_body_terminal + S (pfc_natural_sum_append_coefficient_new) = S ((S (S (i))) * fs_v_pfc_append_coefficient_newsum)) /\\ exists fs_q_pfc_append_coefficient_newsum_body_terminal. fs_u_pfc_append_coefficient_newsum = fs_q_pfc_append_coefficient_newsum_body_terminal * S ((S (S (i))) * fs_v_pfc_append_coefficient_newsum) + (pfc_natural_sum_append_coefficient_new))) /\\ forall fs_i_pfc_append_coefficient_newsum_body_steps. (exists fs_lt_pfc_append_coefficient_newsum_body_steps_bound. fs_lt_pfc_append_coefficient_newsum_body_steps_bound + S fs_i_pfc_append_coefficient_newsum_body_steps = S (i)) -> exists fs_a_pfc_append_coefficient_newsum_body_steps fs_r_pfc_append_coefficient_newsum_body_steps fs_s_pfc_append_coefficient_newsum_body_steps. ((((exists fs_h_pfc_append_coefficient_newsum_body_steps_summand. fs_h_pfc_append_coefficient_newsum_body_steps_summand + S (fs_a_pfc_append_coefficient_newsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_newsum_body_steps)) * pfc_terms_scale_append_coefficient_new)) /\\ exists fs_q_pfc_append_coefficient_newsum_body_steps_summand. pfc_terms_code_append_coefficient_new = fs_q_pfc_append_coefficient_newsum_body_steps_summand * S ((S (fs_i_pfc_append_coefficient_newsum_body_steps)) * pfc_terms_scale_append_coefficient_new) + (fs_a_pfc_append_coefficient_newsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_newsum_body_steps_partial. fs_h_pfc_append_coefficient_newsum_body_steps_partial + S (fs_r_pfc_append_coefficient_newsum_body_steps) = S ((S (fs_i_pfc_append_coefficient_newsum_body_steps)) * fs_v_pfc_append_coefficient_newsum)) /\\ exists fs_q_pfc_append_coefficient_newsum_body_steps_partial. fs_u_pfc_append_coefficient_newsum = fs_q_pfc_append_coefficient_newsum_body_steps_partial * S ((S (fs_i_pfc_append_coefficient_newsum_body_steps)) * fs_v_pfc_append_coefficient_newsum) + (fs_r_pfc_append_coefficient_newsum_body_steps))) /\\ ((((exists fs_h_pfc_append_coefficient_newsum_body_steps_successor. fs_h_pfc_append_coefficient_newsum_body_steps_successor + S (fs_s_pfc_append_coefficient_newsum_body_steps) = S ((S (S fs_i_pfc_append_coefficient_newsum_body_steps)) * fs_v_pfc_append_coefficient_newsum)) /\\ exists fs_q_pfc_append_coefficient_newsum_body_steps_successor. fs_u_pfc_append_coefficient_newsum = fs_q_pfc_append_coefficient_newsum_body_steps_successor * S ((S (S fs_i_pfc_append_coefficient_newsum_body_steps)) * fs_v_pfc_append_coefficient_newsum) + (fs_s_pfc_append_coefficient_newsum_body_steps))) /\\ fs_s_pfc_append_coefficient_newsum_body_steps = fs_r_pfc_append_coefficient_newsum_body_steps + fs_a_pfc_append_coefficient_newsum_body_steps)))))) /\\ ((((exists pfa_gap_append_coefficient_newresiduebound. pfa_gap_append_coefficient_newresiduebound + S (w) = (p)) /\\ ((exists pfa_offset_left_append_coefficient_newresiduecongruence pfa_offset_right_append_coefficient_newresiduecongruence. (pfc_natural_sum_append_coefficient_new) + (p) * pfa_offset_left_append_coefficient_newresiduecongruence = (w) + (p) * pfa_offset_right_append_coefficient_newresiduecongruence))))))))) -> (((exists pfa_gap_append_coefficient_resultleft. pfa_gap_append_coefficient_resultleft + S (u) = (p)) /\\ (((exists pfa_gap_append_coefficient_resultright. pfa_gap_append_coefficient_resultright + S (v) = (p)) /\\ ((((exists pfa_gap_append_coefficient_resultresultbound. pfa_gap_append_coefficient_resultresultbound + S (w) = (p)) /\\ ((exists pfa_offset_left_append_coefficient_resultresultcongruence pfa_offset_right_append_coefficient_resultresultcongruence. ((u) + (v)) + (p) * pfa_offset_left_append_coefficient_resultresultcongruence = (w) + (p) * pfa_offset_right_append_coefficient_resultresultcongruence)))))))))", "statement_sha256": "a11e1f29b31ae9076959706b6b5d0813689194a2ab57a1a4e879e6a6c3ad69bd", "summary": "At every natural coefficient index, an actual right append is the actual field sum of the old convolution coefficient and the product with the padded singleton constant, using genuine diagonal sums and residues rather than a finite-evaluation identity.", "summary_sha256": "88d9d17103ce22bb8249357f65656e4bee4a5370977d417857fe77c2f2e96ad8" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "07e26b2116c6263acbdbb97eaae52c80ace53eda692a8341693cac69987c7532", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 261, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "271845bfffc7e513fdb0bd0c3666dcccace8436d4d3a0f4db64b67bcd4b87042", "test_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85" }, "body_checked": true, "body_receipt": { "command_count": 134, "dependency_count": 9, "dne_command_count": 0, "name": "prime_field_polynomial_shift_scale_aligned_sum_exists", "proof_depth": 46, "proof_edges": 157, "proof_nodes": 158, "proof_objects": 158, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "prime_nonzero", "prime_field_polynomial_shift_exists", "prime_field_polynomial_scale_exists", "prime_field_polynomial_left_pad_exists", "prime_field_polynomial_left_pad_bounded", "prime_field_polynomial_shift_bounded", "prime_field_polynomial_scale_bounded", "add_comm", "prime_field_polynomial_add_exists" ], 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"research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=261]" }, { "document_sha256": "81943a545e7b9f667ad758c24de14bb47280e8cf3fbb3ca714efec55fb096408", "kind": "alpha_v34_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "c13ff61c82cd068bf3638c4296ce2390def48989274db52aeb6205647515ef3a", "membership": "alpha_only", "name": "prime_field_polynomial_shift_scale_aligned_sum_exists", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro c", "intro ab", "intro ac", "intro L", "intro pb", "intro pc", "intro N", "intro hp", "intro hc", "intro ha", "intro hb", "have hp0 : ~(p=0)", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hz", "have hu : exists ub uc. ((forall mdr_i_pfp_append_alignment_shiftprefix mdr_a_pfp_append_alignment_shiftprefix. (exists mdr_gap_pfp_append_alignment_shiftprefixb. mdr_gap_pfp_append_alignment_shiftprefixb + S (mdr_i_pfp_append_alignment_shiftprefix) = (N)) -> (((exists ff_h_mdr_pfp_append_alignment_shiftprefixo. ff_h_mdr_pfp_append_alignment_shiftprefixo + S (mdr_a_pfp_append_alignment_shiftprefix) = S ((S (mdr_i_pfp_append_alignment_shiftprefix)) * pc)) /\\ exists ff_q_mdr_pfp_append_alignment_shiftprefixo. pb = ff_q_mdr_pfp_append_alignment_shiftprefixo * S ((S (mdr_i_pfp_append_alignment_shiftprefix)) * pc) + (mdr_a_pfp_append_alignment_shiftprefix))) -> (((exists ff_h_mdr_pfp_append_alignment_shiftprefixn. ff_h_mdr_pfp_append_alignment_shiftprefixn + S (mdr_a_pfp_append_alignment_shiftprefix) = S ((S (mdr_i_pfp_append_alignment_shiftprefix)) * uc)) /\\ exists ff_q_mdr_pfp_append_alignment_shiftprefixn. ub = ff_q_mdr_pfp_append_alignment_shiftprefixn * S ((S (mdr_i_pfp_append_alignment_shiftprefix)) * uc) + (mdr_a_pfp_append_alignment_shiftprefix)))) /\\ ((((exists ff_h_pfp_append_alignment_shiftlast. ff_h_pfp_append_alignment_shiftlast + S (0) = S ((S (N)) * uc)) /\\ exists ff_q_pfp_append_alignment_shiftlast. ub = ff_q_pfp_append_alignment_shiftlast * S ((S (N)) * uc) + (0)))))", "specialize prime_field_polynomial_shift_exists (pb)", "specialize prime_field_polynomial_shift_exists (pc)", "specialize prime_field_polynomial_shift_exists (N)", "apply prime_field_polynomial_shift_exists", "cases hu", "cases hu_witness", "have hv : exists vb vc. ((exists pfa_gap_append_alignment_scalescalar. pfa_gap_append_alignment_scalescalar + S (c) = (p)) /\\ ((forall pfp_index_append_alignment_scale. (exists pfa_gap_append_alignment_scaleindex. pfa_gap_append_alignment_scaleindex + S (pfp_index_append_alignment_scale) = (L)) -> exists pfp_source_append_alignment_scale pfp_value_append_alignment_scale. ((((exists ff_h_pfp_append_alignment_scalesource. ff_h_pfp_append_alignment_scalesource + S (pfp_source_append_alignment_scale) = S ((S (pfp_index_append_alignment_scale)) * ac)) /\\ exists ff_q_pfp_append_alignment_scalesource. ab = ff_q_pfp_append_alignment_scalesource * S ((S (pfp_index_append_alignment_scale)) * ac) + (pfp_source_append_alignment_scale))) /\\ (((((exists ff_h_pfp_append_alignment_scaletarget. ff_h_pfp_append_alignment_scaletarget + S (pfp_value_append_alignment_scale) = S ((S (pfp_index_append_alignment_scale)) * vc)) /\\ exists ff_q_pfp_append_alignment_scaletarget. vb = ff_q_pfp_append_alignment_scaletarget * S ((S (pfp_index_append_alignment_scale)) * vc) + (pfp_value_append_alignment_scale))) /\\ ((((exists pfa_gap_append_alignment_scaleoperationleft. pfa_gap_append_alignment_scaleoperationleft + S (c) = (p)) /\\ (((exists pfa_gap_append_alignment_scaleoperationright. pfa_gap_append_alignment_scaleoperationright + S (pfp_source_append_alignment_scale) = (p)) /\\ ((((exists pfa_gap_append_alignment_scaleoperationresultbound. pfa_gap_append_alignment_scaleoperationresultbound + S (pfp_value_append_alignment_scale) = (p)) /\\ ((exists pfa_offset_left_append_alignment_scaleoperationresultcongruence pfa_offset_right_append_alignment_scaleoperationresultcongruence. ((c) * (pfp_source_append_alignment_scale)) + (p) * pfa_offset_left_append_alignment_scaleoperationresultcongruence = (pfp_value_append_alignment_scale) + (p) * pfa_offset_right_append_alignment_scaleoperationresultcongruence))))))))))))))))", "specialize prime_field_polynomial_scale_exists (p)", "specialize prime_field_polynomial_scale_exists (c)", "specialize prime_field_polynomial_scale_exists (ab)", "specialize prime_field_polynomial_scale_exists (ac)", "specialize prime_field_polynomial_scale_exists (L)", "apply prime_field_polynomial_scale_exists", "exact hp0", "exact hc", "exact ha", "cases hv", "cases hv_witness", "have hleft : exists UB UC. ((forall pfp_repeat_index_append_alignment_chosen_leftzeros. (exists pfa_gap_append_alignment_chosen_leftzerosindex. pfa_gap_append_alignment_chosen_leftzerosindex + S (pfp_repeat_index_append_alignment_chosen_leftzeros) = (L)) -> (((exists ff_h_pfp_append_alignment_chosen_leftzerosentry. ff_h_pfp_append_alignment_chosen_leftzerosentry + S (0) = S ((S (pfp_repeat_index_append_alignment_chosen_leftzeros)) * UC)) /\\ exists ff_q_pfp_append_alignment_chosen_leftzerosentry. UB = ff_q_pfp_append_alignment_chosen_leftzerosentry * S ((S (pfp_repeat_index_append_alignment_chosen_leftzeros)) * UC) + (0)))) /\\ ((forall pfrep_index_append_alignment_chosen_left pfrep_value_append_alignment_chosen_left. (exists pfa_gap_append_alignment_chosen_leftbound. pfa_gap_append_alignment_chosen_leftbound + S (pfrep_index_append_alignment_chosen_left) = (S N)) -> (((exists ff_h_pfp_append_alignment_chosen_leftinput. ff_h_pfp_append_alignment_chosen_leftinput + S (pfrep_value_append_alignment_chosen_left) = S ((S (pfrep_index_append_alignment_chosen_left)) * x1)) /\\ exists ff_q_pfp_append_alignment_chosen_leftinput. x = ff_q_pfp_append_alignment_chosen_leftinput * S ((S (pfrep_index_append_alignment_chosen_left)) * x1) + (pfrep_value_append_alignment_chosen_left))) -> (((exists ff_h_pfp_append_alignment_chosen_leftoutput. ff_h_pfp_append_alignment_chosen_leftoutput + S (pfrep_value_append_alignment_chosen_left) = S ((S ((L)+pfrep_index_append_alignment_chosen_left)) * UC)) /\\ exists ff_q_pfp_append_alignment_chosen_leftoutput. UB = ff_q_pfp_append_alignment_chosen_leftoutput * S ((S ((L)+pfrep_index_append_alignment_chosen_left)) * UC) + (pfrep_value_append_alignment_chosen_left))))))", "specialize prime_field_polynomial_left_pad_exists (x)", "specialize prime_field_polynomial_left_pad_exists (x1)", "specialize prime_field_polynomial_left_pad_exists (L)", "specialize prime_field_polynomial_left_pad_exists (S N)", "apply prime_field_polynomial_left_pad_exists", "cases hleft", "cases hleft_witness", "have hright : exists VB VC. ((forall pfp_repeat_index_append_alignment_chosen_rightzeros. (exists pfa_gap_append_alignment_chosen_rightzerosindex. pfa_gap_append_alignment_chosen_rightzerosindex + S (pfp_repeat_index_append_alignment_chosen_rightzeros) = (S N)) -> (((exists ff_h_pfp_append_alignment_chosen_rightzerosentry. ff_h_pfp_append_alignment_chosen_rightzerosentry + S (0) = S ((S (pfp_repeat_index_append_alignment_chosen_rightzeros)) * VC)) /\\ exists ff_q_pfp_append_alignment_chosen_rightzerosentry. VB = ff_q_pfp_append_alignment_chosen_rightzerosentry * S ((S (pfp_repeat_index_append_alignment_chosen_rightzeros)) * VC) + (0)))) /\\ ((forall pfrep_index_append_alignment_chosen_right pfrep_value_append_alignment_chosen_right. (exists pfa_gap_append_alignment_chosen_rightbound. pfa_gap_append_alignment_chosen_rightbound + S (pfrep_index_append_alignment_chosen_right) = (L)) -> (((exists ff_h_pfp_append_alignment_chosen_rightinput. ff_h_pfp_append_alignment_chosen_rightinput + S (pfrep_value_append_alignment_chosen_right) = S ((S (pfrep_index_append_alignment_chosen_right)) * x3)) /\\ exists ff_q_pfp_append_alignment_chosen_rightinput. x2 = ff_q_pfp_append_alignment_chosen_rightinput * S ((S (pfrep_index_append_alignment_chosen_right)) * x3) + (pfrep_value_append_alignment_chosen_right))) -> (((exists ff_h_pfp_append_alignment_chosen_rightoutput. ff_h_pfp_append_alignment_chosen_rightoutput + S (pfrep_value_append_alignment_chosen_right) = S ((S ((S N)+pfrep_index_append_alignment_chosen_right)) * VC)) /\\ exists ff_q_pfp_append_alignment_chosen_rightoutput. VB = ff_q_pfp_append_alignment_chosen_rightoutput * S ((S ((S N)+pfrep_index_append_alignment_chosen_right)) * VC) + (pfrep_value_append_alignment_chosen_right))))))", "specialize prime_field_polynomial_left_pad_exists (x2)", "specialize prime_field_polynomial_left_pad_exists (x3)", "specialize prime_field_polynomial_left_pad_exists (S N)", "specialize prime_field_polynomial_left_pad_exists (L)", "apply prime_field_polynomial_left_pad_exists", "cases hright", "cases hright_witness", "have hleft_bound : forall fom_index_pfp_append_alignment_left_bound. (exists fom_gap_pfp_append_alignment_left_bound_index_bound. fom_gap_pfp_append_alignment_left_bound_index_bound + S (fom_index_pfp_append_alignment_left_bound) = L+S N) -> exists fom_value_pfp_append_alignment_left_bound. ((((exists fom_beta_height_pfp_append_alignment_left_bound_entry. fom_beta_height_pfp_append_alignment_left_bound_entry + S (fom_value_pfp_append_alignment_left_bound) = S ((S (fom_index_pfp_append_alignment_left_bound)) * x5)) /\\ exists fom_beta_quotient_pfp_append_alignment_left_bound_entry. x4 = fom_beta_quotient_pfp_append_alignment_left_bound_entry * S ((S (fom_index_pfp_append_alignment_left_bound)) * x5) + (fom_value_pfp_append_alignment_left_bound))) /\\ (exists fom_gap_pfp_append_alignment_left_bound_value_bound. fom_gap_pfp_append_alignment_left_bound_value_bound + S (fom_value_pfp_append_alignment_left_bound) = p))", "specialize prime_field_polynomial_left_pad_bounded (p)", "specialize prime_field_polynomial_left_pad_bounded (x)", "specialize prime_field_polynomial_left_pad_bounded (x1)", "specialize prime_field_polynomial_left_pad_bounded (S N)", "specialize prime_field_polynomial_left_pad_bounded (L)", "specialize prime_field_polynomial_left_pad_bounded (x4)", "specialize prime_field_polynomial_left_pad_bounded (x5)", "apply prime_field_polynomial_left_pad_bounded", "exact hp", "specialize prime_field_polynomial_shift_bounded (p)", "specialize prime_field_polynomial_shift_bounded (pb)", "specialize prime_field_polynomial_shift_bounded (pc)", "specialize prime_field_polynomial_shift_bounded (N)", "specialize prime_field_polynomial_shift_bounded (x)", "specialize prime_field_polynomial_shift_bounded (x1)", "apply prime_field_polynomial_shift_bounded", "exact hp", "exact hb", "exact hu_witness_witness", "exact hleft_witness_witness", "have hscale_bounds : ((forall fom_index_pfp_append_alignment_source_bound. (exists fom_gap_pfp_append_alignment_source_bound_index_bound. fom_gap_pfp_append_alignment_source_bound_index_bound + S (fom_index_pfp_append_alignment_source_bound) = L) -> exists fom_value_pfp_append_alignment_source_bound. ((((exists fom_beta_height_pfp_append_alignment_source_bound_entry. fom_beta_height_pfp_append_alignment_source_bound_entry + S (fom_value_pfp_append_alignment_source_bound) = S ((S (fom_index_pfp_append_alignment_source_bound)) * ac)) /\\ exists fom_beta_quotient_pfp_append_alignment_source_bound_entry. ab = fom_beta_quotient_pfp_append_alignment_source_bound_entry * S ((S (fom_index_pfp_append_alignment_source_bound)) * ac) + (fom_value_pfp_append_alignment_source_bound))) /\\ (exists fom_gap_pfp_append_alignment_source_bound_value_bound. fom_gap_pfp_append_alignment_source_bound_value_bound + S (fom_value_pfp_append_alignment_source_bound) = p))) /\\ ((forall fom_index_pfp_append_alignment_scale_bound. (exists fom_gap_pfp_append_alignment_scale_bound_index_bound. fom_gap_pfp_append_alignment_scale_bound_index_bound + S (fom_index_pfp_append_alignment_scale_bound) = L) -> exists fom_value_pfp_append_alignment_scale_bound. ((((exists fom_beta_height_pfp_append_alignment_scale_bound_entry. fom_beta_height_pfp_append_alignment_scale_bound_entry + S (fom_value_pfp_append_alignment_scale_bound) = S ((S (fom_index_pfp_append_alignment_scale_bound)) * x3)) /\\ exists fom_beta_quotient_pfp_append_alignment_scale_bound_entry. x2 = fom_beta_quotient_pfp_append_alignment_scale_bound_entry * S ((S (fom_index_pfp_append_alignment_scale_bound)) * x3) + (fom_value_pfp_append_alignment_scale_bound))) /\\ (exists fom_gap_pfp_append_alignment_scale_bound_value_bound. fom_gap_pfp_append_alignment_scale_bound_value_bound + S (fom_value_pfp_append_alignment_scale_bound) = p)))))", "specialize prime_field_polynomial_scale_bounded (p)", "specialize prime_field_polynomial_scale_bounded (c)", "specialize prime_field_polynomial_scale_bounded (ab)", "specialize prime_field_polynomial_scale_bounded (ac)", "specialize prime_field_polynomial_scale_bounded (x2)", "specialize prime_field_polynomial_scale_bounded (x3)", "specialize prime_field_polynomial_scale_bounded (L)", "apply prime_field_polynomial_scale_bounded", "exact hv_witness_witness", "cases hscale_bounds", "have hright_bound : forall fom_index_pfp_append_alignment_right_bound. (exists fom_gap_pfp_append_alignment_right_bound_index_bound. fom_gap_pfp_append_alignment_right_bound_index_bound + S (fom_index_pfp_append_alignment_right_bound) = S N+L) -> exists fom_value_pfp_append_alignment_right_bound. ((((exists fom_beta_height_pfp_append_alignment_right_bound_entry. fom_beta_height_pfp_append_alignment_right_bound_entry + S (fom_value_pfp_append_alignment_right_bound) = S ((S (fom_index_pfp_append_alignment_right_bound)) * x7)) /\\ exists fom_beta_quotient_pfp_append_alignment_right_bound_entry. x6 = fom_beta_quotient_pfp_append_alignment_right_bound_entry * S ((S (fom_index_pfp_append_alignment_right_bound)) * x7) + (fom_value_pfp_append_alignment_right_bound))) /\\ (exists fom_gap_pfp_append_alignment_right_bound_value_bound. fom_gap_pfp_append_alignment_right_bound_value_bound + S (fom_value_pfp_append_alignment_right_bound) = p))", "specialize prime_field_polynomial_left_pad_bounded (p)", "specialize prime_field_polynomial_left_pad_bounded (x2)", "specialize prime_field_polynomial_left_pad_bounded (x3)", "specialize prime_field_polynomial_left_pad_bounded (L)", "specialize prime_field_polynomial_left_pad_bounded (S N)", "specialize prime_field_polynomial_left_pad_bounded (x6)", "specialize prime_field_polynomial_left_pad_bounded (x7)", "apply prime_field_polynomial_left_pad_bounded", "exact hp", "exact hscale_bounds_right", "exact hright_witness_witness", "have hcomm : S N+L=L+S N", "specialize add_comm (S N)", "specialize add_comm (L)", "apply add_comm", "rewrite hcomm at hright_bound", "have hsum : exists rb rc. forall pfp_index_append_alignment_chosen_sum. (exists pfa_gap_append_alignment_chosen_sumindex. pfa_gap_append_alignment_chosen_sumindex + S (pfp_index_append_alignment_chosen_sum) = (L+S N)) -> exists pfp_left_append_alignment_chosen_sum pfp_right_append_alignment_chosen_sum pfp_value_append_alignment_chosen_sum. ((((exists ff_h_pfp_append_alignment_chosen_sumleft. ff_h_pfp_append_alignment_chosen_sumleft + S (pfp_left_append_alignment_chosen_sum) = S ((S (pfp_index_append_alignment_chosen_sum)) * x5)) /\\ exists ff_q_pfp_append_alignment_chosen_sumleft. x4 = ff_q_pfp_append_alignment_chosen_sumleft * S ((S (pfp_index_append_alignment_chosen_sum)) * x5) + (pfp_left_append_alignment_chosen_sum))) /\\ (((((exists ff_h_pfp_append_alignment_chosen_sumright. ff_h_pfp_append_alignment_chosen_sumright + S (pfp_right_append_alignment_chosen_sum) = S ((S (pfp_index_append_alignment_chosen_sum)) * x7)) /\\ exists ff_q_pfp_append_alignment_chosen_sumright. x6 = ff_q_pfp_append_alignment_chosen_sumright * S ((S (pfp_index_append_alignment_chosen_sum)) * x7) + (pfp_right_append_alignment_chosen_sum))) /\\ (((((exists ff_h_pfp_append_alignment_chosen_sumtarget. ff_h_pfp_append_alignment_chosen_sumtarget + S (pfp_value_append_alignment_chosen_sum) = S ((S (pfp_index_append_alignment_chosen_sum)) * rc)) /\\ exists ff_q_pfp_append_alignment_chosen_sumtarget. rb = ff_q_pfp_append_alignment_chosen_sumtarget * S ((S (pfp_index_append_alignment_chosen_sum)) * rc) + (pfp_value_append_alignment_chosen_sum))) /\\ ((((exists pfa_gap_append_alignment_chosen_sumoperationleft. pfa_gap_append_alignment_chosen_sumoperationleft + S (pfp_left_append_alignment_chosen_sum) = (p)) /\\ (((exists pfa_gap_append_alignment_chosen_sumoperationright. pfa_gap_append_alignment_chosen_sumoperationright + S (pfp_right_append_alignment_chosen_sum) = (p)) /\\ ((((exists pfa_gap_append_alignment_chosen_sumoperationresultbound. pfa_gap_append_alignment_chosen_sumoperationresultbound + S (pfp_value_append_alignment_chosen_sum) = (p)) /\\ ((exists pfa_offset_left_append_alignment_chosen_sumoperationresultcongruence pfa_offset_right_append_alignment_chosen_sumoperationresultcongruence. ((pfp_left_append_alignment_chosen_sum) + (pfp_right_append_alignment_chosen_sum)) + (p) * pfa_offset_left_append_alignment_chosen_sumoperationresultcongruence = (pfp_value_append_alignment_chosen_sum) + (p) * pfa_offset_right_append_alignment_chosen_sumoperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_add_exists (p)", "specialize prime_field_polynomial_add_exists (x4)", "specialize prime_field_polynomial_add_exists (x5)", "specialize prime_field_polynomial_add_exists (x6)", "specialize prime_field_polynomial_add_exists (x7)", "specialize prime_field_polynomial_add_exists (L+S N)", "apply prime_field_polynomial_add_exists", "exact hp0", "exact hleft_bound", "exact hright_bound", "cases hsum", "cases hsum_witness", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "exists x6", "exists x7", "exists x8", "exists x9", "split", "exact hu_witness_witness", "split", "exact hv_witness_witness", "split", "exact hleft_witness_witness", "split", "exact hright_witness_witness", "exact hsum_witness_witness" ], "script_sha256": "af22187ea7b816af4d5021820f8fc6edcd09bb9af321875e63fbf518166f44b6", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_append_candidate.py", "sha256": "271845bfffc7e513fdb0bd0c3666dcccace8436d4d3a0f4db64b67bcd4b87042" }, "statement": "forall p c ab ac L pb pc N. (~((p) = 1) /\\ forall pfa_factor_left_append_alignment_prime pfa_factor_right_append_alignment_prime. (p) = pfa_factor_left_append_alignment_prime * pfa_factor_right_append_alignment_prime -> pfa_factor_left_append_alignment_prime = 1 \\/ pfa_factor_right_append_alignment_prime = 1) -> (exists pfa_gap_append_alignment_scalar. pfa_gap_append_alignment_scalar + S (c) = (p)) -> (forall fom_index_pfp_append_alignment_A. (exists fom_gap_pfp_append_alignment_A_index_bound. fom_gap_pfp_append_alignment_A_index_bound + S (fom_index_pfp_append_alignment_A) = L) -> exists fom_value_pfp_append_alignment_A. ((((exists fom_beta_height_pfp_append_alignment_A_entry. fom_beta_height_pfp_append_alignment_A_entry + S (fom_value_pfp_append_alignment_A) = S ((S (fom_index_pfp_append_alignment_A)) * ac)) /\\ exists fom_beta_quotient_pfp_append_alignment_A_entry. ab = fom_beta_quotient_pfp_append_alignment_A_entry * S ((S (fom_index_pfp_append_alignment_A)) * ac) + (fom_value_pfp_append_alignment_A))) /\\ (exists fom_gap_pfp_append_alignment_A_value_bound. fom_gap_pfp_append_alignment_A_value_bound + S (fom_value_pfp_append_alignment_A) = p))) -> (forall fom_index_pfp_append_alignment_P. (exists fom_gap_pfp_append_alignment_P_index_bound. fom_gap_pfp_append_alignment_P_index_bound + S (fom_index_pfp_append_alignment_P) = N) -> exists fom_value_pfp_append_alignment_P. ((((exists fom_beta_height_pfp_append_alignment_P_entry. fom_beta_height_pfp_append_alignment_P_entry + S (fom_value_pfp_append_alignment_P) = S ((S (fom_index_pfp_append_alignment_P)) * pc)) /\\ exists fom_beta_quotient_pfp_append_alignment_P_entry. pb = fom_beta_quotient_pfp_append_alignment_P_entry * S ((S (fom_index_pfp_append_alignment_P)) * pc) + (fom_value_pfp_append_alignment_P))) /\\ (exists fom_gap_pfp_append_alignment_P_value_bound. fom_gap_pfp_append_alignment_P_value_bound + S (fom_value_pfp_append_alignment_P) = p))) -> (exists ub uc vb vc UB UC VB VC rb rc. ((((forall mdr_i_pfp_append_alignment_result_shiftprefix mdr_a_pfp_append_alignment_result_shiftprefix. (exists mdr_gap_pfp_append_alignment_result_shiftprefixb. mdr_gap_pfp_append_alignment_result_shiftprefixb + S (mdr_i_pfp_append_alignment_result_shiftprefix) = (N)) -> (((exists ff_h_mdr_pfp_append_alignment_result_shiftprefixo. ff_h_mdr_pfp_append_alignment_result_shiftprefixo + S (mdr_a_pfp_append_alignment_result_shiftprefix) = S ((S (mdr_i_pfp_append_alignment_result_shiftprefix)) * pc)) /\\ exists ff_q_mdr_pfp_append_alignment_result_shiftprefixo. pb = ff_q_mdr_pfp_append_alignment_result_shiftprefixo * S ((S (mdr_i_pfp_append_alignment_result_shiftprefix)) * pc) + (mdr_a_pfp_append_alignment_result_shiftprefix))) -> (((exists ff_h_mdr_pfp_append_alignment_result_shiftprefixn. ff_h_mdr_pfp_append_alignment_result_shiftprefixn + S (mdr_a_pfp_append_alignment_result_shiftprefix) = S ((S (mdr_i_pfp_append_alignment_result_shiftprefix)) * uc)) /\\ exists ff_q_mdr_pfp_append_alignment_result_shiftprefixn. ub = ff_q_mdr_pfp_append_alignment_result_shiftprefixn * S ((S (mdr_i_pfp_append_alignment_result_shiftprefix)) * uc) + (mdr_a_pfp_append_alignment_result_shiftprefix)))) /\\ ((((exists ff_h_pfp_append_alignment_result_shiftlast. ff_h_pfp_append_alignment_result_shiftlast + S (0) = S ((S (N)) * uc)) /\\ exists ff_q_pfp_append_alignment_result_shiftlast. ub = ff_q_pfp_append_alignment_result_shiftlast * S ((S (N)) * uc) + (0)))))) /\\ (((((exists pfa_gap_append_alignment_result_scalescalar. pfa_gap_append_alignment_result_scalescalar + S (c) = (p)) /\\ ((forall pfp_index_append_alignment_result_scale. (exists pfa_gap_append_alignment_result_scaleindex. pfa_gap_append_alignment_result_scaleindex + S (pfp_index_append_alignment_result_scale) = (L)) -> exists pfp_source_append_alignment_result_scale pfp_value_append_alignment_result_scale. ((((exists ff_h_pfp_append_alignment_result_scalesource. ff_h_pfp_append_alignment_result_scalesource + S (pfp_source_append_alignment_result_scale) = S ((S (pfp_index_append_alignment_result_scale)) * ac)) /\\ exists ff_q_pfp_append_alignment_result_scalesource. ab = ff_q_pfp_append_alignment_result_scalesource * S ((S (pfp_index_append_alignment_result_scale)) * ac) + (pfp_source_append_alignment_result_scale))) /\\ (((((exists ff_h_pfp_append_alignment_result_scaletarget. ff_h_pfp_append_alignment_result_scaletarget + S (pfp_value_append_alignment_result_scale) = S ((S (pfp_index_append_alignment_result_scale)) * vc)) /\\ exists ff_q_pfp_append_alignment_result_scaletarget. vb = ff_q_pfp_append_alignment_result_scaletarget * S ((S (pfp_index_append_alignment_result_scale)) * vc) + (pfp_value_append_alignment_result_scale))) /\\ ((((exists pfa_gap_append_alignment_result_scaleoperationleft. pfa_gap_append_alignment_result_scaleoperationleft + S (c) = (p)) /\\ (((exists pfa_gap_append_alignment_result_scaleoperationright. pfa_gap_append_alignment_result_scaleoperationright + S (pfp_source_append_alignment_result_scale) = (p)) /\\ ((((exists pfa_gap_append_alignment_result_scaleoperationresultbound. pfa_gap_append_alignment_result_scaleoperationresultbound + S (pfp_value_append_alignment_result_scale) = (p)) /\\ ((exists pfa_offset_left_append_alignment_result_scaleoperationresultcongruence pfa_offset_right_append_alignment_result_scaleoperationresultcongruence. ((c) * (pfp_source_append_alignment_result_scale)) + (p) * pfa_offset_left_append_alignment_result_scaleoperationresultcongruence = (pfp_value_append_alignment_result_scale) + (p) * pfa_offset_right_append_alignment_result_scaleoperationresultcongruence))))))))))))))))) /\\ (((((forall pfp_repeat_index_append_alignment_result_leftzeros. (exists pfa_gap_append_alignment_result_leftzerosindex. pfa_gap_append_alignment_result_leftzerosindex + S (pfp_repeat_index_append_alignment_result_leftzeros) = (L)) -> (((exists ff_h_pfp_append_alignment_result_leftzerosentry. ff_h_pfp_append_alignment_result_leftzerosentry + S (0) = S ((S (pfp_repeat_index_append_alignment_result_leftzeros)) * UC)) /\\ exists ff_q_pfp_append_alignment_result_leftzerosentry. UB = ff_q_pfp_append_alignment_result_leftzerosentry * S ((S (pfp_repeat_index_append_alignment_result_leftzeros)) * UC) + (0)))) /\\ ((forall pfrep_index_append_alignment_result_left pfrep_value_append_alignment_result_left. (exists pfa_gap_append_alignment_result_leftbound. pfa_gap_append_alignment_result_leftbound + S (pfrep_index_append_alignment_result_left) = (S N)) -> (((exists ff_h_pfp_append_alignment_result_leftinput. ff_h_pfp_append_alignment_result_leftinput + S (pfrep_value_append_alignment_result_left) = S ((S (pfrep_index_append_alignment_result_left)) * uc)) /\\ exists ff_q_pfp_append_alignment_result_leftinput. ub = ff_q_pfp_append_alignment_result_leftinput * S ((S (pfrep_index_append_alignment_result_left)) * uc) + (pfrep_value_append_alignment_result_left))) -> (((exists ff_h_pfp_append_alignment_result_leftoutput. ff_h_pfp_append_alignment_result_leftoutput + S (pfrep_value_append_alignment_result_left) = S ((S ((L)+pfrep_index_append_alignment_result_left)) * UC)) /\\ exists ff_q_pfp_append_alignment_result_leftoutput. UB = ff_q_pfp_append_alignment_result_leftoutput * S ((S ((L)+pfrep_index_append_alignment_result_left)) * UC) + (pfrep_value_append_alignment_result_left))))))) /\\ (((((forall pfp_repeat_index_append_alignment_result_rightzeros. (exists pfa_gap_append_alignment_result_rightzerosindex. pfa_gap_append_alignment_result_rightzerosindex + S (pfp_repeat_index_append_alignment_result_rightzeros) = (S N)) -> (((exists ff_h_pfp_append_alignment_result_rightzerosentry. ff_h_pfp_append_alignment_result_rightzerosentry + S (0) = S ((S (pfp_repeat_index_append_alignment_result_rightzeros)) * VC)) /\\ exists ff_q_pfp_append_alignment_result_rightzerosentry. VB = ff_q_pfp_append_alignment_result_rightzerosentry * S ((S (pfp_repeat_index_append_alignment_result_rightzeros)) * VC) + (0)))) /\\ ((forall pfrep_index_append_alignment_result_right pfrep_value_append_alignment_result_right. (exists pfa_gap_append_alignment_result_rightbound. pfa_gap_append_alignment_result_rightbound + S (pfrep_index_append_alignment_result_right) = (L)) -> (((exists ff_h_pfp_append_alignment_result_rightinput. ff_h_pfp_append_alignment_result_rightinput + S (pfrep_value_append_alignment_result_right) = S ((S (pfrep_index_append_alignment_result_right)) * vc)) /\\ exists ff_q_pfp_append_alignment_result_rightinput. vb = ff_q_pfp_append_alignment_result_rightinput * S ((S (pfrep_index_append_alignment_result_right)) * vc) + (pfrep_value_append_alignment_result_right))) -> (((exists ff_h_pfp_append_alignment_result_rightoutput. ff_h_pfp_append_alignment_result_rightoutput + S (pfrep_value_append_alignment_result_right) = S ((S ((S N)+pfrep_index_append_alignment_result_right)) * VC)) /\\ exists ff_q_pfp_append_alignment_result_rightoutput. VB = ff_q_pfp_append_alignment_result_rightoutput * S ((S ((S N)+pfrep_index_append_alignment_result_right)) * VC) + (pfrep_value_append_alignment_result_right))))))) /\\ ((forall pfp_index_append_alignment_result_sum. (exists pfa_gap_append_alignment_result_sumindex. pfa_gap_append_alignment_result_sumindex + S (pfp_index_append_alignment_result_sum) = (L+S N)) -> exists pfp_left_append_alignment_result_sum pfp_right_append_alignment_result_sum pfp_value_append_alignment_result_sum. ((((exists ff_h_pfp_append_alignment_result_sumleft. ff_h_pfp_append_alignment_result_sumleft + S (pfp_left_append_alignment_result_sum) = S ((S (pfp_index_append_alignment_result_sum)) * UC)) /\\ exists ff_q_pfp_append_alignment_result_sumleft. UB = ff_q_pfp_append_alignment_result_sumleft * S ((S (pfp_index_append_alignment_result_sum)) * UC) + (pfp_left_append_alignment_result_sum))) /\\ (((((exists ff_h_pfp_append_alignment_result_sumright. ff_h_pfp_append_alignment_result_sumright + S (pfp_right_append_alignment_result_sum) = S ((S (pfp_index_append_alignment_result_sum)) * VC)) /\\ exists ff_q_pfp_append_alignment_result_sumright. VB = ff_q_pfp_append_alignment_result_sumright * S ((S (pfp_index_append_alignment_result_sum)) * VC) + (pfp_right_append_alignment_result_sum))) /\\ (((((exists ff_h_pfp_append_alignment_result_sumtarget. ff_h_pfp_append_alignment_result_sumtarget + S (pfp_value_append_alignment_result_sum) = S ((S (pfp_index_append_alignment_result_sum)) * rc)) /\\ exists ff_q_pfp_append_alignment_result_sumtarget. rb = ff_q_pfp_append_alignment_result_sumtarget * S ((S (pfp_index_append_alignment_result_sum)) * rc) + (pfp_value_append_alignment_result_sum))) /\\ ((((exists pfa_gap_append_alignment_result_sumoperationleft. pfa_gap_append_alignment_result_sumoperationleft + S (pfp_left_append_alignment_result_sum) = (p)) /\\ (((exists pfa_gap_append_alignment_result_sumoperationright. pfa_gap_append_alignment_result_sumoperationright + S (pfp_right_append_alignment_result_sum) = (p)) /\\ ((((exists pfa_gap_append_alignment_result_sumoperationresultbound. pfa_gap_append_alignment_result_sumoperationresultbound + S (pfp_value_append_alignment_result_sum) = (p)) /\\ ((exists pfa_offset_left_append_alignment_result_sumoperationresultcongruence pfa_offset_right_append_alignment_result_sumoperationresultcongruence. 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hp0 : ~(p=0)", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hz", "have hold : ((forall fom_index_pfp_append_recurrence_oldleft. (exists fom_gap_pfp_append_recurrence_oldleft_index_bound. fom_gap_pfp_append_recurrence_oldleft_index_bound + S (fom_index_pfp_append_recurrence_oldleft) = L) -> exists fom_value_pfp_append_recurrence_oldleft. ((((exists fom_beta_height_pfp_append_recurrence_oldleft_entry. fom_beta_height_pfp_append_recurrence_oldleft_entry + S (fom_value_pfp_append_recurrence_oldleft) = S ((S (fom_index_pfp_append_recurrence_oldleft)) * ac)) /\\ exists fom_beta_quotient_pfp_append_recurrence_oldleft_entry. ab = fom_beta_quotient_pfp_append_recurrence_oldleft_entry * S ((S (fom_index_pfp_append_recurrence_oldleft)) * ac) + (fom_value_pfp_append_recurrence_oldleft))) /\\ (exists fom_gap_pfp_append_recurrence_oldleft_value_bound. fom_gap_pfp_append_recurrence_oldleft_value_bound + S (fom_value_pfp_append_recurrence_oldleft) = p))) /\\ (((forall fom_index_pfp_append_recurrence_oldright. (exists fom_gap_pfp_append_recurrence_oldright_index_bound. fom_gap_pfp_append_recurrence_oldright_index_bound + S (fom_index_pfp_append_recurrence_oldright) = M) -> exists fom_value_pfp_append_recurrence_oldright. ((((exists fom_beta_height_pfp_append_recurrence_oldright_entry. fom_beta_height_pfp_append_recurrence_oldright_entry + S (fom_value_pfp_append_recurrence_oldright) = S ((S (fom_index_pfp_append_recurrence_oldright)) * bc)) /\\ exists fom_beta_quotient_pfp_append_recurrence_oldright_entry. bb = fom_beta_quotient_pfp_append_recurrence_oldright_entry * S ((S (fom_index_pfp_append_recurrence_oldright)) * bc) + (fom_value_pfp_append_recurrence_oldright))) /\\ (exists fom_gap_pfp_append_recurrence_oldright_value_bound. fom_gap_pfp_append_recurrence_oldright_value_bound + S (fom_value_pfp_append_recurrence_oldright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_append_recurrence_oldcoefficients. (exists pfa_gap_append_recurrence_oldcoefficientsbound. pfa_gap_append_recurrence_oldcoefficientsbound + S (pfc_index_append_recurrence_oldcoefficients) = (N)) -> exists pfc_value_append_recurrence_oldcoefficients. ((((exists ff_h_pfp_append_recurrence_oldcoefficientsentry. ff_h_pfp_append_recurrence_oldcoefficientsentry + S (pfc_value_append_recurrence_oldcoefficients) = S ((S (pfc_index_append_recurrence_oldcoefficients)) * pc)) /\\ exists ff_q_pfp_append_recurrence_oldcoefficientsentry. pb = ff_q_pfp_append_recurrence_oldcoefficientsentry * S ((S (pfc_index_append_recurrence_oldcoefficients)) * pc) + (pfc_value_append_recurrence_oldcoefficients))) /\\ ((exists pfc_terms_code_append_recurrence_oldcoefficientscoefficient pfc_terms_scale_append_recurrence_oldcoefficientscoefficient pfc_natural_sum_append_recurrence_oldcoefficientscoefficient. ((forall pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal. (exists pfa_gap_append_recurrence_oldcoefficientscoefficientdiagonalbound. pfa_gap_append_recurrence_oldcoefficientscoefficientdiagonalbound + S (pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal) = (S (pfc_index_append_recurrence_oldcoefficients))) -> exists pfc_value_append_recurrence_oldcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_recurrence_oldcoefficientscoefficientdiagonalentry. ff_h_pfp_append_recurrence_oldcoefficientscoefficientdiagonalentry + S (pfc_value_append_recurrence_oldcoefficientscoefficientdiagonal) = S ((S (pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_recurrence_oldcoefficientscoefficient)) /\\ exists ff_q_pfp_append_recurrence_oldcoefficientscoefficientdiagonalentry. pfc_terms_code_append_recurrence_oldcoefficientscoefficient = ff_q_pfp_append_recurrence_oldcoefficientscoefficientdiagonalentry * S ((S (pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_recurrence_oldcoefficientscoefficient) + (pfc_value_append_recurrence_oldcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_recurrence_oldcoefficientscoefficientdiagonalterm pfc_left_append_recurrence_oldcoefficientscoefficientdiagonalterm pfc_right_append_recurrence_oldcoefficientscoefficientdiagonalterm. (((pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal)+pfc_complement_append_recurrence_oldcoefficientscoefficientdiagonalterm=(pfc_index_append_recurrence_oldcoefficients)) /\\ ((((((exists pfa_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermleftinside. pfa_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_recurrence_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal)) * ac) + (pfc_left_append_recurrence_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal)) /\\ (((pfc_left_append_recurrence_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermrightinside. pfa_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_recurrence_oldcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_recurrence_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_recurrence_oldcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_recurrence_oldcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_append_recurrence_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_append_recurrence_oldcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_recurrence_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_recurrence_oldcoefficientscoefficientdiagonal)=pfc_left_append_recurrence_oldcoefficientscoefficientdiagonalterm*pfc_right_append_recurrence_oldcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_recurrence_oldcoefficientscoefficientsum fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum. ((((exists fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_start. fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_start. fs_u_pfc_append_recurrence_oldcoefficientscoefficientsum = fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_terminal. fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_recurrence_oldcoefficientscoefficient) = S ((S (S (pfc_index_append_recurrence_oldcoefficients))) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_terminal. fs_u_pfc_append_recurrence_oldcoefficientscoefficientsum = fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_recurrence_oldcoefficients))) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum) + (pfc_natural_sum_append_recurrence_oldcoefficientscoefficient))) /\\ forall fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps = S (pfc_index_append_recurrence_oldcoefficients)) -> exists fs_a_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps fs_r_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps fs_s_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_recurrence_oldcoefficientscoefficient)) /\\ exists fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_recurrence_oldcoefficientscoefficient = fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_recurrence_oldcoefficientscoefficient) + (fs_a_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_recurrence_oldcoefficientscoefficientsum = fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum) + (fs_r_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_recurrence_oldcoefficientscoefficientsum = fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum) + (fs_s_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps = fs_r_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps + fs_a_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_recurrence_oldcoefficientscoefficientresiduebound. pfa_gap_append_recurrence_oldcoefficientscoefficientresiduebound + S (pfc_value_append_recurrence_oldcoefficients) = (p)) /\\ ((exists pfa_offset_left_append_recurrence_oldcoefficientscoefficientresiduecongruence pfa_offset_right_append_recurrence_oldcoefficientscoefficientresiduecongruence. (pfc_natural_sum_append_recurrence_oldcoefficientscoefficient) + (p) * pfa_offset_left_append_recurrence_oldcoefficientscoefficientresiduecongruence = (pfc_value_append_recurrence_oldcoefficients) + (p) * pfa_offset_right_append_recurrence_oldcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hP", "cases hold", "cases hold_right", "cases hold_right_right", "have hnew : ((forall fom_index_pfp_append_recurrence_newleft. (exists fom_gap_pfp_append_recurrence_newleft_index_bound. fom_gap_pfp_append_recurrence_newleft_index_bound + S (fom_index_pfp_append_recurrence_newleft) = L) -> exists fom_value_pfp_append_recurrence_newleft. ((((exists fom_beta_height_pfp_append_recurrence_newleft_entry. fom_beta_height_pfp_append_recurrence_newleft_entry + S (fom_value_pfp_append_recurrence_newleft) = S ((S (fom_index_pfp_append_recurrence_newleft)) * ac)) /\\ exists fom_beta_quotient_pfp_append_recurrence_newleft_entry. ab = fom_beta_quotient_pfp_append_recurrence_newleft_entry * S ((S (fom_index_pfp_append_recurrence_newleft)) * ac) + (fom_value_pfp_append_recurrence_newleft))) /\\ (exists fom_gap_pfp_append_recurrence_newleft_value_bound. fom_gap_pfp_append_recurrence_newleft_value_bound + S (fom_value_pfp_append_recurrence_newleft) = p))) /\\ (((forall fom_index_pfp_append_recurrence_newright. (exists fom_gap_pfp_append_recurrence_newright_index_bound. fom_gap_pfp_append_recurrence_newright_index_bound + S (fom_index_pfp_append_recurrence_newright) = S M) -> exists fom_value_pfp_append_recurrence_newright. ((((exists fom_beta_height_pfp_append_recurrence_newright_entry. fom_beta_height_pfp_append_recurrence_newright_entry + S (fom_value_pfp_append_recurrence_newright) = S ((S (fom_index_pfp_append_recurrence_newright)) * dc)) /\\ exists fom_beta_quotient_pfp_append_recurrence_newright_entry. db = fom_beta_quotient_pfp_append_recurrence_newright_entry * S ((S (fom_index_pfp_append_recurrence_newright)) * dc) + (fom_value_pfp_append_recurrence_newright))) /\\ (exists fom_gap_pfp_append_recurrence_newright_value_bound. fom_gap_pfp_append_recurrence_newright_value_bound + S (fom_value_pfp_append_recurrence_newright) = p))) /\\ (((((((L)=0 \\/ (S M)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((S M)=0)) /\\ (((L)+(S M)=S (K)))))))) /\\ ((forall pfc_index_append_recurrence_newcoefficients. (exists pfa_gap_append_recurrence_newcoefficientsbound. pfa_gap_append_recurrence_newcoefficientsbound + S (pfc_index_append_recurrence_newcoefficients) = (K)) -> exists pfc_value_append_recurrence_newcoefficients. ((((exists ff_h_pfp_append_recurrence_newcoefficientsentry. ff_h_pfp_append_recurrence_newcoefficientsentry + S (pfc_value_append_recurrence_newcoefficients) = S ((S (pfc_index_append_recurrence_newcoefficients)) * qc)) /\\ exists ff_q_pfp_append_recurrence_newcoefficientsentry. qb = ff_q_pfp_append_recurrence_newcoefficientsentry * S ((S (pfc_index_append_recurrence_newcoefficients)) * qc) + (pfc_value_append_recurrence_newcoefficients))) /\\ ((exists pfc_terms_code_append_recurrence_newcoefficientscoefficient pfc_terms_scale_append_recurrence_newcoefficientscoefficient pfc_natural_sum_append_recurrence_newcoefficientscoefficient. ((forall pfc_index_append_recurrence_newcoefficientscoefficientdiagonal. (exists pfa_gap_append_recurrence_newcoefficientscoefficientdiagonalbound. pfa_gap_append_recurrence_newcoefficientscoefficientdiagonalbound + S (pfc_index_append_recurrence_newcoefficientscoefficientdiagonal) = (S (pfc_index_append_recurrence_newcoefficients))) -> exists pfc_value_append_recurrence_newcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_recurrence_newcoefficientscoefficientdiagonalentry. ff_h_pfp_append_recurrence_newcoefficientscoefficientdiagonalentry + S (pfc_value_append_recurrence_newcoefficientscoefficientdiagonal) = S ((S (pfc_index_append_recurrence_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_recurrence_newcoefficientscoefficient)) /\\ exists ff_q_pfp_append_recurrence_newcoefficientscoefficientdiagonalentry. pfc_terms_code_append_recurrence_newcoefficientscoefficient = ff_q_pfp_append_recurrence_newcoefficientscoefficientdiagonalentry * S ((S (pfc_index_append_recurrence_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_recurrence_newcoefficientscoefficient) + (pfc_value_append_recurrence_newcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_recurrence_newcoefficientscoefficientdiagonalterm pfc_left_append_recurrence_newcoefficientscoefficientdiagonalterm pfc_right_append_recurrence_newcoefficientscoefficientdiagonalterm. (((pfc_index_append_recurrence_newcoefficientscoefficientdiagonal)+pfc_complement_append_recurrence_newcoefficientscoefficientdiagonalterm=(pfc_index_append_recurrence_newcoefficients)) /\\ ((((((exists pfa_gap_append_recurrence_newcoefficientscoefficientdiagonaltermleftinside. pfa_gap_append_recurrence_newcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_recurrence_newcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_recurrence_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_recurrence_newcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_recurrence_newcoefficientscoefficientdiagonal)) * ac) + (pfc_left_append_recurrence_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_recurrence_newcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_recurrence_newcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_recurrence_newcoefficientscoefficientdiagonal)) /\\ (((pfc_left_append_recurrence_newcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_recurrence_newcoefficientscoefficientdiagonaltermrightinside. pfa_gap_append_recurrence_newcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_recurrence_newcoefficientscoefficientdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_recurrence_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_recurrence_newcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_recurrence_newcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_append_recurrence_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_recurrence_newcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_recurrence_newcoefficientscoefficientdiagonaltermrightoutside+(S M)=(pfc_complement_append_recurrence_newcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_recurrence_newcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_recurrence_newcoefficientscoefficientdiagonal)=pfc_left_append_recurrence_newcoefficientscoefficientdiagonalterm*pfc_right_append_recurrence_newcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_recurrence_newcoefficientscoefficientsum fs_v_pfc_append_recurrence_newcoefficientscoefficientsum. ((((exists fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_start. fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_start. fs_u_pfc_append_recurrence_newcoefficientscoefficientsum = fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_terminal. fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_recurrence_newcoefficientscoefficient) = S ((S (S (pfc_index_append_recurrence_newcoefficients))) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_terminal. fs_u_pfc_append_recurrence_newcoefficientscoefficientsum = fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_recurrence_newcoefficients))) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum) + (pfc_natural_sum_append_recurrence_newcoefficientscoefficient))) /\\ forall fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps = S (pfc_index_append_recurrence_newcoefficients)) -> exists fs_a_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps fs_r_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps fs_s_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_recurrence_newcoefficientscoefficient)) /\\ exists fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_recurrence_newcoefficientscoefficient = fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_recurrence_newcoefficientscoefficient) + (fs_a_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_recurrence_newcoefficientscoefficientsum = fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum) + (fs_r_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_recurrence_newcoefficientscoefficientsum = fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum) + (fs_s_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps = fs_r_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps + fs_a_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_recurrence_newcoefficientscoefficientresiduebound. pfa_gap_append_recurrence_newcoefficientscoefficientresiduebound + S (pfc_value_append_recurrence_newcoefficients) = (p)) /\\ ((exists pfa_offset_left_append_recurrence_newcoefficientscoefficientresiduecongruence pfa_offset_right_append_recurrence_newcoefficientscoefficientresiduecongruence. (pfc_natural_sum_append_recurrence_newcoefficientscoefficient) + (p) * pfa_offset_left_append_recurrence_newcoefficientscoefficientresiduecongruence = (pfc_value_append_recurrence_newcoefficients) + (p) * pfa_offset_right_append_recurrence_newcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hQ", "cases hnew", "cases hnew_right", "cases hnew_right_right", "have hv_copy : ((exists pfa_gap_append_recurrence_scalar_copyscalar. pfa_gap_append_recurrence_scalar_copyscalar + S (c) = (p)) /\\ ((forall pfp_index_append_recurrence_scalar_copy. (exists pfa_gap_append_recurrence_scalar_copyindex. pfa_gap_append_recurrence_scalar_copyindex + S (pfp_index_append_recurrence_scalar_copy) = (L)) -> exists pfp_source_append_recurrence_scalar_copy pfp_value_append_recurrence_scalar_copy. ((((exists ff_h_pfp_append_recurrence_scalar_copysource. ff_h_pfp_append_recurrence_scalar_copysource + S (pfp_source_append_recurrence_scalar_copy) = S ((S (pfp_index_append_recurrence_scalar_copy)) * ac)) /\\ exists ff_q_pfp_append_recurrence_scalar_copysource. ab = ff_q_pfp_append_recurrence_scalar_copysource * S ((S (pfp_index_append_recurrence_scalar_copy)) * ac) + (pfp_source_append_recurrence_scalar_copy))) /\\ (((((exists ff_h_pfp_append_recurrence_scalar_copytarget. ff_h_pfp_append_recurrence_scalar_copytarget + S (pfp_value_append_recurrence_scalar_copy) = S ((S (pfp_index_append_recurrence_scalar_copy)) * vc)) /\\ exists ff_q_pfp_append_recurrence_scalar_copytarget. vb = ff_q_pfp_append_recurrence_scalar_copytarget * S ((S (pfp_index_append_recurrence_scalar_copy)) * vc) + (pfp_value_append_recurrence_scalar_copy))) /\\ ((((exists pfa_gap_append_recurrence_scalar_copyoperationleft. pfa_gap_append_recurrence_scalar_copyoperationleft + S (c) = (p)) /\\ (((exists pfa_gap_append_recurrence_scalar_copyoperationright. pfa_gap_append_recurrence_scalar_copyoperationright + S (pfp_source_append_recurrence_scalar_copy) = (p)) /\\ ((((exists pfa_gap_append_recurrence_scalar_copyoperationresultbound. pfa_gap_append_recurrence_scalar_copyoperationresultbound + S (pfp_value_append_recurrence_scalar_copy) = (p)) /\\ ((exists pfa_offset_left_append_recurrence_scalar_copyoperationresultcongruence pfa_offset_right_append_recurrence_scalar_copyoperationresultcongruence. ((c) * (pfp_source_append_recurrence_scalar_copy)) + (p) * pfa_offset_left_append_recurrence_scalar_copyoperationresultcongruence = (pfp_value_append_recurrence_scalar_copy) + (p) * pfa_offset_right_append_recurrence_scalar_copyoperationresultcongruence))))))))))))))))", "exact hV", "cases hv_copy", "have hdecomp : exists sb sc kb kc tb tc. ((((forall mdr_i_pfp_append_recurrence_decomposition_shiftprefix mdr_a_pfp_append_recurrence_decomposition_shiftprefix. (exists mdr_gap_pfp_append_recurrence_decomposition_shiftprefixb. mdr_gap_pfp_append_recurrence_decomposition_shiftprefixb + S (mdr_i_pfp_append_recurrence_decomposition_shiftprefix) = (M)) -> (((exists ff_h_mdr_pfp_append_recurrence_decomposition_shiftprefixo. ff_h_mdr_pfp_append_recurrence_decomposition_shiftprefixo + S (mdr_a_pfp_append_recurrence_decomposition_shiftprefix) = S ((S (mdr_i_pfp_append_recurrence_decomposition_shiftprefix)) * bc)) /\\ exists ff_q_mdr_pfp_append_recurrence_decomposition_shiftprefixo. bb = ff_q_mdr_pfp_append_recurrence_decomposition_shiftprefixo * S ((S (mdr_i_pfp_append_recurrence_decomposition_shiftprefix)) * bc) + (mdr_a_pfp_append_recurrence_decomposition_shiftprefix))) -> (((exists ff_h_mdr_pfp_append_recurrence_decomposition_shiftprefixn. ff_h_mdr_pfp_append_recurrence_decomposition_shiftprefixn + S (mdr_a_pfp_append_recurrence_decomposition_shiftprefix) = S ((S (mdr_i_pfp_append_recurrence_decomposition_shiftprefix)) * sc)) /\\ exists ff_q_mdr_pfp_append_recurrence_decomposition_shiftprefixn. sb = ff_q_mdr_pfp_append_recurrence_decomposition_shiftprefixn * S ((S (mdr_i_pfp_append_recurrence_decomposition_shiftprefix)) * sc) + (mdr_a_pfp_append_recurrence_decomposition_shiftprefix)))) /\\ ((((exists ff_h_pfp_append_recurrence_decomposition_shiftlast. ff_h_pfp_append_recurrence_decomposition_shiftlast + S (0) = S ((S (M)) * sc)) /\\ exists ff_q_pfp_append_recurrence_decomposition_shiftlast. sb = ff_q_pfp_append_recurrence_decomposition_shiftlast * S ((S (M)) * sc) + (0)))))) /\\ (((forall fom_index_pfp_append_recurrence_decomposition_bound. (exists fom_gap_pfp_append_recurrence_decomposition_bound_index_bound. fom_gap_pfp_append_recurrence_decomposition_bound_index_bound + S (fom_index_pfp_append_recurrence_decomposition_bound) = 1) -> exists fom_value_pfp_append_recurrence_decomposition_bound. ((((exists fom_beta_height_pfp_append_recurrence_decomposition_bound_entry. fom_beta_height_pfp_append_recurrence_decomposition_bound_entry + S (fom_value_pfp_append_recurrence_decomposition_bound) = S ((S (fom_index_pfp_append_recurrence_decomposition_bound)) * kc)) /\\ exists fom_beta_quotient_pfp_append_recurrence_decomposition_bound_entry. kb = fom_beta_quotient_pfp_append_recurrence_decomposition_bound_entry * S ((S (fom_index_pfp_append_recurrence_decomposition_bound)) * kc) + (fom_value_pfp_append_recurrence_decomposition_bound))) /\\ (exists fom_gap_pfp_append_recurrence_decomposition_bound_value_bound. fom_gap_pfp_append_recurrence_decomposition_bound_value_bound + S (fom_value_pfp_append_recurrence_decomposition_bound) = p))) /\\ (((((exists ff_h_pfp_append_recurrence_decomposition_constant. ff_h_pfp_append_recurrence_decomposition_constant + S (c) = S ((S (0)) * kc)) /\\ exists ff_q_pfp_append_recurrence_decomposition_constant. kb = ff_q_pfp_append_recurrence_decomposition_constant * S ((S (0)) * kc) + (c))) /\\ (((((forall pfp_repeat_index_append_recurrence_decomposition_padzeros. (exists pfa_gap_append_recurrence_decomposition_padzerosindex. pfa_gap_append_recurrence_decomposition_padzerosindex + S (pfp_repeat_index_append_recurrence_decomposition_padzeros) = (M)) -> (((exists ff_h_pfp_append_recurrence_decomposition_padzerosentry. ff_h_pfp_append_recurrence_decomposition_padzerosentry + S (0) = S ((S (pfp_repeat_index_append_recurrence_decomposition_padzeros)) * tc)) /\\ exists ff_q_pfp_append_recurrence_decomposition_padzerosentry. tb = ff_q_pfp_append_recurrence_decomposition_padzerosentry * S ((S (pfp_repeat_index_append_recurrence_decomposition_padzeros)) * tc) + (0)))) /\\ ((forall pfrep_index_append_recurrence_decomposition_pad pfrep_value_append_recurrence_decomposition_pad. (exists pfa_gap_append_recurrence_decomposition_padbound. pfa_gap_append_recurrence_decomposition_padbound + S (pfrep_index_append_recurrence_decomposition_pad) = (1)) -> (((exists ff_h_pfp_append_recurrence_decomposition_padinput. ff_h_pfp_append_recurrence_decomposition_padinput + S (pfrep_value_append_recurrence_decomposition_pad) = S ((S (pfrep_index_append_recurrence_decomposition_pad)) * kc)) /\\ exists ff_q_pfp_append_recurrence_decomposition_padinput. kb = ff_q_pfp_append_recurrence_decomposition_padinput * S ((S (pfrep_index_append_recurrence_decomposition_pad)) * kc) + (pfrep_value_append_recurrence_decomposition_pad))) -> (((exists ff_h_pfp_append_recurrence_decomposition_padoutput. ff_h_pfp_append_recurrence_decomposition_padoutput + S (pfrep_value_append_recurrence_decomposition_pad) = S ((S ((M)+pfrep_index_append_recurrence_decomposition_pad)) * tc)) /\\ exists ff_q_pfp_append_recurrence_decomposition_padoutput. tb = ff_q_pfp_append_recurrence_decomposition_padoutput * S ((S ((M)+pfrep_index_append_recurrence_decomposition_pad)) * tc) + (pfrep_value_append_recurrence_decomposition_pad))))))) /\\ ((forall pfp_index_append_recurrence_decomposition_sum. (exists pfa_gap_append_recurrence_decomposition_sumindex. pfa_gap_append_recurrence_decomposition_sumindex + S (pfp_index_append_recurrence_decomposition_sum) = (S M)) -> exists pfp_left_append_recurrence_decomposition_sum pfp_right_append_recurrence_decomposition_sum pfp_value_append_recurrence_decomposition_sum. ((((exists ff_h_pfp_append_recurrence_decomposition_sumleft. ff_h_pfp_append_recurrence_decomposition_sumleft + S (pfp_left_append_recurrence_decomposition_sum) = S ((S (pfp_index_append_recurrence_decomposition_sum)) * sc)) /\\ exists ff_q_pfp_append_recurrence_decomposition_sumleft. sb = ff_q_pfp_append_recurrence_decomposition_sumleft * S ((S (pfp_index_append_recurrence_decomposition_sum)) * sc) + (pfp_left_append_recurrence_decomposition_sum))) /\\ (((((exists ff_h_pfp_append_recurrence_decomposition_sumright. ff_h_pfp_append_recurrence_decomposition_sumright + S (pfp_right_append_recurrence_decomposition_sum) = S ((S (pfp_index_append_recurrence_decomposition_sum)) * tc)) /\\ exists ff_q_pfp_append_recurrence_decomposition_sumright. tb = ff_q_pfp_append_recurrence_decomposition_sumright * S ((S (pfp_index_append_recurrence_decomposition_sum)) * tc) + (pfp_right_append_recurrence_decomposition_sum))) /\\ (((((exists ff_h_pfp_append_recurrence_decomposition_sumtarget. ff_h_pfp_append_recurrence_decomposition_sumtarget + S (pfp_value_append_recurrence_decomposition_sum) = S ((S (pfp_index_append_recurrence_decomposition_sum)) * dc)) /\\ exists ff_q_pfp_append_recurrence_decomposition_sumtarget. db = ff_q_pfp_append_recurrence_decomposition_sumtarget * S ((S (pfp_index_append_recurrence_decomposition_sum)) * dc) + (pfp_value_append_recurrence_decomposition_sum))) /\\ ((((exists pfa_gap_append_recurrence_decomposition_sumoperationleft. pfa_gap_append_recurrence_decomposition_sumoperationleft + S (pfp_left_append_recurrence_decomposition_sum) = (p)) /\\ (((exists pfa_gap_append_recurrence_decomposition_sumoperationright. pfa_gap_append_recurrence_decomposition_sumoperationright + S (pfp_right_append_recurrence_decomposition_sum) = (p)) /\\ ((((exists pfa_gap_append_recurrence_decomposition_sumoperationresultbound. pfa_gap_append_recurrence_decomposition_sumoperationresultbound + S (pfp_value_append_recurrence_decomposition_sum) = (p)) /\\ ((exists pfa_offset_left_append_recurrence_decomposition_sumoperationresultcongruence pfa_offset_right_append_recurrence_decomposition_sumoperationresultcongruence. ((pfp_left_append_recurrence_decomposition_sum) + (pfp_right_append_recurrence_decomposition_sum)) + (p) * pfa_offset_left_append_recurrence_decomposition_sumoperationresultcongruence = (pfp_value_append_recurrence_decomposition_sum) + (p) * pfa_offset_right_append_recurrence_decomposition_sumoperationresultcongruence))))))))))))))))))))))))", "specialize prime_field_polynomial_append_shift_constant_decomposition_exists (p)", "specialize prime_field_polynomial_append_shift_constant_decomposition_exists (bb)", "specialize prime_field_polynomial_append_shift_constant_decomposition_exists (bc)", "specialize prime_field_polynomial_append_shift_constant_decomposition_exists (M)", "specialize prime_field_polynomial_append_shift_constant_decomposition_exists (c)", "specialize prime_field_polynomial_append_shift_constant_decomposition_exists (db)", "specialize prime_field_polynomial_append_shift_constant_decomposition_exists (dc)", "apply prime_field_polynomial_append_shift_constant_decomposition_exists", "exact hp", "exact hold_right_left", "exact hv_copy_left", "exact he", "exact hlast", "cases hdecomp", "cases hdecomp_witness", "cases hdecomp_witness_witness", "cases hdecomp_witness_witness_witness", "cases hdecomp_witness_witness_witness_witness", "cases hdecomp_witness_witness_witness_witness_witness", "cases hdecomp_witness_witness_witness_witness_witness_witness", "cases hdecomp_witness_witness_witness_witness_witness_witness_right", "cases hdecomp_witness_witness_witness_witness_witness_witness_right_right", "cases hdecomp_witness_witness_witness_witness_witness_witness_right_right_right", "have hbounds : ((forall fom_index_pfp_append_recurrence_shift_bound. (exists fom_gap_pfp_append_recurrence_shift_bound_index_bound. fom_gap_pfp_append_recurrence_shift_bound_index_bound + S (fom_index_pfp_append_recurrence_shift_bound) = S M) -> exists fom_value_pfp_append_recurrence_shift_bound. ((((exists fom_beta_height_pfp_append_recurrence_shift_bound_entry. fom_beta_height_pfp_append_recurrence_shift_bound_entry + S (fom_value_pfp_append_recurrence_shift_bound) = S ((S (fom_index_pfp_append_recurrence_shift_bound)) * x1)) /\\ exists fom_beta_quotient_pfp_append_recurrence_shift_bound_entry. x = fom_beta_quotient_pfp_append_recurrence_shift_bound_entry * S ((S (fom_index_pfp_append_recurrence_shift_bound)) * x1) + (fom_value_pfp_append_recurrence_shift_bound))) /\\ (exists fom_gap_pfp_append_recurrence_shift_bound_value_bound. fom_gap_pfp_append_recurrence_shift_bound_value_bound + S (fom_value_pfp_append_recurrence_shift_bound) = p))) /\\ (((forall fom_index_pfp_append_recurrence_padded_constant_bound. (exists fom_gap_pfp_append_recurrence_padded_constant_bound_index_bound. fom_gap_pfp_append_recurrence_padded_constant_bound_index_bound + S (fom_index_pfp_append_recurrence_padded_constant_bound) = S M) -> exists fom_value_pfp_append_recurrence_padded_constant_bound. ((((exists fom_beta_height_pfp_append_recurrence_padded_constant_bound_entry. fom_beta_height_pfp_append_recurrence_padded_constant_bound_entry + S (fom_value_pfp_append_recurrence_padded_constant_bound) = S ((S (fom_index_pfp_append_recurrence_padded_constant_bound)) * x5)) /\\ exists fom_beta_quotient_pfp_append_recurrence_padded_constant_bound_entry. x4 = fom_beta_quotient_pfp_append_recurrence_padded_constant_bound_entry * S ((S (fom_index_pfp_append_recurrence_padded_constant_bound)) * x5) + (fom_value_pfp_append_recurrence_padded_constant_bound))) /\\ (exists fom_gap_pfp_append_recurrence_padded_constant_bound_value_bound. fom_gap_pfp_append_recurrence_padded_constant_bound_value_bound + S (fom_value_pfp_append_recurrence_padded_constant_bound) = p))) /\\ ((forall fom_index_pfp_append_recurrence_append_bound. (exists fom_gap_pfp_append_recurrence_append_bound_index_bound. fom_gap_pfp_append_recurrence_append_bound_index_bound + S (fom_index_pfp_append_recurrence_append_bound) = S M) -> exists fom_value_pfp_append_recurrence_append_bound. ((((exists fom_beta_height_pfp_append_recurrence_append_bound_entry. fom_beta_height_pfp_append_recurrence_append_bound_entry + S (fom_value_pfp_append_recurrence_append_bound) = S ((S (fom_index_pfp_append_recurrence_append_bound)) * dc)) /\\ exists fom_beta_quotient_pfp_append_recurrence_append_bound_entry. db = fom_beta_quotient_pfp_append_recurrence_append_bound_entry * S ((S (fom_index_pfp_append_recurrence_append_bound)) * dc) + (fom_value_pfp_append_recurrence_append_bound))) /\\ (exists fom_gap_pfp_append_recurrence_append_bound_value_bound. fom_gap_pfp_append_recurrence_append_bound_value_bound + S (fom_value_pfp_append_recurrence_append_bound) = p)))))))", "specialize prime_field_polynomial_add_bounded (p)", "specialize prime_field_polynomial_add_bounded (x)", "specialize prime_field_polynomial_add_bounded (x1)", "specialize prime_field_polynomial_add_bounded (x4)", "specialize prime_field_polynomial_add_bounded (x5)", "specialize prime_field_polynomial_add_bounded (db)", "specialize prime_field_polynomial_add_bounded (dc)", "specialize prime_field_polynomial_add_bounded (S M)", "apply prime_field_polynomial_add_bounded", "exact hdecomp_witness_witness_witness_witness_witness_witness_right_right_right_right", "cases hbounds", "cases hbounds_right", "have hfirst : exists fb fc. ((forall fom_index_pfp_append_recurrence_hfirstleft. (exists fom_gap_pfp_append_recurrence_hfirstleft_index_bound. fom_gap_pfp_append_recurrence_hfirstleft_index_bound + S (fom_index_pfp_append_recurrence_hfirstleft) = L) -> exists fom_value_pfp_append_recurrence_hfirstleft. ((((exists fom_beta_height_pfp_append_recurrence_hfirstleft_entry. fom_beta_height_pfp_append_recurrence_hfirstleft_entry + S (fom_value_pfp_append_recurrence_hfirstleft) = S ((S (fom_index_pfp_append_recurrence_hfirstleft)) * ac)) /\\ exists fom_beta_quotient_pfp_append_recurrence_hfirstleft_entry. ab = fom_beta_quotient_pfp_append_recurrence_hfirstleft_entry * S ((S (fom_index_pfp_append_recurrence_hfirstleft)) * ac) + (fom_value_pfp_append_recurrence_hfirstleft))) /\\ (exists fom_gap_pfp_append_recurrence_hfirstleft_value_bound. fom_gap_pfp_append_recurrence_hfirstleft_value_bound + S (fom_value_pfp_append_recurrence_hfirstleft) = p))) /\\ (((forall fom_index_pfp_append_recurrence_hfirstright. (exists fom_gap_pfp_append_recurrence_hfirstright_index_bound. fom_gap_pfp_append_recurrence_hfirstright_index_bound + S (fom_index_pfp_append_recurrence_hfirstright) = S M) -> exists fom_value_pfp_append_recurrence_hfirstright. ((((exists fom_beta_height_pfp_append_recurrence_hfirstright_entry. fom_beta_height_pfp_append_recurrence_hfirstright_entry + S (fom_value_pfp_append_recurrence_hfirstright) = S ((S (fom_index_pfp_append_recurrence_hfirstright)) * x1)) /\\ exists fom_beta_quotient_pfp_append_recurrence_hfirstright_entry. x = fom_beta_quotient_pfp_append_recurrence_hfirstright_entry * S ((S (fom_index_pfp_append_recurrence_hfirstright)) * x1) + (fom_value_pfp_append_recurrence_hfirstright))) /\\ (exists fom_gap_pfp_append_recurrence_hfirstright_value_bound. fom_gap_pfp_append_recurrence_hfirstright_value_bound + S (fom_value_pfp_append_recurrence_hfirstright) = p))) /\\ (((((((L)=0 \\/ (S M)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((S M)=0)) /\\ (((L)+(S M)=S (K)))))))) /\\ ((forall pfc_index_append_recurrence_hfirstcoefficients. (exists pfa_gap_append_recurrence_hfirstcoefficientsbound. pfa_gap_append_recurrence_hfirstcoefficientsbound + S (pfc_index_append_recurrence_hfirstcoefficients) = (K)) -> exists pfc_value_append_recurrence_hfirstcoefficients. ((((exists ff_h_pfp_append_recurrence_hfirstcoefficientsentry. ff_h_pfp_append_recurrence_hfirstcoefficientsentry + S (pfc_value_append_recurrence_hfirstcoefficients) = S ((S (pfc_index_append_recurrence_hfirstcoefficients)) * fc)) /\\ exists ff_q_pfp_append_recurrence_hfirstcoefficientsentry. fb = ff_q_pfp_append_recurrence_hfirstcoefficientsentry * S ((S (pfc_index_append_recurrence_hfirstcoefficients)) * fc) + (pfc_value_append_recurrence_hfirstcoefficients))) /\\ ((exists pfc_terms_code_append_recurrence_hfirstcoefficientscoefficient pfc_terms_scale_append_recurrence_hfirstcoefficientscoefficient pfc_natural_sum_append_recurrence_hfirstcoefficientscoefficient. ((forall pfc_index_append_recurrence_hfirstcoefficientscoefficientdiagonal. (exists pfa_gap_append_recurrence_hfirstcoefficientscoefficientdiagonalbound. pfa_gap_append_recurrence_hfirstcoefficientscoefficientdiagonalbound + S (pfc_index_append_recurrence_hfirstcoefficientscoefficientdiagonal) = (S (pfc_index_append_recurrence_hfirstcoefficients))) -> exists pfc_value_append_recurrence_hfirstcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_recurrence_hfirstcoefficientscoefficientdiagonalentry. ff_h_pfp_append_recurrence_hfirstcoefficientscoefficientdiagonalentry + S (pfc_value_append_recurrence_hfirstcoefficientscoefficientdiagonal) = S ((S (pfc_index_append_recurrence_hfirstcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_recurrence_hfirstcoefficientscoefficient)) /\\ exists ff_q_pfp_append_recurrence_hfirstcoefficientscoefficientdiagonalentry. pfc_terms_code_append_recurrence_hfirstcoefficientscoefficient = ff_q_pfp_append_recurrence_hfirstcoefficientscoefficientdiagonalentry * S ((S (pfc_index_append_recurrence_hfirstcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_recurrence_hfirstcoefficientscoefficient) + (pfc_value_append_recurrence_hfirstcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_recurrence_hfirstcoefficientscoefficientdiagonalterm pfc_left_append_recurrence_hfirstcoefficientscoefficientdiagonalterm pfc_right_append_recurrence_hfirstcoefficientscoefficientdiagonalterm. (((pfc_index_append_recurrence_hfirstcoefficientscoefficientdiagonal)+pfc_complement_append_recurrence_hfirstcoefficientscoefficientdiagonalterm=(pfc_index_append_recurrence_hfirstcoefficients)) /\\ ((((((exists pfa_gap_append_recurrence_hfirstcoefficientscoefficientdiagonaltermleftinside. pfa_gap_append_recurrence_hfirstcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_recurrence_hfirstcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_recurrence_hfirstcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_recurrence_hfirstcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_recurrence_hfirstcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_recurrence_hfirstcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_recurrence_hfirstcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_recurrence_hfirstcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_recurrence_hfirstcoefficientscoefficientdiagonal)) * ac) + (pfc_left_append_recurrence_hfirstcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_recurrence_hfirstcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_recurrence_hfirstcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_recurrence_hfirstcoefficientscoefficientdiagonal)) /\\ (((pfc_left_append_recurrence_hfirstcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_recurrence_hfirstcoefficientscoefficientdiagonaltermrightinside. pfa_gap_append_recurrence_hfirstcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_recurrence_hfirstcoefficientscoefficientdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_append_recurrence_hfirstcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_recurrence_hfirstcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_recurrence_hfirstcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_recurrence_hfirstcoefficientscoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_append_recurrence_hfirstcoefficientscoefficientdiagonaltermrightentry. x = ff_q_pfp_append_recurrence_hfirstcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_recurrence_hfirstcoefficientscoefficientdiagonalterm)) * x1) + (pfc_right_append_recurrence_hfirstcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_recurrence_hfirstcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_recurrence_hfirstcoefficientscoefficientdiagonaltermrightoutside+(S M)=(pfc_complement_append_recurrence_hfirstcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_recurrence_hfirstcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_recurrence_hfirstcoefficientscoefficientdiagonal)=pfc_left_append_recurrence_hfirstcoefficientscoefficientdiagonalterm*pfc_right_append_recurrence_hfirstcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_recurrence_hfirstcoefficientscoefficientsum fs_v_pfc_append_recurrence_hfirstcoefficientscoefficientsum. ((((exists fs_h_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_start. fs_h_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_recurrence_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_start. fs_u_pfc_append_recurrence_hfirstcoefficientscoefficientsum = fs_q_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_recurrence_hfirstcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_terminal. fs_h_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_recurrence_hfirstcoefficientscoefficient) = S ((S (S (pfc_index_append_recurrence_hfirstcoefficients))) * fs_v_pfc_append_recurrence_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_terminal. fs_u_pfc_append_recurrence_hfirstcoefficientscoefficientsum = fs_q_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_recurrence_hfirstcoefficients))) * fs_v_pfc_append_recurrence_hfirstcoefficientscoefficientsum) + (pfc_natural_sum_append_recurrence_hfirstcoefficientscoefficient))) /\\ forall fs_i_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps = S (pfc_index_append_recurrence_hfirstcoefficients)) -> exists fs_a_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps fs_r_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps fs_s_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_recurrence_hfirstcoefficientscoefficient)) /\\ exists fs_q_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_recurrence_hfirstcoefficientscoefficient = fs_q_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_recurrence_hfirstcoefficientscoefficient) + (fs_a_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_recurrence_hfirstcoefficientscoefficientsum = fs_q_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_hfirstcoefficientscoefficientsum) + (fs_r_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_recurrence_hfirstcoefficientscoefficientsum = fs_q_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_hfirstcoefficientscoefficientsum) + (fs_s_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps = fs_r_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps + fs_a_pfc_append_recurrence_hfirstcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_recurrence_hfirstcoefficientscoefficientresiduebound. pfa_gap_append_recurrence_hfirstcoefficientscoefficientresiduebound + S (pfc_value_append_recurrence_hfirstcoefficients) = (p)) /\\ ((exists pfa_offset_left_append_recurrence_hfirstcoefficientscoefficientresiduecongruence pfa_offset_right_append_recurrence_hfirstcoefficientscoefficientresiduecongruence. (pfc_natural_sum_append_recurrence_hfirstcoefficientscoefficient) + (p) * pfa_offset_left_append_recurrence_hfirstcoefficientscoefficientresiduecongruence = (pfc_value_append_recurrence_hfirstcoefficients) + (p) * pfa_offset_right_append_recurrence_hfirstcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (ab)", "specialize prime_field_polynomial_convolution_at_length_exists (ac)", "specialize prime_field_polynomial_convolution_at_length_exists (L)", "specialize prime_field_polynomial_convolution_at_length_exists (x)", "specialize prime_field_polynomial_convolution_at_length_exists (x1)", "specialize prime_field_polynomial_convolution_at_length_exists (S M)", "specialize prime_field_polynomial_convolution_at_length_exists (K)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hold_left", "exact hbounds_left", "exact hnew_right_right_left", "cases hfirst", "cases hfirst_witness", "have hsecond : exists fb fc. ((forall fom_index_pfp_append_recurrence_hsecondleft. (exists fom_gap_pfp_append_recurrence_hsecondleft_index_bound. fom_gap_pfp_append_recurrence_hsecondleft_index_bound + S (fom_index_pfp_append_recurrence_hsecondleft) = L) -> exists fom_value_pfp_append_recurrence_hsecondleft. ((((exists fom_beta_height_pfp_append_recurrence_hsecondleft_entry. fom_beta_height_pfp_append_recurrence_hsecondleft_entry + S (fom_value_pfp_append_recurrence_hsecondleft) = S ((S (fom_index_pfp_append_recurrence_hsecondleft)) * ac)) /\\ exists fom_beta_quotient_pfp_append_recurrence_hsecondleft_entry. ab = fom_beta_quotient_pfp_append_recurrence_hsecondleft_entry * S ((S (fom_index_pfp_append_recurrence_hsecondleft)) * ac) + (fom_value_pfp_append_recurrence_hsecondleft))) /\\ (exists fom_gap_pfp_append_recurrence_hsecondleft_value_bound. fom_gap_pfp_append_recurrence_hsecondleft_value_bound + S (fom_value_pfp_append_recurrence_hsecondleft) = p))) /\\ (((forall fom_index_pfp_append_recurrence_hsecondright. (exists fom_gap_pfp_append_recurrence_hsecondright_index_bound. fom_gap_pfp_append_recurrence_hsecondright_index_bound + S (fom_index_pfp_append_recurrence_hsecondright) = S M) -> exists fom_value_pfp_append_recurrence_hsecondright. ((((exists fom_beta_height_pfp_append_recurrence_hsecondright_entry. fom_beta_height_pfp_append_recurrence_hsecondright_entry + S (fom_value_pfp_append_recurrence_hsecondright) = S ((S (fom_index_pfp_append_recurrence_hsecondright)) * x5)) /\\ exists fom_beta_quotient_pfp_append_recurrence_hsecondright_entry. x4 = fom_beta_quotient_pfp_append_recurrence_hsecondright_entry * S ((S (fom_index_pfp_append_recurrence_hsecondright)) * x5) + (fom_value_pfp_append_recurrence_hsecondright))) /\\ (exists fom_gap_pfp_append_recurrence_hsecondright_value_bound. fom_gap_pfp_append_recurrence_hsecondright_value_bound + S (fom_value_pfp_append_recurrence_hsecondright) = p))) /\\ (((((((L)=0 \\/ (S M)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((S M)=0)) /\\ (((L)+(S M)=S (K)))))))) /\\ ((forall pfc_index_append_recurrence_hsecondcoefficients. (exists pfa_gap_append_recurrence_hsecondcoefficientsbound. pfa_gap_append_recurrence_hsecondcoefficientsbound + S (pfc_index_append_recurrence_hsecondcoefficients) = (K)) -> exists pfc_value_append_recurrence_hsecondcoefficients. ((((exists ff_h_pfp_append_recurrence_hsecondcoefficientsentry. ff_h_pfp_append_recurrence_hsecondcoefficientsentry + S (pfc_value_append_recurrence_hsecondcoefficients) = S ((S (pfc_index_append_recurrence_hsecondcoefficients)) * fc)) /\\ exists ff_q_pfp_append_recurrence_hsecondcoefficientsentry. fb = ff_q_pfp_append_recurrence_hsecondcoefficientsentry * S ((S (pfc_index_append_recurrence_hsecondcoefficients)) * fc) + (pfc_value_append_recurrence_hsecondcoefficients))) /\\ ((exists pfc_terms_code_append_recurrence_hsecondcoefficientscoefficient pfc_terms_scale_append_recurrence_hsecondcoefficientscoefficient pfc_natural_sum_append_recurrence_hsecondcoefficientscoefficient. ((forall pfc_index_append_recurrence_hsecondcoefficientscoefficientdiagonal. (exists pfa_gap_append_recurrence_hsecondcoefficientscoefficientdiagonalbound. pfa_gap_append_recurrence_hsecondcoefficientscoefficientdiagonalbound + S (pfc_index_append_recurrence_hsecondcoefficientscoefficientdiagonal) = (S (pfc_index_append_recurrence_hsecondcoefficients))) -> exists pfc_value_append_recurrence_hsecondcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_recurrence_hsecondcoefficientscoefficientdiagonalentry. ff_h_pfp_append_recurrence_hsecondcoefficientscoefficientdiagonalentry + S (pfc_value_append_recurrence_hsecondcoefficientscoefficientdiagonal) = S ((S (pfc_index_append_recurrence_hsecondcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_recurrence_hsecondcoefficientscoefficient)) /\\ exists ff_q_pfp_append_recurrence_hsecondcoefficientscoefficientdiagonalentry. pfc_terms_code_append_recurrence_hsecondcoefficientscoefficient = ff_q_pfp_append_recurrence_hsecondcoefficientscoefficientdiagonalentry * S ((S (pfc_index_append_recurrence_hsecondcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_recurrence_hsecondcoefficientscoefficient) + (pfc_value_append_recurrence_hsecondcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_recurrence_hsecondcoefficientscoefficientdiagonalterm pfc_left_append_recurrence_hsecondcoefficientscoefficientdiagonalterm pfc_right_append_recurrence_hsecondcoefficientscoefficientdiagonalterm. (((pfc_index_append_recurrence_hsecondcoefficientscoefficientdiagonal)+pfc_complement_append_recurrence_hsecondcoefficientscoefficientdiagonalterm=(pfc_index_append_recurrence_hsecondcoefficients)) /\\ ((((((exists pfa_gap_append_recurrence_hsecondcoefficientscoefficientdiagonaltermleftinside. pfa_gap_append_recurrence_hsecondcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_recurrence_hsecondcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_recurrence_hsecondcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_recurrence_hsecondcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_recurrence_hsecondcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_recurrence_hsecondcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_recurrence_hsecondcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_recurrence_hsecondcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_recurrence_hsecondcoefficientscoefficientdiagonal)) * ac) + (pfc_left_append_recurrence_hsecondcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_recurrence_hsecondcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_recurrence_hsecondcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_recurrence_hsecondcoefficientscoefficientdiagonal)) /\\ (((pfc_left_append_recurrence_hsecondcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_recurrence_hsecondcoefficientscoefficientdiagonaltermrightinside. pfa_gap_append_recurrence_hsecondcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_recurrence_hsecondcoefficientscoefficientdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_append_recurrence_hsecondcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_recurrence_hsecondcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_recurrence_hsecondcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_recurrence_hsecondcoefficientscoefficientdiagonalterm)) * x5)) /\\ exists ff_q_pfp_append_recurrence_hsecondcoefficientscoefficientdiagonaltermrightentry. x4 = ff_q_pfp_append_recurrence_hsecondcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_recurrence_hsecondcoefficientscoefficientdiagonalterm)) * x5) + (pfc_right_append_recurrence_hsecondcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_recurrence_hsecondcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_recurrence_hsecondcoefficientscoefficientdiagonaltermrightoutside+(S M)=(pfc_complement_append_recurrence_hsecondcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_recurrence_hsecondcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_recurrence_hsecondcoefficientscoefficientdiagonal)=pfc_left_append_recurrence_hsecondcoefficientscoefficientdiagonalterm*pfc_right_append_recurrence_hsecondcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_recurrence_hsecondcoefficientscoefficientsum fs_v_pfc_append_recurrence_hsecondcoefficientscoefficientsum. ((((exists fs_h_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_start. fs_h_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_recurrence_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_start. fs_u_pfc_append_recurrence_hsecondcoefficientscoefficientsum = fs_q_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_recurrence_hsecondcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_terminal. fs_h_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_recurrence_hsecondcoefficientscoefficient) = S ((S (S (pfc_index_append_recurrence_hsecondcoefficients))) * fs_v_pfc_append_recurrence_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_terminal. fs_u_pfc_append_recurrence_hsecondcoefficientscoefficientsum = fs_q_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_recurrence_hsecondcoefficients))) * fs_v_pfc_append_recurrence_hsecondcoefficientscoefficientsum) + (pfc_natural_sum_append_recurrence_hsecondcoefficientscoefficient))) /\\ forall fs_i_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps = S (pfc_index_append_recurrence_hsecondcoefficients)) -> exists fs_a_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps fs_r_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps fs_s_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_recurrence_hsecondcoefficientscoefficient)) /\\ exists fs_q_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_recurrence_hsecondcoefficientscoefficient = fs_q_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_recurrence_hsecondcoefficientscoefficient) + (fs_a_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_recurrence_hsecondcoefficientscoefficientsum = fs_q_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_hsecondcoefficientscoefficientsum) + (fs_r_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_recurrence_hsecondcoefficientscoefficientsum = fs_q_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_hsecondcoefficientscoefficientsum) + (fs_s_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps = fs_r_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps + fs_a_pfc_append_recurrence_hsecondcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_recurrence_hsecondcoefficientscoefficientresiduebound. pfa_gap_append_recurrence_hsecondcoefficientscoefficientresiduebound + S (pfc_value_append_recurrence_hsecondcoefficients) = (p)) /\\ ((exists pfa_offset_left_append_recurrence_hsecondcoefficientscoefficientresiduecongruence pfa_offset_right_append_recurrence_hsecondcoefficientscoefficientresiduecongruence. (pfc_natural_sum_append_recurrence_hsecondcoefficientscoefficient) + (p) * pfa_offset_left_append_recurrence_hsecondcoefficientscoefficientresiduecongruence = (pfc_value_append_recurrence_hsecondcoefficients) + (p) * pfa_offset_right_append_recurrence_hsecondcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (ab)", "specialize prime_field_polynomial_convolution_at_length_exists (ac)", "specialize prime_field_polynomial_convolution_at_length_exists (L)", "specialize prime_field_polynomial_convolution_at_length_exists (x4)", "specialize prime_field_polynomial_convolution_at_length_exists (x5)", "specialize prime_field_polynomial_convolution_at_length_exists (S M)", "specialize prime_field_polynomial_convolution_at_length_exists (K)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hold_left", "exact hbounds_right_left", "exact hnew_right_right_left", "cases hsecond", "cases hsecond_witness", "have hdistributed : forall pfp_index_append_recurrence_distributed. (exists pfa_gap_append_recurrence_distributedindex. pfa_gap_append_recurrence_distributedindex + S (pfp_index_append_recurrence_distributed) = (K)) -> exists pfp_left_append_recurrence_distributed pfp_right_append_recurrence_distributed pfp_value_append_recurrence_distributed. ((((exists ff_h_pfp_append_recurrence_distributedleft. ff_h_pfp_append_recurrence_distributedleft + S (pfp_left_append_recurrence_distributed) = S ((S (pfp_index_append_recurrence_distributed)) * x7)) /\\ exists ff_q_pfp_append_recurrence_distributedleft. x6 = ff_q_pfp_append_recurrence_distributedleft * S ((S (pfp_index_append_recurrence_distributed)) * x7) + (pfp_left_append_recurrence_distributed))) /\\ (((((exists ff_h_pfp_append_recurrence_distributedright. ff_h_pfp_append_recurrence_distributedright + S (pfp_right_append_recurrence_distributed) = S ((S (pfp_index_append_recurrence_distributed)) * x9)) /\\ exists ff_q_pfp_append_recurrence_distributedright. x8 = ff_q_pfp_append_recurrence_distributedright * S ((S (pfp_index_append_recurrence_distributed)) * x9) + (pfp_right_append_recurrence_distributed))) /\\ (((((exists ff_h_pfp_append_recurrence_distributedtarget. ff_h_pfp_append_recurrence_distributedtarget + S (pfp_value_append_recurrence_distributed) = S ((S (pfp_index_append_recurrence_distributed)) * qc)) /\\ exists ff_q_pfp_append_recurrence_distributedtarget. qb = ff_q_pfp_append_recurrence_distributedtarget * S ((S (pfp_index_append_recurrence_distributed)) * qc) + (pfp_value_append_recurrence_distributed))) /\\ ((((exists pfa_gap_append_recurrence_distributedoperationleft. pfa_gap_append_recurrence_distributedoperationleft + S (pfp_left_append_recurrence_distributed) = (p)) /\\ (((exists pfa_gap_append_recurrence_distributedoperationright. pfa_gap_append_recurrence_distributedoperationright + S (pfp_right_append_recurrence_distributed) = (p)) /\\ ((((exists pfa_gap_append_recurrence_distributedoperationresultbound. pfa_gap_append_recurrence_distributedoperationresultbound + S (pfp_value_append_recurrence_distributed) = (p)) /\\ ((exists pfa_offset_left_append_recurrence_distributedoperationresultcongruence pfa_offset_right_append_recurrence_distributedoperationresultcongruence. ((pfp_left_append_recurrence_distributed) + (pfp_right_append_recurrence_distributed)) + (p) * pfa_offset_left_append_recurrence_distributedoperationresultcongruence = (pfp_value_append_recurrence_distributed) + (p) * pfa_offset_right_append_recurrence_distributedoperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_convolution_left_add (p)", "specialize prime_field_polynomial_convolution_left_add (x)", "specialize prime_field_polynomial_convolution_left_add (x1)", "specialize prime_field_polynomial_convolution_left_add (x4)", "specialize prime_field_polynomial_convolution_left_add (x5)", "specialize prime_field_polynomial_convolution_left_add (db)", "specialize prime_field_polynomial_convolution_left_add (dc)", "specialize prime_field_polynomial_convolution_left_add (S M)", "specialize prime_field_polynomial_convolution_left_add (ab)", "specialize prime_field_polynomial_convolution_left_add (ac)", "specialize prime_field_polynomial_convolution_left_add (L)", "specialize prime_field_polynomial_convolution_left_add (x6)", "specialize prime_field_polynomial_convolution_left_add (x7)", "specialize prime_field_polynomial_convolution_left_add (x8)", "specialize prime_field_polynomial_convolution_left_add (x9)", "specialize prime_field_polynomial_convolution_left_add (qb)", "specialize prime_field_polynomial_convolution_left_add (qc)", "specialize prime_field_polynomial_convolution_left_add (K)", "apply prime_field_polynomial_convolution_left_add", "exact hdecomp_witness_witness_witness_witness_witness_witness_right_right_right_right", "exact hfirst_witness_witness", "exact hsecond_witness_witness", "exact hQ", "have hshifted_equal : forall pfrep_power_append_recurrence_shift_equivalent pfrep_left_append_recurrence_shift_equivalent pfrep_right_append_recurrence_shift_equivalent. ((exists pfrep_position_append_recurrence_shift_equivalentfirst. ((pfrep_position_append_recurrence_shift_equivalentfirst+S (pfrep_power_append_recurrence_shift_equivalent)=(K)) /\\ ((((exists ff_h_pfp_append_recurrence_shift_equivalentfirstentry. ff_h_pfp_append_recurrence_shift_equivalentfirstentry + S (pfrep_left_append_recurrence_shift_equivalent) = S ((S (pfrep_position_append_recurrence_shift_equivalentfirst)) * x7)) /\\ exists ff_q_pfp_append_recurrence_shift_equivalentfirstentry. x6 = ff_q_pfp_append_recurrence_shift_equivalentfirstentry * S ((S (pfrep_position_append_recurrence_shift_equivalentfirst)) * x7) + (pfrep_left_append_recurrence_shift_equivalent)))))) \\/ (((exists pfrep_gap_append_recurrence_shift_equivalentfirstoutside. pfrep_gap_append_recurrence_shift_equivalentfirstoutside+(K)=(pfrep_power_append_recurrence_shift_equivalent)) /\\ (((pfrep_left_append_recurrence_shift_equivalent)=0))))) -> ((exists pfrep_position_append_recurrence_shift_equivalentsecond. ((pfrep_position_append_recurrence_shift_equivalentsecond+S (pfrep_power_append_recurrence_shift_equivalent)=(S N)) /\\ ((((exists ff_h_pfp_append_recurrence_shift_equivalentsecondentry. ff_h_pfp_append_recurrence_shift_equivalentsecondentry + S (pfrep_right_append_recurrence_shift_equivalent) = S ((S (pfrep_position_append_recurrence_shift_equivalentsecond)) * uc)) /\\ exists ff_q_pfp_append_recurrence_shift_equivalentsecondentry. ub = ff_q_pfp_append_recurrence_shift_equivalentsecondentry * S ((S (pfrep_position_append_recurrence_shift_equivalentsecond)) * uc) + (pfrep_right_append_recurrence_shift_equivalent)))))) \\/ (((exists pfrep_gap_append_recurrence_shift_equivalentsecondoutside. pfrep_gap_append_recurrence_shift_equivalentsecondoutside+(S N)=(pfrep_power_append_recurrence_shift_equivalent)) /\\ (((pfrep_right_append_recurrence_shift_equivalent)=0))))) -> pfrep_left_append_recurrence_shift_equivalent=pfrep_right_append_recurrence_shift_equivalent", "specialize prime_field_polynomial_convolution_shift_right_equivalent (p)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (ab)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (ac)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (L)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (bb)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (bc)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (M)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (pb)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (pc)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (N)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (x)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (x1)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (x6)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (x7)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (K)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (ub)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (uc)", "apply prime_field_polynomial_convolution_shift_right_equivalent", "exact hp0", "exact hdecomp_witness_witness_witness_witness_witness_witness_left", "exact hP", "exact hfirst_witness_witness", "exact hU", "have hfirst_equal : forall pfrep_power_append_recurrence_aligned_first pfrep_left_append_recurrence_aligned_first pfrep_right_append_recurrence_aligned_first. ((exists pfrep_position_append_recurrence_aligned_firstfirst. ((pfrep_position_append_recurrence_aligned_firstfirst+S (pfrep_power_append_recurrence_aligned_first)=(K)) /\\ ((((exists ff_h_pfp_append_recurrence_aligned_firstfirstentry. ff_h_pfp_append_recurrence_aligned_firstfirstentry + S (pfrep_left_append_recurrence_aligned_first) = S ((S (pfrep_position_append_recurrence_aligned_firstfirst)) * x7)) /\\ exists ff_q_pfp_append_recurrence_aligned_firstfirstentry. x6 = ff_q_pfp_append_recurrence_aligned_firstfirstentry * S ((S (pfrep_position_append_recurrence_aligned_firstfirst)) * x7) + (pfrep_left_append_recurrence_aligned_first)))))) \\/ (((exists pfrep_gap_append_recurrence_aligned_firstfirstoutside. pfrep_gap_append_recurrence_aligned_firstfirstoutside+(K)=(pfrep_power_append_recurrence_aligned_first)) /\\ (((pfrep_left_append_recurrence_aligned_first)=0))))) -> ((exists pfrep_position_append_recurrence_aligned_firstsecond. ((pfrep_position_append_recurrence_aligned_firstsecond+S (pfrep_power_append_recurrence_aligned_first)=(L+S N)) /\\ ((((exists ff_h_pfp_append_recurrence_aligned_firstsecondentry. ff_h_pfp_append_recurrence_aligned_firstsecondentry + S (pfrep_right_append_recurrence_aligned_first) = S ((S (pfrep_position_append_recurrence_aligned_firstsecond)) * UC)) /\\ exists ff_q_pfp_append_recurrence_aligned_firstsecondentry. UB = ff_q_pfp_append_recurrence_aligned_firstsecondentry * S ((S (pfrep_position_append_recurrence_aligned_firstsecond)) * UC) + (pfrep_right_append_recurrence_aligned_first)))))) \\/ (((exists pfrep_gap_append_recurrence_aligned_firstsecondoutside. pfrep_gap_append_recurrence_aligned_firstsecondoutside+(L+S N)=(pfrep_power_append_recurrence_aligned_first)) /\\ (((pfrep_right_append_recurrence_aligned_first)=0))))) -> pfrep_left_append_recurrence_aligned_first=pfrep_right_append_recurrence_aligned_first", "specialize prime_field_polynomial_equivalent_transitive (x6)", "specialize prime_field_polynomial_equivalent_transitive (x7)", "specialize prime_field_polynomial_equivalent_transitive (K)", "specialize prime_field_polynomial_equivalent_transitive (ub)", "specialize prime_field_polynomial_equivalent_transitive (uc)", "specialize prime_field_polynomial_equivalent_transitive (S N)", "specialize prime_field_polynomial_equivalent_transitive (UB)", "specialize prime_field_polynomial_equivalent_transitive (UC)", "specialize prime_field_polynomial_equivalent_transitive (L+S N)", "apply prime_field_polynomial_equivalent_transitive", "exact hshifted_equal", "specialize prime_field_polynomial_left_pad_equivalent (ub)", "specialize prime_field_polynomial_left_pad_equivalent (uc)", "specialize prime_field_polynomial_left_pad_equivalent (S N)", "specialize prime_field_polynomial_left_pad_equivalent (L)", "specialize prime_field_polynomial_left_pad_equivalent (UB)", "specialize prime_field_polynomial_left_pad_equivalent (UC)", "apply prime_field_polynomial_left_pad_equivalent", "exact hUP", "have hconstant_product : ((forall fom_index_pfp_append_recurrence_constant_productleft. (exists fom_gap_pfp_append_recurrence_constant_productleft_index_bound. fom_gap_pfp_append_recurrence_constant_productleft_index_bound + S (fom_index_pfp_append_recurrence_constant_productleft) = L) -> exists fom_value_pfp_append_recurrence_constant_productleft. ((((exists fom_beta_height_pfp_append_recurrence_constant_productleft_entry. fom_beta_height_pfp_append_recurrence_constant_productleft_entry + S (fom_value_pfp_append_recurrence_constant_productleft) = S ((S (fom_index_pfp_append_recurrence_constant_productleft)) * ac)) /\\ exists fom_beta_quotient_pfp_append_recurrence_constant_productleft_entry. ab = fom_beta_quotient_pfp_append_recurrence_constant_productleft_entry * S ((S (fom_index_pfp_append_recurrence_constant_productleft)) * ac) + (fom_value_pfp_append_recurrence_constant_productleft))) /\\ (exists fom_gap_pfp_append_recurrence_constant_productleft_value_bound. fom_gap_pfp_append_recurrence_constant_productleft_value_bound + S (fom_value_pfp_append_recurrence_constant_productleft) = p))) /\\ (((forall fom_index_pfp_append_recurrence_constant_productright. (exists fom_gap_pfp_append_recurrence_constant_productright_index_bound. fom_gap_pfp_append_recurrence_constant_productright_index_bound + S (fom_index_pfp_append_recurrence_constant_productright) = 1) -> exists fom_value_pfp_append_recurrence_constant_productright. ((((exists fom_beta_height_pfp_append_recurrence_constant_productright_entry. fom_beta_height_pfp_append_recurrence_constant_productright_entry + S (fom_value_pfp_append_recurrence_constant_productright) = S ((S (fom_index_pfp_append_recurrence_constant_productright)) * x3)) /\\ exists fom_beta_quotient_pfp_append_recurrence_constant_productright_entry. x2 = fom_beta_quotient_pfp_append_recurrence_constant_productright_entry * S ((S (fom_index_pfp_append_recurrence_constant_productright)) * x3) + (fom_value_pfp_append_recurrence_constant_productright))) /\\ (exists fom_gap_pfp_append_recurrence_constant_productright_value_bound. fom_gap_pfp_append_recurrence_constant_productright_value_bound + S (fom_value_pfp_append_recurrence_constant_productright) = p))) /\\ (((((((L)=0 \\/ (1)=0) /\\ (((L)=0)))) \\/ (((~((L)=0)) /\\ (((~((1)=0)) /\\ (((L)+(1)=S (L)))))))) /\\ ((forall pfc_index_append_recurrence_constant_productcoefficients. (exists pfa_gap_append_recurrence_constant_productcoefficientsbound. pfa_gap_append_recurrence_constant_productcoefficientsbound + S (pfc_index_append_recurrence_constant_productcoefficients) = (L)) -> exists pfc_value_append_recurrence_constant_productcoefficients. ((((exists ff_h_pfp_append_recurrence_constant_productcoefficientsentry. ff_h_pfp_append_recurrence_constant_productcoefficientsentry + S (pfc_value_append_recurrence_constant_productcoefficients) = S ((S (pfc_index_append_recurrence_constant_productcoefficients)) * vc)) /\\ exists ff_q_pfp_append_recurrence_constant_productcoefficientsentry. vb = ff_q_pfp_append_recurrence_constant_productcoefficientsentry * S ((S (pfc_index_append_recurrence_constant_productcoefficients)) * vc) + (pfc_value_append_recurrence_constant_productcoefficients))) /\\ ((exists pfc_terms_code_append_recurrence_constant_productcoefficientscoefficient pfc_terms_scale_append_recurrence_constant_productcoefficientscoefficient pfc_natural_sum_append_recurrence_constant_productcoefficientscoefficient. ((forall pfc_index_append_recurrence_constant_productcoefficientscoefficientdiagonal. (exists pfa_gap_append_recurrence_constant_productcoefficientscoefficientdiagonalbound. pfa_gap_append_recurrence_constant_productcoefficientscoefficientdiagonalbound + S (pfc_index_append_recurrence_constant_productcoefficientscoefficientdiagonal) = (S (pfc_index_append_recurrence_constant_productcoefficients))) -> exists pfc_value_append_recurrence_constant_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_recurrence_constant_productcoefficientscoefficientdiagonalentry. ff_h_pfp_append_recurrence_constant_productcoefficientscoefficientdiagonalentry + S (pfc_value_append_recurrence_constant_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_append_recurrence_constant_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_recurrence_constant_productcoefficientscoefficient)) /\\ exists ff_q_pfp_append_recurrence_constant_productcoefficientscoefficientdiagonalentry. pfc_terms_code_append_recurrence_constant_productcoefficientscoefficient = ff_q_pfp_append_recurrence_constant_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_append_recurrence_constant_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_recurrence_constant_productcoefficientscoefficient) + (pfc_value_append_recurrence_constant_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_recurrence_constant_productcoefficientscoefficientdiagonalterm pfc_left_append_recurrence_constant_productcoefficientscoefficientdiagonalterm pfc_right_append_recurrence_constant_productcoefficientscoefficientdiagonalterm. (((pfc_index_append_recurrence_constant_productcoefficientscoefficientdiagonal)+pfc_complement_append_recurrence_constant_productcoefficientscoefficientdiagonalterm=(pfc_index_append_recurrence_constant_productcoefficients)) /\\ ((((((exists pfa_gap_append_recurrence_constant_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_append_recurrence_constant_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_recurrence_constant_productcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_recurrence_constant_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_recurrence_constant_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_recurrence_constant_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_recurrence_constant_productcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_recurrence_constant_productcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_recurrence_constant_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_recurrence_constant_productcoefficientscoefficientdiagonal)) * ac) + (pfc_left_append_recurrence_constant_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_recurrence_constant_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_recurrence_constant_productcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_recurrence_constant_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_append_recurrence_constant_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_recurrence_constant_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_append_recurrence_constant_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_recurrence_constant_productcoefficientscoefficientdiagonalterm) = (1)) /\\ ((((exists ff_h_pfp_append_recurrence_constant_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_recurrence_constant_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_recurrence_constant_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_recurrence_constant_productcoefficientscoefficientdiagonalterm)) * x3)) /\\ exists ff_q_pfp_append_recurrence_constant_productcoefficientscoefficientdiagonaltermrightentry. x2 = ff_q_pfp_append_recurrence_constant_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_recurrence_constant_productcoefficientscoefficientdiagonalterm)) * x3) + (pfc_right_append_recurrence_constant_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_recurrence_constant_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_recurrence_constant_productcoefficientscoefficientdiagonaltermrightoutside+(1)=(pfc_complement_append_recurrence_constant_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_recurrence_constant_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_recurrence_constant_productcoefficientscoefficientdiagonal)=pfc_left_append_recurrence_constant_productcoefficientscoefficientdiagonalterm*pfc_right_append_recurrence_constant_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_recurrence_constant_productcoefficientscoefficientsum fs_v_pfc_append_recurrence_constant_productcoefficientscoefficientsum. ((((exists fs_h_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_start. fs_h_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_recurrence_constant_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_start. fs_u_pfc_append_recurrence_constant_productcoefficientscoefficientsum = fs_q_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_recurrence_constant_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_recurrence_constant_productcoefficientscoefficient) = S ((S (S (pfc_index_append_recurrence_constant_productcoefficients))) * fs_v_pfc_append_recurrence_constant_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_append_recurrence_constant_productcoefficientscoefficientsum = fs_q_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_recurrence_constant_productcoefficients))) * fs_v_pfc_append_recurrence_constant_productcoefficientscoefficientsum) + (pfc_natural_sum_append_recurrence_constant_productcoefficientscoefficient))) /\\ forall fs_i_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps = S (pfc_index_append_recurrence_constant_productcoefficients)) -> exists fs_a_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps fs_r_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps fs_s_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_recurrence_constant_productcoefficientscoefficient)) /\\ exists fs_q_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_recurrence_constant_productcoefficientscoefficient = fs_q_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_recurrence_constant_productcoefficientscoefficient) + (fs_a_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_constant_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_recurrence_constant_productcoefficientscoefficientsum = fs_q_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_constant_productcoefficientscoefficientsum) + (fs_r_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_constant_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_recurrence_constant_productcoefficientscoefficientsum = fs_q_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_constant_productcoefficientscoefficientsum) + (fs_s_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps = fs_r_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps + fs_a_pfc_append_recurrence_constant_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_recurrence_constant_productcoefficientscoefficientresiduebound. pfa_gap_append_recurrence_constant_productcoefficientscoefficientresiduebound + S (pfc_value_append_recurrence_constant_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_append_recurrence_constant_productcoefficientscoefficientresiduecongruence pfa_offset_right_append_recurrence_constant_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_append_recurrence_constant_productcoefficientscoefficient) + (p) * pfa_offset_left_append_recurrence_constant_productcoefficientscoefficientresiduecongruence = (pfc_value_append_recurrence_constant_productcoefficients) + (p) * pfa_offset_right_append_recurrence_constant_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_scale_to_constant_product (p)", "specialize prime_field_polynomial_scale_to_constant_product (c)", "specialize prime_field_polynomial_scale_to_constant_product (ab)", "specialize prime_field_polynomial_scale_to_constant_product (ac)", "specialize prime_field_polynomial_scale_to_constant_product (x2)", "specialize prime_field_polynomial_scale_to_constant_product (x3)", "specialize prime_field_polynomial_scale_to_constant_product (vb)", "specialize prime_field_polynomial_scale_to_constant_product (vc)", "specialize prime_field_polynomial_scale_to_constant_product (L)", "apply prime_field_polynomial_scale_to_constant_product", "exact hp", "exact hdecomp_witness_witness_witness_witness_witness_witness_right_left", "exact hdecomp_witness_witness_witness_witness_witness_witness_right_right_left", "exact hV", "have hreverse_equal : forall pfrep_power_append_recurrence_constant_equivalent pfrep_left_append_recurrence_constant_equivalent pfrep_right_append_recurrence_constant_equivalent. ((exists pfrep_position_append_recurrence_constant_equivalentfirst. ((pfrep_position_append_recurrence_constant_equivalentfirst+S (pfrep_power_append_recurrence_constant_equivalent)=(L)) /\\ ((((exists ff_h_pfp_append_recurrence_constant_equivalentfirstentry. ff_h_pfp_append_recurrence_constant_equivalentfirstentry + S (pfrep_left_append_recurrence_constant_equivalent) = S ((S (pfrep_position_append_recurrence_constant_equivalentfirst)) * vc)) /\\ exists ff_q_pfp_append_recurrence_constant_equivalentfirstentry. vb = ff_q_pfp_append_recurrence_constant_equivalentfirstentry * S ((S (pfrep_position_append_recurrence_constant_equivalentfirst)) * vc) + (pfrep_left_append_recurrence_constant_equivalent)))))) \\/ (((exists pfrep_gap_append_recurrence_constant_equivalentfirstoutside. pfrep_gap_append_recurrence_constant_equivalentfirstoutside+(L)=(pfrep_power_append_recurrence_constant_equivalent)) /\\ (((pfrep_left_append_recurrence_constant_equivalent)=0))))) -> ((exists pfrep_position_append_recurrence_constant_equivalentsecond. ((pfrep_position_append_recurrence_constant_equivalentsecond+S (pfrep_power_append_recurrence_constant_equivalent)=(K)) /\\ ((((exists ff_h_pfp_append_recurrence_constant_equivalentsecondentry. ff_h_pfp_append_recurrence_constant_equivalentsecondentry + S (pfrep_right_append_recurrence_constant_equivalent) = S ((S (pfrep_position_append_recurrence_constant_equivalentsecond)) * x9)) /\\ exists ff_q_pfp_append_recurrence_constant_equivalentsecondentry. x8 = ff_q_pfp_append_recurrence_constant_equivalentsecondentry * S ((S (pfrep_position_append_recurrence_constant_equivalentsecond)) * x9) + (pfrep_right_append_recurrence_constant_equivalent)))))) \\/ (((exists pfrep_gap_append_recurrence_constant_equivalentsecondoutside. pfrep_gap_append_recurrence_constant_equivalentsecondoutside+(K)=(pfrep_power_append_recurrence_constant_equivalent)) /\\ (((pfrep_right_append_recurrence_constant_equivalent)=0))))) -> pfrep_left_append_recurrence_constant_equivalent=pfrep_right_append_recurrence_constant_equivalent", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (p)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (ab)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (ac)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (L)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x2)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x3)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (1)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (vb)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (vc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (L)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x4)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x5)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (M)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x8)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x9)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (K)", "apply prime_field_polynomial_convolution_left_padding_equivalent_right", "exact hp0", "exact hdecomp_witness_witness_witness_witness_witness_witness_right_right_right_left", "exact hconstant_product", "have hlength : M+1=S M", "simp", "rewrite hlength", "rewrite hlength", "rewrite hlength", "rewrite hlength", "rewrite hlength", "rewrite hlength", "exact hsecond_witness_witness", "have hsecond_equal : forall pfrep_power_append_recurrence_aligned_second pfrep_left_append_recurrence_aligned_second pfrep_right_append_recurrence_aligned_second. ((exists pfrep_position_append_recurrence_aligned_secondfirst. ((pfrep_position_append_recurrence_aligned_secondfirst+S (pfrep_power_append_recurrence_aligned_second)=(K)) /\\ ((((exists ff_h_pfp_append_recurrence_aligned_secondfirstentry. ff_h_pfp_append_recurrence_aligned_secondfirstentry + S (pfrep_left_append_recurrence_aligned_second) = S ((S (pfrep_position_append_recurrence_aligned_secondfirst)) * x9)) /\\ exists ff_q_pfp_append_recurrence_aligned_secondfirstentry. x8 = ff_q_pfp_append_recurrence_aligned_secondfirstentry * S ((S (pfrep_position_append_recurrence_aligned_secondfirst)) * x9) + (pfrep_left_append_recurrence_aligned_second)))))) \\/ (((exists pfrep_gap_append_recurrence_aligned_secondfirstoutside. pfrep_gap_append_recurrence_aligned_secondfirstoutside+(K)=(pfrep_power_append_recurrence_aligned_second)) /\\ (((pfrep_left_append_recurrence_aligned_second)=0))))) -> ((exists pfrep_position_append_recurrence_aligned_secondsecond. ((pfrep_position_append_recurrence_aligned_secondsecond+S (pfrep_power_append_recurrence_aligned_second)=(L+S N)) /\\ ((((exists ff_h_pfp_append_recurrence_aligned_secondsecondentry. ff_h_pfp_append_recurrence_aligned_secondsecondentry + S (pfrep_right_append_recurrence_aligned_second) = S ((S (pfrep_position_append_recurrence_aligned_secondsecond)) * VC)) /\\ exists ff_q_pfp_append_recurrence_aligned_secondsecondentry. VB = ff_q_pfp_append_recurrence_aligned_secondsecondentry * S ((S (pfrep_position_append_recurrence_aligned_secondsecond)) * VC) + (pfrep_right_append_recurrence_aligned_second)))))) \\/ (((exists pfrep_gap_append_recurrence_aligned_secondsecondoutside. pfrep_gap_append_recurrence_aligned_secondsecondoutside+(L+S N)=(pfrep_power_append_recurrence_aligned_second)) /\\ (((pfrep_right_append_recurrence_aligned_second)=0))))) -> pfrep_left_append_recurrence_aligned_second=pfrep_right_append_recurrence_aligned_second", "specialize prime_field_polynomial_equivalent_transitive (x8)", "specialize prime_field_polynomial_equivalent_transitive (x9)", "specialize prime_field_polynomial_equivalent_transitive (K)", "specialize prime_field_polynomial_equivalent_transitive (vb)", "specialize prime_field_polynomial_equivalent_transitive (vc)", "specialize prime_field_polynomial_equivalent_transitive (L)", "specialize prime_field_polynomial_equivalent_transitive (VB)", "specialize prime_field_polynomial_equivalent_transitive (VC)", "specialize prime_field_polynomial_equivalent_transitive (L+S N)", "apply prime_field_polynomial_equivalent_transitive", "specialize prime_field_polynomial_equivalent_symmetric (vb)", "specialize prime_field_polynomial_equivalent_symmetric (vc)", "specialize prime_field_polynomial_equivalent_symmetric (L)", "specialize prime_field_polynomial_equivalent_symmetric (x8)", "specialize prime_field_polynomial_equivalent_symmetric (x9)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "apply prime_field_polynomial_equivalent_symmetric", "exact hreverse_equal", "have hpad_equal : forall pfrep_power_append_recurrence_commuted_padding pfrep_left_append_recurrence_commuted_padding pfrep_right_append_recurrence_commuted_padding. ((exists pfrep_position_append_recurrence_commuted_paddingfirst. ((pfrep_position_append_recurrence_commuted_paddingfirst+S (pfrep_power_append_recurrence_commuted_padding)=(L)) /\\ ((((exists ff_h_pfp_append_recurrence_commuted_paddingfirstentry. ff_h_pfp_append_recurrence_commuted_paddingfirstentry + S (pfrep_left_append_recurrence_commuted_padding) = S ((S (pfrep_position_append_recurrence_commuted_paddingfirst)) * vc)) /\\ exists ff_q_pfp_append_recurrence_commuted_paddingfirstentry. vb = ff_q_pfp_append_recurrence_commuted_paddingfirstentry * S ((S (pfrep_position_append_recurrence_commuted_paddingfirst)) * vc) + (pfrep_left_append_recurrence_commuted_padding)))))) \\/ (((exists pfrep_gap_append_recurrence_commuted_paddingfirstoutside. pfrep_gap_append_recurrence_commuted_paddingfirstoutside+(L)=(pfrep_power_append_recurrence_commuted_padding)) /\\ (((pfrep_left_append_recurrence_commuted_padding)=0))))) -> ((exists pfrep_position_append_recurrence_commuted_paddingsecond. ((pfrep_position_append_recurrence_commuted_paddingsecond+S (pfrep_power_append_recurrence_commuted_padding)=(S N+L)) /\\ ((((exists ff_h_pfp_append_recurrence_commuted_paddingsecondentry. ff_h_pfp_append_recurrence_commuted_paddingsecondentry + S (pfrep_right_append_recurrence_commuted_padding) = S ((S (pfrep_position_append_recurrence_commuted_paddingsecond)) * VC)) /\\ exists ff_q_pfp_append_recurrence_commuted_paddingsecondentry. VB = ff_q_pfp_append_recurrence_commuted_paddingsecondentry * S ((S (pfrep_position_append_recurrence_commuted_paddingsecond)) * VC) + (pfrep_right_append_recurrence_commuted_padding)))))) \\/ (((exists pfrep_gap_append_recurrence_commuted_paddingsecondoutside. pfrep_gap_append_recurrence_commuted_paddingsecondoutside+(S N+L)=(pfrep_power_append_recurrence_commuted_padding)) /\\ (((pfrep_right_append_recurrence_commuted_padding)=0))))) -> pfrep_left_append_recurrence_commuted_padding=pfrep_right_append_recurrence_commuted_padding", "specialize prime_field_polynomial_left_pad_equivalent (vb)", "specialize prime_field_polynomial_left_pad_equivalent (vc)", "specialize prime_field_polynomial_left_pad_equivalent (L)", "specialize prime_field_polynomial_left_pad_equivalent (S N)", "specialize prime_field_polynomial_left_pad_equivalent (VB)", "specialize prime_field_polynomial_left_pad_equivalent (VC)", "apply prime_field_polynomial_left_pad_equivalent", "exact hVP", "have hcomm : S N+L=L+S N", "specialize add_comm (S N)", "specialize add_comm (L)", "apply add_comm", "rewrite hcomm at hpad_equal", "rewrite hcomm at hpad_equal", "exact hpad_equal", "specialize prime_field_polynomial_add_equivalent_congruent (p)", "specialize prime_field_polynomial_add_equivalent_congruent (x6)", "specialize prime_field_polynomial_add_equivalent_congruent (x7)", "specialize prime_field_polynomial_add_equivalent_congruent (x8)", "specialize prime_field_polynomial_add_equivalent_congruent (x9)", "specialize prime_field_polynomial_add_equivalent_congruent (qb)", "specialize prime_field_polynomial_add_equivalent_congruent (qc)", "specialize prime_field_polynomial_add_equivalent_congruent (K)", "specialize prime_field_polynomial_add_equivalent_congruent (UB)", "specialize prime_field_polynomial_add_equivalent_congruent (UC)", "specialize prime_field_polynomial_add_equivalent_congruent (VB)", "specialize prime_field_polynomial_add_equivalent_congruent (VC)", "specialize prime_field_polynomial_add_equivalent_congruent (rb)", "specialize prime_field_polynomial_add_equivalent_congruent (rc)", "specialize prime_field_polynomial_add_equivalent_congruent (L+S N)", "apply prime_field_polynomial_add_equivalent_congruent", "exact hp", "exact hfirst_equal", "exact hsecond_equal", "exact hdistributed", "exact hR" ], "script_sha256": "6621b2e92b99c2aacd5160d318fbe6a5b5694faaacafb40d52f974ec02de569c", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_append_candidate.py", "sha256": "271845bfffc7e513fdb0bd0c3666dcccace8436d4d3a0f4db64b67bcd4b87042" }, "statement": "forall p ab ac L bb bc M c db dc pb pc N qb qc K ub uc vb vc UB UC VB VC rb rc. (~((p) = 1) /\\ forall pfa_factor_left_append_recurrence_prime pfa_factor_right_append_recurrence_prime. (p) = pfa_factor_left_append_recurrence_prime * pfa_factor_right_append_recurrence_prime -> pfa_factor_left_append_recurrence_prime = 1 \\/ pfa_factor_right_append_recurrence_prime = 1) -> (forall mdr_i_pfp_append_recurrence_preserve mdr_a_pfp_append_recurrence_preserve. (exists mdr_gap_pfp_append_recurrence_preserveb. mdr_gap_pfp_append_recurrence_preserveb + S (mdr_i_pfp_append_recurrence_preserve) = (M)) -> (((exists ff_h_mdr_pfp_append_recurrence_preserveo. ff_h_mdr_pfp_append_recurrence_preserveo + S (mdr_a_pfp_append_recurrence_preserve) = S ((S (mdr_i_pfp_append_recurrence_preserve)) * bc)) /\\ exists ff_q_mdr_pfp_append_recurrence_preserveo. bb = ff_q_mdr_pfp_append_recurrence_preserveo * S ((S (mdr_i_pfp_append_recurrence_preserve)) * bc) + (mdr_a_pfp_append_recurrence_preserve))) -> (((exists ff_h_mdr_pfp_append_recurrence_preserven. ff_h_mdr_pfp_append_recurrence_preserven + S (mdr_a_pfp_append_recurrence_preserve) = S ((S (mdr_i_pfp_append_recurrence_preserve)) * dc)) /\\ exists ff_q_mdr_pfp_append_recurrence_preserven. db = ff_q_mdr_pfp_append_recurrence_preserven * S ((S (mdr_i_pfp_append_recurrence_preserve)) * dc) + (mdr_a_pfp_append_recurrence_preserve)))) -> (((exists ff_h_pfp_append_recurrence_last. ff_h_pfp_append_recurrence_last + S (c) = S ((S (M)) * dc)) /\\ exists ff_q_pfp_append_recurrence_last. db = ff_q_pfp_append_recurrence_last * S ((S (M)) * dc) + (c))) -> (((forall fom_index_pfp_append_recurrence_oldleft. (exists fom_gap_pfp_append_recurrence_oldleft_index_bound. fom_gap_pfp_append_recurrence_oldleft_index_bound + S (fom_index_pfp_append_recurrence_oldleft) = L) -> exists fom_value_pfp_append_recurrence_oldleft. ((((exists fom_beta_height_pfp_append_recurrence_oldleft_entry. fom_beta_height_pfp_append_recurrence_oldleft_entry + S (fom_value_pfp_append_recurrence_oldleft) = S ((S (fom_index_pfp_append_recurrence_oldleft)) * ac)) /\\ exists fom_beta_quotient_pfp_append_recurrence_oldleft_entry. ab = fom_beta_quotient_pfp_append_recurrence_oldleft_entry * S ((S (fom_index_pfp_append_recurrence_oldleft)) * ac) + (fom_value_pfp_append_recurrence_oldleft))) /\\ (exists fom_gap_pfp_append_recurrence_oldleft_value_bound. fom_gap_pfp_append_recurrence_oldleft_value_bound + S (fom_value_pfp_append_recurrence_oldleft) = p))) /\\ (((forall fom_index_pfp_append_recurrence_oldright. (exists fom_gap_pfp_append_recurrence_oldright_index_bound. fom_gap_pfp_append_recurrence_oldright_index_bound + S (fom_index_pfp_append_recurrence_oldright) = M) -> exists fom_value_pfp_append_recurrence_oldright. ((((exists fom_beta_height_pfp_append_recurrence_oldright_entry. fom_beta_height_pfp_append_recurrence_oldright_entry + S (fom_value_pfp_append_recurrence_oldright) = S ((S (fom_index_pfp_append_recurrence_oldright)) * bc)) /\\ exists fom_beta_quotient_pfp_append_recurrence_oldright_entry. bb = fom_beta_quotient_pfp_append_recurrence_oldright_entry * S ((S (fom_index_pfp_append_recurrence_oldright)) * bc) + (fom_value_pfp_append_recurrence_oldright))) /\\ (exists fom_gap_pfp_append_recurrence_oldright_value_bound. fom_gap_pfp_append_recurrence_oldright_value_bound + S (fom_value_pfp_append_recurrence_oldright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_append_recurrence_oldcoefficients. (exists pfa_gap_append_recurrence_oldcoefficientsbound. pfa_gap_append_recurrence_oldcoefficientsbound + S (pfc_index_append_recurrence_oldcoefficients) = (N)) -> exists pfc_value_append_recurrence_oldcoefficients. ((((exists ff_h_pfp_append_recurrence_oldcoefficientsentry. ff_h_pfp_append_recurrence_oldcoefficientsentry + S (pfc_value_append_recurrence_oldcoefficients) = S ((S (pfc_index_append_recurrence_oldcoefficients)) * pc)) /\\ exists ff_q_pfp_append_recurrence_oldcoefficientsentry. pb = ff_q_pfp_append_recurrence_oldcoefficientsentry * S ((S (pfc_index_append_recurrence_oldcoefficients)) * pc) + (pfc_value_append_recurrence_oldcoefficients))) /\\ ((exists pfc_terms_code_append_recurrence_oldcoefficientscoefficient pfc_terms_scale_append_recurrence_oldcoefficientscoefficient pfc_natural_sum_append_recurrence_oldcoefficientscoefficient. ((forall pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal. (exists pfa_gap_append_recurrence_oldcoefficientscoefficientdiagonalbound. pfa_gap_append_recurrence_oldcoefficientscoefficientdiagonalbound + S (pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal) = (S (pfc_index_append_recurrence_oldcoefficients))) -> exists pfc_value_append_recurrence_oldcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_recurrence_oldcoefficientscoefficientdiagonalentry. ff_h_pfp_append_recurrence_oldcoefficientscoefficientdiagonalentry + S (pfc_value_append_recurrence_oldcoefficientscoefficientdiagonal) = S ((S (pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_recurrence_oldcoefficientscoefficient)) /\\ exists ff_q_pfp_append_recurrence_oldcoefficientscoefficientdiagonalentry. pfc_terms_code_append_recurrence_oldcoefficientscoefficient = ff_q_pfp_append_recurrence_oldcoefficientscoefficientdiagonalentry * S ((S (pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_recurrence_oldcoefficientscoefficient) + (pfc_value_append_recurrence_oldcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_recurrence_oldcoefficientscoefficientdiagonalterm pfc_left_append_recurrence_oldcoefficientscoefficientdiagonalterm pfc_right_append_recurrence_oldcoefficientscoefficientdiagonalterm. (((pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal)+pfc_complement_append_recurrence_oldcoefficientscoefficientdiagonalterm=(pfc_index_append_recurrence_oldcoefficients)) /\\ ((((((exists pfa_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermleftinside. pfa_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_recurrence_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal)) * ac) + (pfc_left_append_recurrence_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_recurrence_oldcoefficientscoefficientdiagonal)) /\\ (((pfc_left_append_recurrence_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermrightinside. pfa_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_recurrence_oldcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_recurrence_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_recurrence_oldcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_append_recurrence_oldcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_recurrence_oldcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_append_recurrence_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_recurrence_oldcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_append_recurrence_oldcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_recurrence_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_recurrence_oldcoefficientscoefficientdiagonal)=pfc_left_append_recurrence_oldcoefficientscoefficientdiagonalterm*pfc_right_append_recurrence_oldcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_recurrence_oldcoefficientscoefficientsum fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum. ((((exists fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_start. fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_start. fs_u_pfc_append_recurrence_oldcoefficientscoefficientsum = fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_terminal. fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_recurrence_oldcoefficientscoefficient) = S ((S (S (pfc_index_append_recurrence_oldcoefficients))) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_terminal. fs_u_pfc_append_recurrence_oldcoefficientscoefficientsum = fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_recurrence_oldcoefficients))) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum) + (pfc_natural_sum_append_recurrence_oldcoefficientscoefficient))) /\\ forall fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps = S (pfc_index_append_recurrence_oldcoefficients)) -> exists fs_a_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps fs_r_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps fs_s_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_recurrence_oldcoefficientscoefficient)) /\\ exists fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_recurrence_oldcoefficientscoefficient = fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_recurrence_oldcoefficientscoefficient) + (fs_a_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_recurrence_oldcoefficientscoefficientsum = fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum) + (fs_r_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_recurrence_oldcoefficientscoefficientsum = fs_q_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_oldcoefficientscoefficientsum) + (fs_s_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps = fs_r_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps + fs_a_pfc_append_recurrence_oldcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_recurrence_oldcoefficientscoefficientresiduebound. pfa_gap_append_recurrence_oldcoefficientscoefficientresiduebound + S (pfc_value_append_recurrence_oldcoefficients) = (p)) /\\ ((exists pfa_offset_left_append_recurrence_oldcoefficientscoefficientresiduecongruence pfa_offset_right_append_recurrence_oldcoefficientscoefficientresiduecongruence. (pfc_natural_sum_append_recurrence_oldcoefficientscoefficient) + (p) * pfa_offset_left_append_recurrence_oldcoefficientscoefficientresiduecongruence = (pfc_value_append_recurrence_oldcoefficients) + (p) * pfa_offset_right_append_recurrence_oldcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_append_recurrence_newleft. (exists fom_gap_pfp_append_recurrence_newleft_index_bound. fom_gap_pfp_append_recurrence_newleft_index_bound + S (fom_index_pfp_append_recurrence_newleft) = L) -> exists fom_value_pfp_append_recurrence_newleft. ((((exists fom_beta_height_pfp_append_recurrence_newleft_entry. fom_beta_height_pfp_append_recurrence_newleft_entry + S (fom_value_pfp_append_recurrence_newleft) = S ((S (fom_index_pfp_append_recurrence_newleft)) * ac)) /\\ exists fom_beta_quotient_pfp_append_recurrence_newleft_entry. ab = fom_beta_quotient_pfp_append_recurrence_newleft_entry * S ((S (fom_index_pfp_append_recurrence_newleft)) * ac) + (fom_value_pfp_append_recurrence_newleft))) /\\ (exists fom_gap_pfp_append_recurrence_newleft_value_bound. fom_gap_pfp_append_recurrence_newleft_value_bound + S (fom_value_pfp_append_recurrence_newleft) = p))) /\\ (((forall fom_index_pfp_append_recurrence_newright. (exists fom_gap_pfp_append_recurrence_newright_index_bound. fom_gap_pfp_append_recurrence_newright_index_bound + S (fom_index_pfp_append_recurrence_newright) = S M) -> exists fom_value_pfp_append_recurrence_newright. ((((exists fom_beta_height_pfp_append_recurrence_newright_entry. fom_beta_height_pfp_append_recurrence_newright_entry + S (fom_value_pfp_append_recurrence_newright) = S ((S (fom_index_pfp_append_recurrence_newright)) * dc)) /\\ exists fom_beta_quotient_pfp_append_recurrence_newright_entry. db = fom_beta_quotient_pfp_append_recurrence_newright_entry * S ((S (fom_index_pfp_append_recurrence_newright)) * dc) + (fom_value_pfp_append_recurrence_newright))) /\\ (exists fom_gap_pfp_append_recurrence_newright_value_bound. fom_gap_pfp_append_recurrence_newright_value_bound + S (fom_value_pfp_append_recurrence_newright) = p))) /\\ (((((((L)=0 \\/ (S M)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((S M)=0)) /\\ (((L)+(S M)=S (K)))))))) /\\ ((forall pfc_index_append_recurrence_newcoefficients. (exists pfa_gap_append_recurrence_newcoefficientsbound. pfa_gap_append_recurrence_newcoefficientsbound + S (pfc_index_append_recurrence_newcoefficients) = (K)) -> exists pfc_value_append_recurrence_newcoefficients. ((((exists ff_h_pfp_append_recurrence_newcoefficientsentry. ff_h_pfp_append_recurrence_newcoefficientsentry + S (pfc_value_append_recurrence_newcoefficients) = S ((S (pfc_index_append_recurrence_newcoefficients)) * qc)) /\\ exists ff_q_pfp_append_recurrence_newcoefficientsentry. qb = ff_q_pfp_append_recurrence_newcoefficientsentry * S ((S (pfc_index_append_recurrence_newcoefficients)) * qc) + (pfc_value_append_recurrence_newcoefficients))) /\\ ((exists pfc_terms_code_append_recurrence_newcoefficientscoefficient pfc_terms_scale_append_recurrence_newcoefficientscoefficient pfc_natural_sum_append_recurrence_newcoefficientscoefficient. ((forall pfc_index_append_recurrence_newcoefficientscoefficientdiagonal. (exists pfa_gap_append_recurrence_newcoefficientscoefficientdiagonalbound. pfa_gap_append_recurrence_newcoefficientscoefficientdiagonalbound + S (pfc_index_append_recurrence_newcoefficientscoefficientdiagonal) = (S (pfc_index_append_recurrence_newcoefficients))) -> exists pfc_value_append_recurrence_newcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_recurrence_newcoefficientscoefficientdiagonalentry. ff_h_pfp_append_recurrence_newcoefficientscoefficientdiagonalentry + S (pfc_value_append_recurrence_newcoefficientscoefficientdiagonal) = S ((S (pfc_index_append_recurrence_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_recurrence_newcoefficientscoefficient)) /\\ exists ff_q_pfp_append_recurrence_newcoefficientscoefficientdiagonalentry. pfc_terms_code_append_recurrence_newcoefficientscoefficient = ff_q_pfp_append_recurrence_newcoefficientscoefficientdiagonalentry * S ((S (pfc_index_append_recurrence_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_recurrence_newcoefficientscoefficient) + (pfc_value_append_recurrence_newcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_recurrence_newcoefficientscoefficientdiagonalterm pfc_left_append_recurrence_newcoefficientscoefficientdiagonalterm pfc_right_append_recurrence_newcoefficientscoefficientdiagonalterm. (((pfc_index_append_recurrence_newcoefficientscoefficientdiagonal)+pfc_complement_append_recurrence_newcoefficientscoefficientdiagonalterm=(pfc_index_append_recurrence_newcoefficients)) /\\ ((((((exists pfa_gap_append_recurrence_newcoefficientscoefficientdiagonaltermleftinside. pfa_gap_append_recurrence_newcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_recurrence_newcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_recurrence_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_recurrence_newcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_recurrence_newcoefficientscoefficientdiagonal)) * ac) + (pfc_left_append_recurrence_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_recurrence_newcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_recurrence_newcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_recurrence_newcoefficientscoefficientdiagonal)) /\\ (((pfc_left_append_recurrence_newcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_recurrence_newcoefficientscoefficientdiagonaltermrightinside. pfa_gap_append_recurrence_newcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_recurrence_newcoefficientscoefficientdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_recurrence_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_recurrence_newcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_append_recurrence_newcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_recurrence_newcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_append_recurrence_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_recurrence_newcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_recurrence_newcoefficientscoefficientdiagonaltermrightoutside+(S M)=(pfc_complement_append_recurrence_newcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_recurrence_newcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_recurrence_newcoefficientscoefficientdiagonal)=pfc_left_append_recurrence_newcoefficientscoefficientdiagonalterm*pfc_right_append_recurrence_newcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_recurrence_newcoefficientscoefficientsum fs_v_pfc_append_recurrence_newcoefficientscoefficientsum. ((((exists fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_start. fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_start. fs_u_pfc_append_recurrence_newcoefficientscoefficientsum = fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_terminal. fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_recurrence_newcoefficientscoefficient) = S ((S (S (pfc_index_append_recurrence_newcoefficients))) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_terminal. fs_u_pfc_append_recurrence_newcoefficientscoefficientsum = fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_recurrence_newcoefficients))) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum) + (pfc_natural_sum_append_recurrence_newcoefficientscoefficient))) /\\ forall fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps = S (pfc_index_append_recurrence_newcoefficients)) -> exists fs_a_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps fs_r_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps fs_s_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_recurrence_newcoefficientscoefficient)) /\\ exists fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_recurrence_newcoefficientscoefficient = fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_recurrence_newcoefficientscoefficient) + (fs_a_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_recurrence_newcoefficientscoefficientsum = fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum) + (fs_r_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_recurrence_newcoefficientscoefficientsum = fs_q_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_recurrence_newcoefficientscoefficientsum) + (fs_s_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps = fs_r_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps + fs_a_pfc_append_recurrence_newcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_recurrence_newcoefficientscoefficientresiduebound. pfa_gap_append_recurrence_newcoefficientscoefficientresiduebound + S (pfc_value_append_recurrence_newcoefficients) = (p)) /\\ ((exists pfa_offset_left_append_recurrence_newcoefficientscoefficientresiduecongruence pfa_offset_right_append_recurrence_newcoefficientscoefficientresiduecongruence. (pfc_natural_sum_append_recurrence_newcoefficientscoefficient) + (p) * pfa_offset_left_append_recurrence_newcoefficientscoefficientresiduecongruence = (pfc_value_append_recurrence_newcoefficients) + (p) * pfa_offset_right_append_recurrence_newcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall mdr_i_pfp_append_recurrence_shiftprefix mdr_a_pfp_append_recurrence_shiftprefix. (exists mdr_gap_pfp_append_recurrence_shiftprefixb. mdr_gap_pfp_append_recurrence_shiftprefixb + S (mdr_i_pfp_append_recurrence_shiftprefix) = (N)) -> (((exists ff_h_mdr_pfp_append_recurrence_shiftprefixo. ff_h_mdr_pfp_append_recurrence_shiftprefixo + S (mdr_a_pfp_append_recurrence_shiftprefix) = S ((S (mdr_i_pfp_append_recurrence_shiftprefix)) * pc)) /\\ exists ff_q_mdr_pfp_append_recurrence_shiftprefixo. pb = ff_q_mdr_pfp_append_recurrence_shiftprefixo * S ((S (mdr_i_pfp_append_recurrence_shiftprefix)) * pc) + (mdr_a_pfp_append_recurrence_shiftprefix))) -> (((exists ff_h_mdr_pfp_append_recurrence_shiftprefixn. ff_h_mdr_pfp_append_recurrence_shiftprefixn + S (mdr_a_pfp_append_recurrence_shiftprefix) = S ((S (mdr_i_pfp_append_recurrence_shiftprefix)) * uc)) /\\ exists ff_q_mdr_pfp_append_recurrence_shiftprefixn. ub = ff_q_mdr_pfp_append_recurrence_shiftprefixn * S ((S (mdr_i_pfp_append_recurrence_shiftprefix)) * uc) + (mdr_a_pfp_append_recurrence_shiftprefix)))) /\\ ((((exists ff_h_pfp_append_recurrence_shiftlast. ff_h_pfp_append_recurrence_shiftlast + S (0) = S ((S (N)) * uc)) /\\ exists ff_q_pfp_append_recurrence_shiftlast. ub = ff_q_pfp_append_recurrence_shiftlast * S ((S (N)) * uc) + (0)))))) -> (((exists pfa_gap_append_recurrence_scalescalar. pfa_gap_append_recurrence_scalescalar + S (c) = (p)) /\\ ((forall pfp_index_append_recurrence_scale. (exists pfa_gap_append_recurrence_scaleindex. pfa_gap_append_recurrence_scaleindex + S (pfp_index_append_recurrence_scale) = (L)) -> exists pfp_source_append_recurrence_scale pfp_value_append_recurrence_scale. ((((exists ff_h_pfp_append_recurrence_scalesource. ff_h_pfp_append_recurrence_scalesource + S (pfp_source_append_recurrence_scale) = S ((S (pfp_index_append_recurrence_scale)) * ac)) /\\ exists ff_q_pfp_append_recurrence_scalesource. ab = ff_q_pfp_append_recurrence_scalesource * S ((S (pfp_index_append_recurrence_scale)) * ac) + (pfp_source_append_recurrence_scale))) /\\ (((((exists ff_h_pfp_append_recurrence_scaletarget. ff_h_pfp_append_recurrence_scaletarget + S (pfp_value_append_recurrence_scale) = S ((S (pfp_index_append_recurrence_scale)) * vc)) /\\ exists ff_q_pfp_append_recurrence_scaletarget. vb = ff_q_pfp_append_recurrence_scaletarget * S ((S (pfp_index_append_recurrence_scale)) * vc) + (pfp_value_append_recurrence_scale))) /\\ ((((exists pfa_gap_append_recurrence_scaleoperationleft. pfa_gap_append_recurrence_scaleoperationleft + S (c) = (p)) /\\ (((exists pfa_gap_append_recurrence_scaleoperationright. pfa_gap_append_recurrence_scaleoperationright + S (pfp_source_append_recurrence_scale) = (p)) /\\ ((((exists pfa_gap_append_recurrence_scaleoperationresultbound. pfa_gap_append_recurrence_scaleoperationresultbound + S (pfp_value_append_recurrence_scale) = (p)) /\\ ((exists pfa_offset_left_append_recurrence_scaleoperationresultcongruence pfa_offset_right_append_recurrence_scaleoperationresultcongruence. ((c) * (pfp_source_append_recurrence_scale)) + (p) * pfa_offset_left_append_recurrence_scaleoperationresultcongruence = (pfp_value_append_recurrence_scale) + (p) * pfa_offset_right_append_recurrence_scaleoperationresultcongruence))))))))))))))))) -> (((forall pfp_repeat_index_append_recurrence_leftzeros. (exists pfa_gap_append_recurrence_leftzerosindex. pfa_gap_append_recurrence_leftzerosindex + S (pfp_repeat_index_append_recurrence_leftzeros) = (L)) -> (((exists ff_h_pfp_append_recurrence_leftzerosentry. ff_h_pfp_append_recurrence_leftzerosentry + S (0) = S ((S (pfp_repeat_index_append_recurrence_leftzeros)) * UC)) /\\ exists ff_q_pfp_append_recurrence_leftzerosentry. UB = ff_q_pfp_append_recurrence_leftzerosentry * S ((S (pfp_repeat_index_append_recurrence_leftzeros)) * UC) + (0)))) /\\ ((forall pfrep_index_append_recurrence_left pfrep_value_append_recurrence_left. (exists pfa_gap_append_recurrence_leftbound. pfa_gap_append_recurrence_leftbound + S (pfrep_index_append_recurrence_left) = (S N)) -> (((exists ff_h_pfp_append_recurrence_leftinput. ff_h_pfp_append_recurrence_leftinput + S (pfrep_value_append_recurrence_left) = S ((S (pfrep_index_append_recurrence_left)) * uc)) /\\ exists ff_q_pfp_append_recurrence_leftinput. ub = ff_q_pfp_append_recurrence_leftinput * S ((S (pfrep_index_append_recurrence_left)) * uc) + (pfrep_value_append_recurrence_left))) -> (((exists ff_h_pfp_append_recurrence_leftoutput. ff_h_pfp_append_recurrence_leftoutput + S (pfrep_value_append_recurrence_left) = S ((S ((L)+pfrep_index_append_recurrence_left)) * UC)) /\\ exists ff_q_pfp_append_recurrence_leftoutput. UB = ff_q_pfp_append_recurrence_leftoutput * S ((S ((L)+pfrep_index_append_recurrence_left)) * UC) + (pfrep_value_append_recurrence_left))))))) -> (((forall pfp_repeat_index_append_recurrence_rightzeros. (exists pfa_gap_append_recurrence_rightzerosindex. pfa_gap_append_recurrence_rightzerosindex + S (pfp_repeat_index_append_recurrence_rightzeros) = (S N)) -> (((exists ff_h_pfp_append_recurrence_rightzerosentry. ff_h_pfp_append_recurrence_rightzerosentry + S (0) = S ((S (pfp_repeat_index_append_recurrence_rightzeros)) * VC)) /\\ exists ff_q_pfp_append_recurrence_rightzerosentry. VB = ff_q_pfp_append_recurrence_rightzerosentry * S ((S (pfp_repeat_index_append_recurrence_rightzeros)) * VC) + (0)))) /\\ ((forall pfrep_index_append_recurrence_right pfrep_value_append_recurrence_right. (exists pfa_gap_append_recurrence_rightbound. pfa_gap_append_recurrence_rightbound + S (pfrep_index_append_recurrence_right) = (L)) -> (((exists ff_h_pfp_append_recurrence_rightinput. ff_h_pfp_append_recurrence_rightinput + S (pfrep_value_append_recurrence_right) = S ((S (pfrep_index_append_recurrence_right)) * vc)) /\\ exists ff_q_pfp_append_recurrence_rightinput. vb = ff_q_pfp_append_recurrence_rightinput * S ((S (pfrep_index_append_recurrence_right)) * vc) + (pfrep_value_append_recurrence_right))) -> (((exists ff_h_pfp_append_recurrence_rightoutput. ff_h_pfp_append_recurrence_rightoutput + S (pfrep_value_append_recurrence_right) = S ((S ((S N)+pfrep_index_append_recurrence_right)) * VC)) /\\ exists ff_q_pfp_append_recurrence_rightoutput. VB = ff_q_pfp_append_recurrence_rightoutput * S ((S ((S N)+pfrep_index_append_recurrence_right)) * VC) + (pfrep_value_append_recurrence_right))))))) -> (forall pfp_index_append_recurrence_sum. (exists pfa_gap_append_recurrence_sumindex. pfa_gap_append_recurrence_sumindex + S (pfp_index_append_recurrence_sum) = (L+S N)) -> exists pfp_left_append_recurrence_sum pfp_right_append_recurrence_sum pfp_value_append_recurrence_sum. ((((exists ff_h_pfp_append_recurrence_sumleft. ff_h_pfp_append_recurrence_sumleft + S (pfp_left_append_recurrence_sum) = S ((S (pfp_index_append_recurrence_sum)) * UC)) /\\ exists ff_q_pfp_append_recurrence_sumleft. UB = ff_q_pfp_append_recurrence_sumleft * S ((S (pfp_index_append_recurrence_sum)) * UC) + (pfp_left_append_recurrence_sum))) /\\ (((((exists ff_h_pfp_append_recurrence_sumright. ff_h_pfp_append_recurrence_sumright + S (pfp_right_append_recurrence_sum) = S ((S (pfp_index_append_recurrence_sum)) * VC)) /\\ exists ff_q_pfp_append_recurrence_sumright. VB = ff_q_pfp_append_recurrence_sumright * S ((S (pfp_index_append_recurrence_sum)) * VC) + (pfp_right_append_recurrence_sum))) /\\ (((((exists ff_h_pfp_append_recurrence_sumtarget. ff_h_pfp_append_recurrence_sumtarget + S (pfp_value_append_recurrence_sum) = S ((S (pfp_index_append_recurrence_sum)) * rc)) /\\ exists ff_q_pfp_append_recurrence_sumtarget. rb = ff_q_pfp_append_recurrence_sumtarget * S ((S (pfp_index_append_recurrence_sum)) * rc) + (pfp_value_append_recurrence_sum))) /\\ ((((exists pfa_gap_append_recurrence_sumoperationleft. pfa_gap_append_recurrence_sumoperationleft + S (pfp_left_append_recurrence_sum) = (p)) /\\ (((exists pfa_gap_append_recurrence_sumoperationright. pfa_gap_append_recurrence_sumoperationright + S (pfp_right_append_recurrence_sum) = (p)) /\\ ((((exists pfa_gap_append_recurrence_sumoperationresultbound. pfa_gap_append_recurrence_sumoperationresultbound + S (pfp_value_append_recurrence_sum) = (p)) /\\ ((exists pfa_offset_left_append_recurrence_sumoperationresultcongruence pfa_offset_right_append_recurrence_sumoperationresultcongruence. ((pfp_left_append_recurrence_sum) + (pfp_right_append_recurrence_sum)) + (p) * pfa_offset_left_append_recurrence_sumoperationresultcongruence = (pfp_value_append_recurrence_sum) + (p) * pfa_offset_right_append_recurrence_sumoperationresultcongruence)))))))))))))))) -> (forall pfrep_power_append_recurrence_result pfrep_left_append_recurrence_result pfrep_right_append_recurrence_result. ((exists pfrep_position_append_recurrence_resultfirst. ((pfrep_position_append_recurrence_resultfirst+S (pfrep_power_append_recurrence_result)=(K)) /\\ ((((exists ff_h_pfp_append_recurrence_resultfirstentry. ff_h_pfp_append_recurrence_resultfirstentry + S (pfrep_left_append_recurrence_result) = S ((S (pfrep_position_append_recurrence_resultfirst)) * qc)) /\\ exists ff_q_pfp_append_recurrence_resultfirstentry. qb = ff_q_pfp_append_recurrence_resultfirstentry * S ((S (pfrep_position_append_recurrence_resultfirst)) * qc) + (pfrep_left_append_recurrence_result)))))) \\/ (((exists pfrep_gap_append_recurrence_resultfirstoutside. pfrep_gap_append_recurrence_resultfirstoutside+(K)=(pfrep_power_append_recurrence_result)) /\\ (((pfrep_left_append_recurrence_result)=0))))) -> ((exists pfrep_position_append_recurrence_resultsecond. ((pfrep_position_append_recurrence_resultsecond+S (pfrep_power_append_recurrence_result)=(L+S N)) /\\ ((((exists ff_h_pfp_append_recurrence_resultsecondentry. ff_h_pfp_append_recurrence_resultsecondentry + S (pfrep_right_append_recurrence_result) = S ((S (pfrep_position_append_recurrence_resultsecond)) * rc)) /\\ exists ff_q_pfp_append_recurrence_resultsecondentry. rb = ff_q_pfp_append_recurrence_resultsecondentry * S ((S (pfrep_position_append_recurrence_resultsecond)) * rc) + (pfrep_right_append_recurrence_result)))))) \\/ (((exists pfrep_gap_append_recurrence_resultsecondoutside. pfrep_gap_append_recurrence_resultsecondoutside+(L+S N)=(pfrep_power_append_recurrence_result)) /\\ (((pfrep_right_append_recurrence_result)=0))))) -> pfrep_left_append_recurrence_result=pfrep_right_append_recurrence_result)", "statement_sha256": "5faf710c359ab888ce853bfa86c5c14bcf11e1b8c4fb09b0c834b5911bce96a3", "summary": "An actual right-factor append satisfies A*append(C,c) formally equivalent to X*(A*C)+c*A through genuine products and arbitrary actual aligned sum outputs. 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prime_nonzero", "exact hp", "exact hz", "have hold : ((forall fom_index_pfp_append_exists_oldleft. (exists fom_gap_pfp_append_exists_oldleft_index_bound. fom_gap_pfp_append_exists_oldleft_index_bound + S (fom_index_pfp_append_exists_oldleft) = L) -> exists fom_value_pfp_append_exists_oldleft. ((((exists fom_beta_height_pfp_append_exists_oldleft_entry. fom_beta_height_pfp_append_exists_oldleft_entry + S (fom_value_pfp_append_exists_oldleft) = S ((S (fom_index_pfp_append_exists_oldleft)) * ac)) /\\ exists fom_beta_quotient_pfp_append_exists_oldleft_entry. ab = fom_beta_quotient_pfp_append_exists_oldleft_entry * S ((S (fom_index_pfp_append_exists_oldleft)) * ac) + (fom_value_pfp_append_exists_oldleft))) /\\ (exists fom_gap_pfp_append_exists_oldleft_value_bound. fom_gap_pfp_append_exists_oldleft_value_bound + S (fom_value_pfp_append_exists_oldleft) = p))) /\\ (((forall fom_index_pfp_append_exists_oldright. (exists fom_gap_pfp_append_exists_oldright_index_bound. fom_gap_pfp_append_exists_oldright_index_bound + S (fom_index_pfp_append_exists_oldright) = M) -> exists fom_value_pfp_append_exists_oldright. ((((exists fom_beta_height_pfp_append_exists_oldright_entry. fom_beta_height_pfp_append_exists_oldright_entry + S (fom_value_pfp_append_exists_oldright) = S ((S (fom_index_pfp_append_exists_oldright)) * bc)) /\\ exists fom_beta_quotient_pfp_append_exists_oldright_entry. bb = fom_beta_quotient_pfp_append_exists_oldright_entry * S ((S (fom_index_pfp_append_exists_oldright)) * bc) + (fom_value_pfp_append_exists_oldright))) /\\ (exists fom_gap_pfp_append_exists_oldright_value_bound. fom_gap_pfp_append_exists_oldright_value_bound + S (fom_value_pfp_append_exists_oldright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_append_exists_oldcoefficients. (exists pfa_gap_append_exists_oldcoefficientsbound. pfa_gap_append_exists_oldcoefficientsbound + S (pfc_index_append_exists_oldcoefficients) = (N)) -> exists pfc_value_append_exists_oldcoefficients. ((((exists ff_h_pfp_append_exists_oldcoefficientsentry. ff_h_pfp_append_exists_oldcoefficientsentry + S (pfc_value_append_exists_oldcoefficients) = S ((S (pfc_index_append_exists_oldcoefficients)) * pc)) /\\ exists ff_q_pfp_append_exists_oldcoefficientsentry. pb = ff_q_pfp_append_exists_oldcoefficientsentry * S ((S (pfc_index_append_exists_oldcoefficients)) * pc) + (pfc_value_append_exists_oldcoefficients))) /\\ ((exists pfc_terms_code_append_exists_oldcoefficientscoefficient pfc_terms_scale_append_exists_oldcoefficientscoefficient pfc_natural_sum_append_exists_oldcoefficientscoefficient. ((forall pfc_index_append_exists_oldcoefficientscoefficientdiagonal. (exists pfa_gap_append_exists_oldcoefficientscoefficientdiagonalbound. pfa_gap_append_exists_oldcoefficientscoefficientdiagonalbound + S (pfc_index_append_exists_oldcoefficientscoefficientdiagonal) = (S (pfc_index_append_exists_oldcoefficients))) -> exists pfc_value_append_exists_oldcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_exists_oldcoefficientscoefficientdiagonalentry. ff_h_pfp_append_exists_oldcoefficientscoefficientdiagonalentry + S (pfc_value_append_exists_oldcoefficientscoefficientdiagonal) = S ((S (pfc_index_append_exists_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_exists_oldcoefficientscoefficient)) /\\ exists ff_q_pfp_append_exists_oldcoefficientscoefficientdiagonalentry. pfc_terms_code_append_exists_oldcoefficientscoefficient = ff_q_pfp_append_exists_oldcoefficientscoefficientdiagonalentry * S ((S (pfc_index_append_exists_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_exists_oldcoefficientscoefficient) + (pfc_value_append_exists_oldcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_exists_oldcoefficientscoefficientdiagonalterm pfc_left_append_exists_oldcoefficientscoefficientdiagonalterm pfc_right_append_exists_oldcoefficientscoefficientdiagonalterm. (((pfc_index_append_exists_oldcoefficientscoefficientdiagonal)+pfc_complement_append_exists_oldcoefficientscoefficientdiagonalterm=(pfc_index_append_exists_oldcoefficients)) /\\ ((((((exists pfa_gap_append_exists_oldcoefficientscoefficientdiagonaltermleftinside. pfa_gap_append_exists_oldcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_exists_oldcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_exists_oldcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_exists_oldcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_exists_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_exists_oldcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_exists_oldcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_exists_oldcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_exists_oldcoefficientscoefficientdiagonal)) * ac) + (pfc_left_append_exists_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_exists_oldcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_exists_oldcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_exists_oldcoefficientscoefficientdiagonal)) /\\ (((pfc_left_append_exists_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_exists_oldcoefficientscoefficientdiagonaltermrightinside. pfa_gap_append_exists_oldcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_exists_oldcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_append_exists_oldcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_exists_oldcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_exists_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_exists_oldcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_append_exists_oldcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_append_exists_oldcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_exists_oldcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_append_exists_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_exists_oldcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_exists_oldcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_append_exists_oldcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_exists_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_exists_oldcoefficientscoefficientdiagonal)=pfc_left_append_exists_oldcoefficientscoefficientdiagonalterm*pfc_right_append_exists_oldcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_exists_oldcoefficientscoefficientsum fs_v_pfc_append_exists_oldcoefficientscoefficientsum. ((((exists fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_start. fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_start. fs_u_pfc_append_exists_oldcoefficientscoefficientsum = fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_terminal. fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_exists_oldcoefficientscoefficient) = S ((S (S (pfc_index_append_exists_oldcoefficients))) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_terminal. fs_u_pfc_append_exists_oldcoefficientscoefficientsum = fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_exists_oldcoefficients))) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum) + (pfc_natural_sum_append_exists_oldcoefficientscoefficient))) /\\ forall fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps = S (pfc_index_append_exists_oldcoefficients)) -> exists fs_a_pfc_append_exists_oldcoefficientscoefficientsum_body_steps fs_r_pfc_append_exists_oldcoefficientscoefficientsum_body_steps fs_s_pfc_append_exists_oldcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_exists_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_exists_oldcoefficientscoefficient)) /\\ exists fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_exists_oldcoefficientscoefficient = fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_exists_oldcoefficientscoefficient) + (fs_a_pfc_append_exists_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_exists_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_exists_oldcoefficientscoefficientsum = fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum) + (fs_r_pfc_append_exists_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_exists_oldcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_exists_oldcoefficientscoefficientsum = fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum) + (fs_s_pfc_append_exists_oldcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_exists_oldcoefficientscoefficientsum_body_steps = fs_r_pfc_append_exists_oldcoefficientscoefficientsum_body_steps + fs_a_pfc_append_exists_oldcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_exists_oldcoefficientscoefficientresiduebound. pfa_gap_append_exists_oldcoefficientscoefficientresiduebound + S (pfc_value_append_exists_oldcoefficients) = (p)) /\\ ((exists pfa_offset_left_append_exists_oldcoefficientscoefficientresiduecongruence pfa_offset_right_append_exists_oldcoefficientscoefficientresiduecongruence. (pfc_natural_sum_append_exists_oldcoefficientscoefficient) + (p) * pfa_offset_left_append_exists_oldcoefficientscoefficientresiduecongruence = (pfc_value_append_exists_oldcoefficients) + (p) * pfa_offset_right_append_exists_oldcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hP", "cases hold", "cases hold_right", "cases hold_right_right", "have hd : exists db dc. ((((exists ff_h_pfp_append_exists_last. ff_h_pfp_append_exists_last + S (c) = S ((S (M)) * dc)) /\\ exists ff_q_pfp_append_exists_last. db = ff_q_pfp_append_exists_last * S ((S (M)) * dc) + (c))) /\\ ((forall mdr_i_pfp_append_exists_preserve mdr_a_pfp_append_exists_preserve. (exists mdr_gap_pfp_append_exists_preserveb. mdr_gap_pfp_append_exists_preserveb + S (mdr_i_pfp_append_exists_preserve) = (M)) -> (((exists ff_h_mdr_pfp_append_exists_preserveo. ff_h_mdr_pfp_append_exists_preserveo + S (mdr_a_pfp_append_exists_preserve) = S ((S (mdr_i_pfp_append_exists_preserve)) * bc)) /\\ exists ff_q_mdr_pfp_append_exists_preserveo. bb = ff_q_mdr_pfp_append_exists_preserveo * S ((S (mdr_i_pfp_append_exists_preserve)) * bc) + (mdr_a_pfp_append_exists_preserve))) -> (((exists ff_h_mdr_pfp_append_exists_preserven. ff_h_mdr_pfp_append_exists_preserven + S (mdr_a_pfp_append_exists_preserve) = S ((S (mdr_i_pfp_append_exists_preserve)) * dc)) /\\ exists ff_q_mdr_pfp_append_exists_preserven. db = ff_q_mdr_pfp_append_exists_preserven * S ((S (mdr_i_pfp_append_exists_preserve)) * dc) + (mdr_a_pfp_append_exists_preserve))))))", "specialize beta_prefix_extend (M)", "specialize beta_prefix_extend (bb)", "specialize beta_prefix_extend (bc)", "specialize beta_prefix_extend (c)", "apply beta_prefix_extend", "cases hd", "cases hd_witness", "cases hd_witness_witness", "have hbounded : forall fom_index_pfp_append_exists_new_bound. (exists fom_gap_pfp_append_exists_new_bound_index_bound. fom_gap_pfp_append_exists_new_bound_index_bound + S (fom_index_pfp_append_exists_new_bound) = S M) -> exists fom_value_pfp_append_exists_new_bound. ((((exists fom_beta_height_pfp_append_exists_new_bound_entry. fom_beta_height_pfp_append_exists_new_bound_entry + S (fom_value_pfp_append_exists_new_bound) = S ((S (fom_index_pfp_append_exists_new_bound)) * x1)) /\\ exists fom_beta_quotient_pfp_append_exists_new_bound_entry. x = fom_beta_quotient_pfp_append_exists_new_bound_entry * S ((S (fom_index_pfp_append_exists_new_bound)) * x1) + (fom_value_pfp_append_exists_new_bound))) /\\ (exists fom_gap_pfp_append_exists_new_bound_value_bound. fom_gap_pfp_append_exists_new_bound_value_bound + S (fom_value_pfp_append_exists_new_bound) = p))", "specialize matrix_rank_bounded_prefix_extend (x)", "specialize matrix_rank_bounded_prefix_extend (x1)", "specialize matrix_rank_bounded_prefix_extend (M)", "specialize matrix_rank_bounded_prefix_extend (p)", "specialize matrix_rank_bounded_prefix_extend (c)", "apply matrix_rank_bounded_prefix_extend", "specialize matrix_rank_bounded_prefix_transport (bb)", "specialize matrix_rank_bounded_prefix_transport (bc)", "specialize matrix_rank_bounded_prefix_transport (x)", "specialize matrix_rank_bounded_prefix_transport (x1)", "specialize matrix_rank_bounded_prefix_transport (M)", "specialize matrix_rank_bounded_prefix_transport (p)", "apply matrix_rank_bounded_prefix_transport", "exact hd_witness_witness_right", "exact hold_right_left", "exact hd_witness_witness_left", "exact hc", "have hl : exists K. ((((L)=0 \\/ (S M)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((S M)=0)) /\\ (((L)+(S M)=S (K)))))))", "specialize polynomial_product_length_exists (L)", "specialize polynomial_product_length_exists (S M)", "apply polynomial_product_length_exists", "cases hl", "have hQ : exists qb qc. ((forall fom_index_pfp_append_exists_chosen_productleft. (exists fom_gap_pfp_append_exists_chosen_productleft_index_bound. fom_gap_pfp_append_exists_chosen_productleft_index_bound + S (fom_index_pfp_append_exists_chosen_productleft) = L) -> exists fom_value_pfp_append_exists_chosen_productleft. ((((exists fom_beta_height_pfp_append_exists_chosen_productleft_entry. fom_beta_height_pfp_append_exists_chosen_productleft_entry + S (fom_value_pfp_append_exists_chosen_productleft) = S ((S (fom_index_pfp_append_exists_chosen_productleft)) * ac)) /\\ exists fom_beta_quotient_pfp_append_exists_chosen_productleft_entry. ab = fom_beta_quotient_pfp_append_exists_chosen_productleft_entry * S ((S (fom_index_pfp_append_exists_chosen_productleft)) * ac) + (fom_value_pfp_append_exists_chosen_productleft))) /\\ (exists fom_gap_pfp_append_exists_chosen_productleft_value_bound. fom_gap_pfp_append_exists_chosen_productleft_value_bound + S (fom_value_pfp_append_exists_chosen_productleft) = p))) /\\ (((forall fom_index_pfp_append_exists_chosen_productright. (exists fom_gap_pfp_append_exists_chosen_productright_index_bound. fom_gap_pfp_append_exists_chosen_productright_index_bound + S (fom_index_pfp_append_exists_chosen_productright) = S M) -> exists fom_value_pfp_append_exists_chosen_productright. ((((exists fom_beta_height_pfp_append_exists_chosen_productright_entry. fom_beta_height_pfp_append_exists_chosen_productright_entry + S (fom_value_pfp_append_exists_chosen_productright) = S ((S (fom_index_pfp_append_exists_chosen_productright)) * x1)) /\\ exists fom_beta_quotient_pfp_append_exists_chosen_productright_entry. x = fom_beta_quotient_pfp_append_exists_chosen_productright_entry * S ((S (fom_index_pfp_append_exists_chosen_productright)) * x1) + (fom_value_pfp_append_exists_chosen_productright))) /\\ (exists fom_gap_pfp_append_exists_chosen_productright_value_bound. fom_gap_pfp_append_exists_chosen_productright_value_bound + S (fom_value_pfp_append_exists_chosen_productright) = p))) /\\ (((((((L)=0 \\/ (S M)=0) /\\ (((x2)=0)))) \\/ (((~((L)=0)) /\\ (((~((S M)=0)) /\\ (((L)+(S M)=S (x2)))))))) /\\ ((forall pfc_index_append_exists_chosen_productcoefficients. (exists pfa_gap_append_exists_chosen_productcoefficientsbound. pfa_gap_append_exists_chosen_productcoefficientsbound + S (pfc_index_append_exists_chosen_productcoefficients) = (x2)) -> exists pfc_value_append_exists_chosen_productcoefficients. ((((exists ff_h_pfp_append_exists_chosen_productcoefficientsentry. ff_h_pfp_append_exists_chosen_productcoefficientsentry + S (pfc_value_append_exists_chosen_productcoefficients) = S ((S (pfc_index_append_exists_chosen_productcoefficients)) * qc)) /\\ exists ff_q_pfp_append_exists_chosen_productcoefficientsentry. qb = ff_q_pfp_append_exists_chosen_productcoefficientsentry * S ((S (pfc_index_append_exists_chosen_productcoefficients)) * qc) + (pfc_value_append_exists_chosen_productcoefficients))) /\\ ((exists pfc_terms_code_append_exists_chosen_productcoefficientscoefficient pfc_terms_scale_append_exists_chosen_productcoefficientscoefficient pfc_natural_sum_append_exists_chosen_productcoefficientscoefficient. ((forall pfc_index_append_exists_chosen_productcoefficientscoefficientdiagonal. (exists pfa_gap_append_exists_chosen_productcoefficientscoefficientdiagonalbound. pfa_gap_append_exists_chosen_productcoefficientscoefficientdiagonalbound + S (pfc_index_append_exists_chosen_productcoefficientscoefficientdiagonal) = (S (pfc_index_append_exists_chosen_productcoefficients))) -> exists pfc_value_append_exists_chosen_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_exists_chosen_productcoefficientscoefficientdiagonalentry. ff_h_pfp_append_exists_chosen_productcoefficientscoefficientdiagonalentry + S (pfc_value_append_exists_chosen_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_append_exists_chosen_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_exists_chosen_productcoefficientscoefficient)) /\\ exists ff_q_pfp_append_exists_chosen_productcoefficientscoefficientdiagonalentry. pfc_terms_code_append_exists_chosen_productcoefficientscoefficient = ff_q_pfp_append_exists_chosen_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_append_exists_chosen_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_exists_chosen_productcoefficientscoefficient) + (pfc_value_append_exists_chosen_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_exists_chosen_productcoefficientscoefficientdiagonalterm pfc_left_append_exists_chosen_productcoefficientscoefficientdiagonalterm pfc_right_append_exists_chosen_productcoefficientscoefficientdiagonalterm. (((pfc_index_append_exists_chosen_productcoefficientscoefficientdiagonal)+pfc_complement_append_exists_chosen_productcoefficientscoefficientdiagonalterm=(pfc_index_append_exists_chosen_productcoefficients)) /\\ ((((((exists pfa_gap_append_exists_chosen_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_append_exists_chosen_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_exists_chosen_productcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_exists_chosen_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_exists_chosen_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_exists_chosen_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_exists_chosen_productcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_exists_chosen_productcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_exists_chosen_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_exists_chosen_productcoefficientscoefficientdiagonal)) * ac) + (pfc_left_append_exists_chosen_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_exists_chosen_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_exists_chosen_productcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_exists_chosen_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_append_exists_chosen_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_exists_chosen_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_append_exists_chosen_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_exists_chosen_productcoefficientscoefficientdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_append_exists_chosen_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_exists_chosen_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_exists_chosen_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_exists_chosen_productcoefficientscoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_append_exists_chosen_productcoefficientscoefficientdiagonaltermrightentry. x = ff_q_pfp_append_exists_chosen_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_exists_chosen_productcoefficientscoefficientdiagonalterm)) * x1) + (pfc_right_append_exists_chosen_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_exists_chosen_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_exists_chosen_productcoefficientscoefficientdiagonaltermrightoutside+(S M)=(pfc_complement_append_exists_chosen_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_exists_chosen_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_exists_chosen_productcoefficientscoefficientdiagonal)=pfc_left_append_exists_chosen_productcoefficientscoefficientdiagonalterm*pfc_right_append_exists_chosen_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_exists_chosen_productcoefficientscoefficientsum fs_v_pfc_append_exists_chosen_productcoefficientscoefficientsum. ((((exists fs_h_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_start. fs_h_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_exists_chosen_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_start. fs_u_pfc_append_exists_chosen_productcoefficientscoefficientsum = fs_q_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_exists_chosen_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_exists_chosen_productcoefficientscoefficient) = S ((S (S (pfc_index_append_exists_chosen_productcoefficients))) * fs_v_pfc_append_exists_chosen_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_append_exists_chosen_productcoefficientscoefficientsum = fs_q_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_exists_chosen_productcoefficients))) * fs_v_pfc_append_exists_chosen_productcoefficientscoefficientsum) + (pfc_natural_sum_append_exists_chosen_productcoefficientscoefficient))) /\\ forall fs_i_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps = S (pfc_index_append_exists_chosen_productcoefficients)) -> exists fs_a_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps fs_r_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps fs_s_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_exists_chosen_productcoefficientscoefficient)) /\\ exists fs_q_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_exists_chosen_productcoefficientscoefficient = fs_q_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_exists_chosen_productcoefficientscoefficient) + (fs_a_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_chosen_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_exists_chosen_productcoefficientscoefficientsum = fs_q_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_chosen_productcoefficientscoefficientsum) + (fs_r_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_chosen_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_exists_chosen_productcoefficientscoefficientsum = fs_q_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_chosen_productcoefficientscoefficientsum) + (fs_s_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps = fs_r_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps + fs_a_pfc_append_exists_chosen_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_exists_chosen_productcoefficientscoefficientresiduebound. pfa_gap_append_exists_chosen_productcoefficientscoefficientresiduebound + S (pfc_value_append_exists_chosen_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_append_exists_chosen_productcoefficientscoefficientresiduecongruence pfa_offset_right_append_exists_chosen_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_append_exists_chosen_productcoefficientscoefficient) + (p) * pfa_offset_left_append_exists_chosen_productcoefficientscoefficientresiduecongruence = (pfc_value_append_exists_chosen_productcoefficients) + (p) * pfa_offset_right_append_exists_chosen_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (ab)", "specialize prime_field_polynomial_convolution_at_length_exists (ac)", "specialize prime_field_polynomial_convolution_at_length_exists (L)", "specialize prime_field_polynomial_convolution_at_length_exists (x)", "specialize prime_field_polynomial_convolution_at_length_exists (x1)", "specialize prime_field_polynomial_convolution_at_length_exists (S M)", "specialize prime_field_polynomial_convolution_at_length_exists (x2)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hold_left", "exact hbounded", "exact hl_witness", "cases hQ", "cases hQ_witness", "have hA : exists ub uc vb vc UB UC VB VC rb rc. ((((forall mdr_i_pfp_append_exists_alignment_shiftprefix mdr_a_pfp_append_exists_alignment_shiftprefix. (exists mdr_gap_pfp_append_exists_alignment_shiftprefixb. mdr_gap_pfp_append_exists_alignment_shiftprefixb + S (mdr_i_pfp_append_exists_alignment_shiftprefix) = (N)) -> (((exists ff_h_mdr_pfp_append_exists_alignment_shiftprefixo. ff_h_mdr_pfp_append_exists_alignment_shiftprefixo + S (mdr_a_pfp_append_exists_alignment_shiftprefix) = S ((S (mdr_i_pfp_append_exists_alignment_shiftprefix)) * pc)) /\\ exists ff_q_mdr_pfp_append_exists_alignment_shiftprefixo. pb = ff_q_mdr_pfp_append_exists_alignment_shiftprefixo * S ((S (mdr_i_pfp_append_exists_alignment_shiftprefix)) * pc) + (mdr_a_pfp_append_exists_alignment_shiftprefix))) -> (((exists ff_h_mdr_pfp_append_exists_alignment_shiftprefixn. ff_h_mdr_pfp_append_exists_alignment_shiftprefixn + S (mdr_a_pfp_append_exists_alignment_shiftprefix) = S ((S (mdr_i_pfp_append_exists_alignment_shiftprefix)) * uc)) /\\ exists ff_q_mdr_pfp_append_exists_alignment_shiftprefixn. ub = ff_q_mdr_pfp_append_exists_alignment_shiftprefixn * S ((S (mdr_i_pfp_append_exists_alignment_shiftprefix)) * uc) + (mdr_a_pfp_append_exists_alignment_shiftprefix)))) /\\ ((((exists ff_h_pfp_append_exists_alignment_shiftlast. ff_h_pfp_append_exists_alignment_shiftlast + S (0) = S ((S (N)) * uc)) /\\ exists ff_q_pfp_append_exists_alignment_shiftlast. ub = ff_q_pfp_append_exists_alignment_shiftlast * S ((S (N)) * uc) + (0)))))) /\\ (((((exists pfa_gap_append_exists_alignment_scalescalar. pfa_gap_append_exists_alignment_scalescalar + S (c) = (p)) /\\ ((forall pfp_index_append_exists_alignment_scale. (exists pfa_gap_append_exists_alignment_scaleindex. pfa_gap_append_exists_alignment_scaleindex + S (pfp_index_append_exists_alignment_scale) = (L)) -> exists pfp_source_append_exists_alignment_scale pfp_value_append_exists_alignment_scale. ((((exists ff_h_pfp_append_exists_alignment_scalesource. ff_h_pfp_append_exists_alignment_scalesource + S (pfp_source_append_exists_alignment_scale) = S ((S (pfp_index_append_exists_alignment_scale)) * ac)) /\\ exists ff_q_pfp_append_exists_alignment_scalesource. ab = ff_q_pfp_append_exists_alignment_scalesource * S ((S (pfp_index_append_exists_alignment_scale)) * ac) + (pfp_source_append_exists_alignment_scale))) /\\ (((((exists ff_h_pfp_append_exists_alignment_scaletarget. ff_h_pfp_append_exists_alignment_scaletarget + S (pfp_value_append_exists_alignment_scale) = S ((S (pfp_index_append_exists_alignment_scale)) * vc)) /\\ exists ff_q_pfp_append_exists_alignment_scaletarget. vb = ff_q_pfp_append_exists_alignment_scaletarget * S ((S (pfp_index_append_exists_alignment_scale)) * vc) + (pfp_value_append_exists_alignment_scale))) /\\ ((((exists pfa_gap_append_exists_alignment_scaleoperationleft. pfa_gap_append_exists_alignment_scaleoperationleft + S (c) = (p)) /\\ (((exists pfa_gap_append_exists_alignment_scaleoperationright. pfa_gap_append_exists_alignment_scaleoperationright + S (pfp_source_append_exists_alignment_scale) = (p)) /\\ ((((exists pfa_gap_append_exists_alignment_scaleoperationresultbound. pfa_gap_append_exists_alignment_scaleoperationresultbound + S (pfp_value_append_exists_alignment_scale) = (p)) /\\ ((exists pfa_offset_left_append_exists_alignment_scaleoperationresultcongruence pfa_offset_right_append_exists_alignment_scaleoperationresultcongruence. ((c) * (pfp_source_append_exists_alignment_scale)) + (p) * pfa_offset_left_append_exists_alignment_scaleoperationresultcongruence = (pfp_value_append_exists_alignment_scale) + (p) * pfa_offset_right_append_exists_alignment_scaleoperationresultcongruence))))))))))))))))) /\\ (((((forall pfp_repeat_index_append_exists_alignment_leftzeros. (exists pfa_gap_append_exists_alignment_leftzerosindex. pfa_gap_append_exists_alignment_leftzerosindex + S (pfp_repeat_index_append_exists_alignment_leftzeros) = (L)) -> (((exists ff_h_pfp_append_exists_alignment_leftzerosentry. ff_h_pfp_append_exists_alignment_leftzerosentry + S (0) = S ((S (pfp_repeat_index_append_exists_alignment_leftzeros)) * UC)) /\\ exists ff_q_pfp_append_exists_alignment_leftzerosentry. UB = ff_q_pfp_append_exists_alignment_leftzerosentry * S ((S (pfp_repeat_index_append_exists_alignment_leftzeros)) * UC) + (0)))) /\\ ((forall pfrep_index_append_exists_alignment_left pfrep_value_append_exists_alignment_left. (exists pfa_gap_append_exists_alignment_leftbound. pfa_gap_append_exists_alignment_leftbound + S (pfrep_index_append_exists_alignment_left) = (S N)) -> (((exists ff_h_pfp_append_exists_alignment_leftinput. ff_h_pfp_append_exists_alignment_leftinput + S (pfrep_value_append_exists_alignment_left) = S ((S (pfrep_index_append_exists_alignment_left)) * uc)) /\\ exists ff_q_pfp_append_exists_alignment_leftinput. ub = ff_q_pfp_append_exists_alignment_leftinput * S ((S (pfrep_index_append_exists_alignment_left)) * uc) + (pfrep_value_append_exists_alignment_left))) -> (((exists ff_h_pfp_append_exists_alignment_leftoutput. ff_h_pfp_append_exists_alignment_leftoutput + S (pfrep_value_append_exists_alignment_left) = S ((S ((L)+pfrep_index_append_exists_alignment_left)) * UC)) /\\ exists ff_q_pfp_append_exists_alignment_leftoutput. UB = ff_q_pfp_append_exists_alignment_leftoutput * S ((S ((L)+pfrep_index_append_exists_alignment_left)) * UC) + (pfrep_value_append_exists_alignment_left))))))) /\\ (((((forall pfp_repeat_index_append_exists_alignment_rightzeros. (exists pfa_gap_append_exists_alignment_rightzerosindex. pfa_gap_append_exists_alignment_rightzerosindex + S (pfp_repeat_index_append_exists_alignment_rightzeros) = (S N)) -> (((exists ff_h_pfp_append_exists_alignment_rightzerosentry. ff_h_pfp_append_exists_alignment_rightzerosentry + S (0) = S ((S (pfp_repeat_index_append_exists_alignment_rightzeros)) * VC)) /\\ exists ff_q_pfp_append_exists_alignment_rightzerosentry. VB = ff_q_pfp_append_exists_alignment_rightzerosentry * S ((S (pfp_repeat_index_append_exists_alignment_rightzeros)) * VC) + (0)))) /\\ ((forall pfrep_index_append_exists_alignment_right pfrep_value_append_exists_alignment_right. (exists pfa_gap_append_exists_alignment_rightbound. pfa_gap_append_exists_alignment_rightbound + S (pfrep_index_append_exists_alignment_right) = (L)) -> (((exists ff_h_pfp_append_exists_alignment_rightinput. ff_h_pfp_append_exists_alignment_rightinput + S (pfrep_value_append_exists_alignment_right) = S ((S (pfrep_index_append_exists_alignment_right)) * vc)) /\\ exists ff_q_pfp_append_exists_alignment_rightinput. vb = ff_q_pfp_append_exists_alignment_rightinput * S ((S (pfrep_index_append_exists_alignment_right)) * vc) + (pfrep_value_append_exists_alignment_right))) -> (((exists ff_h_pfp_append_exists_alignment_rightoutput. ff_h_pfp_append_exists_alignment_rightoutput + S (pfrep_value_append_exists_alignment_right) = S ((S ((S N)+pfrep_index_append_exists_alignment_right)) * VC)) /\\ exists ff_q_pfp_append_exists_alignment_rightoutput. VB = ff_q_pfp_append_exists_alignment_rightoutput * S ((S ((S N)+pfrep_index_append_exists_alignment_right)) * VC) + (pfrep_value_append_exists_alignment_right))))))) /\\ ((forall pfp_index_append_exists_alignment_sum. (exists pfa_gap_append_exists_alignment_sumindex. pfa_gap_append_exists_alignment_sumindex + S (pfp_index_append_exists_alignment_sum) = (L+S N)) -> exists pfp_left_append_exists_alignment_sum pfp_right_append_exists_alignment_sum pfp_value_append_exists_alignment_sum. ((((exists ff_h_pfp_append_exists_alignment_sumleft. ff_h_pfp_append_exists_alignment_sumleft + S (pfp_left_append_exists_alignment_sum) = S ((S (pfp_index_append_exists_alignment_sum)) * UC)) /\\ exists ff_q_pfp_append_exists_alignment_sumleft. UB = ff_q_pfp_append_exists_alignment_sumleft * S ((S (pfp_index_append_exists_alignment_sum)) * UC) + (pfp_left_append_exists_alignment_sum))) /\\ (((((exists ff_h_pfp_append_exists_alignment_sumright. ff_h_pfp_append_exists_alignment_sumright + S (pfp_right_append_exists_alignment_sum) = S ((S (pfp_index_append_exists_alignment_sum)) * VC)) /\\ exists ff_q_pfp_append_exists_alignment_sumright. VB = ff_q_pfp_append_exists_alignment_sumright * S ((S (pfp_index_append_exists_alignment_sum)) * VC) + (pfp_right_append_exists_alignment_sum))) /\\ (((((exists ff_h_pfp_append_exists_alignment_sumtarget. ff_h_pfp_append_exists_alignment_sumtarget + S (pfp_value_append_exists_alignment_sum) = S ((S (pfp_index_append_exists_alignment_sum)) * rc)) /\\ exists ff_q_pfp_append_exists_alignment_sumtarget. rb = ff_q_pfp_append_exists_alignment_sumtarget * S ((S (pfp_index_append_exists_alignment_sum)) * rc) + (pfp_value_append_exists_alignment_sum))) /\\ ((((exists pfa_gap_append_exists_alignment_sumoperationleft. pfa_gap_append_exists_alignment_sumoperationleft + S (pfp_left_append_exists_alignment_sum) = (p)) /\\ (((exists pfa_gap_append_exists_alignment_sumoperationright. pfa_gap_append_exists_alignment_sumoperationright + S (pfp_right_append_exists_alignment_sum) = (p)) /\\ ((((exists pfa_gap_append_exists_alignment_sumoperationresultbound. pfa_gap_append_exists_alignment_sumoperationresultbound + S (pfp_value_append_exists_alignment_sum) = (p)) /\\ ((exists pfa_offset_left_append_exists_alignment_sumoperationresultcongruence pfa_offset_right_append_exists_alignment_sumoperationresultcongruence. ((pfp_left_append_exists_alignment_sum) + (pfp_right_append_exists_alignment_sum)) + (p) * pfa_offset_left_append_exists_alignment_sumoperationresultcongruence = (pfp_value_append_exists_alignment_sum) + (p) * pfa_offset_right_append_exists_alignment_sumoperationresultcongruence))))))))))))))))))))))))", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (p)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (c)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (ab)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (ac)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (L)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (pb)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (pc)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (N)", "apply prime_field_polynomial_shift_scale_aligned_sum_exists", "exact hp", "exact hc", "exact hold_left", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (ab)", "specialize prime_field_polynomial_convolution_bounded (ac)", "specialize prime_field_polynomial_convolution_bounded (L)", "specialize prime_field_polynomial_convolution_bounded (bb)", "specialize prime_field_polynomial_convolution_bounded (bc)", "specialize prime_field_polynomial_convolution_bounded (M)", "specialize prime_field_polynomial_convolution_bounded (pb)", "specialize prime_field_polynomial_convolution_bounded (pc)", "specialize prime_field_polynomial_convolution_bounded (N)", "apply prime_field_polynomial_convolution_bounded", "exact hP", "cases hA", "cases hA_witness", "cases hA_witness_witness", "cases hA_witness_witness_witness", "cases hA_witness_witness_witness_witness", "cases hA_witness_witness_witness_witness_witness", "cases hA_witness_witness_witness_witness_witness_witness", "cases hA_witness_witness_witness_witness_witness_witness_witness", "cases hA_witness_witness_witness_witness_witness_witness_witness_witness", "cases hA_witness_witness_witness_witness_witness_witness_witness_witness_witness", "cases hA_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness", "cases hA_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right", "cases hA_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right", "cases hA_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "exists x6", "exists x7", "exists x8", "exists x9", "exists x10", "exists x11", "exists x12", "exists x13", "exists x14", "split", "exact hd_witness_witness_right", "split", "exact hd_witness_witness_left", "split", "exact hQ_witness_witness", "split", "exact hA_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_left", "split", "exact hA_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "split", "exact hA_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left", "split", "exact hA_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "split", "exact hA_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "specialize prime_field_polynomial_convolution_right_append_equivalent (p)", "specialize prime_field_polynomial_convolution_right_append_equivalent (ab)", "specialize prime_field_polynomial_convolution_right_append_equivalent (ac)", "specialize prime_field_polynomial_convolution_right_append_equivalent (L)", "specialize prime_field_polynomial_convolution_right_append_equivalent (bb)", "specialize prime_field_polynomial_convolution_right_append_equivalent (bc)", "specialize prime_field_polynomial_convolution_right_append_equivalent (M)", "specialize prime_field_polynomial_convolution_right_append_equivalent (c)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x1)", "specialize prime_field_polynomial_convolution_right_append_equivalent (pb)", "specialize prime_field_polynomial_convolution_right_append_equivalent (pc)", "specialize prime_field_polynomial_convolution_right_append_equivalent (N)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x3)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x4)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x2)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x5)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x6)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x7)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x8)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x9)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x10)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x11)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x12)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x13)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x14)", "apply prime_field_polynomial_convolution_right_append_equivalent", "exact hp", "exact hd_witness_witness_right", "exact hd_witness_witness_left", "exact hP", "exact hQ_witness_witness", "exact hA_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_left", "exact hA_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "exact hA_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left", "exact hA_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "exact hA_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right" ], "script_sha256": "ab00fef15c3a9433413763176d17e1a3fe00d546071fc4cb7de06340e5e2fb96", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_append_candidate.py", "sha256": "271845bfffc7e513fdb0bd0c3666dcccace8436d4d3a0f4db64b67bcd4b87042" }, "statement": "forall p ab ac L bb bc M c pb pc N. (~((p) = 1) /\\ forall pfa_factor_left_append_exists_prime pfa_factor_right_append_exists_prime. (p) = pfa_factor_left_append_exists_prime * pfa_factor_right_append_exists_prime -> pfa_factor_left_append_exists_prime = 1 \\/ pfa_factor_right_append_exists_prime = 1) -> (exists pfa_gap_append_exists_scalar. pfa_gap_append_exists_scalar + S (c) = (p)) -> (((forall fom_index_pfp_append_exists_oldleft. (exists fom_gap_pfp_append_exists_oldleft_index_bound. fom_gap_pfp_append_exists_oldleft_index_bound + S (fom_index_pfp_append_exists_oldleft) = L) -> exists fom_value_pfp_append_exists_oldleft. ((((exists fom_beta_height_pfp_append_exists_oldleft_entry. fom_beta_height_pfp_append_exists_oldleft_entry + S (fom_value_pfp_append_exists_oldleft) = S ((S (fom_index_pfp_append_exists_oldleft)) * ac)) /\\ exists fom_beta_quotient_pfp_append_exists_oldleft_entry. ab = fom_beta_quotient_pfp_append_exists_oldleft_entry * S ((S (fom_index_pfp_append_exists_oldleft)) * ac) + (fom_value_pfp_append_exists_oldleft))) /\\ (exists fom_gap_pfp_append_exists_oldleft_value_bound. fom_gap_pfp_append_exists_oldleft_value_bound + S (fom_value_pfp_append_exists_oldleft) = p))) /\\ (((forall fom_index_pfp_append_exists_oldright. (exists fom_gap_pfp_append_exists_oldright_index_bound. fom_gap_pfp_append_exists_oldright_index_bound + S (fom_index_pfp_append_exists_oldright) = M) -> exists fom_value_pfp_append_exists_oldright. ((((exists fom_beta_height_pfp_append_exists_oldright_entry. fom_beta_height_pfp_append_exists_oldright_entry + S (fom_value_pfp_append_exists_oldright) = S ((S (fom_index_pfp_append_exists_oldright)) * bc)) /\\ exists fom_beta_quotient_pfp_append_exists_oldright_entry. bb = fom_beta_quotient_pfp_append_exists_oldright_entry * S ((S (fom_index_pfp_append_exists_oldright)) * bc) + (fom_value_pfp_append_exists_oldright))) /\\ (exists fom_gap_pfp_append_exists_oldright_value_bound. fom_gap_pfp_append_exists_oldright_value_bound + S (fom_value_pfp_append_exists_oldright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_append_exists_oldcoefficients. (exists pfa_gap_append_exists_oldcoefficientsbound. pfa_gap_append_exists_oldcoefficientsbound + S (pfc_index_append_exists_oldcoefficients) = (N)) -> exists pfc_value_append_exists_oldcoefficients. ((((exists ff_h_pfp_append_exists_oldcoefficientsentry. ff_h_pfp_append_exists_oldcoefficientsentry + S (pfc_value_append_exists_oldcoefficients) = S ((S (pfc_index_append_exists_oldcoefficients)) * pc)) /\\ exists ff_q_pfp_append_exists_oldcoefficientsentry. pb = ff_q_pfp_append_exists_oldcoefficientsentry * S ((S (pfc_index_append_exists_oldcoefficients)) * pc) + (pfc_value_append_exists_oldcoefficients))) /\\ ((exists pfc_terms_code_append_exists_oldcoefficientscoefficient pfc_terms_scale_append_exists_oldcoefficientscoefficient pfc_natural_sum_append_exists_oldcoefficientscoefficient. ((forall pfc_index_append_exists_oldcoefficientscoefficientdiagonal. (exists pfa_gap_append_exists_oldcoefficientscoefficientdiagonalbound. pfa_gap_append_exists_oldcoefficientscoefficientdiagonalbound + S (pfc_index_append_exists_oldcoefficientscoefficientdiagonal) = (S (pfc_index_append_exists_oldcoefficients))) -> exists pfc_value_append_exists_oldcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_exists_oldcoefficientscoefficientdiagonalentry. ff_h_pfp_append_exists_oldcoefficientscoefficientdiagonalentry + S (pfc_value_append_exists_oldcoefficientscoefficientdiagonal) = S ((S (pfc_index_append_exists_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_exists_oldcoefficientscoefficient)) /\\ exists ff_q_pfp_append_exists_oldcoefficientscoefficientdiagonalentry. pfc_terms_code_append_exists_oldcoefficientscoefficient = ff_q_pfp_append_exists_oldcoefficientscoefficientdiagonalentry * S ((S (pfc_index_append_exists_oldcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_exists_oldcoefficientscoefficient) + (pfc_value_append_exists_oldcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_exists_oldcoefficientscoefficientdiagonalterm pfc_left_append_exists_oldcoefficientscoefficientdiagonalterm pfc_right_append_exists_oldcoefficientscoefficientdiagonalterm. (((pfc_index_append_exists_oldcoefficientscoefficientdiagonal)+pfc_complement_append_exists_oldcoefficientscoefficientdiagonalterm=(pfc_index_append_exists_oldcoefficients)) /\\ ((((((exists pfa_gap_append_exists_oldcoefficientscoefficientdiagonaltermleftinside. pfa_gap_append_exists_oldcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_exists_oldcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_exists_oldcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_exists_oldcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_exists_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_exists_oldcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_exists_oldcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_exists_oldcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_exists_oldcoefficientscoefficientdiagonal)) * ac) + (pfc_left_append_exists_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_exists_oldcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_exists_oldcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_exists_oldcoefficientscoefficientdiagonal)) /\\ (((pfc_left_append_exists_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_exists_oldcoefficientscoefficientdiagonaltermrightinside. pfa_gap_append_exists_oldcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_exists_oldcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_append_exists_oldcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_exists_oldcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_exists_oldcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_exists_oldcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_append_exists_oldcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_append_exists_oldcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_exists_oldcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_append_exists_oldcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_exists_oldcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_exists_oldcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_append_exists_oldcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_exists_oldcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_exists_oldcoefficientscoefficientdiagonal)=pfc_left_append_exists_oldcoefficientscoefficientdiagonalterm*pfc_right_append_exists_oldcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_exists_oldcoefficientscoefficientsum fs_v_pfc_append_exists_oldcoefficientscoefficientsum. ((((exists fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_start. fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_start. fs_u_pfc_append_exists_oldcoefficientscoefficientsum = fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_terminal. fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_exists_oldcoefficientscoefficient) = S ((S (S (pfc_index_append_exists_oldcoefficients))) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_terminal. fs_u_pfc_append_exists_oldcoefficientscoefficientsum = fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_exists_oldcoefficients))) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum) + (pfc_natural_sum_append_exists_oldcoefficientscoefficient))) /\\ forall fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps = S (pfc_index_append_exists_oldcoefficients)) -> exists fs_a_pfc_append_exists_oldcoefficientscoefficientsum_body_steps fs_r_pfc_append_exists_oldcoefficientscoefficientsum_body_steps fs_s_pfc_append_exists_oldcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_exists_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_exists_oldcoefficientscoefficient)) /\\ exists fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_exists_oldcoefficientscoefficient = fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_exists_oldcoefficientscoefficient) + (fs_a_pfc_append_exists_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_exists_oldcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_exists_oldcoefficientscoefficientsum = fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum) + (fs_r_pfc_append_exists_oldcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_exists_oldcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_exists_oldcoefficientscoefficientsum = fs_q_pfc_append_exists_oldcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_exists_oldcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_oldcoefficientscoefficientsum) + (fs_s_pfc_append_exists_oldcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_exists_oldcoefficientscoefficientsum_body_steps = fs_r_pfc_append_exists_oldcoefficientscoefficientsum_body_steps + fs_a_pfc_append_exists_oldcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_exists_oldcoefficientscoefficientresiduebound. pfa_gap_append_exists_oldcoefficientscoefficientresiduebound + S (pfc_value_append_exists_oldcoefficients) = (p)) /\\ ((exists pfa_offset_left_append_exists_oldcoefficientscoefficientresiduecongruence pfa_offset_right_append_exists_oldcoefficientscoefficientresiduecongruence. (pfc_natural_sum_append_exists_oldcoefficientscoefficient) + (p) * pfa_offset_left_append_exists_oldcoefficientscoefficientresiduecongruence = (pfc_value_append_exists_oldcoefficients) + (p) * pfa_offset_right_append_exists_oldcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (exists db dc K qb qc ub uc vb vc UB UC VB VC rb rc. ((forall mdr_i_pfp_append_exists_preserve mdr_a_pfp_append_exists_preserve. (exists mdr_gap_pfp_append_exists_preserveb. mdr_gap_pfp_append_exists_preserveb + S (mdr_i_pfp_append_exists_preserve) = (M)) -> (((exists ff_h_mdr_pfp_append_exists_preserveo. ff_h_mdr_pfp_append_exists_preserveo + S (mdr_a_pfp_append_exists_preserve) = S ((S (mdr_i_pfp_append_exists_preserve)) * bc)) /\\ exists ff_q_mdr_pfp_append_exists_preserveo. bb = ff_q_mdr_pfp_append_exists_preserveo * S ((S (mdr_i_pfp_append_exists_preserve)) * bc) + (mdr_a_pfp_append_exists_preserve))) -> (((exists ff_h_mdr_pfp_append_exists_preserven. ff_h_mdr_pfp_append_exists_preserven + S (mdr_a_pfp_append_exists_preserve) = S ((S (mdr_i_pfp_append_exists_preserve)) * dc)) /\\ exists ff_q_mdr_pfp_append_exists_preserven. db = ff_q_mdr_pfp_append_exists_preserven * S ((S (mdr_i_pfp_append_exists_preserve)) * dc) + (mdr_a_pfp_append_exists_preserve)))) /\\ (((((exists ff_h_pfp_append_exists_last. ff_h_pfp_append_exists_last + S (c) = S ((S (M)) * dc)) /\\ exists ff_q_pfp_append_exists_last. db = ff_q_pfp_append_exists_last * S ((S (M)) * dc) + (c))) /\\ (((((forall fom_index_pfp_append_exists_newleft. (exists fom_gap_pfp_append_exists_newleft_index_bound. fom_gap_pfp_append_exists_newleft_index_bound + S (fom_index_pfp_append_exists_newleft) = L) -> exists fom_value_pfp_append_exists_newleft. ((((exists fom_beta_height_pfp_append_exists_newleft_entry. fom_beta_height_pfp_append_exists_newleft_entry + S (fom_value_pfp_append_exists_newleft) = S ((S (fom_index_pfp_append_exists_newleft)) * ac)) /\\ exists fom_beta_quotient_pfp_append_exists_newleft_entry. ab = fom_beta_quotient_pfp_append_exists_newleft_entry * S ((S (fom_index_pfp_append_exists_newleft)) * ac) + (fom_value_pfp_append_exists_newleft))) /\\ (exists fom_gap_pfp_append_exists_newleft_value_bound. fom_gap_pfp_append_exists_newleft_value_bound + S (fom_value_pfp_append_exists_newleft) = p))) /\\ (((forall fom_index_pfp_append_exists_newright. (exists fom_gap_pfp_append_exists_newright_index_bound. fom_gap_pfp_append_exists_newright_index_bound + S (fom_index_pfp_append_exists_newright) = S M) -> exists fom_value_pfp_append_exists_newright. ((((exists fom_beta_height_pfp_append_exists_newright_entry. fom_beta_height_pfp_append_exists_newright_entry + S (fom_value_pfp_append_exists_newright) = S ((S (fom_index_pfp_append_exists_newright)) * dc)) /\\ exists fom_beta_quotient_pfp_append_exists_newright_entry. db = fom_beta_quotient_pfp_append_exists_newright_entry * S ((S (fom_index_pfp_append_exists_newright)) * dc) + (fom_value_pfp_append_exists_newright))) /\\ (exists fom_gap_pfp_append_exists_newright_value_bound. fom_gap_pfp_append_exists_newright_value_bound + S (fom_value_pfp_append_exists_newright) = p))) /\\ (((((((L)=0 \\/ (S M)=0) /\\ (((K)=0)))) \\/ (((~((L)=0)) /\\ (((~((S M)=0)) /\\ (((L)+(S M)=S (K)))))))) /\\ ((forall pfc_index_append_exists_newcoefficients. (exists pfa_gap_append_exists_newcoefficientsbound. pfa_gap_append_exists_newcoefficientsbound + S (pfc_index_append_exists_newcoefficients) = (K)) -> exists pfc_value_append_exists_newcoefficients. ((((exists ff_h_pfp_append_exists_newcoefficientsentry. ff_h_pfp_append_exists_newcoefficientsentry + S (pfc_value_append_exists_newcoefficients) = S ((S (pfc_index_append_exists_newcoefficients)) * qc)) /\\ exists ff_q_pfp_append_exists_newcoefficientsentry. qb = ff_q_pfp_append_exists_newcoefficientsentry * S ((S (pfc_index_append_exists_newcoefficients)) * qc) + (pfc_value_append_exists_newcoefficients))) /\\ ((exists pfc_terms_code_append_exists_newcoefficientscoefficient pfc_terms_scale_append_exists_newcoefficientscoefficient pfc_natural_sum_append_exists_newcoefficientscoefficient. ((forall pfc_index_append_exists_newcoefficientscoefficientdiagonal. (exists pfa_gap_append_exists_newcoefficientscoefficientdiagonalbound. pfa_gap_append_exists_newcoefficientscoefficientdiagonalbound + S (pfc_index_append_exists_newcoefficientscoefficientdiagonal) = (S (pfc_index_append_exists_newcoefficients))) -> exists pfc_value_append_exists_newcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_exists_newcoefficientscoefficientdiagonalentry. ff_h_pfp_append_exists_newcoefficientscoefficientdiagonalentry + S (pfc_value_append_exists_newcoefficientscoefficientdiagonal) = S ((S (pfc_index_append_exists_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_exists_newcoefficientscoefficient)) /\\ exists ff_q_pfp_append_exists_newcoefficientscoefficientdiagonalentry. pfc_terms_code_append_exists_newcoefficientscoefficient = ff_q_pfp_append_exists_newcoefficientscoefficientdiagonalentry * S ((S (pfc_index_append_exists_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_exists_newcoefficientscoefficient) + (pfc_value_append_exists_newcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_exists_newcoefficientscoefficientdiagonalterm pfc_left_append_exists_newcoefficientscoefficientdiagonalterm pfc_right_append_exists_newcoefficientscoefficientdiagonalterm. (((pfc_index_append_exists_newcoefficientscoefficientdiagonal)+pfc_complement_append_exists_newcoefficientscoefficientdiagonalterm=(pfc_index_append_exists_newcoefficients)) /\\ ((((((exists pfa_gap_append_exists_newcoefficientscoefficientdiagonaltermleftinside. pfa_gap_append_exists_newcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_exists_newcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_exists_newcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_exists_newcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_exists_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_exists_newcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_exists_newcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_exists_newcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_exists_newcoefficientscoefficientdiagonal)) * ac) + (pfc_left_append_exists_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_exists_newcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_exists_newcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_exists_newcoefficientscoefficientdiagonal)) /\\ (((pfc_left_append_exists_newcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_exists_newcoefficientscoefficientdiagonaltermrightinside. pfa_gap_append_exists_newcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_exists_newcoefficientscoefficientdiagonalterm) = (S M)) /\\ ((((exists ff_h_pfp_append_exists_newcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_exists_newcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_exists_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_exists_newcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_append_exists_newcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_append_exists_newcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_exists_newcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_append_exists_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_exists_newcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_exists_newcoefficientscoefficientdiagonaltermrightoutside+(S M)=(pfc_complement_append_exists_newcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_exists_newcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_exists_newcoefficientscoefficientdiagonal)=pfc_left_append_exists_newcoefficientscoefficientdiagonalterm*pfc_right_append_exists_newcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_exists_newcoefficientscoefficientsum fs_v_pfc_append_exists_newcoefficientscoefficientsum. ((((exists fs_h_pfc_append_exists_newcoefficientscoefficientsum_body_start. fs_h_pfc_append_exists_newcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_exists_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_newcoefficientscoefficientsum_body_start. fs_u_pfc_append_exists_newcoefficientscoefficientsum = fs_q_pfc_append_exists_newcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_exists_newcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_exists_newcoefficientscoefficientsum_body_terminal. fs_h_pfc_append_exists_newcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_exists_newcoefficientscoefficient) = S ((S (S (pfc_index_append_exists_newcoefficients))) * fs_v_pfc_append_exists_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_newcoefficientscoefficientsum_body_terminal. fs_u_pfc_append_exists_newcoefficientscoefficientsum = fs_q_pfc_append_exists_newcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_exists_newcoefficients))) * fs_v_pfc_append_exists_newcoefficientscoefficientsum) + (pfc_natural_sum_append_exists_newcoefficientscoefficient))) /\\ forall fs_i_pfc_append_exists_newcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_exists_newcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_exists_newcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_exists_newcoefficientscoefficientsum_body_steps = S (pfc_index_append_exists_newcoefficients)) -> exists fs_a_pfc_append_exists_newcoefficientscoefficientsum_body_steps fs_r_pfc_append_exists_newcoefficientscoefficientsum_body_steps fs_s_pfc_append_exists_newcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_exists_newcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_exists_newcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_exists_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_exists_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_exists_newcoefficientscoefficient)) /\\ exists fs_q_pfc_append_exists_newcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_exists_newcoefficientscoefficient = fs_q_pfc_append_exists_newcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_exists_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_exists_newcoefficientscoefficient) + (fs_a_pfc_append_exists_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_exists_newcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_exists_newcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_exists_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_exists_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_newcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_exists_newcoefficientscoefficientsum = fs_q_pfc_append_exists_newcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_exists_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_newcoefficientscoefficientsum) + (fs_r_pfc_append_exists_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_exists_newcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_exists_newcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_exists_newcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_exists_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_exists_newcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_exists_newcoefficientscoefficientsum = fs_q_pfc_append_exists_newcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_exists_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_exists_newcoefficientscoefficientsum) + (fs_s_pfc_append_exists_newcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_exists_newcoefficientscoefficientsum_body_steps = fs_r_pfc_append_exists_newcoefficientscoefficientsum_body_steps + fs_a_pfc_append_exists_newcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_exists_newcoefficientscoefficientresiduebound. pfa_gap_append_exists_newcoefficientscoefficientresiduebound + S (pfc_value_append_exists_newcoefficients) = (p)) /\\ ((exists pfa_offset_left_append_exists_newcoefficientscoefficientresiduecongruence pfa_offset_right_append_exists_newcoefficientscoefficientresiduecongruence. (pfc_natural_sum_append_exists_newcoefficientscoefficient) + (p) * pfa_offset_left_append_exists_newcoefficientscoefficientresiduecongruence = (pfc_value_append_exists_newcoefficients) + (p) * pfa_offset_right_append_exists_newcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall mdr_i_pfp_append_exists_result_shiftprefix mdr_a_pfp_append_exists_result_shiftprefix. (exists mdr_gap_pfp_append_exists_result_shiftprefixb. mdr_gap_pfp_append_exists_result_shiftprefixb + S (mdr_i_pfp_append_exists_result_shiftprefix) = (N)) -> (((exists ff_h_mdr_pfp_append_exists_result_shiftprefixo. ff_h_mdr_pfp_append_exists_result_shiftprefixo + S (mdr_a_pfp_append_exists_result_shiftprefix) = S ((S (mdr_i_pfp_append_exists_result_shiftprefix)) * pc)) /\\ exists ff_q_mdr_pfp_append_exists_result_shiftprefixo. pb = ff_q_mdr_pfp_append_exists_result_shiftprefixo * S ((S (mdr_i_pfp_append_exists_result_shiftprefix)) * pc) + (mdr_a_pfp_append_exists_result_shiftprefix))) -> (((exists ff_h_mdr_pfp_append_exists_result_shiftprefixn. ff_h_mdr_pfp_append_exists_result_shiftprefixn + S (mdr_a_pfp_append_exists_result_shiftprefix) = S ((S (mdr_i_pfp_append_exists_result_shiftprefix)) * uc)) /\\ exists ff_q_mdr_pfp_append_exists_result_shiftprefixn. ub = ff_q_mdr_pfp_append_exists_result_shiftprefixn * S ((S (mdr_i_pfp_append_exists_result_shiftprefix)) * uc) + (mdr_a_pfp_append_exists_result_shiftprefix)))) /\\ ((((exists ff_h_pfp_append_exists_result_shiftlast. ff_h_pfp_append_exists_result_shiftlast + S (0) = S ((S (N)) * uc)) /\\ exists ff_q_pfp_append_exists_result_shiftlast. ub = ff_q_pfp_append_exists_result_shiftlast * S ((S (N)) * uc) + (0)))))) /\\ (((((exists pfa_gap_append_exists_result_scalescalar. pfa_gap_append_exists_result_scalescalar + S (c) = (p)) /\\ ((forall pfp_index_append_exists_result_scale. (exists pfa_gap_append_exists_result_scaleindex. pfa_gap_append_exists_result_scaleindex + S (pfp_index_append_exists_result_scale) = (L)) -> exists pfp_source_append_exists_result_scale pfp_value_append_exists_result_scale. ((((exists ff_h_pfp_append_exists_result_scalesource. ff_h_pfp_append_exists_result_scalesource + S (pfp_source_append_exists_result_scale) = S ((S (pfp_index_append_exists_result_scale)) * ac)) /\\ exists ff_q_pfp_append_exists_result_scalesource. ab = ff_q_pfp_append_exists_result_scalesource * S ((S (pfp_index_append_exists_result_scale)) * ac) + (pfp_source_append_exists_result_scale))) /\\ (((((exists ff_h_pfp_append_exists_result_scaletarget. ff_h_pfp_append_exists_result_scaletarget + S (pfp_value_append_exists_result_scale) = S ((S (pfp_index_append_exists_result_scale)) * vc)) /\\ exists ff_q_pfp_append_exists_result_scaletarget. vb = ff_q_pfp_append_exists_result_scaletarget * S ((S (pfp_index_append_exists_result_scale)) * vc) + (pfp_value_append_exists_result_scale))) /\\ ((((exists pfa_gap_append_exists_result_scaleoperationleft. pfa_gap_append_exists_result_scaleoperationleft + S (c) = (p)) /\\ (((exists pfa_gap_append_exists_result_scaleoperationright. pfa_gap_append_exists_result_scaleoperationright + S (pfp_source_append_exists_result_scale) = (p)) /\\ ((((exists pfa_gap_append_exists_result_scaleoperationresultbound. pfa_gap_append_exists_result_scaleoperationresultbound + S (pfp_value_append_exists_result_scale) = (p)) /\\ ((exists pfa_offset_left_append_exists_result_scaleoperationresultcongruence pfa_offset_right_append_exists_result_scaleoperationresultcongruence. ((c) * (pfp_source_append_exists_result_scale)) + (p) * pfa_offset_left_append_exists_result_scaleoperationresultcongruence = (pfp_value_append_exists_result_scale) + (p) * pfa_offset_right_append_exists_result_scaleoperationresultcongruence))))))))))))))))) /\\ (((((forall pfp_repeat_index_append_exists_result_leftzeros. (exists pfa_gap_append_exists_result_leftzerosindex. pfa_gap_append_exists_result_leftzerosindex + S (pfp_repeat_index_append_exists_result_leftzeros) = (L)) -> (((exists ff_h_pfp_append_exists_result_leftzerosentry. ff_h_pfp_append_exists_result_leftzerosentry + S (0) = S ((S (pfp_repeat_index_append_exists_result_leftzeros)) * UC)) /\\ exists ff_q_pfp_append_exists_result_leftzerosentry. UB = ff_q_pfp_append_exists_result_leftzerosentry * S ((S (pfp_repeat_index_append_exists_result_leftzeros)) * UC) + (0)))) /\\ ((forall pfrep_index_append_exists_result_left pfrep_value_append_exists_result_left. (exists pfa_gap_append_exists_result_leftbound. pfa_gap_append_exists_result_leftbound + S (pfrep_index_append_exists_result_left) = (S N)) -> (((exists ff_h_pfp_append_exists_result_leftinput. ff_h_pfp_append_exists_result_leftinput + S (pfrep_value_append_exists_result_left) = S ((S (pfrep_index_append_exists_result_left)) * uc)) /\\ exists ff_q_pfp_append_exists_result_leftinput. ub = ff_q_pfp_append_exists_result_leftinput * S ((S (pfrep_index_append_exists_result_left)) * uc) + (pfrep_value_append_exists_result_left))) -> (((exists ff_h_pfp_append_exists_result_leftoutput. ff_h_pfp_append_exists_result_leftoutput + S (pfrep_value_append_exists_result_left) = S ((S ((L)+pfrep_index_append_exists_result_left)) * UC)) /\\ exists ff_q_pfp_append_exists_result_leftoutput. UB = ff_q_pfp_append_exists_result_leftoutput * S ((S ((L)+pfrep_index_append_exists_result_left)) * UC) + (pfrep_value_append_exists_result_left))))))) /\\ (((((forall pfp_repeat_index_append_exists_result_rightzeros. (exists pfa_gap_append_exists_result_rightzerosindex. pfa_gap_append_exists_result_rightzerosindex + S (pfp_repeat_index_append_exists_result_rightzeros) = (S N)) -> (((exists ff_h_pfp_append_exists_result_rightzerosentry. ff_h_pfp_append_exists_result_rightzerosentry + S (0) = S ((S (pfp_repeat_index_append_exists_result_rightzeros)) * VC)) /\\ exists ff_q_pfp_append_exists_result_rightzerosentry. VB = ff_q_pfp_append_exists_result_rightzerosentry * S ((S (pfp_repeat_index_append_exists_result_rightzeros)) * VC) + (0)))) /\\ ((forall pfrep_index_append_exists_result_right pfrep_value_append_exists_result_right. (exists pfa_gap_append_exists_result_rightbound. pfa_gap_append_exists_result_rightbound + S (pfrep_index_append_exists_result_right) = (L)) -> (((exists ff_h_pfp_append_exists_result_rightinput. ff_h_pfp_append_exists_result_rightinput + S (pfrep_value_append_exists_result_right) = S ((S (pfrep_index_append_exists_result_right)) * vc)) /\\ exists ff_q_pfp_append_exists_result_rightinput. vb = ff_q_pfp_append_exists_result_rightinput * S ((S (pfrep_index_append_exists_result_right)) * vc) + (pfrep_value_append_exists_result_right))) -> (((exists ff_h_pfp_append_exists_result_rightoutput. ff_h_pfp_append_exists_result_rightoutput + S (pfrep_value_append_exists_result_right) = S ((S ((S N)+pfrep_index_append_exists_result_right)) * VC)) /\\ exists ff_q_pfp_append_exists_result_rightoutput. VB = ff_q_pfp_append_exists_result_rightoutput * S ((S ((S N)+pfrep_index_append_exists_result_right)) * VC) + (pfrep_value_append_exists_result_right))))))) /\\ (((forall pfp_index_append_exists_result_sum. (exists pfa_gap_append_exists_result_sumindex. pfa_gap_append_exists_result_sumindex + S (pfp_index_append_exists_result_sum) = (L+S N)) -> exists pfp_left_append_exists_result_sum pfp_right_append_exists_result_sum pfp_value_append_exists_result_sum. ((((exists ff_h_pfp_append_exists_result_sumleft. ff_h_pfp_append_exists_result_sumleft + S (pfp_left_append_exists_result_sum) = S ((S (pfp_index_append_exists_result_sum)) * UC)) /\\ exists ff_q_pfp_append_exists_result_sumleft. UB = ff_q_pfp_append_exists_result_sumleft * S ((S (pfp_index_append_exists_result_sum)) * UC) + (pfp_left_append_exists_result_sum))) /\\ (((((exists ff_h_pfp_append_exists_result_sumright. ff_h_pfp_append_exists_result_sumright + S (pfp_right_append_exists_result_sum) = S ((S (pfp_index_append_exists_result_sum)) * VC)) /\\ exists ff_q_pfp_append_exists_result_sumright. VB = ff_q_pfp_append_exists_result_sumright * S ((S (pfp_index_append_exists_result_sum)) * VC) + (pfp_right_append_exists_result_sum))) /\\ (((((exists ff_h_pfp_append_exists_result_sumtarget. ff_h_pfp_append_exists_result_sumtarget + S (pfp_value_append_exists_result_sum) = S ((S (pfp_index_append_exists_result_sum)) * rc)) /\\ exists ff_q_pfp_append_exists_result_sumtarget. rb = ff_q_pfp_append_exists_result_sumtarget * S ((S (pfp_index_append_exists_result_sum)) * rc) + (pfp_value_append_exists_result_sum))) /\\ ((((exists pfa_gap_append_exists_result_sumoperationleft. pfa_gap_append_exists_result_sumoperationleft + S (pfp_left_append_exists_result_sum) = (p)) /\\ (((exists pfa_gap_append_exists_result_sumoperationright. pfa_gap_append_exists_result_sumoperationright + S (pfp_right_append_exists_result_sum) = (p)) /\\ ((((exists pfa_gap_append_exists_result_sumoperationresultbound. pfa_gap_append_exists_result_sumoperationresultbound + S (pfp_value_append_exists_result_sum) = (p)) /\\ ((exists pfa_offset_left_append_exists_result_sumoperationresultcongruence pfa_offset_right_append_exists_result_sumoperationresultcongruence. ((pfp_left_append_exists_result_sum) + (pfp_right_append_exists_result_sum)) + (p) * pfa_offset_left_append_exists_result_sumoperationresultcongruence = (pfp_value_append_exists_result_sum) + (p) * pfa_offset_right_append_exists_result_sumoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_append_exists_equivalence pfrep_left_append_exists_equivalence pfrep_right_append_exists_equivalence. ((exists pfrep_position_append_exists_equivalencefirst. ((pfrep_position_append_exists_equivalencefirst+S (pfrep_power_append_exists_equivalence)=(K)) /\\ ((((exists ff_h_pfp_append_exists_equivalencefirstentry. ff_h_pfp_append_exists_equivalencefirstentry + S (pfrep_left_append_exists_equivalence) = S ((S (pfrep_position_append_exists_equivalencefirst)) * qc)) /\\ exists ff_q_pfp_append_exists_equivalencefirstentry. qb = ff_q_pfp_append_exists_equivalencefirstentry * S ((S (pfrep_position_append_exists_equivalencefirst)) * qc) + (pfrep_left_append_exists_equivalence)))))) \\/ (((exists pfrep_gap_append_exists_equivalencefirstoutside. pfrep_gap_append_exists_equivalencefirstoutside+(K)=(pfrep_power_append_exists_equivalence)) /\\ (((pfrep_left_append_exists_equivalence)=0))))) -> ((exists pfrep_position_append_exists_equivalencesecond. 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No output existence or polynomial identity is assumed.", "summary_sha256": "b2277f08a06d3891c988910b6344e3e1adda0ce716b6bbae9c0917a41ebbc7e3" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "83bf5a8b9dea2347bd7e803856aac23bd33a5f06a9d37d41d3e52c911b8635b2", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 301, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "8846224923876a4f57ad8d6f31020838ccc86c86a683ec78a7c7c23c35b92068", "test_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85" }, "body_checked": true, "body_receipt": { "command_count": 138, "dependency_count": 5, "dne_command_count": 0, "name": "prime_field_polynomial_shift_equivalent_congruent", "proof_depth": 41, "proof_edges": 146, "proof_nodes": 147, "proof_objects": 147, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "zero_or_succ", "prime_field_polynomial_shift_power_zero", "prime_field_polynomial_shift_power_successor", "prime_field_polynomial_power_coefficient_exists", "prime_field_polynomial_power_coefficient_functional" ], "dependencies_sha256": "4abe009c22f83c459a23f82a5865ca37ef89a0d3a29cc264903e1a177286306c", "empty_context_closure": { "body_proof_depth": 41, "body_proof_nodes": 147, "bundle_campaign": "polynomial-gcd-bezout", "bundle_dependency_edge_count": 1578, "bundle_node_count": 493, "bundle_node_id": 301, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "bundle_root_id": 492, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "7378d8a98b894a47bd3d88b64bf0899879b88a4854c913a28910802486972365", "status": "checked" }, "enrollment_index": 4123, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "8846224923876a4f57ad8d6f31020838ccc86c86a683ec78a7c7c23c35b92068", "kind": "alpha_v34_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_shift_equivalence_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85", "kind": "alpha_v34_frontier_executable_audit", "path": "peano-lab/py/tests/test_campaign_research_v34_closure.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "kind": "alpha_v34_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-field-polynomial-gcd-bezout-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "kind": "alpha_v34_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=301]" }, { "document_sha256": "81943a545e7b9f667ad758c24de14bb47280e8cf3fbb3ca714efec55fb096408", "kind": "alpha_v34_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "6bc5ab4da90f6bc935d6ead09fd652c62c940ee335fa555a214e5012cb6f3565", "membership": "alpha_only", "name": "prime_field_polynomial_shift_equivalent_congruent", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro b", "intro c", "intro L", "intro d", "intro e", "intro M", "intro ub", "intro uc", "intro vb", "intro vc", "intro he", "intro hu", "intro hv", "intro k", "intro a", "intro r", "intro hleft", "intro hright", "have hcases : k=0 \\/ exists j. k=S j", "specialize zero_or_succ (k)", "apply zero_or_succ", "cases hcases", "rewrite hcases_left at hleft", "rewrite hcases_left at hleft", "rewrite hcases_left at hright", "rewrite hcases_left at hright", "have hleft_zero : (exists pfrep_position_shift_congruent_left_zero. ((pfrep_position_shift_congruent_left_zero+S (0)=(S L)) /\\ ((((exists ff_h_pfp_shift_congruent_left_zeroentry. ff_h_pfp_shift_congruent_left_zeroentry + S (0) = S ((S (pfrep_position_shift_congruent_left_zero)) * uc)) /\\ exists ff_q_pfp_shift_congruent_left_zeroentry. ub = ff_q_pfp_shift_congruent_left_zeroentry * S ((S (pfrep_position_shift_congruent_left_zero)) * uc) + (0)))))) \\/ (((exists pfrep_gap_shift_congruent_left_zerooutside. pfrep_gap_shift_congruent_left_zerooutside+(S L)=(0)) /\\ (((0)=0))))", "specialize prime_field_polynomial_shift_power_zero (b)", "specialize prime_field_polynomial_shift_power_zero (c)", "specialize prime_field_polynomial_shift_power_zero (L)", "specialize prime_field_polynomial_shift_power_zero (ub)", "specialize prime_field_polynomial_shift_power_zero (uc)", "apply prime_field_polynomial_shift_power_zero", "exact hu", "have hright_zero : (exists pfrep_position_shift_congruent_right_zero. ((pfrep_position_shift_congruent_right_zero+S (0)=(S M)) /\\ ((((exists ff_h_pfp_shift_congruent_right_zeroentry. ff_h_pfp_shift_congruent_right_zeroentry + S (0) = S ((S (pfrep_position_shift_congruent_right_zero)) * vc)) /\\ exists ff_q_pfp_shift_congruent_right_zeroentry. vb = ff_q_pfp_shift_congruent_right_zeroentry * S ((S (pfrep_position_shift_congruent_right_zero)) * vc) + (0)))))) \\/ (((exists pfrep_gap_shift_congruent_right_zerooutside. pfrep_gap_shift_congruent_right_zerooutside+(S M)=(0)) /\\ (((0)=0))))", "specialize prime_field_polynomial_shift_power_zero (d)", "specialize prime_field_polynomial_shift_power_zero (e)", "specialize prime_field_polynomial_shift_power_zero (M)", "specialize prime_field_polynomial_shift_power_zero (vb)", "specialize prime_field_polynomial_shift_power_zero (vc)", "apply prime_field_polynomial_shift_power_zero", "exact hv", "have hleft_value : a=0", "specialize prime_field_polynomial_power_coefficient_functional (ub)", "specialize prime_field_polynomial_power_coefficient_functional (uc)", "specialize prime_field_polynomial_power_coefficient_functional (S L)", "specialize prime_field_polynomial_power_coefficient_functional (0)", "specialize prime_field_polynomial_power_coefficient_functional (a)", "specialize prime_field_polynomial_power_coefficient_functional (0)", "apply prime_field_polynomial_power_coefficient_functional", "exact hleft", "exact hleft_zero", "have hright_value : 0=r", "specialize prime_field_polynomial_power_coefficient_functional (vb)", "specialize prime_field_polynomial_power_coefficient_functional (vc)", "specialize prime_field_polynomial_power_coefficient_functional (S M)", "specialize prime_field_polynomial_power_coefficient_functional (0)", "specialize prime_field_polynomial_power_coefficient_functional (0)", "specialize prime_field_polynomial_power_coefficient_functional (r)", "apply prime_field_polynomial_power_coefficient_functional", "exact hright_zero", "exact hright", "trans 0", "exact hleft_value", "exact hright_value", "cases hcases_right", "rewrite hcases_right_witness at hleft", "rewrite hcases_right_witness at hleft", "rewrite hcases_right_witness at hright", "rewrite hcases_right_witness at hright", "have hleft_previous : exists s. ((exists pfrep_position_shift_congruent_previous_left. ((pfrep_position_shift_congruent_previous_left+S (x)=(L)) /\\ ((((exists ff_h_pfp_shift_congruent_previous_leftentry. ff_h_pfp_shift_congruent_previous_leftentry + S (s) = S ((S (pfrep_position_shift_congruent_previous_left)) * c)) /\\ exists ff_q_pfp_shift_congruent_previous_leftentry. b = ff_q_pfp_shift_congruent_previous_leftentry * S ((S (pfrep_position_shift_congruent_previous_left)) * c) + (s)))))) \\/ (((exists pfrep_gap_shift_congruent_previous_leftoutside. pfrep_gap_shift_congruent_previous_leftoutside+(L)=(x)) /\\ (((s)=0)))))", "specialize prime_field_polynomial_power_coefficient_exists (b)", "specialize prime_field_polynomial_power_coefficient_exists (c)", "specialize prime_field_polynomial_power_coefficient_exists (L)", "specialize prime_field_polynomial_power_coefficient_exists (x)", "apply prime_field_polynomial_power_coefficient_exists", "cases hleft_previous", "have hright_previous : exists s. ((exists pfrep_position_shift_congruent_previous_right. ((pfrep_position_shift_congruent_previous_right+S (x)=(M)) /\\ ((((exists ff_h_pfp_shift_congruent_previous_rightentry. ff_h_pfp_shift_congruent_previous_rightentry + S (s) = S ((S (pfrep_position_shift_congruent_previous_right)) * e)) /\\ exists ff_q_pfp_shift_congruent_previous_rightentry. d = ff_q_pfp_shift_congruent_previous_rightentry * S ((S (pfrep_position_shift_congruent_previous_right)) * e) + (s)))))) \\/ (((exists pfrep_gap_shift_congruent_previous_rightoutside. pfrep_gap_shift_congruent_previous_rightoutside+(M)=(x)) /\\ (((s)=0)))))", "specialize prime_field_polynomial_power_coefficient_exists (d)", "specialize prime_field_polynomial_power_coefficient_exists (e)", "specialize prime_field_polynomial_power_coefficient_exists (M)", "specialize prime_field_polynomial_power_coefficient_exists (x)", "apply prime_field_polynomial_power_coefficient_exists", "cases hright_previous", "have hleft_shifted : (exists pfrep_position_shift_congruent_shifted_left. ((pfrep_position_shift_congruent_shifted_left+S (S x)=(S L)) /\\ ((((exists ff_h_pfp_shift_congruent_shifted_leftentry. ff_h_pfp_shift_congruent_shifted_leftentry + S (x1) = S ((S (pfrep_position_shift_congruent_shifted_left)) * uc)) /\\ exists ff_q_pfp_shift_congruent_shifted_leftentry. ub = ff_q_pfp_shift_congruent_shifted_leftentry * S ((S (pfrep_position_shift_congruent_shifted_left)) * uc) + (x1)))))) \\/ (((exists pfrep_gap_shift_congruent_shifted_leftoutside. pfrep_gap_shift_congruent_shifted_leftoutside+(S L)=(S x)) /\\ (((x1)=0))))", "specialize prime_field_polynomial_shift_power_successor (b)", "specialize prime_field_polynomial_shift_power_successor (c)", "specialize prime_field_polynomial_shift_power_successor (L)", "specialize prime_field_polynomial_shift_power_successor (ub)", "specialize prime_field_polynomial_shift_power_successor (uc)", "specialize prime_field_polynomial_shift_power_successor (x)", "specialize prime_field_polynomial_shift_power_successor (x1)", "apply prime_field_polynomial_shift_power_successor", "exact hu", "exact hleft_previous_witness", "have hright_shifted : (exists pfrep_position_shift_congruent_shifted_right. ((pfrep_position_shift_congruent_shifted_right+S (S x)=(S M)) /\\ ((((exists ff_h_pfp_shift_congruent_shifted_rightentry. ff_h_pfp_shift_congruent_shifted_rightentry + S (x2) = S ((S (pfrep_position_shift_congruent_shifted_right)) * vc)) /\\ exists ff_q_pfp_shift_congruent_shifted_rightentry. vb = ff_q_pfp_shift_congruent_shifted_rightentry * S ((S (pfrep_position_shift_congruent_shifted_right)) * vc) + (x2)))))) \\/ (((exists pfrep_gap_shift_congruent_shifted_rightoutside. pfrep_gap_shift_congruent_shifted_rightoutside+(S M)=(S x)) /\\ (((x2)=0))))", "specialize prime_field_polynomial_shift_power_successor (d)", "specialize prime_field_polynomial_shift_power_successor (e)", "specialize prime_field_polynomial_shift_power_successor (M)", "specialize prime_field_polynomial_shift_power_successor (vb)", "specialize prime_field_polynomial_shift_power_successor (vc)", "specialize prime_field_polynomial_shift_power_successor (x)", "specialize prime_field_polynomial_shift_power_successor (x2)", "apply prime_field_polynomial_shift_power_successor", "exact hv", "exact hright_previous_witness", "have hleft_value : a=x1", "specialize prime_field_polynomial_power_coefficient_functional (ub)", "specialize prime_field_polynomial_power_coefficient_functional (uc)", "specialize prime_field_polynomial_power_coefficient_functional (S L)", "specialize prime_field_polynomial_power_coefficient_functional (S x)", "specialize prime_field_polynomial_power_coefficient_functional (a)", "specialize prime_field_polynomial_power_coefficient_functional (x1)", "apply prime_field_polynomial_power_coefficient_functional", "exact hleft", "exact hleft_shifted", "have hright_value : x2=r", "specialize prime_field_polynomial_power_coefficient_functional (vb)", "specialize prime_field_polynomial_power_coefficient_functional (vc)", "specialize prime_field_polynomial_power_coefficient_functional (S M)", "specialize prime_field_polynomial_power_coefficient_functional (S x)", "specialize prime_field_polynomial_power_coefficient_functional (x2)", "specialize prime_field_polynomial_power_coefficient_functional (r)", "apply prime_field_polynomial_power_coefficient_functional", "exact hright_shifted", "exact hright", "have hprevious_equal : x1=x2", "specialize he (x)", "specialize he (x1)", "specialize he (x2)", "apply he", "exact hleft_previous_witness", "exact hright_previous_witness", "trans x1", "exact hleft_value", "trans x2", "exact hprevious_equal", "exact hright_value" ], "script_sha256": "1ee0ec9d06fcf879e8754246494d3cbfe0e0050ce05bda40fe3e820d346c6510", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_shift_equivalence_candidate.py", "sha256": "8846224923876a4f57ad8d6f31020838ccc86c86a683ec78a7c7c23c35b92068" }, "statement": "forall b c L d e M ub uc vb vc. (forall pfrep_power_shift_congruent_input pfrep_left_shift_congruent_input pfrep_right_shift_congruent_input. ((exists pfrep_position_shift_congruent_inputfirst. ((pfrep_position_shift_congruent_inputfirst+S (pfrep_power_shift_congruent_input)=(L)) /\\ ((((exists ff_h_pfp_shift_congruent_inputfirstentry. ff_h_pfp_shift_congruent_inputfirstentry + S (pfrep_left_shift_congruent_input) = S ((S (pfrep_position_shift_congruent_inputfirst)) * c)) /\\ exists ff_q_pfp_shift_congruent_inputfirstentry. b = ff_q_pfp_shift_congruent_inputfirstentry * S ((S (pfrep_position_shift_congruent_inputfirst)) * c) + (pfrep_left_shift_congruent_input)))))) \\/ (((exists pfrep_gap_shift_congruent_inputfirstoutside. pfrep_gap_shift_congruent_inputfirstoutside+(L)=(pfrep_power_shift_congruent_input)) /\\ (((pfrep_left_shift_congruent_input)=0))))) -> ((exists pfrep_position_shift_congruent_inputsecond. ((pfrep_position_shift_congruent_inputsecond+S (pfrep_power_shift_congruent_input)=(M)) /\\ ((((exists ff_h_pfp_shift_congruent_inputsecondentry. ff_h_pfp_shift_congruent_inputsecondentry + S (pfrep_right_shift_congruent_input) = S ((S (pfrep_position_shift_congruent_inputsecond)) * e)) /\\ exists ff_q_pfp_shift_congruent_inputsecondentry. d = ff_q_pfp_shift_congruent_inputsecondentry * S ((S (pfrep_position_shift_congruent_inputsecond)) * e) + (pfrep_right_shift_congruent_input)))))) \\/ (((exists pfrep_gap_shift_congruent_inputsecondoutside. pfrep_gap_shift_congruent_inputsecondoutside+(M)=(pfrep_power_shift_congruent_input)) /\\ (((pfrep_right_shift_congruent_input)=0))))) -> pfrep_left_shift_congruent_input=pfrep_right_shift_congruent_input) -> (((forall mdr_i_pfp_shift_congruent_leftprefix mdr_a_pfp_shift_congruent_leftprefix. (exists mdr_gap_pfp_shift_congruent_leftprefixb. mdr_gap_pfp_shift_congruent_leftprefixb + S (mdr_i_pfp_shift_congruent_leftprefix) = (L)) -> (((exists ff_h_mdr_pfp_shift_congruent_leftprefixo. ff_h_mdr_pfp_shift_congruent_leftprefixo + S (mdr_a_pfp_shift_congruent_leftprefix) = S ((S (mdr_i_pfp_shift_congruent_leftprefix)) * c)) /\\ exists ff_q_mdr_pfp_shift_congruent_leftprefixo. b = ff_q_mdr_pfp_shift_congruent_leftprefixo * S ((S (mdr_i_pfp_shift_congruent_leftprefix)) * c) + (mdr_a_pfp_shift_congruent_leftprefix))) -> (((exists ff_h_mdr_pfp_shift_congruent_leftprefixn. ff_h_mdr_pfp_shift_congruent_leftprefixn + S (mdr_a_pfp_shift_congruent_leftprefix) = S ((S (mdr_i_pfp_shift_congruent_leftprefix)) * uc)) /\\ exists ff_q_mdr_pfp_shift_congruent_leftprefixn. ub = ff_q_mdr_pfp_shift_congruent_leftprefixn * S ((S (mdr_i_pfp_shift_congruent_leftprefix)) * uc) + (mdr_a_pfp_shift_congruent_leftprefix)))) /\\ ((((exists ff_h_pfp_shift_congruent_leftlast. ff_h_pfp_shift_congruent_leftlast + S (0) = S ((S (L)) * uc)) /\\ exists ff_q_pfp_shift_congruent_leftlast. ub = ff_q_pfp_shift_congruent_leftlast * S ((S (L)) * uc) + (0)))))) -> (((forall mdr_i_pfp_shift_congruent_rightprefix mdr_a_pfp_shift_congruent_rightprefix. (exists mdr_gap_pfp_shift_congruent_rightprefixb. mdr_gap_pfp_shift_congruent_rightprefixb + S (mdr_i_pfp_shift_congruent_rightprefix) = (M)) -> (((exists ff_h_mdr_pfp_shift_congruent_rightprefixo. ff_h_mdr_pfp_shift_congruent_rightprefixo + S (mdr_a_pfp_shift_congruent_rightprefix) = S ((S (mdr_i_pfp_shift_congruent_rightprefix)) * e)) /\\ exists ff_q_mdr_pfp_shift_congruent_rightprefixo. d = ff_q_mdr_pfp_shift_congruent_rightprefixo * S ((S (mdr_i_pfp_shift_congruent_rightprefix)) * e) + (mdr_a_pfp_shift_congruent_rightprefix))) -> (((exists ff_h_mdr_pfp_shift_congruent_rightprefixn. ff_h_mdr_pfp_shift_congruent_rightprefixn + S (mdr_a_pfp_shift_congruent_rightprefix) = S ((S (mdr_i_pfp_shift_congruent_rightprefix)) * vc)) /\\ exists ff_q_mdr_pfp_shift_congruent_rightprefixn. vb = ff_q_mdr_pfp_shift_congruent_rightprefixn * S ((S (mdr_i_pfp_shift_congruent_rightprefix)) * vc) + (mdr_a_pfp_shift_congruent_rightprefix)))) /\\ ((((exists ff_h_pfp_shift_congruent_rightlast. ff_h_pfp_shift_congruent_rightlast + S (0) = S ((S (M)) * vc)) /\\ exists ff_q_pfp_shift_congruent_rightlast. vb = ff_q_pfp_shift_congruent_rightlast * S ((S (M)) * vc) + (0)))))) -> (forall pfrep_power_shift_congruent_result pfrep_left_shift_congruent_result pfrep_right_shift_congruent_result. ((exists pfrep_position_shift_congruent_resultfirst. ((pfrep_position_shift_congruent_resultfirst+S (pfrep_power_shift_congruent_result)=(S L)) /\\ ((((exists ff_h_pfp_shift_congruent_resultfirstentry. ff_h_pfp_shift_congruent_resultfirstentry + S (pfrep_left_shift_congruent_result) = S ((S (pfrep_position_shift_congruent_resultfirst)) * uc)) /\\ exists ff_q_pfp_shift_congruent_resultfirstentry. ub = ff_q_pfp_shift_congruent_resultfirstentry * S ((S (pfrep_position_shift_congruent_resultfirst)) * uc) + (pfrep_left_shift_congruent_result)))))) \\/ (((exists pfrep_gap_shift_congruent_resultfirstoutside. pfrep_gap_shift_congruent_resultfirstoutside+(S L)=(pfrep_power_shift_congruent_result)) /\\ (((pfrep_left_shift_congruent_result)=0))))) -> ((exists pfrep_position_shift_congruent_resultsecond. ((pfrep_position_shift_congruent_resultsecond+S (pfrep_power_shift_congruent_result)=(S M)) /\\ ((((exists ff_h_pfp_shift_congruent_resultsecondentry. ff_h_pfp_shift_congruent_resultsecondentry + S (pfrep_right_shift_congruent_result) = S ((S (pfrep_position_shift_congruent_resultsecond)) * vc)) /\\ exists ff_q_pfp_shift_congruent_resultsecondentry. vb = ff_q_pfp_shift_congruent_resultsecondentry * S ((S (pfrep_position_shift_congruent_resultsecond)) * vc) + (pfrep_right_shift_congruent_result)))))) \\/ (((exists pfrep_gap_shift_congruent_resultsecondoutside. pfrep_gap_shift_congruent_resultsecondoutside+(S M)=(pfrep_power_shift_congruent_result)) /\\ (((pfrep_right_shift_congruent_result)=0))))) -> pfrep_left_shift_congruent_result=pfrep_right_shift_congruent_result)", "statement_sha256": "7378d8a98b894a47bd3d88b64bf0899879b88a4854c913a28910802486972365", "summary": "Two actual trailing-zero shifts preserve formal coefficient equivalence across arbitrary represented lengths, including empty prefixes. The proof obtains actual predecessor-power coefficients and compares decoded values, without any modulus, primality, coefficient-bound, raw-code identity, or evaluation-equality assumption.", "summary_sha256": "4630a4d41b4b210f8f4546c38aa49bf98a03bad4e45fbd406b27a32da687b441" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "0c6f2c23ed3e1329be0b6785a06177e56a4e71e6f04f03954a2662d222c96a2c", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 302, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "dd85dbd1bd87143715a4286724ac7c87f280a909dac6759f00a6cb7dff7c85f1", "test_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85" }, "body_checked": true, 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"8cafa50eeaa87547256744332189a40fe086cbce3500b022d05af433e8e72267", "membership": "alpha_only", "name": "prime_field_polynomial_convolution_shift_scale_aligned_equivalent", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro c", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro M", "intro pb", "intro pc", "intro N", "intro qb", "intro qc", "intro K", "intro rb", "intro rc", "intro T", "intro ub", "intro uc", "intro vb", "intro vc", "intro UPb", "intro UPc", "intro VPb", "intro VPc", "intro zb", "intro zc", "intro sb", "intro sc", "intro W", "intro eb", "intro ec", "intro fb", "intro fc", "intro EPb", "intro EPc", "intro FPb", "intro FPc", "intro yb", "intro yc", "intro hp", "intro hAB", "intro hAQ", "intro hU", "intro hV", "intro hUP", "intro hVP", "intro hZ", "intro hAZ", "intro hE", "intro hF", "intro hEP", "intro hFP", "intro hY", "have hp0 : ~(p=0)", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hz", "have hABcopy : ((forall fom_index_pfp_step_helper_ABleft. (exists fom_gap_pfp_step_helper_ABleft_index_bound. fom_gap_pfp_step_helper_ABleft_index_bound + S (fom_index_pfp_step_helper_ABleft) = L) -> exists fom_value_pfp_step_helper_ABleft. ((((exists fom_beta_height_pfp_step_helper_ABleft_entry. fom_beta_height_pfp_step_helper_ABleft_entry + S (fom_value_pfp_step_helper_ABleft) = S ((S (fom_index_pfp_step_helper_ABleft)) * ac)) /\\ exists fom_beta_quotient_pfp_step_helper_ABleft_entry. ab = fom_beta_quotient_pfp_step_helper_ABleft_entry * S ((S (fom_index_pfp_step_helper_ABleft)) * ac) + (fom_value_pfp_step_helper_ABleft))) /\\ (exists fom_gap_pfp_step_helper_ABleft_value_bound. fom_gap_pfp_step_helper_ABleft_value_bound + S (fom_value_pfp_step_helper_ABleft) = p))) /\\ (((forall fom_index_pfp_step_helper_ABright. (exists fom_gap_pfp_step_helper_ABright_index_bound. fom_gap_pfp_step_helper_ABright_index_bound + S (fom_index_pfp_step_helper_ABright) = M) -> exists fom_value_pfp_step_helper_ABright. ((((exists fom_beta_height_pfp_step_helper_ABright_entry. fom_beta_height_pfp_step_helper_ABright_entry + S (fom_value_pfp_step_helper_ABright) = S ((S (fom_index_pfp_step_helper_ABright)) * bc)) /\\ exists fom_beta_quotient_pfp_step_helper_ABright_entry. bb = fom_beta_quotient_pfp_step_helper_ABright_entry * S ((S (fom_index_pfp_step_helper_ABright)) * bc) + (fom_value_pfp_step_helper_ABright))) /\\ (exists fom_gap_pfp_step_helper_ABright_value_bound. fom_gap_pfp_step_helper_ABright_value_bound + S (fom_value_pfp_step_helper_ABright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_step_helper_ABcoefficients. (exists pfa_gap_step_helper_ABcoefficientsbound. pfa_gap_step_helper_ABcoefficientsbound + S (pfc_index_step_helper_ABcoefficients) = (N)) -> exists pfc_value_step_helper_ABcoefficients. ((((exists ff_h_pfp_step_helper_ABcoefficientsentry. ff_h_pfp_step_helper_ABcoefficientsentry + S (pfc_value_step_helper_ABcoefficients) = S ((S (pfc_index_step_helper_ABcoefficients)) * pc)) /\\ exists ff_q_pfp_step_helper_ABcoefficientsentry. pb = ff_q_pfp_step_helper_ABcoefficientsentry * S ((S (pfc_index_step_helper_ABcoefficients)) * pc) + (pfc_value_step_helper_ABcoefficients))) /\\ ((exists pfc_terms_code_step_helper_ABcoefficientscoefficient pfc_terms_scale_step_helper_ABcoefficientscoefficient pfc_natural_sum_step_helper_ABcoefficientscoefficient. ((forall pfc_index_step_helper_ABcoefficientscoefficientdiagonal. (exists pfa_gap_step_helper_ABcoefficientscoefficientdiagonalbound. pfa_gap_step_helper_ABcoefficientscoefficientdiagonalbound + S (pfc_index_step_helper_ABcoefficientscoefficientdiagonal) = (S (pfc_index_step_helper_ABcoefficients))) -> exists pfc_value_step_helper_ABcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_step_helper_ABcoefficientscoefficientdiagonalentry. ff_h_pfp_step_helper_ABcoefficientscoefficientdiagonalentry + S (pfc_value_step_helper_ABcoefficientscoefficientdiagonal) = S ((S (pfc_index_step_helper_ABcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_ABcoefficientscoefficient)) /\\ exists ff_q_pfp_step_helper_ABcoefficientscoefficientdiagonalentry. pfc_terms_code_step_helper_ABcoefficientscoefficient = ff_q_pfp_step_helper_ABcoefficientscoefficientdiagonalentry * S ((S (pfc_index_step_helper_ABcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_ABcoefficientscoefficient) + (pfc_value_step_helper_ABcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_step_helper_ABcoefficientscoefficientdiagonalterm pfc_left_step_helper_ABcoefficientscoefficientdiagonalterm pfc_right_step_helper_ABcoefficientscoefficientdiagonalterm. (((pfc_index_step_helper_ABcoefficientscoefficientdiagonal)+pfc_complement_step_helper_ABcoefficientscoefficientdiagonalterm=(pfc_index_step_helper_ABcoefficients)) /\\ ((((((exists pfa_gap_step_helper_ABcoefficientscoefficientdiagonaltermleftinside. pfa_gap_step_helper_ABcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_step_helper_ABcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_step_helper_ABcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_step_helper_ABcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_step_helper_ABcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_step_helper_ABcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_step_helper_ABcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_step_helper_ABcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_step_helper_ABcoefficientscoefficientdiagonal)) * ac) + (pfc_left_step_helper_ABcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_ABcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_step_helper_ABcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_step_helper_ABcoefficientscoefficientdiagonal)) /\\ (((pfc_left_step_helper_ABcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_step_helper_ABcoefficientscoefficientdiagonaltermrightinside. pfa_gap_step_helper_ABcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_step_helper_ABcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_step_helper_ABcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_step_helper_ABcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_step_helper_ABcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_step_helper_ABcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_step_helper_ABcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_step_helper_ABcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_step_helper_ABcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_step_helper_ABcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_ABcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_step_helper_ABcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_step_helper_ABcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_step_helper_ABcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_step_helper_ABcoefficientscoefficientdiagonal)=pfc_left_step_helper_ABcoefficientscoefficientdiagonalterm*pfc_right_step_helper_ABcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_step_helper_ABcoefficientscoefficientsum fs_v_pfc_step_helper_ABcoefficientscoefficientsum. ((((exists fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_start. fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_start. fs_u_pfc_step_helper_ABcoefficientscoefficientsum = fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_terminal. fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_step_helper_ABcoefficientscoefficient) = S ((S (S (pfc_index_step_helper_ABcoefficients))) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_terminal. fs_u_pfc_step_helper_ABcoefficientscoefficientsum = fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_step_helper_ABcoefficients))) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum) + (pfc_natural_sum_step_helper_ABcoefficientscoefficient))) /\\ forall fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps = S (pfc_index_step_helper_ABcoefficients)) -> exists fs_a_pfc_step_helper_ABcoefficientscoefficientsum_body_steps fs_r_pfc_step_helper_ABcoefficientscoefficientsum_body_steps fs_s_pfc_step_helper_ABcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_step_helper_ABcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_ABcoefficientscoefficient)) /\\ exists fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_step_helper_ABcoefficientscoefficient = fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_ABcoefficientscoefficient) + (fs_a_pfc_step_helper_ABcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_step_helper_ABcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_step_helper_ABcoefficientscoefficientsum = fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum) + (fs_r_pfc_step_helper_ABcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_step_helper_ABcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_step_helper_ABcoefficientscoefficientsum = fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum) + (fs_s_pfc_step_helper_ABcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_step_helper_ABcoefficientscoefficientsum_body_steps = fs_r_pfc_step_helper_ABcoefficientscoefficientsum_body_steps + fs_a_pfc_step_helper_ABcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_step_helper_ABcoefficientscoefficientresiduebound. pfa_gap_step_helper_ABcoefficientscoefficientresiduebound + S (pfc_value_step_helper_ABcoefficients) = (p)) /\\ ((exists pfa_offset_left_step_helper_ABcoefficientscoefficientresiduecongruence pfa_offset_right_step_helper_ABcoefficientscoefficientresiduecongruence. (pfc_natural_sum_step_helper_ABcoefficientscoefficient) + (p) * pfa_offset_left_step_helper_ABcoefficientscoefficientresiduecongruence = (pfc_value_step_helper_ABcoefficients) + (p) * pfa_offset_right_step_helper_ABcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hAB", "cases hABcopy", "cases hABcopy_right", "cases hABcopy_right_right", "have hAQcopy : ((forall fom_index_pfp_step_helper_AQleft. (exists fom_gap_pfp_step_helper_AQleft_index_bound. fom_gap_pfp_step_helper_AQleft_index_bound + S (fom_index_pfp_step_helper_AQleft) = L) -> exists fom_value_pfp_step_helper_AQleft. ((((exists fom_beta_height_pfp_step_helper_AQleft_entry. fom_beta_height_pfp_step_helper_AQleft_entry + S (fom_value_pfp_step_helper_AQleft) = S ((S (fom_index_pfp_step_helper_AQleft)) * ac)) /\\ exists fom_beta_quotient_pfp_step_helper_AQleft_entry. ab = fom_beta_quotient_pfp_step_helper_AQleft_entry * S ((S (fom_index_pfp_step_helper_AQleft)) * ac) + (fom_value_pfp_step_helper_AQleft))) /\\ (exists fom_gap_pfp_step_helper_AQleft_value_bound. fom_gap_pfp_step_helper_AQleft_value_bound + S (fom_value_pfp_step_helper_AQleft) = p))) /\\ (((forall fom_index_pfp_step_helper_AQright. (exists fom_gap_pfp_step_helper_AQright_index_bound. fom_gap_pfp_step_helper_AQright_index_bound + S (fom_index_pfp_step_helper_AQright) = K) -> exists fom_value_pfp_step_helper_AQright. ((((exists fom_beta_height_pfp_step_helper_AQright_entry. fom_beta_height_pfp_step_helper_AQright_entry + S (fom_value_pfp_step_helper_AQright) = S ((S (fom_index_pfp_step_helper_AQright)) * qc)) /\\ exists fom_beta_quotient_pfp_step_helper_AQright_entry. qb = fom_beta_quotient_pfp_step_helper_AQright_entry * S ((S (fom_index_pfp_step_helper_AQright)) * qc) + (fom_value_pfp_step_helper_AQright))) /\\ (exists fom_gap_pfp_step_helper_AQright_value_bound. fom_gap_pfp_step_helper_AQright_value_bound + S (fom_value_pfp_step_helper_AQright) = p))) /\\ (((((((L)=0 \\/ (K)=0) /\\ (((T)=0)))) \\/ (((~((L)=0)) /\\ (((~((K)=0)) /\\ (((L)+(K)=S (T)))))))) /\\ ((forall pfc_index_step_helper_AQcoefficients. (exists pfa_gap_step_helper_AQcoefficientsbound. pfa_gap_step_helper_AQcoefficientsbound + S (pfc_index_step_helper_AQcoefficients) = (T)) -> exists pfc_value_step_helper_AQcoefficients. ((((exists ff_h_pfp_step_helper_AQcoefficientsentry. ff_h_pfp_step_helper_AQcoefficientsentry + S (pfc_value_step_helper_AQcoefficients) = S ((S (pfc_index_step_helper_AQcoefficients)) * rc)) /\\ exists ff_q_pfp_step_helper_AQcoefficientsentry. rb = ff_q_pfp_step_helper_AQcoefficientsentry * S ((S (pfc_index_step_helper_AQcoefficients)) * rc) + (pfc_value_step_helper_AQcoefficients))) /\\ ((exists pfc_terms_code_step_helper_AQcoefficientscoefficient pfc_terms_scale_step_helper_AQcoefficientscoefficient pfc_natural_sum_step_helper_AQcoefficientscoefficient. ((forall pfc_index_step_helper_AQcoefficientscoefficientdiagonal. (exists pfa_gap_step_helper_AQcoefficientscoefficientdiagonalbound. pfa_gap_step_helper_AQcoefficientscoefficientdiagonalbound + S (pfc_index_step_helper_AQcoefficientscoefficientdiagonal) = (S (pfc_index_step_helper_AQcoefficients))) -> exists pfc_value_step_helper_AQcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_step_helper_AQcoefficientscoefficientdiagonalentry. ff_h_pfp_step_helper_AQcoefficientscoefficientdiagonalentry + S (pfc_value_step_helper_AQcoefficientscoefficientdiagonal) = S ((S (pfc_index_step_helper_AQcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_AQcoefficientscoefficient)) /\\ exists ff_q_pfp_step_helper_AQcoefficientscoefficientdiagonalentry. pfc_terms_code_step_helper_AQcoefficientscoefficient = ff_q_pfp_step_helper_AQcoefficientscoefficientdiagonalentry * S ((S (pfc_index_step_helper_AQcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_AQcoefficientscoefficient) + (pfc_value_step_helper_AQcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_step_helper_AQcoefficientscoefficientdiagonalterm pfc_left_step_helper_AQcoefficientscoefficientdiagonalterm pfc_right_step_helper_AQcoefficientscoefficientdiagonalterm. (((pfc_index_step_helper_AQcoefficientscoefficientdiagonal)+pfc_complement_step_helper_AQcoefficientscoefficientdiagonalterm=(pfc_index_step_helper_AQcoefficients)) /\\ ((((((exists pfa_gap_step_helper_AQcoefficientscoefficientdiagonaltermleftinside. pfa_gap_step_helper_AQcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_step_helper_AQcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_step_helper_AQcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_step_helper_AQcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_step_helper_AQcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_step_helper_AQcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_step_helper_AQcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_step_helper_AQcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_step_helper_AQcoefficientscoefficientdiagonal)) * ac) + (pfc_left_step_helper_AQcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_AQcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_step_helper_AQcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_step_helper_AQcoefficientscoefficientdiagonal)) /\\ (((pfc_left_step_helper_AQcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_step_helper_AQcoefficientscoefficientdiagonaltermrightinside. pfa_gap_step_helper_AQcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_step_helper_AQcoefficientscoefficientdiagonalterm) = (K)) /\\ ((((exists ff_h_pfp_step_helper_AQcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_step_helper_AQcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_step_helper_AQcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_step_helper_AQcoefficientscoefficientdiagonalterm)) * qc)) /\\ exists ff_q_pfp_step_helper_AQcoefficientscoefficientdiagonaltermrightentry. qb = ff_q_pfp_step_helper_AQcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_step_helper_AQcoefficientscoefficientdiagonalterm)) * qc) + (pfc_right_step_helper_AQcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_AQcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_step_helper_AQcoefficientscoefficientdiagonaltermrightoutside+(K)=(pfc_complement_step_helper_AQcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_step_helper_AQcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_step_helper_AQcoefficientscoefficientdiagonal)=pfc_left_step_helper_AQcoefficientscoefficientdiagonalterm*pfc_right_step_helper_AQcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_step_helper_AQcoefficientscoefficientsum fs_v_pfc_step_helper_AQcoefficientscoefficientsum. ((((exists fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_start. fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_start. fs_u_pfc_step_helper_AQcoefficientscoefficientsum = fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_terminal. fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_step_helper_AQcoefficientscoefficient) = S ((S (S (pfc_index_step_helper_AQcoefficients))) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_terminal. fs_u_pfc_step_helper_AQcoefficientscoefficientsum = fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_step_helper_AQcoefficients))) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum) + (pfc_natural_sum_step_helper_AQcoefficientscoefficient))) /\\ forall fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps = S (pfc_index_step_helper_AQcoefficients)) -> exists fs_a_pfc_step_helper_AQcoefficientscoefficientsum_body_steps fs_r_pfc_step_helper_AQcoefficientscoefficientsum_body_steps fs_s_pfc_step_helper_AQcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_step_helper_AQcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_AQcoefficientscoefficient)) /\\ exists fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_step_helper_AQcoefficientscoefficient = fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_AQcoefficientscoefficient) + (fs_a_pfc_step_helper_AQcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_step_helper_AQcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_step_helper_AQcoefficientscoefficientsum = fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum) + (fs_r_pfc_step_helper_AQcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_step_helper_AQcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_step_helper_AQcoefficientscoefficientsum = fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum) + (fs_s_pfc_step_helper_AQcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_step_helper_AQcoefficientscoefficientsum_body_steps = fs_r_pfc_step_helper_AQcoefficientscoefficientsum_body_steps + fs_a_pfc_step_helper_AQcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_step_helper_AQcoefficientscoefficientresiduebound. pfa_gap_step_helper_AQcoefficientscoefficientresiduebound + S (pfc_value_step_helper_AQcoefficients) = (p)) /\\ ((exists pfa_offset_left_step_helper_AQcoefficientscoefficientresiduecongruence pfa_offset_right_step_helper_AQcoefficientscoefficientresiduecongruence. (pfc_natural_sum_step_helper_AQcoefficientscoefficient) + (p) * pfa_offset_left_step_helper_AQcoefficientscoefficientresiduecongruence = (pfc_value_step_helper_AQcoefficients) + (p) * pfa_offset_right_step_helper_AQcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hAQ", "cases hAQcopy", "cases hAQcopy_right", "cases hAQcopy_right_right", "have hAZcopy : ((forall fom_index_pfp_step_helper_AZleft. (exists fom_gap_pfp_step_helper_AZleft_index_bound. fom_gap_pfp_step_helper_AZleft_index_bound + S (fom_index_pfp_step_helper_AZleft) = L) -> exists fom_value_pfp_step_helper_AZleft. ((((exists fom_beta_height_pfp_step_helper_AZleft_entry. fom_beta_height_pfp_step_helper_AZleft_entry + S (fom_value_pfp_step_helper_AZleft) = S ((S (fom_index_pfp_step_helper_AZleft)) * ac)) /\\ exists fom_beta_quotient_pfp_step_helper_AZleft_entry. ab = fom_beta_quotient_pfp_step_helper_AZleft_entry * S ((S (fom_index_pfp_step_helper_AZleft)) * ac) + (fom_value_pfp_step_helper_AZleft))) /\\ (exists fom_gap_pfp_step_helper_AZleft_value_bound. fom_gap_pfp_step_helper_AZleft_value_bound + S (fom_value_pfp_step_helper_AZleft) = p))) /\\ (((forall fom_index_pfp_step_helper_AZright. (exists fom_gap_pfp_step_helper_AZright_index_bound. fom_gap_pfp_step_helper_AZright_index_bound + S (fom_index_pfp_step_helper_AZright) = M+S K) -> exists fom_value_pfp_step_helper_AZright. ((((exists fom_beta_height_pfp_step_helper_AZright_entry. fom_beta_height_pfp_step_helper_AZright_entry + S (fom_value_pfp_step_helper_AZright) = S ((S (fom_index_pfp_step_helper_AZright)) * zc)) /\\ exists fom_beta_quotient_pfp_step_helper_AZright_entry. zb = fom_beta_quotient_pfp_step_helper_AZright_entry * S ((S (fom_index_pfp_step_helper_AZright)) * zc) + (fom_value_pfp_step_helper_AZright))) /\\ (exists fom_gap_pfp_step_helper_AZright_value_bound. fom_gap_pfp_step_helper_AZright_value_bound + S (fom_value_pfp_step_helper_AZright) = p))) /\\ (((((((L)=0 \\/ (M+S K)=0) /\\ (((W)=0)))) \\/ (((~((L)=0)) /\\ (((~((M+S K)=0)) /\\ (((L)+(M+S K)=S (W)))))))) /\\ ((forall pfc_index_step_helper_AZcoefficients. (exists pfa_gap_step_helper_AZcoefficientsbound. pfa_gap_step_helper_AZcoefficientsbound + S (pfc_index_step_helper_AZcoefficients) = (W)) -> exists pfc_value_step_helper_AZcoefficients. ((((exists ff_h_pfp_step_helper_AZcoefficientsentry. ff_h_pfp_step_helper_AZcoefficientsentry + S (pfc_value_step_helper_AZcoefficients) = S ((S (pfc_index_step_helper_AZcoefficients)) * sc)) /\\ exists ff_q_pfp_step_helper_AZcoefficientsentry. sb = ff_q_pfp_step_helper_AZcoefficientsentry * S ((S (pfc_index_step_helper_AZcoefficients)) * sc) + (pfc_value_step_helper_AZcoefficients))) /\\ ((exists pfc_terms_code_step_helper_AZcoefficientscoefficient pfc_terms_scale_step_helper_AZcoefficientscoefficient pfc_natural_sum_step_helper_AZcoefficientscoefficient. ((forall pfc_index_step_helper_AZcoefficientscoefficientdiagonal. (exists pfa_gap_step_helper_AZcoefficientscoefficientdiagonalbound. pfa_gap_step_helper_AZcoefficientscoefficientdiagonalbound + S (pfc_index_step_helper_AZcoefficientscoefficientdiagonal) = (S (pfc_index_step_helper_AZcoefficients))) -> exists pfc_value_step_helper_AZcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_step_helper_AZcoefficientscoefficientdiagonalentry. ff_h_pfp_step_helper_AZcoefficientscoefficientdiagonalentry + S (pfc_value_step_helper_AZcoefficientscoefficientdiagonal) = S ((S (pfc_index_step_helper_AZcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_AZcoefficientscoefficient)) /\\ exists ff_q_pfp_step_helper_AZcoefficientscoefficientdiagonalentry. pfc_terms_code_step_helper_AZcoefficientscoefficient = ff_q_pfp_step_helper_AZcoefficientscoefficientdiagonalentry * S ((S (pfc_index_step_helper_AZcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_AZcoefficientscoefficient) + (pfc_value_step_helper_AZcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_step_helper_AZcoefficientscoefficientdiagonalterm pfc_left_step_helper_AZcoefficientscoefficientdiagonalterm pfc_right_step_helper_AZcoefficientscoefficientdiagonalterm. (((pfc_index_step_helper_AZcoefficientscoefficientdiagonal)+pfc_complement_step_helper_AZcoefficientscoefficientdiagonalterm=(pfc_index_step_helper_AZcoefficients)) /\\ ((((((exists pfa_gap_step_helper_AZcoefficientscoefficientdiagonaltermleftinside. pfa_gap_step_helper_AZcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_step_helper_AZcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_step_helper_AZcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_step_helper_AZcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_step_helper_AZcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_step_helper_AZcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_step_helper_AZcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_step_helper_AZcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_step_helper_AZcoefficientscoefficientdiagonal)) * ac) + (pfc_left_step_helper_AZcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_AZcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_step_helper_AZcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_step_helper_AZcoefficientscoefficientdiagonal)) /\\ (((pfc_left_step_helper_AZcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_step_helper_AZcoefficientscoefficientdiagonaltermrightinside. pfa_gap_step_helper_AZcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_step_helper_AZcoefficientscoefficientdiagonalterm) = (M+S K)) /\\ ((((exists ff_h_pfp_step_helper_AZcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_step_helper_AZcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_step_helper_AZcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_step_helper_AZcoefficientscoefficientdiagonalterm)) * zc)) /\\ exists ff_q_pfp_step_helper_AZcoefficientscoefficientdiagonaltermrightentry. zb = ff_q_pfp_step_helper_AZcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_step_helper_AZcoefficientscoefficientdiagonalterm)) * zc) + (pfc_right_step_helper_AZcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_AZcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_step_helper_AZcoefficientscoefficientdiagonaltermrightoutside+(M+S K)=(pfc_complement_step_helper_AZcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_step_helper_AZcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_step_helper_AZcoefficientscoefficientdiagonal)=pfc_left_step_helper_AZcoefficientscoefficientdiagonalterm*pfc_right_step_helper_AZcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_step_helper_AZcoefficientscoefficientsum fs_v_pfc_step_helper_AZcoefficientscoefficientsum. ((((exists fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_start. fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_start. fs_u_pfc_step_helper_AZcoefficientscoefficientsum = fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_terminal. fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_step_helper_AZcoefficientscoefficient) = S ((S (S (pfc_index_step_helper_AZcoefficients))) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_terminal. fs_u_pfc_step_helper_AZcoefficientscoefficientsum = fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_step_helper_AZcoefficients))) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum) + (pfc_natural_sum_step_helper_AZcoefficientscoefficient))) /\\ forall fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps = S (pfc_index_step_helper_AZcoefficients)) -> exists fs_a_pfc_step_helper_AZcoefficientscoefficientsum_body_steps fs_r_pfc_step_helper_AZcoefficientscoefficientsum_body_steps fs_s_pfc_step_helper_AZcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_step_helper_AZcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_AZcoefficientscoefficient)) /\\ exists fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_step_helper_AZcoefficientscoefficient = fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_AZcoefficientscoefficient) + (fs_a_pfc_step_helper_AZcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_step_helper_AZcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_step_helper_AZcoefficientscoefficientsum = fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum) + (fs_r_pfc_step_helper_AZcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_step_helper_AZcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_step_helper_AZcoefficientscoefficientsum = fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum) + (fs_s_pfc_step_helper_AZcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_step_helper_AZcoefficientscoefficientsum_body_steps = fs_r_pfc_step_helper_AZcoefficientscoefficientsum_body_steps + fs_a_pfc_step_helper_AZcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_step_helper_AZcoefficientscoefficientresiduebound. pfa_gap_step_helper_AZcoefficientscoefficientresiduebound + S (pfc_value_step_helper_AZcoefficients) = (p)) /\\ ((exists pfa_offset_left_step_helper_AZcoefficientscoefficientresiduecongruence pfa_offset_right_step_helper_AZcoefficientscoefficientresiduecongruence. (pfc_natural_sum_step_helper_AZcoefficientscoefficient) + (p) * pfa_offset_left_step_helper_AZcoefficientscoefficientresiduecongruence = (pfc_value_step_helper_AZcoefficients) + (p) * pfa_offset_right_step_helper_AZcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hAZ", "cases hAZcopy", "cases hAZcopy_right", "cases hAZcopy_right_right", "have hscale_bounds : ((forall fom_index_pfp_step_helper_B_bound. (exists fom_gap_pfp_step_helper_B_bound_index_bound. fom_gap_pfp_step_helper_B_bound_index_bound + S (fom_index_pfp_step_helper_B_bound) = M) -> exists fom_value_pfp_step_helper_B_bound. ((((exists fom_beta_height_pfp_step_helper_B_bound_entry. fom_beta_height_pfp_step_helper_B_bound_entry + S (fom_value_pfp_step_helper_B_bound) = S ((S (fom_index_pfp_step_helper_B_bound)) * bc)) /\\ exists fom_beta_quotient_pfp_step_helper_B_bound_entry. bb = fom_beta_quotient_pfp_step_helper_B_bound_entry * S ((S (fom_index_pfp_step_helper_B_bound)) * bc) + (fom_value_pfp_step_helper_B_bound))) /\\ (exists fom_gap_pfp_step_helper_B_bound_value_bound. fom_gap_pfp_step_helper_B_bound_value_bound + S (fom_value_pfp_step_helper_B_bound) = p))) /\\ ((forall fom_index_pfp_step_helper_scaled_B_bound. (exists fom_gap_pfp_step_helper_scaled_B_bound_index_bound. fom_gap_pfp_step_helper_scaled_B_bound_index_bound + S (fom_index_pfp_step_helper_scaled_B_bound) = M) -> exists fom_value_pfp_step_helper_scaled_B_bound. ((((exists fom_beta_height_pfp_step_helper_scaled_B_bound_entry. fom_beta_height_pfp_step_helper_scaled_B_bound_entry + S (fom_value_pfp_step_helper_scaled_B_bound) = S ((S (fom_index_pfp_step_helper_scaled_B_bound)) * vc)) /\\ exists fom_beta_quotient_pfp_step_helper_scaled_B_bound_entry. vb = fom_beta_quotient_pfp_step_helper_scaled_B_bound_entry * S ((S (fom_index_pfp_step_helper_scaled_B_bound)) * vc) + (fom_value_pfp_step_helper_scaled_B_bound))) /\\ (exists fom_gap_pfp_step_helper_scaled_B_bound_value_bound. fom_gap_pfp_step_helper_scaled_B_bound_value_bound + S (fom_value_pfp_step_helper_scaled_B_bound) = p)))))", "specialize prime_field_polynomial_scale_bounded (p)", "specialize prime_field_polynomial_scale_bounded (c)", "specialize prime_field_polynomial_scale_bounded (bb)", "specialize prime_field_polynomial_scale_bounded (bc)", "specialize prime_field_polynomial_scale_bounded (vb)", "specialize prime_field_polynomial_scale_bounded (vc)", "specialize prime_field_polynomial_scale_bounded (M)", "apply prime_field_polynomial_scale_bounded", "exact hV", "cases hscale_bounds", "have hadd_bounds : ((forall fom_index_pfp_step_helper_input_left_bound. (exists fom_gap_pfp_step_helper_input_left_bound_index_bound. fom_gap_pfp_step_helper_input_left_bound_index_bound + S (fom_index_pfp_step_helper_input_left_bound) = M+S K) -> exists fom_value_pfp_step_helper_input_left_bound. ((((exists fom_beta_height_pfp_step_helper_input_left_bound_entry. fom_beta_height_pfp_step_helper_input_left_bound_entry + S (fom_value_pfp_step_helper_input_left_bound) = S ((S (fom_index_pfp_step_helper_input_left_bound)) * UPc)) /\\ exists fom_beta_quotient_pfp_step_helper_input_left_bound_entry. UPb = fom_beta_quotient_pfp_step_helper_input_left_bound_entry * S ((S (fom_index_pfp_step_helper_input_left_bound)) * UPc) + (fom_value_pfp_step_helper_input_left_bound))) /\\ (exists fom_gap_pfp_step_helper_input_left_bound_value_bound. fom_gap_pfp_step_helper_input_left_bound_value_bound + S (fom_value_pfp_step_helper_input_left_bound) = p))) /\\ (((forall fom_index_pfp_step_helper_input_right_bound. (exists fom_gap_pfp_step_helper_input_right_bound_index_bound. fom_gap_pfp_step_helper_input_right_bound_index_bound + S (fom_index_pfp_step_helper_input_right_bound) = M+S K) -> exists fom_value_pfp_step_helper_input_right_bound. ((((exists fom_beta_height_pfp_step_helper_input_right_bound_entry. fom_beta_height_pfp_step_helper_input_right_bound_entry + S (fom_value_pfp_step_helper_input_right_bound) = S ((S (fom_index_pfp_step_helper_input_right_bound)) * VPc)) /\\ exists fom_beta_quotient_pfp_step_helper_input_right_bound_entry. VPb = fom_beta_quotient_pfp_step_helper_input_right_bound_entry * S ((S (fom_index_pfp_step_helper_input_right_bound)) * VPc) + (fom_value_pfp_step_helper_input_right_bound))) /\\ (exists fom_gap_pfp_step_helper_input_right_bound_value_bound. fom_gap_pfp_step_helper_input_right_bound_value_bound + S (fom_value_pfp_step_helper_input_right_bound) = p))) /\\ ((forall fom_index_pfp_step_helper_input_sum_bound. (exists fom_gap_pfp_step_helper_input_sum_bound_index_bound. fom_gap_pfp_step_helper_input_sum_bound_index_bound + S (fom_index_pfp_step_helper_input_sum_bound) = M+S K) -> exists fom_value_pfp_step_helper_input_sum_bound. ((((exists fom_beta_height_pfp_step_helper_input_sum_bound_entry. fom_beta_height_pfp_step_helper_input_sum_bound_entry + S (fom_value_pfp_step_helper_input_sum_bound) = S ((S (fom_index_pfp_step_helper_input_sum_bound)) * zc)) /\\ exists fom_beta_quotient_pfp_step_helper_input_sum_bound_entry. zb = fom_beta_quotient_pfp_step_helper_input_sum_bound_entry * S ((S (fom_index_pfp_step_helper_input_sum_bound)) * zc) + (fom_value_pfp_step_helper_input_sum_bound))) /\\ (exists fom_gap_pfp_step_helper_input_sum_bound_value_bound. fom_gap_pfp_step_helper_input_sum_bound_value_bound + S (fom_value_pfp_step_helper_input_sum_bound) = p)))))))", "specialize prime_field_polynomial_add_bounded (p)", "specialize prime_field_polynomial_add_bounded (UPb)", "specialize prime_field_polynomial_add_bounded (UPc)", "specialize prime_field_polynomial_add_bounded (VPb)", "specialize prime_field_polynomial_add_bounded (VPc)", "specialize prime_field_polynomial_add_bounded (zb)", "specialize prime_field_polynomial_add_bounded (zc)", "specialize prime_field_polynomial_add_bounded (M+S K)", "apply prime_field_polynomial_add_bounded", "exact hZ", "cases hadd_bounds", "cases hadd_bounds_right", "have hlength : exists J. (((((L)=0 \\/ (S K)=0) /\\ (((J)=0)))) \\/ (((~((L)=0)) /\\ (((~((S K)=0)) /\\ (((L)+(S K)=S (J))))))))", "specialize polynomial_product_length_exists (L)", "specialize polynomial_product_length_exists (S K)", "apply polynomial_product_length_exists", "cases hlength", "have hshifted_product : exists mb mc. ((forall fom_index_pfp_step_helper_shifted_productleft. (exists fom_gap_pfp_step_helper_shifted_productleft_index_bound. fom_gap_pfp_step_helper_shifted_productleft_index_bound + S (fom_index_pfp_step_helper_shifted_productleft) = L) -> exists fom_value_pfp_step_helper_shifted_productleft. ((((exists fom_beta_height_pfp_step_helper_shifted_productleft_entry. fom_beta_height_pfp_step_helper_shifted_productleft_entry + S (fom_value_pfp_step_helper_shifted_productleft) = S ((S (fom_index_pfp_step_helper_shifted_productleft)) * ac)) /\\ exists fom_beta_quotient_pfp_step_helper_shifted_productleft_entry. ab = fom_beta_quotient_pfp_step_helper_shifted_productleft_entry * S ((S (fom_index_pfp_step_helper_shifted_productleft)) * ac) + (fom_value_pfp_step_helper_shifted_productleft))) /\\ (exists fom_gap_pfp_step_helper_shifted_productleft_value_bound. fom_gap_pfp_step_helper_shifted_productleft_value_bound + S (fom_value_pfp_step_helper_shifted_productleft) = p))) /\\ (((forall fom_index_pfp_step_helper_shifted_productright. (exists fom_gap_pfp_step_helper_shifted_productright_index_bound. fom_gap_pfp_step_helper_shifted_productright_index_bound + S (fom_index_pfp_step_helper_shifted_productright) = S K) -> exists fom_value_pfp_step_helper_shifted_productright. ((((exists fom_beta_height_pfp_step_helper_shifted_productright_entry. fom_beta_height_pfp_step_helper_shifted_productright_entry + S (fom_value_pfp_step_helper_shifted_productright) = S ((S (fom_index_pfp_step_helper_shifted_productright)) * uc)) /\\ exists fom_beta_quotient_pfp_step_helper_shifted_productright_entry. ub = fom_beta_quotient_pfp_step_helper_shifted_productright_entry * S ((S (fom_index_pfp_step_helper_shifted_productright)) * uc) + (fom_value_pfp_step_helper_shifted_productright))) /\\ (exists fom_gap_pfp_step_helper_shifted_productright_value_bound. fom_gap_pfp_step_helper_shifted_productright_value_bound + S (fom_value_pfp_step_helper_shifted_productright) = p))) /\\ (((((((L)=0 \\/ (S K)=0) /\\ (((x)=0)))) \\/ (((~((L)=0)) /\\ (((~((S K)=0)) /\\ (((L)+(S K)=S (x)))))))) /\\ ((forall pfc_index_step_helper_shifted_productcoefficients. (exists pfa_gap_step_helper_shifted_productcoefficientsbound. pfa_gap_step_helper_shifted_productcoefficientsbound + S (pfc_index_step_helper_shifted_productcoefficients) = (x)) -> exists pfc_value_step_helper_shifted_productcoefficients. ((((exists ff_h_pfp_step_helper_shifted_productcoefficientsentry. ff_h_pfp_step_helper_shifted_productcoefficientsentry + S (pfc_value_step_helper_shifted_productcoefficients) = S ((S (pfc_index_step_helper_shifted_productcoefficients)) * mc)) /\\ exists ff_q_pfp_step_helper_shifted_productcoefficientsentry. mb = ff_q_pfp_step_helper_shifted_productcoefficientsentry * S ((S (pfc_index_step_helper_shifted_productcoefficients)) * mc) + (pfc_value_step_helper_shifted_productcoefficients))) /\\ ((exists pfc_terms_code_step_helper_shifted_productcoefficientscoefficient pfc_terms_scale_step_helper_shifted_productcoefficientscoefficient pfc_natural_sum_step_helper_shifted_productcoefficientscoefficient. ((forall pfc_index_step_helper_shifted_productcoefficientscoefficientdiagonal. (exists pfa_gap_step_helper_shifted_productcoefficientscoefficientdiagonalbound. pfa_gap_step_helper_shifted_productcoefficientscoefficientdiagonalbound + S (pfc_index_step_helper_shifted_productcoefficientscoefficientdiagonal) = (S (pfc_index_step_helper_shifted_productcoefficients))) -> exists pfc_value_step_helper_shifted_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_step_helper_shifted_productcoefficientscoefficientdiagonalentry. ff_h_pfp_step_helper_shifted_productcoefficientscoefficientdiagonalentry + S (pfc_value_step_helper_shifted_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_step_helper_shifted_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_shifted_productcoefficientscoefficient)) /\\ exists ff_q_pfp_step_helper_shifted_productcoefficientscoefficientdiagonalentry. pfc_terms_code_step_helper_shifted_productcoefficientscoefficient = ff_q_pfp_step_helper_shifted_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_step_helper_shifted_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_shifted_productcoefficientscoefficient) + (pfc_value_step_helper_shifted_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_step_helper_shifted_productcoefficientscoefficientdiagonalterm pfc_left_step_helper_shifted_productcoefficientscoefficientdiagonalterm pfc_right_step_helper_shifted_productcoefficientscoefficientdiagonalterm. (((pfc_index_step_helper_shifted_productcoefficientscoefficientdiagonal)+pfc_complement_step_helper_shifted_productcoefficientscoefficientdiagonalterm=(pfc_index_step_helper_shifted_productcoefficients)) /\\ ((((((exists pfa_gap_step_helper_shifted_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_step_helper_shifted_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_step_helper_shifted_productcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_step_helper_shifted_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_step_helper_shifted_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_step_helper_shifted_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_step_helper_shifted_productcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_step_helper_shifted_productcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_step_helper_shifted_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_step_helper_shifted_productcoefficientscoefficientdiagonal)) * ac) + (pfc_left_step_helper_shifted_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_shifted_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_step_helper_shifted_productcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_step_helper_shifted_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_step_helper_shifted_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_step_helper_shifted_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_step_helper_shifted_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_step_helper_shifted_productcoefficientscoefficientdiagonalterm) = (S K)) /\\ ((((exists ff_h_pfp_step_helper_shifted_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_step_helper_shifted_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_step_helper_shifted_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_step_helper_shifted_productcoefficientscoefficientdiagonalterm)) * uc)) /\\ exists ff_q_pfp_step_helper_shifted_productcoefficientscoefficientdiagonaltermrightentry. ub = ff_q_pfp_step_helper_shifted_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_step_helper_shifted_productcoefficientscoefficientdiagonalterm)) * uc) + (pfc_right_step_helper_shifted_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_shifted_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_step_helper_shifted_productcoefficientscoefficientdiagonaltermrightoutside+(S K)=(pfc_complement_step_helper_shifted_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_step_helper_shifted_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_step_helper_shifted_productcoefficientscoefficientdiagonal)=pfc_left_step_helper_shifted_productcoefficientscoefficientdiagonalterm*pfc_right_step_helper_shifted_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_step_helper_shifted_productcoefficientscoefficientsum fs_v_pfc_step_helper_shifted_productcoefficientscoefficientsum. ((((exists fs_h_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_start. fs_h_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_helper_shifted_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_start. fs_u_pfc_step_helper_shifted_productcoefficientscoefficientsum = fs_q_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_step_helper_shifted_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_step_helper_shifted_productcoefficientscoefficient) = S ((S (S (pfc_index_step_helper_shifted_productcoefficients))) * fs_v_pfc_step_helper_shifted_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_step_helper_shifted_productcoefficientscoefficientsum = fs_q_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_step_helper_shifted_productcoefficients))) * fs_v_pfc_step_helper_shifted_productcoefficientscoefficientsum) + (pfc_natural_sum_step_helper_shifted_productcoefficientscoefficient))) /\\ forall fs_i_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps = S (pfc_index_step_helper_shifted_productcoefficients)) -> exists fs_a_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps fs_r_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps fs_s_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_shifted_productcoefficientscoefficient)) /\\ exists fs_q_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_step_helper_shifted_productcoefficientscoefficient = fs_q_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_shifted_productcoefficientscoefficient) + (fs_a_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_shifted_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_step_helper_shifted_productcoefficientscoefficientsum = fs_q_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_shifted_productcoefficientscoefficientsum) + (fs_r_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_shifted_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_step_helper_shifted_productcoefficientscoefficientsum = fs_q_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_shifted_productcoefficientscoefficientsum) + (fs_s_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps = fs_r_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps + fs_a_pfc_step_helper_shifted_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_step_helper_shifted_productcoefficientscoefficientresiduebound. pfa_gap_step_helper_shifted_productcoefficientscoefficientresiduebound + S (pfc_value_step_helper_shifted_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_step_helper_shifted_productcoefficientscoefficientresiduecongruence pfa_offset_right_step_helper_shifted_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_step_helper_shifted_productcoefficientscoefficient) + (p) * pfa_offset_left_step_helper_shifted_productcoefficientscoefficientresiduecongruence = (pfc_value_step_helper_shifted_productcoefficients) + (p) * pfa_offset_right_step_helper_shifted_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (ab)", "specialize prime_field_polynomial_convolution_at_length_exists (ac)", "specialize prime_field_polynomial_convolution_at_length_exists (L)", "specialize prime_field_polynomial_convolution_at_length_exists (ub)", "specialize prime_field_polynomial_convolution_at_length_exists (uc)", "specialize prime_field_polynomial_convolution_at_length_exists (S K)", "specialize prime_field_polynomial_convolution_at_length_exists (x)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hABcopy_left", "specialize prime_field_polynomial_shift_bounded (p)", "specialize prime_field_polynomial_shift_bounded (qb)", "specialize prime_field_polynomial_shift_bounded (qc)", "specialize prime_field_polynomial_shift_bounded (K)", "specialize prime_field_polynomial_shift_bounded (ub)", "specialize prime_field_polynomial_shift_bounded (uc)", "apply prime_field_polynomial_shift_bounded", "exact hp", "exact hAQcopy_right_left", "exact hU", "exact hlength_witness", "cases hshifted_product", "cases hshifted_product_witness", "have hscaled_product : exists mb mc. ((forall fom_index_pfp_step_helper_scaled_productleft. (exists fom_gap_pfp_step_helper_scaled_productleft_index_bound. fom_gap_pfp_step_helper_scaled_productleft_index_bound + S (fom_index_pfp_step_helper_scaled_productleft) = L) -> exists fom_value_pfp_step_helper_scaled_productleft. ((((exists fom_beta_height_pfp_step_helper_scaled_productleft_entry. fom_beta_height_pfp_step_helper_scaled_productleft_entry + S (fom_value_pfp_step_helper_scaled_productleft) = S ((S (fom_index_pfp_step_helper_scaled_productleft)) * ac)) /\\ exists fom_beta_quotient_pfp_step_helper_scaled_productleft_entry. ab = fom_beta_quotient_pfp_step_helper_scaled_productleft_entry * S ((S (fom_index_pfp_step_helper_scaled_productleft)) * ac) + (fom_value_pfp_step_helper_scaled_productleft))) /\\ (exists fom_gap_pfp_step_helper_scaled_productleft_value_bound. fom_gap_pfp_step_helper_scaled_productleft_value_bound + S (fom_value_pfp_step_helper_scaled_productleft) = p))) /\\ (((forall fom_index_pfp_step_helper_scaled_productright. (exists fom_gap_pfp_step_helper_scaled_productright_index_bound. fom_gap_pfp_step_helper_scaled_productright_index_bound + S (fom_index_pfp_step_helper_scaled_productright) = M) -> exists fom_value_pfp_step_helper_scaled_productright. ((((exists fom_beta_height_pfp_step_helper_scaled_productright_entry. fom_beta_height_pfp_step_helper_scaled_productright_entry + S (fom_value_pfp_step_helper_scaled_productright) = S ((S (fom_index_pfp_step_helper_scaled_productright)) * vc)) /\\ exists fom_beta_quotient_pfp_step_helper_scaled_productright_entry. vb = fom_beta_quotient_pfp_step_helper_scaled_productright_entry * S ((S (fom_index_pfp_step_helper_scaled_productright)) * vc) + (fom_value_pfp_step_helper_scaled_productright))) /\\ (exists fom_gap_pfp_step_helper_scaled_productright_value_bound. fom_gap_pfp_step_helper_scaled_productright_value_bound + S (fom_value_pfp_step_helper_scaled_productright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_step_helper_scaled_productcoefficients. (exists pfa_gap_step_helper_scaled_productcoefficientsbound. pfa_gap_step_helper_scaled_productcoefficientsbound + S (pfc_index_step_helper_scaled_productcoefficients) = (N)) -> exists pfc_value_step_helper_scaled_productcoefficients. ((((exists ff_h_pfp_step_helper_scaled_productcoefficientsentry. ff_h_pfp_step_helper_scaled_productcoefficientsentry + S (pfc_value_step_helper_scaled_productcoefficients) = S ((S (pfc_index_step_helper_scaled_productcoefficients)) * mc)) /\\ exists ff_q_pfp_step_helper_scaled_productcoefficientsentry. mb = ff_q_pfp_step_helper_scaled_productcoefficientsentry * S ((S (pfc_index_step_helper_scaled_productcoefficients)) * mc) + (pfc_value_step_helper_scaled_productcoefficients))) /\\ ((exists pfc_terms_code_step_helper_scaled_productcoefficientscoefficient pfc_terms_scale_step_helper_scaled_productcoefficientscoefficient pfc_natural_sum_step_helper_scaled_productcoefficientscoefficient. ((forall pfc_index_step_helper_scaled_productcoefficientscoefficientdiagonal. (exists pfa_gap_step_helper_scaled_productcoefficientscoefficientdiagonalbound. pfa_gap_step_helper_scaled_productcoefficientscoefficientdiagonalbound + S (pfc_index_step_helper_scaled_productcoefficientscoefficientdiagonal) = (S (pfc_index_step_helper_scaled_productcoefficients))) -> exists pfc_value_step_helper_scaled_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_step_helper_scaled_productcoefficientscoefficientdiagonalentry. ff_h_pfp_step_helper_scaled_productcoefficientscoefficientdiagonalentry + S (pfc_value_step_helper_scaled_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_step_helper_scaled_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_scaled_productcoefficientscoefficient)) /\\ exists ff_q_pfp_step_helper_scaled_productcoefficientscoefficientdiagonalentry. pfc_terms_code_step_helper_scaled_productcoefficientscoefficient = ff_q_pfp_step_helper_scaled_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_step_helper_scaled_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_scaled_productcoefficientscoefficient) + (pfc_value_step_helper_scaled_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_step_helper_scaled_productcoefficientscoefficientdiagonalterm pfc_left_step_helper_scaled_productcoefficientscoefficientdiagonalterm pfc_right_step_helper_scaled_productcoefficientscoefficientdiagonalterm. (((pfc_index_step_helper_scaled_productcoefficientscoefficientdiagonal)+pfc_complement_step_helper_scaled_productcoefficientscoefficientdiagonalterm=(pfc_index_step_helper_scaled_productcoefficients)) /\\ ((((((exists pfa_gap_step_helper_scaled_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_step_helper_scaled_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_step_helper_scaled_productcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_step_helper_scaled_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_step_helper_scaled_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_step_helper_scaled_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_step_helper_scaled_productcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_step_helper_scaled_productcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_step_helper_scaled_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_step_helper_scaled_productcoefficientscoefficientdiagonal)) * ac) + (pfc_left_step_helper_scaled_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_scaled_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_step_helper_scaled_productcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_step_helper_scaled_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_step_helper_scaled_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_step_helper_scaled_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_step_helper_scaled_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_step_helper_scaled_productcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_step_helper_scaled_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_step_helper_scaled_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_step_helper_scaled_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_step_helper_scaled_productcoefficientscoefficientdiagonalterm)) * vc)) /\\ exists ff_q_pfp_step_helper_scaled_productcoefficientscoefficientdiagonaltermrightentry. vb = ff_q_pfp_step_helper_scaled_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_step_helper_scaled_productcoefficientscoefficientdiagonalterm)) * vc) + (pfc_right_step_helper_scaled_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_scaled_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_step_helper_scaled_productcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_step_helper_scaled_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_step_helper_scaled_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_step_helper_scaled_productcoefficientscoefficientdiagonal)=pfc_left_step_helper_scaled_productcoefficientscoefficientdiagonalterm*pfc_right_step_helper_scaled_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_step_helper_scaled_productcoefficientscoefficientsum fs_v_pfc_step_helper_scaled_productcoefficientscoefficientsum. ((((exists fs_h_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_start. fs_h_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_helper_scaled_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_start. fs_u_pfc_step_helper_scaled_productcoefficientscoefficientsum = fs_q_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_step_helper_scaled_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_step_helper_scaled_productcoefficientscoefficient) = S ((S (S (pfc_index_step_helper_scaled_productcoefficients))) * fs_v_pfc_step_helper_scaled_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_step_helper_scaled_productcoefficientscoefficientsum = fs_q_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_step_helper_scaled_productcoefficients))) * fs_v_pfc_step_helper_scaled_productcoefficientscoefficientsum) + (pfc_natural_sum_step_helper_scaled_productcoefficientscoefficient))) /\\ forall fs_i_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps = S (pfc_index_step_helper_scaled_productcoefficients)) -> exists fs_a_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps fs_r_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps fs_s_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_scaled_productcoefficientscoefficient)) /\\ exists fs_q_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_step_helper_scaled_productcoefficientscoefficient = fs_q_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_scaled_productcoefficientscoefficient) + (fs_a_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_scaled_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_step_helper_scaled_productcoefficientscoefficientsum = fs_q_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_scaled_productcoefficientscoefficientsum) + (fs_r_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_scaled_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_step_helper_scaled_productcoefficientscoefficientsum = fs_q_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_scaled_productcoefficientscoefficientsum) + (fs_s_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps = fs_r_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps + fs_a_pfc_step_helper_scaled_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_step_helper_scaled_productcoefficientscoefficientresiduebound. pfa_gap_step_helper_scaled_productcoefficientscoefficientresiduebound + S (pfc_value_step_helper_scaled_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_step_helper_scaled_productcoefficientscoefficientresiduecongruence pfa_offset_right_step_helper_scaled_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_step_helper_scaled_productcoefficientscoefficient) + (p) * pfa_offset_left_step_helper_scaled_productcoefficientscoefficientresiduecongruence = (pfc_value_step_helper_scaled_productcoefficients) + (p) * pfa_offset_right_step_helper_scaled_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (ab)", "specialize prime_field_polynomial_convolution_at_length_exists (ac)", "specialize prime_field_polynomial_convolution_at_length_exists (L)", "specialize prime_field_polynomial_convolution_at_length_exists (vb)", "specialize prime_field_polynomial_convolution_at_length_exists (vc)", "specialize prime_field_polynomial_convolution_at_length_exists (M)", "specialize prime_field_polynomial_convolution_at_length_exists (N)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hABcopy_left", "exact hscale_bounds_right", "exact hABcopy_right_right_left", "cases hscaled_product", "cases hscaled_product_witness", "have hfirst : exists mb mc. ((forall fom_index_pfp_step_helper_hfirstleft. (exists fom_gap_pfp_step_helper_hfirstleft_index_bound. fom_gap_pfp_step_helper_hfirstleft_index_bound + S (fom_index_pfp_step_helper_hfirstleft) = L) -> exists fom_value_pfp_step_helper_hfirstleft. ((((exists fom_beta_height_pfp_step_helper_hfirstleft_entry. fom_beta_height_pfp_step_helper_hfirstleft_entry + S (fom_value_pfp_step_helper_hfirstleft) = S ((S (fom_index_pfp_step_helper_hfirstleft)) * ac)) /\\ exists fom_beta_quotient_pfp_step_helper_hfirstleft_entry. ab = fom_beta_quotient_pfp_step_helper_hfirstleft_entry * S ((S (fom_index_pfp_step_helper_hfirstleft)) * ac) + (fom_value_pfp_step_helper_hfirstleft))) /\\ (exists fom_gap_pfp_step_helper_hfirstleft_value_bound. fom_gap_pfp_step_helper_hfirstleft_value_bound + S (fom_value_pfp_step_helper_hfirstleft) = p))) /\\ (((forall fom_index_pfp_step_helper_hfirstright. (exists fom_gap_pfp_step_helper_hfirstright_index_bound. fom_gap_pfp_step_helper_hfirstright_index_bound + S (fom_index_pfp_step_helper_hfirstright) = M+S K) -> exists fom_value_pfp_step_helper_hfirstright. ((((exists fom_beta_height_pfp_step_helper_hfirstright_entry. fom_beta_height_pfp_step_helper_hfirstright_entry + S (fom_value_pfp_step_helper_hfirstright) = S ((S (fom_index_pfp_step_helper_hfirstright)) * UPc)) /\\ exists fom_beta_quotient_pfp_step_helper_hfirstright_entry. UPb = fom_beta_quotient_pfp_step_helper_hfirstright_entry * S ((S (fom_index_pfp_step_helper_hfirstright)) * UPc) + (fom_value_pfp_step_helper_hfirstright))) /\\ (exists fom_gap_pfp_step_helper_hfirstright_value_bound. fom_gap_pfp_step_helper_hfirstright_value_bound + S (fom_value_pfp_step_helper_hfirstright) = p))) /\\ (((((((L)=0 \\/ (M+S K)=0) /\\ (((W)=0)))) \\/ (((~((L)=0)) /\\ (((~((M+S K)=0)) /\\ (((L)+(M+S K)=S (W)))))))) /\\ ((forall pfc_index_step_helper_hfirstcoefficients. (exists pfa_gap_step_helper_hfirstcoefficientsbound. pfa_gap_step_helper_hfirstcoefficientsbound + S (pfc_index_step_helper_hfirstcoefficients) = (W)) -> exists pfc_value_step_helper_hfirstcoefficients. ((((exists ff_h_pfp_step_helper_hfirstcoefficientsentry. ff_h_pfp_step_helper_hfirstcoefficientsentry + S (pfc_value_step_helper_hfirstcoefficients) = S ((S (pfc_index_step_helper_hfirstcoefficients)) * mc)) /\\ exists ff_q_pfp_step_helper_hfirstcoefficientsentry. mb = ff_q_pfp_step_helper_hfirstcoefficientsentry * S ((S (pfc_index_step_helper_hfirstcoefficients)) * mc) + (pfc_value_step_helper_hfirstcoefficients))) /\\ ((exists pfc_terms_code_step_helper_hfirstcoefficientscoefficient pfc_terms_scale_step_helper_hfirstcoefficientscoefficient pfc_natural_sum_step_helper_hfirstcoefficientscoefficient. ((forall pfc_index_step_helper_hfirstcoefficientscoefficientdiagonal. (exists pfa_gap_step_helper_hfirstcoefficientscoefficientdiagonalbound. pfa_gap_step_helper_hfirstcoefficientscoefficientdiagonalbound + S (pfc_index_step_helper_hfirstcoefficientscoefficientdiagonal) = (S (pfc_index_step_helper_hfirstcoefficients))) -> exists pfc_value_step_helper_hfirstcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_step_helper_hfirstcoefficientscoefficientdiagonalentry. ff_h_pfp_step_helper_hfirstcoefficientscoefficientdiagonalentry + S (pfc_value_step_helper_hfirstcoefficientscoefficientdiagonal) = S ((S (pfc_index_step_helper_hfirstcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_hfirstcoefficientscoefficient)) /\\ exists ff_q_pfp_step_helper_hfirstcoefficientscoefficientdiagonalentry. pfc_terms_code_step_helper_hfirstcoefficientscoefficient = ff_q_pfp_step_helper_hfirstcoefficientscoefficientdiagonalentry * S ((S (pfc_index_step_helper_hfirstcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_hfirstcoefficientscoefficient) + (pfc_value_step_helper_hfirstcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_step_helper_hfirstcoefficientscoefficientdiagonalterm pfc_left_step_helper_hfirstcoefficientscoefficientdiagonalterm pfc_right_step_helper_hfirstcoefficientscoefficientdiagonalterm. (((pfc_index_step_helper_hfirstcoefficientscoefficientdiagonal)+pfc_complement_step_helper_hfirstcoefficientscoefficientdiagonalterm=(pfc_index_step_helper_hfirstcoefficients)) /\\ ((((((exists pfa_gap_step_helper_hfirstcoefficientscoefficientdiagonaltermleftinside. pfa_gap_step_helper_hfirstcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_step_helper_hfirstcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_step_helper_hfirstcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_step_helper_hfirstcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_step_helper_hfirstcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_step_helper_hfirstcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_step_helper_hfirstcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_step_helper_hfirstcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_step_helper_hfirstcoefficientscoefficientdiagonal)) * ac) + (pfc_left_step_helper_hfirstcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_hfirstcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_step_helper_hfirstcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_step_helper_hfirstcoefficientscoefficientdiagonal)) /\\ (((pfc_left_step_helper_hfirstcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_step_helper_hfirstcoefficientscoefficientdiagonaltermrightinside. pfa_gap_step_helper_hfirstcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_step_helper_hfirstcoefficientscoefficientdiagonalterm) = (M+S K)) /\\ ((((exists ff_h_pfp_step_helper_hfirstcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_step_helper_hfirstcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_step_helper_hfirstcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_step_helper_hfirstcoefficientscoefficientdiagonalterm)) * UPc)) /\\ exists ff_q_pfp_step_helper_hfirstcoefficientscoefficientdiagonaltermrightentry. UPb = ff_q_pfp_step_helper_hfirstcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_step_helper_hfirstcoefficientscoefficientdiagonalterm)) * UPc) + (pfc_right_step_helper_hfirstcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_hfirstcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_step_helper_hfirstcoefficientscoefficientdiagonaltermrightoutside+(M+S K)=(pfc_complement_step_helper_hfirstcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_step_helper_hfirstcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_step_helper_hfirstcoefficientscoefficientdiagonal)=pfc_left_step_helper_hfirstcoefficientscoefficientdiagonalterm*pfc_right_step_helper_hfirstcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_step_helper_hfirstcoefficientscoefficientsum fs_v_pfc_step_helper_hfirstcoefficientscoefficientsum. ((((exists fs_h_pfc_step_helper_hfirstcoefficientscoefficientsum_body_start. fs_h_pfc_step_helper_hfirstcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_helper_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_hfirstcoefficientscoefficientsum_body_start. fs_u_pfc_step_helper_hfirstcoefficientscoefficientsum = fs_q_pfc_step_helper_hfirstcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_step_helper_hfirstcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_step_helper_hfirstcoefficientscoefficientsum_body_terminal. fs_h_pfc_step_helper_hfirstcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_step_helper_hfirstcoefficientscoefficient) = S ((S (S (pfc_index_step_helper_hfirstcoefficients))) * fs_v_pfc_step_helper_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_hfirstcoefficientscoefficientsum_body_terminal. fs_u_pfc_step_helper_hfirstcoefficientscoefficientsum = fs_q_pfc_step_helper_hfirstcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_step_helper_hfirstcoefficients))) * fs_v_pfc_step_helper_hfirstcoefficientscoefficientsum) + (pfc_natural_sum_step_helper_hfirstcoefficientscoefficient))) /\\ forall fs_i_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps = S (pfc_index_step_helper_hfirstcoefficients)) -> exists fs_a_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps fs_r_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps fs_s_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_hfirstcoefficientscoefficient)) /\\ exists fs_q_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_step_helper_hfirstcoefficientscoefficient = fs_q_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_hfirstcoefficientscoefficient) + (fs_a_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_step_helper_hfirstcoefficientscoefficientsum = fs_q_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_hfirstcoefficientscoefficientsum) + (fs_r_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_step_helper_hfirstcoefficientscoefficientsum = fs_q_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_hfirstcoefficientscoefficientsum) + (fs_s_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps = fs_r_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps + fs_a_pfc_step_helper_hfirstcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_step_helper_hfirstcoefficientscoefficientresiduebound. pfa_gap_step_helper_hfirstcoefficientscoefficientresiduebound + S (pfc_value_step_helper_hfirstcoefficients) = (p)) /\\ ((exists pfa_offset_left_step_helper_hfirstcoefficientscoefficientresiduecongruence pfa_offset_right_step_helper_hfirstcoefficientscoefficientresiduecongruence. (pfc_natural_sum_step_helper_hfirstcoefficientscoefficient) + (p) * pfa_offset_left_step_helper_hfirstcoefficientscoefficientresiduecongruence = (pfc_value_step_helper_hfirstcoefficients) + (p) * pfa_offset_right_step_helper_hfirstcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (ab)", "specialize prime_field_polynomial_convolution_at_length_exists (ac)", "specialize prime_field_polynomial_convolution_at_length_exists (L)", "specialize prime_field_polynomial_convolution_at_length_exists (UPb)", "specialize prime_field_polynomial_convolution_at_length_exists (UPc)", "specialize prime_field_polynomial_convolution_at_length_exists (M+S K)", "specialize prime_field_polynomial_convolution_at_length_exists (W)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hABcopy_left", "exact hadd_bounds_left", "exact hAZcopy_right_right_left", "cases hfirst", "cases hfirst_witness", "have hsecond : exists mb mc. ((forall fom_index_pfp_step_helper_hsecondleft. (exists fom_gap_pfp_step_helper_hsecondleft_index_bound. fom_gap_pfp_step_helper_hsecondleft_index_bound + S (fom_index_pfp_step_helper_hsecondleft) = L) -> exists fom_value_pfp_step_helper_hsecondleft. ((((exists fom_beta_height_pfp_step_helper_hsecondleft_entry. fom_beta_height_pfp_step_helper_hsecondleft_entry + S (fom_value_pfp_step_helper_hsecondleft) = S ((S (fom_index_pfp_step_helper_hsecondleft)) * ac)) /\\ exists fom_beta_quotient_pfp_step_helper_hsecondleft_entry. ab = fom_beta_quotient_pfp_step_helper_hsecondleft_entry * S ((S (fom_index_pfp_step_helper_hsecondleft)) * ac) + (fom_value_pfp_step_helper_hsecondleft))) /\\ (exists fom_gap_pfp_step_helper_hsecondleft_value_bound. fom_gap_pfp_step_helper_hsecondleft_value_bound + S (fom_value_pfp_step_helper_hsecondleft) = p))) /\\ (((forall fom_index_pfp_step_helper_hsecondright. (exists fom_gap_pfp_step_helper_hsecondright_index_bound. fom_gap_pfp_step_helper_hsecondright_index_bound + S (fom_index_pfp_step_helper_hsecondright) = M+S K) -> exists fom_value_pfp_step_helper_hsecondright. ((((exists fom_beta_height_pfp_step_helper_hsecondright_entry. fom_beta_height_pfp_step_helper_hsecondright_entry + S (fom_value_pfp_step_helper_hsecondright) = S ((S (fom_index_pfp_step_helper_hsecondright)) * VPc)) /\\ exists fom_beta_quotient_pfp_step_helper_hsecondright_entry. VPb = fom_beta_quotient_pfp_step_helper_hsecondright_entry * S ((S (fom_index_pfp_step_helper_hsecondright)) * VPc) + (fom_value_pfp_step_helper_hsecondright))) /\\ (exists fom_gap_pfp_step_helper_hsecondright_value_bound. fom_gap_pfp_step_helper_hsecondright_value_bound + S (fom_value_pfp_step_helper_hsecondright) = p))) /\\ (((((((L)=0 \\/ (M+S K)=0) /\\ (((W)=0)))) \\/ (((~((L)=0)) /\\ (((~((M+S K)=0)) /\\ (((L)+(M+S K)=S (W)))))))) /\\ ((forall pfc_index_step_helper_hsecondcoefficients. (exists pfa_gap_step_helper_hsecondcoefficientsbound. pfa_gap_step_helper_hsecondcoefficientsbound + S (pfc_index_step_helper_hsecondcoefficients) = (W)) -> exists pfc_value_step_helper_hsecondcoefficients. ((((exists ff_h_pfp_step_helper_hsecondcoefficientsentry. ff_h_pfp_step_helper_hsecondcoefficientsentry + S (pfc_value_step_helper_hsecondcoefficients) = S ((S (pfc_index_step_helper_hsecondcoefficients)) * mc)) /\\ exists ff_q_pfp_step_helper_hsecondcoefficientsentry. mb = ff_q_pfp_step_helper_hsecondcoefficientsentry * S ((S (pfc_index_step_helper_hsecondcoefficients)) * mc) + (pfc_value_step_helper_hsecondcoefficients))) /\\ ((exists pfc_terms_code_step_helper_hsecondcoefficientscoefficient pfc_terms_scale_step_helper_hsecondcoefficientscoefficient pfc_natural_sum_step_helper_hsecondcoefficientscoefficient. ((forall pfc_index_step_helper_hsecondcoefficientscoefficientdiagonal. (exists pfa_gap_step_helper_hsecondcoefficientscoefficientdiagonalbound. pfa_gap_step_helper_hsecondcoefficientscoefficientdiagonalbound + S (pfc_index_step_helper_hsecondcoefficientscoefficientdiagonal) = (S (pfc_index_step_helper_hsecondcoefficients))) -> exists pfc_value_step_helper_hsecondcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_step_helper_hsecondcoefficientscoefficientdiagonalentry. ff_h_pfp_step_helper_hsecondcoefficientscoefficientdiagonalentry + S (pfc_value_step_helper_hsecondcoefficientscoefficientdiagonal) = S ((S (pfc_index_step_helper_hsecondcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_hsecondcoefficientscoefficient)) /\\ exists ff_q_pfp_step_helper_hsecondcoefficientscoefficientdiagonalentry. pfc_terms_code_step_helper_hsecondcoefficientscoefficient = ff_q_pfp_step_helper_hsecondcoefficientscoefficientdiagonalentry * S ((S (pfc_index_step_helper_hsecondcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_hsecondcoefficientscoefficient) + (pfc_value_step_helper_hsecondcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_step_helper_hsecondcoefficientscoefficientdiagonalterm pfc_left_step_helper_hsecondcoefficientscoefficientdiagonalterm pfc_right_step_helper_hsecondcoefficientscoefficientdiagonalterm. (((pfc_index_step_helper_hsecondcoefficientscoefficientdiagonal)+pfc_complement_step_helper_hsecondcoefficientscoefficientdiagonalterm=(pfc_index_step_helper_hsecondcoefficients)) /\\ ((((((exists pfa_gap_step_helper_hsecondcoefficientscoefficientdiagonaltermleftinside. pfa_gap_step_helper_hsecondcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_step_helper_hsecondcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_step_helper_hsecondcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_step_helper_hsecondcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_step_helper_hsecondcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_step_helper_hsecondcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_step_helper_hsecondcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_step_helper_hsecondcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_step_helper_hsecondcoefficientscoefficientdiagonal)) * ac) + (pfc_left_step_helper_hsecondcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_hsecondcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_step_helper_hsecondcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_step_helper_hsecondcoefficientscoefficientdiagonal)) /\\ (((pfc_left_step_helper_hsecondcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_step_helper_hsecondcoefficientscoefficientdiagonaltermrightinside. pfa_gap_step_helper_hsecondcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_step_helper_hsecondcoefficientscoefficientdiagonalterm) = (M+S K)) /\\ ((((exists ff_h_pfp_step_helper_hsecondcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_step_helper_hsecondcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_step_helper_hsecondcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_step_helper_hsecondcoefficientscoefficientdiagonalterm)) * VPc)) /\\ exists ff_q_pfp_step_helper_hsecondcoefficientscoefficientdiagonaltermrightentry. VPb = ff_q_pfp_step_helper_hsecondcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_step_helper_hsecondcoefficientscoefficientdiagonalterm)) * VPc) + (pfc_right_step_helper_hsecondcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_hsecondcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_step_helper_hsecondcoefficientscoefficientdiagonaltermrightoutside+(M+S K)=(pfc_complement_step_helper_hsecondcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_step_helper_hsecondcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_step_helper_hsecondcoefficientscoefficientdiagonal)=pfc_left_step_helper_hsecondcoefficientscoefficientdiagonalterm*pfc_right_step_helper_hsecondcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_step_helper_hsecondcoefficientscoefficientsum fs_v_pfc_step_helper_hsecondcoefficientscoefficientsum. ((((exists fs_h_pfc_step_helper_hsecondcoefficientscoefficientsum_body_start. fs_h_pfc_step_helper_hsecondcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_helper_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_hsecondcoefficientscoefficientsum_body_start. fs_u_pfc_step_helper_hsecondcoefficientscoefficientsum = fs_q_pfc_step_helper_hsecondcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_step_helper_hsecondcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_step_helper_hsecondcoefficientscoefficientsum_body_terminal. fs_h_pfc_step_helper_hsecondcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_step_helper_hsecondcoefficientscoefficient) = S ((S (S (pfc_index_step_helper_hsecondcoefficients))) * fs_v_pfc_step_helper_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_hsecondcoefficientscoefficientsum_body_terminal. fs_u_pfc_step_helper_hsecondcoefficientscoefficientsum = fs_q_pfc_step_helper_hsecondcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_step_helper_hsecondcoefficients))) * fs_v_pfc_step_helper_hsecondcoefficientscoefficientsum) + (pfc_natural_sum_step_helper_hsecondcoefficientscoefficient))) /\\ forall fs_i_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps = S (pfc_index_step_helper_hsecondcoefficients)) -> exists fs_a_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps fs_r_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps fs_s_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_hsecondcoefficientscoefficient)) /\\ exists fs_q_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_step_helper_hsecondcoefficientscoefficient = fs_q_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_hsecondcoefficientscoefficient) + (fs_a_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_step_helper_hsecondcoefficientscoefficientsum = fs_q_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_hsecondcoefficientscoefficientsum) + (fs_r_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_step_helper_hsecondcoefficientscoefficientsum = fs_q_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_hsecondcoefficientscoefficientsum) + (fs_s_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps = fs_r_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps + fs_a_pfc_step_helper_hsecondcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_step_helper_hsecondcoefficientscoefficientresiduebound. pfa_gap_step_helper_hsecondcoefficientscoefficientresiduebound + S (pfc_value_step_helper_hsecondcoefficients) = (p)) /\\ ((exists pfa_offset_left_step_helper_hsecondcoefficientscoefficientresiduecongruence pfa_offset_right_step_helper_hsecondcoefficientscoefficientresiduecongruence. (pfc_natural_sum_step_helper_hsecondcoefficientscoefficient) + (p) * pfa_offset_left_step_helper_hsecondcoefficientscoefficientresiduecongruence = (pfc_value_step_helper_hsecondcoefficients) + (p) * pfa_offset_right_step_helper_hsecondcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (ab)", "specialize prime_field_polynomial_convolution_at_length_exists (ac)", "specialize prime_field_polynomial_convolution_at_length_exists (L)", "specialize prime_field_polynomial_convolution_at_length_exists (VPb)", "specialize prime_field_polynomial_convolution_at_length_exists (VPc)", "specialize prime_field_polynomial_convolution_at_length_exists (M+S K)", "specialize prime_field_polynomial_convolution_at_length_exists (W)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hABcopy_left", "exact hadd_bounds_right_left", "exact hAZcopy_right_right_left", "cases hsecond", "cases hsecond_witness", "have hdistributed : forall pfp_index_step_helper_distributed. (exists pfa_gap_step_helper_distributedindex. pfa_gap_step_helper_distributedindex + S (pfp_index_step_helper_distributed) = (W)) -> exists pfp_left_step_helper_distributed pfp_right_step_helper_distributed pfp_value_step_helper_distributed. ((((exists ff_h_pfp_step_helper_distributedleft. ff_h_pfp_step_helper_distributedleft + S (pfp_left_step_helper_distributed) = S ((S (pfp_index_step_helper_distributed)) * x6)) /\\ exists ff_q_pfp_step_helper_distributedleft. x5 = ff_q_pfp_step_helper_distributedleft * S ((S (pfp_index_step_helper_distributed)) * x6) + (pfp_left_step_helper_distributed))) /\\ (((((exists ff_h_pfp_step_helper_distributedright. ff_h_pfp_step_helper_distributedright + S (pfp_right_step_helper_distributed) = S ((S (pfp_index_step_helper_distributed)) * x8)) /\\ exists ff_q_pfp_step_helper_distributedright. x7 = ff_q_pfp_step_helper_distributedright * S ((S (pfp_index_step_helper_distributed)) * x8) + (pfp_right_step_helper_distributed))) /\\ (((((exists ff_h_pfp_step_helper_distributedtarget. ff_h_pfp_step_helper_distributedtarget + S (pfp_value_step_helper_distributed) = S ((S (pfp_index_step_helper_distributed)) * sc)) /\\ exists ff_q_pfp_step_helper_distributedtarget. sb = ff_q_pfp_step_helper_distributedtarget * S ((S (pfp_index_step_helper_distributed)) * sc) + (pfp_value_step_helper_distributed))) /\\ ((((exists pfa_gap_step_helper_distributedoperationleft. pfa_gap_step_helper_distributedoperationleft + S (pfp_left_step_helper_distributed) = (p)) /\\ (((exists pfa_gap_step_helper_distributedoperationright. pfa_gap_step_helper_distributedoperationright + S (pfp_right_step_helper_distributed) = (p)) /\\ ((((exists pfa_gap_step_helper_distributedoperationresultbound. pfa_gap_step_helper_distributedoperationresultbound + S (pfp_value_step_helper_distributed) = (p)) /\\ ((exists pfa_offset_left_step_helper_distributedoperationresultcongruence pfa_offset_right_step_helper_distributedoperationresultcongruence. ((pfp_left_step_helper_distributed) + (pfp_right_step_helper_distributed)) + (p) * pfa_offset_left_step_helper_distributedoperationresultcongruence = (pfp_value_step_helper_distributed) + (p) * pfa_offset_right_step_helper_distributedoperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_convolution_left_add (p)", "specialize prime_field_polynomial_convolution_left_add (UPb)", "specialize prime_field_polynomial_convolution_left_add (UPc)", "specialize prime_field_polynomial_convolution_left_add (VPb)", "specialize prime_field_polynomial_convolution_left_add (VPc)", "specialize prime_field_polynomial_convolution_left_add (zb)", "specialize prime_field_polynomial_convolution_left_add (zc)", "specialize prime_field_polynomial_convolution_left_add (M+S K)", "specialize prime_field_polynomial_convolution_left_add (ab)", "specialize prime_field_polynomial_convolution_left_add (ac)", "specialize prime_field_polynomial_convolution_left_add (L)", "specialize prime_field_polynomial_convolution_left_add (x5)", "specialize prime_field_polynomial_convolution_left_add (x6)", "specialize prime_field_polynomial_convolution_left_add (x7)", "specialize prime_field_polynomial_convolution_left_add (x8)", "specialize prime_field_polynomial_convolution_left_add (sb)", "specialize prime_field_polynomial_convolution_left_add (sc)", "specialize prime_field_polynomial_convolution_left_add (W)", "apply prime_field_polynomial_convolution_left_add", "exact hZ", "exact hfirst_witness_witness", "exact hsecond_witness_witness", "exact hAZ", "have hshift_equal : forall pfrep_power_step_helper_shift_equal pfrep_left_step_helper_shift_equal pfrep_right_step_helper_shift_equal. ((exists pfrep_position_step_helper_shift_equalfirst. ((pfrep_position_step_helper_shift_equalfirst+S (pfrep_power_step_helper_shift_equal)=(x)) /\\ ((((exists ff_h_pfp_step_helper_shift_equalfirstentry. ff_h_pfp_step_helper_shift_equalfirstentry + S (pfrep_left_step_helper_shift_equal) = S ((S (pfrep_position_step_helper_shift_equalfirst)) * x2)) /\\ exists ff_q_pfp_step_helper_shift_equalfirstentry. x1 = ff_q_pfp_step_helper_shift_equalfirstentry * S ((S (pfrep_position_step_helper_shift_equalfirst)) * x2) + (pfrep_left_step_helper_shift_equal)))))) \\/ (((exists pfrep_gap_step_helper_shift_equalfirstoutside. pfrep_gap_step_helper_shift_equalfirstoutside+(x)=(pfrep_power_step_helper_shift_equal)) /\\ (((pfrep_left_step_helper_shift_equal)=0))))) -> ((exists pfrep_position_step_helper_shift_equalsecond. ((pfrep_position_step_helper_shift_equalsecond+S (pfrep_power_step_helper_shift_equal)=(S T)) /\\ ((((exists ff_h_pfp_step_helper_shift_equalsecondentry. ff_h_pfp_step_helper_shift_equalsecondentry + S (pfrep_right_step_helper_shift_equal) = S ((S (pfrep_position_step_helper_shift_equalsecond)) * ec)) /\\ exists ff_q_pfp_step_helper_shift_equalsecondentry. eb = ff_q_pfp_step_helper_shift_equalsecondentry * S ((S (pfrep_position_step_helper_shift_equalsecond)) * ec) + (pfrep_right_step_helper_shift_equal)))))) \\/ (((exists pfrep_gap_step_helper_shift_equalsecondoutside. pfrep_gap_step_helper_shift_equalsecondoutside+(S T)=(pfrep_power_step_helper_shift_equal)) /\\ (((pfrep_right_step_helper_shift_equal)=0))))) -> pfrep_left_step_helper_shift_equal=pfrep_right_step_helper_shift_equal", "specialize prime_field_polynomial_convolution_shift_right_equivalent (p)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (ab)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (ac)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (L)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (qb)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (qc)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (K)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (rb)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (rc)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (T)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (ub)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (uc)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (x1)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (x2)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (x)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (eb)", "specialize prime_field_polynomial_convolution_shift_right_equivalent (ec)", "apply prime_field_polynomial_convolution_shift_right_equivalent", "exact hp0", "exact hU", "exact hAQ", "exact hshifted_product_witness_witness", "exact hE", "have hfirst_pad : forall pfrep_power_step_helper_first_pad pfrep_left_step_helper_first_pad pfrep_right_step_helper_first_pad. ((exists pfrep_position_step_helper_first_padfirst. ((pfrep_position_step_helper_first_padfirst+S (pfrep_power_step_helper_first_pad)=(x)) /\\ ((((exists ff_h_pfp_step_helper_first_padfirstentry. ff_h_pfp_step_helper_first_padfirstentry + S (pfrep_left_step_helper_first_pad) = S ((S (pfrep_position_step_helper_first_padfirst)) * x2)) /\\ exists ff_q_pfp_step_helper_first_padfirstentry. x1 = ff_q_pfp_step_helper_first_padfirstentry * S ((S (pfrep_position_step_helper_first_padfirst)) * x2) + (pfrep_left_step_helper_first_pad)))))) \\/ (((exists pfrep_gap_step_helper_first_padfirstoutside. pfrep_gap_step_helper_first_padfirstoutside+(x)=(pfrep_power_step_helper_first_pad)) /\\ (((pfrep_left_step_helper_first_pad)=0))))) -> ((exists pfrep_position_step_helper_first_padsecond. ((pfrep_position_step_helper_first_padsecond+S (pfrep_power_step_helper_first_pad)=(W)) /\\ ((((exists ff_h_pfp_step_helper_first_padsecondentry. ff_h_pfp_step_helper_first_padsecondentry + S (pfrep_right_step_helper_first_pad) = S ((S (pfrep_position_step_helper_first_padsecond)) * x6)) /\\ exists ff_q_pfp_step_helper_first_padsecondentry. x5 = ff_q_pfp_step_helper_first_padsecondentry * S ((S (pfrep_position_step_helper_first_padsecond)) * x6) + (pfrep_right_step_helper_first_pad)))))) \\/ (((exists pfrep_gap_step_helper_first_padsecondoutside. pfrep_gap_step_helper_first_padsecondoutside+(W)=(pfrep_power_step_helper_first_pad)) /\\ (((pfrep_right_step_helper_first_pad)=0))))) -> pfrep_left_step_helper_first_pad=pfrep_right_step_helper_first_pad", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (p)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (ab)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (ac)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (L)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (ub)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (uc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (S K)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x1)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x2)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (UPb)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (UPc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (M)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x5)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x6)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (W)", "apply prime_field_polynomial_convolution_left_padding_equivalent_right", "exact hp0", "exact hUP", "exact hshifted_product_witness_witness", "exact hfirst_witness_witness", "have hfirst_base : forall pfrep_power_step_helper_first_base pfrep_left_step_helper_first_base pfrep_right_step_helper_first_base. ((exists pfrep_position_step_helper_first_basefirst. ((pfrep_position_step_helper_first_basefirst+S (pfrep_power_step_helper_first_base)=(W)) /\\ ((((exists ff_h_pfp_step_helper_first_basefirstentry. ff_h_pfp_step_helper_first_basefirstentry + S (pfrep_left_step_helper_first_base) = S ((S (pfrep_position_step_helper_first_basefirst)) * x6)) /\\ exists ff_q_pfp_step_helper_first_basefirstentry. x5 = ff_q_pfp_step_helper_first_basefirstentry * S ((S (pfrep_position_step_helper_first_basefirst)) * x6) + (pfrep_left_step_helper_first_base)))))) \\/ (((exists pfrep_gap_step_helper_first_basefirstoutside. pfrep_gap_step_helper_first_basefirstoutside+(W)=(pfrep_power_step_helper_first_base)) /\\ (((pfrep_left_step_helper_first_base)=0))))) -> ((exists pfrep_position_step_helper_first_basesecond. ((pfrep_position_step_helper_first_basesecond+S (pfrep_power_step_helper_first_base)=(S T)) /\\ ((((exists ff_h_pfp_step_helper_first_basesecondentry. ff_h_pfp_step_helper_first_basesecondentry + S (pfrep_right_step_helper_first_base) = S ((S (pfrep_position_step_helper_first_basesecond)) * ec)) /\\ exists ff_q_pfp_step_helper_first_basesecondentry. eb = ff_q_pfp_step_helper_first_basesecondentry * S ((S (pfrep_position_step_helper_first_basesecond)) * ec) + (pfrep_right_step_helper_first_base)))))) \\/ (((exists pfrep_gap_step_helper_first_basesecondoutside. pfrep_gap_step_helper_first_basesecondoutside+(S T)=(pfrep_power_step_helper_first_base)) /\\ (((pfrep_right_step_helper_first_base)=0))))) -> pfrep_left_step_helper_first_base=pfrep_right_step_helper_first_base", "specialize prime_field_polynomial_equivalent_transitive (x5)", "specialize prime_field_polynomial_equivalent_transitive (x6)", "specialize prime_field_polynomial_equivalent_transitive (W)", "specialize prime_field_polynomial_equivalent_transitive (x1)", "specialize prime_field_polynomial_equivalent_transitive (x2)", "specialize prime_field_polynomial_equivalent_transitive (x)", "specialize prime_field_polynomial_equivalent_transitive (eb)", "specialize prime_field_polynomial_equivalent_transitive (ec)", "specialize prime_field_polynomial_equivalent_transitive (S T)", "apply prime_field_polynomial_equivalent_transitive", "specialize prime_field_polynomial_equivalent_symmetric (x1)", "specialize prime_field_polynomial_equivalent_symmetric (x2)", "specialize prime_field_polynomial_equivalent_symmetric (x)", "specialize prime_field_polynomial_equivalent_symmetric (x5)", "specialize prime_field_polynomial_equivalent_symmetric (x6)", "specialize prime_field_polynomial_equivalent_symmetric (W)", "apply prime_field_polynomial_equivalent_symmetric", "exact hfirst_pad", "exact hshift_equal", "have hfirst_equal : forall pfrep_power_step_helper_first_equal pfrep_left_step_helper_first_equal pfrep_right_step_helper_first_equal. ((exists pfrep_position_step_helper_first_equalfirst. ((pfrep_position_step_helper_first_equalfirst+S (pfrep_power_step_helper_first_equal)=(W)) /\\ ((((exists ff_h_pfp_step_helper_first_equalfirstentry. ff_h_pfp_step_helper_first_equalfirstentry + S (pfrep_left_step_helper_first_equal) = S ((S (pfrep_position_step_helper_first_equalfirst)) * x6)) /\\ exists ff_q_pfp_step_helper_first_equalfirstentry. x5 = ff_q_pfp_step_helper_first_equalfirstentry * S ((S (pfrep_position_step_helper_first_equalfirst)) * x6) + (pfrep_left_step_helper_first_equal)))))) \\/ (((exists pfrep_gap_step_helper_first_equalfirstoutside. pfrep_gap_step_helper_first_equalfirstoutside+(W)=(pfrep_power_step_helper_first_equal)) /\\ (((pfrep_left_step_helper_first_equal)=0))))) -> ((exists pfrep_position_step_helper_first_equalsecond. ((pfrep_position_step_helper_first_equalsecond+S (pfrep_power_step_helper_first_equal)=(N+S T)) /\\ ((((exists ff_h_pfp_step_helper_first_equalsecondentry. ff_h_pfp_step_helper_first_equalsecondentry + S (pfrep_right_step_helper_first_equal) = S ((S (pfrep_position_step_helper_first_equalsecond)) * EPc)) /\\ exists ff_q_pfp_step_helper_first_equalsecondentry. EPb = ff_q_pfp_step_helper_first_equalsecondentry * S ((S (pfrep_position_step_helper_first_equalsecond)) * EPc) + (pfrep_right_step_helper_first_equal)))))) \\/ (((exists pfrep_gap_step_helper_first_equalsecondoutside. pfrep_gap_step_helper_first_equalsecondoutside+(N+S T)=(pfrep_power_step_helper_first_equal)) /\\ (((pfrep_right_step_helper_first_equal)=0))))) -> pfrep_left_step_helper_first_equal=pfrep_right_step_helper_first_equal", "specialize prime_field_polynomial_equivalent_transitive (x5)", "specialize prime_field_polynomial_equivalent_transitive (x6)", "specialize prime_field_polynomial_equivalent_transitive (W)", "specialize prime_field_polynomial_equivalent_transitive (eb)", "specialize prime_field_polynomial_equivalent_transitive (ec)", "specialize prime_field_polynomial_equivalent_transitive (S T)", "specialize prime_field_polynomial_equivalent_transitive (EPb)", "specialize prime_field_polynomial_equivalent_transitive (EPc)", "specialize prime_field_polynomial_equivalent_transitive (N+S T)", "apply prime_field_polynomial_equivalent_transitive", "exact hfirst_base", "specialize prime_field_polynomial_left_pad_equivalent (eb)", "specialize prime_field_polynomial_left_pad_equivalent (ec)", "specialize prime_field_polynomial_left_pad_equivalent (S T)", "specialize prime_field_polynomial_left_pad_equivalent (N)", "specialize prime_field_polynomial_left_pad_equivalent (EPb)", "specialize prime_field_polynomial_left_pad_equivalent (EPc)", "apply prime_field_polynomial_left_pad_equivalent", "exact hEP", "have hscaled_equal : ((N=N) /\\ ((forall mdr_i_pfp_step_helper_scalar_equal mdr_a_pfp_step_helper_scalar_equal. (exists mdr_gap_pfp_step_helper_scalar_equalb. mdr_gap_pfp_step_helper_scalar_equalb + S (mdr_i_pfp_step_helper_scalar_equal) = (N)) -> (((exists ff_h_mdr_pfp_step_helper_scalar_equalo. ff_h_mdr_pfp_step_helper_scalar_equalo + S (mdr_a_pfp_step_helper_scalar_equal) = S ((S (mdr_i_pfp_step_helper_scalar_equal)) * x4)) /\\ exists ff_q_mdr_pfp_step_helper_scalar_equalo. x3 = ff_q_mdr_pfp_step_helper_scalar_equalo * S ((S (mdr_i_pfp_step_helper_scalar_equal)) * x4) + (mdr_a_pfp_step_helper_scalar_equal))) -> (((exists ff_h_mdr_pfp_step_helper_scalar_equaln. ff_h_mdr_pfp_step_helper_scalar_equaln + S (mdr_a_pfp_step_helper_scalar_equal) = S ((S (mdr_i_pfp_step_helper_scalar_equal)) * fc)) /\\ exists ff_q_mdr_pfp_step_helper_scalar_equaln. fb = ff_q_mdr_pfp_step_helper_scalar_equaln * S ((S (mdr_i_pfp_step_helper_scalar_equal)) * fc) + (mdr_a_pfp_step_helper_scalar_equal))))))", "specialize prime_field_polynomial_convolution_right_scale_equal (p)", "specialize prime_field_polynomial_convolution_right_scale_equal (c)", "specialize prime_field_polynomial_convolution_right_scale_equal (ab)", "specialize prime_field_polynomial_convolution_right_scale_equal (ac)", "specialize prime_field_polynomial_convolution_right_scale_equal (L)", "specialize prime_field_polynomial_convolution_right_scale_equal (bb)", "specialize prime_field_polynomial_convolution_right_scale_equal (bc)", "specialize prime_field_polynomial_convolution_right_scale_equal (M)", "specialize prime_field_polynomial_convolution_right_scale_equal (vb)", "specialize prime_field_polynomial_convolution_right_scale_equal (vc)", "specialize prime_field_polynomial_convolution_right_scale_equal (pb)", "specialize prime_field_polynomial_convolution_right_scale_equal (pc)", "specialize prime_field_polynomial_convolution_right_scale_equal (N)", "specialize prime_field_polynomial_convolution_right_scale_equal (x3)", "specialize prime_field_polynomial_convolution_right_scale_equal (x4)", "specialize prime_field_polynomial_convolution_right_scale_equal (N)", "specialize prime_field_polynomial_convolution_right_scale_equal (fb)", "specialize prime_field_polynomial_convolution_right_scale_equal (fc)", "apply prime_field_polynomial_convolution_right_scale_equal", "exact hV", "exact hAB", "exact hscaled_product_witness_witness", "exact hF", "cases hscaled_equal", "have hscalar_base : forall pfrep_power_step_helper_scalar_base pfrep_left_step_helper_scalar_base pfrep_right_step_helper_scalar_base. ((exists pfrep_position_step_helper_scalar_basefirst. ((pfrep_position_step_helper_scalar_basefirst+S (pfrep_power_step_helper_scalar_base)=(N)) /\\ ((((exists ff_h_pfp_step_helper_scalar_basefirstentry. ff_h_pfp_step_helper_scalar_basefirstentry + S (pfrep_left_step_helper_scalar_base) = S ((S (pfrep_position_step_helper_scalar_basefirst)) * x4)) /\\ exists ff_q_pfp_step_helper_scalar_basefirstentry. x3 = ff_q_pfp_step_helper_scalar_basefirstentry * S ((S (pfrep_position_step_helper_scalar_basefirst)) * x4) + (pfrep_left_step_helper_scalar_base)))))) \\/ (((exists pfrep_gap_step_helper_scalar_basefirstoutside. pfrep_gap_step_helper_scalar_basefirstoutside+(N)=(pfrep_power_step_helper_scalar_base)) /\\ (((pfrep_left_step_helper_scalar_base)=0))))) -> ((exists pfrep_position_step_helper_scalar_basesecond. ((pfrep_position_step_helper_scalar_basesecond+S (pfrep_power_step_helper_scalar_base)=(N)) /\\ ((((exists ff_h_pfp_step_helper_scalar_basesecondentry. ff_h_pfp_step_helper_scalar_basesecondentry + S (pfrep_right_step_helper_scalar_base) = S ((S (pfrep_position_step_helper_scalar_basesecond)) * fc)) /\\ exists ff_q_pfp_step_helper_scalar_basesecondentry. fb = ff_q_pfp_step_helper_scalar_basesecondentry * S ((S (pfrep_position_step_helper_scalar_basesecond)) * fc) + (pfrep_right_step_helper_scalar_base)))))) \\/ (((exists pfrep_gap_step_helper_scalar_basesecondoutside. pfrep_gap_step_helper_scalar_basesecondoutside+(N)=(pfrep_power_step_helper_scalar_base)) /\\ (((pfrep_right_step_helper_scalar_base)=0))))) -> pfrep_left_step_helper_scalar_base=pfrep_right_step_helper_scalar_base", "specialize prime_field_polynomial_equal_implies_equivalent (x3)", "specialize prime_field_polynomial_equal_implies_equivalent (x4)", "specialize prime_field_polynomial_equal_implies_equivalent (fb)", "specialize prime_field_polynomial_equal_implies_equivalent (fc)", "specialize prime_field_polynomial_equal_implies_equivalent (N)", "apply prime_field_polynomial_equal_implies_equivalent", "exact hscaled_equal_right", "have hsecond_pad : forall pfrep_power_step_helper_second_pad pfrep_left_step_helper_second_pad pfrep_right_step_helper_second_pad. ((exists pfrep_position_step_helper_second_padfirst. ((pfrep_position_step_helper_second_padfirst+S (pfrep_power_step_helper_second_pad)=(N)) /\\ ((((exists ff_h_pfp_step_helper_second_padfirstentry. ff_h_pfp_step_helper_second_padfirstentry + S (pfrep_left_step_helper_second_pad) = S ((S (pfrep_position_step_helper_second_padfirst)) * x4)) /\\ exists ff_q_pfp_step_helper_second_padfirstentry. x3 = ff_q_pfp_step_helper_second_padfirstentry * S ((S (pfrep_position_step_helper_second_padfirst)) * x4) + (pfrep_left_step_helper_second_pad)))))) \\/ (((exists pfrep_gap_step_helper_second_padfirstoutside. pfrep_gap_step_helper_second_padfirstoutside+(N)=(pfrep_power_step_helper_second_pad)) /\\ (((pfrep_left_step_helper_second_pad)=0))))) -> ((exists pfrep_position_step_helper_second_padsecond. ((pfrep_position_step_helper_second_padsecond+S (pfrep_power_step_helper_second_pad)=(W)) /\\ ((((exists ff_h_pfp_step_helper_second_padsecondentry. ff_h_pfp_step_helper_second_padsecondentry + S (pfrep_right_step_helper_second_pad) = S ((S (pfrep_position_step_helper_second_padsecond)) * x8)) /\\ exists ff_q_pfp_step_helper_second_padsecondentry. x7 = ff_q_pfp_step_helper_second_padsecondentry * S ((S (pfrep_position_step_helper_second_padsecond)) * x8) + (pfrep_right_step_helper_second_pad)))))) \\/ (((exists pfrep_gap_step_helper_second_padsecondoutside. pfrep_gap_step_helper_second_padsecondoutside+(W)=(pfrep_power_step_helper_second_pad)) /\\ (((pfrep_right_step_helper_second_pad)=0))))) -> pfrep_left_step_helper_second_pad=pfrep_right_step_helper_second_pad", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (p)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (ab)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (ac)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (L)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (vb)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (vc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (M)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x3)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x4)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (N)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (VPb)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (VPc)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (S K)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x7)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (x8)", "specialize prime_field_polynomial_convolution_left_padding_equivalent_right (W)", "apply prime_field_polynomial_convolution_left_padding_equivalent_right", "exact hp0", "exact hVP", "exact hscaled_product_witness_witness", "have hcomm_input : S K+M=M+S K", "specialize add_comm (S K)", "specialize add_comm (M)", "apply add_comm", "rewrite hcomm_input", "rewrite hcomm_input", "rewrite hcomm_input", "rewrite hcomm_input", "rewrite hcomm_input", "rewrite hcomm_input", "exact hsecond_witness_witness", "have hsecond_base : forall pfrep_power_step_helper_second_base pfrep_left_step_helper_second_base pfrep_right_step_helper_second_base. ((exists pfrep_position_step_helper_second_basefirst. ((pfrep_position_step_helper_second_basefirst+S (pfrep_power_step_helper_second_base)=(W)) /\\ ((((exists ff_h_pfp_step_helper_second_basefirstentry. ff_h_pfp_step_helper_second_basefirstentry + S (pfrep_left_step_helper_second_base) = S ((S (pfrep_position_step_helper_second_basefirst)) * x8)) /\\ exists ff_q_pfp_step_helper_second_basefirstentry. x7 = ff_q_pfp_step_helper_second_basefirstentry * S ((S (pfrep_position_step_helper_second_basefirst)) * x8) + (pfrep_left_step_helper_second_base)))))) \\/ (((exists pfrep_gap_step_helper_second_basefirstoutside. pfrep_gap_step_helper_second_basefirstoutside+(W)=(pfrep_power_step_helper_second_base)) /\\ (((pfrep_left_step_helper_second_base)=0))))) -> ((exists pfrep_position_step_helper_second_basesecond. ((pfrep_position_step_helper_second_basesecond+S (pfrep_power_step_helper_second_base)=(N)) /\\ ((((exists ff_h_pfp_step_helper_second_basesecondentry. ff_h_pfp_step_helper_second_basesecondentry + S (pfrep_right_step_helper_second_base) = S ((S (pfrep_position_step_helper_second_basesecond)) * fc)) /\\ exists ff_q_pfp_step_helper_second_basesecondentry. fb = ff_q_pfp_step_helper_second_basesecondentry * S ((S (pfrep_position_step_helper_second_basesecond)) * fc) + (pfrep_right_step_helper_second_base)))))) \\/ (((exists pfrep_gap_step_helper_second_basesecondoutside. pfrep_gap_step_helper_second_basesecondoutside+(N)=(pfrep_power_step_helper_second_base)) /\\ (((pfrep_right_step_helper_second_base)=0))))) -> pfrep_left_step_helper_second_base=pfrep_right_step_helper_second_base", "specialize prime_field_polynomial_equivalent_transitive (x7)", "specialize prime_field_polynomial_equivalent_transitive (x8)", "specialize prime_field_polynomial_equivalent_transitive (W)", "specialize prime_field_polynomial_equivalent_transitive (x3)", "specialize prime_field_polynomial_equivalent_transitive (x4)", "specialize prime_field_polynomial_equivalent_transitive (N)", "specialize prime_field_polynomial_equivalent_transitive (fb)", "specialize prime_field_polynomial_equivalent_transitive (fc)", "specialize prime_field_polynomial_equivalent_transitive (N)", "apply prime_field_polynomial_equivalent_transitive", "specialize prime_field_polynomial_equivalent_symmetric (x3)", "specialize prime_field_polynomial_equivalent_symmetric (x4)", "specialize prime_field_polynomial_equivalent_symmetric (N)", "specialize prime_field_polynomial_equivalent_symmetric (x7)", "specialize prime_field_polynomial_equivalent_symmetric (x8)", "specialize prime_field_polynomial_equivalent_symmetric (W)", "apply prime_field_polynomial_equivalent_symmetric", "exact hsecond_pad", "exact hscalar_base", "have hsecond_equal : forall pfrep_power_step_helper_second_equal pfrep_left_step_helper_second_equal pfrep_right_step_helper_second_equal. ((exists pfrep_position_step_helper_second_equalfirst. ((pfrep_position_step_helper_second_equalfirst+S (pfrep_power_step_helper_second_equal)=(W)) /\\ ((((exists ff_h_pfp_step_helper_second_equalfirstentry. ff_h_pfp_step_helper_second_equalfirstentry + S (pfrep_left_step_helper_second_equal) = S ((S (pfrep_position_step_helper_second_equalfirst)) * x8)) /\\ exists ff_q_pfp_step_helper_second_equalfirstentry. x7 = ff_q_pfp_step_helper_second_equalfirstentry * S ((S (pfrep_position_step_helper_second_equalfirst)) * x8) + (pfrep_left_step_helper_second_equal)))))) \\/ (((exists pfrep_gap_step_helper_second_equalfirstoutside. pfrep_gap_step_helper_second_equalfirstoutside+(W)=(pfrep_power_step_helper_second_equal)) /\\ (((pfrep_left_step_helper_second_equal)=0))))) -> ((exists pfrep_position_step_helper_second_equalsecond. ((pfrep_position_step_helper_second_equalsecond+S (pfrep_power_step_helper_second_equal)=(N+S T)) /\\ ((((exists ff_h_pfp_step_helper_second_equalsecondentry. ff_h_pfp_step_helper_second_equalsecondentry + S (pfrep_right_step_helper_second_equal) = S ((S (pfrep_position_step_helper_second_equalsecond)) * FPc)) /\\ exists ff_q_pfp_step_helper_second_equalsecondentry. FPb = ff_q_pfp_step_helper_second_equalsecondentry * S ((S (pfrep_position_step_helper_second_equalsecond)) * FPc) + (pfrep_right_step_helper_second_equal)))))) \\/ (((exists pfrep_gap_step_helper_second_equalsecondoutside. pfrep_gap_step_helper_second_equalsecondoutside+(N+S T)=(pfrep_power_step_helper_second_equal)) /\\ (((pfrep_right_step_helper_second_equal)=0))))) -> pfrep_left_step_helper_second_equal=pfrep_right_step_helper_second_equal", "specialize prime_field_polynomial_equivalent_transitive (x7)", "specialize prime_field_polynomial_equivalent_transitive (x8)", "specialize prime_field_polynomial_equivalent_transitive (W)", "specialize prime_field_polynomial_equivalent_transitive (fb)", "specialize prime_field_polynomial_equivalent_transitive (fc)", "specialize prime_field_polynomial_equivalent_transitive (N)", "specialize prime_field_polynomial_equivalent_transitive (FPb)", "specialize prime_field_polynomial_equivalent_transitive (FPc)", "specialize prime_field_polynomial_equivalent_transitive (N+S T)", "apply prime_field_polynomial_equivalent_transitive", "exact hsecond_base", "have houtput_pad : forall pfrep_power_step_helper_output_commute pfrep_left_step_helper_output_commute pfrep_right_step_helper_output_commute. ((exists pfrep_position_step_helper_output_commutefirst. ((pfrep_position_step_helper_output_commutefirst+S (pfrep_power_step_helper_output_commute)=(N)) /\\ ((((exists ff_h_pfp_step_helper_output_commutefirstentry. ff_h_pfp_step_helper_output_commutefirstentry + S (pfrep_left_step_helper_output_commute) = S ((S (pfrep_position_step_helper_output_commutefirst)) * fc)) /\\ exists ff_q_pfp_step_helper_output_commutefirstentry. fb = ff_q_pfp_step_helper_output_commutefirstentry * S ((S (pfrep_position_step_helper_output_commutefirst)) * fc) + (pfrep_left_step_helper_output_commute)))))) \\/ (((exists pfrep_gap_step_helper_output_commutefirstoutside. pfrep_gap_step_helper_output_commutefirstoutside+(N)=(pfrep_power_step_helper_output_commute)) /\\ (((pfrep_left_step_helper_output_commute)=0))))) -> ((exists pfrep_position_step_helper_output_commutesecond. ((pfrep_position_step_helper_output_commutesecond+S (pfrep_power_step_helper_output_commute)=(S T+N)) /\\ ((((exists ff_h_pfp_step_helper_output_commutesecondentry. ff_h_pfp_step_helper_output_commutesecondentry + S (pfrep_right_step_helper_output_commute) = S ((S (pfrep_position_step_helper_output_commutesecond)) * FPc)) /\\ exists ff_q_pfp_step_helper_output_commutesecondentry. FPb = ff_q_pfp_step_helper_output_commutesecondentry * S ((S (pfrep_position_step_helper_output_commutesecond)) * FPc) + (pfrep_right_step_helper_output_commute)))))) \\/ (((exists pfrep_gap_step_helper_output_commutesecondoutside. pfrep_gap_step_helper_output_commutesecondoutside+(S T+N)=(pfrep_power_step_helper_output_commute)) /\\ (((pfrep_right_step_helper_output_commute)=0))))) -> pfrep_left_step_helper_output_commute=pfrep_right_step_helper_output_commute", "specialize prime_field_polynomial_left_pad_equivalent (fb)", "specialize prime_field_polynomial_left_pad_equivalent (fc)", "specialize prime_field_polynomial_left_pad_equivalent (N)", "specialize prime_field_polynomial_left_pad_equivalent (S T)", "specialize prime_field_polynomial_left_pad_equivalent (FPb)", "specialize prime_field_polynomial_left_pad_equivalent (FPc)", "apply prime_field_polynomial_left_pad_equivalent", "exact hFP", "have hcomm_output : S T+N=N+S T", "specialize add_comm (S T)", "specialize add_comm (N)", "apply add_comm", "rewrite hcomm_output at houtput_pad", "rewrite hcomm_output at houtput_pad", "exact houtput_pad", "specialize prime_field_polynomial_add_equivalent_congruent (p)", "specialize prime_field_polynomial_add_equivalent_congruent (x5)", "specialize prime_field_polynomial_add_equivalent_congruent (x6)", "specialize prime_field_polynomial_add_equivalent_congruent (x7)", "specialize prime_field_polynomial_add_equivalent_congruent (x8)", "specialize prime_field_polynomial_add_equivalent_congruent (sb)", "specialize prime_field_polynomial_add_equivalent_congruent (sc)", "specialize prime_field_polynomial_add_equivalent_congruent (W)", "specialize prime_field_polynomial_add_equivalent_congruent (EPb)", "specialize prime_field_polynomial_add_equivalent_congruent (EPc)", "specialize prime_field_polynomial_add_equivalent_congruent (FPb)", "specialize prime_field_polynomial_add_equivalent_congruent (FPc)", "specialize prime_field_polynomial_add_equivalent_congruent (yb)", "specialize prime_field_polynomial_add_equivalent_congruent (yc)", "specialize prime_field_polynomial_add_equivalent_congruent (N+S T)", "apply prime_field_polynomial_add_equivalent_congruent", "exact hp", "exact hfirst_equal", "exact hsecond_equal", "exact hdistributed", "exact hY" ], "script_sha256": "65a8f80b7f4418736692cc5ece3164b53b4b707872cc4081ad3a230134df6ec2", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_associativity_step_candidate.py", "sha256": "dd85dbd1bd87143715a4286724ac7c87f280a909dac6759f00a6cb7dff7c85f1" }, "statement": "forall p c ab ac L bb bc M pb pc N qb qc K rb rc T ub uc vb vc UPb UPc VPb VPc zb zc sb sc W eb ec fb fc EPb EPc FPb FPc yb yc. (~((p) = 1) /\\ forall pfa_factor_left_step_helper_prime pfa_factor_right_step_helper_prime. (p) = pfa_factor_left_step_helper_prime * pfa_factor_right_step_helper_prime -> pfa_factor_left_step_helper_prime = 1 \\/ pfa_factor_right_step_helper_prime = 1) -> (((forall fom_index_pfp_step_helper_ABleft. (exists fom_gap_pfp_step_helper_ABleft_index_bound. fom_gap_pfp_step_helper_ABleft_index_bound + S (fom_index_pfp_step_helper_ABleft) = L) -> exists fom_value_pfp_step_helper_ABleft. ((((exists fom_beta_height_pfp_step_helper_ABleft_entry. fom_beta_height_pfp_step_helper_ABleft_entry + S (fom_value_pfp_step_helper_ABleft) = S ((S (fom_index_pfp_step_helper_ABleft)) * ac)) /\\ exists fom_beta_quotient_pfp_step_helper_ABleft_entry. ab = fom_beta_quotient_pfp_step_helper_ABleft_entry * S ((S (fom_index_pfp_step_helper_ABleft)) * ac) + (fom_value_pfp_step_helper_ABleft))) /\\ (exists fom_gap_pfp_step_helper_ABleft_value_bound. fom_gap_pfp_step_helper_ABleft_value_bound + S (fom_value_pfp_step_helper_ABleft) = p))) /\\ (((forall fom_index_pfp_step_helper_ABright. (exists fom_gap_pfp_step_helper_ABright_index_bound. fom_gap_pfp_step_helper_ABright_index_bound + S (fom_index_pfp_step_helper_ABright) = M) -> exists fom_value_pfp_step_helper_ABright. ((((exists fom_beta_height_pfp_step_helper_ABright_entry. fom_beta_height_pfp_step_helper_ABright_entry + S (fom_value_pfp_step_helper_ABright) = S ((S (fom_index_pfp_step_helper_ABright)) * bc)) /\\ exists fom_beta_quotient_pfp_step_helper_ABright_entry. bb = fom_beta_quotient_pfp_step_helper_ABright_entry * S ((S (fom_index_pfp_step_helper_ABright)) * bc) + (fom_value_pfp_step_helper_ABright))) /\\ (exists fom_gap_pfp_step_helper_ABright_value_bound. fom_gap_pfp_step_helper_ABright_value_bound + S (fom_value_pfp_step_helper_ABright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_step_helper_ABcoefficients. (exists pfa_gap_step_helper_ABcoefficientsbound. pfa_gap_step_helper_ABcoefficientsbound + S (pfc_index_step_helper_ABcoefficients) = (N)) -> exists pfc_value_step_helper_ABcoefficients. ((((exists ff_h_pfp_step_helper_ABcoefficientsentry. ff_h_pfp_step_helper_ABcoefficientsentry + S (pfc_value_step_helper_ABcoefficients) = S ((S (pfc_index_step_helper_ABcoefficients)) * pc)) /\\ exists ff_q_pfp_step_helper_ABcoefficientsentry. pb = ff_q_pfp_step_helper_ABcoefficientsentry * S ((S (pfc_index_step_helper_ABcoefficients)) * pc) + (pfc_value_step_helper_ABcoefficients))) /\\ ((exists pfc_terms_code_step_helper_ABcoefficientscoefficient pfc_terms_scale_step_helper_ABcoefficientscoefficient pfc_natural_sum_step_helper_ABcoefficientscoefficient. ((forall pfc_index_step_helper_ABcoefficientscoefficientdiagonal. (exists pfa_gap_step_helper_ABcoefficientscoefficientdiagonalbound. pfa_gap_step_helper_ABcoefficientscoefficientdiagonalbound + S (pfc_index_step_helper_ABcoefficientscoefficientdiagonal) = (S (pfc_index_step_helper_ABcoefficients))) -> exists pfc_value_step_helper_ABcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_step_helper_ABcoefficientscoefficientdiagonalentry. ff_h_pfp_step_helper_ABcoefficientscoefficientdiagonalentry + S (pfc_value_step_helper_ABcoefficientscoefficientdiagonal) = S ((S (pfc_index_step_helper_ABcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_ABcoefficientscoefficient)) /\\ exists ff_q_pfp_step_helper_ABcoefficientscoefficientdiagonalentry. pfc_terms_code_step_helper_ABcoefficientscoefficient = ff_q_pfp_step_helper_ABcoefficientscoefficientdiagonalentry * S ((S (pfc_index_step_helper_ABcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_ABcoefficientscoefficient) + (pfc_value_step_helper_ABcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_step_helper_ABcoefficientscoefficientdiagonalterm pfc_left_step_helper_ABcoefficientscoefficientdiagonalterm pfc_right_step_helper_ABcoefficientscoefficientdiagonalterm. (((pfc_index_step_helper_ABcoefficientscoefficientdiagonal)+pfc_complement_step_helper_ABcoefficientscoefficientdiagonalterm=(pfc_index_step_helper_ABcoefficients)) /\\ ((((((exists pfa_gap_step_helper_ABcoefficientscoefficientdiagonaltermleftinside. pfa_gap_step_helper_ABcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_step_helper_ABcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_step_helper_ABcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_step_helper_ABcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_step_helper_ABcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_step_helper_ABcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_step_helper_ABcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_step_helper_ABcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_step_helper_ABcoefficientscoefficientdiagonal)) * ac) + (pfc_left_step_helper_ABcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_ABcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_step_helper_ABcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_step_helper_ABcoefficientscoefficientdiagonal)) /\\ (((pfc_left_step_helper_ABcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_step_helper_ABcoefficientscoefficientdiagonaltermrightinside. pfa_gap_step_helper_ABcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_step_helper_ABcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_step_helper_ABcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_step_helper_ABcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_step_helper_ABcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_step_helper_ABcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_step_helper_ABcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_step_helper_ABcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_step_helper_ABcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_step_helper_ABcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_ABcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_step_helper_ABcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_step_helper_ABcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_step_helper_ABcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_step_helper_ABcoefficientscoefficientdiagonal)=pfc_left_step_helper_ABcoefficientscoefficientdiagonalterm*pfc_right_step_helper_ABcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_step_helper_ABcoefficientscoefficientsum fs_v_pfc_step_helper_ABcoefficientscoefficientsum. ((((exists fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_start. fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_start. fs_u_pfc_step_helper_ABcoefficientscoefficientsum = fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_terminal. fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_step_helper_ABcoefficientscoefficient) = S ((S (S (pfc_index_step_helper_ABcoefficients))) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_terminal. fs_u_pfc_step_helper_ABcoefficientscoefficientsum = fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_step_helper_ABcoefficients))) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum) + (pfc_natural_sum_step_helper_ABcoefficientscoefficient))) /\\ forall fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps = S (pfc_index_step_helper_ABcoefficients)) -> exists fs_a_pfc_step_helper_ABcoefficientscoefficientsum_body_steps fs_r_pfc_step_helper_ABcoefficientscoefficientsum_body_steps fs_s_pfc_step_helper_ABcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_step_helper_ABcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_ABcoefficientscoefficient)) /\\ exists fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_step_helper_ABcoefficientscoefficient = fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_ABcoefficientscoefficient) + (fs_a_pfc_step_helper_ABcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_step_helper_ABcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_step_helper_ABcoefficientscoefficientsum = fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum) + (fs_r_pfc_step_helper_ABcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_step_helper_ABcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_step_helper_ABcoefficientscoefficientsum = fs_q_pfc_step_helper_ABcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_step_helper_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_ABcoefficientscoefficientsum) + (fs_s_pfc_step_helper_ABcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_step_helper_ABcoefficientscoefficientsum_body_steps = fs_r_pfc_step_helper_ABcoefficientscoefficientsum_body_steps + fs_a_pfc_step_helper_ABcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_step_helper_ABcoefficientscoefficientresiduebound. pfa_gap_step_helper_ABcoefficientscoefficientresiduebound + S (pfc_value_step_helper_ABcoefficients) = (p)) /\\ ((exists pfa_offset_left_step_helper_ABcoefficientscoefficientresiduecongruence pfa_offset_right_step_helper_ABcoefficientscoefficientresiduecongruence. (pfc_natural_sum_step_helper_ABcoefficientscoefficient) + (p) * pfa_offset_left_step_helper_ABcoefficientscoefficientresiduecongruence = (pfc_value_step_helper_ABcoefficients) + (p) * pfa_offset_right_step_helper_ABcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_step_helper_AQleft. (exists fom_gap_pfp_step_helper_AQleft_index_bound. fom_gap_pfp_step_helper_AQleft_index_bound + S (fom_index_pfp_step_helper_AQleft) = L) -> exists fom_value_pfp_step_helper_AQleft. ((((exists fom_beta_height_pfp_step_helper_AQleft_entry. fom_beta_height_pfp_step_helper_AQleft_entry + S (fom_value_pfp_step_helper_AQleft) = S ((S (fom_index_pfp_step_helper_AQleft)) * ac)) /\\ exists fom_beta_quotient_pfp_step_helper_AQleft_entry. ab = fom_beta_quotient_pfp_step_helper_AQleft_entry * S ((S (fom_index_pfp_step_helper_AQleft)) * ac) + (fom_value_pfp_step_helper_AQleft))) /\\ (exists fom_gap_pfp_step_helper_AQleft_value_bound. fom_gap_pfp_step_helper_AQleft_value_bound + S (fom_value_pfp_step_helper_AQleft) = p))) /\\ (((forall fom_index_pfp_step_helper_AQright. (exists fom_gap_pfp_step_helper_AQright_index_bound. fom_gap_pfp_step_helper_AQright_index_bound + S (fom_index_pfp_step_helper_AQright) = K) -> exists fom_value_pfp_step_helper_AQright. ((((exists fom_beta_height_pfp_step_helper_AQright_entry. fom_beta_height_pfp_step_helper_AQright_entry + S (fom_value_pfp_step_helper_AQright) = S ((S (fom_index_pfp_step_helper_AQright)) * qc)) /\\ exists fom_beta_quotient_pfp_step_helper_AQright_entry. qb = fom_beta_quotient_pfp_step_helper_AQright_entry * S ((S (fom_index_pfp_step_helper_AQright)) * qc) + (fom_value_pfp_step_helper_AQright))) /\\ (exists fom_gap_pfp_step_helper_AQright_value_bound. fom_gap_pfp_step_helper_AQright_value_bound + S (fom_value_pfp_step_helper_AQright) = p))) /\\ (((((((L)=0 \\/ (K)=0) /\\ (((T)=0)))) \\/ (((~((L)=0)) /\\ (((~((K)=0)) /\\ (((L)+(K)=S (T)))))))) /\\ ((forall pfc_index_step_helper_AQcoefficients. (exists pfa_gap_step_helper_AQcoefficientsbound. pfa_gap_step_helper_AQcoefficientsbound + S (pfc_index_step_helper_AQcoefficients) = (T)) -> exists pfc_value_step_helper_AQcoefficients. ((((exists ff_h_pfp_step_helper_AQcoefficientsentry. ff_h_pfp_step_helper_AQcoefficientsentry + S (pfc_value_step_helper_AQcoefficients) = S ((S (pfc_index_step_helper_AQcoefficients)) * rc)) /\\ exists ff_q_pfp_step_helper_AQcoefficientsentry. rb = ff_q_pfp_step_helper_AQcoefficientsentry * S ((S (pfc_index_step_helper_AQcoefficients)) * rc) + (pfc_value_step_helper_AQcoefficients))) /\\ ((exists pfc_terms_code_step_helper_AQcoefficientscoefficient pfc_terms_scale_step_helper_AQcoefficientscoefficient pfc_natural_sum_step_helper_AQcoefficientscoefficient. ((forall pfc_index_step_helper_AQcoefficientscoefficientdiagonal. (exists pfa_gap_step_helper_AQcoefficientscoefficientdiagonalbound. pfa_gap_step_helper_AQcoefficientscoefficientdiagonalbound + S (pfc_index_step_helper_AQcoefficientscoefficientdiagonal) = (S (pfc_index_step_helper_AQcoefficients))) -> exists pfc_value_step_helper_AQcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_step_helper_AQcoefficientscoefficientdiagonalentry. ff_h_pfp_step_helper_AQcoefficientscoefficientdiagonalentry + S (pfc_value_step_helper_AQcoefficientscoefficientdiagonal) = S ((S (pfc_index_step_helper_AQcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_AQcoefficientscoefficient)) /\\ exists ff_q_pfp_step_helper_AQcoefficientscoefficientdiagonalentry. pfc_terms_code_step_helper_AQcoefficientscoefficient = ff_q_pfp_step_helper_AQcoefficientscoefficientdiagonalentry * S ((S (pfc_index_step_helper_AQcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_AQcoefficientscoefficient) + (pfc_value_step_helper_AQcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_step_helper_AQcoefficientscoefficientdiagonalterm pfc_left_step_helper_AQcoefficientscoefficientdiagonalterm pfc_right_step_helper_AQcoefficientscoefficientdiagonalterm. (((pfc_index_step_helper_AQcoefficientscoefficientdiagonal)+pfc_complement_step_helper_AQcoefficientscoefficientdiagonalterm=(pfc_index_step_helper_AQcoefficients)) /\\ ((((((exists pfa_gap_step_helper_AQcoefficientscoefficientdiagonaltermleftinside. pfa_gap_step_helper_AQcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_step_helper_AQcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_step_helper_AQcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_step_helper_AQcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_step_helper_AQcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_step_helper_AQcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_step_helper_AQcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_step_helper_AQcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_step_helper_AQcoefficientscoefficientdiagonal)) * ac) + (pfc_left_step_helper_AQcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_AQcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_step_helper_AQcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_step_helper_AQcoefficientscoefficientdiagonal)) /\\ (((pfc_left_step_helper_AQcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_step_helper_AQcoefficientscoefficientdiagonaltermrightinside. pfa_gap_step_helper_AQcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_step_helper_AQcoefficientscoefficientdiagonalterm) = (K)) /\\ ((((exists ff_h_pfp_step_helper_AQcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_step_helper_AQcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_step_helper_AQcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_step_helper_AQcoefficientscoefficientdiagonalterm)) * qc)) /\\ exists ff_q_pfp_step_helper_AQcoefficientscoefficientdiagonaltermrightentry. qb = ff_q_pfp_step_helper_AQcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_step_helper_AQcoefficientscoefficientdiagonalterm)) * qc) + (pfc_right_step_helper_AQcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_AQcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_step_helper_AQcoefficientscoefficientdiagonaltermrightoutside+(K)=(pfc_complement_step_helper_AQcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_step_helper_AQcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_step_helper_AQcoefficientscoefficientdiagonal)=pfc_left_step_helper_AQcoefficientscoefficientdiagonalterm*pfc_right_step_helper_AQcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_step_helper_AQcoefficientscoefficientsum fs_v_pfc_step_helper_AQcoefficientscoefficientsum. ((((exists fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_start. fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_start. fs_u_pfc_step_helper_AQcoefficientscoefficientsum = fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_terminal. fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_step_helper_AQcoefficientscoefficient) = S ((S (S (pfc_index_step_helper_AQcoefficients))) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_terminal. fs_u_pfc_step_helper_AQcoefficientscoefficientsum = fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_step_helper_AQcoefficients))) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum) + (pfc_natural_sum_step_helper_AQcoefficientscoefficient))) /\\ forall fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps = S (pfc_index_step_helper_AQcoefficients)) -> exists fs_a_pfc_step_helper_AQcoefficientscoefficientsum_body_steps fs_r_pfc_step_helper_AQcoefficientscoefficientsum_body_steps fs_s_pfc_step_helper_AQcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_step_helper_AQcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_AQcoefficientscoefficient)) /\\ exists fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_step_helper_AQcoefficientscoefficient = fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_AQcoefficientscoefficient) + (fs_a_pfc_step_helper_AQcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_step_helper_AQcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_step_helper_AQcoefficientscoefficientsum = fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum) + (fs_r_pfc_step_helper_AQcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_step_helper_AQcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_step_helper_AQcoefficientscoefficientsum = fs_q_pfc_step_helper_AQcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_step_helper_AQcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AQcoefficientscoefficientsum) + (fs_s_pfc_step_helper_AQcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_step_helper_AQcoefficientscoefficientsum_body_steps = fs_r_pfc_step_helper_AQcoefficientscoefficientsum_body_steps + fs_a_pfc_step_helper_AQcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_step_helper_AQcoefficientscoefficientresiduebound. pfa_gap_step_helper_AQcoefficientscoefficientresiduebound + S (pfc_value_step_helper_AQcoefficients) = (p)) /\\ ((exists pfa_offset_left_step_helper_AQcoefficientscoefficientresiduecongruence pfa_offset_right_step_helper_AQcoefficientscoefficientresiduecongruence. (pfc_natural_sum_step_helper_AQcoefficientscoefficient) + (p) * pfa_offset_left_step_helper_AQcoefficientscoefficientresiduecongruence = (pfc_value_step_helper_AQcoefficients) + (p) * pfa_offset_right_step_helper_AQcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall mdr_i_pfp_step_helper_left_shiftprefix mdr_a_pfp_step_helper_left_shiftprefix. (exists mdr_gap_pfp_step_helper_left_shiftprefixb. mdr_gap_pfp_step_helper_left_shiftprefixb + S (mdr_i_pfp_step_helper_left_shiftprefix) = (K)) -> (((exists ff_h_mdr_pfp_step_helper_left_shiftprefixo. ff_h_mdr_pfp_step_helper_left_shiftprefixo + S (mdr_a_pfp_step_helper_left_shiftprefix) = S ((S (mdr_i_pfp_step_helper_left_shiftprefix)) * qc)) /\\ exists ff_q_mdr_pfp_step_helper_left_shiftprefixo. qb = ff_q_mdr_pfp_step_helper_left_shiftprefixo * S ((S (mdr_i_pfp_step_helper_left_shiftprefix)) * qc) + (mdr_a_pfp_step_helper_left_shiftprefix))) -> (((exists ff_h_mdr_pfp_step_helper_left_shiftprefixn. ff_h_mdr_pfp_step_helper_left_shiftprefixn + S (mdr_a_pfp_step_helper_left_shiftprefix) = S ((S (mdr_i_pfp_step_helper_left_shiftprefix)) * uc)) /\\ exists ff_q_mdr_pfp_step_helper_left_shiftprefixn. ub = ff_q_mdr_pfp_step_helper_left_shiftprefixn * S ((S (mdr_i_pfp_step_helper_left_shiftprefix)) * uc) + (mdr_a_pfp_step_helper_left_shiftprefix)))) /\\ ((((exists ff_h_pfp_step_helper_left_shiftzero. ff_h_pfp_step_helper_left_shiftzero + S (0) = S ((S (K)) * uc)) /\\ exists ff_q_pfp_step_helper_left_shiftzero. ub = ff_q_pfp_step_helper_left_shiftzero * S ((S (K)) * uc) + (0)))))) -> (((exists pfa_gap_step_helper_left_scalescalar. pfa_gap_step_helper_left_scalescalar + S (c) = (p)) /\\ ((forall pfp_index_step_helper_left_scale. (exists pfa_gap_step_helper_left_scaleindex. pfa_gap_step_helper_left_scaleindex + S (pfp_index_step_helper_left_scale) = (M)) -> exists pfp_source_step_helper_left_scale pfp_value_step_helper_left_scale. ((((exists ff_h_pfp_step_helper_left_scalesource. ff_h_pfp_step_helper_left_scalesource + S (pfp_source_step_helper_left_scale) = S ((S (pfp_index_step_helper_left_scale)) * bc)) /\\ exists ff_q_pfp_step_helper_left_scalesource. bb = ff_q_pfp_step_helper_left_scalesource * S ((S (pfp_index_step_helper_left_scale)) * bc) + (pfp_source_step_helper_left_scale))) /\\ (((((exists ff_h_pfp_step_helper_left_scaletarget. ff_h_pfp_step_helper_left_scaletarget + S (pfp_value_step_helper_left_scale) = S ((S (pfp_index_step_helper_left_scale)) * vc)) /\\ exists ff_q_pfp_step_helper_left_scaletarget. vb = ff_q_pfp_step_helper_left_scaletarget * S ((S (pfp_index_step_helper_left_scale)) * vc) + (pfp_value_step_helper_left_scale))) /\\ ((((exists pfa_gap_step_helper_left_scaleoperationleft. pfa_gap_step_helper_left_scaleoperationleft + S (c) = (p)) /\\ (((exists pfa_gap_step_helper_left_scaleoperationright. pfa_gap_step_helper_left_scaleoperationright + S (pfp_source_step_helper_left_scale) = (p)) /\\ ((((exists pfa_gap_step_helper_left_scaleoperationresultbound. pfa_gap_step_helper_left_scaleoperationresultbound + S (pfp_value_step_helper_left_scale) = (p)) /\\ ((exists pfa_offset_left_step_helper_left_scaleoperationresultcongruence pfa_offset_right_step_helper_left_scaleoperationresultcongruence. ((c) * (pfp_source_step_helper_left_scale)) + (p) * pfa_offset_left_step_helper_left_scaleoperationresultcongruence = (pfp_value_step_helper_left_scale) + (p) * pfa_offset_right_step_helper_left_scaleoperationresultcongruence))))))))))))))))) -> (((forall pfp_repeat_index_step_helper_left_leftzeros. (exists pfa_gap_step_helper_left_leftzerosindex. pfa_gap_step_helper_left_leftzerosindex + S (pfp_repeat_index_step_helper_left_leftzeros) = (M)) -> (((exists ff_h_pfp_step_helper_left_leftzerosentry. ff_h_pfp_step_helper_left_leftzerosentry + S (0) = S ((S (pfp_repeat_index_step_helper_left_leftzeros)) * UPc)) /\\ exists ff_q_pfp_step_helper_left_leftzerosentry. UPb = ff_q_pfp_step_helper_left_leftzerosentry * S ((S (pfp_repeat_index_step_helper_left_leftzeros)) * UPc) + (0)))) /\\ ((forall pfrep_index_step_helper_left_left pfrep_value_step_helper_left_left. (exists pfa_gap_step_helper_left_leftbound. pfa_gap_step_helper_left_leftbound + S (pfrep_index_step_helper_left_left) = (S K)) -> (((exists ff_h_pfp_step_helper_left_leftinput. ff_h_pfp_step_helper_left_leftinput + S (pfrep_value_step_helper_left_left) = S ((S (pfrep_index_step_helper_left_left)) * uc)) /\\ exists ff_q_pfp_step_helper_left_leftinput. ub = ff_q_pfp_step_helper_left_leftinput * S ((S (pfrep_index_step_helper_left_left)) * uc) + (pfrep_value_step_helper_left_left))) -> (((exists ff_h_pfp_step_helper_left_leftoutput. ff_h_pfp_step_helper_left_leftoutput + S (pfrep_value_step_helper_left_left) = S ((S ((M)+pfrep_index_step_helper_left_left)) * UPc)) /\\ exists ff_q_pfp_step_helper_left_leftoutput. UPb = ff_q_pfp_step_helper_left_leftoutput * S ((S ((M)+pfrep_index_step_helper_left_left)) * UPc) + (pfrep_value_step_helper_left_left))))))) -> (((forall pfp_repeat_index_step_helper_left_rightzeros. (exists pfa_gap_step_helper_left_rightzerosindex. pfa_gap_step_helper_left_rightzerosindex + S (pfp_repeat_index_step_helper_left_rightzeros) = (S K)) -> (((exists ff_h_pfp_step_helper_left_rightzerosentry. ff_h_pfp_step_helper_left_rightzerosentry + S (0) = S ((S (pfp_repeat_index_step_helper_left_rightzeros)) * VPc)) /\\ exists ff_q_pfp_step_helper_left_rightzerosentry. VPb = ff_q_pfp_step_helper_left_rightzerosentry * S ((S (pfp_repeat_index_step_helper_left_rightzeros)) * VPc) + (0)))) /\\ ((forall pfrep_index_step_helper_left_right pfrep_value_step_helper_left_right. (exists pfa_gap_step_helper_left_rightbound. pfa_gap_step_helper_left_rightbound + S (pfrep_index_step_helper_left_right) = (M)) -> (((exists ff_h_pfp_step_helper_left_rightinput. ff_h_pfp_step_helper_left_rightinput + S (pfrep_value_step_helper_left_right) = S ((S (pfrep_index_step_helper_left_right)) * vc)) /\\ exists ff_q_pfp_step_helper_left_rightinput. vb = ff_q_pfp_step_helper_left_rightinput * S ((S (pfrep_index_step_helper_left_right)) * vc) + (pfrep_value_step_helper_left_right))) -> (((exists ff_h_pfp_step_helper_left_rightoutput. ff_h_pfp_step_helper_left_rightoutput + S (pfrep_value_step_helper_left_right) = S ((S ((S K)+pfrep_index_step_helper_left_right)) * VPc)) /\\ exists ff_q_pfp_step_helper_left_rightoutput. VPb = ff_q_pfp_step_helper_left_rightoutput * S ((S ((S K)+pfrep_index_step_helper_left_right)) * VPc) + (pfrep_value_step_helper_left_right))))))) -> (forall pfp_index_step_helper_left_sum. (exists pfa_gap_step_helper_left_sumindex. pfa_gap_step_helper_left_sumindex + S (pfp_index_step_helper_left_sum) = (M+S K)) -> exists pfp_left_step_helper_left_sum pfp_right_step_helper_left_sum pfp_value_step_helper_left_sum. ((((exists ff_h_pfp_step_helper_left_sumleft. ff_h_pfp_step_helper_left_sumleft + S (pfp_left_step_helper_left_sum) = S ((S (pfp_index_step_helper_left_sum)) * UPc)) /\\ exists ff_q_pfp_step_helper_left_sumleft. UPb = ff_q_pfp_step_helper_left_sumleft * S ((S (pfp_index_step_helper_left_sum)) * UPc) + (pfp_left_step_helper_left_sum))) /\\ (((((exists ff_h_pfp_step_helper_left_sumright. ff_h_pfp_step_helper_left_sumright + S (pfp_right_step_helper_left_sum) = S ((S (pfp_index_step_helper_left_sum)) * VPc)) /\\ exists ff_q_pfp_step_helper_left_sumright. VPb = ff_q_pfp_step_helper_left_sumright * S ((S (pfp_index_step_helper_left_sum)) * VPc) + (pfp_right_step_helper_left_sum))) /\\ (((((exists ff_h_pfp_step_helper_left_sumtarget. ff_h_pfp_step_helper_left_sumtarget + S (pfp_value_step_helper_left_sum) = S ((S (pfp_index_step_helper_left_sum)) * zc)) /\\ exists ff_q_pfp_step_helper_left_sumtarget. zb = ff_q_pfp_step_helper_left_sumtarget * S ((S (pfp_index_step_helper_left_sum)) * zc) + (pfp_value_step_helper_left_sum))) /\\ ((((exists pfa_gap_step_helper_left_sumoperationleft. pfa_gap_step_helper_left_sumoperationleft + S (pfp_left_step_helper_left_sum) = (p)) /\\ (((exists pfa_gap_step_helper_left_sumoperationright. pfa_gap_step_helper_left_sumoperationright + S (pfp_right_step_helper_left_sum) = (p)) /\\ ((((exists pfa_gap_step_helper_left_sumoperationresultbound. pfa_gap_step_helper_left_sumoperationresultbound + S (pfp_value_step_helper_left_sum) = (p)) /\\ ((exists pfa_offset_left_step_helper_left_sumoperationresultcongruence pfa_offset_right_step_helper_left_sumoperationresultcongruence. ((pfp_left_step_helper_left_sum) + (pfp_right_step_helper_left_sum)) + (p) * pfa_offset_left_step_helper_left_sumoperationresultcongruence = (pfp_value_step_helper_left_sum) + (p) * pfa_offset_right_step_helper_left_sumoperationresultcongruence)))))))))))))))) -> (((forall fom_index_pfp_step_helper_AZleft. (exists fom_gap_pfp_step_helper_AZleft_index_bound. fom_gap_pfp_step_helper_AZleft_index_bound + S (fom_index_pfp_step_helper_AZleft) = L) -> exists fom_value_pfp_step_helper_AZleft. ((((exists fom_beta_height_pfp_step_helper_AZleft_entry. fom_beta_height_pfp_step_helper_AZleft_entry + S (fom_value_pfp_step_helper_AZleft) = S ((S (fom_index_pfp_step_helper_AZleft)) * ac)) /\\ exists fom_beta_quotient_pfp_step_helper_AZleft_entry. ab = fom_beta_quotient_pfp_step_helper_AZleft_entry * S ((S (fom_index_pfp_step_helper_AZleft)) * ac) + (fom_value_pfp_step_helper_AZleft))) /\\ (exists fom_gap_pfp_step_helper_AZleft_value_bound. fom_gap_pfp_step_helper_AZleft_value_bound + S (fom_value_pfp_step_helper_AZleft) = p))) /\\ (((forall fom_index_pfp_step_helper_AZright. (exists fom_gap_pfp_step_helper_AZright_index_bound. fom_gap_pfp_step_helper_AZright_index_bound + S (fom_index_pfp_step_helper_AZright) = M+S K) -> exists fom_value_pfp_step_helper_AZright. ((((exists fom_beta_height_pfp_step_helper_AZright_entry. fom_beta_height_pfp_step_helper_AZright_entry + S (fom_value_pfp_step_helper_AZright) = S ((S (fom_index_pfp_step_helper_AZright)) * zc)) /\\ exists fom_beta_quotient_pfp_step_helper_AZright_entry. zb = fom_beta_quotient_pfp_step_helper_AZright_entry * S ((S (fom_index_pfp_step_helper_AZright)) * zc) + (fom_value_pfp_step_helper_AZright))) /\\ (exists fom_gap_pfp_step_helper_AZright_value_bound. fom_gap_pfp_step_helper_AZright_value_bound + S (fom_value_pfp_step_helper_AZright) = p))) /\\ (((((((L)=0 \\/ (M+S K)=0) /\\ (((W)=0)))) \\/ (((~((L)=0)) /\\ (((~((M+S K)=0)) /\\ (((L)+(M+S K)=S (W)))))))) /\\ ((forall pfc_index_step_helper_AZcoefficients. (exists pfa_gap_step_helper_AZcoefficientsbound. pfa_gap_step_helper_AZcoefficientsbound + S (pfc_index_step_helper_AZcoefficients) = (W)) -> exists pfc_value_step_helper_AZcoefficients. ((((exists ff_h_pfp_step_helper_AZcoefficientsentry. ff_h_pfp_step_helper_AZcoefficientsentry + S (pfc_value_step_helper_AZcoefficients) = S ((S (pfc_index_step_helper_AZcoefficients)) * sc)) /\\ exists ff_q_pfp_step_helper_AZcoefficientsentry. sb = ff_q_pfp_step_helper_AZcoefficientsentry * S ((S (pfc_index_step_helper_AZcoefficients)) * sc) + (pfc_value_step_helper_AZcoefficients))) /\\ ((exists pfc_terms_code_step_helper_AZcoefficientscoefficient pfc_terms_scale_step_helper_AZcoefficientscoefficient pfc_natural_sum_step_helper_AZcoefficientscoefficient. ((forall pfc_index_step_helper_AZcoefficientscoefficientdiagonal. (exists pfa_gap_step_helper_AZcoefficientscoefficientdiagonalbound. pfa_gap_step_helper_AZcoefficientscoefficientdiagonalbound + S (pfc_index_step_helper_AZcoefficientscoefficientdiagonal) = (S (pfc_index_step_helper_AZcoefficients))) -> exists pfc_value_step_helper_AZcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_step_helper_AZcoefficientscoefficientdiagonalentry. ff_h_pfp_step_helper_AZcoefficientscoefficientdiagonalentry + S (pfc_value_step_helper_AZcoefficientscoefficientdiagonal) = S ((S (pfc_index_step_helper_AZcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_AZcoefficientscoefficient)) /\\ exists ff_q_pfp_step_helper_AZcoefficientscoefficientdiagonalentry. pfc_terms_code_step_helper_AZcoefficientscoefficient = ff_q_pfp_step_helper_AZcoefficientscoefficientdiagonalentry * S ((S (pfc_index_step_helper_AZcoefficientscoefficientdiagonal)) * pfc_terms_scale_step_helper_AZcoefficientscoefficient) + (pfc_value_step_helper_AZcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_step_helper_AZcoefficientscoefficientdiagonalterm pfc_left_step_helper_AZcoefficientscoefficientdiagonalterm pfc_right_step_helper_AZcoefficientscoefficientdiagonalterm. (((pfc_index_step_helper_AZcoefficientscoefficientdiagonal)+pfc_complement_step_helper_AZcoefficientscoefficientdiagonalterm=(pfc_index_step_helper_AZcoefficients)) /\\ ((((((exists pfa_gap_step_helper_AZcoefficientscoefficientdiagonaltermleftinside. pfa_gap_step_helper_AZcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_step_helper_AZcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_step_helper_AZcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_step_helper_AZcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_step_helper_AZcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_step_helper_AZcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_step_helper_AZcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_step_helper_AZcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_step_helper_AZcoefficientscoefficientdiagonal)) * ac) + (pfc_left_step_helper_AZcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_AZcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_step_helper_AZcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_step_helper_AZcoefficientscoefficientdiagonal)) /\\ (((pfc_left_step_helper_AZcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_step_helper_AZcoefficientscoefficientdiagonaltermrightinside. pfa_gap_step_helper_AZcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_step_helper_AZcoefficientscoefficientdiagonalterm) = (M+S K)) /\\ ((((exists ff_h_pfp_step_helper_AZcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_step_helper_AZcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_step_helper_AZcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_step_helper_AZcoefficientscoefficientdiagonalterm)) * zc)) /\\ exists ff_q_pfp_step_helper_AZcoefficientscoefficientdiagonaltermrightentry. zb = ff_q_pfp_step_helper_AZcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_step_helper_AZcoefficientscoefficientdiagonalterm)) * zc) + (pfc_right_step_helper_AZcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_step_helper_AZcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_step_helper_AZcoefficientscoefficientdiagonaltermrightoutside+(M+S K)=(pfc_complement_step_helper_AZcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_step_helper_AZcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_step_helper_AZcoefficientscoefficientdiagonal)=pfc_left_step_helper_AZcoefficientscoefficientdiagonalterm*pfc_right_step_helper_AZcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_step_helper_AZcoefficientscoefficientsum fs_v_pfc_step_helper_AZcoefficientscoefficientsum. ((((exists fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_start. fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_start. fs_u_pfc_step_helper_AZcoefficientscoefficientsum = fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_terminal. fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_step_helper_AZcoefficientscoefficient) = S ((S (S (pfc_index_step_helper_AZcoefficients))) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_terminal. fs_u_pfc_step_helper_AZcoefficientscoefficientsum = fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_step_helper_AZcoefficients))) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum) + (pfc_natural_sum_step_helper_AZcoefficientscoefficient))) /\\ forall fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps = S (pfc_index_step_helper_AZcoefficients)) -> exists fs_a_pfc_step_helper_AZcoefficientscoefficientsum_body_steps fs_r_pfc_step_helper_AZcoefficientscoefficientsum_body_steps fs_s_pfc_step_helper_AZcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_step_helper_AZcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_AZcoefficientscoefficient)) /\\ exists fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_step_helper_AZcoefficientscoefficient = fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_step_helper_AZcoefficientscoefficient) + (fs_a_pfc_step_helper_AZcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_step_helper_AZcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_step_helper_AZcoefficientscoefficientsum = fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum) + (fs_r_pfc_step_helper_AZcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_step_helper_AZcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum)) /\\ exists fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_step_helper_AZcoefficientscoefficientsum = fs_q_pfc_step_helper_AZcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_step_helper_AZcoefficientscoefficientsum_body_steps)) * fs_v_pfc_step_helper_AZcoefficientscoefficientsum) + (fs_s_pfc_step_helper_AZcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_step_helper_AZcoefficientscoefficientsum_body_steps = fs_r_pfc_step_helper_AZcoefficientscoefficientsum_body_steps + fs_a_pfc_step_helper_AZcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_step_helper_AZcoefficientscoefficientresiduebound. pfa_gap_step_helper_AZcoefficientscoefficientresiduebound + S (pfc_value_step_helper_AZcoefficients) = (p)) /\\ ((exists pfa_offset_left_step_helper_AZcoefficientscoefficientresiduecongruence pfa_offset_right_step_helper_AZcoefficientscoefficientresiduecongruence. (pfc_natural_sum_step_helper_AZcoefficientscoefficient) + (p) * pfa_offset_left_step_helper_AZcoefficientscoefficientresiduecongruence = (pfc_value_step_helper_AZcoefficients) + (p) * pfa_offset_right_step_helper_AZcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall mdr_i_pfp_step_helper_right_shiftprefix mdr_a_pfp_step_helper_right_shiftprefix. (exists mdr_gap_pfp_step_helper_right_shiftprefixb. mdr_gap_pfp_step_helper_right_shiftprefixb + S (mdr_i_pfp_step_helper_right_shiftprefix) = (T)) -> (((exists ff_h_mdr_pfp_step_helper_right_shiftprefixo. ff_h_mdr_pfp_step_helper_right_shiftprefixo + S (mdr_a_pfp_step_helper_right_shiftprefix) = S ((S (mdr_i_pfp_step_helper_right_shiftprefix)) * rc)) /\\ exists ff_q_mdr_pfp_step_helper_right_shiftprefixo. rb = ff_q_mdr_pfp_step_helper_right_shiftprefixo * S ((S (mdr_i_pfp_step_helper_right_shiftprefix)) * rc) + (mdr_a_pfp_step_helper_right_shiftprefix))) -> (((exists ff_h_mdr_pfp_step_helper_right_shiftprefixn. ff_h_mdr_pfp_step_helper_right_shiftprefixn + S (mdr_a_pfp_step_helper_right_shiftprefix) = S ((S (mdr_i_pfp_step_helper_right_shiftprefix)) * ec)) /\\ exists ff_q_mdr_pfp_step_helper_right_shiftprefixn. eb = ff_q_mdr_pfp_step_helper_right_shiftprefixn * S ((S (mdr_i_pfp_step_helper_right_shiftprefix)) * ec) + (mdr_a_pfp_step_helper_right_shiftprefix)))) /\\ ((((exists ff_h_pfp_step_helper_right_shiftzero. ff_h_pfp_step_helper_right_shiftzero + S (0) = S ((S (T)) * ec)) /\\ exists ff_q_pfp_step_helper_right_shiftzero. eb = ff_q_pfp_step_helper_right_shiftzero * S ((S (T)) * ec) + (0)))))) -> (((exists pfa_gap_step_helper_right_scalescalar. pfa_gap_step_helper_right_scalescalar + S (c) = (p)) /\\ ((forall pfp_index_step_helper_right_scale. (exists pfa_gap_step_helper_right_scaleindex. pfa_gap_step_helper_right_scaleindex + S (pfp_index_step_helper_right_scale) = (N)) -> exists pfp_source_step_helper_right_scale pfp_value_step_helper_right_scale. ((((exists ff_h_pfp_step_helper_right_scalesource. ff_h_pfp_step_helper_right_scalesource + S (pfp_source_step_helper_right_scale) = S ((S (pfp_index_step_helper_right_scale)) * pc)) /\\ exists ff_q_pfp_step_helper_right_scalesource. pb = ff_q_pfp_step_helper_right_scalesource * S ((S (pfp_index_step_helper_right_scale)) * pc) + (pfp_source_step_helper_right_scale))) /\\ (((((exists ff_h_pfp_step_helper_right_scaletarget. ff_h_pfp_step_helper_right_scaletarget + S (pfp_value_step_helper_right_scale) = S ((S (pfp_index_step_helper_right_scale)) * fc)) /\\ exists ff_q_pfp_step_helper_right_scaletarget. fb = ff_q_pfp_step_helper_right_scaletarget * S ((S (pfp_index_step_helper_right_scale)) * fc) + (pfp_value_step_helper_right_scale))) /\\ ((((exists pfa_gap_step_helper_right_scaleoperationleft. pfa_gap_step_helper_right_scaleoperationleft + S (c) = (p)) /\\ (((exists pfa_gap_step_helper_right_scaleoperationright. pfa_gap_step_helper_right_scaleoperationright + S (pfp_source_step_helper_right_scale) = (p)) /\\ ((((exists pfa_gap_step_helper_right_scaleoperationresultbound. pfa_gap_step_helper_right_scaleoperationresultbound + S (pfp_value_step_helper_right_scale) = (p)) /\\ ((exists pfa_offset_left_step_helper_right_scaleoperationresultcongruence pfa_offset_right_step_helper_right_scaleoperationresultcongruence. ((c) * (pfp_source_step_helper_right_scale)) + (p) * pfa_offset_left_step_helper_right_scaleoperationresultcongruence = (pfp_value_step_helper_right_scale) + (p) * pfa_offset_right_step_helper_right_scaleoperationresultcongruence))))))))))))))))) -> (((forall pfp_repeat_index_step_helper_right_leftzeros. (exists pfa_gap_step_helper_right_leftzerosindex. pfa_gap_step_helper_right_leftzerosindex + S (pfp_repeat_index_step_helper_right_leftzeros) = (N)) -> (((exists ff_h_pfp_step_helper_right_leftzerosentry. ff_h_pfp_step_helper_right_leftzerosentry + S (0) = S ((S (pfp_repeat_index_step_helper_right_leftzeros)) * EPc)) /\\ exists ff_q_pfp_step_helper_right_leftzerosentry. EPb = ff_q_pfp_step_helper_right_leftzerosentry * S ((S (pfp_repeat_index_step_helper_right_leftzeros)) * EPc) + (0)))) /\\ ((forall pfrep_index_step_helper_right_left pfrep_value_step_helper_right_left. (exists pfa_gap_step_helper_right_leftbound. pfa_gap_step_helper_right_leftbound + S (pfrep_index_step_helper_right_left) = (S T)) -> (((exists ff_h_pfp_step_helper_right_leftinput. ff_h_pfp_step_helper_right_leftinput + S (pfrep_value_step_helper_right_left) = S ((S (pfrep_index_step_helper_right_left)) * ec)) /\\ exists ff_q_pfp_step_helper_right_leftinput. eb = ff_q_pfp_step_helper_right_leftinput * S ((S (pfrep_index_step_helper_right_left)) * ec) + (pfrep_value_step_helper_right_left))) -> (((exists ff_h_pfp_step_helper_right_leftoutput. ff_h_pfp_step_helper_right_leftoutput + S (pfrep_value_step_helper_right_left) = S ((S ((N)+pfrep_index_step_helper_right_left)) * EPc)) /\\ exists ff_q_pfp_step_helper_right_leftoutput. EPb = ff_q_pfp_step_helper_right_leftoutput * S ((S ((N)+pfrep_index_step_helper_right_left)) * EPc) + (pfrep_value_step_helper_right_left))))))) -> (((forall pfp_repeat_index_step_helper_right_rightzeros. (exists pfa_gap_step_helper_right_rightzerosindex. pfa_gap_step_helper_right_rightzerosindex + S (pfp_repeat_index_step_helper_right_rightzeros) = (S T)) -> (((exists ff_h_pfp_step_helper_right_rightzerosentry. ff_h_pfp_step_helper_right_rightzerosentry + S (0) = S ((S (pfp_repeat_index_step_helper_right_rightzeros)) * FPc)) /\\ exists ff_q_pfp_step_helper_right_rightzerosentry. FPb = ff_q_pfp_step_helper_right_rightzerosentry * S ((S (pfp_repeat_index_step_helper_right_rightzeros)) * FPc) + (0)))) /\\ ((forall pfrep_index_step_helper_right_right pfrep_value_step_helper_right_right. (exists pfa_gap_step_helper_right_rightbound. pfa_gap_step_helper_right_rightbound + S (pfrep_index_step_helper_right_right) = (N)) -> (((exists ff_h_pfp_step_helper_right_rightinput. ff_h_pfp_step_helper_right_rightinput + S (pfrep_value_step_helper_right_right) = S ((S (pfrep_index_step_helper_right_right)) * fc)) /\\ exists ff_q_pfp_step_helper_right_rightinput. fb = ff_q_pfp_step_helper_right_rightinput * S ((S (pfrep_index_step_helper_right_right)) * fc) + (pfrep_value_step_helper_right_right))) -> (((exists ff_h_pfp_step_helper_right_rightoutput. ff_h_pfp_step_helper_right_rightoutput + S (pfrep_value_step_helper_right_right) = S ((S ((S T)+pfrep_index_step_helper_right_right)) * FPc)) /\\ exists ff_q_pfp_step_helper_right_rightoutput. FPb = ff_q_pfp_step_helper_right_rightoutput * S ((S ((S T)+pfrep_index_step_helper_right_right)) * FPc) + (pfrep_value_step_helper_right_right))))))) -> (forall pfp_index_step_helper_right_sum. (exists pfa_gap_step_helper_right_sumindex. pfa_gap_step_helper_right_sumindex + S (pfp_index_step_helper_right_sum) = (N+S T)) -> exists pfp_left_step_helper_right_sum pfp_right_step_helper_right_sum pfp_value_step_helper_right_sum. ((((exists ff_h_pfp_step_helper_right_sumleft. ff_h_pfp_step_helper_right_sumleft + S (pfp_left_step_helper_right_sum) = S ((S (pfp_index_step_helper_right_sum)) * EPc)) /\\ exists ff_q_pfp_step_helper_right_sumleft. EPb = ff_q_pfp_step_helper_right_sumleft * S ((S (pfp_index_step_helper_right_sum)) * EPc) + (pfp_left_step_helper_right_sum))) /\\ (((((exists ff_h_pfp_step_helper_right_sumright. ff_h_pfp_step_helper_right_sumright + S (pfp_right_step_helper_right_sum) = S ((S (pfp_index_step_helper_right_sum)) * FPc)) /\\ exists ff_q_pfp_step_helper_right_sumright. FPb = ff_q_pfp_step_helper_right_sumright * S ((S (pfp_index_step_helper_right_sum)) * FPc) + (pfp_right_step_helper_right_sum))) /\\ (((((exists ff_h_pfp_step_helper_right_sumtarget. ff_h_pfp_step_helper_right_sumtarget + S (pfp_value_step_helper_right_sum) = S ((S (pfp_index_step_helper_right_sum)) * yc)) /\\ exists ff_q_pfp_step_helper_right_sumtarget. yb = ff_q_pfp_step_helper_right_sumtarget * S ((S (pfp_index_step_helper_right_sum)) * yc) + (pfp_value_step_helper_right_sum))) /\\ ((((exists pfa_gap_step_helper_right_sumoperationleft. pfa_gap_step_helper_right_sumoperationleft + S (pfp_left_step_helper_right_sum) = (p)) /\\ (((exists pfa_gap_step_helper_right_sumoperationright. pfa_gap_step_helper_right_sumoperationright + S (pfp_right_step_helper_right_sum) = (p)) /\\ ((((exists pfa_gap_step_helper_right_sumoperationresultbound. pfa_gap_step_helper_right_sumoperationresultbound + S (pfp_value_step_helper_right_sum) = (p)) /\\ ((exists pfa_offset_left_step_helper_right_sumoperationresultcongruence pfa_offset_right_step_helper_right_sumoperationresultcongruence. ((pfp_left_step_helper_right_sum) + (pfp_right_step_helper_right_sum)) + (p) * pfa_offset_left_step_helper_right_sumoperationresultcongruence = (pfp_value_step_helper_right_sum) + (p) * pfa_offset_right_step_helper_right_sumoperationresultcongruence)))))))))))))))) -> (forall pfrep_power_step_helper_result pfrep_left_step_helper_result pfrep_right_step_helper_result. ((exists pfrep_position_step_helper_resultfirst. ((pfrep_position_step_helper_resultfirst+S (pfrep_power_step_helper_result)=(W)) /\\ ((((exists ff_h_pfp_step_helper_resultfirstentry. ff_h_pfp_step_helper_resultfirstentry + S (pfrep_left_step_helper_result) = S ((S (pfrep_position_step_helper_resultfirst)) * sc)) /\\ exists ff_q_pfp_step_helper_resultfirstentry. sb = ff_q_pfp_step_helper_resultfirstentry * S ((S (pfrep_position_step_helper_resultfirst)) * sc) + (pfrep_left_step_helper_result)))))) \\/ (((exists pfrep_gap_step_helper_resultfirstoutside. pfrep_gap_step_helper_resultfirstoutside+(W)=(pfrep_power_step_helper_result)) /\\ (((pfrep_left_step_helper_result)=0))))) -> ((exists pfrep_position_step_helper_resultsecond. ((pfrep_position_step_helper_resultsecond+S (pfrep_power_step_helper_result)=(N+S T)) /\\ ((((exists ff_h_pfp_step_helper_resultsecondentry. ff_h_pfp_step_helper_resultsecondentry + S (pfrep_right_step_helper_result) = S ((S (pfrep_position_step_helper_resultsecond)) * yc)) /\\ exists ff_q_pfp_step_helper_resultsecondentry. yb = ff_q_pfp_step_helper_resultsecondentry * S ((S (pfrep_position_step_helper_resultsecond)) * yc) + (pfrep_right_step_helper_result)))))) \\/ (((exists pfrep_gap_step_helper_resultsecondoutside. pfrep_gap_step_helper_resultsecondoutside+(N+S T)=(pfrep_power_step_helper_result)) /\\ (((pfrep_right_step_helper_result)=0))))) -> pfrep_left_step_helper_result=pfrep_right_step_helper_result)", "statement_sha256": "f2e6ea8a698155ba880bb8d12d5d2e68dfcd6a8e9d0dd4864c77caff82e75bd1", "summary": "For actual AB and AQ, multiplying an actual leading-pad-aligned sum XQ+cB by A is formally equivalent to every actual aligned sum X(AQ)+c(AB). 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"alpha_v34_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "3c2fbeb86a3e3c444c0046723710941a602e168cdcffc316f45cddc3edd70c5c", "membership": "alpha_only", "name": "prime_field_polynomial_shift_scale_aligned_congruent", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro c", "intro ab", "intro ac", "intro L", "intro b0", "intro c0", "intro N0", "intro b1", "intro c1", "intro N1", "intro u0b", "intro u0c", "intro v0b", "intro v0c", "intro UP0b", "intro UP0c", "intro VP0b", "intro VP0c", "intro z0b", "intro z0c", "intro u1b", "intro u1c", "intro v1b", "intro v1c", "intro UP1b", "intro UP1c", "intro VP1b", "intro VP1c", "intro z1b", "intro z1c", "intro hp", "intro he", "intro hU0", "intro hV0", "intro hUP0", "intro hVP0", "intro hZ0", "intro hU1", "intro hV1", "intro hUP1", "intro hVP1", "intro hZ1", "have hshifted : forall pfrep_power_step_alignment_shifted pfrep_left_step_alignment_shifted pfrep_right_step_alignment_shifted. ((exists pfrep_position_step_alignment_shiftedfirst. ((pfrep_position_step_alignment_shiftedfirst+S (pfrep_power_step_alignment_shifted)=(S N0)) /\\ ((((exists ff_h_pfp_step_alignment_shiftedfirstentry. ff_h_pfp_step_alignment_shiftedfirstentry + S (pfrep_left_step_alignment_shifted) = S ((S (pfrep_position_step_alignment_shiftedfirst)) * u0c)) /\\ exists ff_q_pfp_step_alignment_shiftedfirstentry. u0b = ff_q_pfp_step_alignment_shiftedfirstentry * S ((S (pfrep_position_step_alignment_shiftedfirst)) * u0c) + (pfrep_left_step_alignment_shifted)))))) \\/ (((exists pfrep_gap_step_alignment_shiftedfirstoutside. pfrep_gap_step_alignment_shiftedfirstoutside+(S N0)=(pfrep_power_step_alignment_shifted)) /\\ (((pfrep_left_step_alignment_shifted)=0))))) -> ((exists pfrep_position_step_alignment_shiftedsecond. ((pfrep_position_step_alignment_shiftedsecond+S (pfrep_power_step_alignment_shifted)=(S N1)) /\\ ((((exists ff_h_pfp_step_alignment_shiftedsecondentry. ff_h_pfp_step_alignment_shiftedsecondentry + S (pfrep_right_step_alignment_shifted) = S ((S (pfrep_position_step_alignment_shiftedsecond)) * u1c)) /\\ exists ff_q_pfp_step_alignment_shiftedsecondentry. u1b = ff_q_pfp_step_alignment_shiftedsecondentry * S ((S (pfrep_position_step_alignment_shiftedsecond)) * u1c) + (pfrep_right_step_alignment_shifted)))))) \\/ (((exists pfrep_gap_step_alignment_shiftedsecondoutside. pfrep_gap_step_alignment_shiftedsecondoutside+(S N1)=(pfrep_power_step_alignment_shifted)) /\\ (((pfrep_right_step_alignment_shifted)=0))))) -> pfrep_left_step_alignment_shifted=pfrep_right_step_alignment_shifted", "specialize prime_field_polynomial_shift_equivalent_congruent (b0)", "specialize prime_field_polynomial_shift_equivalent_congruent (c0)", "specialize prime_field_polynomial_shift_equivalent_congruent (N0)", "specialize prime_field_polynomial_shift_equivalent_congruent (b1)", "specialize prime_field_polynomial_shift_equivalent_congruent (c1)", "specialize prime_field_polynomial_shift_equivalent_congruent (N1)", "specialize prime_field_polynomial_shift_equivalent_congruent (u0b)", "specialize prime_field_polynomial_shift_equivalent_congruent (u0c)", "specialize prime_field_polynomial_shift_equivalent_congruent (u1b)", "specialize prime_field_polynomial_shift_equivalent_congruent (u1c)", "apply prime_field_polynomial_shift_equivalent_congruent", "exact he", "exact hU0", "exact hU1", "have hscaled : forall mdr_i_pfp_step_alignment_scaled mdr_a_pfp_step_alignment_scaled. (exists mdr_gap_pfp_step_alignment_scaledb. mdr_gap_pfp_step_alignment_scaledb + S (mdr_i_pfp_step_alignment_scaled) = (L)) -> (((exists ff_h_mdr_pfp_step_alignment_scaledo. ff_h_mdr_pfp_step_alignment_scaledo + S (mdr_a_pfp_step_alignment_scaled) = S ((S (mdr_i_pfp_step_alignment_scaled)) * v0c)) /\\ exists ff_q_mdr_pfp_step_alignment_scaledo. v0b = ff_q_mdr_pfp_step_alignment_scaledo * S ((S (mdr_i_pfp_step_alignment_scaled)) * v0c) + (mdr_a_pfp_step_alignment_scaled))) -> (((exists ff_h_mdr_pfp_step_alignment_scaledn. ff_h_mdr_pfp_step_alignment_scaledn + S (mdr_a_pfp_step_alignment_scaled) = S ((S (mdr_i_pfp_step_alignment_scaled)) * v1c)) /\\ exists ff_q_mdr_pfp_step_alignment_scaledn. v1b = ff_q_mdr_pfp_step_alignment_scaledn * S ((S (mdr_i_pfp_step_alignment_scaled)) * v1c) + (mdr_a_pfp_step_alignment_scaled)))", "specialize prime_field_polynomial_scale_functional (p)", "specialize prime_field_polynomial_scale_functional (c)", "specialize prime_field_polynomial_scale_functional (ab)", "specialize prime_field_polynomial_scale_functional (ac)", "specialize prime_field_polynomial_scale_functional (v0b)", "specialize prime_field_polynomial_scale_functional (v0c)", "specialize prime_field_polynomial_scale_functional (v1b)", "specialize prime_field_polynomial_scale_functional (v1c)", "specialize prime_field_polynomial_scale_functional (L)", "apply prime_field_polynomial_scale_functional", "exact hV0", "exact hV1", "have hscalar_equal : forall pfrep_power_step_alignment_scalar_equal pfrep_left_step_alignment_scalar_equal pfrep_right_step_alignment_scalar_equal. ((exists pfrep_position_step_alignment_scalar_equalfirst. ((pfrep_position_step_alignment_scalar_equalfirst+S (pfrep_power_step_alignment_scalar_equal)=(L)) /\\ ((((exists ff_h_pfp_step_alignment_scalar_equalfirstentry. ff_h_pfp_step_alignment_scalar_equalfirstentry + S (pfrep_left_step_alignment_scalar_equal) = S ((S (pfrep_position_step_alignment_scalar_equalfirst)) * v0c)) /\\ exists ff_q_pfp_step_alignment_scalar_equalfirstentry. v0b = ff_q_pfp_step_alignment_scalar_equalfirstentry * S ((S (pfrep_position_step_alignment_scalar_equalfirst)) * v0c) + (pfrep_left_step_alignment_scalar_equal)))))) \\/ (((exists pfrep_gap_step_alignment_scalar_equalfirstoutside. pfrep_gap_step_alignment_scalar_equalfirstoutside+(L)=(pfrep_power_step_alignment_scalar_equal)) /\\ (((pfrep_left_step_alignment_scalar_equal)=0))))) -> ((exists pfrep_position_step_alignment_scalar_equalsecond. ((pfrep_position_step_alignment_scalar_equalsecond+S (pfrep_power_step_alignment_scalar_equal)=(L)) /\\ ((((exists ff_h_pfp_step_alignment_scalar_equalsecondentry. ff_h_pfp_step_alignment_scalar_equalsecondentry + S (pfrep_right_step_alignment_scalar_equal) = S ((S (pfrep_position_step_alignment_scalar_equalsecond)) * v1c)) /\\ exists ff_q_pfp_step_alignment_scalar_equalsecondentry. v1b = ff_q_pfp_step_alignment_scalar_equalsecondentry * S ((S (pfrep_position_step_alignment_scalar_equalsecond)) * v1c) + (pfrep_right_step_alignment_scalar_equal)))))) \\/ (((exists pfrep_gap_step_alignment_scalar_equalsecondoutside. pfrep_gap_step_alignment_scalar_equalsecondoutside+(L)=(pfrep_power_step_alignment_scalar_equal)) /\\ (((pfrep_right_step_alignment_scalar_equal)=0))))) -> pfrep_left_step_alignment_scalar_equal=pfrep_right_step_alignment_scalar_equal", "specialize prime_field_polynomial_equal_implies_equivalent (v0b)", "specialize prime_field_polynomial_equal_implies_equivalent (v0c)", "specialize prime_field_polynomial_equal_implies_equivalent (v1b)", "specialize prime_field_polynomial_equal_implies_equivalent (v1c)", "specialize prime_field_polynomial_equal_implies_equivalent (L)", "apply prime_field_polynomial_equal_implies_equivalent", "exact hscaled", "have hpad_left0 : forall pfrep_power_step_alignment_pad_left0 pfrep_left_step_alignment_pad_left0 pfrep_right_step_alignment_pad_left0. ((exists pfrep_position_step_alignment_pad_left0first. ((pfrep_position_step_alignment_pad_left0first+S (pfrep_power_step_alignment_pad_left0)=(S N0)) /\\ ((((exists ff_h_pfp_step_alignment_pad_left0firstentry. ff_h_pfp_step_alignment_pad_left0firstentry + S (pfrep_left_step_alignment_pad_left0) = S ((S (pfrep_position_step_alignment_pad_left0first)) * u0c)) /\\ exists ff_q_pfp_step_alignment_pad_left0firstentry. u0b = ff_q_pfp_step_alignment_pad_left0firstentry * S ((S (pfrep_position_step_alignment_pad_left0first)) * u0c) + (pfrep_left_step_alignment_pad_left0)))))) \\/ (((exists pfrep_gap_step_alignment_pad_left0firstoutside. pfrep_gap_step_alignment_pad_left0firstoutside+(S N0)=(pfrep_power_step_alignment_pad_left0)) /\\ (((pfrep_left_step_alignment_pad_left0)=0))))) -> ((exists pfrep_position_step_alignment_pad_left0second. ((pfrep_position_step_alignment_pad_left0second+S (pfrep_power_step_alignment_pad_left0)=(L+S N0)) /\\ ((((exists ff_h_pfp_step_alignment_pad_left0secondentry. ff_h_pfp_step_alignment_pad_left0secondentry + S (pfrep_right_step_alignment_pad_left0) = S ((S (pfrep_position_step_alignment_pad_left0second)) * UP0c)) /\\ exists ff_q_pfp_step_alignment_pad_left0secondentry. UP0b = ff_q_pfp_step_alignment_pad_left0secondentry * S ((S (pfrep_position_step_alignment_pad_left0second)) * UP0c) + (pfrep_right_step_alignment_pad_left0)))))) \\/ (((exists pfrep_gap_step_alignment_pad_left0secondoutside. pfrep_gap_step_alignment_pad_left0secondoutside+(L+S N0)=(pfrep_power_step_alignment_pad_left0)) /\\ (((pfrep_right_step_alignment_pad_left0)=0))))) -> pfrep_left_step_alignment_pad_left0=pfrep_right_step_alignment_pad_left0", "specialize prime_field_polynomial_left_pad_equivalent (u0b)", "specialize prime_field_polynomial_left_pad_equivalent (u0c)", "specialize prime_field_polynomial_left_pad_equivalent (S N0)", "specialize prime_field_polynomial_left_pad_equivalent (L)", "specialize prime_field_polynomial_left_pad_equivalent (UP0b)", "specialize prime_field_polynomial_left_pad_equivalent (UP0c)", "apply prime_field_polynomial_left_pad_equivalent", "exact hUP0", "have hpad_left1 : forall pfrep_power_step_alignment_pad_left1 pfrep_left_step_alignment_pad_left1 pfrep_right_step_alignment_pad_left1. ((exists pfrep_position_step_alignment_pad_left1first. ((pfrep_position_step_alignment_pad_left1first+S (pfrep_power_step_alignment_pad_left1)=(S N1)) /\\ ((((exists ff_h_pfp_step_alignment_pad_left1firstentry. ff_h_pfp_step_alignment_pad_left1firstentry + S (pfrep_left_step_alignment_pad_left1) = S ((S (pfrep_position_step_alignment_pad_left1first)) * u1c)) /\\ exists ff_q_pfp_step_alignment_pad_left1firstentry. u1b = ff_q_pfp_step_alignment_pad_left1firstentry * S ((S (pfrep_position_step_alignment_pad_left1first)) * u1c) + (pfrep_left_step_alignment_pad_left1)))))) \\/ (((exists pfrep_gap_step_alignment_pad_left1firstoutside. pfrep_gap_step_alignment_pad_left1firstoutside+(S N1)=(pfrep_power_step_alignment_pad_left1)) /\\ (((pfrep_left_step_alignment_pad_left1)=0))))) -> ((exists pfrep_position_step_alignment_pad_left1second. ((pfrep_position_step_alignment_pad_left1second+S (pfrep_power_step_alignment_pad_left1)=(L+S N1)) /\\ ((((exists ff_h_pfp_step_alignment_pad_left1secondentry. ff_h_pfp_step_alignment_pad_left1secondentry + S (pfrep_right_step_alignment_pad_left1) = S ((S (pfrep_position_step_alignment_pad_left1second)) * UP1c)) /\\ exists ff_q_pfp_step_alignment_pad_left1secondentry. UP1b = ff_q_pfp_step_alignment_pad_left1secondentry * S ((S (pfrep_position_step_alignment_pad_left1second)) * UP1c) + (pfrep_right_step_alignment_pad_left1)))))) \\/ (((exists pfrep_gap_step_alignment_pad_left1secondoutside. pfrep_gap_step_alignment_pad_left1secondoutside+(L+S N1)=(pfrep_power_step_alignment_pad_left1)) /\\ (((pfrep_right_step_alignment_pad_left1)=0))))) -> pfrep_left_step_alignment_pad_left1=pfrep_right_step_alignment_pad_left1", "specialize prime_field_polynomial_left_pad_equivalent (u1b)", "specialize prime_field_polynomial_left_pad_equivalent (u1c)", "specialize prime_field_polynomial_left_pad_equivalent (S N1)", "specialize prime_field_polynomial_left_pad_equivalent (L)", "specialize prime_field_polynomial_left_pad_equivalent (UP1b)", "specialize prime_field_polynomial_left_pad_equivalent (UP1c)", "apply prime_field_polynomial_left_pad_equivalent", "exact hUP1", "have hpad_right0 : forall pfrep_power_step_alignment_pad_right0 pfrep_left_step_alignment_pad_right0 pfrep_right_step_alignment_pad_right0. ((exists pfrep_position_step_alignment_pad_right0first. ((pfrep_position_step_alignment_pad_right0first+S (pfrep_power_step_alignment_pad_right0)=(L)) /\\ ((((exists ff_h_pfp_step_alignment_pad_right0firstentry. ff_h_pfp_step_alignment_pad_right0firstentry + S (pfrep_left_step_alignment_pad_right0) = S ((S (pfrep_position_step_alignment_pad_right0first)) * v0c)) /\\ exists ff_q_pfp_step_alignment_pad_right0firstentry. v0b = ff_q_pfp_step_alignment_pad_right0firstentry * S ((S (pfrep_position_step_alignment_pad_right0first)) * v0c) + (pfrep_left_step_alignment_pad_right0)))))) \\/ (((exists pfrep_gap_step_alignment_pad_right0firstoutside. pfrep_gap_step_alignment_pad_right0firstoutside+(L)=(pfrep_power_step_alignment_pad_right0)) /\\ (((pfrep_left_step_alignment_pad_right0)=0))))) -> ((exists pfrep_position_step_alignment_pad_right0second. ((pfrep_position_step_alignment_pad_right0second+S (pfrep_power_step_alignment_pad_right0)=(S N0+L)) /\\ ((((exists ff_h_pfp_step_alignment_pad_right0secondentry. ff_h_pfp_step_alignment_pad_right0secondentry + S (pfrep_right_step_alignment_pad_right0) = S ((S (pfrep_position_step_alignment_pad_right0second)) * VP0c)) /\\ exists ff_q_pfp_step_alignment_pad_right0secondentry. VP0b = ff_q_pfp_step_alignment_pad_right0secondentry * S ((S (pfrep_position_step_alignment_pad_right0second)) * VP0c) + (pfrep_right_step_alignment_pad_right0)))))) \\/ (((exists pfrep_gap_step_alignment_pad_right0secondoutside. pfrep_gap_step_alignment_pad_right0secondoutside+(S N0+L)=(pfrep_power_step_alignment_pad_right0)) /\\ (((pfrep_right_step_alignment_pad_right0)=0))))) -> pfrep_left_step_alignment_pad_right0=pfrep_right_step_alignment_pad_right0", "specialize prime_field_polynomial_left_pad_equivalent (v0b)", "specialize prime_field_polynomial_left_pad_equivalent (v0c)", "specialize prime_field_polynomial_left_pad_equivalent (L)", "specialize prime_field_polynomial_left_pad_equivalent (S N0)", "specialize prime_field_polynomial_left_pad_equivalent (VP0b)", "specialize prime_field_polynomial_left_pad_equivalent (VP0c)", "apply prime_field_polynomial_left_pad_equivalent", "exact hVP0", "have hcomm_right0 : S N0+L=L+S N0", "specialize add_comm (S N0)", "specialize add_comm (L)", "apply add_comm", "rewrite hcomm_right0 at hpad_right0", "rewrite hcomm_right0 at hpad_right0", "have hpad_right1 : forall pfrep_power_step_alignment_pad_right1 pfrep_left_step_alignment_pad_right1 pfrep_right_step_alignment_pad_right1. ((exists pfrep_position_step_alignment_pad_right1first. ((pfrep_position_step_alignment_pad_right1first+S (pfrep_power_step_alignment_pad_right1)=(L)) /\\ ((((exists ff_h_pfp_step_alignment_pad_right1firstentry. ff_h_pfp_step_alignment_pad_right1firstentry + S (pfrep_left_step_alignment_pad_right1) = S ((S (pfrep_position_step_alignment_pad_right1first)) * v1c)) /\\ exists ff_q_pfp_step_alignment_pad_right1firstentry. v1b = ff_q_pfp_step_alignment_pad_right1firstentry * S ((S (pfrep_position_step_alignment_pad_right1first)) * v1c) + (pfrep_left_step_alignment_pad_right1)))))) \\/ (((exists pfrep_gap_step_alignment_pad_right1firstoutside. pfrep_gap_step_alignment_pad_right1firstoutside+(L)=(pfrep_power_step_alignment_pad_right1)) /\\ (((pfrep_left_step_alignment_pad_right1)=0))))) -> ((exists pfrep_position_step_alignment_pad_right1second. ((pfrep_position_step_alignment_pad_right1second+S (pfrep_power_step_alignment_pad_right1)=(S N1+L)) /\\ ((((exists ff_h_pfp_step_alignment_pad_right1secondentry. ff_h_pfp_step_alignment_pad_right1secondentry + S (pfrep_right_step_alignment_pad_right1) = S ((S (pfrep_position_step_alignment_pad_right1second)) * VP1c)) /\\ exists ff_q_pfp_step_alignment_pad_right1secondentry. VP1b = ff_q_pfp_step_alignment_pad_right1secondentry * S ((S (pfrep_position_step_alignment_pad_right1second)) * VP1c) + (pfrep_right_step_alignment_pad_right1)))))) \\/ (((exists pfrep_gap_step_alignment_pad_right1secondoutside. pfrep_gap_step_alignment_pad_right1secondoutside+(S N1+L)=(pfrep_power_step_alignment_pad_right1)) /\\ (((pfrep_right_step_alignment_pad_right1)=0))))) -> pfrep_left_step_alignment_pad_right1=pfrep_right_step_alignment_pad_right1", "specialize prime_field_polynomial_left_pad_equivalent (v1b)", "specialize prime_field_polynomial_left_pad_equivalent (v1c)", "specialize prime_field_polynomial_left_pad_equivalent (L)", "specialize prime_field_polynomial_left_pad_equivalent (S N1)", "specialize prime_field_polynomial_left_pad_equivalent (VP1b)", "specialize prime_field_polynomial_left_pad_equivalent (VP1c)", "apply prime_field_polynomial_left_pad_equivalent", "exact hVP1", "have hcomm_right1 : S N1+L=L+S N1", "specialize add_comm (S N1)", "specialize add_comm (L)", "apply add_comm", "rewrite hcomm_right1 at hpad_right1", "rewrite hcomm_right1 at hpad_right1", "have hmiddle_left : forall pfrep_power_step_alignment_middle_left pfrep_left_step_alignment_middle_left pfrep_right_step_alignment_middle_left. ((exists pfrep_position_step_alignment_middle_leftfirst. ((pfrep_position_step_alignment_middle_leftfirst+S (pfrep_power_step_alignment_middle_left)=(L+S N0)) /\\ ((((exists ff_h_pfp_step_alignment_middle_leftfirstentry. ff_h_pfp_step_alignment_middle_leftfirstentry + S (pfrep_left_step_alignment_middle_left) = S ((S (pfrep_position_step_alignment_middle_leftfirst)) * UP0c)) /\\ exists ff_q_pfp_step_alignment_middle_leftfirstentry. UP0b = ff_q_pfp_step_alignment_middle_leftfirstentry * S ((S (pfrep_position_step_alignment_middle_leftfirst)) * UP0c) + (pfrep_left_step_alignment_middle_left)))))) \\/ (((exists pfrep_gap_step_alignment_middle_leftfirstoutside. pfrep_gap_step_alignment_middle_leftfirstoutside+(L+S N0)=(pfrep_power_step_alignment_middle_left)) /\\ (((pfrep_left_step_alignment_middle_left)=0))))) -> ((exists pfrep_position_step_alignment_middle_leftsecond. ((pfrep_position_step_alignment_middle_leftsecond+S (pfrep_power_step_alignment_middle_left)=(S N1)) /\\ ((((exists ff_h_pfp_step_alignment_middle_leftsecondentry. ff_h_pfp_step_alignment_middle_leftsecondentry + S (pfrep_right_step_alignment_middle_left) = S ((S (pfrep_position_step_alignment_middle_leftsecond)) * u1c)) /\\ exists ff_q_pfp_step_alignment_middle_leftsecondentry. u1b = ff_q_pfp_step_alignment_middle_leftsecondentry * S ((S (pfrep_position_step_alignment_middle_leftsecond)) * u1c) + (pfrep_right_step_alignment_middle_left)))))) \\/ (((exists pfrep_gap_step_alignment_middle_leftsecondoutside. pfrep_gap_step_alignment_middle_leftsecondoutside+(S N1)=(pfrep_power_step_alignment_middle_left)) /\\ (((pfrep_right_step_alignment_middle_left)=0))))) -> pfrep_left_step_alignment_middle_left=pfrep_right_step_alignment_middle_left", "specialize prime_field_polynomial_equivalent_transitive (UP0b)", "specialize prime_field_polynomial_equivalent_transitive (UP0c)", "specialize prime_field_polynomial_equivalent_transitive (L+S N0)", "specialize prime_field_polynomial_equivalent_transitive (u0b)", "specialize prime_field_polynomial_equivalent_transitive (u0c)", "specialize prime_field_polynomial_equivalent_transitive (S N0)", "specialize prime_field_polynomial_equivalent_transitive (u1b)", "specialize prime_field_polynomial_equivalent_transitive (u1c)", "specialize prime_field_polynomial_equivalent_transitive (S N1)", "apply prime_field_polynomial_equivalent_transitive", "specialize prime_field_polynomial_equivalent_symmetric (u0b)", "specialize prime_field_polynomial_equivalent_symmetric (u0c)", "specialize prime_field_polynomial_equivalent_symmetric (S N0)", "specialize prime_field_polynomial_equivalent_symmetric (UP0b)", "specialize prime_field_polynomial_equivalent_symmetric (UP0c)", "specialize prime_field_polynomial_equivalent_symmetric (L+S N0)", "apply prime_field_polynomial_equivalent_symmetric", "exact hpad_left0", "exact hshifted", "have hequal_left : forall pfrep_power_step_alignment_result_left pfrep_left_step_alignment_result_left pfrep_right_step_alignment_result_left. ((exists pfrep_position_step_alignment_result_leftfirst. ((pfrep_position_step_alignment_result_leftfirst+S (pfrep_power_step_alignment_result_left)=(L+S N0)) /\\ ((((exists ff_h_pfp_step_alignment_result_leftfirstentry. ff_h_pfp_step_alignment_result_leftfirstentry + S (pfrep_left_step_alignment_result_left) = S ((S (pfrep_position_step_alignment_result_leftfirst)) * UP0c)) /\\ exists ff_q_pfp_step_alignment_result_leftfirstentry. UP0b = ff_q_pfp_step_alignment_result_leftfirstentry * S ((S (pfrep_position_step_alignment_result_leftfirst)) * UP0c) + (pfrep_left_step_alignment_result_left)))))) \\/ (((exists pfrep_gap_step_alignment_result_leftfirstoutside. pfrep_gap_step_alignment_result_leftfirstoutside+(L+S N0)=(pfrep_power_step_alignment_result_left)) /\\ (((pfrep_left_step_alignment_result_left)=0))))) -> ((exists pfrep_position_step_alignment_result_leftsecond. ((pfrep_position_step_alignment_result_leftsecond+S (pfrep_power_step_alignment_result_left)=(L+S N1)) /\\ ((((exists ff_h_pfp_step_alignment_result_leftsecondentry. ff_h_pfp_step_alignment_result_leftsecondentry + S (pfrep_right_step_alignment_result_left) = S ((S (pfrep_position_step_alignment_result_leftsecond)) * UP1c)) /\\ exists ff_q_pfp_step_alignment_result_leftsecondentry. UP1b = ff_q_pfp_step_alignment_result_leftsecondentry * S ((S (pfrep_position_step_alignment_result_leftsecond)) * UP1c) + (pfrep_right_step_alignment_result_left)))))) \\/ (((exists pfrep_gap_step_alignment_result_leftsecondoutside. pfrep_gap_step_alignment_result_leftsecondoutside+(L+S N1)=(pfrep_power_step_alignment_result_left)) /\\ (((pfrep_right_step_alignment_result_left)=0))))) -> pfrep_left_step_alignment_result_left=pfrep_right_step_alignment_result_left", "specialize prime_field_polynomial_equivalent_transitive (UP0b)", "specialize prime_field_polynomial_equivalent_transitive (UP0c)", "specialize prime_field_polynomial_equivalent_transitive (L+S N0)", "specialize prime_field_polynomial_equivalent_transitive (u1b)", "specialize prime_field_polynomial_equivalent_transitive (u1c)", "specialize prime_field_polynomial_equivalent_transitive (S N1)", "specialize prime_field_polynomial_equivalent_transitive (UP1b)", "specialize prime_field_polynomial_equivalent_transitive (UP1c)", "specialize prime_field_polynomial_equivalent_transitive (L+S N1)", "apply prime_field_polynomial_equivalent_transitive", "exact hmiddle_left", "exact hpad_left1", "have hmiddle_right : forall pfrep_power_step_alignment_middle_right pfrep_left_step_alignment_middle_right pfrep_right_step_alignment_middle_right. ((exists pfrep_position_step_alignment_middle_rightfirst. ((pfrep_position_step_alignment_middle_rightfirst+S (pfrep_power_step_alignment_middle_right)=(L+S N0)) /\\ ((((exists ff_h_pfp_step_alignment_middle_rightfirstentry. ff_h_pfp_step_alignment_middle_rightfirstentry + S (pfrep_left_step_alignment_middle_right) = S ((S (pfrep_position_step_alignment_middle_rightfirst)) * VP0c)) /\\ exists ff_q_pfp_step_alignment_middle_rightfirstentry. VP0b = ff_q_pfp_step_alignment_middle_rightfirstentry * S ((S (pfrep_position_step_alignment_middle_rightfirst)) * VP0c) + (pfrep_left_step_alignment_middle_right)))))) \\/ (((exists pfrep_gap_step_alignment_middle_rightfirstoutside. pfrep_gap_step_alignment_middle_rightfirstoutside+(L+S N0)=(pfrep_power_step_alignment_middle_right)) /\\ (((pfrep_left_step_alignment_middle_right)=0))))) -> ((exists pfrep_position_step_alignment_middle_rightsecond. ((pfrep_position_step_alignment_middle_rightsecond+S (pfrep_power_step_alignment_middle_right)=(L)) /\\ ((((exists ff_h_pfp_step_alignment_middle_rightsecondentry. ff_h_pfp_step_alignment_middle_rightsecondentry + S (pfrep_right_step_alignment_middle_right) = S ((S (pfrep_position_step_alignment_middle_rightsecond)) * v1c)) /\\ exists ff_q_pfp_step_alignment_middle_rightsecondentry. v1b = ff_q_pfp_step_alignment_middle_rightsecondentry * S ((S (pfrep_position_step_alignment_middle_rightsecond)) * v1c) + (pfrep_right_step_alignment_middle_right)))))) \\/ (((exists pfrep_gap_step_alignment_middle_rightsecondoutside. pfrep_gap_step_alignment_middle_rightsecondoutside+(L)=(pfrep_power_step_alignment_middle_right)) /\\ (((pfrep_right_step_alignment_middle_right)=0))))) -> pfrep_left_step_alignment_middle_right=pfrep_right_step_alignment_middle_right", "specialize prime_field_polynomial_equivalent_transitive (VP0b)", "specialize prime_field_polynomial_equivalent_transitive (VP0c)", "specialize prime_field_polynomial_equivalent_transitive (L+S N0)", "specialize prime_field_polynomial_equivalent_transitive (v0b)", "specialize prime_field_polynomial_equivalent_transitive (v0c)", "specialize prime_field_polynomial_equivalent_transitive (L)", "specialize prime_field_polynomial_equivalent_transitive (v1b)", "specialize prime_field_polynomial_equivalent_transitive (v1c)", "specialize prime_field_polynomial_equivalent_transitive (L)", "apply prime_field_polynomial_equivalent_transitive", "specialize prime_field_polynomial_equivalent_symmetric (v0b)", "specialize prime_field_polynomial_equivalent_symmetric (v0c)", "specialize prime_field_polynomial_equivalent_symmetric (L)", "specialize prime_field_polynomial_equivalent_symmetric (VP0b)", "specialize prime_field_polynomial_equivalent_symmetric (VP0c)", "specialize prime_field_polynomial_equivalent_symmetric (L+S N0)", "apply prime_field_polynomial_equivalent_symmetric", "exact hpad_right0", "exact hscalar_equal", "have hequal_right : forall pfrep_power_step_alignment_result_right pfrep_left_step_alignment_result_right pfrep_right_step_alignment_result_right. ((exists pfrep_position_step_alignment_result_rightfirst. ((pfrep_position_step_alignment_result_rightfirst+S (pfrep_power_step_alignment_result_right)=(L+S N0)) /\\ ((((exists ff_h_pfp_step_alignment_result_rightfirstentry. ff_h_pfp_step_alignment_result_rightfirstentry + S (pfrep_left_step_alignment_result_right) = S ((S (pfrep_position_step_alignment_result_rightfirst)) * VP0c)) /\\ exists ff_q_pfp_step_alignment_result_rightfirstentry. VP0b = ff_q_pfp_step_alignment_result_rightfirstentry * S ((S (pfrep_position_step_alignment_result_rightfirst)) * VP0c) + (pfrep_left_step_alignment_result_right)))))) \\/ (((exists pfrep_gap_step_alignment_result_rightfirstoutside. pfrep_gap_step_alignment_result_rightfirstoutside+(L+S N0)=(pfrep_power_step_alignment_result_right)) /\\ (((pfrep_left_step_alignment_result_right)=0))))) -> ((exists pfrep_position_step_alignment_result_rightsecond. ((pfrep_position_step_alignment_result_rightsecond+S (pfrep_power_step_alignment_result_right)=(L+S N1)) /\\ ((((exists ff_h_pfp_step_alignment_result_rightsecondentry. ff_h_pfp_step_alignment_result_rightsecondentry + S (pfrep_right_step_alignment_result_right) = S ((S (pfrep_position_step_alignment_result_rightsecond)) * VP1c)) /\\ exists ff_q_pfp_step_alignment_result_rightsecondentry. VP1b = ff_q_pfp_step_alignment_result_rightsecondentry * S ((S (pfrep_position_step_alignment_result_rightsecond)) * VP1c) + (pfrep_right_step_alignment_result_right)))))) \\/ (((exists pfrep_gap_step_alignment_result_rightsecondoutside. pfrep_gap_step_alignment_result_rightsecondoutside+(L+S N1)=(pfrep_power_step_alignment_result_right)) /\\ (((pfrep_right_step_alignment_result_right)=0))))) -> pfrep_left_step_alignment_result_right=pfrep_right_step_alignment_result_right", "specialize prime_field_polynomial_equivalent_transitive (VP0b)", "specialize prime_field_polynomial_equivalent_transitive (VP0c)", "specialize prime_field_polynomial_equivalent_transitive (L+S N0)", "specialize prime_field_polynomial_equivalent_transitive (v1b)", "specialize prime_field_polynomial_equivalent_transitive (v1c)", "specialize prime_field_polynomial_equivalent_transitive (L)", "specialize prime_field_polynomial_equivalent_transitive (VP1b)", "specialize prime_field_polynomial_equivalent_transitive (VP1c)", "specialize prime_field_polynomial_equivalent_transitive (L+S N1)", "apply prime_field_polynomial_equivalent_transitive", "exact hmiddle_right", "exact hpad_right1", "specialize prime_field_polynomial_add_equivalent_congruent (p)", "specialize prime_field_polynomial_add_equivalent_congruent (UP0b)", "specialize prime_field_polynomial_add_equivalent_congruent (UP0c)", "specialize prime_field_polynomial_add_equivalent_congruent (VP0b)", "specialize prime_field_polynomial_add_equivalent_congruent (VP0c)", "specialize prime_field_polynomial_add_equivalent_congruent (z0b)", "specialize prime_field_polynomial_add_equivalent_congruent (z0c)", "specialize prime_field_polynomial_add_equivalent_congruent (L+S N0)", "specialize prime_field_polynomial_add_equivalent_congruent (UP1b)", "specialize prime_field_polynomial_add_equivalent_congruent (UP1c)", "specialize prime_field_polynomial_add_equivalent_congruent (VP1b)", "specialize prime_field_polynomial_add_equivalent_congruent (VP1c)", "specialize prime_field_polynomial_add_equivalent_congruent (z1b)", "specialize prime_field_polynomial_add_equivalent_congruent (z1c)", "specialize prime_field_polynomial_add_equivalent_congruent (L+S N1)", "apply prime_field_polynomial_add_equivalent_congruent", "exact hp", "exact hequal_left", "exact hequal_right", "exact hZ0", "exact hZ1" ], "script_sha256": "446c8fa57d14639ec7c3aaa15001b5d47ddd2c203064fb3f741f628a95a56635", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_associativity_step_candidate.py", "sha256": "dd85dbd1bd87143715a4286724ac7c87f280a909dac6759f00a6cb7dff7c85f1" }, "statement": "forall p c ab ac L b0 c0 N0 b1 c1 N1 u0b u0c v0b v0c UP0b UP0c VP0b VP0c z0b z0c u1b u1c v1b v1c UP1b UP1c VP1b VP1c z1b z1c. (~((p) = 1) /\\ forall pfa_factor_left_step_alignment_prime pfa_factor_right_step_alignment_prime. (p) = pfa_factor_left_step_alignment_prime * pfa_factor_right_step_alignment_prime -> pfa_factor_left_step_alignment_prime = 1 \\/ pfa_factor_right_step_alignment_prime = 1) -> (forall pfrep_power_step_alignment_old_equal pfrep_left_step_alignment_old_equal pfrep_right_step_alignment_old_equal. ((exists pfrep_position_step_alignment_old_equalfirst. ((pfrep_position_step_alignment_old_equalfirst+S (pfrep_power_step_alignment_old_equal)=(N0)) /\\ ((((exists ff_h_pfp_step_alignment_old_equalfirstentry. ff_h_pfp_step_alignment_old_equalfirstentry + S (pfrep_left_step_alignment_old_equal) = S ((S (pfrep_position_step_alignment_old_equalfirst)) * c0)) /\\ exists ff_q_pfp_step_alignment_old_equalfirstentry. b0 = ff_q_pfp_step_alignment_old_equalfirstentry * S ((S (pfrep_position_step_alignment_old_equalfirst)) * c0) + (pfrep_left_step_alignment_old_equal)))))) \\/ (((exists pfrep_gap_step_alignment_old_equalfirstoutside. pfrep_gap_step_alignment_old_equalfirstoutside+(N0)=(pfrep_power_step_alignment_old_equal)) /\\ (((pfrep_left_step_alignment_old_equal)=0))))) -> ((exists pfrep_position_step_alignment_old_equalsecond. ((pfrep_position_step_alignment_old_equalsecond+S (pfrep_power_step_alignment_old_equal)=(N1)) /\\ ((((exists ff_h_pfp_step_alignment_old_equalsecondentry. ff_h_pfp_step_alignment_old_equalsecondentry + S (pfrep_right_step_alignment_old_equal) = S ((S (pfrep_position_step_alignment_old_equalsecond)) * c1)) /\\ exists ff_q_pfp_step_alignment_old_equalsecondentry. b1 = ff_q_pfp_step_alignment_old_equalsecondentry * S ((S (pfrep_position_step_alignment_old_equalsecond)) * c1) + (pfrep_right_step_alignment_old_equal)))))) \\/ (((exists pfrep_gap_step_alignment_old_equalsecondoutside. pfrep_gap_step_alignment_old_equalsecondoutside+(N1)=(pfrep_power_step_alignment_old_equal)) /\\ (((pfrep_right_step_alignment_old_equal)=0))))) -> pfrep_left_step_alignment_old_equal=pfrep_right_step_alignment_old_equal) -> (((forall mdr_i_pfp_step_alignment_old_shiftprefix mdr_a_pfp_step_alignment_old_shiftprefix. (exists mdr_gap_pfp_step_alignment_old_shiftprefixb. mdr_gap_pfp_step_alignment_old_shiftprefixb + S (mdr_i_pfp_step_alignment_old_shiftprefix) = (N0)) -> (((exists ff_h_mdr_pfp_step_alignment_old_shiftprefixo. ff_h_mdr_pfp_step_alignment_old_shiftprefixo + S (mdr_a_pfp_step_alignment_old_shiftprefix) = S ((S (mdr_i_pfp_step_alignment_old_shiftprefix)) * c0)) /\\ exists ff_q_mdr_pfp_step_alignment_old_shiftprefixo. b0 = ff_q_mdr_pfp_step_alignment_old_shiftprefixo * S ((S (mdr_i_pfp_step_alignment_old_shiftprefix)) * c0) + (mdr_a_pfp_step_alignment_old_shiftprefix))) -> (((exists ff_h_mdr_pfp_step_alignment_old_shiftprefixn. ff_h_mdr_pfp_step_alignment_old_shiftprefixn + S (mdr_a_pfp_step_alignment_old_shiftprefix) = S ((S (mdr_i_pfp_step_alignment_old_shiftprefix)) * u0c)) /\\ exists ff_q_mdr_pfp_step_alignment_old_shiftprefixn. u0b = ff_q_mdr_pfp_step_alignment_old_shiftprefixn * S ((S (mdr_i_pfp_step_alignment_old_shiftprefix)) * u0c) + (mdr_a_pfp_step_alignment_old_shiftprefix)))) /\\ ((((exists ff_h_pfp_step_alignment_old_shiftzero. ff_h_pfp_step_alignment_old_shiftzero + S (0) = S ((S (N0)) * u0c)) /\\ exists ff_q_pfp_step_alignment_old_shiftzero. u0b = ff_q_pfp_step_alignment_old_shiftzero * S ((S (N0)) * u0c) + (0)))))) -> (((exists pfa_gap_step_alignment_old_scalescalar. pfa_gap_step_alignment_old_scalescalar + S (c) = (p)) /\\ ((forall pfp_index_step_alignment_old_scale. (exists pfa_gap_step_alignment_old_scaleindex. pfa_gap_step_alignment_old_scaleindex + S (pfp_index_step_alignment_old_scale) = (L)) -> exists pfp_source_step_alignment_old_scale pfp_value_step_alignment_old_scale. ((((exists ff_h_pfp_step_alignment_old_scalesource. ff_h_pfp_step_alignment_old_scalesource + S (pfp_source_step_alignment_old_scale) = S ((S (pfp_index_step_alignment_old_scale)) * ac)) /\\ exists ff_q_pfp_step_alignment_old_scalesource. ab = ff_q_pfp_step_alignment_old_scalesource * S ((S (pfp_index_step_alignment_old_scale)) * ac) + (pfp_source_step_alignment_old_scale))) /\\ (((((exists ff_h_pfp_step_alignment_old_scaletarget. ff_h_pfp_step_alignment_old_scaletarget + S (pfp_value_step_alignment_old_scale) = S ((S (pfp_index_step_alignment_old_scale)) * v0c)) /\\ exists ff_q_pfp_step_alignment_old_scaletarget. v0b = ff_q_pfp_step_alignment_old_scaletarget * S ((S (pfp_index_step_alignment_old_scale)) * v0c) + (pfp_value_step_alignment_old_scale))) /\\ ((((exists pfa_gap_step_alignment_old_scaleoperationleft. pfa_gap_step_alignment_old_scaleoperationleft + S (c) = (p)) /\\ (((exists pfa_gap_step_alignment_old_scaleoperationright. pfa_gap_step_alignment_old_scaleoperationright + S (pfp_source_step_alignment_old_scale) = (p)) /\\ ((((exists pfa_gap_step_alignment_old_scaleoperationresultbound. pfa_gap_step_alignment_old_scaleoperationresultbound + S (pfp_value_step_alignment_old_scale) = (p)) /\\ ((exists pfa_offset_left_step_alignment_old_scaleoperationresultcongruence pfa_offset_right_step_alignment_old_scaleoperationresultcongruence. ((c) * (pfp_source_step_alignment_old_scale)) + (p) * pfa_offset_left_step_alignment_old_scaleoperationresultcongruence = (pfp_value_step_alignment_old_scale) + (p) * pfa_offset_right_step_alignment_old_scaleoperationresultcongruence))))))))))))))))) -> (((forall pfp_repeat_index_step_alignment_old_leftzeros. (exists pfa_gap_step_alignment_old_leftzerosindex. pfa_gap_step_alignment_old_leftzerosindex + S (pfp_repeat_index_step_alignment_old_leftzeros) = (L)) -> (((exists ff_h_pfp_step_alignment_old_leftzerosentry. ff_h_pfp_step_alignment_old_leftzerosentry + S (0) = S ((S (pfp_repeat_index_step_alignment_old_leftzeros)) * UP0c)) /\\ exists ff_q_pfp_step_alignment_old_leftzerosentry. UP0b = ff_q_pfp_step_alignment_old_leftzerosentry * S ((S (pfp_repeat_index_step_alignment_old_leftzeros)) * UP0c) + (0)))) /\\ ((forall pfrep_index_step_alignment_old_left pfrep_value_step_alignment_old_left. (exists pfa_gap_step_alignment_old_leftbound. pfa_gap_step_alignment_old_leftbound + S (pfrep_index_step_alignment_old_left) = (S N0)) -> (((exists ff_h_pfp_step_alignment_old_leftinput. ff_h_pfp_step_alignment_old_leftinput + S (pfrep_value_step_alignment_old_left) = S ((S (pfrep_index_step_alignment_old_left)) * u0c)) /\\ exists ff_q_pfp_step_alignment_old_leftinput. u0b = ff_q_pfp_step_alignment_old_leftinput * S ((S (pfrep_index_step_alignment_old_left)) * u0c) + (pfrep_value_step_alignment_old_left))) -> (((exists ff_h_pfp_step_alignment_old_leftoutput. ff_h_pfp_step_alignment_old_leftoutput + S (pfrep_value_step_alignment_old_left) = S ((S ((L)+pfrep_index_step_alignment_old_left)) * UP0c)) /\\ exists ff_q_pfp_step_alignment_old_leftoutput. UP0b = ff_q_pfp_step_alignment_old_leftoutput * S ((S ((L)+pfrep_index_step_alignment_old_left)) * UP0c) + (pfrep_value_step_alignment_old_left))))))) -> (((forall pfp_repeat_index_step_alignment_old_rightzeros. (exists pfa_gap_step_alignment_old_rightzerosindex. pfa_gap_step_alignment_old_rightzerosindex + S (pfp_repeat_index_step_alignment_old_rightzeros) = (S N0)) -> (((exists ff_h_pfp_step_alignment_old_rightzerosentry. ff_h_pfp_step_alignment_old_rightzerosentry + S (0) = S ((S (pfp_repeat_index_step_alignment_old_rightzeros)) * VP0c)) /\\ exists ff_q_pfp_step_alignment_old_rightzerosentry. VP0b = ff_q_pfp_step_alignment_old_rightzerosentry * S ((S (pfp_repeat_index_step_alignment_old_rightzeros)) * VP0c) + (0)))) /\\ ((forall pfrep_index_step_alignment_old_right pfrep_value_step_alignment_old_right. (exists pfa_gap_step_alignment_old_rightbound. pfa_gap_step_alignment_old_rightbound + S (pfrep_index_step_alignment_old_right) = (L)) -> (((exists ff_h_pfp_step_alignment_old_rightinput. ff_h_pfp_step_alignment_old_rightinput + S (pfrep_value_step_alignment_old_right) = S ((S (pfrep_index_step_alignment_old_right)) * v0c)) /\\ exists ff_q_pfp_step_alignment_old_rightinput. v0b = ff_q_pfp_step_alignment_old_rightinput * S ((S (pfrep_index_step_alignment_old_right)) * v0c) + (pfrep_value_step_alignment_old_right))) -> (((exists ff_h_pfp_step_alignment_old_rightoutput. ff_h_pfp_step_alignment_old_rightoutput + S (pfrep_value_step_alignment_old_right) = S ((S ((S N0)+pfrep_index_step_alignment_old_right)) * VP0c)) /\\ exists ff_q_pfp_step_alignment_old_rightoutput. VP0b = ff_q_pfp_step_alignment_old_rightoutput * S ((S ((S N0)+pfrep_index_step_alignment_old_right)) * VP0c) + (pfrep_value_step_alignment_old_right))))))) -> (forall pfp_index_step_alignment_old_sum. (exists pfa_gap_step_alignment_old_sumindex. pfa_gap_step_alignment_old_sumindex + S (pfp_index_step_alignment_old_sum) = (L+S N0)) -> exists pfp_left_step_alignment_old_sum pfp_right_step_alignment_old_sum pfp_value_step_alignment_old_sum. ((((exists ff_h_pfp_step_alignment_old_sumleft. ff_h_pfp_step_alignment_old_sumleft + S (pfp_left_step_alignment_old_sum) = S ((S (pfp_index_step_alignment_old_sum)) * UP0c)) /\\ exists ff_q_pfp_step_alignment_old_sumleft. UP0b = ff_q_pfp_step_alignment_old_sumleft * S ((S (pfp_index_step_alignment_old_sum)) * UP0c) + (pfp_left_step_alignment_old_sum))) /\\ (((((exists ff_h_pfp_step_alignment_old_sumright. ff_h_pfp_step_alignment_old_sumright + S (pfp_right_step_alignment_old_sum) = S ((S (pfp_index_step_alignment_old_sum)) * VP0c)) /\\ exists ff_q_pfp_step_alignment_old_sumright. VP0b = ff_q_pfp_step_alignment_old_sumright * S ((S (pfp_index_step_alignment_old_sum)) * VP0c) + (pfp_right_step_alignment_old_sum))) /\\ (((((exists ff_h_pfp_step_alignment_old_sumtarget. ff_h_pfp_step_alignment_old_sumtarget + S (pfp_value_step_alignment_old_sum) = S ((S (pfp_index_step_alignment_old_sum)) * z0c)) /\\ exists ff_q_pfp_step_alignment_old_sumtarget. z0b = ff_q_pfp_step_alignment_old_sumtarget * S ((S (pfp_index_step_alignment_old_sum)) * z0c) + (pfp_value_step_alignment_old_sum))) /\\ ((((exists pfa_gap_step_alignment_old_sumoperationleft. pfa_gap_step_alignment_old_sumoperationleft + S (pfp_left_step_alignment_old_sum) = (p)) /\\ (((exists pfa_gap_step_alignment_old_sumoperationright. pfa_gap_step_alignment_old_sumoperationright + S (pfp_right_step_alignment_old_sum) = (p)) /\\ ((((exists pfa_gap_step_alignment_old_sumoperationresultbound. pfa_gap_step_alignment_old_sumoperationresultbound + S (pfp_value_step_alignment_old_sum) = (p)) /\\ ((exists pfa_offset_left_step_alignment_old_sumoperationresultcongruence pfa_offset_right_step_alignment_old_sumoperationresultcongruence. ((pfp_left_step_alignment_old_sum) + (pfp_right_step_alignment_old_sum)) + (p) * pfa_offset_left_step_alignment_old_sumoperationresultcongruence = (pfp_value_step_alignment_old_sum) + (p) * pfa_offset_right_step_alignment_old_sumoperationresultcongruence)))))))))))))))) -> (((forall mdr_i_pfp_step_alignment_new_shiftprefix mdr_a_pfp_step_alignment_new_shiftprefix. (exists mdr_gap_pfp_step_alignment_new_shiftprefixb. mdr_gap_pfp_step_alignment_new_shiftprefixb + S (mdr_i_pfp_step_alignment_new_shiftprefix) = (N1)) -> (((exists ff_h_mdr_pfp_step_alignment_new_shiftprefixo. ff_h_mdr_pfp_step_alignment_new_shiftprefixo + S (mdr_a_pfp_step_alignment_new_shiftprefix) = S ((S (mdr_i_pfp_step_alignment_new_shiftprefix)) * c1)) /\\ exists ff_q_mdr_pfp_step_alignment_new_shiftprefixo. b1 = ff_q_mdr_pfp_step_alignment_new_shiftprefixo * S ((S (mdr_i_pfp_step_alignment_new_shiftprefix)) * c1) + (mdr_a_pfp_step_alignment_new_shiftprefix))) -> (((exists ff_h_mdr_pfp_step_alignment_new_shiftprefixn. ff_h_mdr_pfp_step_alignment_new_shiftprefixn + S (mdr_a_pfp_step_alignment_new_shiftprefix) = S ((S (mdr_i_pfp_step_alignment_new_shiftprefix)) * u1c)) /\\ exists ff_q_mdr_pfp_step_alignment_new_shiftprefixn. u1b = ff_q_mdr_pfp_step_alignment_new_shiftprefixn * S ((S (mdr_i_pfp_step_alignment_new_shiftprefix)) * u1c) + (mdr_a_pfp_step_alignment_new_shiftprefix)))) /\\ ((((exists ff_h_pfp_step_alignment_new_shiftzero. ff_h_pfp_step_alignment_new_shiftzero + S (0) = S ((S (N1)) * u1c)) /\\ exists ff_q_pfp_step_alignment_new_shiftzero. u1b = ff_q_pfp_step_alignment_new_shiftzero * S ((S (N1)) * u1c) + (0)))))) -> (((exists pfa_gap_step_alignment_new_scalescalar. pfa_gap_step_alignment_new_scalescalar + S (c) = (p)) /\\ ((forall pfp_index_step_alignment_new_scale. (exists pfa_gap_step_alignment_new_scaleindex. pfa_gap_step_alignment_new_scaleindex + S (pfp_index_step_alignment_new_scale) = (L)) -> exists pfp_source_step_alignment_new_scale pfp_value_step_alignment_new_scale. ((((exists ff_h_pfp_step_alignment_new_scalesource. ff_h_pfp_step_alignment_new_scalesource + S (pfp_source_step_alignment_new_scale) = S ((S (pfp_index_step_alignment_new_scale)) * ac)) /\\ exists ff_q_pfp_step_alignment_new_scalesource. ab = ff_q_pfp_step_alignment_new_scalesource * S ((S (pfp_index_step_alignment_new_scale)) * ac) + (pfp_source_step_alignment_new_scale))) /\\ (((((exists ff_h_pfp_step_alignment_new_scaletarget. ff_h_pfp_step_alignment_new_scaletarget + S (pfp_value_step_alignment_new_scale) = S ((S (pfp_index_step_alignment_new_scale)) * v1c)) /\\ exists ff_q_pfp_step_alignment_new_scaletarget. v1b = ff_q_pfp_step_alignment_new_scaletarget * S ((S (pfp_index_step_alignment_new_scale)) * v1c) + (pfp_value_step_alignment_new_scale))) /\\ ((((exists pfa_gap_step_alignment_new_scaleoperationleft. pfa_gap_step_alignment_new_scaleoperationleft + S (c) = (p)) /\\ (((exists pfa_gap_step_alignment_new_scaleoperationright. pfa_gap_step_alignment_new_scaleoperationright + S (pfp_source_step_alignment_new_scale) = (p)) /\\ ((((exists pfa_gap_step_alignment_new_scaleoperationresultbound. pfa_gap_step_alignment_new_scaleoperationresultbound + S (pfp_value_step_alignment_new_scale) = (p)) /\\ ((exists pfa_offset_left_step_alignment_new_scaleoperationresultcongruence pfa_offset_right_step_alignment_new_scaleoperationresultcongruence. ((c) * (pfp_source_step_alignment_new_scale)) + (p) * pfa_offset_left_step_alignment_new_scaleoperationresultcongruence = (pfp_value_step_alignment_new_scale) + (p) * pfa_offset_right_step_alignment_new_scaleoperationresultcongruence))))))))))))))))) -> (((forall pfp_repeat_index_step_alignment_new_leftzeros. (exists pfa_gap_step_alignment_new_leftzerosindex. pfa_gap_step_alignment_new_leftzerosindex + S (pfp_repeat_index_step_alignment_new_leftzeros) = (L)) -> (((exists ff_h_pfp_step_alignment_new_leftzerosentry. ff_h_pfp_step_alignment_new_leftzerosentry + S (0) = S ((S (pfp_repeat_index_step_alignment_new_leftzeros)) * UP1c)) /\\ exists ff_q_pfp_step_alignment_new_leftzerosentry. UP1b = ff_q_pfp_step_alignment_new_leftzerosentry * S ((S (pfp_repeat_index_step_alignment_new_leftzeros)) * UP1c) + (0)))) /\\ ((forall pfrep_index_step_alignment_new_left pfrep_value_step_alignment_new_left. (exists pfa_gap_step_alignment_new_leftbound. pfa_gap_step_alignment_new_leftbound + S (pfrep_index_step_alignment_new_left) = (S N1)) -> (((exists ff_h_pfp_step_alignment_new_leftinput. ff_h_pfp_step_alignment_new_leftinput + S (pfrep_value_step_alignment_new_left) = S ((S (pfrep_index_step_alignment_new_left)) * u1c)) /\\ exists ff_q_pfp_step_alignment_new_leftinput. u1b = ff_q_pfp_step_alignment_new_leftinput * S ((S (pfrep_index_step_alignment_new_left)) * u1c) + (pfrep_value_step_alignment_new_left))) -> (((exists ff_h_pfp_step_alignment_new_leftoutput. ff_h_pfp_step_alignment_new_leftoutput + S (pfrep_value_step_alignment_new_left) = S ((S ((L)+pfrep_index_step_alignment_new_left)) * UP1c)) /\\ exists ff_q_pfp_step_alignment_new_leftoutput. UP1b = ff_q_pfp_step_alignment_new_leftoutput * S ((S ((L)+pfrep_index_step_alignment_new_left)) * UP1c) + (pfrep_value_step_alignment_new_left))))))) -> (((forall pfp_repeat_index_step_alignment_new_rightzeros. (exists pfa_gap_step_alignment_new_rightzerosindex. pfa_gap_step_alignment_new_rightzerosindex + S (pfp_repeat_index_step_alignment_new_rightzeros) = (S N1)) -> (((exists ff_h_pfp_step_alignment_new_rightzerosentry. ff_h_pfp_step_alignment_new_rightzerosentry + S (0) = S ((S (pfp_repeat_index_step_alignment_new_rightzeros)) * VP1c)) /\\ exists ff_q_pfp_step_alignment_new_rightzerosentry. VP1b = ff_q_pfp_step_alignment_new_rightzerosentry * S ((S (pfp_repeat_index_step_alignment_new_rightzeros)) * VP1c) + (0)))) /\\ ((forall pfrep_index_step_alignment_new_right pfrep_value_step_alignment_new_right. (exists pfa_gap_step_alignment_new_rightbound. pfa_gap_step_alignment_new_rightbound + S (pfrep_index_step_alignment_new_right) = (L)) -> (((exists ff_h_pfp_step_alignment_new_rightinput. ff_h_pfp_step_alignment_new_rightinput + S (pfrep_value_step_alignment_new_right) = S ((S (pfrep_index_step_alignment_new_right)) * v1c)) /\\ exists ff_q_pfp_step_alignment_new_rightinput. v1b = ff_q_pfp_step_alignment_new_rightinput * S ((S (pfrep_index_step_alignment_new_right)) * v1c) + (pfrep_value_step_alignment_new_right))) -> (((exists ff_h_pfp_step_alignment_new_rightoutput. ff_h_pfp_step_alignment_new_rightoutput + S (pfrep_value_step_alignment_new_right) = S ((S ((S N1)+pfrep_index_step_alignment_new_right)) * VP1c)) /\\ exists ff_q_pfp_step_alignment_new_rightoutput. VP1b = ff_q_pfp_step_alignment_new_rightoutput * S ((S ((S N1)+pfrep_index_step_alignment_new_right)) * VP1c) + (pfrep_value_step_alignment_new_right))))))) -> (forall pfp_index_step_alignment_new_sum. (exists pfa_gap_step_alignment_new_sumindex. pfa_gap_step_alignment_new_sumindex + S (pfp_index_step_alignment_new_sum) = (L+S N1)) -> exists pfp_left_step_alignment_new_sum pfp_right_step_alignment_new_sum pfp_value_step_alignment_new_sum. ((((exists ff_h_pfp_step_alignment_new_sumleft. ff_h_pfp_step_alignment_new_sumleft + S (pfp_left_step_alignment_new_sum) = S ((S (pfp_index_step_alignment_new_sum)) * UP1c)) /\\ exists ff_q_pfp_step_alignment_new_sumleft. UP1b = ff_q_pfp_step_alignment_new_sumleft * S ((S (pfp_index_step_alignment_new_sum)) * UP1c) + (pfp_left_step_alignment_new_sum))) /\\ (((((exists ff_h_pfp_step_alignment_new_sumright. ff_h_pfp_step_alignment_new_sumright + S (pfp_right_step_alignment_new_sum) = S ((S (pfp_index_step_alignment_new_sum)) * VP1c)) /\\ exists ff_q_pfp_step_alignment_new_sumright. VP1b = ff_q_pfp_step_alignment_new_sumright * S ((S (pfp_index_step_alignment_new_sum)) * VP1c) + (pfp_right_step_alignment_new_sum))) /\\ (((((exists ff_h_pfp_step_alignment_new_sumtarget. ff_h_pfp_step_alignment_new_sumtarget + S (pfp_value_step_alignment_new_sum) = S ((S (pfp_index_step_alignment_new_sum)) * z1c)) /\\ exists ff_q_pfp_step_alignment_new_sumtarget. z1b = ff_q_pfp_step_alignment_new_sumtarget * S ((S (pfp_index_step_alignment_new_sum)) * z1c) + (pfp_value_step_alignment_new_sum))) /\\ ((((exists pfa_gap_step_alignment_new_sumoperationleft. pfa_gap_step_alignment_new_sumoperationleft + S (pfp_left_step_alignment_new_sum) = (p)) /\\ (((exists pfa_gap_step_alignment_new_sumoperationright. pfa_gap_step_alignment_new_sumoperationright + S (pfp_right_step_alignment_new_sum) = (p)) /\\ ((((exists pfa_gap_step_alignment_new_sumoperationresultbound. pfa_gap_step_alignment_new_sumoperationresultbound + S (pfp_value_step_alignment_new_sum) = (p)) /\\ ((exists pfa_offset_left_step_alignment_new_sumoperationresultcongruence pfa_offset_right_step_alignment_new_sumoperationresultcongruence. ((pfp_left_step_alignment_new_sum) + (pfp_right_step_alignment_new_sum)) + (p) * pfa_offset_left_step_alignment_new_sumoperationresultcongruence = (pfp_value_step_alignment_new_sum) + (p) * pfa_offset_right_step_alignment_new_sumoperationresultcongruence)))))))))))))))) -> (forall pfrep_power_step_alignment_result pfrep_left_step_alignment_result pfrep_right_step_alignment_result. ((exists pfrep_position_step_alignment_resultfirst. ((pfrep_position_step_alignment_resultfirst+S (pfrep_power_step_alignment_result)=(L+S N0)) /\\ ((((exists ff_h_pfp_step_alignment_resultfirstentry. ff_h_pfp_step_alignment_resultfirstentry + S (pfrep_left_step_alignment_result) = S ((S (pfrep_position_step_alignment_resultfirst)) * z0c)) /\\ exists ff_q_pfp_step_alignment_resultfirstentry. z0b = ff_q_pfp_step_alignment_resultfirstentry * S ((S (pfrep_position_step_alignment_resultfirst)) * z0c) + (pfrep_left_step_alignment_result)))))) \\/ (((exists pfrep_gap_step_alignment_resultfirstoutside. pfrep_gap_step_alignment_resultfirstoutside+(L+S N0)=(pfrep_power_step_alignment_result)) /\\ (((pfrep_left_step_alignment_result)=0))))) -> ((exists pfrep_position_step_alignment_resultsecond. ((pfrep_position_step_alignment_resultsecond+S (pfrep_power_step_alignment_result)=(L+S N1)) /\\ ((((exists ff_h_pfp_step_alignment_resultsecondentry. ff_h_pfp_step_alignment_resultsecondentry + S (pfrep_right_step_alignment_result) = S ((S (pfrep_position_step_alignment_resultsecond)) * z1c)) /\\ exists ff_q_pfp_step_alignment_resultsecondentry. z1b = ff_q_pfp_step_alignment_resultsecondentry * S ((S (pfrep_position_step_alignment_resultsecond)) * z1c) + (pfrep_right_step_alignment_result)))))) \\/ (((exists pfrep_gap_step_alignment_resultsecondoutside. pfrep_gap_step_alignment_resultsecondoutside+(L+S N1)=(pfrep_power_step_alignment_result)) /\\ (((pfrep_right_step_alignment_result)=0))))) -> pfrep_left_step_alignment_result=pfrep_right_step_alignment_result)", "statement_sha256": "24592e5c2ea3c7cd6c0f0ed29b97acc4f512726a4880abb43828982e59216497", "summary": "Formal equivalence of two old prefixes preserves every actual aligned sum of their trailing-zero shifts with a fixed actual scalar multiple of the same source. 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"prime_field_polynomial_convolution_associativity_append_step", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro M", "intro pb", "intro pc", "intro N", "intro cb", "intro cc", "intro J", "intro q0b", "intro q0c", "intro K0", "intro r0b", "intro r0c", "intro U0", "intro s0b", "intro s0c", "intro V0", "intro c", "intro db", "intro dc", "intro q1b", "intro q1c", "intro K1", "intro r1b", "intro r1c", "intro U1", "intro s1b", "intro s1c", "intro V1", "intro hp", "intro hAB", "intro hQ0", "intro hR0", "intro hS0", "intro hIH", "intro hprefix", "intro hlast", "intro hQ1", "intro hR1", "intro hS1", "have hp0 : ~(p=0)", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hz", "have hABcopy : ((forall fom_index_pfp_append_step_ABleft. (exists fom_gap_pfp_append_step_ABleft_index_bound. fom_gap_pfp_append_step_ABleft_index_bound + S (fom_index_pfp_append_step_ABleft) = L) -> exists fom_value_pfp_append_step_ABleft. ((((exists fom_beta_height_pfp_append_step_ABleft_entry. fom_beta_height_pfp_append_step_ABleft_entry + S (fom_value_pfp_append_step_ABleft) = S ((S (fom_index_pfp_append_step_ABleft)) * ac)) /\\ exists fom_beta_quotient_pfp_append_step_ABleft_entry. ab = fom_beta_quotient_pfp_append_step_ABleft_entry * S ((S (fom_index_pfp_append_step_ABleft)) * ac) + (fom_value_pfp_append_step_ABleft))) /\\ (exists fom_gap_pfp_append_step_ABleft_value_bound. fom_gap_pfp_append_step_ABleft_value_bound + S (fom_value_pfp_append_step_ABleft) = p))) /\\ (((forall fom_index_pfp_append_step_ABright. (exists fom_gap_pfp_append_step_ABright_index_bound. fom_gap_pfp_append_step_ABright_index_bound + S (fom_index_pfp_append_step_ABright) = M) -> exists fom_value_pfp_append_step_ABright. ((((exists fom_beta_height_pfp_append_step_ABright_entry. fom_beta_height_pfp_append_step_ABright_entry + S (fom_value_pfp_append_step_ABright) = S ((S (fom_index_pfp_append_step_ABright)) * bc)) /\\ exists fom_beta_quotient_pfp_append_step_ABright_entry. bb = fom_beta_quotient_pfp_append_step_ABright_entry * S ((S (fom_index_pfp_append_step_ABright)) * bc) + (fom_value_pfp_append_step_ABright))) /\\ (exists fom_gap_pfp_append_step_ABright_value_bound. fom_gap_pfp_append_step_ABright_value_bound + S (fom_value_pfp_append_step_ABright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_append_step_ABcoefficients. (exists pfa_gap_append_step_ABcoefficientsbound. pfa_gap_append_step_ABcoefficientsbound + S (pfc_index_append_step_ABcoefficients) = (N)) -> exists pfc_value_append_step_ABcoefficients. ((((exists ff_h_pfp_append_step_ABcoefficientsentry. ff_h_pfp_append_step_ABcoefficientsentry + S (pfc_value_append_step_ABcoefficients) = S ((S (pfc_index_append_step_ABcoefficients)) * pc)) /\\ exists ff_q_pfp_append_step_ABcoefficientsentry. pb = ff_q_pfp_append_step_ABcoefficientsentry * S ((S (pfc_index_append_step_ABcoefficients)) * pc) + (pfc_value_append_step_ABcoefficients))) /\\ ((exists pfc_terms_code_append_step_ABcoefficientscoefficient pfc_terms_scale_append_step_ABcoefficientscoefficient pfc_natural_sum_append_step_ABcoefficientscoefficient. ((forall pfc_index_append_step_ABcoefficientscoefficientdiagonal. (exists pfa_gap_append_step_ABcoefficientscoefficientdiagonalbound. pfa_gap_append_step_ABcoefficientscoefficientdiagonalbound + S (pfc_index_append_step_ABcoefficientscoefficientdiagonal) = (S (pfc_index_append_step_ABcoefficients))) -> exists pfc_value_append_step_ABcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_step_ABcoefficientscoefficientdiagonalentry. ff_h_pfp_append_step_ABcoefficientscoefficientdiagonalentry + S (pfc_value_append_step_ABcoefficientscoefficientdiagonal) = S ((S (pfc_index_append_step_ABcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_ABcoefficientscoefficient)) /\\ exists ff_q_pfp_append_step_ABcoefficientscoefficientdiagonalentry. pfc_terms_code_append_step_ABcoefficientscoefficient = ff_q_pfp_append_step_ABcoefficientscoefficientdiagonalentry * S ((S (pfc_index_append_step_ABcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_ABcoefficientscoefficient) + (pfc_value_append_step_ABcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_step_ABcoefficientscoefficientdiagonalterm pfc_left_append_step_ABcoefficientscoefficientdiagonalterm pfc_right_append_step_ABcoefficientscoefficientdiagonalterm. (((pfc_index_append_step_ABcoefficientscoefficientdiagonal)+pfc_complement_append_step_ABcoefficientscoefficientdiagonalterm=(pfc_index_append_step_ABcoefficients)) /\\ ((((((exists pfa_gap_append_step_ABcoefficientscoefficientdiagonaltermleftinside. pfa_gap_append_step_ABcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_step_ABcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_step_ABcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_step_ABcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_step_ABcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_step_ABcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_step_ABcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_step_ABcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_step_ABcoefficientscoefficientdiagonal)) * ac) + (pfc_left_append_step_ABcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_ABcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_step_ABcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_step_ABcoefficientscoefficientdiagonal)) /\\ (((pfc_left_append_step_ABcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_step_ABcoefficientscoefficientdiagonaltermrightinside. pfa_gap_append_step_ABcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_step_ABcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_append_step_ABcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_step_ABcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_step_ABcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_step_ABcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_append_step_ABcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_append_step_ABcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_step_ABcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_append_step_ABcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_ABcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_step_ABcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_append_step_ABcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_step_ABcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_step_ABcoefficientscoefficientdiagonal)=pfc_left_append_step_ABcoefficientscoefficientdiagonalterm*pfc_right_append_step_ABcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_step_ABcoefficientscoefficientsum fs_v_pfc_append_step_ABcoefficientscoefficientsum. ((((exists fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_start. fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_step_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_start. fs_u_pfc_append_step_ABcoefficientscoefficientsum = fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_step_ABcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_terminal. fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_step_ABcoefficientscoefficient) = S ((S (S (pfc_index_append_step_ABcoefficients))) * fs_v_pfc_append_step_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_terminal. fs_u_pfc_append_step_ABcoefficientscoefficientsum = fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_step_ABcoefficients))) * fs_v_pfc_append_step_ABcoefficientscoefficientsum) + (pfc_natural_sum_append_step_ABcoefficientscoefficient))) /\\ forall fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_step_ABcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_step_ABcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps = S (pfc_index_append_step_ABcoefficients)) -> exists fs_a_pfc_append_step_ABcoefficientscoefficientsum_body_steps fs_r_pfc_append_step_ABcoefficientscoefficientsum_body_steps fs_s_pfc_append_step_ABcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_step_ABcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_ABcoefficientscoefficient)) /\\ exists fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_step_ABcoefficientscoefficient = fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_ABcoefficientscoefficient) + (fs_a_pfc_append_step_ABcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_step_ABcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_step_ABcoefficientscoefficientsum = fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_ABcoefficientscoefficientsum) + (fs_r_pfc_append_step_ABcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_step_ABcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_step_ABcoefficientscoefficientsum = fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_ABcoefficientscoefficientsum) + (fs_s_pfc_append_step_ABcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_step_ABcoefficientscoefficientsum_body_steps = fs_r_pfc_append_step_ABcoefficientscoefficientsum_body_steps + fs_a_pfc_append_step_ABcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_step_ABcoefficientscoefficientresiduebound. pfa_gap_append_step_ABcoefficientscoefficientresiduebound + S (pfc_value_append_step_ABcoefficients) = (p)) /\\ ((exists pfa_offset_left_append_step_ABcoefficientscoefficientresiduecongruence pfa_offset_right_append_step_ABcoefficientscoefficientresiduecongruence. (pfc_natural_sum_append_step_ABcoefficientscoefficient) + (p) * pfa_offset_left_append_step_ABcoefficientscoefficientresiduecongruence = (pfc_value_append_step_ABcoefficients) + (p) * pfa_offset_right_append_step_ABcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hAB", "cases hABcopy", "cases hABcopy_right", "cases hABcopy_right_right", "have hQ1copy : ((forall fom_index_pfp_append_step_Q1left. (exists fom_gap_pfp_append_step_Q1left_index_bound. fom_gap_pfp_append_step_Q1left_index_bound + S (fom_index_pfp_append_step_Q1left) = M) -> exists fom_value_pfp_append_step_Q1left. ((((exists fom_beta_height_pfp_append_step_Q1left_entry. fom_beta_height_pfp_append_step_Q1left_entry + S (fom_value_pfp_append_step_Q1left) = S ((S (fom_index_pfp_append_step_Q1left)) * bc)) /\\ exists fom_beta_quotient_pfp_append_step_Q1left_entry. bb = fom_beta_quotient_pfp_append_step_Q1left_entry * S ((S (fom_index_pfp_append_step_Q1left)) * bc) + (fom_value_pfp_append_step_Q1left))) /\\ (exists fom_gap_pfp_append_step_Q1left_value_bound. fom_gap_pfp_append_step_Q1left_value_bound + S (fom_value_pfp_append_step_Q1left) = p))) /\\ (((forall fom_index_pfp_append_step_Q1right. (exists fom_gap_pfp_append_step_Q1right_index_bound. fom_gap_pfp_append_step_Q1right_index_bound + S (fom_index_pfp_append_step_Q1right) = S J) -> exists fom_value_pfp_append_step_Q1right. ((((exists fom_beta_height_pfp_append_step_Q1right_entry. fom_beta_height_pfp_append_step_Q1right_entry + S (fom_value_pfp_append_step_Q1right) = S ((S (fom_index_pfp_append_step_Q1right)) * dc)) /\\ exists fom_beta_quotient_pfp_append_step_Q1right_entry. db = fom_beta_quotient_pfp_append_step_Q1right_entry * S ((S (fom_index_pfp_append_step_Q1right)) * dc) + (fom_value_pfp_append_step_Q1right))) /\\ (exists fom_gap_pfp_append_step_Q1right_value_bound. fom_gap_pfp_append_step_Q1right_value_bound + S (fom_value_pfp_append_step_Q1right) = p))) /\\ (((((((M)=0 \\/ (S J)=0) /\\ (((K1)=0)))) \\/ (((~((M)=0)) /\\ (((~((S J)=0)) /\\ (((M)+(S J)=S (K1)))))))) /\\ ((forall pfc_index_append_step_Q1coefficients. (exists pfa_gap_append_step_Q1coefficientsbound. pfa_gap_append_step_Q1coefficientsbound + S (pfc_index_append_step_Q1coefficients) = (K1)) -> exists pfc_value_append_step_Q1coefficients. ((((exists ff_h_pfp_append_step_Q1coefficientsentry. ff_h_pfp_append_step_Q1coefficientsentry + S (pfc_value_append_step_Q1coefficients) = S ((S (pfc_index_append_step_Q1coefficients)) * q1c)) /\\ exists ff_q_pfp_append_step_Q1coefficientsentry. q1b = ff_q_pfp_append_step_Q1coefficientsentry * S ((S (pfc_index_append_step_Q1coefficients)) * q1c) + (pfc_value_append_step_Q1coefficients))) /\\ ((exists pfc_terms_code_append_step_Q1coefficientscoefficient pfc_terms_scale_append_step_Q1coefficientscoefficient pfc_natural_sum_append_step_Q1coefficientscoefficient. ((forall pfc_index_append_step_Q1coefficientscoefficientdiagonal. (exists pfa_gap_append_step_Q1coefficientscoefficientdiagonalbound. pfa_gap_append_step_Q1coefficientscoefficientdiagonalbound + S (pfc_index_append_step_Q1coefficientscoefficientdiagonal) = (S (pfc_index_append_step_Q1coefficients))) -> exists pfc_value_append_step_Q1coefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_step_Q1coefficientscoefficientdiagonalentry. ff_h_pfp_append_step_Q1coefficientscoefficientdiagonalentry + S (pfc_value_append_step_Q1coefficientscoefficientdiagonal) = S ((S (pfc_index_append_step_Q1coefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_Q1coefficientscoefficient)) /\\ exists ff_q_pfp_append_step_Q1coefficientscoefficientdiagonalentry. pfc_terms_code_append_step_Q1coefficientscoefficient = ff_q_pfp_append_step_Q1coefficientscoefficientdiagonalentry * S ((S (pfc_index_append_step_Q1coefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_Q1coefficientscoefficient) + (pfc_value_append_step_Q1coefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_step_Q1coefficientscoefficientdiagonalterm pfc_left_append_step_Q1coefficientscoefficientdiagonalterm pfc_right_append_step_Q1coefficientscoefficientdiagonalterm. (((pfc_index_append_step_Q1coefficientscoefficientdiagonal)+pfc_complement_append_step_Q1coefficientscoefficientdiagonalterm=(pfc_index_append_step_Q1coefficients)) /\\ ((((((exists pfa_gap_append_step_Q1coefficientscoefficientdiagonaltermleftinside. pfa_gap_append_step_Q1coefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_step_Q1coefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_append_step_Q1coefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_step_Q1coefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_step_Q1coefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_step_Q1coefficientscoefficientdiagonal)) * bc)) /\\ exists ff_q_pfp_append_step_Q1coefficientscoefficientdiagonaltermleftentry. bb = ff_q_pfp_append_step_Q1coefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_step_Q1coefficientscoefficientdiagonal)) * bc) + (pfc_left_append_step_Q1coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_Q1coefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_step_Q1coefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_append_step_Q1coefficientscoefficientdiagonal)) /\\ (((pfc_left_append_step_Q1coefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_step_Q1coefficientscoefficientdiagonaltermrightinside. pfa_gap_append_step_Q1coefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_step_Q1coefficientscoefficientdiagonalterm) = (S J)) /\\ ((((exists ff_h_pfp_append_step_Q1coefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_step_Q1coefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_step_Q1coefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_step_Q1coefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_append_step_Q1coefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_append_step_Q1coefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_step_Q1coefficientscoefficientdiagonalterm)) * dc) + (pfc_right_append_step_Q1coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_Q1coefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_step_Q1coefficientscoefficientdiagonaltermrightoutside+(S J)=(pfc_complement_append_step_Q1coefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_step_Q1coefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_step_Q1coefficientscoefficientdiagonal)=pfc_left_append_step_Q1coefficientscoefficientdiagonalterm*pfc_right_append_step_Q1coefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_step_Q1coefficientscoefficientsum fs_v_pfc_append_step_Q1coefficientscoefficientsum. ((((exists fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_start. fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_step_Q1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_start. fs_u_pfc_append_step_Q1coefficientscoefficientsum = fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_step_Q1coefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_terminal. fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_step_Q1coefficientscoefficient) = S ((S (S (pfc_index_append_step_Q1coefficients))) * fs_v_pfc_append_step_Q1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_terminal. fs_u_pfc_append_step_Q1coefficientscoefficientsum = fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_step_Q1coefficients))) * fs_v_pfc_append_step_Q1coefficientscoefficientsum) + (pfc_natural_sum_append_step_Q1coefficientscoefficient))) /\\ forall fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_step_Q1coefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_step_Q1coefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps = S (pfc_index_append_step_Q1coefficients)) -> exists fs_a_pfc_append_step_Q1coefficientscoefficientsum_body_steps fs_r_pfc_append_step_Q1coefficientscoefficientsum_body_steps fs_s_pfc_append_step_Q1coefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_step_Q1coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_Q1coefficientscoefficient)) /\\ exists fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_step_Q1coefficientscoefficient = fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_Q1coefficientscoefficient) + (fs_a_pfc_append_step_Q1coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_step_Q1coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_Q1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_step_Q1coefficientscoefficientsum = fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_Q1coefficientscoefficientsum) + (fs_r_pfc_append_step_Q1coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_step_Q1coefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_Q1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_step_Q1coefficientscoefficientsum = fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_Q1coefficientscoefficientsum) + (fs_s_pfc_append_step_Q1coefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_step_Q1coefficientscoefficientsum_body_steps = fs_r_pfc_append_step_Q1coefficientscoefficientsum_body_steps + fs_a_pfc_append_step_Q1coefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_step_Q1coefficientscoefficientresiduebound. pfa_gap_append_step_Q1coefficientscoefficientresiduebound + S (pfc_value_append_step_Q1coefficients) = (p)) /\\ ((exists pfa_offset_left_append_step_Q1coefficientscoefficientresiduecongruence pfa_offset_right_append_step_Q1coefficientscoefficientresiduecongruence. (pfc_natural_sum_append_step_Q1coefficientscoefficient) + (p) * pfa_offset_left_append_step_Q1coefficientscoefficientresiduecongruence = (pfc_value_append_step_Q1coefficients) + (p) * pfa_offset_right_append_step_Q1coefficientscoefficientresiduecongruence))))))))))))))))))", "exact hQ1", "cases hQ1copy", "cases hQ1copy_right", "cases hQ1copy_right_right", "have hc : exists pfa_gap_append_step_scalar_bound. pfa_gap_append_step_scalar_bound + S (c) = (p)", "specialize matrix_rank_bounded_prefix_value (db)", "specialize matrix_rank_bounded_prefix_value (dc)", "specialize matrix_rank_bounded_prefix_value (S J)", "specialize matrix_rank_bounded_prefix_value (p)", "specialize matrix_rank_bounded_prefix_value (J)", "specialize matrix_rank_bounded_prefix_value (c)", "apply matrix_rank_bounded_prefix_value", "exact hQ1copy_right_left", "specialize le_refl (S J)", "apply le_refl", "exact hlast", "have hPbound : forall fom_index_pfp_append_step_hPbound. (exists fom_gap_pfp_append_step_hPbound_index_bound. fom_gap_pfp_append_step_hPbound_index_bound + S (fom_index_pfp_append_step_hPbound) = N) -> exists fom_value_pfp_append_step_hPbound. ((((exists fom_beta_height_pfp_append_step_hPbound_entry. fom_beta_height_pfp_append_step_hPbound_entry + S (fom_value_pfp_append_step_hPbound) = S ((S (fom_index_pfp_append_step_hPbound)) * pc)) /\\ exists fom_beta_quotient_pfp_append_step_hPbound_entry. pb = fom_beta_quotient_pfp_append_step_hPbound_entry * S ((S (fom_index_pfp_append_step_hPbound)) * pc) + (fom_value_pfp_append_step_hPbound))) /\\ (exists fom_gap_pfp_append_step_hPbound_value_bound. fom_gap_pfp_append_step_hPbound_value_bound + S (fom_value_pfp_append_step_hPbound) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (ab)", "specialize prime_field_polynomial_convolution_bounded (ac)", "specialize prime_field_polynomial_convolution_bounded (L)", "specialize prime_field_polynomial_convolution_bounded (bb)", "specialize prime_field_polynomial_convolution_bounded (bc)", "specialize prime_field_polynomial_convolution_bounded (M)", "specialize prime_field_polynomial_convolution_bounded (pb)", "specialize prime_field_polynomial_convolution_bounded (pc)", "specialize prime_field_polynomial_convolution_bounded (N)", "apply prime_field_polynomial_convolution_bounded", "exact hAB", "have hQ0bound : forall fom_index_pfp_append_step_hQ0bound. (exists fom_gap_pfp_append_step_hQ0bound_index_bound. fom_gap_pfp_append_step_hQ0bound_index_bound + S (fom_index_pfp_append_step_hQ0bound) = K0) -> exists fom_value_pfp_append_step_hQ0bound. ((((exists fom_beta_height_pfp_append_step_hQ0bound_entry. fom_beta_height_pfp_append_step_hQ0bound_entry + S (fom_value_pfp_append_step_hQ0bound) = S ((S (fom_index_pfp_append_step_hQ0bound)) * q0c)) /\\ exists fom_beta_quotient_pfp_append_step_hQ0bound_entry. q0b = fom_beta_quotient_pfp_append_step_hQ0bound_entry * S ((S (fom_index_pfp_append_step_hQ0bound)) * q0c) + (fom_value_pfp_append_step_hQ0bound))) /\\ (exists fom_gap_pfp_append_step_hQ0bound_value_bound. fom_gap_pfp_append_step_hQ0bound_value_bound + S (fom_value_pfp_append_step_hQ0bound) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (bb)", "specialize prime_field_polynomial_convolution_bounded (bc)", "specialize prime_field_polynomial_convolution_bounded (M)", "specialize prime_field_polynomial_convolution_bounded (cb)", "specialize prime_field_polynomial_convolution_bounded (cc)", "specialize prime_field_polynomial_convolution_bounded (J)", "specialize prime_field_polynomial_convolution_bounded (q0b)", "specialize prime_field_polynomial_convolution_bounded (q0c)", "specialize prime_field_polynomial_convolution_bounded (K0)", "apply prime_field_polynomial_convolution_bounded", "exact hQ0", "have hR0bound : forall fom_index_pfp_append_step_hR0bound. (exists fom_gap_pfp_append_step_hR0bound_index_bound. fom_gap_pfp_append_step_hR0bound_index_bound + S (fom_index_pfp_append_step_hR0bound) = U0) -> exists fom_value_pfp_append_step_hR0bound. ((((exists fom_beta_height_pfp_append_step_hR0bound_entry. fom_beta_height_pfp_append_step_hR0bound_entry + S (fom_value_pfp_append_step_hR0bound) = S ((S (fom_index_pfp_append_step_hR0bound)) * r0c)) /\\ exists fom_beta_quotient_pfp_append_step_hR0bound_entry. r0b = fom_beta_quotient_pfp_append_step_hR0bound_entry * S ((S (fom_index_pfp_append_step_hR0bound)) * r0c) + (fom_value_pfp_append_step_hR0bound))) /\\ (exists fom_gap_pfp_append_step_hR0bound_value_bound. fom_gap_pfp_append_step_hR0bound_value_bound + S (fom_value_pfp_append_step_hR0bound) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (pb)", "specialize prime_field_polynomial_convolution_bounded (pc)", "specialize prime_field_polynomial_convolution_bounded (N)", "specialize prime_field_polynomial_convolution_bounded (cb)", "specialize prime_field_polynomial_convolution_bounded (cc)", "specialize prime_field_polynomial_convolution_bounded (J)", "specialize prime_field_polynomial_convolution_bounded (r0b)", "specialize prime_field_polynomial_convolution_bounded (r0c)", "specialize prime_field_polynomial_convolution_bounded (U0)", "apply prime_field_polynomial_convolution_bounded", "exact hR0", "have hS0bound : forall fom_index_pfp_append_step_hS0bound. (exists fom_gap_pfp_append_step_hS0bound_index_bound. fom_gap_pfp_append_step_hS0bound_index_bound + S (fom_index_pfp_append_step_hS0bound) = V0) -> exists fom_value_pfp_append_step_hS0bound. ((((exists fom_beta_height_pfp_append_step_hS0bound_entry. fom_beta_height_pfp_append_step_hS0bound_entry + S (fom_value_pfp_append_step_hS0bound) = S ((S (fom_index_pfp_append_step_hS0bound)) * s0c)) /\\ exists fom_beta_quotient_pfp_append_step_hS0bound_entry. s0b = fom_beta_quotient_pfp_append_step_hS0bound_entry * S ((S (fom_index_pfp_append_step_hS0bound)) * s0c) + (fom_value_pfp_append_step_hS0bound))) /\\ (exists fom_gap_pfp_append_step_hS0bound_value_bound. fom_gap_pfp_append_step_hS0bound_value_bound + S (fom_value_pfp_append_step_hS0bound) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (ab)", "specialize prime_field_polynomial_convolution_bounded (ac)", "specialize prime_field_polynomial_convolution_bounded (L)", "specialize prime_field_polynomial_convolution_bounded (q0b)", "specialize prime_field_polynomial_convolution_bounded (q0c)", "specialize prime_field_polynomial_convolution_bounded (K0)", "specialize prime_field_polynomial_convolution_bounded (s0b)", "specialize prime_field_polynomial_convolution_bounded (s0c)", "specialize prime_field_polynomial_convolution_bounded (V0)", "apply prime_field_polynomial_convolution_bounded", "exact hS0", "have hYalign : exists ub uc vb vc UPb UPc VPb VPc zb zc. ((((forall mdr_i_pfp_append_step_hYalign_shiftprefix mdr_a_pfp_append_step_hYalign_shiftprefix. (exists mdr_gap_pfp_append_step_hYalign_shiftprefixb. mdr_gap_pfp_append_step_hYalign_shiftprefixb + S (mdr_i_pfp_append_step_hYalign_shiftprefix) = (V0)) -> (((exists ff_h_mdr_pfp_append_step_hYalign_shiftprefixo. ff_h_mdr_pfp_append_step_hYalign_shiftprefixo + S (mdr_a_pfp_append_step_hYalign_shiftprefix) = S ((S (mdr_i_pfp_append_step_hYalign_shiftprefix)) * s0c)) /\\ exists ff_q_mdr_pfp_append_step_hYalign_shiftprefixo. s0b = ff_q_mdr_pfp_append_step_hYalign_shiftprefixo * S ((S (mdr_i_pfp_append_step_hYalign_shiftprefix)) * s0c) + (mdr_a_pfp_append_step_hYalign_shiftprefix))) -> (((exists ff_h_mdr_pfp_append_step_hYalign_shiftprefixn. ff_h_mdr_pfp_append_step_hYalign_shiftprefixn + S (mdr_a_pfp_append_step_hYalign_shiftprefix) = S ((S (mdr_i_pfp_append_step_hYalign_shiftprefix)) * uc)) /\\ exists ff_q_mdr_pfp_append_step_hYalign_shiftprefixn. ub = ff_q_mdr_pfp_append_step_hYalign_shiftprefixn * S ((S (mdr_i_pfp_append_step_hYalign_shiftprefix)) * uc) + (mdr_a_pfp_append_step_hYalign_shiftprefix)))) /\\ ((((exists ff_h_pfp_append_step_hYalign_shiftzero. ff_h_pfp_append_step_hYalign_shiftzero + S (0) = S ((S (V0)) * uc)) /\\ exists ff_q_pfp_append_step_hYalign_shiftzero. ub = ff_q_pfp_append_step_hYalign_shiftzero * S ((S (V0)) * uc) + (0)))))) /\\ (((((exists pfa_gap_append_step_hYalign_scalescalar. pfa_gap_append_step_hYalign_scalescalar + S (c) = (p)) /\\ ((forall pfp_index_append_step_hYalign_scale. (exists pfa_gap_append_step_hYalign_scaleindex. pfa_gap_append_step_hYalign_scaleindex + S (pfp_index_append_step_hYalign_scale) = (N)) -> exists pfp_source_append_step_hYalign_scale pfp_value_append_step_hYalign_scale. ((((exists ff_h_pfp_append_step_hYalign_scalesource. ff_h_pfp_append_step_hYalign_scalesource + S (pfp_source_append_step_hYalign_scale) = S ((S (pfp_index_append_step_hYalign_scale)) * pc)) /\\ exists ff_q_pfp_append_step_hYalign_scalesource. pb = ff_q_pfp_append_step_hYalign_scalesource * S ((S (pfp_index_append_step_hYalign_scale)) * pc) + (pfp_source_append_step_hYalign_scale))) /\\ (((((exists ff_h_pfp_append_step_hYalign_scaletarget. ff_h_pfp_append_step_hYalign_scaletarget + S (pfp_value_append_step_hYalign_scale) = S ((S (pfp_index_append_step_hYalign_scale)) * vc)) /\\ exists ff_q_pfp_append_step_hYalign_scaletarget. vb = ff_q_pfp_append_step_hYalign_scaletarget * S ((S (pfp_index_append_step_hYalign_scale)) * vc) + (pfp_value_append_step_hYalign_scale))) /\\ ((((exists pfa_gap_append_step_hYalign_scaleoperationleft. pfa_gap_append_step_hYalign_scaleoperationleft + S (c) = (p)) /\\ (((exists pfa_gap_append_step_hYalign_scaleoperationright. pfa_gap_append_step_hYalign_scaleoperationright + S (pfp_source_append_step_hYalign_scale) = (p)) /\\ ((((exists pfa_gap_append_step_hYalign_scaleoperationresultbound. pfa_gap_append_step_hYalign_scaleoperationresultbound + S (pfp_value_append_step_hYalign_scale) = (p)) /\\ ((exists pfa_offset_left_append_step_hYalign_scaleoperationresultcongruence pfa_offset_right_append_step_hYalign_scaleoperationresultcongruence. ((c) * (pfp_source_append_step_hYalign_scale)) + (p) * pfa_offset_left_append_step_hYalign_scaleoperationresultcongruence = (pfp_value_append_step_hYalign_scale) + (p) * pfa_offset_right_append_step_hYalign_scaleoperationresultcongruence))))))))))))))))) /\\ (((((forall pfp_repeat_index_append_step_hYalign_leftzeros. (exists pfa_gap_append_step_hYalign_leftzerosindex. pfa_gap_append_step_hYalign_leftzerosindex + S (pfp_repeat_index_append_step_hYalign_leftzeros) = (N)) -> (((exists ff_h_pfp_append_step_hYalign_leftzerosentry. ff_h_pfp_append_step_hYalign_leftzerosentry + S (0) = S ((S (pfp_repeat_index_append_step_hYalign_leftzeros)) * UPc)) /\\ exists ff_q_pfp_append_step_hYalign_leftzerosentry. UPb = ff_q_pfp_append_step_hYalign_leftzerosentry * S ((S (pfp_repeat_index_append_step_hYalign_leftzeros)) * UPc) + (0)))) /\\ ((forall pfrep_index_append_step_hYalign_left pfrep_value_append_step_hYalign_left. (exists pfa_gap_append_step_hYalign_leftbound. pfa_gap_append_step_hYalign_leftbound + S (pfrep_index_append_step_hYalign_left) = (S V0)) -> (((exists ff_h_pfp_append_step_hYalign_leftinput. ff_h_pfp_append_step_hYalign_leftinput + S (pfrep_value_append_step_hYalign_left) = S ((S (pfrep_index_append_step_hYalign_left)) * uc)) /\\ exists ff_q_pfp_append_step_hYalign_leftinput. ub = ff_q_pfp_append_step_hYalign_leftinput * S ((S (pfrep_index_append_step_hYalign_left)) * uc) + (pfrep_value_append_step_hYalign_left))) -> (((exists ff_h_pfp_append_step_hYalign_leftoutput. ff_h_pfp_append_step_hYalign_leftoutput + S (pfrep_value_append_step_hYalign_left) = S ((S ((N)+pfrep_index_append_step_hYalign_left)) * UPc)) /\\ exists ff_q_pfp_append_step_hYalign_leftoutput. UPb = ff_q_pfp_append_step_hYalign_leftoutput * S ((S ((N)+pfrep_index_append_step_hYalign_left)) * UPc) + (pfrep_value_append_step_hYalign_left))))))) /\\ (((((forall pfp_repeat_index_append_step_hYalign_rightzeros. (exists pfa_gap_append_step_hYalign_rightzerosindex. pfa_gap_append_step_hYalign_rightzerosindex + S (pfp_repeat_index_append_step_hYalign_rightzeros) = (S V0)) -> (((exists ff_h_pfp_append_step_hYalign_rightzerosentry. ff_h_pfp_append_step_hYalign_rightzerosentry + S (0) = S ((S (pfp_repeat_index_append_step_hYalign_rightzeros)) * VPc)) /\\ exists ff_q_pfp_append_step_hYalign_rightzerosentry. VPb = ff_q_pfp_append_step_hYalign_rightzerosentry * S ((S (pfp_repeat_index_append_step_hYalign_rightzeros)) * VPc) + (0)))) /\\ ((forall pfrep_index_append_step_hYalign_right pfrep_value_append_step_hYalign_right. (exists pfa_gap_append_step_hYalign_rightbound. pfa_gap_append_step_hYalign_rightbound + S (pfrep_index_append_step_hYalign_right) = (N)) -> (((exists ff_h_pfp_append_step_hYalign_rightinput. ff_h_pfp_append_step_hYalign_rightinput + S (pfrep_value_append_step_hYalign_right) = S ((S (pfrep_index_append_step_hYalign_right)) * vc)) /\\ exists ff_q_pfp_append_step_hYalign_rightinput. vb = ff_q_pfp_append_step_hYalign_rightinput * S ((S (pfrep_index_append_step_hYalign_right)) * vc) + (pfrep_value_append_step_hYalign_right))) -> (((exists ff_h_pfp_append_step_hYalign_rightoutput. ff_h_pfp_append_step_hYalign_rightoutput + S (pfrep_value_append_step_hYalign_right) = S ((S ((S V0)+pfrep_index_append_step_hYalign_right)) * VPc)) /\\ exists ff_q_pfp_append_step_hYalign_rightoutput. VPb = ff_q_pfp_append_step_hYalign_rightoutput * S ((S ((S V0)+pfrep_index_append_step_hYalign_right)) * VPc) + (pfrep_value_append_step_hYalign_right))))))) /\\ ((forall pfp_index_append_step_hYalign_sum. (exists pfa_gap_append_step_hYalign_sumindex. pfa_gap_append_step_hYalign_sumindex + S (pfp_index_append_step_hYalign_sum) = (N+S V0)) -> exists pfp_left_append_step_hYalign_sum pfp_right_append_step_hYalign_sum pfp_value_append_step_hYalign_sum. ((((exists ff_h_pfp_append_step_hYalign_sumleft. ff_h_pfp_append_step_hYalign_sumleft + S (pfp_left_append_step_hYalign_sum) = S ((S (pfp_index_append_step_hYalign_sum)) * UPc)) /\\ exists ff_q_pfp_append_step_hYalign_sumleft. UPb = ff_q_pfp_append_step_hYalign_sumleft * S ((S (pfp_index_append_step_hYalign_sum)) * UPc) + (pfp_left_append_step_hYalign_sum))) /\\ (((((exists ff_h_pfp_append_step_hYalign_sumright. ff_h_pfp_append_step_hYalign_sumright + S (pfp_right_append_step_hYalign_sum) = S ((S (pfp_index_append_step_hYalign_sum)) * VPc)) /\\ exists ff_q_pfp_append_step_hYalign_sumright. VPb = ff_q_pfp_append_step_hYalign_sumright * S ((S (pfp_index_append_step_hYalign_sum)) * VPc) + (pfp_right_append_step_hYalign_sum))) /\\ (((((exists ff_h_pfp_append_step_hYalign_sumtarget. ff_h_pfp_append_step_hYalign_sumtarget + S (pfp_value_append_step_hYalign_sum) = S ((S (pfp_index_append_step_hYalign_sum)) * zc)) /\\ exists ff_q_pfp_append_step_hYalign_sumtarget. zb = ff_q_pfp_append_step_hYalign_sumtarget * S ((S (pfp_index_append_step_hYalign_sum)) * zc) + (pfp_value_append_step_hYalign_sum))) /\\ ((((exists pfa_gap_append_step_hYalign_sumoperationleft. pfa_gap_append_step_hYalign_sumoperationleft + S (pfp_left_append_step_hYalign_sum) = (p)) /\\ (((exists pfa_gap_append_step_hYalign_sumoperationright. pfa_gap_append_step_hYalign_sumoperationright + S (pfp_right_append_step_hYalign_sum) = (p)) /\\ ((((exists pfa_gap_append_step_hYalign_sumoperationresultbound. pfa_gap_append_step_hYalign_sumoperationresultbound + S (pfp_value_append_step_hYalign_sum) = (p)) /\\ ((exists pfa_offset_left_append_step_hYalign_sumoperationresultcongruence pfa_offset_right_append_step_hYalign_sumoperationresultcongruence. ((pfp_left_append_step_hYalign_sum) + (pfp_right_append_step_hYalign_sum)) + (p) * pfa_offset_left_append_step_hYalign_sumoperationresultcongruence = (pfp_value_append_step_hYalign_sum) + (p) * pfa_offset_right_append_step_hYalign_sumoperationresultcongruence))))))))))))))))))))))))", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (p)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (c)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (pb)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (pc)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (N)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (s0b)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (s0c)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (V0)", "apply prime_field_polynomial_shift_scale_aligned_sum_exists", "exact hp", "exact hc", "exact hPbound", "exact hS0bound", "cases hYalign", "cases hYalign_witness", "cases hYalign_witness_witness", "cases hYalign_witness_witness_witness", "cases hYalign_witness_witness_witness_witness", "cases hYalign_witness_witness_witness_witness_witness", "cases hYalign_witness_witness_witness_witness_witness_witness", "cases hYalign_witness_witness_witness_witness_witness_witness_witness", "cases hYalign_witness_witness_witness_witness_witness_witness_witness_witness", "cases hYalign_witness_witness_witness_witness_witness_witness_witness_witness_witness", "cases hYalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness", "cases hYalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right", "cases hYalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right", "cases hYalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right", "have hS_total : forall pfrep_power_append_step_S_total pfrep_left_append_step_S_total pfrep_right_append_step_S_total. ((exists pfrep_position_append_step_S_totalfirst. ((pfrep_position_append_step_S_totalfirst+S (pfrep_power_append_step_S_total)=(V1)) /\\ ((((exists ff_h_pfp_append_step_S_totalfirstentry. ff_h_pfp_append_step_S_totalfirstentry + S (pfrep_left_append_step_S_total) = S ((S (pfrep_position_append_step_S_totalfirst)) * s1c)) /\\ exists ff_q_pfp_append_step_S_totalfirstentry. s1b = ff_q_pfp_append_step_S_totalfirstentry * S ((S (pfrep_position_append_step_S_totalfirst)) * s1c) + (pfrep_left_append_step_S_total)))))) \\/ (((exists pfrep_gap_append_step_S_totalfirstoutside. pfrep_gap_append_step_S_totalfirstoutside+(V1)=(pfrep_power_append_step_S_total)) /\\ (((pfrep_left_append_step_S_total)=0))))) -> ((exists pfrep_position_append_step_S_totalsecond. ((pfrep_position_append_step_S_totalsecond+S (pfrep_power_append_step_S_total)=(N+S V0)) /\\ ((((exists ff_h_pfp_append_step_S_totalsecondentry. ff_h_pfp_append_step_S_totalsecondentry + S (pfrep_right_append_step_S_total) = S ((S (pfrep_position_append_step_S_totalsecond)) * x9)) /\\ exists ff_q_pfp_append_step_S_totalsecondentry. x8 = ff_q_pfp_append_step_S_totalsecondentry * S ((S (pfrep_position_append_step_S_totalsecond)) * x9) + (pfrep_right_append_step_S_total)))))) \\/ (((exists pfrep_gap_append_step_S_totalsecondoutside. pfrep_gap_append_step_S_totalsecondoutside+(N+S V0)=(pfrep_power_append_step_S_total)) /\\ (((pfrep_right_append_step_S_total)=0))))) -> pfrep_left_append_step_S_total=pfrep_right_append_step_S_total", "have hZalign : exists ub uc vb vc UPb UPc VPb VPc zb zc. ((((forall mdr_i_pfp_append_step_hZalign_shiftprefix mdr_a_pfp_append_step_hZalign_shiftprefix. (exists mdr_gap_pfp_append_step_hZalign_shiftprefixb. mdr_gap_pfp_append_step_hZalign_shiftprefixb + S (mdr_i_pfp_append_step_hZalign_shiftprefix) = (K0)) -> (((exists ff_h_mdr_pfp_append_step_hZalign_shiftprefixo. ff_h_mdr_pfp_append_step_hZalign_shiftprefixo + S (mdr_a_pfp_append_step_hZalign_shiftprefix) = S ((S (mdr_i_pfp_append_step_hZalign_shiftprefix)) * q0c)) /\\ exists ff_q_mdr_pfp_append_step_hZalign_shiftprefixo. q0b = ff_q_mdr_pfp_append_step_hZalign_shiftprefixo * S ((S (mdr_i_pfp_append_step_hZalign_shiftprefix)) * q0c) + (mdr_a_pfp_append_step_hZalign_shiftprefix))) -> (((exists ff_h_mdr_pfp_append_step_hZalign_shiftprefixn. ff_h_mdr_pfp_append_step_hZalign_shiftprefixn + S (mdr_a_pfp_append_step_hZalign_shiftprefix) = S ((S (mdr_i_pfp_append_step_hZalign_shiftprefix)) * uc)) /\\ exists ff_q_mdr_pfp_append_step_hZalign_shiftprefixn. ub = ff_q_mdr_pfp_append_step_hZalign_shiftprefixn * S ((S (mdr_i_pfp_append_step_hZalign_shiftprefix)) * uc) + (mdr_a_pfp_append_step_hZalign_shiftprefix)))) /\\ ((((exists ff_h_pfp_append_step_hZalign_shiftzero. ff_h_pfp_append_step_hZalign_shiftzero + S (0) = S ((S (K0)) * uc)) /\\ exists ff_q_pfp_append_step_hZalign_shiftzero. ub = ff_q_pfp_append_step_hZalign_shiftzero * S ((S (K0)) * uc) + (0)))))) /\\ (((((exists pfa_gap_append_step_hZalign_scalescalar. pfa_gap_append_step_hZalign_scalescalar + S (c) = (p)) /\\ ((forall pfp_index_append_step_hZalign_scale. (exists pfa_gap_append_step_hZalign_scaleindex. pfa_gap_append_step_hZalign_scaleindex + S (pfp_index_append_step_hZalign_scale) = (M)) -> exists pfp_source_append_step_hZalign_scale pfp_value_append_step_hZalign_scale. ((((exists ff_h_pfp_append_step_hZalign_scalesource. ff_h_pfp_append_step_hZalign_scalesource + S (pfp_source_append_step_hZalign_scale) = S ((S (pfp_index_append_step_hZalign_scale)) * bc)) /\\ exists ff_q_pfp_append_step_hZalign_scalesource. bb = ff_q_pfp_append_step_hZalign_scalesource * S ((S (pfp_index_append_step_hZalign_scale)) * bc) + (pfp_source_append_step_hZalign_scale))) /\\ (((((exists ff_h_pfp_append_step_hZalign_scaletarget. ff_h_pfp_append_step_hZalign_scaletarget + S (pfp_value_append_step_hZalign_scale) = S ((S (pfp_index_append_step_hZalign_scale)) * vc)) /\\ exists ff_q_pfp_append_step_hZalign_scaletarget. vb = ff_q_pfp_append_step_hZalign_scaletarget * S ((S (pfp_index_append_step_hZalign_scale)) * vc) + (pfp_value_append_step_hZalign_scale))) /\\ ((((exists pfa_gap_append_step_hZalign_scaleoperationleft. pfa_gap_append_step_hZalign_scaleoperationleft + S (c) = (p)) /\\ (((exists pfa_gap_append_step_hZalign_scaleoperationright. pfa_gap_append_step_hZalign_scaleoperationright + S (pfp_source_append_step_hZalign_scale) = (p)) /\\ ((((exists pfa_gap_append_step_hZalign_scaleoperationresultbound. pfa_gap_append_step_hZalign_scaleoperationresultbound + S (pfp_value_append_step_hZalign_scale) = (p)) /\\ ((exists pfa_offset_left_append_step_hZalign_scaleoperationresultcongruence pfa_offset_right_append_step_hZalign_scaleoperationresultcongruence. ((c) * (pfp_source_append_step_hZalign_scale)) + (p) * pfa_offset_left_append_step_hZalign_scaleoperationresultcongruence = (pfp_value_append_step_hZalign_scale) + (p) * pfa_offset_right_append_step_hZalign_scaleoperationresultcongruence))))))))))))))))) /\\ (((((forall pfp_repeat_index_append_step_hZalign_leftzeros. (exists pfa_gap_append_step_hZalign_leftzerosindex. pfa_gap_append_step_hZalign_leftzerosindex + S (pfp_repeat_index_append_step_hZalign_leftzeros) = (M)) -> (((exists ff_h_pfp_append_step_hZalign_leftzerosentry. ff_h_pfp_append_step_hZalign_leftzerosentry + S (0) = S ((S (pfp_repeat_index_append_step_hZalign_leftzeros)) * UPc)) /\\ exists ff_q_pfp_append_step_hZalign_leftzerosentry. UPb = ff_q_pfp_append_step_hZalign_leftzerosentry * S ((S (pfp_repeat_index_append_step_hZalign_leftzeros)) * UPc) + (0)))) /\\ ((forall pfrep_index_append_step_hZalign_left pfrep_value_append_step_hZalign_left. (exists pfa_gap_append_step_hZalign_leftbound. pfa_gap_append_step_hZalign_leftbound + S (pfrep_index_append_step_hZalign_left) = (S K0)) -> (((exists ff_h_pfp_append_step_hZalign_leftinput. ff_h_pfp_append_step_hZalign_leftinput + S (pfrep_value_append_step_hZalign_left) = S ((S (pfrep_index_append_step_hZalign_left)) * uc)) /\\ exists ff_q_pfp_append_step_hZalign_leftinput. ub = ff_q_pfp_append_step_hZalign_leftinput * S ((S (pfrep_index_append_step_hZalign_left)) * uc) + (pfrep_value_append_step_hZalign_left))) -> (((exists ff_h_pfp_append_step_hZalign_leftoutput. ff_h_pfp_append_step_hZalign_leftoutput + S (pfrep_value_append_step_hZalign_left) = S ((S ((M)+pfrep_index_append_step_hZalign_left)) * UPc)) /\\ exists ff_q_pfp_append_step_hZalign_leftoutput. UPb = ff_q_pfp_append_step_hZalign_leftoutput * S ((S ((M)+pfrep_index_append_step_hZalign_left)) * UPc) + (pfrep_value_append_step_hZalign_left))))))) /\\ (((((forall pfp_repeat_index_append_step_hZalign_rightzeros. (exists pfa_gap_append_step_hZalign_rightzerosindex. pfa_gap_append_step_hZalign_rightzerosindex + S (pfp_repeat_index_append_step_hZalign_rightzeros) = (S K0)) -> (((exists ff_h_pfp_append_step_hZalign_rightzerosentry. ff_h_pfp_append_step_hZalign_rightzerosentry + S (0) = S ((S (pfp_repeat_index_append_step_hZalign_rightzeros)) * VPc)) /\\ exists ff_q_pfp_append_step_hZalign_rightzerosentry. VPb = ff_q_pfp_append_step_hZalign_rightzerosentry * S ((S (pfp_repeat_index_append_step_hZalign_rightzeros)) * VPc) + (0)))) /\\ ((forall pfrep_index_append_step_hZalign_right pfrep_value_append_step_hZalign_right. (exists pfa_gap_append_step_hZalign_rightbound. pfa_gap_append_step_hZalign_rightbound + S (pfrep_index_append_step_hZalign_right) = (M)) -> (((exists ff_h_pfp_append_step_hZalign_rightinput. ff_h_pfp_append_step_hZalign_rightinput + S (pfrep_value_append_step_hZalign_right) = S ((S (pfrep_index_append_step_hZalign_right)) * vc)) /\\ exists ff_q_pfp_append_step_hZalign_rightinput. vb = ff_q_pfp_append_step_hZalign_rightinput * S ((S (pfrep_index_append_step_hZalign_right)) * vc) + (pfrep_value_append_step_hZalign_right))) -> (((exists ff_h_pfp_append_step_hZalign_rightoutput. ff_h_pfp_append_step_hZalign_rightoutput + S (pfrep_value_append_step_hZalign_right) = S ((S ((S K0)+pfrep_index_append_step_hZalign_right)) * VPc)) /\\ exists ff_q_pfp_append_step_hZalign_rightoutput. VPb = ff_q_pfp_append_step_hZalign_rightoutput * S ((S ((S K0)+pfrep_index_append_step_hZalign_right)) * VPc) + (pfrep_value_append_step_hZalign_right))))))) /\\ ((forall pfp_index_append_step_hZalign_sum. (exists pfa_gap_append_step_hZalign_sumindex. pfa_gap_append_step_hZalign_sumindex + S (pfp_index_append_step_hZalign_sum) = (M+S K0)) -> exists pfp_left_append_step_hZalign_sum pfp_right_append_step_hZalign_sum pfp_value_append_step_hZalign_sum. ((((exists ff_h_pfp_append_step_hZalign_sumleft. ff_h_pfp_append_step_hZalign_sumleft + S (pfp_left_append_step_hZalign_sum) = S ((S (pfp_index_append_step_hZalign_sum)) * UPc)) /\\ exists ff_q_pfp_append_step_hZalign_sumleft. UPb = ff_q_pfp_append_step_hZalign_sumleft * S ((S (pfp_index_append_step_hZalign_sum)) * UPc) + (pfp_left_append_step_hZalign_sum))) /\\ (((((exists ff_h_pfp_append_step_hZalign_sumright. ff_h_pfp_append_step_hZalign_sumright + S (pfp_right_append_step_hZalign_sum) = S ((S (pfp_index_append_step_hZalign_sum)) * VPc)) /\\ exists ff_q_pfp_append_step_hZalign_sumright. VPb = ff_q_pfp_append_step_hZalign_sumright * S ((S (pfp_index_append_step_hZalign_sum)) * VPc) + (pfp_right_append_step_hZalign_sum))) /\\ (((((exists ff_h_pfp_append_step_hZalign_sumtarget. ff_h_pfp_append_step_hZalign_sumtarget + S (pfp_value_append_step_hZalign_sum) = S ((S (pfp_index_append_step_hZalign_sum)) * zc)) /\\ exists ff_q_pfp_append_step_hZalign_sumtarget. zb = ff_q_pfp_append_step_hZalign_sumtarget * S ((S (pfp_index_append_step_hZalign_sum)) * zc) + (pfp_value_append_step_hZalign_sum))) /\\ ((((exists pfa_gap_append_step_hZalign_sumoperationleft. pfa_gap_append_step_hZalign_sumoperationleft + S (pfp_left_append_step_hZalign_sum) = (p)) /\\ (((exists pfa_gap_append_step_hZalign_sumoperationright. pfa_gap_append_step_hZalign_sumoperationright + S (pfp_right_append_step_hZalign_sum) = (p)) /\\ ((((exists pfa_gap_append_step_hZalign_sumoperationresultbound. pfa_gap_append_step_hZalign_sumoperationresultbound + S (pfp_value_append_step_hZalign_sum) = (p)) /\\ ((exists pfa_offset_left_append_step_hZalign_sumoperationresultcongruence pfa_offset_right_append_step_hZalign_sumoperationresultcongruence. ((pfp_left_append_step_hZalign_sum) + (pfp_right_append_step_hZalign_sum)) + (p) * pfa_offset_left_append_step_hZalign_sumoperationresultcongruence = (pfp_value_append_step_hZalign_sum) + (p) * pfa_offset_right_append_step_hZalign_sumoperationresultcongruence))))))))))))))))))))))))", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (p)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (c)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (bb)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (bc)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (M)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (q0b)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (q0c)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (K0)", "apply prime_field_polynomial_shift_scale_aligned_sum_exists", "exact hp", "exact hc", "exact hABcopy_right_left", "exact hQ0bound", "cases hZalign", "cases hZalign_witness", "cases hZalign_witness_witness", "cases hZalign_witness_witness_witness", "cases hZalign_witness_witness_witness_witness", "cases hZalign_witness_witness_witness_witness_witness", "cases hZalign_witness_witness_witness_witness_witness_witness", "cases hZalign_witness_witness_witness_witness_witness_witness_witness", "cases hZalign_witness_witness_witness_witness_witness_witness_witness_witness", "cases hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness", "cases hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness", "cases hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right", "cases hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right", "cases hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right", "have hZbounds : ((forall fom_index_pfp_append_step_Z_left_bound. (exists fom_gap_pfp_append_step_Z_left_bound_index_bound. fom_gap_pfp_append_step_Z_left_bound_index_bound + S (fom_index_pfp_append_step_Z_left_bound) = M+S K0) -> exists fom_value_pfp_append_step_Z_left_bound. ((((exists fom_beta_height_pfp_append_step_Z_left_bound_entry. fom_beta_height_pfp_append_step_Z_left_bound_entry + S (fom_value_pfp_append_step_Z_left_bound) = S ((S (fom_index_pfp_append_step_Z_left_bound)) * x15)) /\\ exists fom_beta_quotient_pfp_append_step_Z_left_bound_entry. x14 = fom_beta_quotient_pfp_append_step_Z_left_bound_entry * S ((S (fom_index_pfp_append_step_Z_left_bound)) * x15) + (fom_value_pfp_append_step_Z_left_bound))) /\\ (exists fom_gap_pfp_append_step_Z_left_bound_value_bound. fom_gap_pfp_append_step_Z_left_bound_value_bound + S (fom_value_pfp_append_step_Z_left_bound) = p))) /\\ (((forall fom_index_pfp_append_step_Z_right_bound. (exists fom_gap_pfp_append_step_Z_right_bound_index_bound. fom_gap_pfp_append_step_Z_right_bound_index_bound + S (fom_index_pfp_append_step_Z_right_bound) = M+S K0) -> exists fom_value_pfp_append_step_Z_right_bound. ((((exists fom_beta_height_pfp_append_step_Z_right_bound_entry. fom_beta_height_pfp_append_step_Z_right_bound_entry + S (fom_value_pfp_append_step_Z_right_bound) = S ((S (fom_index_pfp_append_step_Z_right_bound)) * x17)) /\\ exists fom_beta_quotient_pfp_append_step_Z_right_bound_entry. x16 = fom_beta_quotient_pfp_append_step_Z_right_bound_entry * S ((S (fom_index_pfp_append_step_Z_right_bound)) * x17) + (fom_value_pfp_append_step_Z_right_bound))) /\\ (exists fom_gap_pfp_append_step_Z_right_bound_value_bound. fom_gap_pfp_append_step_Z_right_bound_value_bound + S (fom_value_pfp_append_step_Z_right_bound) = p))) /\\ ((forall fom_index_pfp_append_step_Z_output_bound. (exists fom_gap_pfp_append_step_Z_output_bound_index_bound. fom_gap_pfp_append_step_Z_output_bound_index_bound + S (fom_index_pfp_append_step_Z_output_bound) = M+S K0) -> exists fom_value_pfp_append_step_Z_output_bound. ((((exists fom_beta_height_pfp_append_step_Z_output_bound_entry. fom_beta_height_pfp_append_step_Z_output_bound_entry + S (fom_value_pfp_append_step_Z_output_bound) = S ((S (fom_index_pfp_append_step_Z_output_bound)) * x19)) /\\ exists fom_beta_quotient_pfp_append_step_Z_output_bound_entry. x18 = fom_beta_quotient_pfp_append_step_Z_output_bound_entry * S ((S (fom_index_pfp_append_step_Z_output_bound)) * x19) + (fom_value_pfp_append_step_Z_output_bound))) /\\ (exists fom_gap_pfp_append_step_Z_output_bound_value_bound. fom_gap_pfp_append_step_Z_output_bound_value_bound + S (fom_value_pfp_append_step_Z_output_bound) = p)))))))", "specialize prime_field_polynomial_add_bounded (p)", "specialize prime_field_polynomial_add_bounded (x14)", "specialize prime_field_polynomial_add_bounded (x15)", "specialize prime_field_polynomial_add_bounded (x16)", "specialize prime_field_polynomial_add_bounded (x17)", "specialize prime_field_polynomial_add_bounded (x18)", "specialize prime_field_polynomial_add_bounded (x19)", "specialize prime_field_polynomial_add_bounded (M+S K0)", "apply prime_field_polynomial_add_bounded", "exact hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "cases hZbounds", "cases hZbounds_right", "have hZlength : exists W. (((((L)=0 \\/ (M+S K0)=0) /\\ (((W)=0)))) \\/ (((~((L)=0)) /\\ (((~((M+S K0)=0)) /\\ (((L)+(M+S K0)=S (W))))))))", "specialize polynomial_product_length_exists (L)", "specialize polynomial_product_length_exists (M+S K0)", "apply polynomial_product_length_exists", "cases hZlength", "have hAZ : exists mb mc. ((forall fom_index_pfp_append_step_AZleft. (exists fom_gap_pfp_append_step_AZleft_index_bound. fom_gap_pfp_append_step_AZleft_index_bound + S (fom_index_pfp_append_step_AZleft) = L) -> exists fom_value_pfp_append_step_AZleft. ((((exists fom_beta_height_pfp_append_step_AZleft_entry. fom_beta_height_pfp_append_step_AZleft_entry + S (fom_value_pfp_append_step_AZleft) = S ((S (fom_index_pfp_append_step_AZleft)) * ac)) /\\ exists fom_beta_quotient_pfp_append_step_AZleft_entry. ab = fom_beta_quotient_pfp_append_step_AZleft_entry * S ((S (fom_index_pfp_append_step_AZleft)) * ac) + (fom_value_pfp_append_step_AZleft))) /\\ (exists fom_gap_pfp_append_step_AZleft_value_bound. fom_gap_pfp_append_step_AZleft_value_bound + S (fom_value_pfp_append_step_AZleft) = p))) /\\ (((forall fom_index_pfp_append_step_AZright. (exists fom_gap_pfp_append_step_AZright_index_bound. fom_gap_pfp_append_step_AZright_index_bound + S (fom_index_pfp_append_step_AZright) = M+S K0) -> exists fom_value_pfp_append_step_AZright. ((((exists fom_beta_height_pfp_append_step_AZright_entry. fom_beta_height_pfp_append_step_AZright_entry + S (fom_value_pfp_append_step_AZright) = S ((S (fom_index_pfp_append_step_AZright)) * x19)) /\\ exists fom_beta_quotient_pfp_append_step_AZright_entry. x18 = fom_beta_quotient_pfp_append_step_AZright_entry * S ((S (fom_index_pfp_append_step_AZright)) * x19) + (fom_value_pfp_append_step_AZright))) /\\ (exists fom_gap_pfp_append_step_AZright_value_bound. fom_gap_pfp_append_step_AZright_value_bound + S (fom_value_pfp_append_step_AZright) = p))) /\\ (((((((L)=0 \\/ (M+S K0)=0) /\\ (((x20)=0)))) \\/ (((~((L)=0)) /\\ (((~((M+S K0)=0)) /\\ (((L)+(M+S K0)=S (x20)))))))) /\\ ((forall pfc_index_append_step_AZcoefficients. (exists pfa_gap_append_step_AZcoefficientsbound. pfa_gap_append_step_AZcoefficientsbound + S (pfc_index_append_step_AZcoefficients) = (x20)) -> exists pfc_value_append_step_AZcoefficients. ((((exists ff_h_pfp_append_step_AZcoefficientsentry. ff_h_pfp_append_step_AZcoefficientsentry + S (pfc_value_append_step_AZcoefficients) = S ((S (pfc_index_append_step_AZcoefficients)) * mc)) /\\ exists ff_q_pfp_append_step_AZcoefficientsentry. mb = ff_q_pfp_append_step_AZcoefficientsentry * S ((S (pfc_index_append_step_AZcoefficients)) * mc) + (pfc_value_append_step_AZcoefficients))) /\\ ((exists pfc_terms_code_append_step_AZcoefficientscoefficient pfc_terms_scale_append_step_AZcoefficientscoefficient pfc_natural_sum_append_step_AZcoefficientscoefficient. ((forall pfc_index_append_step_AZcoefficientscoefficientdiagonal. (exists pfa_gap_append_step_AZcoefficientscoefficientdiagonalbound. pfa_gap_append_step_AZcoefficientscoefficientdiagonalbound + S (pfc_index_append_step_AZcoefficientscoefficientdiagonal) = (S (pfc_index_append_step_AZcoefficients))) -> exists pfc_value_append_step_AZcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_step_AZcoefficientscoefficientdiagonalentry. ff_h_pfp_append_step_AZcoefficientscoefficientdiagonalentry + S (pfc_value_append_step_AZcoefficientscoefficientdiagonal) = S ((S (pfc_index_append_step_AZcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_AZcoefficientscoefficient)) /\\ exists ff_q_pfp_append_step_AZcoefficientscoefficientdiagonalentry. pfc_terms_code_append_step_AZcoefficientscoefficient = ff_q_pfp_append_step_AZcoefficientscoefficientdiagonalentry * S ((S (pfc_index_append_step_AZcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_AZcoefficientscoefficient) + (pfc_value_append_step_AZcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_step_AZcoefficientscoefficientdiagonalterm pfc_left_append_step_AZcoefficientscoefficientdiagonalterm pfc_right_append_step_AZcoefficientscoefficientdiagonalterm. (((pfc_index_append_step_AZcoefficientscoefficientdiagonal)+pfc_complement_append_step_AZcoefficientscoefficientdiagonalterm=(pfc_index_append_step_AZcoefficients)) /\\ ((((((exists pfa_gap_append_step_AZcoefficientscoefficientdiagonaltermleftinside. pfa_gap_append_step_AZcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_step_AZcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_step_AZcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_step_AZcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_step_AZcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_step_AZcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_step_AZcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_step_AZcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_step_AZcoefficientscoefficientdiagonal)) * ac) + (pfc_left_append_step_AZcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_AZcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_step_AZcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_step_AZcoefficientscoefficientdiagonal)) /\\ (((pfc_left_append_step_AZcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_step_AZcoefficientscoefficientdiagonaltermrightinside. pfa_gap_append_step_AZcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_step_AZcoefficientscoefficientdiagonalterm) = (M+S K0)) /\\ ((((exists ff_h_pfp_append_step_AZcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_step_AZcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_step_AZcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_step_AZcoefficientscoefficientdiagonalterm)) * x19)) /\\ exists ff_q_pfp_append_step_AZcoefficientscoefficientdiagonaltermrightentry. x18 = ff_q_pfp_append_step_AZcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_step_AZcoefficientscoefficientdiagonalterm)) * x19) + (pfc_right_append_step_AZcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_AZcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_step_AZcoefficientscoefficientdiagonaltermrightoutside+(M+S K0)=(pfc_complement_append_step_AZcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_step_AZcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_step_AZcoefficientscoefficientdiagonal)=pfc_left_append_step_AZcoefficientscoefficientdiagonalterm*pfc_right_append_step_AZcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_step_AZcoefficientscoefficientsum fs_v_pfc_append_step_AZcoefficientscoefficientsum. ((((exists fs_h_pfc_append_step_AZcoefficientscoefficientsum_body_start. fs_h_pfc_append_step_AZcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_step_AZcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_AZcoefficientscoefficientsum_body_start. fs_u_pfc_append_step_AZcoefficientscoefficientsum = fs_q_pfc_append_step_AZcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_step_AZcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_step_AZcoefficientscoefficientsum_body_terminal. fs_h_pfc_append_step_AZcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_step_AZcoefficientscoefficient) = S ((S (S (pfc_index_append_step_AZcoefficients))) * fs_v_pfc_append_step_AZcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_AZcoefficientscoefficientsum_body_terminal. fs_u_pfc_append_step_AZcoefficientscoefficientsum = fs_q_pfc_append_step_AZcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_step_AZcoefficients))) * fs_v_pfc_append_step_AZcoefficientscoefficientsum) + (pfc_natural_sum_append_step_AZcoefficientscoefficient))) /\\ forall fs_i_pfc_append_step_AZcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_step_AZcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_step_AZcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_step_AZcoefficientscoefficientsum_body_steps = S (pfc_index_append_step_AZcoefficients)) -> exists fs_a_pfc_append_step_AZcoefficientscoefficientsum_body_steps fs_r_pfc_append_step_AZcoefficientscoefficientsum_body_steps fs_s_pfc_append_step_AZcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_step_AZcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_step_AZcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_step_AZcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_AZcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_AZcoefficientscoefficient)) /\\ exists fs_q_pfc_append_step_AZcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_step_AZcoefficientscoefficient = fs_q_pfc_append_step_AZcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_step_AZcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_AZcoefficientscoefficient) + (fs_a_pfc_append_step_AZcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_AZcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_step_AZcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_step_AZcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_AZcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_AZcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_AZcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_step_AZcoefficientscoefficientsum = fs_q_pfc_append_step_AZcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_step_AZcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_AZcoefficientscoefficientsum) + (fs_r_pfc_append_step_AZcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_AZcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_step_AZcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_step_AZcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_step_AZcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_AZcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_AZcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_step_AZcoefficientscoefficientsum = fs_q_pfc_append_step_AZcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_step_AZcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_AZcoefficientscoefficientsum) + (fs_s_pfc_append_step_AZcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_step_AZcoefficientscoefficientsum_body_steps = fs_r_pfc_append_step_AZcoefficientscoefficientsum_body_steps + fs_a_pfc_append_step_AZcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_step_AZcoefficientscoefficientresiduebound. pfa_gap_append_step_AZcoefficientscoefficientresiduebound + S (pfc_value_append_step_AZcoefficients) = (p)) /\\ ((exists pfa_offset_left_append_step_AZcoefficientscoefficientresiduecongruence pfa_offset_right_append_step_AZcoefficientscoefficientresiduecongruence. (pfc_natural_sum_append_step_AZcoefficientscoefficient) + (p) * pfa_offset_left_append_step_AZcoefficientscoefficientresiduecongruence = (pfc_value_append_step_AZcoefficients) + (p) * pfa_offset_right_append_step_AZcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (ab)", "specialize prime_field_polynomial_convolution_at_length_exists (ac)", "specialize prime_field_polynomial_convolution_at_length_exists (L)", "specialize prime_field_polynomial_convolution_at_length_exists (x18)", "specialize prime_field_polynomial_convolution_at_length_exists (x19)", "specialize prime_field_polynomial_convolution_at_length_exists (M+S K0)", "specialize prime_field_polynomial_convolution_at_length_exists (x20)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hABcopy_left", "exact hZbounds_right_right", "exact hZlength_witness", "cases hAZ", "cases hAZ_witness", "have hhelper_equal : forall pfrep_power_append_step_helper_equal pfrep_left_append_step_helper_equal pfrep_right_append_step_helper_equal. ((exists pfrep_position_append_step_helper_equalfirst. ((pfrep_position_append_step_helper_equalfirst+S (pfrep_power_append_step_helper_equal)=(x20)) /\\ ((((exists ff_h_pfp_append_step_helper_equalfirstentry. ff_h_pfp_append_step_helper_equalfirstentry + S (pfrep_left_append_step_helper_equal) = S ((S (pfrep_position_append_step_helper_equalfirst)) * x22)) /\\ exists ff_q_pfp_append_step_helper_equalfirstentry. x21 = ff_q_pfp_append_step_helper_equalfirstentry * S ((S (pfrep_position_append_step_helper_equalfirst)) * x22) + (pfrep_left_append_step_helper_equal)))))) \\/ (((exists pfrep_gap_append_step_helper_equalfirstoutside. pfrep_gap_append_step_helper_equalfirstoutside+(x20)=(pfrep_power_append_step_helper_equal)) /\\ (((pfrep_left_append_step_helper_equal)=0))))) -> ((exists pfrep_position_append_step_helper_equalsecond. ((pfrep_position_append_step_helper_equalsecond+S (pfrep_power_append_step_helper_equal)=(N+S V0)) /\\ ((((exists ff_h_pfp_append_step_helper_equalsecondentry. ff_h_pfp_append_step_helper_equalsecondentry + S (pfrep_right_append_step_helper_equal) = S ((S (pfrep_position_append_step_helper_equalsecond)) * x9)) /\\ exists ff_q_pfp_append_step_helper_equalsecondentry. x8 = ff_q_pfp_append_step_helper_equalsecondentry * S ((S (pfrep_position_append_step_helper_equalsecond)) * x9) + (pfrep_right_append_step_helper_equal)))))) \\/ (((exists pfrep_gap_append_step_helper_equalsecondoutside. pfrep_gap_append_step_helper_equalsecondoutside+(N+S V0)=(pfrep_power_append_step_helper_equal)) /\\ (((pfrep_right_append_step_helper_equal)=0))))) -> pfrep_left_append_step_helper_equal=pfrep_right_append_step_helper_equal", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (p)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (c)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (ab)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (ac)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (L)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (bb)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (bc)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (M)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (pb)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (pc)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (N)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (q0b)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (q0c)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (K0)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (s0b)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (s0c)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (V0)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x10)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x11)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x12)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x13)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x14)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x15)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x16)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x17)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x18)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x19)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x21)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x22)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x20)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x1)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x2)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x3)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x4)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x5)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x6)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x7)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x8)", "specialize prime_field_polynomial_convolution_shift_scale_aligned_equivalent (x9)", "apply prime_field_polynomial_convolution_shift_scale_aligned_equivalent", "exact hp", "exact hAB", "exact hS0", "exact hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_left", "exact hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "exact hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left", "exact hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "exact hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "exact hAZ_witness_witness", "exact hYalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_left", "exact hYalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "exact hYalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left", "exact hYalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "exact hYalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "have hQ_append : forall pfrep_power_append_step_Q_append pfrep_left_append_step_Q_append pfrep_right_append_step_Q_append. ((exists pfrep_position_append_step_Q_appendfirst. ((pfrep_position_append_step_Q_appendfirst+S (pfrep_power_append_step_Q_append)=(K1)) /\\ ((((exists ff_h_pfp_append_step_Q_appendfirstentry. ff_h_pfp_append_step_Q_appendfirstentry + S (pfrep_left_append_step_Q_append) = S ((S (pfrep_position_append_step_Q_appendfirst)) * q1c)) /\\ exists ff_q_pfp_append_step_Q_appendfirstentry. q1b = ff_q_pfp_append_step_Q_appendfirstentry * S ((S (pfrep_position_append_step_Q_appendfirst)) * q1c) + (pfrep_left_append_step_Q_append)))))) \\/ (((exists pfrep_gap_append_step_Q_appendfirstoutside. pfrep_gap_append_step_Q_appendfirstoutside+(K1)=(pfrep_power_append_step_Q_append)) /\\ (((pfrep_left_append_step_Q_append)=0))))) -> ((exists pfrep_position_append_step_Q_appendsecond. ((pfrep_position_append_step_Q_appendsecond+S (pfrep_power_append_step_Q_append)=(M+S K0)) /\\ ((((exists ff_h_pfp_append_step_Q_appendsecondentry. ff_h_pfp_append_step_Q_appendsecondentry + S (pfrep_right_append_step_Q_append) = S ((S (pfrep_position_append_step_Q_appendsecond)) * x19)) /\\ exists ff_q_pfp_append_step_Q_appendsecondentry. x18 = ff_q_pfp_append_step_Q_appendsecondentry * S ((S (pfrep_position_append_step_Q_appendsecond)) * x19) + (pfrep_right_append_step_Q_append)))))) \\/ (((exists pfrep_gap_append_step_Q_appendsecondoutside. pfrep_gap_append_step_Q_appendsecondoutside+(M+S K0)=(pfrep_power_append_step_Q_append)) /\\ (((pfrep_right_append_step_Q_append)=0))))) -> pfrep_left_append_step_Q_append=pfrep_right_append_step_Q_append", "specialize prime_field_polynomial_convolution_right_append_equivalent (p)", "specialize prime_field_polynomial_convolution_right_append_equivalent (bb)", "specialize prime_field_polynomial_convolution_right_append_equivalent (bc)", "specialize prime_field_polynomial_convolution_right_append_equivalent (M)", "specialize prime_field_polynomial_convolution_right_append_equivalent (cb)", "specialize prime_field_polynomial_convolution_right_append_equivalent (cc)", "specialize prime_field_polynomial_convolution_right_append_equivalent (J)", "specialize prime_field_polynomial_convolution_right_append_equivalent (c)", "specialize prime_field_polynomial_convolution_right_append_equivalent (db)", "specialize prime_field_polynomial_convolution_right_append_equivalent (dc)", "specialize prime_field_polynomial_convolution_right_append_equivalent (q0b)", "specialize prime_field_polynomial_convolution_right_append_equivalent (q0c)", "specialize prime_field_polynomial_convolution_right_append_equivalent (K0)", "specialize prime_field_polynomial_convolution_right_append_equivalent (q1b)", "specialize prime_field_polynomial_convolution_right_append_equivalent (q1c)", "specialize prime_field_polynomial_convolution_right_append_equivalent (K1)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x10)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x11)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x12)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x13)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x14)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x15)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x16)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x17)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x18)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x19)", "apply prime_field_polynomial_convolution_right_append_equivalent", "exact hp", "exact hprefix", "exact hlast", "exact hQ0", "exact hQ1", "exact hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_left", "exact hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "exact hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left", "exact hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "exact hZalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "have hS_transport : forall pfrep_power_append_step_S_transport pfrep_left_append_step_S_transport pfrep_right_append_step_S_transport. ((exists pfrep_position_append_step_S_transportfirst. ((pfrep_position_append_step_S_transportfirst+S (pfrep_power_append_step_S_transport)=(V1)) /\\ ((((exists ff_h_pfp_append_step_S_transportfirstentry. ff_h_pfp_append_step_S_transportfirstentry + S (pfrep_left_append_step_S_transport) = S ((S (pfrep_position_append_step_S_transportfirst)) * s1c)) /\\ exists ff_q_pfp_append_step_S_transportfirstentry. s1b = ff_q_pfp_append_step_S_transportfirstentry * S ((S (pfrep_position_append_step_S_transportfirst)) * s1c) + (pfrep_left_append_step_S_transport)))))) \\/ (((exists pfrep_gap_append_step_S_transportfirstoutside. pfrep_gap_append_step_S_transportfirstoutside+(V1)=(pfrep_power_append_step_S_transport)) /\\ (((pfrep_left_append_step_S_transport)=0))))) -> ((exists pfrep_position_append_step_S_transportsecond. ((pfrep_position_append_step_S_transportsecond+S (pfrep_power_append_step_S_transport)=(x20)) /\\ ((((exists ff_h_pfp_append_step_S_transportsecondentry. ff_h_pfp_append_step_S_transportsecondentry + S (pfrep_right_append_step_S_transport) = S ((S (pfrep_position_append_step_S_transportsecond)) * x22)) /\\ exists ff_q_pfp_append_step_S_transportsecondentry. x21 = ff_q_pfp_append_step_S_transportsecondentry * S ((S (pfrep_position_append_step_S_transportsecond)) * x22) + (pfrep_right_append_step_S_transport)))))) \\/ (((exists pfrep_gap_append_step_S_transportsecondoutside. pfrep_gap_append_step_S_transportsecondoutside+(x20)=(pfrep_power_append_step_S_transport)) /\\ (((pfrep_right_append_step_S_transport)=0))))) -> pfrep_left_append_step_S_transport=pfrep_right_append_step_S_transport", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (p)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (ab)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (ac)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (L)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (q1b)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (q1c)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (K1)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (s1b)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (s1c)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (V1)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x18)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x19)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (M+S K0)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x21)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x22)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x20)", "apply prime_field_polynomial_convolution_equivalent_congruent_right", "exact hp0", "exact hQ_append", "exact hS1", "exact hAZ_witness_witness", "specialize prime_field_polynomial_equivalent_transitive (s1b)", "specialize prime_field_polynomial_equivalent_transitive (s1c)", "specialize prime_field_polynomial_equivalent_transitive (V1)", "specialize prime_field_polynomial_equivalent_transitive (x21)", "specialize prime_field_polynomial_equivalent_transitive (x22)", "specialize prime_field_polynomial_equivalent_transitive (x20)", "specialize prime_field_polynomial_equivalent_transitive (x8)", "specialize prime_field_polynomial_equivalent_transitive (x9)", "specialize prime_field_polynomial_equivalent_transitive (N+S V0)", "apply prime_field_polynomial_equivalent_transitive", "exact hS_transport", "exact hhelper_equal", "have hR_total : forall pfrep_power_append_step_R_total pfrep_left_append_step_R_total pfrep_right_append_step_R_total. ((exists pfrep_position_append_step_R_totalfirst. ((pfrep_position_append_step_R_totalfirst+S (pfrep_power_append_step_R_total)=(U1)) /\\ ((((exists ff_h_pfp_append_step_R_totalfirstentry. ff_h_pfp_append_step_R_totalfirstentry + S (pfrep_left_append_step_R_total) = S ((S (pfrep_position_append_step_R_totalfirst)) * r1c)) /\\ exists ff_q_pfp_append_step_R_totalfirstentry. r1b = ff_q_pfp_append_step_R_totalfirstentry * S ((S (pfrep_position_append_step_R_totalfirst)) * r1c) + (pfrep_left_append_step_R_total)))))) \\/ (((exists pfrep_gap_append_step_R_totalfirstoutside. pfrep_gap_append_step_R_totalfirstoutside+(U1)=(pfrep_power_append_step_R_total)) /\\ (((pfrep_left_append_step_R_total)=0))))) -> ((exists pfrep_position_append_step_R_totalsecond. ((pfrep_position_append_step_R_totalsecond+S (pfrep_power_append_step_R_total)=(N+S V0)) /\\ ((((exists ff_h_pfp_append_step_R_totalsecondentry. ff_h_pfp_append_step_R_totalsecondentry + S (pfrep_right_append_step_R_total) = S ((S (pfrep_position_append_step_R_totalsecond)) * x9)) /\\ exists ff_q_pfp_append_step_R_totalsecondentry. x8 = ff_q_pfp_append_step_R_totalsecondentry * S ((S (pfrep_position_append_step_R_totalsecond)) * x9) + (pfrep_right_append_step_R_total)))))) \\/ (((exists pfrep_gap_append_step_R_totalsecondoutside. pfrep_gap_append_step_R_totalsecondoutside+(N+S V0)=(pfrep_power_append_step_R_total)) /\\ (((pfrep_right_append_step_R_total)=0))))) -> pfrep_left_append_step_R_total=pfrep_right_append_step_R_total", "have hY0align : exists ub uc vb vc UPb UPc VPb VPc zb zc. ((((forall mdr_i_pfp_append_step_hY0align_shiftprefix mdr_a_pfp_append_step_hY0align_shiftprefix. (exists mdr_gap_pfp_append_step_hY0align_shiftprefixb. mdr_gap_pfp_append_step_hY0align_shiftprefixb + S (mdr_i_pfp_append_step_hY0align_shiftprefix) = (U0)) -> (((exists ff_h_mdr_pfp_append_step_hY0align_shiftprefixo. ff_h_mdr_pfp_append_step_hY0align_shiftprefixo + S (mdr_a_pfp_append_step_hY0align_shiftprefix) = S ((S (mdr_i_pfp_append_step_hY0align_shiftprefix)) * r0c)) /\\ exists ff_q_mdr_pfp_append_step_hY0align_shiftprefixo. r0b = ff_q_mdr_pfp_append_step_hY0align_shiftprefixo * S ((S (mdr_i_pfp_append_step_hY0align_shiftprefix)) * r0c) + (mdr_a_pfp_append_step_hY0align_shiftprefix))) -> (((exists ff_h_mdr_pfp_append_step_hY0align_shiftprefixn. ff_h_mdr_pfp_append_step_hY0align_shiftprefixn + S (mdr_a_pfp_append_step_hY0align_shiftprefix) = S ((S (mdr_i_pfp_append_step_hY0align_shiftprefix)) * uc)) /\\ exists ff_q_mdr_pfp_append_step_hY0align_shiftprefixn. ub = ff_q_mdr_pfp_append_step_hY0align_shiftprefixn * S ((S (mdr_i_pfp_append_step_hY0align_shiftprefix)) * uc) + (mdr_a_pfp_append_step_hY0align_shiftprefix)))) /\\ ((((exists ff_h_pfp_append_step_hY0align_shiftzero. ff_h_pfp_append_step_hY0align_shiftzero + S (0) = S ((S (U0)) * uc)) /\\ exists ff_q_pfp_append_step_hY0align_shiftzero. ub = ff_q_pfp_append_step_hY0align_shiftzero * S ((S (U0)) * uc) + (0)))))) /\\ (((((exists pfa_gap_append_step_hY0align_scalescalar. pfa_gap_append_step_hY0align_scalescalar + S (c) = (p)) /\\ ((forall pfp_index_append_step_hY0align_scale. (exists pfa_gap_append_step_hY0align_scaleindex. pfa_gap_append_step_hY0align_scaleindex + S (pfp_index_append_step_hY0align_scale) = (N)) -> exists pfp_source_append_step_hY0align_scale pfp_value_append_step_hY0align_scale. ((((exists ff_h_pfp_append_step_hY0align_scalesource. ff_h_pfp_append_step_hY0align_scalesource + S (pfp_source_append_step_hY0align_scale) = S ((S (pfp_index_append_step_hY0align_scale)) * pc)) /\\ exists ff_q_pfp_append_step_hY0align_scalesource. pb = ff_q_pfp_append_step_hY0align_scalesource * S ((S (pfp_index_append_step_hY0align_scale)) * pc) + (pfp_source_append_step_hY0align_scale))) /\\ (((((exists ff_h_pfp_append_step_hY0align_scaletarget. ff_h_pfp_append_step_hY0align_scaletarget + S (pfp_value_append_step_hY0align_scale) = S ((S (pfp_index_append_step_hY0align_scale)) * vc)) /\\ exists ff_q_pfp_append_step_hY0align_scaletarget. vb = ff_q_pfp_append_step_hY0align_scaletarget * S ((S (pfp_index_append_step_hY0align_scale)) * vc) + (pfp_value_append_step_hY0align_scale))) /\\ ((((exists pfa_gap_append_step_hY0align_scaleoperationleft. pfa_gap_append_step_hY0align_scaleoperationleft + S (c) = (p)) /\\ (((exists pfa_gap_append_step_hY0align_scaleoperationright. pfa_gap_append_step_hY0align_scaleoperationright + S (pfp_source_append_step_hY0align_scale) = (p)) /\\ ((((exists pfa_gap_append_step_hY0align_scaleoperationresultbound. pfa_gap_append_step_hY0align_scaleoperationresultbound + S (pfp_value_append_step_hY0align_scale) = (p)) /\\ ((exists pfa_offset_left_append_step_hY0align_scaleoperationresultcongruence pfa_offset_right_append_step_hY0align_scaleoperationresultcongruence. ((c) * (pfp_source_append_step_hY0align_scale)) + (p) * pfa_offset_left_append_step_hY0align_scaleoperationresultcongruence = (pfp_value_append_step_hY0align_scale) + (p) * pfa_offset_right_append_step_hY0align_scaleoperationresultcongruence))))))))))))))))) /\\ (((((forall pfp_repeat_index_append_step_hY0align_leftzeros. (exists pfa_gap_append_step_hY0align_leftzerosindex. pfa_gap_append_step_hY0align_leftzerosindex + S (pfp_repeat_index_append_step_hY0align_leftzeros) = (N)) -> (((exists ff_h_pfp_append_step_hY0align_leftzerosentry. ff_h_pfp_append_step_hY0align_leftzerosentry + S (0) = S ((S (pfp_repeat_index_append_step_hY0align_leftzeros)) * UPc)) /\\ exists ff_q_pfp_append_step_hY0align_leftzerosentry. UPb = ff_q_pfp_append_step_hY0align_leftzerosentry * S ((S (pfp_repeat_index_append_step_hY0align_leftzeros)) * UPc) + (0)))) /\\ ((forall pfrep_index_append_step_hY0align_left pfrep_value_append_step_hY0align_left. (exists pfa_gap_append_step_hY0align_leftbound. pfa_gap_append_step_hY0align_leftbound + S (pfrep_index_append_step_hY0align_left) = (S U0)) -> (((exists ff_h_pfp_append_step_hY0align_leftinput. ff_h_pfp_append_step_hY0align_leftinput + S (pfrep_value_append_step_hY0align_left) = S ((S (pfrep_index_append_step_hY0align_left)) * uc)) /\\ exists ff_q_pfp_append_step_hY0align_leftinput. ub = ff_q_pfp_append_step_hY0align_leftinput * S ((S (pfrep_index_append_step_hY0align_left)) * uc) + (pfrep_value_append_step_hY0align_left))) -> (((exists ff_h_pfp_append_step_hY0align_leftoutput. ff_h_pfp_append_step_hY0align_leftoutput + S (pfrep_value_append_step_hY0align_left) = S ((S ((N)+pfrep_index_append_step_hY0align_left)) * UPc)) /\\ exists ff_q_pfp_append_step_hY0align_leftoutput. UPb = ff_q_pfp_append_step_hY0align_leftoutput * S ((S ((N)+pfrep_index_append_step_hY0align_left)) * UPc) + (pfrep_value_append_step_hY0align_left))))))) /\\ (((((forall pfp_repeat_index_append_step_hY0align_rightzeros. (exists pfa_gap_append_step_hY0align_rightzerosindex. pfa_gap_append_step_hY0align_rightzerosindex + S (pfp_repeat_index_append_step_hY0align_rightzeros) = (S U0)) -> (((exists ff_h_pfp_append_step_hY0align_rightzerosentry. ff_h_pfp_append_step_hY0align_rightzerosentry + S (0) = S ((S (pfp_repeat_index_append_step_hY0align_rightzeros)) * VPc)) /\\ exists ff_q_pfp_append_step_hY0align_rightzerosentry. VPb = ff_q_pfp_append_step_hY0align_rightzerosentry * S ((S (pfp_repeat_index_append_step_hY0align_rightzeros)) * VPc) + (0)))) /\\ ((forall pfrep_index_append_step_hY0align_right pfrep_value_append_step_hY0align_right. (exists pfa_gap_append_step_hY0align_rightbound. pfa_gap_append_step_hY0align_rightbound + S (pfrep_index_append_step_hY0align_right) = (N)) -> (((exists ff_h_pfp_append_step_hY0align_rightinput. ff_h_pfp_append_step_hY0align_rightinput + S (pfrep_value_append_step_hY0align_right) = S ((S (pfrep_index_append_step_hY0align_right)) * vc)) /\\ exists ff_q_pfp_append_step_hY0align_rightinput. vb = ff_q_pfp_append_step_hY0align_rightinput * S ((S (pfrep_index_append_step_hY0align_right)) * vc) + (pfrep_value_append_step_hY0align_right))) -> (((exists ff_h_pfp_append_step_hY0align_rightoutput. ff_h_pfp_append_step_hY0align_rightoutput + S (pfrep_value_append_step_hY0align_right) = S ((S ((S U0)+pfrep_index_append_step_hY0align_right)) * VPc)) /\\ exists ff_q_pfp_append_step_hY0align_rightoutput. VPb = ff_q_pfp_append_step_hY0align_rightoutput * S ((S ((S U0)+pfrep_index_append_step_hY0align_right)) * VPc) + (pfrep_value_append_step_hY0align_right))))))) /\\ ((forall pfp_index_append_step_hY0align_sum. (exists pfa_gap_append_step_hY0align_sumindex. pfa_gap_append_step_hY0align_sumindex + S (pfp_index_append_step_hY0align_sum) = (N+S U0)) -> exists pfp_left_append_step_hY0align_sum pfp_right_append_step_hY0align_sum pfp_value_append_step_hY0align_sum. ((((exists ff_h_pfp_append_step_hY0align_sumleft. ff_h_pfp_append_step_hY0align_sumleft + S (pfp_left_append_step_hY0align_sum) = S ((S (pfp_index_append_step_hY0align_sum)) * UPc)) /\\ exists ff_q_pfp_append_step_hY0align_sumleft. UPb = ff_q_pfp_append_step_hY0align_sumleft * S ((S (pfp_index_append_step_hY0align_sum)) * UPc) + (pfp_left_append_step_hY0align_sum))) /\\ (((((exists ff_h_pfp_append_step_hY0align_sumright. ff_h_pfp_append_step_hY0align_sumright + S (pfp_right_append_step_hY0align_sum) = S ((S (pfp_index_append_step_hY0align_sum)) * VPc)) /\\ exists ff_q_pfp_append_step_hY0align_sumright. VPb = ff_q_pfp_append_step_hY0align_sumright * S ((S (pfp_index_append_step_hY0align_sum)) * VPc) + (pfp_right_append_step_hY0align_sum))) /\\ (((((exists ff_h_pfp_append_step_hY0align_sumtarget. ff_h_pfp_append_step_hY0align_sumtarget + S (pfp_value_append_step_hY0align_sum) = S ((S (pfp_index_append_step_hY0align_sum)) * zc)) /\\ exists ff_q_pfp_append_step_hY0align_sumtarget. zb = ff_q_pfp_append_step_hY0align_sumtarget * S ((S (pfp_index_append_step_hY0align_sum)) * zc) + (pfp_value_append_step_hY0align_sum))) /\\ ((((exists pfa_gap_append_step_hY0align_sumoperationleft. pfa_gap_append_step_hY0align_sumoperationleft + S (pfp_left_append_step_hY0align_sum) = (p)) /\\ (((exists pfa_gap_append_step_hY0align_sumoperationright. pfa_gap_append_step_hY0align_sumoperationright + S (pfp_right_append_step_hY0align_sum) = (p)) /\\ ((((exists pfa_gap_append_step_hY0align_sumoperationresultbound. pfa_gap_append_step_hY0align_sumoperationresultbound + S (pfp_value_append_step_hY0align_sum) = (p)) /\\ ((exists pfa_offset_left_append_step_hY0align_sumoperationresultcongruence pfa_offset_right_append_step_hY0align_sumoperationresultcongruence. ((pfp_left_append_step_hY0align_sum) + (pfp_right_append_step_hY0align_sum)) + (p) * pfa_offset_left_append_step_hY0align_sumoperationresultcongruence = (pfp_value_append_step_hY0align_sum) + (p) * pfa_offset_right_append_step_hY0align_sumoperationresultcongruence))))))))))))))))))))))))", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (p)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (c)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (pb)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (pc)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (N)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (r0b)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (r0c)", "specialize prime_field_polynomial_shift_scale_aligned_sum_exists (U0)", "apply prime_field_polynomial_shift_scale_aligned_sum_exists", "exact hp", "exact hc", "exact hPbound", "exact hR0bound", "cases hY0align", "cases hY0align_witness", "cases hY0align_witness_witness", "cases hY0align_witness_witness_witness", "cases hY0align_witness_witness_witness_witness", "cases hY0align_witness_witness_witness_witness_witness", "cases hY0align_witness_witness_witness_witness_witness_witness", "cases hY0align_witness_witness_witness_witness_witness_witness_witness", "cases hY0align_witness_witness_witness_witness_witness_witness_witness_witness", "cases hY0align_witness_witness_witness_witness_witness_witness_witness_witness_witness", "cases hY0align_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness", "cases hY0align_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right", "cases hY0align_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right", "cases hY0align_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right", "have hR_append : forall pfrep_power_append_step_R_append pfrep_left_append_step_R_append pfrep_right_append_step_R_append. ((exists pfrep_position_append_step_R_appendfirst. ((pfrep_position_append_step_R_appendfirst+S (pfrep_power_append_step_R_append)=(U1)) /\\ ((((exists ff_h_pfp_append_step_R_appendfirstentry. ff_h_pfp_append_step_R_appendfirstentry + S (pfrep_left_append_step_R_append) = S ((S (pfrep_position_append_step_R_appendfirst)) * r1c)) /\\ exists ff_q_pfp_append_step_R_appendfirstentry. r1b = ff_q_pfp_append_step_R_appendfirstentry * S ((S (pfrep_position_append_step_R_appendfirst)) * r1c) + (pfrep_left_append_step_R_append)))))) \\/ (((exists pfrep_gap_append_step_R_appendfirstoutside. pfrep_gap_append_step_R_appendfirstoutside+(U1)=(pfrep_power_append_step_R_append)) /\\ (((pfrep_left_append_step_R_append)=0))))) -> ((exists pfrep_position_append_step_R_appendsecond. ((pfrep_position_append_step_R_appendsecond+S (pfrep_power_append_step_R_append)=(N+S U0)) /\\ ((((exists ff_h_pfp_append_step_R_appendsecondentry. ff_h_pfp_append_step_R_appendsecondentry + S (pfrep_right_append_step_R_append) = S ((S (pfrep_position_append_step_R_appendsecond)) * x19)) /\\ exists ff_q_pfp_append_step_R_appendsecondentry. x18 = ff_q_pfp_append_step_R_appendsecondentry * S ((S (pfrep_position_append_step_R_appendsecond)) * x19) + (pfrep_right_append_step_R_append)))))) \\/ (((exists pfrep_gap_append_step_R_appendsecondoutside. pfrep_gap_append_step_R_appendsecondoutside+(N+S U0)=(pfrep_power_append_step_R_append)) /\\ (((pfrep_right_append_step_R_append)=0))))) -> pfrep_left_append_step_R_append=pfrep_right_append_step_R_append", "specialize prime_field_polynomial_convolution_right_append_equivalent (p)", "specialize prime_field_polynomial_convolution_right_append_equivalent (pb)", "specialize prime_field_polynomial_convolution_right_append_equivalent (pc)", "specialize prime_field_polynomial_convolution_right_append_equivalent (N)", "specialize prime_field_polynomial_convolution_right_append_equivalent (cb)", "specialize prime_field_polynomial_convolution_right_append_equivalent (cc)", "specialize prime_field_polynomial_convolution_right_append_equivalent (J)", "specialize prime_field_polynomial_convolution_right_append_equivalent (c)", "specialize prime_field_polynomial_convolution_right_append_equivalent (db)", "specialize prime_field_polynomial_convolution_right_append_equivalent (dc)", "specialize prime_field_polynomial_convolution_right_append_equivalent (r0b)", "specialize prime_field_polynomial_convolution_right_append_equivalent (r0c)", "specialize prime_field_polynomial_convolution_right_append_equivalent (U0)", "specialize prime_field_polynomial_convolution_right_append_equivalent (r1b)", "specialize prime_field_polynomial_convolution_right_append_equivalent (r1c)", "specialize prime_field_polynomial_convolution_right_append_equivalent (U1)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x10)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x11)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x12)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x13)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x14)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x15)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x16)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x17)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x18)", "specialize prime_field_polynomial_convolution_right_append_equivalent (x19)", "apply prime_field_polynomial_convolution_right_append_equivalent", "exact hp", "exact hprefix", "exact hlast", "exact hR0", "exact hR1", "exact hY0align_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_left", "exact hY0align_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "exact hY0align_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left", "exact hY0align_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "exact hY0align_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "have haligned_equal : forall pfrep_power_append_step_aligned_induction pfrep_left_append_step_aligned_induction pfrep_right_append_step_aligned_induction. ((exists pfrep_position_append_step_aligned_inductionfirst. ((pfrep_position_append_step_aligned_inductionfirst+S (pfrep_power_append_step_aligned_induction)=(N+S U0)) /\\ ((((exists ff_h_pfp_append_step_aligned_inductionfirstentry. ff_h_pfp_append_step_aligned_inductionfirstentry + S (pfrep_left_append_step_aligned_induction) = S ((S (pfrep_position_append_step_aligned_inductionfirst)) * x19)) /\\ exists ff_q_pfp_append_step_aligned_inductionfirstentry. x18 = ff_q_pfp_append_step_aligned_inductionfirstentry * S ((S (pfrep_position_append_step_aligned_inductionfirst)) * x19) + (pfrep_left_append_step_aligned_induction)))))) \\/ (((exists pfrep_gap_append_step_aligned_inductionfirstoutside. pfrep_gap_append_step_aligned_inductionfirstoutside+(N+S U0)=(pfrep_power_append_step_aligned_induction)) /\\ (((pfrep_left_append_step_aligned_induction)=0))))) -> ((exists pfrep_position_append_step_aligned_inductionsecond. ((pfrep_position_append_step_aligned_inductionsecond+S (pfrep_power_append_step_aligned_induction)=(N+S V0)) /\\ ((((exists ff_h_pfp_append_step_aligned_inductionsecondentry. ff_h_pfp_append_step_aligned_inductionsecondentry + S (pfrep_right_append_step_aligned_induction) = S ((S (pfrep_position_append_step_aligned_inductionsecond)) * x9)) /\\ exists ff_q_pfp_append_step_aligned_inductionsecondentry. x8 = ff_q_pfp_append_step_aligned_inductionsecondentry * S ((S (pfrep_position_append_step_aligned_inductionsecond)) * x9) + (pfrep_right_append_step_aligned_induction)))))) \\/ (((exists pfrep_gap_append_step_aligned_inductionsecondoutside. pfrep_gap_append_step_aligned_inductionsecondoutside+(N+S V0)=(pfrep_power_append_step_aligned_induction)) /\\ (((pfrep_right_append_step_aligned_induction)=0))))) -> pfrep_left_append_step_aligned_induction=pfrep_right_append_step_aligned_induction", "specialize prime_field_polynomial_shift_scale_aligned_congruent (p)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (c)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (pb)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (pc)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (N)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (r0b)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (r0c)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (U0)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (s0b)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (s0c)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (V0)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x10)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x11)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x12)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x13)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x14)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x15)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x16)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x17)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x18)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x19)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x1)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x2)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x3)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x4)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x5)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x6)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x7)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x8)", "specialize prime_field_polynomial_shift_scale_aligned_congruent (x9)", "apply prime_field_polynomial_shift_scale_aligned_congruent", "exact hp", "exact hIH", "exact hY0align_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_left", "exact hY0align_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "exact hY0align_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left", "exact hY0align_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "exact hY0align_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "exact hYalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_left", "exact hYalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "exact hYalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left", "exact hYalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "exact hYalign_witness_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "specialize prime_field_polynomial_equivalent_transitive (r1b)", "specialize prime_field_polynomial_equivalent_transitive (r1c)", "specialize prime_field_polynomial_equivalent_transitive (U1)", "specialize prime_field_polynomial_equivalent_transitive (x18)", "specialize prime_field_polynomial_equivalent_transitive (x19)", "specialize prime_field_polynomial_equivalent_transitive (N+S U0)", "specialize prime_field_polynomial_equivalent_transitive (x8)", "specialize prime_field_polynomial_equivalent_transitive (x9)", "specialize prime_field_polynomial_equivalent_transitive (N+S V0)", "apply prime_field_polynomial_equivalent_transitive", "exact hR_append", "exact haligned_equal", "specialize prime_field_polynomial_equivalent_transitive (r1b)", "specialize prime_field_polynomial_equivalent_transitive (r1c)", "specialize prime_field_polynomial_equivalent_transitive (U1)", "specialize prime_field_polynomial_equivalent_transitive (x8)", "specialize prime_field_polynomial_equivalent_transitive (x9)", "specialize prime_field_polynomial_equivalent_transitive (N+S V0)", "specialize prime_field_polynomial_equivalent_transitive (s1b)", "specialize prime_field_polynomial_equivalent_transitive (s1c)", "specialize prime_field_polynomial_equivalent_transitive (V1)", "apply prime_field_polynomial_equivalent_transitive", "exact hR_total", "specialize prime_field_polynomial_equivalent_symmetric (s1b)", "specialize prime_field_polynomial_equivalent_symmetric (s1c)", "specialize prime_field_polynomial_equivalent_symmetric (V1)", "specialize prime_field_polynomial_equivalent_symmetric (x8)", "specialize prime_field_polynomial_equivalent_symmetric (x9)", "specialize prime_field_polynomial_equivalent_symmetric (N+S V0)", "apply prime_field_polynomial_equivalent_symmetric", "exact hS_total" ], "script_sha256": "7eb1d85cf7fb0c2cb5ecd5e43430590aa821d3a18d7755c9769fd54372c4e977", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_associativity_step_candidate.py", "sha256": "dd85dbd1bd87143715a4286724ac7c87f280a909dac6759f00a6cb7dff7c85f1" }, "statement": "forall p ab ac L bb bc M pb pc N cb cc J q0b q0c K0 r0b r0c U0 s0b s0c V0 c db dc q1b q1c K1 r1b r1c U1 s1b s1c V1. (~((p) = 1) /\\ forall pfa_factor_left_append_step_prime pfa_factor_right_append_step_prime. (p) = pfa_factor_left_append_step_prime * pfa_factor_right_append_step_prime -> pfa_factor_left_append_step_prime = 1 \\/ pfa_factor_right_append_step_prime = 1) -> (((forall fom_index_pfp_append_step_ABleft. (exists fom_gap_pfp_append_step_ABleft_index_bound. fom_gap_pfp_append_step_ABleft_index_bound + S (fom_index_pfp_append_step_ABleft) = L) -> exists fom_value_pfp_append_step_ABleft. ((((exists fom_beta_height_pfp_append_step_ABleft_entry. fom_beta_height_pfp_append_step_ABleft_entry + S (fom_value_pfp_append_step_ABleft) = S ((S (fom_index_pfp_append_step_ABleft)) * ac)) /\\ exists fom_beta_quotient_pfp_append_step_ABleft_entry. ab = fom_beta_quotient_pfp_append_step_ABleft_entry * S ((S (fom_index_pfp_append_step_ABleft)) * ac) + (fom_value_pfp_append_step_ABleft))) /\\ (exists fom_gap_pfp_append_step_ABleft_value_bound. fom_gap_pfp_append_step_ABleft_value_bound + S (fom_value_pfp_append_step_ABleft) = p))) /\\ (((forall fom_index_pfp_append_step_ABright. (exists fom_gap_pfp_append_step_ABright_index_bound. fom_gap_pfp_append_step_ABright_index_bound + S (fom_index_pfp_append_step_ABright) = M) -> exists fom_value_pfp_append_step_ABright. ((((exists fom_beta_height_pfp_append_step_ABright_entry. fom_beta_height_pfp_append_step_ABright_entry + S (fom_value_pfp_append_step_ABright) = S ((S (fom_index_pfp_append_step_ABright)) * bc)) /\\ exists fom_beta_quotient_pfp_append_step_ABright_entry. bb = fom_beta_quotient_pfp_append_step_ABright_entry * S ((S (fom_index_pfp_append_step_ABright)) * bc) + (fom_value_pfp_append_step_ABright))) /\\ (exists fom_gap_pfp_append_step_ABright_value_bound. fom_gap_pfp_append_step_ABright_value_bound + S (fom_value_pfp_append_step_ABright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_append_step_ABcoefficients. (exists pfa_gap_append_step_ABcoefficientsbound. pfa_gap_append_step_ABcoefficientsbound + S (pfc_index_append_step_ABcoefficients) = (N)) -> exists pfc_value_append_step_ABcoefficients. ((((exists ff_h_pfp_append_step_ABcoefficientsentry. ff_h_pfp_append_step_ABcoefficientsentry + S (pfc_value_append_step_ABcoefficients) = S ((S (pfc_index_append_step_ABcoefficients)) * pc)) /\\ exists ff_q_pfp_append_step_ABcoefficientsentry. pb = ff_q_pfp_append_step_ABcoefficientsentry * S ((S (pfc_index_append_step_ABcoefficients)) * pc) + (pfc_value_append_step_ABcoefficients))) /\\ ((exists pfc_terms_code_append_step_ABcoefficientscoefficient pfc_terms_scale_append_step_ABcoefficientscoefficient pfc_natural_sum_append_step_ABcoefficientscoefficient. ((forall pfc_index_append_step_ABcoefficientscoefficientdiagonal. (exists pfa_gap_append_step_ABcoefficientscoefficientdiagonalbound. pfa_gap_append_step_ABcoefficientscoefficientdiagonalbound + S (pfc_index_append_step_ABcoefficientscoefficientdiagonal) = (S (pfc_index_append_step_ABcoefficients))) -> exists pfc_value_append_step_ABcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_step_ABcoefficientscoefficientdiagonalentry. ff_h_pfp_append_step_ABcoefficientscoefficientdiagonalentry + S (pfc_value_append_step_ABcoefficientscoefficientdiagonal) = S ((S (pfc_index_append_step_ABcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_ABcoefficientscoefficient)) /\\ exists ff_q_pfp_append_step_ABcoefficientscoefficientdiagonalentry. pfc_terms_code_append_step_ABcoefficientscoefficient = ff_q_pfp_append_step_ABcoefficientscoefficientdiagonalentry * S ((S (pfc_index_append_step_ABcoefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_ABcoefficientscoefficient) + (pfc_value_append_step_ABcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_step_ABcoefficientscoefficientdiagonalterm pfc_left_append_step_ABcoefficientscoefficientdiagonalterm pfc_right_append_step_ABcoefficientscoefficientdiagonalterm. (((pfc_index_append_step_ABcoefficientscoefficientdiagonal)+pfc_complement_append_step_ABcoefficientscoefficientdiagonalterm=(pfc_index_append_step_ABcoefficients)) /\\ ((((((exists pfa_gap_append_step_ABcoefficientscoefficientdiagonaltermleftinside. pfa_gap_append_step_ABcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_step_ABcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_step_ABcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_step_ABcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_step_ABcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_step_ABcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_step_ABcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_step_ABcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_step_ABcoefficientscoefficientdiagonal)) * ac) + (pfc_left_append_step_ABcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_ABcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_step_ABcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_step_ABcoefficientscoefficientdiagonal)) /\\ (((pfc_left_append_step_ABcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_step_ABcoefficientscoefficientdiagonaltermrightinside. pfa_gap_append_step_ABcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_step_ABcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_append_step_ABcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_step_ABcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_step_ABcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_step_ABcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_append_step_ABcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_append_step_ABcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_step_ABcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_append_step_ABcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_ABcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_step_ABcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_append_step_ABcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_step_ABcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_step_ABcoefficientscoefficientdiagonal)=pfc_left_append_step_ABcoefficientscoefficientdiagonalterm*pfc_right_append_step_ABcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_step_ABcoefficientscoefficientsum fs_v_pfc_append_step_ABcoefficientscoefficientsum. ((((exists fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_start. fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_step_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_start. fs_u_pfc_append_step_ABcoefficientscoefficientsum = fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_step_ABcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_terminal. fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_step_ABcoefficientscoefficient) = S ((S (S (pfc_index_append_step_ABcoefficients))) * fs_v_pfc_append_step_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_terminal. fs_u_pfc_append_step_ABcoefficientscoefficientsum = fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_step_ABcoefficients))) * fs_v_pfc_append_step_ABcoefficientscoefficientsum) + (pfc_natural_sum_append_step_ABcoefficientscoefficient))) /\\ forall fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_step_ABcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_step_ABcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps = S (pfc_index_append_step_ABcoefficients)) -> exists fs_a_pfc_append_step_ABcoefficientscoefficientsum_body_steps fs_r_pfc_append_step_ABcoefficientscoefficientsum_body_steps fs_s_pfc_append_step_ABcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_step_ABcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_ABcoefficientscoefficient)) /\\ exists fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_step_ABcoefficientscoefficient = fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_ABcoefficientscoefficient) + (fs_a_pfc_append_step_ABcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_step_ABcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_step_ABcoefficientscoefficientsum = fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_ABcoefficientscoefficientsum) + (fs_r_pfc_append_step_ABcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_step_ABcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_step_ABcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_step_ABcoefficientscoefficientsum = fs_q_pfc_append_step_ABcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_step_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_ABcoefficientscoefficientsum) + (fs_s_pfc_append_step_ABcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_step_ABcoefficientscoefficientsum_body_steps = fs_r_pfc_append_step_ABcoefficientscoefficientsum_body_steps + fs_a_pfc_append_step_ABcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_step_ABcoefficientscoefficientresiduebound. pfa_gap_append_step_ABcoefficientscoefficientresiduebound + S (pfc_value_append_step_ABcoefficients) = (p)) /\\ ((exists pfa_offset_left_append_step_ABcoefficientscoefficientresiduecongruence pfa_offset_right_append_step_ABcoefficientscoefficientresiduecongruence. (pfc_natural_sum_append_step_ABcoefficientscoefficient) + (p) * pfa_offset_left_append_step_ABcoefficientscoefficientresiduecongruence = (pfc_value_append_step_ABcoefficients) + (p) * pfa_offset_right_append_step_ABcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_append_step_Q0left. (exists fom_gap_pfp_append_step_Q0left_index_bound. fom_gap_pfp_append_step_Q0left_index_bound + S (fom_index_pfp_append_step_Q0left) = M) -> exists fom_value_pfp_append_step_Q0left. ((((exists fom_beta_height_pfp_append_step_Q0left_entry. fom_beta_height_pfp_append_step_Q0left_entry + S (fom_value_pfp_append_step_Q0left) = S ((S (fom_index_pfp_append_step_Q0left)) * bc)) /\\ exists fom_beta_quotient_pfp_append_step_Q0left_entry. bb = fom_beta_quotient_pfp_append_step_Q0left_entry * S ((S (fom_index_pfp_append_step_Q0left)) * bc) + (fom_value_pfp_append_step_Q0left))) /\\ (exists fom_gap_pfp_append_step_Q0left_value_bound. fom_gap_pfp_append_step_Q0left_value_bound + S (fom_value_pfp_append_step_Q0left) = p))) /\\ (((forall fom_index_pfp_append_step_Q0right. (exists fom_gap_pfp_append_step_Q0right_index_bound. fom_gap_pfp_append_step_Q0right_index_bound + S (fom_index_pfp_append_step_Q0right) = J) -> exists fom_value_pfp_append_step_Q0right. ((((exists fom_beta_height_pfp_append_step_Q0right_entry. fom_beta_height_pfp_append_step_Q0right_entry + S (fom_value_pfp_append_step_Q0right) = S ((S (fom_index_pfp_append_step_Q0right)) * cc)) /\\ exists fom_beta_quotient_pfp_append_step_Q0right_entry. cb = fom_beta_quotient_pfp_append_step_Q0right_entry * S ((S (fom_index_pfp_append_step_Q0right)) * cc) + (fom_value_pfp_append_step_Q0right))) /\\ (exists fom_gap_pfp_append_step_Q0right_value_bound. fom_gap_pfp_append_step_Q0right_value_bound + S (fom_value_pfp_append_step_Q0right) = p))) /\\ (((((((M)=0 \\/ (J)=0) /\\ (((K0)=0)))) \\/ (((~((M)=0)) /\\ (((~((J)=0)) /\\ (((M)+(J)=S (K0)))))))) /\\ ((forall pfc_index_append_step_Q0coefficients. (exists pfa_gap_append_step_Q0coefficientsbound. pfa_gap_append_step_Q0coefficientsbound + S (pfc_index_append_step_Q0coefficients) = (K0)) -> exists pfc_value_append_step_Q0coefficients. ((((exists ff_h_pfp_append_step_Q0coefficientsentry. ff_h_pfp_append_step_Q0coefficientsentry + S (pfc_value_append_step_Q0coefficients) = S ((S (pfc_index_append_step_Q0coefficients)) * q0c)) /\\ exists ff_q_pfp_append_step_Q0coefficientsentry. q0b = ff_q_pfp_append_step_Q0coefficientsentry * S ((S (pfc_index_append_step_Q0coefficients)) * q0c) + (pfc_value_append_step_Q0coefficients))) /\\ ((exists pfc_terms_code_append_step_Q0coefficientscoefficient pfc_terms_scale_append_step_Q0coefficientscoefficient pfc_natural_sum_append_step_Q0coefficientscoefficient. ((forall pfc_index_append_step_Q0coefficientscoefficientdiagonal. (exists pfa_gap_append_step_Q0coefficientscoefficientdiagonalbound. pfa_gap_append_step_Q0coefficientscoefficientdiagonalbound + S (pfc_index_append_step_Q0coefficientscoefficientdiagonal) = (S (pfc_index_append_step_Q0coefficients))) -> exists pfc_value_append_step_Q0coefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_step_Q0coefficientscoefficientdiagonalentry. ff_h_pfp_append_step_Q0coefficientscoefficientdiagonalentry + S (pfc_value_append_step_Q0coefficientscoefficientdiagonal) = S ((S (pfc_index_append_step_Q0coefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_Q0coefficientscoefficient)) /\\ exists ff_q_pfp_append_step_Q0coefficientscoefficientdiagonalentry. pfc_terms_code_append_step_Q0coefficientscoefficient = ff_q_pfp_append_step_Q0coefficientscoefficientdiagonalentry * S ((S (pfc_index_append_step_Q0coefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_Q0coefficientscoefficient) + (pfc_value_append_step_Q0coefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_step_Q0coefficientscoefficientdiagonalterm pfc_left_append_step_Q0coefficientscoefficientdiagonalterm pfc_right_append_step_Q0coefficientscoefficientdiagonalterm. (((pfc_index_append_step_Q0coefficientscoefficientdiagonal)+pfc_complement_append_step_Q0coefficientscoefficientdiagonalterm=(pfc_index_append_step_Q0coefficients)) /\\ ((((((exists pfa_gap_append_step_Q0coefficientscoefficientdiagonaltermleftinside. pfa_gap_append_step_Q0coefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_step_Q0coefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_append_step_Q0coefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_step_Q0coefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_step_Q0coefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_step_Q0coefficientscoefficientdiagonal)) * bc)) /\\ exists ff_q_pfp_append_step_Q0coefficientscoefficientdiagonaltermleftentry. bb = ff_q_pfp_append_step_Q0coefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_step_Q0coefficientscoefficientdiagonal)) * bc) + (pfc_left_append_step_Q0coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_Q0coefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_step_Q0coefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_append_step_Q0coefficientscoefficientdiagonal)) /\\ (((pfc_left_append_step_Q0coefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_step_Q0coefficientscoefficientdiagonaltermrightinside. pfa_gap_append_step_Q0coefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_step_Q0coefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_append_step_Q0coefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_step_Q0coefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_step_Q0coefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_step_Q0coefficientscoefficientdiagonalterm)) * cc)) /\\ exists ff_q_pfp_append_step_Q0coefficientscoefficientdiagonaltermrightentry. cb = ff_q_pfp_append_step_Q0coefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_step_Q0coefficientscoefficientdiagonalterm)) * cc) + (pfc_right_append_step_Q0coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_Q0coefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_step_Q0coefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_append_step_Q0coefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_step_Q0coefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_step_Q0coefficientscoefficientdiagonal)=pfc_left_append_step_Q0coefficientscoefficientdiagonalterm*pfc_right_append_step_Q0coefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_step_Q0coefficientscoefficientsum fs_v_pfc_append_step_Q0coefficientscoefficientsum. ((((exists fs_h_pfc_append_step_Q0coefficientscoefficientsum_body_start. fs_h_pfc_append_step_Q0coefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_step_Q0coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_Q0coefficientscoefficientsum_body_start. fs_u_pfc_append_step_Q0coefficientscoefficientsum = fs_q_pfc_append_step_Q0coefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_step_Q0coefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_step_Q0coefficientscoefficientsum_body_terminal. fs_h_pfc_append_step_Q0coefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_step_Q0coefficientscoefficient) = S ((S (S (pfc_index_append_step_Q0coefficients))) * fs_v_pfc_append_step_Q0coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_Q0coefficientscoefficientsum_body_terminal. fs_u_pfc_append_step_Q0coefficientscoefficientsum = fs_q_pfc_append_step_Q0coefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_step_Q0coefficients))) * fs_v_pfc_append_step_Q0coefficientscoefficientsum) + (pfc_natural_sum_append_step_Q0coefficientscoefficient))) /\\ forall fs_i_pfc_append_step_Q0coefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_step_Q0coefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_step_Q0coefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_step_Q0coefficientscoefficientsum_body_steps = S (pfc_index_append_step_Q0coefficients)) -> exists fs_a_pfc_append_step_Q0coefficientscoefficientsum_body_steps fs_r_pfc_append_step_Q0coefficientscoefficientsum_body_steps fs_s_pfc_append_step_Q0coefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_step_Q0coefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_step_Q0coefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_step_Q0coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_Q0coefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_Q0coefficientscoefficient)) /\\ exists fs_q_pfc_append_step_Q0coefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_step_Q0coefficientscoefficient = fs_q_pfc_append_step_Q0coefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_step_Q0coefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_Q0coefficientscoefficient) + (fs_a_pfc_append_step_Q0coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_Q0coefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_step_Q0coefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_step_Q0coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_Q0coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_Q0coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_Q0coefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_step_Q0coefficientscoefficientsum = fs_q_pfc_append_step_Q0coefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_step_Q0coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_Q0coefficientscoefficientsum) + (fs_r_pfc_append_step_Q0coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_Q0coefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_step_Q0coefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_step_Q0coefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_step_Q0coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_Q0coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_Q0coefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_step_Q0coefficientscoefficientsum = fs_q_pfc_append_step_Q0coefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_step_Q0coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_Q0coefficientscoefficientsum) + (fs_s_pfc_append_step_Q0coefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_step_Q0coefficientscoefficientsum_body_steps = fs_r_pfc_append_step_Q0coefficientscoefficientsum_body_steps + fs_a_pfc_append_step_Q0coefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_step_Q0coefficientscoefficientresiduebound. pfa_gap_append_step_Q0coefficientscoefficientresiduebound + S (pfc_value_append_step_Q0coefficients) = (p)) /\\ ((exists pfa_offset_left_append_step_Q0coefficientscoefficientresiduecongruence pfa_offset_right_append_step_Q0coefficientscoefficientresiduecongruence. (pfc_natural_sum_append_step_Q0coefficientscoefficient) + (p) * pfa_offset_left_append_step_Q0coefficientscoefficientresiduecongruence = (pfc_value_append_step_Q0coefficients) + (p) * pfa_offset_right_append_step_Q0coefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_append_step_R0left. (exists fom_gap_pfp_append_step_R0left_index_bound. fom_gap_pfp_append_step_R0left_index_bound + S (fom_index_pfp_append_step_R0left) = N) -> exists fom_value_pfp_append_step_R0left. ((((exists fom_beta_height_pfp_append_step_R0left_entry. fom_beta_height_pfp_append_step_R0left_entry + S (fom_value_pfp_append_step_R0left) = S ((S (fom_index_pfp_append_step_R0left)) * pc)) /\\ exists fom_beta_quotient_pfp_append_step_R0left_entry. pb = fom_beta_quotient_pfp_append_step_R0left_entry * S ((S (fom_index_pfp_append_step_R0left)) * pc) + (fom_value_pfp_append_step_R0left))) /\\ (exists fom_gap_pfp_append_step_R0left_value_bound. fom_gap_pfp_append_step_R0left_value_bound + S (fom_value_pfp_append_step_R0left) = p))) /\\ (((forall fom_index_pfp_append_step_R0right. (exists fom_gap_pfp_append_step_R0right_index_bound. fom_gap_pfp_append_step_R0right_index_bound + S (fom_index_pfp_append_step_R0right) = J) -> exists fom_value_pfp_append_step_R0right. ((((exists fom_beta_height_pfp_append_step_R0right_entry. fom_beta_height_pfp_append_step_R0right_entry + S (fom_value_pfp_append_step_R0right) = S ((S (fom_index_pfp_append_step_R0right)) * cc)) /\\ exists fom_beta_quotient_pfp_append_step_R0right_entry. cb = fom_beta_quotient_pfp_append_step_R0right_entry * S ((S (fom_index_pfp_append_step_R0right)) * cc) + (fom_value_pfp_append_step_R0right))) /\\ (exists fom_gap_pfp_append_step_R0right_value_bound. fom_gap_pfp_append_step_R0right_value_bound + S (fom_value_pfp_append_step_R0right) = p))) /\\ (((((((N)=0 \\/ (J)=0) /\\ (((U0)=0)))) \\/ (((~((N)=0)) /\\ (((~((J)=0)) /\\ (((N)+(J)=S (U0)))))))) /\\ ((forall pfc_index_append_step_R0coefficients. (exists pfa_gap_append_step_R0coefficientsbound. pfa_gap_append_step_R0coefficientsbound + S (pfc_index_append_step_R0coefficients) = (U0)) -> exists pfc_value_append_step_R0coefficients. ((((exists ff_h_pfp_append_step_R0coefficientsentry. ff_h_pfp_append_step_R0coefficientsentry + S (pfc_value_append_step_R0coefficients) = S ((S (pfc_index_append_step_R0coefficients)) * r0c)) /\\ exists ff_q_pfp_append_step_R0coefficientsentry. r0b = ff_q_pfp_append_step_R0coefficientsentry * S ((S (pfc_index_append_step_R0coefficients)) * r0c) + (pfc_value_append_step_R0coefficients))) /\\ ((exists pfc_terms_code_append_step_R0coefficientscoefficient pfc_terms_scale_append_step_R0coefficientscoefficient pfc_natural_sum_append_step_R0coefficientscoefficient. ((forall pfc_index_append_step_R0coefficientscoefficientdiagonal. (exists pfa_gap_append_step_R0coefficientscoefficientdiagonalbound. pfa_gap_append_step_R0coefficientscoefficientdiagonalbound + S (pfc_index_append_step_R0coefficientscoefficientdiagonal) = (S (pfc_index_append_step_R0coefficients))) -> exists pfc_value_append_step_R0coefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_step_R0coefficientscoefficientdiagonalentry. ff_h_pfp_append_step_R0coefficientscoefficientdiagonalentry + S (pfc_value_append_step_R0coefficientscoefficientdiagonal) = S ((S (pfc_index_append_step_R0coefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_R0coefficientscoefficient)) /\\ exists ff_q_pfp_append_step_R0coefficientscoefficientdiagonalentry. pfc_terms_code_append_step_R0coefficientscoefficient = ff_q_pfp_append_step_R0coefficientscoefficientdiagonalentry * S ((S (pfc_index_append_step_R0coefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_R0coefficientscoefficient) + (pfc_value_append_step_R0coefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_step_R0coefficientscoefficientdiagonalterm pfc_left_append_step_R0coefficientscoefficientdiagonalterm pfc_right_append_step_R0coefficientscoefficientdiagonalterm. (((pfc_index_append_step_R0coefficientscoefficientdiagonal)+pfc_complement_append_step_R0coefficientscoefficientdiagonalterm=(pfc_index_append_step_R0coefficients)) /\\ ((((((exists pfa_gap_append_step_R0coefficientscoefficientdiagonaltermleftinside. pfa_gap_append_step_R0coefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_step_R0coefficientscoefficientdiagonal) = (N)) /\\ ((((exists ff_h_pfp_append_step_R0coefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_step_R0coefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_step_R0coefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_step_R0coefficientscoefficientdiagonal)) * pc)) /\\ exists ff_q_pfp_append_step_R0coefficientscoefficientdiagonaltermleftentry. pb = ff_q_pfp_append_step_R0coefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_step_R0coefficientscoefficientdiagonal)) * pc) + (pfc_left_append_step_R0coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_R0coefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_step_R0coefficientscoefficientdiagonaltermleftoutside+(N)=(pfc_index_append_step_R0coefficientscoefficientdiagonal)) /\\ (((pfc_left_append_step_R0coefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_step_R0coefficientscoefficientdiagonaltermrightinside. pfa_gap_append_step_R0coefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_step_R0coefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_append_step_R0coefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_step_R0coefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_step_R0coefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_step_R0coefficientscoefficientdiagonalterm)) * cc)) /\\ exists ff_q_pfp_append_step_R0coefficientscoefficientdiagonaltermrightentry. cb = ff_q_pfp_append_step_R0coefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_step_R0coefficientscoefficientdiagonalterm)) * cc) + (pfc_right_append_step_R0coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_R0coefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_step_R0coefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_append_step_R0coefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_step_R0coefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_step_R0coefficientscoefficientdiagonal)=pfc_left_append_step_R0coefficientscoefficientdiagonalterm*pfc_right_append_step_R0coefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_step_R0coefficientscoefficientsum fs_v_pfc_append_step_R0coefficientscoefficientsum. ((((exists fs_h_pfc_append_step_R0coefficientscoefficientsum_body_start. fs_h_pfc_append_step_R0coefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_step_R0coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_R0coefficientscoefficientsum_body_start. fs_u_pfc_append_step_R0coefficientscoefficientsum = fs_q_pfc_append_step_R0coefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_step_R0coefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_step_R0coefficientscoefficientsum_body_terminal. fs_h_pfc_append_step_R0coefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_step_R0coefficientscoefficient) = S ((S (S (pfc_index_append_step_R0coefficients))) * fs_v_pfc_append_step_R0coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_R0coefficientscoefficientsum_body_terminal. fs_u_pfc_append_step_R0coefficientscoefficientsum = fs_q_pfc_append_step_R0coefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_step_R0coefficients))) * fs_v_pfc_append_step_R0coefficientscoefficientsum) + (pfc_natural_sum_append_step_R0coefficientscoefficient))) /\\ forall fs_i_pfc_append_step_R0coefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_step_R0coefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_step_R0coefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_step_R0coefficientscoefficientsum_body_steps = S (pfc_index_append_step_R0coefficients)) -> exists fs_a_pfc_append_step_R0coefficientscoefficientsum_body_steps fs_r_pfc_append_step_R0coefficientscoefficientsum_body_steps fs_s_pfc_append_step_R0coefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_step_R0coefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_step_R0coefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_step_R0coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_R0coefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_R0coefficientscoefficient)) /\\ exists fs_q_pfc_append_step_R0coefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_step_R0coefficientscoefficient = fs_q_pfc_append_step_R0coefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_step_R0coefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_R0coefficientscoefficient) + (fs_a_pfc_append_step_R0coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_R0coefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_step_R0coefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_step_R0coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_R0coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_R0coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_R0coefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_step_R0coefficientscoefficientsum = fs_q_pfc_append_step_R0coefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_step_R0coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_R0coefficientscoefficientsum) + (fs_r_pfc_append_step_R0coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_R0coefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_step_R0coefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_step_R0coefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_step_R0coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_R0coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_R0coefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_step_R0coefficientscoefficientsum = fs_q_pfc_append_step_R0coefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_step_R0coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_R0coefficientscoefficientsum) + (fs_s_pfc_append_step_R0coefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_step_R0coefficientscoefficientsum_body_steps = fs_r_pfc_append_step_R0coefficientscoefficientsum_body_steps + fs_a_pfc_append_step_R0coefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_step_R0coefficientscoefficientresiduebound. pfa_gap_append_step_R0coefficientscoefficientresiduebound + S (pfc_value_append_step_R0coefficients) = (p)) /\\ ((exists pfa_offset_left_append_step_R0coefficientscoefficientresiduecongruence pfa_offset_right_append_step_R0coefficientscoefficientresiduecongruence. (pfc_natural_sum_append_step_R0coefficientscoefficient) + (p) * pfa_offset_left_append_step_R0coefficientscoefficientresiduecongruence = (pfc_value_append_step_R0coefficients) + (p) * pfa_offset_right_append_step_R0coefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_append_step_S0left. (exists fom_gap_pfp_append_step_S0left_index_bound. fom_gap_pfp_append_step_S0left_index_bound + S (fom_index_pfp_append_step_S0left) = L) -> exists fom_value_pfp_append_step_S0left. ((((exists fom_beta_height_pfp_append_step_S0left_entry. fom_beta_height_pfp_append_step_S0left_entry + S (fom_value_pfp_append_step_S0left) = S ((S (fom_index_pfp_append_step_S0left)) * ac)) /\\ exists fom_beta_quotient_pfp_append_step_S0left_entry. ab = fom_beta_quotient_pfp_append_step_S0left_entry * S ((S (fom_index_pfp_append_step_S0left)) * ac) + (fom_value_pfp_append_step_S0left))) /\\ (exists fom_gap_pfp_append_step_S0left_value_bound. fom_gap_pfp_append_step_S0left_value_bound + S (fom_value_pfp_append_step_S0left) = p))) /\\ (((forall fom_index_pfp_append_step_S0right. (exists fom_gap_pfp_append_step_S0right_index_bound. fom_gap_pfp_append_step_S0right_index_bound + S (fom_index_pfp_append_step_S0right) = K0) -> exists fom_value_pfp_append_step_S0right. ((((exists fom_beta_height_pfp_append_step_S0right_entry. fom_beta_height_pfp_append_step_S0right_entry + S (fom_value_pfp_append_step_S0right) = S ((S (fom_index_pfp_append_step_S0right)) * q0c)) /\\ exists fom_beta_quotient_pfp_append_step_S0right_entry. q0b = fom_beta_quotient_pfp_append_step_S0right_entry * S ((S (fom_index_pfp_append_step_S0right)) * q0c) + (fom_value_pfp_append_step_S0right))) /\\ (exists fom_gap_pfp_append_step_S0right_value_bound. fom_gap_pfp_append_step_S0right_value_bound + S (fom_value_pfp_append_step_S0right) = p))) /\\ (((((((L)=0 \\/ (K0)=0) /\\ (((V0)=0)))) \\/ (((~((L)=0)) /\\ (((~((K0)=0)) /\\ (((L)+(K0)=S (V0)))))))) /\\ ((forall pfc_index_append_step_S0coefficients. (exists pfa_gap_append_step_S0coefficientsbound. pfa_gap_append_step_S0coefficientsbound + S (pfc_index_append_step_S0coefficients) = (V0)) -> exists pfc_value_append_step_S0coefficients. ((((exists ff_h_pfp_append_step_S0coefficientsentry. ff_h_pfp_append_step_S0coefficientsentry + S (pfc_value_append_step_S0coefficients) = S ((S (pfc_index_append_step_S0coefficients)) * s0c)) /\\ exists ff_q_pfp_append_step_S0coefficientsentry. s0b = ff_q_pfp_append_step_S0coefficientsentry * S ((S (pfc_index_append_step_S0coefficients)) * s0c) + (pfc_value_append_step_S0coefficients))) /\\ ((exists pfc_terms_code_append_step_S0coefficientscoefficient pfc_terms_scale_append_step_S0coefficientscoefficient pfc_natural_sum_append_step_S0coefficientscoefficient. ((forall pfc_index_append_step_S0coefficientscoefficientdiagonal. (exists pfa_gap_append_step_S0coefficientscoefficientdiagonalbound. pfa_gap_append_step_S0coefficientscoefficientdiagonalbound + S (pfc_index_append_step_S0coefficientscoefficientdiagonal) = (S (pfc_index_append_step_S0coefficients))) -> exists pfc_value_append_step_S0coefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_step_S0coefficientscoefficientdiagonalentry. ff_h_pfp_append_step_S0coefficientscoefficientdiagonalentry + S (pfc_value_append_step_S0coefficientscoefficientdiagonal) = S ((S (pfc_index_append_step_S0coefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_S0coefficientscoefficient)) /\\ exists ff_q_pfp_append_step_S0coefficientscoefficientdiagonalentry. pfc_terms_code_append_step_S0coefficientscoefficient = ff_q_pfp_append_step_S0coefficientscoefficientdiagonalentry * S ((S (pfc_index_append_step_S0coefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_S0coefficientscoefficient) + (pfc_value_append_step_S0coefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_step_S0coefficientscoefficientdiagonalterm pfc_left_append_step_S0coefficientscoefficientdiagonalterm pfc_right_append_step_S0coefficientscoefficientdiagonalterm. (((pfc_index_append_step_S0coefficientscoefficientdiagonal)+pfc_complement_append_step_S0coefficientscoefficientdiagonalterm=(pfc_index_append_step_S0coefficients)) /\\ ((((((exists pfa_gap_append_step_S0coefficientscoefficientdiagonaltermleftinside. pfa_gap_append_step_S0coefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_step_S0coefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_step_S0coefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_step_S0coefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_step_S0coefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_step_S0coefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_step_S0coefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_step_S0coefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_step_S0coefficientscoefficientdiagonal)) * ac) + (pfc_left_append_step_S0coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_S0coefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_step_S0coefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_step_S0coefficientscoefficientdiagonal)) /\\ (((pfc_left_append_step_S0coefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_step_S0coefficientscoefficientdiagonaltermrightinside. pfa_gap_append_step_S0coefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_step_S0coefficientscoefficientdiagonalterm) = (K0)) /\\ ((((exists ff_h_pfp_append_step_S0coefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_step_S0coefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_step_S0coefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_step_S0coefficientscoefficientdiagonalterm)) * q0c)) /\\ exists ff_q_pfp_append_step_S0coefficientscoefficientdiagonaltermrightentry. q0b = ff_q_pfp_append_step_S0coefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_step_S0coefficientscoefficientdiagonalterm)) * q0c) + (pfc_right_append_step_S0coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_S0coefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_step_S0coefficientscoefficientdiagonaltermrightoutside+(K0)=(pfc_complement_append_step_S0coefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_step_S0coefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_step_S0coefficientscoefficientdiagonal)=pfc_left_append_step_S0coefficientscoefficientdiagonalterm*pfc_right_append_step_S0coefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_step_S0coefficientscoefficientsum fs_v_pfc_append_step_S0coefficientscoefficientsum. ((((exists fs_h_pfc_append_step_S0coefficientscoefficientsum_body_start. fs_h_pfc_append_step_S0coefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_step_S0coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_S0coefficientscoefficientsum_body_start. fs_u_pfc_append_step_S0coefficientscoefficientsum = fs_q_pfc_append_step_S0coefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_step_S0coefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_step_S0coefficientscoefficientsum_body_terminal. fs_h_pfc_append_step_S0coefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_step_S0coefficientscoefficient) = S ((S (S (pfc_index_append_step_S0coefficients))) * fs_v_pfc_append_step_S0coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_S0coefficientscoefficientsum_body_terminal. fs_u_pfc_append_step_S0coefficientscoefficientsum = fs_q_pfc_append_step_S0coefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_step_S0coefficients))) * fs_v_pfc_append_step_S0coefficientscoefficientsum) + (pfc_natural_sum_append_step_S0coefficientscoefficient))) /\\ forall fs_i_pfc_append_step_S0coefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_step_S0coefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_step_S0coefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_step_S0coefficientscoefficientsum_body_steps = S (pfc_index_append_step_S0coefficients)) -> exists fs_a_pfc_append_step_S0coefficientscoefficientsum_body_steps fs_r_pfc_append_step_S0coefficientscoefficientsum_body_steps fs_s_pfc_append_step_S0coefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_step_S0coefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_step_S0coefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_step_S0coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_S0coefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_S0coefficientscoefficient)) /\\ exists fs_q_pfc_append_step_S0coefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_step_S0coefficientscoefficient = fs_q_pfc_append_step_S0coefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_step_S0coefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_S0coefficientscoefficient) + (fs_a_pfc_append_step_S0coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_S0coefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_step_S0coefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_step_S0coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_S0coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_S0coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_S0coefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_step_S0coefficientscoefficientsum = fs_q_pfc_append_step_S0coefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_step_S0coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_S0coefficientscoefficientsum) + (fs_r_pfc_append_step_S0coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_S0coefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_step_S0coefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_step_S0coefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_step_S0coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_S0coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_S0coefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_step_S0coefficientscoefficientsum = fs_q_pfc_append_step_S0coefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_step_S0coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_S0coefficientscoefficientsum) + (fs_s_pfc_append_step_S0coefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_step_S0coefficientscoefficientsum_body_steps = fs_r_pfc_append_step_S0coefficientscoefficientsum_body_steps + fs_a_pfc_append_step_S0coefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_step_S0coefficientscoefficientresiduebound. pfa_gap_append_step_S0coefficientscoefficientresiduebound + S (pfc_value_append_step_S0coefficients) = (p)) /\\ ((exists pfa_offset_left_append_step_S0coefficientscoefficientresiduecongruence pfa_offset_right_append_step_S0coefficientscoefficientresiduecongruence. (pfc_natural_sum_append_step_S0coefficientscoefficient) + (p) * pfa_offset_left_append_step_S0coefficientscoefficientresiduecongruence = (pfc_value_append_step_S0coefficients) + (p) * pfa_offset_right_append_step_S0coefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_append_step_actual_induction_hypothesis pfrep_left_append_step_actual_induction_hypothesis pfrep_right_append_step_actual_induction_hypothesis. ((exists pfrep_position_append_step_actual_induction_hypothesisfirst. ((pfrep_position_append_step_actual_induction_hypothesisfirst+S (pfrep_power_append_step_actual_induction_hypothesis)=(U0)) /\\ ((((exists ff_h_pfp_append_step_actual_induction_hypothesisfirstentry. ff_h_pfp_append_step_actual_induction_hypothesisfirstentry + S (pfrep_left_append_step_actual_induction_hypothesis) = S ((S (pfrep_position_append_step_actual_induction_hypothesisfirst)) * r0c)) /\\ exists ff_q_pfp_append_step_actual_induction_hypothesisfirstentry. r0b = ff_q_pfp_append_step_actual_induction_hypothesisfirstentry * S ((S (pfrep_position_append_step_actual_induction_hypothesisfirst)) * r0c) + (pfrep_left_append_step_actual_induction_hypothesis)))))) \\/ (((exists pfrep_gap_append_step_actual_induction_hypothesisfirstoutside. pfrep_gap_append_step_actual_induction_hypothesisfirstoutside+(U0)=(pfrep_power_append_step_actual_induction_hypothesis)) /\\ (((pfrep_left_append_step_actual_induction_hypothesis)=0))))) -> ((exists pfrep_position_append_step_actual_induction_hypothesissecond. ((pfrep_position_append_step_actual_induction_hypothesissecond+S (pfrep_power_append_step_actual_induction_hypothesis)=(V0)) /\\ ((((exists ff_h_pfp_append_step_actual_induction_hypothesissecondentry. ff_h_pfp_append_step_actual_induction_hypothesissecondentry + S (pfrep_right_append_step_actual_induction_hypothesis) = S ((S (pfrep_position_append_step_actual_induction_hypothesissecond)) * s0c)) /\\ exists ff_q_pfp_append_step_actual_induction_hypothesissecondentry. s0b = ff_q_pfp_append_step_actual_induction_hypothesissecondentry * S ((S (pfrep_position_append_step_actual_induction_hypothesissecond)) * s0c) + (pfrep_right_append_step_actual_induction_hypothesis)))))) \\/ (((exists pfrep_gap_append_step_actual_induction_hypothesissecondoutside. pfrep_gap_append_step_actual_induction_hypothesissecondoutside+(V0)=(pfrep_power_append_step_actual_induction_hypothesis)) /\\ (((pfrep_right_append_step_actual_induction_hypothesis)=0))))) -> pfrep_left_append_step_actual_induction_hypothesis=pfrep_right_append_step_actual_induction_hypothesis) -> (forall mdr_i_pfp_append_step_prefix mdr_a_pfp_append_step_prefix. (exists mdr_gap_pfp_append_step_prefixb. mdr_gap_pfp_append_step_prefixb + S (mdr_i_pfp_append_step_prefix) = (J)) -> (((exists ff_h_mdr_pfp_append_step_prefixo. ff_h_mdr_pfp_append_step_prefixo + S (mdr_a_pfp_append_step_prefix) = S ((S (mdr_i_pfp_append_step_prefix)) * cc)) /\\ exists ff_q_mdr_pfp_append_step_prefixo. cb = ff_q_mdr_pfp_append_step_prefixo * S ((S (mdr_i_pfp_append_step_prefix)) * cc) + (mdr_a_pfp_append_step_prefix))) -> (((exists ff_h_mdr_pfp_append_step_prefixn. ff_h_mdr_pfp_append_step_prefixn + S (mdr_a_pfp_append_step_prefix) = S ((S (mdr_i_pfp_append_step_prefix)) * dc)) /\\ exists ff_q_mdr_pfp_append_step_prefixn. db = ff_q_mdr_pfp_append_step_prefixn * S ((S (mdr_i_pfp_append_step_prefix)) * dc) + (mdr_a_pfp_append_step_prefix)))) -> (((exists ff_h_pfp_append_step_last. ff_h_pfp_append_step_last + S (c) = S ((S (J)) * dc)) /\\ exists ff_q_pfp_append_step_last. db = ff_q_pfp_append_step_last * S ((S (J)) * dc) + (c))) -> (((forall fom_index_pfp_append_step_Q1left. (exists fom_gap_pfp_append_step_Q1left_index_bound. fom_gap_pfp_append_step_Q1left_index_bound + S (fom_index_pfp_append_step_Q1left) = M) -> exists fom_value_pfp_append_step_Q1left. ((((exists fom_beta_height_pfp_append_step_Q1left_entry. fom_beta_height_pfp_append_step_Q1left_entry + S (fom_value_pfp_append_step_Q1left) = S ((S (fom_index_pfp_append_step_Q1left)) * bc)) /\\ exists fom_beta_quotient_pfp_append_step_Q1left_entry. bb = fom_beta_quotient_pfp_append_step_Q1left_entry * S ((S (fom_index_pfp_append_step_Q1left)) * bc) + (fom_value_pfp_append_step_Q1left))) /\\ (exists fom_gap_pfp_append_step_Q1left_value_bound. fom_gap_pfp_append_step_Q1left_value_bound + S (fom_value_pfp_append_step_Q1left) = p))) /\\ (((forall fom_index_pfp_append_step_Q1right. (exists fom_gap_pfp_append_step_Q1right_index_bound. fom_gap_pfp_append_step_Q1right_index_bound + S (fom_index_pfp_append_step_Q1right) = S J) -> exists fom_value_pfp_append_step_Q1right. ((((exists fom_beta_height_pfp_append_step_Q1right_entry. fom_beta_height_pfp_append_step_Q1right_entry + S (fom_value_pfp_append_step_Q1right) = S ((S (fom_index_pfp_append_step_Q1right)) * dc)) /\\ exists fom_beta_quotient_pfp_append_step_Q1right_entry. db = fom_beta_quotient_pfp_append_step_Q1right_entry * S ((S (fom_index_pfp_append_step_Q1right)) * dc) + (fom_value_pfp_append_step_Q1right))) /\\ (exists fom_gap_pfp_append_step_Q1right_value_bound. fom_gap_pfp_append_step_Q1right_value_bound + S (fom_value_pfp_append_step_Q1right) = p))) /\\ (((((((M)=0 \\/ (S J)=0) /\\ (((K1)=0)))) \\/ (((~((M)=0)) /\\ (((~((S J)=0)) /\\ (((M)+(S J)=S (K1)))))))) /\\ ((forall pfc_index_append_step_Q1coefficients. (exists pfa_gap_append_step_Q1coefficientsbound. pfa_gap_append_step_Q1coefficientsbound + S (pfc_index_append_step_Q1coefficients) = (K1)) -> exists pfc_value_append_step_Q1coefficients. ((((exists ff_h_pfp_append_step_Q1coefficientsentry. ff_h_pfp_append_step_Q1coefficientsentry + S (pfc_value_append_step_Q1coefficients) = S ((S (pfc_index_append_step_Q1coefficients)) * q1c)) /\\ exists ff_q_pfp_append_step_Q1coefficientsentry. q1b = ff_q_pfp_append_step_Q1coefficientsentry * S ((S (pfc_index_append_step_Q1coefficients)) * q1c) + (pfc_value_append_step_Q1coefficients))) /\\ ((exists pfc_terms_code_append_step_Q1coefficientscoefficient pfc_terms_scale_append_step_Q1coefficientscoefficient pfc_natural_sum_append_step_Q1coefficientscoefficient. ((forall pfc_index_append_step_Q1coefficientscoefficientdiagonal. (exists pfa_gap_append_step_Q1coefficientscoefficientdiagonalbound. pfa_gap_append_step_Q1coefficientscoefficientdiagonalbound + S (pfc_index_append_step_Q1coefficientscoefficientdiagonal) = (S (pfc_index_append_step_Q1coefficients))) -> exists pfc_value_append_step_Q1coefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_step_Q1coefficientscoefficientdiagonalentry. ff_h_pfp_append_step_Q1coefficientscoefficientdiagonalentry + S (pfc_value_append_step_Q1coefficientscoefficientdiagonal) = S ((S (pfc_index_append_step_Q1coefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_Q1coefficientscoefficient)) /\\ exists ff_q_pfp_append_step_Q1coefficientscoefficientdiagonalentry. pfc_terms_code_append_step_Q1coefficientscoefficient = ff_q_pfp_append_step_Q1coefficientscoefficientdiagonalentry * S ((S (pfc_index_append_step_Q1coefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_Q1coefficientscoefficient) + (pfc_value_append_step_Q1coefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_step_Q1coefficientscoefficientdiagonalterm pfc_left_append_step_Q1coefficientscoefficientdiagonalterm pfc_right_append_step_Q1coefficientscoefficientdiagonalterm. (((pfc_index_append_step_Q1coefficientscoefficientdiagonal)+pfc_complement_append_step_Q1coefficientscoefficientdiagonalterm=(pfc_index_append_step_Q1coefficients)) /\\ ((((((exists pfa_gap_append_step_Q1coefficientscoefficientdiagonaltermleftinside. pfa_gap_append_step_Q1coefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_step_Q1coefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_append_step_Q1coefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_step_Q1coefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_step_Q1coefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_step_Q1coefficientscoefficientdiagonal)) * bc)) /\\ exists ff_q_pfp_append_step_Q1coefficientscoefficientdiagonaltermleftentry. bb = ff_q_pfp_append_step_Q1coefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_step_Q1coefficientscoefficientdiagonal)) * bc) + (pfc_left_append_step_Q1coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_Q1coefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_step_Q1coefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_append_step_Q1coefficientscoefficientdiagonal)) /\\ (((pfc_left_append_step_Q1coefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_step_Q1coefficientscoefficientdiagonaltermrightinside. pfa_gap_append_step_Q1coefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_step_Q1coefficientscoefficientdiagonalterm) = (S J)) /\\ ((((exists ff_h_pfp_append_step_Q1coefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_step_Q1coefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_step_Q1coefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_step_Q1coefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_append_step_Q1coefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_append_step_Q1coefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_step_Q1coefficientscoefficientdiagonalterm)) * dc) + (pfc_right_append_step_Q1coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_Q1coefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_step_Q1coefficientscoefficientdiagonaltermrightoutside+(S J)=(pfc_complement_append_step_Q1coefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_step_Q1coefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_step_Q1coefficientscoefficientdiagonal)=pfc_left_append_step_Q1coefficientscoefficientdiagonalterm*pfc_right_append_step_Q1coefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_step_Q1coefficientscoefficientsum fs_v_pfc_append_step_Q1coefficientscoefficientsum. ((((exists fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_start. fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_step_Q1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_start. fs_u_pfc_append_step_Q1coefficientscoefficientsum = fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_step_Q1coefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_terminal. fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_step_Q1coefficientscoefficient) = S ((S (S (pfc_index_append_step_Q1coefficients))) * fs_v_pfc_append_step_Q1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_terminal. fs_u_pfc_append_step_Q1coefficientscoefficientsum = fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_step_Q1coefficients))) * fs_v_pfc_append_step_Q1coefficientscoefficientsum) + (pfc_natural_sum_append_step_Q1coefficientscoefficient))) /\\ forall fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_step_Q1coefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_step_Q1coefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps = S (pfc_index_append_step_Q1coefficients)) -> exists fs_a_pfc_append_step_Q1coefficientscoefficientsum_body_steps fs_r_pfc_append_step_Q1coefficientscoefficientsum_body_steps fs_s_pfc_append_step_Q1coefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_step_Q1coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_Q1coefficientscoefficient)) /\\ exists fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_step_Q1coefficientscoefficient = fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_Q1coefficientscoefficient) + (fs_a_pfc_append_step_Q1coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_step_Q1coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_Q1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_step_Q1coefficientscoefficientsum = fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_Q1coefficientscoefficientsum) + (fs_r_pfc_append_step_Q1coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_step_Q1coefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_step_Q1coefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_Q1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_step_Q1coefficientscoefficientsum = fs_q_pfc_append_step_Q1coefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_step_Q1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_Q1coefficientscoefficientsum) + (fs_s_pfc_append_step_Q1coefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_step_Q1coefficientscoefficientsum_body_steps = fs_r_pfc_append_step_Q1coefficientscoefficientsum_body_steps + fs_a_pfc_append_step_Q1coefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_step_Q1coefficientscoefficientresiduebound. pfa_gap_append_step_Q1coefficientscoefficientresiduebound + S (pfc_value_append_step_Q1coefficients) = (p)) /\\ ((exists pfa_offset_left_append_step_Q1coefficientscoefficientresiduecongruence pfa_offset_right_append_step_Q1coefficientscoefficientresiduecongruence. (pfc_natural_sum_append_step_Q1coefficientscoefficient) + (p) * pfa_offset_left_append_step_Q1coefficientscoefficientresiduecongruence = (pfc_value_append_step_Q1coefficients) + (p) * pfa_offset_right_append_step_Q1coefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_append_step_R1left. (exists fom_gap_pfp_append_step_R1left_index_bound. fom_gap_pfp_append_step_R1left_index_bound + S (fom_index_pfp_append_step_R1left) = N) -> exists fom_value_pfp_append_step_R1left. ((((exists fom_beta_height_pfp_append_step_R1left_entry. fom_beta_height_pfp_append_step_R1left_entry + S (fom_value_pfp_append_step_R1left) = S ((S (fom_index_pfp_append_step_R1left)) * pc)) /\\ exists fom_beta_quotient_pfp_append_step_R1left_entry. pb = fom_beta_quotient_pfp_append_step_R1left_entry * S ((S (fom_index_pfp_append_step_R1left)) * pc) + (fom_value_pfp_append_step_R1left))) /\\ (exists fom_gap_pfp_append_step_R1left_value_bound. fom_gap_pfp_append_step_R1left_value_bound + S (fom_value_pfp_append_step_R1left) = p))) /\\ (((forall fom_index_pfp_append_step_R1right. (exists fom_gap_pfp_append_step_R1right_index_bound. fom_gap_pfp_append_step_R1right_index_bound + S (fom_index_pfp_append_step_R1right) = S J) -> exists fom_value_pfp_append_step_R1right. ((((exists fom_beta_height_pfp_append_step_R1right_entry. fom_beta_height_pfp_append_step_R1right_entry + S (fom_value_pfp_append_step_R1right) = S ((S (fom_index_pfp_append_step_R1right)) * dc)) /\\ exists fom_beta_quotient_pfp_append_step_R1right_entry. db = fom_beta_quotient_pfp_append_step_R1right_entry * S ((S (fom_index_pfp_append_step_R1right)) * dc) + (fom_value_pfp_append_step_R1right))) /\\ (exists fom_gap_pfp_append_step_R1right_value_bound. fom_gap_pfp_append_step_R1right_value_bound + S (fom_value_pfp_append_step_R1right) = p))) /\\ (((((((N)=0 \\/ (S J)=0) /\\ (((U1)=0)))) \\/ (((~((N)=0)) /\\ (((~((S J)=0)) /\\ (((N)+(S J)=S (U1)))))))) /\\ ((forall pfc_index_append_step_R1coefficients. (exists pfa_gap_append_step_R1coefficientsbound. pfa_gap_append_step_R1coefficientsbound + S (pfc_index_append_step_R1coefficients) = (U1)) -> exists pfc_value_append_step_R1coefficients. ((((exists ff_h_pfp_append_step_R1coefficientsentry. ff_h_pfp_append_step_R1coefficientsentry + S (pfc_value_append_step_R1coefficients) = S ((S (pfc_index_append_step_R1coefficients)) * r1c)) /\\ exists ff_q_pfp_append_step_R1coefficientsentry. r1b = ff_q_pfp_append_step_R1coefficientsentry * S ((S (pfc_index_append_step_R1coefficients)) * r1c) + (pfc_value_append_step_R1coefficients))) /\\ ((exists pfc_terms_code_append_step_R1coefficientscoefficient pfc_terms_scale_append_step_R1coefficientscoefficient pfc_natural_sum_append_step_R1coefficientscoefficient. ((forall pfc_index_append_step_R1coefficientscoefficientdiagonal. (exists pfa_gap_append_step_R1coefficientscoefficientdiagonalbound. pfa_gap_append_step_R1coefficientscoefficientdiagonalbound + S (pfc_index_append_step_R1coefficientscoefficientdiagonal) = (S (pfc_index_append_step_R1coefficients))) -> exists pfc_value_append_step_R1coefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_step_R1coefficientscoefficientdiagonalentry. ff_h_pfp_append_step_R1coefficientscoefficientdiagonalentry + S (pfc_value_append_step_R1coefficientscoefficientdiagonal) = S ((S (pfc_index_append_step_R1coefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_R1coefficientscoefficient)) /\\ exists ff_q_pfp_append_step_R1coefficientscoefficientdiagonalentry. pfc_terms_code_append_step_R1coefficientscoefficient = ff_q_pfp_append_step_R1coefficientscoefficientdiagonalentry * S ((S (pfc_index_append_step_R1coefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_R1coefficientscoefficient) + (pfc_value_append_step_R1coefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_step_R1coefficientscoefficientdiagonalterm pfc_left_append_step_R1coefficientscoefficientdiagonalterm pfc_right_append_step_R1coefficientscoefficientdiagonalterm. (((pfc_index_append_step_R1coefficientscoefficientdiagonal)+pfc_complement_append_step_R1coefficientscoefficientdiagonalterm=(pfc_index_append_step_R1coefficients)) /\\ ((((((exists pfa_gap_append_step_R1coefficientscoefficientdiagonaltermleftinside. pfa_gap_append_step_R1coefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_step_R1coefficientscoefficientdiagonal) = (N)) /\\ ((((exists ff_h_pfp_append_step_R1coefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_step_R1coefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_step_R1coefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_step_R1coefficientscoefficientdiagonal)) * pc)) /\\ exists ff_q_pfp_append_step_R1coefficientscoefficientdiagonaltermleftentry. pb = ff_q_pfp_append_step_R1coefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_step_R1coefficientscoefficientdiagonal)) * pc) + (pfc_left_append_step_R1coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_R1coefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_step_R1coefficientscoefficientdiagonaltermleftoutside+(N)=(pfc_index_append_step_R1coefficientscoefficientdiagonal)) /\\ (((pfc_left_append_step_R1coefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_step_R1coefficientscoefficientdiagonaltermrightinside. pfa_gap_append_step_R1coefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_step_R1coefficientscoefficientdiagonalterm) = (S J)) /\\ ((((exists ff_h_pfp_append_step_R1coefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_step_R1coefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_step_R1coefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_step_R1coefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_append_step_R1coefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_append_step_R1coefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_step_R1coefficientscoefficientdiagonalterm)) * dc) + (pfc_right_append_step_R1coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_R1coefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_step_R1coefficientscoefficientdiagonaltermrightoutside+(S J)=(pfc_complement_append_step_R1coefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_step_R1coefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_step_R1coefficientscoefficientdiagonal)=pfc_left_append_step_R1coefficientscoefficientdiagonalterm*pfc_right_append_step_R1coefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_step_R1coefficientscoefficientsum fs_v_pfc_append_step_R1coefficientscoefficientsum. ((((exists fs_h_pfc_append_step_R1coefficientscoefficientsum_body_start. fs_h_pfc_append_step_R1coefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_step_R1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_R1coefficientscoefficientsum_body_start. fs_u_pfc_append_step_R1coefficientscoefficientsum = fs_q_pfc_append_step_R1coefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_step_R1coefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_step_R1coefficientscoefficientsum_body_terminal. fs_h_pfc_append_step_R1coefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_step_R1coefficientscoefficient) = S ((S (S (pfc_index_append_step_R1coefficients))) * fs_v_pfc_append_step_R1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_R1coefficientscoefficientsum_body_terminal. fs_u_pfc_append_step_R1coefficientscoefficientsum = fs_q_pfc_append_step_R1coefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_step_R1coefficients))) * fs_v_pfc_append_step_R1coefficientscoefficientsum) + (pfc_natural_sum_append_step_R1coefficientscoefficient))) /\\ forall fs_i_pfc_append_step_R1coefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_step_R1coefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_step_R1coefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_step_R1coefficientscoefficientsum_body_steps = S (pfc_index_append_step_R1coefficients)) -> exists fs_a_pfc_append_step_R1coefficientscoefficientsum_body_steps fs_r_pfc_append_step_R1coefficientscoefficientsum_body_steps fs_s_pfc_append_step_R1coefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_step_R1coefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_step_R1coefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_step_R1coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_R1coefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_R1coefficientscoefficient)) /\\ exists fs_q_pfc_append_step_R1coefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_step_R1coefficientscoefficient = fs_q_pfc_append_step_R1coefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_step_R1coefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_R1coefficientscoefficient) + (fs_a_pfc_append_step_R1coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_R1coefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_step_R1coefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_step_R1coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_R1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_R1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_R1coefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_step_R1coefficientscoefficientsum = fs_q_pfc_append_step_R1coefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_step_R1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_R1coefficientscoefficientsum) + (fs_r_pfc_append_step_R1coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_R1coefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_step_R1coefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_step_R1coefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_step_R1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_R1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_R1coefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_step_R1coefficientscoefficientsum = fs_q_pfc_append_step_R1coefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_step_R1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_R1coefficientscoefficientsum) + (fs_s_pfc_append_step_R1coefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_step_R1coefficientscoefficientsum_body_steps = fs_r_pfc_append_step_R1coefficientscoefficientsum_body_steps + fs_a_pfc_append_step_R1coefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_step_R1coefficientscoefficientresiduebound. pfa_gap_append_step_R1coefficientscoefficientresiduebound + S (pfc_value_append_step_R1coefficients) = (p)) /\\ ((exists pfa_offset_left_append_step_R1coefficientscoefficientresiduecongruence pfa_offset_right_append_step_R1coefficientscoefficientresiduecongruence. (pfc_natural_sum_append_step_R1coefficientscoefficient) + (p) * pfa_offset_left_append_step_R1coefficientscoefficientresiduecongruence = (pfc_value_append_step_R1coefficients) + (p) * pfa_offset_right_append_step_R1coefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_append_step_S1left. (exists fom_gap_pfp_append_step_S1left_index_bound. fom_gap_pfp_append_step_S1left_index_bound + S (fom_index_pfp_append_step_S1left) = L) -> exists fom_value_pfp_append_step_S1left. ((((exists fom_beta_height_pfp_append_step_S1left_entry. fom_beta_height_pfp_append_step_S1left_entry + S (fom_value_pfp_append_step_S1left) = S ((S (fom_index_pfp_append_step_S1left)) * ac)) /\\ exists fom_beta_quotient_pfp_append_step_S1left_entry. ab = fom_beta_quotient_pfp_append_step_S1left_entry * S ((S (fom_index_pfp_append_step_S1left)) * ac) + (fom_value_pfp_append_step_S1left))) /\\ (exists fom_gap_pfp_append_step_S1left_value_bound. fom_gap_pfp_append_step_S1left_value_bound + S (fom_value_pfp_append_step_S1left) = p))) /\\ (((forall fom_index_pfp_append_step_S1right. (exists fom_gap_pfp_append_step_S1right_index_bound. fom_gap_pfp_append_step_S1right_index_bound + S (fom_index_pfp_append_step_S1right) = K1) -> exists fom_value_pfp_append_step_S1right. ((((exists fom_beta_height_pfp_append_step_S1right_entry. fom_beta_height_pfp_append_step_S1right_entry + S (fom_value_pfp_append_step_S1right) = S ((S (fom_index_pfp_append_step_S1right)) * q1c)) /\\ exists fom_beta_quotient_pfp_append_step_S1right_entry. q1b = fom_beta_quotient_pfp_append_step_S1right_entry * S ((S (fom_index_pfp_append_step_S1right)) * q1c) + (fom_value_pfp_append_step_S1right))) /\\ (exists fom_gap_pfp_append_step_S1right_value_bound. fom_gap_pfp_append_step_S1right_value_bound + S (fom_value_pfp_append_step_S1right) = p))) /\\ (((((((L)=0 \\/ (K1)=0) /\\ (((V1)=0)))) \\/ (((~((L)=0)) /\\ (((~((K1)=0)) /\\ (((L)+(K1)=S (V1)))))))) /\\ ((forall pfc_index_append_step_S1coefficients. (exists pfa_gap_append_step_S1coefficientsbound. pfa_gap_append_step_S1coefficientsbound + S (pfc_index_append_step_S1coefficients) = (V1)) -> exists pfc_value_append_step_S1coefficients. ((((exists ff_h_pfp_append_step_S1coefficientsentry. ff_h_pfp_append_step_S1coefficientsentry + S (pfc_value_append_step_S1coefficients) = S ((S (pfc_index_append_step_S1coefficients)) * s1c)) /\\ exists ff_q_pfp_append_step_S1coefficientsentry. s1b = ff_q_pfp_append_step_S1coefficientsentry * S ((S (pfc_index_append_step_S1coefficients)) * s1c) + (pfc_value_append_step_S1coefficients))) /\\ ((exists pfc_terms_code_append_step_S1coefficientscoefficient pfc_terms_scale_append_step_S1coefficientscoefficient pfc_natural_sum_append_step_S1coefficientscoefficient. ((forall pfc_index_append_step_S1coefficientscoefficientdiagonal. (exists pfa_gap_append_step_S1coefficientscoefficientdiagonalbound. pfa_gap_append_step_S1coefficientscoefficientdiagonalbound + S (pfc_index_append_step_S1coefficientscoefficientdiagonal) = (S (pfc_index_append_step_S1coefficients))) -> exists pfc_value_append_step_S1coefficientscoefficientdiagonal. ((((exists ff_h_pfp_append_step_S1coefficientscoefficientdiagonalentry. ff_h_pfp_append_step_S1coefficientscoefficientdiagonalentry + S (pfc_value_append_step_S1coefficientscoefficientdiagonal) = S ((S (pfc_index_append_step_S1coefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_S1coefficientscoefficient)) /\\ exists ff_q_pfp_append_step_S1coefficientscoefficientdiagonalentry. pfc_terms_code_append_step_S1coefficientscoefficient = ff_q_pfp_append_step_S1coefficientscoefficientdiagonalentry * S ((S (pfc_index_append_step_S1coefficientscoefficientdiagonal)) * pfc_terms_scale_append_step_S1coefficientscoefficient) + (pfc_value_append_step_S1coefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_append_step_S1coefficientscoefficientdiagonalterm pfc_left_append_step_S1coefficientscoefficientdiagonalterm pfc_right_append_step_S1coefficientscoefficientdiagonalterm. (((pfc_index_append_step_S1coefficientscoefficientdiagonal)+pfc_complement_append_step_S1coefficientscoefficientdiagonalterm=(pfc_index_append_step_S1coefficients)) /\\ ((((((exists pfa_gap_append_step_S1coefficientscoefficientdiagonaltermleftinside. pfa_gap_append_step_S1coefficientscoefficientdiagonaltermleftinside + S (pfc_index_append_step_S1coefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_append_step_S1coefficientscoefficientdiagonaltermleftentry. ff_h_pfp_append_step_S1coefficientscoefficientdiagonaltermleftentry + S (pfc_left_append_step_S1coefficientscoefficientdiagonalterm) = S ((S (pfc_index_append_step_S1coefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_append_step_S1coefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_append_step_S1coefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_append_step_S1coefficientscoefficientdiagonal)) * ac) + (pfc_left_append_step_S1coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_S1coefficientscoefficientdiagonaltermleftoutside. pfc_gap_append_step_S1coefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_append_step_S1coefficientscoefficientdiagonal)) /\\ (((pfc_left_append_step_S1coefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_append_step_S1coefficientscoefficientdiagonaltermrightinside. pfa_gap_append_step_S1coefficientscoefficientdiagonaltermrightinside + S (pfc_complement_append_step_S1coefficientscoefficientdiagonalterm) = (K1)) /\\ ((((exists ff_h_pfp_append_step_S1coefficientscoefficientdiagonaltermrightentry. ff_h_pfp_append_step_S1coefficientscoefficientdiagonaltermrightentry + S (pfc_right_append_step_S1coefficientscoefficientdiagonalterm) = S ((S (pfc_complement_append_step_S1coefficientscoefficientdiagonalterm)) * q1c)) /\\ exists ff_q_pfp_append_step_S1coefficientscoefficientdiagonaltermrightentry. q1b = ff_q_pfp_append_step_S1coefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_append_step_S1coefficientscoefficientdiagonalterm)) * q1c) + (pfc_right_append_step_S1coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_append_step_S1coefficientscoefficientdiagonaltermrightoutside. pfc_gap_append_step_S1coefficientscoefficientdiagonaltermrightoutside+(K1)=(pfc_complement_append_step_S1coefficientscoefficientdiagonalterm)) /\\ (((pfc_right_append_step_S1coefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_append_step_S1coefficientscoefficientdiagonal)=pfc_left_append_step_S1coefficientscoefficientdiagonalterm*pfc_right_append_step_S1coefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_append_step_S1coefficientscoefficientsum fs_v_pfc_append_step_S1coefficientscoefficientsum. ((((exists fs_h_pfc_append_step_S1coefficientscoefficientsum_body_start. fs_h_pfc_append_step_S1coefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_append_step_S1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_S1coefficientscoefficientsum_body_start. fs_u_pfc_append_step_S1coefficientscoefficientsum = fs_q_pfc_append_step_S1coefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_append_step_S1coefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_append_step_S1coefficientscoefficientsum_body_terminal. fs_h_pfc_append_step_S1coefficientscoefficientsum_body_terminal + S (pfc_natural_sum_append_step_S1coefficientscoefficient) = S ((S (S (pfc_index_append_step_S1coefficients))) * fs_v_pfc_append_step_S1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_S1coefficientscoefficientsum_body_terminal. fs_u_pfc_append_step_S1coefficientscoefficientsum = fs_q_pfc_append_step_S1coefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_append_step_S1coefficients))) * fs_v_pfc_append_step_S1coefficientscoefficientsum) + (pfc_natural_sum_append_step_S1coefficientscoefficient))) /\\ forall fs_i_pfc_append_step_S1coefficientscoefficientsum_body_steps. (exists fs_lt_pfc_append_step_S1coefficientscoefficientsum_body_steps_bound. fs_lt_pfc_append_step_S1coefficientscoefficientsum_body_steps_bound + S fs_i_pfc_append_step_S1coefficientscoefficientsum_body_steps = S (pfc_index_append_step_S1coefficients)) -> exists fs_a_pfc_append_step_S1coefficientscoefficientsum_body_steps fs_r_pfc_append_step_S1coefficientscoefficientsum_body_steps fs_s_pfc_append_step_S1coefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_append_step_S1coefficientscoefficientsum_body_steps_summand. fs_h_pfc_append_step_S1coefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_append_step_S1coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_S1coefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_S1coefficientscoefficient)) /\\ exists fs_q_pfc_append_step_S1coefficientscoefficientsum_body_steps_summand. pfc_terms_code_append_step_S1coefficientscoefficient = fs_q_pfc_append_step_S1coefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_append_step_S1coefficientscoefficientsum_body_steps)) * pfc_terms_scale_append_step_S1coefficientscoefficient) + (fs_a_pfc_append_step_S1coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_S1coefficientscoefficientsum_body_steps_partial. fs_h_pfc_append_step_S1coefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_append_step_S1coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_append_step_S1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_S1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_S1coefficientscoefficientsum_body_steps_partial. fs_u_pfc_append_step_S1coefficientscoefficientsum = fs_q_pfc_append_step_S1coefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_append_step_S1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_S1coefficientscoefficientsum) + (fs_r_pfc_append_step_S1coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_append_step_S1coefficientscoefficientsum_body_steps_successor. fs_h_pfc_append_step_S1coefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_append_step_S1coefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_append_step_S1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_S1coefficientscoefficientsum)) /\\ exists fs_q_pfc_append_step_S1coefficientscoefficientsum_body_steps_successor. fs_u_pfc_append_step_S1coefficientscoefficientsum = fs_q_pfc_append_step_S1coefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_append_step_S1coefficientscoefficientsum_body_steps)) * fs_v_pfc_append_step_S1coefficientscoefficientsum) + (fs_s_pfc_append_step_S1coefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_append_step_S1coefficientscoefficientsum_body_steps = fs_r_pfc_append_step_S1coefficientscoefficientsum_body_steps + fs_a_pfc_append_step_S1coefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_append_step_S1coefficientscoefficientresiduebound. pfa_gap_append_step_S1coefficientscoefficientresiduebound + S (pfc_value_append_step_S1coefficients) = (p)) /\\ ((exists pfa_offset_left_append_step_S1coefficientscoefficientresiduecongruence pfa_offset_right_append_step_S1coefficientscoefficientresiduecongruence. (pfc_natural_sum_append_step_S1coefficientscoefficient) + (p) * pfa_offset_left_append_step_S1coefficientscoefficientresiduecongruence = (pfc_value_append_step_S1coefficients) + (p) * pfa_offset_right_append_step_S1coefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_append_step_result pfrep_left_append_step_result pfrep_right_append_step_result. ((exists pfrep_position_append_step_resultfirst. ((pfrep_position_append_step_resultfirst+S (pfrep_power_append_step_result)=(U1)) /\\ ((((exists ff_h_pfp_append_step_resultfirstentry. ff_h_pfp_append_step_resultfirstentry + S (pfrep_left_append_step_result) = S ((S (pfrep_position_append_step_resultfirst)) * r1c)) /\\ exists ff_q_pfp_append_step_resultfirstentry. r1b = ff_q_pfp_append_step_resultfirstentry * S ((S (pfrep_position_append_step_resultfirst)) * r1c) + (pfrep_left_append_step_result)))))) \\/ (((exists pfrep_gap_append_step_resultfirstoutside. pfrep_gap_append_step_resultfirstoutside+(U1)=(pfrep_power_append_step_result)) /\\ (((pfrep_left_append_step_result)=0))))) -> ((exists pfrep_position_append_step_resultsecond. ((pfrep_position_append_step_resultsecond+S (pfrep_power_append_step_result)=(V1)) /\\ ((((exists ff_h_pfp_append_step_resultsecondentry. ff_h_pfp_append_step_resultsecondentry + S (pfrep_right_append_step_result) = S ((S (pfrep_position_append_step_resultsecond)) * s1c)) /\\ exists ff_q_pfp_append_step_resultsecondentry. s1b = ff_q_pfp_append_step_resultsecondentry * S ((S (pfrep_position_append_step_resultsecond)) * s1c) + (pfrep_right_append_step_result)))))) \\/ (((exists pfrep_gap_append_step_resultsecondoutside. pfrep_gap_append_step_resultsecondoutside+(V1)=(pfrep_power_append_step_result)) /\\ (((pfrep_right_append_step_result)=0))))) -> pfrep_left_append_step_result=pfrep_right_append_step_result)", "statement_sha256": "f2d971514a76d10991d99754cdb81e84b6096f8fbdc24e7dfd0b6a580dbddaa0", "summary": "An actual formal-coefficient associativity hypothesis for one rightmost prefix extends through one genuine appended coefficient. Every old and new product is an actual proper-length convolution; the proof derives the canonical appended coefficient bound, constructs three real shift/scale/pad/add alignments and a real intermediate product, and concludes only the next formal equivalence. Empty factors are retained, no successor product-length identity is assumed, and the induction step alone is not full associativity.", "summary_sha256": "7871e5f404ee4a4ad63cd2681d97e345882a95c4621ffb612d4ba1fa812e4075" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "54deb87938c6b0565d58979e4127ded41169aa6a5eef67cb344799042df2cd4f", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 306, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "8d276a028764cd08e6eaebbf25bb4e21fcd5076a610d356a77d52ba6603ebe4c", "test_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85" }, "body_checked": true, "body_receipt": { "command_count": 104, "dependency_count": 5, "dne_command_count": 0, "name": "prime_field_polynomial_nested_empty_right_equivalent", "proof_depth": 49, "proof_edges": 121, "proof_nodes": 122, "proof_objects": 122, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "beta_repeat_empty", "prime_field_polynomial_convolution_zero_right", "prime_field_polynomial_equivalent_transitive", "prime_field_polynomial_zero_prefix_equivalent_empty", "prime_field_polynomial_equivalent_symmetric" ], "dependencies_sha256": "1f6ca63cfdbe1f74ba6f79d703b02a5727a2830e1fc840adfa032a1c9a4218f7", "empty_context_closure": { "body_proof_depth": 49, "body_proof_nodes": 122, "bundle_campaign": "polynomial-gcd-bezout", "bundle_dependency_edge_count": 1578, "bundle_node_count": 493, "bundle_node_id": 306, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "bundle_root_id": 492, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "81ce0ee6c24350459ae386643bbb4cca7b311efc17dc3ada524d7d0e58544741", "status": "checked" }, "enrollment_index": 4127, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "8d276a028764cd08e6eaebbf25bb4e21fcd5076a610d356a77d52ba6603ebe4c", "kind": "alpha_v34_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_associativity_induction_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85", "kind": "alpha_v34_frontier_executable_audit", "path": "peano-lab/py/tests/test_campaign_research_v34_closure.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "kind": "alpha_v34_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-field-polynomial-gcd-bezout-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "kind": "alpha_v34_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=306]" }, { "document_sha256": "81943a545e7b9f667ad758c24de14bb47280e8cf3fbb3ca714efec55fb096408", "kind": "alpha_v34_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "e9ea542ef59f166d7a99f28311da979c7178c54c50f8bf25b383503b53adcfca", "membership": "alpha_only", "name": "prime_field_polynomial_nested_empty_right_equivalent", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro M", "intro pb", "intro pc", "intro N", "intro cb", "intro cc", "intro qb", "intro qc", "intro K", "intro rb", "intro rc", "intro U", "intro sb", "intro sc", "intro V", "intro hp", "intro hQ", "intro hR", "intro hS", "have hempty : forall pfp_repeat_index_associativity_empty_prefix. (exists pfa_gap_associativity_empty_prefixindex. pfa_gap_associativity_empty_prefixindex + S (pfp_repeat_index_associativity_empty_prefix) = (0)) -> (((exists ff_h_pfp_associativity_empty_prefixentry. ff_h_pfp_associativity_empty_prefixentry + S (0) = S ((S (pfp_repeat_index_associativity_empty_prefix)) * cc)) /\\ exists ff_q_pfp_associativity_empty_prefixentry. cb = ff_q_pfp_associativity_empty_prefixentry * S ((S (pfp_repeat_index_associativity_empty_prefix)) * cc) + (0)))", "specialize beta_repeat_empty (cb)", "specialize beta_repeat_empty (cc)", "specialize beta_repeat_empty (0)", "specialize beta_repeat_empty (0)", "apply beta_repeat_empty", "refl", "have hQzero : forall pfp_repeat_index_associativity_hQzero. (exists pfa_gap_associativity_hQzeroindex. pfa_gap_associativity_hQzeroindex + S (pfp_repeat_index_associativity_hQzero) = (K)) -> (((exists ff_h_pfp_associativity_hQzeroentry. ff_h_pfp_associativity_hQzeroentry + S (0) = S ((S (pfp_repeat_index_associativity_hQzero)) * qc)) /\\ exists ff_q_pfp_associativity_hQzeroentry. qb = ff_q_pfp_associativity_hQzeroentry * S ((S (pfp_repeat_index_associativity_hQzero)) * qc) + (0)))", "specialize prime_field_polynomial_convolution_zero_right (p)", "specialize prime_field_polynomial_convolution_zero_right (bb)", "specialize prime_field_polynomial_convolution_zero_right (bc)", "specialize prime_field_polynomial_convolution_zero_right (M)", "specialize prime_field_polynomial_convolution_zero_right (cb)", "specialize prime_field_polynomial_convolution_zero_right (cc)", "specialize prime_field_polynomial_convolution_zero_right (0)", "specialize prime_field_polynomial_convolution_zero_right (qb)", "specialize prime_field_polynomial_convolution_zero_right (qc)", "specialize prime_field_polynomial_convolution_zero_right (K)", "apply prime_field_polynomial_convolution_zero_right", "exact hp", "exact hempty", "exact hQ", "have hRzero : forall pfp_repeat_index_associativity_hRzero. (exists pfa_gap_associativity_hRzeroindex. pfa_gap_associativity_hRzeroindex + S (pfp_repeat_index_associativity_hRzero) = (U)) -> (((exists ff_h_pfp_associativity_hRzeroentry. ff_h_pfp_associativity_hRzeroentry + S (0) = S ((S (pfp_repeat_index_associativity_hRzero)) * rc)) /\\ exists ff_q_pfp_associativity_hRzeroentry. rb = ff_q_pfp_associativity_hRzeroentry * S ((S (pfp_repeat_index_associativity_hRzero)) * rc) + (0)))", "specialize prime_field_polynomial_convolution_zero_right (p)", "specialize prime_field_polynomial_convolution_zero_right (pb)", "specialize prime_field_polynomial_convolution_zero_right (pc)", "specialize prime_field_polynomial_convolution_zero_right (N)", "specialize prime_field_polynomial_convolution_zero_right (cb)", "specialize prime_field_polynomial_convolution_zero_right (cc)", "specialize prime_field_polynomial_convolution_zero_right (0)", "specialize prime_field_polynomial_convolution_zero_right (rb)", "specialize prime_field_polynomial_convolution_zero_right (rc)", "specialize prime_field_polynomial_convolution_zero_right (U)", "apply prime_field_polynomial_convolution_zero_right", "exact hp", "exact hempty", "exact hR", "have hSzero : forall pfp_repeat_index_associativity_hSzero. (exists pfa_gap_associativity_hSzeroindex. pfa_gap_associativity_hSzeroindex + S (pfp_repeat_index_associativity_hSzero) = (V)) -> (((exists ff_h_pfp_associativity_hSzeroentry. ff_h_pfp_associativity_hSzeroentry + S (0) = S ((S (pfp_repeat_index_associativity_hSzero)) * sc)) /\\ exists ff_q_pfp_associativity_hSzeroentry. sb = ff_q_pfp_associativity_hSzeroentry * S ((S (pfp_repeat_index_associativity_hSzero)) * sc) + (0)))", "specialize prime_field_polynomial_convolution_zero_right (p)", "specialize prime_field_polynomial_convolution_zero_right (ab)", "specialize prime_field_polynomial_convolution_zero_right (ac)", "specialize prime_field_polynomial_convolution_zero_right (L)", "specialize prime_field_polynomial_convolution_zero_right (qb)", "specialize prime_field_polynomial_convolution_zero_right (qc)", "specialize prime_field_polynomial_convolution_zero_right (K)", "specialize prime_field_polynomial_convolution_zero_right (sb)", "specialize prime_field_polynomial_convolution_zero_right (sc)", "specialize prime_field_polynomial_convolution_zero_right (V)", "apply prime_field_polynomial_convolution_zero_right", "exact hp", "exact hQzero", "exact hS", "specialize prime_field_polynomial_equivalent_transitive (rb)", "specialize prime_field_polynomial_equivalent_transitive (rc)", "specialize prime_field_polynomial_equivalent_transitive (U)", "specialize prime_field_polynomial_equivalent_transitive (0)", "specialize prime_field_polynomial_equivalent_transitive (0)", "specialize prime_field_polynomial_equivalent_transitive (0)", "specialize prime_field_polynomial_equivalent_transitive (sb)", "specialize prime_field_polynomial_equivalent_transitive (sc)", "specialize prime_field_polynomial_equivalent_transitive (V)", "apply prime_field_polynomial_equivalent_transitive", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (rb)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (rc)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (U)", "apply prime_field_polynomial_zero_prefix_equivalent_empty", "exact hRzero", "specialize prime_field_polynomial_equivalent_symmetric (sb)", "specialize prime_field_polynomial_equivalent_symmetric (sc)", "specialize prime_field_polynomial_equivalent_symmetric (V)", "specialize prime_field_polynomial_equivalent_symmetric (0)", "specialize prime_field_polynomial_equivalent_symmetric (0)", "specialize prime_field_polynomial_equivalent_symmetric (0)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (sb)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (sc)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (V)", "apply prime_field_polynomial_zero_prefix_equivalent_empty", "exact hSzero" ], "script_sha256": "d270c78961e3fb6b07a2406519437da7e2696fc6ebecd5f7ab298f7171a1b692", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_associativity_induction_candidate.py", "sha256": "8d276a028764cd08e6eaebbf25bb4e21fcd5076a610d356a77d52ba6603ebe4c" }, "statement": "forall p ab ac L bb bc M pb pc N cb cc qb qc K rb rc U sb sc V. (~(p=0)) -> (((forall fom_index_pfp_associativity_empty_Qleft. (exists fom_gap_pfp_associativity_empty_Qleft_index_bound. fom_gap_pfp_associativity_empty_Qleft_index_bound + S (fom_index_pfp_associativity_empty_Qleft) = M) -> exists fom_value_pfp_associativity_empty_Qleft. ((((exists fom_beta_height_pfp_associativity_empty_Qleft_entry. fom_beta_height_pfp_associativity_empty_Qleft_entry + S (fom_value_pfp_associativity_empty_Qleft) = S ((S (fom_index_pfp_associativity_empty_Qleft)) * bc)) /\\ exists fom_beta_quotient_pfp_associativity_empty_Qleft_entry. bb = fom_beta_quotient_pfp_associativity_empty_Qleft_entry * S ((S (fom_index_pfp_associativity_empty_Qleft)) * bc) + (fom_value_pfp_associativity_empty_Qleft))) /\\ (exists fom_gap_pfp_associativity_empty_Qleft_value_bound. fom_gap_pfp_associativity_empty_Qleft_value_bound + S (fom_value_pfp_associativity_empty_Qleft) = p))) /\\ (((forall fom_index_pfp_associativity_empty_Qright. (exists fom_gap_pfp_associativity_empty_Qright_index_bound. fom_gap_pfp_associativity_empty_Qright_index_bound + S (fom_index_pfp_associativity_empty_Qright) = 0) -> exists fom_value_pfp_associativity_empty_Qright. ((((exists fom_beta_height_pfp_associativity_empty_Qright_entry. fom_beta_height_pfp_associativity_empty_Qright_entry + S (fom_value_pfp_associativity_empty_Qright) = S ((S (fom_index_pfp_associativity_empty_Qright)) * cc)) /\\ exists fom_beta_quotient_pfp_associativity_empty_Qright_entry. cb = fom_beta_quotient_pfp_associativity_empty_Qright_entry * S ((S (fom_index_pfp_associativity_empty_Qright)) * cc) + (fom_value_pfp_associativity_empty_Qright))) /\\ (exists fom_gap_pfp_associativity_empty_Qright_value_bound. fom_gap_pfp_associativity_empty_Qright_value_bound + S (fom_value_pfp_associativity_empty_Qright) = p))) /\\ (((((((M)=0 \\/ (0)=0) /\\ (((K)=0)))) \\/ (((~((M)=0)) /\\ (((~((0)=0)) /\\ (((M)+(0)=S (K)))))))) /\\ ((forall pfc_index_associativity_empty_Qcoefficients. (exists pfa_gap_associativity_empty_Qcoefficientsbound. pfa_gap_associativity_empty_Qcoefficientsbound + S (pfc_index_associativity_empty_Qcoefficients) = (K)) -> exists pfc_value_associativity_empty_Qcoefficients. ((((exists ff_h_pfp_associativity_empty_Qcoefficientsentry. ff_h_pfp_associativity_empty_Qcoefficientsentry + S (pfc_value_associativity_empty_Qcoefficients) = S ((S (pfc_index_associativity_empty_Qcoefficients)) * qc)) /\\ exists ff_q_pfp_associativity_empty_Qcoefficientsentry. qb = ff_q_pfp_associativity_empty_Qcoefficientsentry * S ((S (pfc_index_associativity_empty_Qcoefficients)) * qc) + (pfc_value_associativity_empty_Qcoefficients))) /\\ ((exists pfc_terms_code_associativity_empty_Qcoefficientscoefficient pfc_terms_scale_associativity_empty_Qcoefficientscoefficient pfc_natural_sum_associativity_empty_Qcoefficientscoefficient. ((forall pfc_index_associativity_empty_Qcoefficientscoefficientdiagonal. (exists pfa_gap_associativity_empty_Qcoefficientscoefficientdiagonalbound. pfa_gap_associativity_empty_Qcoefficientscoefficientdiagonalbound + S (pfc_index_associativity_empty_Qcoefficientscoefficientdiagonal) = (S (pfc_index_associativity_empty_Qcoefficients))) -> exists pfc_value_associativity_empty_Qcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associativity_empty_Qcoefficientscoefficientdiagonalentry. ff_h_pfp_associativity_empty_Qcoefficientscoefficientdiagonalentry + S (pfc_value_associativity_empty_Qcoefficientscoefficientdiagonal) = S ((S (pfc_index_associativity_empty_Qcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_empty_Qcoefficientscoefficient)) /\\ exists ff_q_pfp_associativity_empty_Qcoefficientscoefficientdiagonalentry. pfc_terms_code_associativity_empty_Qcoefficientscoefficient = ff_q_pfp_associativity_empty_Qcoefficientscoefficientdiagonalentry * S ((S (pfc_index_associativity_empty_Qcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_empty_Qcoefficientscoefficient) + (pfc_value_associativity_empty_Qcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associativity_empty_Qcoefficientscoefficientdiagonalterm pfc_left_associativity_empty_Qcoefficientscoefficientdiagonalterm pfc_right_associativity_empty_Qcoefficientscoefficientdiagonalterm. (((pfc_index_associativity_empty_Qcoefficientscoefficientdiagonal)+pfc_complement_associativity_empty_Qcoefficientscoefficientdiagonalterm=(pfc_index_associativity_empty_Qcoefficients)) /\\ ((((((exists pfa_gap_associativity_empty_Qcoefficientscoefficientdiagonaltermleftinside. pfa_gap_associativity_empty_Qcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associativity_empty_Qcoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_associativity_empty_Qcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associativity_empty_Qcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associativity_empty_Qcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associativity_empty_Qcoefficientscoefficientdiagonal)) * bc)) /\\ exists ff_q_pfp_associativity_empty_Qcoefficientscoefficientdiagonaltermleftentry. bb = ff_q_pfp_associativity_empty_Qcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associativity_empty_Qcoefficientscoefficientdiagonal)) * bc) + (pfc_left_associativity_empty_Qcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_empty_Qcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associativity_empty_Qcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_associativity_empty_Qcoefficientscoefficientdiagonal)) /\\ (((pfc_left_associativity_empty_Qcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associativity_empty_Qcoefficientscoefficientdiagonaltermrightinside. pfa_gap_associativity_empty_Qcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associativity_empty_Qcoefficientscoefficientdiagonalterm) = (0)) /\\ ((((exists ff_h_pfp_associativity_empty_Qcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associativity_empty_Qcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associativity_empty_Qcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associativity_empty_Qcoefficientscoefficientdiagonalterm)) * cc)) /\\ exists ff_q_pfp_associativity_empty_Qcoefficientscoefficientdiagonaltermrightentry. cb = ff_q_pfp_associativity_empty_Qcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associativity_empty_Qcoefficientscoefficientdiagonalterm)) * cc) + (pfc_right_associativity_empty_Qcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_empty_Qcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associativity_empty_Qcoefficientscoefficientdiagonaltermrightoutside+(0)=(pfc_complement_associativity_empty_Qcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associativity_empty_Qcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associativity_empty_Qcoefficientscoefficientdiagonal)=pfc_left_associativity_empty_Qcoefficientscoefficientdiagonalterm*pfc_right_associativity_empty_Qcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associativity_empty_Qcoefficientscoefficientsum fs_v_pfc_associativity_empty_Qcoefficientscoefficientsum. ((((exists fs_h_pfc_associativity_empty_Qcoefficientscoefficientsum_body_start. fs_h_pfc_associativity_empty_Qcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associativity_empty_Qcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_empty_Qcoefficientscoefficientsum_body_start. fs_u_pfc_associativity_empty_Qcoefficientscoefficientsum = fs_q_pfc_associativity_empty_Qcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associativity_empty_Qcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associativity_empty_Qcoefficientscoefficientsum_body_terminal. fs_h_pfc_associativity_empty_Qcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associativity_empty_Qcoefficientscoefficient) = S ((S (S (pfc_index_associativity_empty_Qcoefficients))) * fs_v_pfc_associativity_empty_Qcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_empty_Qcoefficientscoefficientsum_body_terminal. fs_u_pfc_associativity_empty_Qcoefficientscoefficientsum = fs_q_pfc_associativity_empty_Qcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associativity_empty_Qcoefficients))) * fs_v_pfc_associativity_empty_Qcoefficientscoefficientsum) + (pfc_natural_sum_associativity_empty_Qcoefficientscoefficient))) /\\ forall fs_i_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps = S (pfc_index_associativity_empty_Qcoefficients)) -> exists fs_a_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps fs_r_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps fs_s_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_empty_Qcoefficientscoefficient)) /\\ exists fs_q_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associativity_empty_Qcoefficientscoefficient = fs_q_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_empty_Qcoefficientscoefficient) + (fs_a_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_empty_Qcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associativity_empty_Qcoefficientscoefficientsum = fs_q_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_empty_Qcoefficientscoefficientsum) + (fs_r_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_empty_Qcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associativity_empty_Qcoefficientscoefficientsum = fs_q_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_empty_Qcoefficientscoefficientsum) + (fs_s_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps = fs_r_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps + fs_a_pfc_associativity_empty_Qcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associativity_empty_Qcoefficientscoefficientresiduebound. pfa_gap_associativity_empty_Qcoefficientscoefficientresiduebound + S (pfc_value_associativity_empty_Qcoefficients) = (p)) /\\ ((exists pfa_offset_left_associativity_empty_Qcoefficientscoefficientresiduecongruence pfa_offset_right_associativity_empty_Qcoefficientscoefficientresiduecongruence. (pfc_natural_sum_associativity_empty_Qcoefficientscoefficient) + (p) * pfa_offset_left_associativity_empty_Qcoefficientscoefficientresiduecongruence = (pfc_value_associativity_empty_Qcoefficients) + (p) * pfa_offset_right_associativity_empty_Qcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_associativity_empty_Rleft. (exists fom_gap_pfp_associativity_empty_Rleft_index_bound. fom_gap_pfp_associativity_empty_Rleft_index_bound + S (fom_index_pfp_associativity_empty_Rleft) = N) -> exists fom_value_pfp_associativity_empty_Rleft. ((((exists fom_beta_height_pfp_associativity_empty_Rleft_entry. fom_beta_height_pfp_associativity_empty_Rleft_entry + S (fom_value_pfp_associativity_empty_Rleft) = S ((S (fom_index_pfp_associativity_empty_Rleft)) * pc)) /\\ exists fom_beta_quotient_pfp_associativity_empty_Rleft_entry. pb = fom_beta_quotient_pfp_associativity_empty_Rleft_entry * S ((S (fom_index_pfp_associativity_empty_Rleft)) * pc) + (fom_value_pfp_associativity_empty_Rleft))) /\\ (exists fom_gap_pfp_associativity_empty_Rleft_value_bound. fom_gap_pfp_associativity_empty_Rleft_value_bound + S (fom_value_pfp_associativity_empty_Rleft) = p))) /\\ (((forall fom_index_pfp_associativity_empty_Rright. (exists fom_gap_pfp_associativity_empty_Rright_index_bound. fom_gap_pfp_associativity_empty_Rright_index_bound + S (fom_index_pfp_associativity_empty_Rright) = 0) -> exists fom_value_pfp_associativity_empty_Rright. ((((exists fom_beta_height_pfp_associativity_empty_Rright_entry. fom_beta_height_pfp_associativity_empty_Rright_entry + S (fom_value_pfp_associativity_empty_Rright) = S ((S (fom_index_pfp_associativity_empty_Rright)) * cc)) /\\ exists fom_beta_quotient_pfp_associativity_empty_Rright_entry. cb = fom_beta_quotient_pfp_associativity_empty_Rright_entry * S ((S (fom_index_pfp_associativity_empty_Rright)) * cc) + (fom_value_pfp_associativity_empty_Rright))) /\\ (exists fom_gap_pfp_associativity_empty_Rright_value_bound. fom_gap_pfp_associativity_empty_Rright_value_bound + S (fom_value_pfp_associativity_empty_Rright) = p))) /\\ (((((((N)=0 \\/ (0)=0) /\\ (((U)=0)))) \\/ (((~((N)=0)) /\\ (((~((0)=0)) /\\ (((N)+(0)=S (U)))))))) /\\ ((forall pfc_index_associativity_empty_Rcoefficients. (exists pfa_gap_associativity_empty_Rcoefficientsbound. pfa_gap_associativity_empty_Rcoefficientsbound + S (pfc_index_associativity_empty_Rcoefficients) = (U)) -> exists pfc_value_associativity_empty_Rcoefficients. ((((exists ff_h_pfp_associativity_empty_Rcoefficientsentry. ff_h_pfp_associativity_empty_Rcoefficientsentry + S (pfc_value_associativity_empty_Rcoefficients) = S ((S (pfc_index_associativity_empty_Rcoefficients)) * rc)) /\\ exists ff_q_pfp_associativity_empty_Rcoefficientsentry. rb = ff_q_pfp_associativity_empty_Rcoefficientsentry * S ((S (pfc_index_associativity_empty_Rcoefficients)) * rc) + (pfc_value_associativity_empty_Rcoefficients))) /\\ ((exists pfc_terms_code_associativity_empty_Rcoefficientscoefficient pfc_terms_scale_associativity_empty_Rcoefficientscoefficient pfc_natural_sum_associativity_empty_Rcoefficientscoefficient. ((forall pfc_index_associativity_empty_Rcoefficientscoefficientdiagonal. (exists pfa_gap_associativity_empty_Rcoefficientscoefficientdiagonalbound. pfa_gap_associativity_empty_Rcoefficientscoefficientdiagonalbound + S (pfc_index_associativity_empty_Rcoefficientscoefficientdiagonal) = (S (pfc_index_associativity_empty_Rcoefficients))) -> exists pfc_value_associativity_empty_Rcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associativity_empty_Rcoefficientscoefficientdiagonalentry. ff_h_pfp_associativity_empty_Rcoefficientscoefficientdiagonalentry + S (pfc_value_associativity_empty_Rcoefficientscoefficientdiagonal) = S ((S (pfc_index_associativity_empty_Rcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_empty_Rcoefficientscoefficient)) /\\ exists ff_q_pfp_associativity_empty_Rcoefficientscoefficientdiagonalentry. pfc_terms_code_associativity_empty_Rcoefficientscoefficient = ff_q_pfp_associativity_empty_Rcoefficientscoefficientdiagonalentry * S ((S (pfc_index_associativity_empty_Rcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_empty_Rcoefficientscoefficient) + (pfc_value_associativity_empty_Rcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associativity_empty_Rcoefficientscoefficientdiagonalterm pfc_left_associativity_empty_Rcoefficientscoefficientdiagonalterm pfc_right_associativity_empty_Rcoefficientscoefficientdiagonalterm. (((pfc_index_associativity_empty_Rcoefficientscoefficientdiagonal)+pfc_complement_associativity_empty_Rcoefficientscoefficientdiagonalterm=(pfc_index_associativity_empty_Rcoefficients)) /\\ ((((((exists pfa_gap_associativity_empty_Rcoefficientscoefficientdiagonaltermleftinside. pfa_gap_associativity_empty_Rcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associativity_empty_Rcoefficientscoefficientdiagonal) = (N)) /\\ ((((exists ff_h_pfp_associativity_empty_Rcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associativity_empty_Rcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associativity_empty_Rcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associativity_empty_Rcoefficientscoefficientdiagonal)) * pc)) /\\ exists ff_q_pfp_associativity_empty_Rcoefficientscoefficientdiagonaltermleftentry. pb = ff_q_pfp_associativity_empty_Rcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associativity_empty_Rcoefficientscoefficientdiagonal)) * pc) + (pfc_left_associativity_empty_Rcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_empty_Rcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associativity_empty_Rcoefficientscoefficientdiagonaltermleftoutside+(N)=(pfc_index_associativity_empty_Rcoefficientscoefficientdiagonal)) /\\ (((pfc_left_associativity_empty_Rcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associativity_empty_Rcoefficientscoefficientdiagonaltermrightinside. pfa_gap_associativity_empty_Rcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associativity_empty_Rcoefficientscoefficientdiagonalterm) = (0)) /\\ ((((exists ff_h_pfp_associativity_empty_Rcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associativity_empty_Rcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associativity_empty_Rcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associativity_empty_Rcoefficientscoefficientdiagonalterm)) * cc)) /\\ exists ff_q_pfp_associativity_empty_Rcoefficientscoefficientdiagonaltermrightentry. cb = ff_q_pfp_associativity_empty_Rcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associativity_empty_Rcoefficientscoefficientdiagonalterm)) * cc) + (pfc_right_associativity_empty_Rcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_empty_Rcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associativity_empty_Rcoefficientscoefficientdiagonaltermrightoutside+(0)=(pfc_complement_associativity_empty_Rcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associativity_empty_Rcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associativity_empty_Rcoefficientscoefficientdiagonal)=pfc_left_associativity_empty_Rcoefficientscoefficientdiagonalterm*pfc_right_associativity_empty_Rcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associativity_empty_Rcoefficientscoefficientsum fs_v_pfc_associativity_empty_Rcoefficientscoefficientsum. ((((exists fs_h_pfc_associativity_empty_Rcoefficientscoefficientsum_body_start. fs_h_pfc_associativity_empty_Rcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associativity_empty_Rcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_empty_Rcoefficientscoefficientsum_body_start. fs_u_pfc_associativity_empty_Rcoefficientscoefficientsum = fs_q_pfc_associativity_empty_Rcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associativity_empty_Rcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associativity_empty_Rcoefficientscoefficientsum_body_terminal. fs_h_pfc_associativity_empty_Rcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associativity_empty_Rcoefficientscoefficient) = S ((S (S (pfc_index_associativity_empty_Rcoefficients))) * fs_v_pfc_associativity_empty_Rcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_empty_Rcoefficientscoefficientsum_body_terminal. fs_u_pfc_associativity_empty_Rcoefficientscoefficientsum = fs_q_pfc_associativity_empty_Rcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associativity_empty_Rcoefficients))) * fs_v_pfc_associativity_empty_Rcoefficientscoefficientsum) + (pfc_natural_sum_associativity_empty_Rcoefficientscoefficient))) /\\ forall fs_i_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps = S (pfc_index_associativity_empty_Rcoefficients)) -> exists fs_a_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps fs_r_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps fs_s_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_empty_Rcoefficientscoefficient)) /\\ exists fs_q_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associativity_empty_Rcoefficientscoefficient = fs_q_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_empty_Rcoefficientscoefficient) + (fs_a_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_empty_Rcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associativity_empty_Rcoefficientscoefficientsum = fs_q_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_empty_Rcoefficientscoefficientsum) + (fs_r_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_empty_Rcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associativity_empty_Rcoefficientscoefficientsum = fs_q_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_empty_Rcoefficientscoefficientsum) + (fs_s_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps = fs_r_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps + fs_a_pfc_associativity_empty_Rcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associativity_empty_Rcoefficientscoefficientresiduebound. pfa_gap_associativity_empty_Rcoefficientscoefficientresiduebound + S (pfc_value_associativity_empty_Rcoefficients) = (p)) /\\ ((exists pfa_offset_left_associativity_empty_Rcoefficientscoefficientresiduecongruence pfa_offset_right_associativity_empty_Rcoefficientscoefficientresiduecongruence. (pfc_natural_sum_associativity_empty_Rcoefficientscoefficient) + (p) * pfa_offset_left_associativity_empty_Rcoefficientscoefficientresiduecongruence = (pfc_value_associativity_empty_Rcoefficients) + (p) * pfa_offset_right_associativity_empty_Rcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_associativity_empty_Sleft. (exists fom_gap_pfp_associativity_empty_Sleft_index_bound. fom_gap_pfp_associativity_empty_Sleft_index_bound + S (fom_index_pfp_associativity_empty_Sleft) = L) -> exists fom_value_pfp_associativity_empty_Sleft. ((((exists fom_beta_height_pfp_associativity_empty_Sleft_entry. fom_beta_height_pfp_associativity_empty_Sleft_entry + S (fom_value_pfp_associativity_empty_Sleft) = S ((S (fom_index_pfp_associativity_empty_Sleft)) * ac)) /\\ exists fom_beta_quotient_pfp_associativity_empty_Sleft_entry. ab = fom_beta_quotient_pfp_associativity_empty_Sleft_entry * S ((S (fom_index_pfp_associativity_empty_Sleft)) * ac) + (fom_value_pfp_associativity_empty_Sleft))) /\\ (exists fom_gap_pfp_associativity_empty_Sleft_value_bound. fom_gap_pfp_associativity_empty_Sleft_value_bound + S (fom_value_pfp_associativity_empty_Sleft) = p))) /\\ (((forall fom_index_pfp_associativity_empty_Sright. (exists fom_gap_pfp_associativity_empty_Sright_index_bound. fom_gap_pfp_associativity_empty_Sright_index_bound + S (fom_index_pfp_associativity_empty_Sright) = K) -> exists fom_value_pfp_associativity_empty_Sright. ((((exists fom_beta_height_pfp_associativity_empty_Sright_entry. fom_beta_height_pfp_associativity_empty_Sright_entry + S (fom_value_pfp_associativity_empty_Sright) = S ((S (fom_index_pfp_associativity_empty_Sright)) * qc)) /\\ exists fom_beta_quotient_pfp_associativity_empty_Sright_entry. qb = fom_beta_quotient_pfp_associativity_empty_Sright_entry * S ((S (fom_index_pfp_associativity_empty_Sright)) * qc) + (fom_value_pfp_associativity_empty_Sright))) /\\ (exists fom_gap_pfp_associativity_empty_Sright_value_bound. fom_gap_pfp_associativity_empty_Sright_value_bound + S (fom_value_pfp_associativity_empty_Sright) = p))) /\\ (((((((L)=0 \\/ (K)=0) /\\ (((V)=0)))) \\/ (((~((L)=0)) /\\ (((~((K)=0)) /\\ (((L)+(K)=S (V)))))))) /\\ ((forall pfc_index_associativity_empty_Scoefficients. (exists pfa_gap_associativity_empty_Scoefficientsbound. pfa_gap_associativity_empty_Scoefficientsbound + S (pfc_index_associativity_empty_Scoefficients) = (V)) -> exists pfc_value_associativity_empty_Scoefficients. ((((exists ff_h_pfp_associativity_empty_Scoefficientsentry. ff_h_pfp_associativity_empty_Scoefficientsentry + S (pfc_value_associativity_empty_Scoefficients) = S ((S (pfc_index_associativity_empty_Scoefficients)) * sc)) /\\ exists ff_q_pfp_associativity_empty_Scoefficientsentry. sb = ff_q_pfp_associativity_empty_Scoefficientsentry * S ((S (pfc_index_associativity_empty_Scoefficients)) * sc) + (pfc_value_associativity_empty_Scoefficients))) /\\ ((exists pfc_terms_code_associativity_empty_Scoefficientscoefficient pfc_terms_scale_associativity_empty_Scoefficientscoefficient pfc_natural_sum_associativity_empty_Scoefficientscoefficient. ((forall pfc_index_associativity_empty_Scoefficientscoefficientdiagonal. (exists pfa_gap_associativity_empty_Scoefficientscoefficientdiagonalbound. pfa_gap_associativity_empty_Scoefficientscoefficientdiagonalbound + S (pfc_index_associativity_empty_Scoefficientscoefficientdiagonal) = (S (pfc_index_associativity_empty_Scoefficients))) -> exists pfc_value_associativity_empty_Scoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associativity_empty_Scoefficientscoefficientdiagonalentry. ff_h_pfp_associativity_empty_Scoefficientscoefficientdiagonalentry + S (pfc_value_associativity_empty_Scoefficientscoefficientdiagonal) = S ((S (pfc_index_associativity_empty_Scoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_empty_Scoefficientscoefficient)) /\\ exists ff_q_pfp_associativity_empty_Scoefficientscoefficientdiagonalentry. pfc_terms_code_associativity_empty_Scoefficientscoefficient = ff_q_pfp_associativity_empty_Scoefficientscoefficientdiagonalentry * S ((S (pfc_index_associativity_empty_Scoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_empty_Scoefficientscoefficient) + (pfc_value_associativity_empty_Scoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associativity_empty_Scoefficientscoefficientdiagonalterm pfc_left_associativity_empty_Scoefficientscoefficientdiagonalterm pfc_right_associativity_empty_Scoefficientscoefficientdiagonalterm. (((pfc_index_associativity_empty_Scoefficientscoefficientdiagonal)+pfc_complement_associativity_empty_Scoefficientscoefficientdiagonalterm=(pfc_index_associativity_empty_Scoefficients)) /\\ ((((((exists pfa_gap_associativity_empty_Scoefficientscoefficientdiagonaltermleftinside. pfa_gap_associativity_empty_Scoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associativity_empty_Scoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_associativity_empty_Scoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associativity_empty_Scoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associativity_empty_Scoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associativity_empty_Scoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_associativity_empty_Scoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_associativity_empty_Scoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associativity_empty_Scoefficientscoefficientdiagonal)) * ac) + (pfc_left_associativity_empty_Scoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_empty_Scoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associativity_empty_Scoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_associativity_empty_Scoefficientscoefficientdiagonal)) /\\ (((pfc_left_associativity_empty_Scoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associativity_empty_Scoefficientscoefficientdiagonaltermrightinside. pfa_gap_associativity_empty_Scoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associativity_empty_Scoefficientscoefficientdiagonalterm) = (K)) /\\ ((((exists ff_h_pfp_associativity_empty_Scoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associativity_empty_Scoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associativity_empty_Scoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associativity_empty_Scoefficientscoefficientdiagonalterm)) * qc)) /\\ exists ff_q_pfp_associativity_empty_Scoefficientscoefficientdiagonaltermrightentry. qb = ff_q_pfp_associativity_empty_Scoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associativity_empty_Scoefficientscoefficientdiagonalterm)) * qc) + (pfc_right_associativity_empty_Scoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_empty_Scoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associativity_empty_Scoefficientscoefficientdiagonaltermrightoutside+(K)=(pfc_complement_associativity_empty_Scoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associativity_empty_Scoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associativity_empty_Scoefficientscoefficientdiagonal)=pfc_left_associativity_empty_Scoefficientscoefficientdiagonalterm*pfc_right_associativity_empty_Scoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associativity_empty_Scoefficientscoefficientsum fs_v_pfc_associativity_empty_Scoefficientscoefficientsum. ((((exists fs_h_pfc_associativity_empty_Scoefficientscoefficientsum_body_start. fs_h_pfc_associativity_empty_Scoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associativity_empty_Scoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_empty_Scoefficientscoefficientsum_body_start. fs_u_pfc_associativity_empty_Scoefficientscoefficientsum = fs_q_pfc_associativity_empty_Scoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associativity_empty_Scoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associativity_empty_Scoefficientscoefficientsum_body_terminal. fs_h_pfc_associativity_empty_Scoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associativity_empty_Scoefficientscoefficient) = S ((S (S (pfc_index_associativity_empty_Scoefficients))) * fs_v_pfc_associativity_empty_Scoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_empty_Scoefficientscoefficientsum_body_terminal. fs_u_pfc_associativity_empty_Scoefficientscoefficientsum = fs_q_pfc_associativity_empty_Scoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associativity_empty_Scoefficients))) * fs_v_pfc_associativity_empty_Scoefficientscoefficientsum) + (pfc_natural_sum_associativity_empty_Scoefficientscoefficient))) /\\ forall fs_i_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps = S (pfc_index_associativity_empty_Scoefficients)) -> exists fs_a_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps fs_r_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps fs_s_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_empty_Scoefficientscoefficient)) /\\ exists fs_q_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associativity_empty_Scoefficientscoefficient = fs_q_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_empty_Scoefficientscoefficient) + (fs_a_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_empty_Scoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associativity_empty_Scoefficientscoefficientsum = fs_q_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_empty_Scoefficientscoefficientsum) + (fs_r_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_empty_Scoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associativity_empty_Scoefficientscoefficientsum = fs_q_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_empty_Scoefficientscoefficientsum) + (fs_s_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps = fs_r_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps + fs_a_pfc_associativity_empty_Scoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associativity_empty_Scoefficientscoefficientresiduebound. pfa_gap_associativity_empty_Scoefficientscoefficientresiduebound + S (pfc_value_associativity_empty_Scoefficients) = (p)) /\\ ((exists pfa_offset_left_associativity_empty_Scoefficientscoefficientresiduecongruence pfa_offset_right_associativity_empty_Scoefficientscoefficientresiduecongruence. (pfc_natural_sum_associativity_empty_Scoefficientscoefficient) + (p) * pfa_offset_left_associativity_empty_Scoefficientscoefficientresiduecongruence = (pfc_value_associativity_empty_Scoefficients) + (p) * pfa_offset_right_associativity_empty_Scoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_associativity_empty_result pfrep_left_associativity_empty_result pfrep_right_associativity_empty_result. ((exists pfrep_position_associativity_empty_resultfirst. ((pfrep_position_associativity_empty_resultfirst+S (pfrep_power_associativity_empty_result)=(U)) /\\ ((((exists ff_h_pfp_associativity_empty_resultfirstentry. ff_h_pfp_associativity_empty_resultfirstentry + S (pfrep_left_associativity_empty_result) = S ((S (pfrep_position_associativity_empty_resultfirst)) * rc)) /\\ exists ff_q_pfp_associativity_empty_resultfirstentry. rb = ff_q_pfp_associativity_empty_resultfirstentry * S ((S (pfrep_position_associativity_empty_resultfirst)) * rc) + (pfrep_left_associativity_empty_result)))))) \\/ (((exists pfrep_gap_associativity_empty_resultfirstoutside. pfrep_gap_associativity_empty_resultfirstoutside+(U)=(pfrep_power_associativity_empty_result)) /\\ (((pfrep_left_associativity_empty_result)=0))))) -> ((exists pfrep_position_associativity_empty_resultsecond. 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"intro V", "intro hp", "intro hAB", "intro hBC", "intro hPC", "intro hAQ", "have hp0 : ~(p=0)", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hz", "have hABcopy : ((forall fom_index_pfp_associativity_ABleft. (exists fom_gap_pfp_associativity_ABleft_index_bound. fom_gap_pfp_associativity_ABleft_index_bound + S (fom_index_pfp_associativity_ABleft) = L) -> exists fom_value_pfp_associativity_ABleft. ((((exists fom_beta_height_pfp_associativity_ABleft_entry. fom_beta_height_pfp_associativity_ABleft_entry + S (fom_value_pfp_associativity_ABleft) = S ((S (fom_index_pfp_associativity_ABleft)) * ac)) /\\ exists fom_beta_quotient_pfp_associativity_ABleft_entry. ab = fom_beta_quotient_pfp_associativity_ABleft_entry * S ((S (fom_index_pfp_associativity_ABleft)) * ac) + (fom_value_pfp_associativity_ABleft))) /\\ (exists fom_gap_pfp_associativity_ABleft_value_bound. fom_gap_pfp_associativity_ABleft_value_bound + S (fom_value_pfp_associativity_ABleft) = p))) /\\ (((forall fom_index_pfp_associativity_ABright. (exists fom_gap_pfp_associativity_ABright_index_bound. fom_gap_pfp_associativity_ABright_index_bound + S (fom_index_pfp_associativity_ABright) = M) -> exists fom_value_pfp_associativity_ABright. ((((exists fom_beta_height_pfp_associativity_ABright_entry. fom_beta_height_pfp_associativity_ABright_entry + S (fom_value_pfp_associativity_ABright) = S ((S (fom_index_pfp_associativity_ABright)) * bc)) /\\ exists fom_beta_quotient_pfp_associativity_ABright_entry. bb = fom_beta_quotient_pfp_associativity_ABright_entry * S ((S (fom_index_pfp_associativity_ABright)) * bc) + (fom_value_pfp_associativity_ABright))) /\\ (exists fom_gap_pfp_associativity_ABright_value_bound. fom_gap_pfp_associativity_ABright_value_bound + S (fom_value_pfp_associativity_ABright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_associativity_ABcoefficients. (exists pfa_gap_associativity_ABcoefficientsbound. pfa_gap_associativity_ABcoefficientsbound + S (pfc_index_associativity_ABcoefficients) = (N)) -> exists pfc_value_associativity_ABcoefficients. ((((exists ff_h_pfp_associativity_ABcoefficientsentry. ff_h_pfp_associativity_ABcoefficientsentry + S (pfc_value_associativity_ABcoefficients) = S ((S (pfc_index_associativity_ABcoefficients)) * pc)) /\\ exists ff_q_pfp_associativity_ABcoefficientsentry. pb = ff_q_pfp_associativity_ABcoefficientsentry * S ((S (pfc_index_associativity_ABcoefficients)) * pc) + (pfc_value_associativity_ABcoefficients))) /\\ ((exists pfc_terms_code_associativity_ABcoefficientscoefficient pfc_terms_scale_associativity_ABcoefficientscoefficient pfc_natural_sum_associativity_ABcoefficientscoefficient. ((forall pfc_index_associativity_ABcoefficientscoefficientdiagonal. (exists pfa_gap_associativity_ABcoefficientscoefficientdiagonalbound. pfa_gap_associativity_ABcoefficientscoefficientdiagonalbound + S (pfc_index_associativity_ABcoefficientscoefficientdiagonal) = (S (pfc_index_associativity_ABcoefficients))) -> exists pfc_value_associativity_ABcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associativity_ABcoefficientscoefficientdiagonalentry. ff_h_pfp_associativity_ABcoefficientscoefficientdiagonalentry + S (pfc_value_associativity_ABcoefficientscoefficientdiagonal) = S ((S (pfc_index_associativity_ABcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_ABcoefficientscoefficient)) /\\ exists ff_q_pfp_associativity_ABcoefficientscoefficientdiagonalentry. pfc_terms_code_associativity_ABcoefficientscoefficient = ff_q_pfp_associativity_ABcoefficientscoefficientdiagonalentry * S ((S (pfc_index_associativity_ABcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_ABcoefficientscoefficient) + (pfc_value_associativity_ABcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associativity_ABcoefficientscoefficientdiagonalterm pfc_left_associativity_ABcoefficientscoefficientdiagonalterm pfc_right_associativity_ABcoefficientscoefficientdiagonalterm. (((pfc_index_associativity_ABcoefficientscoefficientdiagonal)+pfc_complement_associativity_ABcoefficientscoefficientdiagonalterm=(pfc_index_associativity_ABcoefficients)) /\\ ((((((exists pfa_gap_associativity_ABcoefficientscoefficientdiagonaltermleftinside. pfa_gap_associativity_ABcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associativity_ABcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_associativity_ABcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associativity_ABcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associativity_ABcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associativity_ABcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_associativity_ABcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_associativity_ABcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associativity_ABcoefficientscoefficientdiagonal)) * ac) + (pfc_left_associativity_ABcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_ABcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associativity_ABcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_associativity_ABcoefficientscoefficientdiagonal)) /\\ (((pfc_left_associativity_ABcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associativity_ABcoefficientscoefficientdiagonaltermrightinside. pfa_gap_associativity_ABcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associativity_ABcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_associativity_ABcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associativity_ABcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associativity_ABcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associativity_ABcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_associativity_ABcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_associativity_ABcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associativity_ABcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_associativity_ABcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_ABcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associativity_ABcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_associativity_ABcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associativity_ABcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associativity_ABcoefficientscoefficientdiagonal)=pfc_left_associativity_ABcoefficientscoefficientdiagonalterm*pfc_right_associativity_ABcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associativity_ABcoefficientscoefficientsum fs_v_pfc_associativity_ABcoefficientscoefficientsum. ((((exists fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_start. fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associativity_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_start. fs_u_pfc_associativity_ABcoefficientscoefficientsum = fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associativity_ABcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_terminal. fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associativity_ABcoefficientscoefficient) = S ((S (S (pfc_index_associativity_ABcoefficients))) * fs_v_pfc_associativity_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_terminal. fs_u_pfc_associativity_ABcoefficientscoefficientsum = fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associativity_ABcoefficients))) * fs_v_pfc_associativity_ABcoefficientscoefficientsum) + (pfc_natural_sum_associativity_ABcoefficientscoefficient))) /\\ forall fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associativity_ABcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associativity_ABcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps = S (pfc_index_associativity_ABcoefficients)) -> exists fs_a_pfc_associativity_ABcoefficientscoefficientsum_body_steps fs_r_pfc_associativity_ABcoefficientscoefficientsum_body_steps fs_s_pfc_associativity_ABcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associativity_ABcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_ABcoefficientscoefficient)) /\\ exists fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associativity_ABcoefficientscoefficient = fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_ABcoefficientscoefficient) + (fs_a_pfc_associativity_ABcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associativity_ABcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associativity_ABcoefficientscoefficientsum = fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_ABcoefficientscoefficientsum) + (fs_r_pfc_associativity_ABcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associativity_ABcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associativity_ABcoefficientscoefficientsum = fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_ABcoefficientscoefficientsum) + (fs_s_pfc_associativity_ABcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associativity_ABcoefficientscoefficientsum_body_steps = fs_r_pfc_associativity_ABcoefficientscoefficientsum_body_steps + fs_a_pfc_associativity_ABcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associativity_ABcoefficientscoefficientresiduebound. pfa_gap_associativity_ABcoefficientscoefficientresiduebound + S (pfc_value_associativity_ABcoefficients) = (p)) /\\ ((exists pfa_offset_left_associativity_ABcoefficientscoefficientresiduecongruence pfa_offset_right_associativity_ABcoefficientscoefficientresiduecongruence. (pfc_natural_sum_associativity_ABcoefficientscoefficient) + (p) * pfa_offset_left_associativity_ABcoefficientscoefficientresiduecongruence = (pfc_value_associativity_ABcoefficients) + (p) * pfa_offset_right_associativity_ABcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hAB", "cases hABcopy", "cases hABcopy_right", "cases hABcopy_right_right", "have hPbound : forall fom_index_pfp_associativity_P_bound. (exists fom_gap_pfp_associativity_P_bound_index_bound. fom_gap_pfp_associativity_P_bound_index_bound + S (fom_index_pfp_associativity_P_bound) = N) -> exists fom_value_pfp_associativity_P_bound. ((((exists fom_beta_height_pfp_associativity_P_bound_entry. fom_beta_height_pfp_associativity_P_bound_entry + S (fom_value_pfp_associativity_P_bound) = S ((S (fom_index_pfp_associativity_P_bound)) * pc)) /\\ exists fom_beta_quotient_pfp_associativity_P_bound_entry. pb = fom_beta_quotient_pfp_associativity_P_bound_entry * S ((S (fom_index_pfp_associativity_P_bound)) * pc) + (fom_value_pfp_associativity_P_bound))) /\\ (exists fom_gap_pfp_associativity_P_bound_value_bound. fom_gap_pfp_associativity_P_bound_value_bound + S (fom_value_pfp_associativity_P_bound) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (ab)", "specialize prime_field_polynomial_convolution_bounded (ac)", "specialize prime_field_polynomial_convolution_bounded (L)", "specialize prime_field_polynomial_convolution_bounded (bb)", "specialize prime_field_polynomial_convolution_bounded (bc)", "specialize prime_field_polynomial_convolution_bounded (M)", "specialize prime_field_polynomial_convolution_bounded (pb)", "specialize prime_field_polynomial_convolution_bounded (pc)", "specialize prime_field_polynomial_convolution_bounded (N)", "apply prime_field_polynomial_convolution_bounded", "exact hAB", "have hall : forall j db dc qxb qxc k rxb rxc u sxb sxc v. (((forall fom_index_pfp_associativity_quantified_Qleft. (exists fom_gap_pfp_associativity_quantified_Qleft_index_bound. fom_gap_pfp_associativity_quantified_Qleft_index_bound + S (fom_index_pfp_associativity_quantified_Qleft) = M) -> exists fom_value_pfp_associativity_quantified_Qleft. ((((exists fom_beta_height_pfp_associativity_quantified_Qleft_entry. fom_beta_height_pfp_associativity_quantified_Qleft_entry + S (fom_value_pfp_associativity_quantified_Qleft) = S ((S (fom_index_pfp_associativity_quantified_Qleft)) * bc)) /\\ exists fom_beta_quotient_pfp_associativity_quantified_Qleft_entry. bb = fom_beta_quotient_pfp_associativity_quantified_Qleft_entry * S ((S (fom_index_pfp_associativity_quantified_Qleft)) * bc) + (fom_value_pfp_associativity_quantified_Qleft))) /\\ (exists fom_gap_pfp_associativity_quantified_Qleft_value_bound. fom_gap_pfp_associativity_quantified_Qleft_value_bound + S (fom_value_pfp_associativity_quantified_Qleft) = p))) /\\ (((forall fom_index_pfp_associativity_quantified_Qright. (exists fom_gap_pfp_associativity_quantified_Qright_index_bound. fom_gap_pfp_associativity_quantified_Qright_index_bound + S (fom_index_pfp_associativity_quantified_Qright) = j) -> exists fom_value_pfp_associativity_quantified_Qright. ((((exists fom_beta_height_pfp_associativity_quantified_Qright_entry. fom_beta_height_pfp_associativity_quantified_Qright_entry + S (fom_value_pfp_associativity_quantified_Qright) = S ((S (fom_index_pfp_associativity_quantified_Qright)) * dc)) /\\ exists fom_beta_quotient_pfp_associativity_quantified_Qright_entry. db = fom_beta_quotient_pfp_associativity_quantified_Qright_entry * S ((S (fom_index_pfp_associativity_quantified_Qright)) * dc) + (fom_value_pfp_associativity_quantified_Qright))) /\\ (exists fom_gap_pfp_associativity_quantified_Qright_value_bound. fom_gap_pfp_associativity_quantified_Qright_value_bound + S (fom_value_pfp_associativity_quantified_Qright) = p))) /\\ (((((((M)=0 \\/ (j)=0) /\\ (((k)=0)))) \\/ (((~((M)=0)) /\\ (((~((j)=0)) /\\ (((M)+(j)=S (k)))))))) /\\ ((forall pfc_index_associativity_quantified_Qcoefficients. (exists pfa_gap_associativity_quantified_Qcoefficientsbound. pfa_gap_associativity_quantified_Qcoefficientsbound + S (pfc_index_associativity_quantified_Qcoefficients) = (k)) -> exists pfc_value_associativity_quantified_Qcoefficients. ((((exists ff_h_pfp_associativity_quantified_Qcoefficientsentry. ff_h_pfp_associativity_quantified_Qcoefficientsentry + S (pfc_value_associativity_quantified_Qcoefficients) = S ((S (pfc_index_associativity_quantified_Qcoefficients)) * qxc)) /\\ exists ff_q_pfp_associativity_quantified_Qcoefficientsentry. qxb = ff_q_pfp_associativity_quantified_Qcoefficientsentry * S ((S (pfc_index_associativity_quantified_Qcoefficients)) * qxc) + (pfc_value_associativity_quantified_Qcoefficients))) /\\ ((exists pfc_terms_code_associativity_quantified_Qcoefficientscoefficient pfc_terms_scale_associativity_quantified_Qcoefficientscoefficient pfc_natural_sum_associativity_quantified_Qcoefficientscoefficient. ((forall pfc_index_associativity_quantified_Qcoefficientscoefficientdiagonal. (exists pfa_gap_associativity_quantified_Qcoefficientscoefficientdiagonalbound. pfa_gap_associativity_quantified_Qcoefficientscoefficientdiagonalbound + S (pfc_index_associativity_quantified_Qcoefficientscoefficientdiagonal) = (S (pfc_index_associativity_quantified_Qcoefficients))) -> exists pfc_value_associativity_quantified_Qcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associativity_quantified_Qcoefficientscoefficientdiagonalentry. ff_h_pfp_associativity_quantified_Qcoefficientscoefficientdiagonalentry + S (pfc_value_associativity_quantified_Qcoefficientscoefficientdiagonal) = S ((S (pfc_index_associativity_quantified_Qcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_quantified_Qcoefficientscoefficient)) /\\ exists ff_q_pfp_associativity_quantified_Qcoefficientscoefficientdiagonalentry. pfc_terms_code_associativity_quantified_Qcoefficientscoefficient = ff_q_pfp_associativity_quantified_Qcoefficientscoefficientdiagonalentry * S ((S (pfc_index_associativity_quantified_Qcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_quantified_Qcoefficientscoefficient) + (pfc_value_associativity_quantified_Qcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associativity_quantified_Qcoefficientscoefficientdiagonalterm pfc_left_associativity_quantified_Qcoefficientscoefficientdiagonalterm pfc_right_associativity_quantified_Qcoefficientscoefficientdiagonalterm. (((pfc_index_associativity_quantified_Qcoefficientscoefficientdiagonal)+pfc_complement_associativity_quantified_Qcoefficientscoefficientdiagonalterm=(pfc_index_associativity_quantified_Qcoefficients)) /\\ ((((((exists pfa_gap_associativity_quantified_Qcoefficientscoefficientdiagonaltermleftinside. pfa_gap_associativity_quantified_Qcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associativity_quantified_Qcoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_associativity_quantified_Qcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associativity_quantified_Qcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associativity_quantified_Qcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associativity_quantified_Qcoefficientscoefficientdiagonal)) * bc)) /\\ exists ff_q_pfp_associativity_quantified_Qcoefficientscoefficientdiagonaltermleftentry. bb = ff_q_pfp_associativity_quantified_Qcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associativity_quantified_Qcoefficientscoefficientdiagonal)) * bc) + (pfc_left_associativity_quantified_Qcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_quantified_Qcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associativity_quantified_Qcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_associativity_quantified_Qcoefficientscoefficientdiagonal)) /\\ (((pfc_left_associativity_quantified_Qcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associativity_quantified_Qcoefficientscoefficientdiagonaltermrightinside. pfa_gap_associativity_quantified_Qcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associativity_quantified_Qcoefficientscoefficientdiagonalterm) = (j)) /\\ ((((exists ff_h_pfp_associativity_quantified_Qcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associativity_quantified_Qcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associativity_quantified_Qcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associativity_quantified_Qcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_associativity_quantified_Qcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_associativity_quantified_Qcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associativity_quantified_Qcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_associativity_quantified_Qcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_quantified_Qcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associativity_quantified_Qcoefficientscoefficientdiagonaltermrightoutside+(j)=(pfc_complement_associativity_quantified_Qcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associativity_quantified_Qcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associativity_quantified_Qcoefficientscoefficientdiagonal)=pfc_left_associativity_quantified_Qcoefficientscoefficientdiagonalterm*pfc_right_associativity_quantified_Qcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associativity_quantified_Qcoefficientscoefficientsum fs_v_pfc_associativity_quantified_Qcoefficientscoefficientsum. ((((exists fs_h_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_start. fs_h_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associativity_quantified_Qcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_start. fs_u_pfc_associativity_quantified_Qcoefficientscoefficientsum = fs_q_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associativity_quantified_Qcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_terminal. fs_h_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associativity_quantified_Qcoefficientscoefficient) = S ((S (S (pfc_index_associativity_quantified_Qcoefficients))) * fs_v_pfc_associativity_quantified_Qcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_terminal. fs_u_pfc_associativity_quantified_Qcoefficientscoefficientsum = fs_q_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associativity_quantified_Qcoefficients))) * fs_v_pfc_associativity_quantified_Qcoefficientscoefficientsum) + (pfc_natural_sum_associativity_quantified_Qcoefficientscoefficient))) /\\ forall fs_i_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps = S (pfc_index_associativity_quantified_Qcoefficients)) -> exists fs_a_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps fs_r_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps fs_s_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_quantified_Qcoefficientscoefficient)) /\\ exists fs_q_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associativity_quantified_Qcoefficientscoefficient = fs_q_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_quantified_Qcoefficientscoefficient) + (fs_a_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_quantified_Qcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associativity_quantified_Qcoefficientscoefficientsum = fs_q_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_quantified_Qcoefficientscoefficientsum) + (fs_r_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_quantified_Qcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associativity_quantified_Qcoefficientscoefficientsum = fs_q_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_quantified_Qcoefficientscoefficientsum) + (fs_s_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps = fs_r_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps + fs_a_pfc_associativity_quantified_Qcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associativity_quantified_Qcoefficientscoefficientresiduebound. pfa_gap_associativity_quantified_Qcoefficientscoefficientresiduebound + S (pfc_value_associativity_quantified_Qcoefficients) = (p)) /\\ ((exists pfa_offset_left_associativity_quantified_Qcoefficientscoefficientresiduecongruence pfa_offset_right_associativity_quantified_Qcoefficientscoefficientresiduecongruence. (pfc_natural_sum_associativity_quantified_Qcoefficientscoefficient) + (p) * pfa_offset_left_associativity_quantified_Qcoefficientscoefficientresiduecongruence = (pfc_value_associativity_quantified_Qcoefficients) + (p) * pfa_offset_right_associativity_quantified_Qcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_associativity_quantified_Rleft. (exists fom_gap_pfp_associativity_quantified_Rleft_index_bound. fom_gap_pfp_associativity_quantified_Rleft_index_bound + S (fom_index_pfp_associativity_quantified_Rleft) = N) -> exists fom_value_pfp_associativity_quantified_Rleft. ((((exists fom_beta_height_pfp_associativity_quantified_Rleft_entry. fom_beta_height_pfp_associativity_quantified_Rleft_entry + S (fom_value_pfp_associativity_quantified_Rleft) = S ((S (fom_index_pfp_associativity_quantified_Rleft)) * pc)) /\\ exists fom_beta_quotient_pfp_associativity_quantified_Rleft_entry. pb = fom_beta_quotient_pfp_associativity_quantified_Rleft_entry * S ((S (fom_index_pfp_associativity_quantified_Rleft)) * pc) + (fom_value_pfp_associativity_quantified_Rleft))) /\\ (exists fom_gap_pfp_associativity_quantified_Rleft_value_bound. fom_gap_pfp_associativity_quantified_Rleft_value_bound + S (fom_value_pfp_associativity_quantified_Rleft) = p))) /\\ (((forall fom_index_pfp_associativity_quantified_Rright. (exists fom_gap_pfp_associativity_quantified_Rright_index_bound. fom_gap_pfp_associativity_quantified_Rright_index_bound + S (fom_index_pfp_associativity_quantified_Rright) = j) -> exists fom_value_pfp_associativity_quantified_Rright. ((((exists fom_beta_height_pfp_associativity_quantified_Rright_entry. fom_beta_height_pfp_associativity_quantified_Rright_entry + S (fom_value_pfp_associativity_quantified_Rright) = S ((S (fom_index_pfp_associativity_quantified_Rright)) * dc)) /\\ exists fom_beta_quotient_pfp_associativity_quantified_Rright_entry. db = fom_beta_quotient_pfp_associativity_quantified_Rright_entry * S ((S (fom_index_pfp_associativity_quantified_Rright)) * dc) + (fom_value_pfp_associativity_quantified_Rright))) /\\ (exists fom_gap_pfp_associativity_quantified_Rright_value_bound. fom_gap_pfp_associativity_quantified_Rright_value_bound + S (fom_value_pfp_associativity_quantified_Rright) = p))) /\\ (((((((N)=0 \\/ (j)=0) /\\ (((u)=0)))) \\/ (((~((N)=0)) /\\ (((~((j)=0)) /\\ (((N)+(j)=S (u)))))))) /\\ ((forall pfc_index_associativity_quantified_Rcoefficients. (exists pfa_gap_associativity_quantified_Rcoefficientsbound. pfa_gap_associativity_quantified_Rcoefficientsbound + S (pfc_index_associativity_quantified_Rcoefficients) = (u)) -> exists pfc_value_associativity_quantified_Rcoefficients. ((((exists ff_h_pfp_associativity_quantified_Rcoefficientsentry. ff_h_pfp_associativity_quantified_Rcoefficientsentry + S (pfc_value_associativity_quantified_Rcoefficients) = S ((S (pfc_index_associativity_quantified_Rcoefficients)) * rxc)) /\\ exists ff_q_pfp_associativity_quantified_Rcoefficientsentry. rxb = ff_q_pfp_associativity_quantified_Rcoefficientsentry * S ((S (pfc_index_associativity_quantified_Rcoefficients)) * rxc) + (pfc_value_associativity_quantified_Rcoefficients))) /\\ ((exists pfc_terms_code_associativity_quantified_Rcoefficientscoefficient pfc_terms_scale_associativity_quantified_Rcoefficientscoefficient pfc_natural_sum_associativity_quantified_Rcoefficientscoefficient. ((forall pfc_index_associativity_quantified_Rcoefficientscoefficientdiagonal. (exists pfa_gap_associativity_quantified_Rcoefficientscoefficientdiagonalbound. pfa_gap_associativity_quantified_Rcoefficientscoefficientdiagonalbound + S (pfc_index_associativity_quantified_Rcoefficientscoefficientdiagonal) = (S (pfc_index_associativity_quantified_Rcoefficients))) -> exists pfc_value_associativity_quantified_Rcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associativity_quantified_Rcoefficientscoefficientdiagonalentry. ff_h_pfp_associativity_quantified_Rcoefficientscoefficientdiagonalentry + S (pfc_value_associativity_quantified_Rcoefficientscoefficientdiagonal) = S ((S (pfc_index_associativity_quantified_Rcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_quantified_Rcoefficientscoefficient)) /\\ exists ff_q_pfp_associativity_quantified_Rcoefficientscoefficientdiagonalentry. pfc_terms_code_associativity_quantified_Rcoefficientscoefficient = ff_q_pfp_associativity_quantified_Rcoefficientscoefficientdiagonalentry * S ((S (pfc_index_associativity_quantified_Rcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_quantified_Rcoefficientscoefficient) + (pfc_value_associativity_quantified_Rcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associativity_quantified_Rcoefficientscoefficientdiagonalterm pfc_left_associativity_quantified_Rcoefficientscoefficientdiagonalterm pfc_right_associativity_quantified_Rcoefficientscoefficientdiagonalterm. (((pfc_index_associativity_quantified_Rcoefficientscoefficientdiagonal)+pfc_complement_associativity_quantified_Rcoefficientscoefficientdiagonalterm=(pfc_index_associativity_quantified_Rcoefficients)) /\\ ((((((exists pfa_gap_associativity_quantified_Rcoefficientscoefficientdiagonaltermleftinside. pfa_gap_associativity_quantified_Rcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associativity_quantified_Rcoefficientscoefficientdiagonal) = (N)) /\\ ((((exists ff_h_pfp_associativity_quantified_Rcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associativity_quantified_Rcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associativity_quantified_Rcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associativity_quantified_Rcoefficientscoefficientdiagonal)) * pc)) /\\ exists ff_q_pfp_associativity_quantified_Rcoefficientscoefficientdiagonaltermleftentry. pb = ff_q_pfp_associativity_quantified_Rcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associativity_quantified_Rcoefficientscoefficientdiagonal)) * pc) + (pfc_left_associativity_quantified_Rcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_quantified_Rcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associativity_quantified_Rcoefficientscoefficientdiagonaltermleftoutside+(N)=(pfc_index_associativity_quantified_Rcoefficientscoefficientdiagonal)) /\\ (((pfc_left_associativity_quantified_Rcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associativity_quantified_Rcoefficientscoefficientdiagonaltermrightinside. pfa_gap_associativity_quantified_Rcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associativity_quantified_Rcoefficientscoefficientdiagonalterm) = (j)) /\\ ((((exists ff_h_pfp_associativity_quantified_Rcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associativity_quantified_Rcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associativity_quantified_Rcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associativity_quantified_Rcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_associativity_quantified_Rcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_associativity_quantified_Rcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associativity_quantified_Rcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_associativity_quantified_Rcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_quantified_Rcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associativity_quantified_Rcoefficientscoefficientdiagonaltermrightoutside+(j)=(pfc_complement_associativity_quantified_Rcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associativity_quantified_Rcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associativity_quantified_Rcoefficientscoefficientdiagonal)=pfc_left_associativity_quantified_Rcoefficientscoefficientdiagonalterm*pfc_right_associativity_quantified_Rcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associativity_quantified_Rcoefficientscoefficientsum fs_v_pfc_associativity_quantified_Rcoefficientscoefficientsum. ((((exists fs_h_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_start. fs_h_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associativity_quantified_Rcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_start. fs_u_pfc_associativity_quantified_Rcoefficientscoefficientsum = fs_q_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associativity_quantified_Rcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_terminal. fs_h_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associativity_quantified_Rcoefficientscoefficient) = S ((S (S (pfc_index_associativity_quantified_Rcoefficients))) * fs_v_pfc_associativity_quantified_Rcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_terminal. fs_u_pfc_associativity_quantified_Rcoefficientscoefficientsum = fs_q_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associativity_quantified_Rcoefficients))) * fs_v_pfc_associativity_quantified_Rcoefficientscoefficientsum) + (pfc_natural_sum_associativity_quantified_Rcoefficientscoefficient))) /\\ forall fs_i_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps = S (pfc_index_associativity_quantified_Rcoefficients)) -> exists fs_a_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps fs_r_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps fs_s_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_quantified_Rcoefficientscoefficient)) /\\ exists fs_q_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associativity_quantified_Rcoefficientscoefficient = fs_q_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_quantified_Rcoefficientscoefficient) + (fs_a_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_quantified_Rcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associativity_quantified_Rcoefficientscoefficientsum = fs_q_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_quantified_Rcoefficientscoefficientsum) + (fs_r_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_quantified_Rcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associativity_quantified_Rcoefficientscoefficientsum = fs_q_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_quantified_Rcoefficientscoefficientsum) + (fs_s_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps = fs_r_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps + fs_a_pfc_associativity_quantified_Rcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associativity_quantified_Rcoefficientscoefficientresiduebound. pfa_gap_associativity_quantified_Rcoefficientscoefficientresiduebound + S (pfc_value_associativity_quantified_Rcoefficients) = (p)) /\\ ((exists pfa_offset_left_associativity_quantified_Rcoefficientscoefficientresiduecongruence pfa_offset_right_associativity_quantified_Rcoefficientscoefficientresiduecongruence. (pfc_natural_sum_associativity_quantified_Rcoefficientscoefficient) + (p) * pfa_offset_left_associativity_quantified_Rcoefficientscoefficientresiduecongruence = (pfc_value_associativity_quantified_Rcoefficients) + (p) * pfa_offset_right_associativity_quantified_Rcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_associativity_quantified_Sleft. (exists fom_gap_pfp_associativity_quantified_Sleft_index_bound. fom_gap_pfp_associativity_quantified_Sleft_index_bound + S (fom_index_pfp_associativity_quantified_Sleft) = L) -> exists fom_value_pfp_associativity_quantified_Sleft. ((((exists fom_beta_height_pfp_associativity_quantified_Sleft_entry. fom_beta_height_pfp_associativity_quantified_Sleft_entry + S (fom_value_pfp_associativity_quantified_Sleft) = S ((S (fom_index_pfp_associativity_quantified_Sleft)) * ac)) /\\ exists fom_beta_quotient_pfp_associativity_quantified_Sleft_entry. ab = fom_beta_quotient_pfp_associativity_quantified_Sleft_entry * S ((S (fom_index_pfp_associativity_quantified_Sleft)) * ac) + (fom_value_pfp_associativity_quantified_Sleft))) /\\ (exists fom_gap_pfp_associativity_quantified_Sleft_value_bound. fom_gap_pfp_associativity_quantified_Sleft_value_bound + S (fom_value_pfp_associativity_quantified_Sleft) = p))) /\\ (((forall fom_index_pfp_associativity_quantified_Sright. (exists fom_gap_pfp_associativity_quantified_Sright_index_bound. fom_gap_pfp_associativity_quantified_Sright_index_bound + S (fom_index_pfp_associativity_quantified_Sright) = k) -> exists fom_value_pfp_associativity_quantified_Sright. ((((exists fom_beta_height_pfp_associativity_quantified_Sright_entry. fom_beta_height_pfp_associativity_quantified_Sright_entry + S (fom_value_pfp_associativity_quantified_Sright) = S ((S (fom_index_pfp_associativity_quantified_Sright)) * qxc)) /\\ exists fom_beta_quotient_pfp_associativity_quantified_Sright_entry. qxb = fom_beta_quotient_pfp_associativity_quantified_Sright_entry * S ((S (fom_index_pfp_associativity_quantified_Sright)) * qxc) + (fom_value_pfp_associativity_quantified_Sright))) /\\ (exists fom_gap_pfp_associativity_quantified_Sright_value_bound. fom_gap_pfp_associativity_quantified_Sright_value_bound + S (fom_value_pfp_associativity_quantified_Sright) = p))) /\\ (((((((L)=0 \\/ (k)=0) /\\ (((v)=0)))) \\/ (((~((L)=0)) /\\ (((~((k)=0)) /\\ (((L)+(k)=S (v)))))))) /\\ ((forall pfc_index_associativity_quantified_Scoefficients. (exists pfa_gap_associativity_quantified_Scoefficientsbound. pfa_gap_associativity_quantified_Scoefficientsbound + S (pfc_index_associativity_quantified_Scoefficients) = (v)) -> exists pfc_value_associativity_quantified_Scoefficients. ((((exists ff_h_pfp_associativity_quantified_Scoefficientsentry. ff_h_pfp_associativity_quantified_Scoefficientsentry + S (pfc_value_associativity_quantified_Scoefficients) = S ((S (pfc_index_associativity_quantified_Scoefficients)) * sxc)) /\\ exists ff_q_pfp_associativity_quantified_Scoefficientsentry. sxb = ff_q_pfp_associativity_quantified_Scoefficientsentry * S ((S (pfc_index_associativity_quantified_Scoefficients)) * sxc) + (pfc_value_associativity_quantified_Scoefficients))) /\\ ((exists pfc_terms_code_associativity_quantified_Scoefficientscoefficient pfc_terms_scale_associativity_quantified_Scoefficientscoefficient pfc_natural_sum_associativity_quantified_Scoefficientscoefficient. ((forall pfc_index_associativity_quantified_Scoefficientscoefficientdiagonal. (exists pfa_gap_associativity_quantified_Scoefficientscoefficientdiagonalbound. pfa_gap_associativity_quantified_Scoefficientscoefficientdiagonalbound + S (pfc_index_associativity_quantified_Scoefficientscoefficientdiagonal) = (S (pfc_index_associativity_quantified_Scoefficients))) -> exists pfc_value_associativity_quantified_Scoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associativity_quantified_Scoefficientscoefficientdiagonalentry. ff_h_pfp_associativity_quantified_Scoefficientscoefficientdiagonalentry + S (pfc_value_associativity_quantified_Scoefficientscoefficientdiagonal) = S ((S (pfc_index_associativity_quantified_Scoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_quantified_Scoefficientscoefficient)) /\\ exists ff_q_pfp_associativity_quantified_Scoefficientscoefficientdiagonalentry. pfc_terms_code_associativity_quantified_Scoefficientscoefficient = ff_q_pfp_associativity_quantified_Scoefficientscoefficientdiagonalentry * S ((S (pfc_index_associativity_quantified_Scoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_quantified_Scoefficientscoefficient) + (pfc_value_associativity_quantified_Scoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associativity_quantified_Scoefficientscoefficientdiagonalterm pfc_left_associativity_quantified_Scoefficientscoefficientdiagonalterm pfc_right_associativity_quantified_Scoefficientscoefficientdiagonalterm. (((pfc_index_associativity_quantified_Scoefficientscoefficientdiagonal)+pfc_complement_associativity_quantified_Scoefficientscoefficientdiagonalterm=(pfc_index_associativity_quantified_Scoefficients)) /\\ ((((((exists pfa_gap_associativity_quantified_Scoefficientscoefficientdiagonaltermleftinside. pfa_gap_associativity_quantified_Scoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associativity_quantified_Scoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_associativity_quantified_Scoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associativity_quantified_Scoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associativity_quantified_Scoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associativity_quantified_Scoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_associativity_quantified_Scoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_associativity_quantified_Scoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associativity_quantified_Scoefficientscoefficientdiagonal)) * ac) + (pfc_left_associativity_quantified_Scoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_quantified_Scoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associativity_quantified_Scoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_associativity_quantified_Scoefficientscoefficientdiagonal)) /\\ (((pfc_left_associativity_quantified_Scoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associativity_quantified_Scoefficientscoefficientdiagonaltermrightinside. pfa_gap_associativity_quantified_Scoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associativity_quantified_Scoefficientscoefficientdiagonalterm) = (k)) /\\ ((((exists ff_h_pfp_associativity_quantified_Scoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associativity_quantified_Scoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associativity_quantified_Scoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associativity_quantified_Scoefficientscoefficientdiagonalterm)) * qxc)) /\\ exists ff_q_pfp_associativity_quantified_Scoefficientscoefficientdiagonaltermrightentry. qxb = ff_q_pfp_associativity_quantified_Scoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associativity_quantified_Scoefficientscoefficientdiagonalterm)) * qxc) + (pfc_right_associativity_quantified_Scoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_quantified_Scoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associativity_quantified_Scoefficientscoefficientdiagonaltermrightoutside+(k)=(pfc_complement_associativity_quantified_Scoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associativity_quantified_Scoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associativity_quantified_Scoefficientscoefficientdiagonal)=pfc_left_associativity_quantified_Scoefficientscoefficientdiagonalterm*pfc_right_associativity_quantified_Scoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associativity_quantified_Scoefficientscoefficientsum fs_v_pfc_associativity_quantified_Scoefficientscoefficientsum. ((((exists fs_h_pfc_associativity_quantified_Scoefficientscoefficientsum_body_start. fs_h_pfc_associativity_quantified_Scoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associativity_quantified_Scoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_quantified_Scoefficientscoefficientsum_body_start. fs_u_pfc_associativity_quantified_Scoefficientscoefficientsum = fs_q_pfc_associativity_quantified_Scoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associativity_quantified_Scoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associativity_quantified_Scoefficientscoefficientsum_body_terminal. fs_h_pfc_associativity_quantified_Scoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associativity_quantified_Scoefficientscoefficient) = S ((S (S (pfc_index_associativity_quantified_Scoefficients))) * fs_v_pfc_associativity_quantified_Scoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_quantified_Scoefficientscoefficientsum_body_terminal. fs_u_pfc_associativity_quantified_Scoefficientscoefficientsum = fs_q_pfc_associativity_quantified_Scoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associativity_quantified_Scoefficients))) * fs_v_pfc_associativity_quantified_Scoefficientscoefficientsum) + (pfc_natural_sum_associativity_quantified_Scoefficientscoefficient))) /\\ forall fs_i_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps = S (pfc_index_associativity_quantified_Scoefficients)) -> exists fs_a_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps fs_r_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps fs_s_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_quantified_Scoefficientscoefficient)) /\\ exists fs_q_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associativity_quantified_Scoefficientscoefficient = fs_q_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_quantified_Scoefficientscoefficient) + (fs_a_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_quantified_Scoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associativity_quantified_Scoefficientscoefficientsum = fs_q_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_quantified_Scoefficientscoefficientsum) + (fs_r_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_quantified_Scoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associativity_quantified_Scoefficientscoefficientsum = fs_q_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_quantified_Scoefficientscoefficientsum) + (fs_s_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps = fs_r_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps + fs_a_pfc_associativity_quantified_Scoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associativity_quantified_Scoefficientscoefficientresiduebound. pfa_gap_associativity_quantified_Scoefficientscoefficientresiduebound + S (pfc_value_associativity_quantified_Scoefficients) = (p)) /\\ ((exists pfa_offset_left_associativity_quantified_Scoefficientscoefficientresiduecongruence pfa_offset_right_associativity_quantified_Scoefficientscoefficientresiduecongruence. (pfc_natural_sum_associativity_quantified_Scoefficientscoefficient) + (p) * pfa_offset_left_associativity_quantified_Scoefficientscoefficientresiduecongruence = (pfc_value_associativity_quantified_Scoefficients) + (p) * pfa_offset_right_associativity_quantified_Scoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_associativity_quantified_result pfrep_left_associativity_quantified_result pfrep_right_associativity_quantified_result. ((exists pfrep_position_associativity_quantified_resultfirst. ((pfrep_position_associativity_quantified_resultfirst+S (pfrep_power_associativity_quantified_result)=(u)) /\\ ((((exists ff_h_pfp_associativity_quantified_resultfirstentry. ff_h_pfp_associativity_quantified_resultfirstentry + S (pfrep_left_associativity_quantified_result) = S ((S (pfrep_position_associativity_quantified_resultfirst)) * rxc)) /\\ exists ff_q_pfp_associativity_quantified_resultfirstentry. rxb = ff_q_pfp_associativity_quantified_resultfirstentry * S ((S (pfrep_position_associativity_quantified_resultfirst)) * rxc) + (pfrep_left_associativity_quantified_result)))))) \\/ (((exists pfrep_gap_associativity_quantified_resultfirstoutside. pfrep_gap_associativity_quantified_resultfirstoutside+(u)=(pfrep_power_associativity_quantified_result)) /\\ (((pfrep_left_associativity_quantified_result)=0))))) -> ((exists pfrep_position_associativity_quantified_resultsecond. ((pfrep_position_associativity_quantified_resultsecond+S (pfrep_power_associativity_quantified_result)=(v)) /\\ ((((exists ff_h_pfp_associativity_quantified_resultsecondentry. ff_h_pfp_associativity_quantified_resultsecondentry + S (pfrep_right_associativity_quantified_result) = S ((S (pfrep_position_associativity_quantified_resultsecond)) * sxc)) /\\ exists ff_q_pfp_associativity_quantified_resultsecondentry. sxb = ff_q_pfp_associativity_quantified_resultsecondentry * S ((S (pfrep_position_associativity_quantified_resultsecond)) * sxc) + (pfrep_right_associativity_quantified_result)))))) \\/ (((exists pfrep_gap_associativity_quantified_resultsecondoutside. pfrep_gap_associativity_quantified_resultsecondoutside+(v)=(pfrep_power_associativity_quantified_result)) /\\ (((pfrep_right_associativity_quantified_result)=0))))) -> pfrep_left_associativity_quantified_result=pfrep_right_associativity_quantified_result)", "induction j", "intro db", "intro dc", "intro qxb", "intro qxc", "intro k", "intro rxb", "intro rxc", "intro u", "intro sxb", "intro sxc", "intro v", "intro hQ", "intro hR", "intro hS", "specialize prime_field_polynomial_nested_empty_right_equivalent (p)", "specialize prime_field_polynomial_nested_empty_right_equivalent (ab)", "specialize prime_field_polynomial_nested_empty_right_equivalent (ac)", "specialize prime_field_polynomial_nested_empty_right_equivalent (L)", "specialize prime_field_polynomial_nested_empty_right_equivalent (bb)", "specialize prime_field_polynomial_nested_empty_right_equivalent (bc)", "specialize prime_field_polynomial_nested_empty_right_equivalent (M)", "specialize prime_field_polynomial_nested_empty_right_equivalent (pb)", "specialize prime_field_polynomial_nested_empty_right_equivalent (pc)", "specialize prime_field_polynomial_nested_empty_right_equivalent (N)", "specialize prime_field_polynomial_nested_empty_right_equivalent (db)", "specialize prime_field_polynomial_nested_empty_right_equivalent (dc)", "specialize prime_field_polynomial_nested_empty_right_equivalent (qxb)", "specialize prime_field_polynomial_nested_empty_right_equivalent (qxc)", "specialize prime_field_polynomial_nested_empty_right_equivalent (k)", "specialize prime_field_polynomial_nested_empty_right_equivalent (rxb)", "specialize prime_field_polynomial_nested_empty_right_equivalent (rxc)", "specialize prime_field_polynomial_nested_empty_right_equivalent (u)", "specialize prime_field_polynomial_nested_empty_right_equivalent (sxb)", "specialize prime_field_polynomial_nested_empty_right_equivalent (sxc)", "specialize prime_field_polynomial_nested_empty_right_equivalent (v)", "apply prime_field_polynomial_nested_empty_right_equivalent", "exact hp0", "exact hQ", "exact hR", "exact hS", "intro db", "intro dc", "intro qxb", "intro qxc", "intro k", "intro rxb", "intro rxc", "intro u", "intro sxb", "intro sxc", "intro v", "intro hQ", "intro hR", "intro hS", "have hQcopy : ((forall fom_index_pfp_associativity_step_Q_copyleft. (exists fom_gap_pfp_associativity_step_Q_copyleft_index_bound. fom_gap_pfp_associativity_step_Q_copyleft_index_bound + S (fom_index_pfp_associativity_step_Q_copyleft) = M) -> exists fom_value_pfp_associativity_step_Q_copyleft. ((((exists fom_beta_height_pfp_associativity_step_Q_copyleft_entry. fom_beta_height_pfp_associativity_step_Q_copyleft_entry + S (fom_value_pfp_associativity_step_Q_copyleft) = S ((S (fom_index_pfp_associativity_step_Q_copyleft)) * bc)) /\\ exists fom_beta_quotient_pfp_associativity_step_Q_copyleft_entry. bb = fom_beta_quotient_pfp_associativity_step_Q_copyleft_entry * S ((S (fom_index_pfp_associativity_step_Q_copyleft)) * bc) + (fom_value_pfp_associativity_step_Q_copyleft))) /\\ (exists fom_gap_pfp_associativity_step_Q_copyleft_value_bound. fom_gap_pfp_associativity_step_Q_copyleft_value_bound + S (fom_value_pfp_associativity_step_Q_copyleft) = p))) /\\ (((forall fom_index_pfp_associativity_step_Q_copyright. (exists fom_gap_pfp_associativity_step_Q_copyright_index_bound. fom_gap_pfp_associativity_step_Q_copyright_index_bound + S (fom_index_pfp_associativity_step_Q_copyright) = S j) -> exists fom_value_pfp_associativity_step_Q_copyright. ((((exists fom_beta_height_pfp_associativity_step_Q_copyright_entry. fom_beta_height_pfp_associativity_step_Q_copyright_entry + S (fom_value_pfp_associativity_step_Q_copyright) = S ((S (fom_index_pfp_associativity_step_Q_copyright)) * dc)) /\\ exists fom_beta_quotient_pfp_associativity_step_Q_copyright_entry. db = fom_beta_quotient_pfp_associativity_step_Q_copyright_entry * S ((S (fom_index_pfp_associativity_step_Q_copyright)) * dc) + (fom_value_pfp_associativity_step_Q_copyright))) /\\ (exists fom_gap_pfp_associativity_step_Q_copyright_value_bound. fom_gap_pfp_associativity_step_Q_copyright_value_bound + S (fom_value_pfp_associativity_step_Q_copyright) = p))) /\\ (((((((M)=0 \\/ (S j)=0) /\\ (((k)=0)))) \\/ (((~((M)=0)) /\\ (((~((S j)=0)) /\\ (((M)+(S j)=S (k)))))))) /\\ ((forall pfc_index_associativity_step_Q_copycoefficients. (exists pfa_gap_associativity_step_Q_copycoefficientsbound. pfa_gap_associativity_step_Q_copycoefficientsbound + S (pfc_index_associativity_step_Q_copycoefficients) = (k)) -> exists pfc_value_associativity_step_Q_copycoefficients. ((((exists ff_h_pfp_associativity_step_Q_copycoefficientsentry. ff_h_pfp_associativity_step_Q_copycoefficientsentry + S (pfc_value_associativity_step_Q_copycoefficients) = S ((S (pfc_index_associativity_step_Q_copycoefficients)) * qxc)) /\\ exists ff_q_pfp_associativity_step_Q_copycoefficientsentry. qxb = ff_q_pfp_associativity_step_Q_copycoefficientsentry * S ((S (pfc_index_associativity_step_Q_copycoefficients)) * qxc) + (pfc_value_associativity_step_Q_copycoefficients))) /\\ ((exists pfc_terms_code_associativity_step_Q_copycoefficientscoefficient pfc_terms_scale_associativity_step_Q_copycoefficientscoefficient pfc_natural_sum_associativity_step_Q_copycoefficientscoefficient. ((forall pfc_index_associativity_step_Q_copycoefficientscoefficientdiagonal. (exists pfa_gap_associativity_step_Q_copycoefficientscoefficientdiagonalbound. pfa_gap_associativity_step_Q_copycoefficientscoefficientdiagonalbound + S (pfc_index_associativity_step_Q_copycoefficientscoefficientdiagonal) = (S (pfc_index_associativity_step_Q_copycoefficients))) -> exists pfc_value_associativity_step_Q_copycoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associativity_step_Q_copycoefficientscoefficientdiagonalentry. ff_h_pfp_associativity_step_Q_copycoefficientscoefficientdiagonalentry + S (pfc_value_associativity_step_Q_copycoefficientscoefficientdiagonal) = S ((S (pfc_index_associativity_step_Q_copycoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_step_Q_copycoefficientscoefficient)) /\\ exists ff_q_pfp_associativity_step_Q_copycoefficientscoefficientdiagonalentry. pfc_terms_code_associativity_step_Q_copycoefficientscoefficient = ff_q_pfp_associativity_step_Q_copycoefficientscoefficientdiagonalentry * S ((S (pfc_index_associativity_step_Q_copycoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_step_Q_copycoefficientscoefficient) + (pfc_value_associativity_step_Q_copycoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associativity_step_Q_copycoefficientscoefficientdiagonalterm pfc_left_associativity_step_Q_copycoefficientscoefficientdiagonalterm pfc_right_associativity_step_Q_copycoefficientscoefficientdiagonalterm. (((pfc_index_associativity_step_Q_copycoefficientscoefficientdiagonal)+pfc_complement_associativity_step_Q_copycoefficientscoefficientdiagonalterm=(pfc_index_associativity_step_Q_copycoefficients)) /\\ ((((((exists pfa_gap_associativity_step_Q_copycoefficientscoefficientdiagonaltermleftinside. pfa_gap_associativity_step_Q_copycoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associativity_step_Q_copycoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_associativity_step_Q_copycoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associativity_step_Q_copycoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associativity_step_Q_copycoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associativity_step_Q_copycoefficientscoefficientdiagonal)) * bc)) /\\ exists ff_q_pfp_associativity_step_Q_copycoefficientscoefficientdiagonaltermleftentry. bb = ff_q_pfp_associativity_step_Q_copycoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associativity_step_Q_copycoefficientscoefficientdiagonal)) * bc) + (pfc_left_associativity_step_Q_copycoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_step_Q_copycoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associativity_step_Q_copycoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_associativity_step_Q_copycoefficientscoefficientdiagonal)) /\\ (((pfc_left_associativity_step_Q_copycoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associativity_step_Q_copycoefficientscoefficientdiagonaltermrightinside. pfa_gap_associativity_step_Q_copycoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associativity_step_Q_copycoefficientscoefficientdiagonalterm) = (S j)) /\\ ((((exists ff_h_pfp_associativity_step_Q_copycoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associativity_step_Q_copycoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associativity_step_Q_copycoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associativity_step_Q_copycoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_associativity_step_Q_copycoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_associativity_step_Q_copycoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associativity_step_Q_copycoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_associativity_step_Q_copycoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_step_Q_copycoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associativity_step_Q_copycoefficientscoefficientdiagonaltermrightoutside+(S j)=(pfc_complement_associativity_step_Q_copycoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associativity_step_Q_copycoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associativity_step_Q_copycoefficientscoefficientdiagonal)=pfc_left_associativity_step_Q_copycoefficientscoefficientdiagonalterm*pfc_right_associativity_step_Q_copycoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associativity_step_Q_copycoefficientscoefficientsum fs_v_pfc_associativity_step_Q_copycoefficientscoefficientsum. ((((exists fs_h_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_start. fs_h_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associativity_step_Q_copycoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_start. fs_u_pfc_associativity_step_Q_copycoefficientscoefficientsum = fs_q_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associativity_step_Q_copycoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_terminal. fs_h_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associativity_step_Q_copycoefficientscoefficient) = S ((S (S (pfc_index_associativity_step_Q_copycoefficients))) * fs_v_pfc_associativity_step_Q_copycoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_terminal. fs_u_pfc_associativity_step_Q_copycoefficientscoefficientsum = fs_q_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associativity_step_Q_copycoefficients))) * fs_v_pfc_associativity_step_Q_copycoefficientscoefficientsum) + (pfc_natural_sum_associativity_step_Q_copycoefficientscoefficient))) /\\ forall fs_i_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps = S (pfc_index_associativity_step_Q_copycoefficients)) -> exists fs_a_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps fs_r_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps fs_s_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_step_Q_copycoefficientscoefficient)) /\\ exists fs_q_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associativity_step_Q_copycoefficientscoefficient = fs_q_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_step_Q_copycoefficientscoefficient) + (fs_a_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_step_Q_copycoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associativity_step_Q_copycoefficientscoefficientsum = fs_q_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_step_Q_copycoefficientscoefficientsum) + (fs_r_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_step_Q_copycoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associativity_step_Q_copycoefficientscoefficientsum = fs_q_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_step_Q_copycoefficientscoefficientsum) + (fs_s_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps = fs_r_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps + fs_a_pfc_associativity_step_Q_copycoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associativity_step_Q_copycoefficientscoefficientresiduebound. pfa_gap_associativity_step_Q_copycoefficientscoefficientresiduebound + S (pfc_value_associativity_step_Q_copycoefficients) = (p)) /\\ ((exists pfa_offset_left_associativity_step_Q_copycoefficientscoefficientresiduecongruence pfa_offset_right_associativity_step_Q_copycoefficientscoefficientresiduecongruence. (pfc_natural_sum_associativity_step_Q_copycoefficientscoefficient) + (p) * pfa_offset_left_associativity_step_Q_copycoefficientscoefficientresiduecongruence = (pfc_value_associativity_step_Q_copycoefficients) + (p) * pfa_offset_right_associativity_step_Q_copycoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hQ", "cases hQcopy", "cases hQcopy_right", "cases hQcopy_right_right", "have hprefix_bound : forall fom_index_pfp_associativity_prefix_bound. (exists fom_gap_pfp_associativity_prefix_bound_index_bound. fom_gap_pfp_associativity_prefix_bound_index_bound + S (fom_index_pfp_associativity_prefix_bound) = j) -> exists fom_value_pfp_associativity_prefix_bound. ((((exists fom_beta_height_pfp_associativity_prefix_bound_entry. fom_beta_height_pfp_associativity_prefix_bound_entry + S (fom_value_pfp_associativity_prefix_bound) = S ((S (fom_index_pfp_associativity_prefix_bound)) * dc)) /\\ exists fom_beta_quotient_pfp_associativity_prefix_bound_entry. db = fom_beta_quotient_pfp_associativity_prefix_bound_entry * S ((S (fom_index_pfp_associativity_prefix_bound)) * dc) + (fom_value_pfp_associativity_prefix_bound))) /\\ (exists fom_gap_pfp_associativity_prefix_bound_value_bound. fom_gap_pfp_associativity_prefix_bound_value_bound + S (fom_value_pfp_associativity_prefix_bound) = p))", "specialize matrix_rank_bounded_prefix_drop_last (db)", "specialize matrix_rank_bounded_prefix_drop_last (dc)", "specialize matrix_rank_bounded_prefix_drop_last (j)", "specialize matrix_rank_bounded_prefix_drop_last (p)", "apply matrix_rank_bounded_prefix_drop_last", "exact hQcopy_right_left", "have hlast : exists a. (((exists ff_h_pfp_associativity_actual_last. ff_h_pfp_associativity_actual_last + S (a) = S ((S (j)) * dc)) /\\ exists ff_q_pfp_associativity_actual_last. db = ff_q_pfp_associativity_actual_last * S ((S (j)) * dc) + (a)))", "specialize beta_at_exists (db)", "specialize beta_at_exists (dc)", "specialize beta_at_exists (j)", "apply beta_at_exists", "cases hlast", "have hQlength : exists k0. (((((M)=0 \\/ (j)=0) /\\ (((k0)=0)))) \\/ (((~((M)=0)) /\\ (((~((j)=0)) /\\ (((M)+(j)=S (k0))))))))", "specialize polynomial_product_length_exists (M)", "specialize polynomial_product_length_exists (j)", "apply polynomial_product_length_exists", "cases hQlength", "have hQ0 : exists q0b q0c. ((forall fom_index_pfp_associativity_Q0left. (exists fom_gap_pfp_associativity_Q0left_index_bound. fom_gap_pfp_associativity_Q0left_index_bound + S (fom_index_pfp_associativity_Q0left) = M) -> exists fom_value_pfp_associativity_Q0left. ((((exists fom_beta_height_pfp_associativity_Q0left_entry. fom_beta_height_pfp_associativity_Q0left_entry + S (fom_value_pfp_associativity_Q0left) = S ((S (fom_index_pfp_associativity_Q0left)) * bc)) /\\ exists fom_beta_quotient_pfp_associativity_Q0left_entry. bb = fom_beta_quotient_pfp_associativity_Q0left_entry * S ((S (fom_index_pfp_associativity_Q0left)) * bc) + (fom_value_pfp_associativity_Q0left))) /\\ (exists fom_gap_pfp_associativity_Q0left_value_bound. fom_gap_pfp_associativity_Q0left_value_bound + S (fom_value_pfp_associativity_Q0left) = p))) /\\ (((forall fom_index_pfp_associativity_Q0right. (exists fom_gap_pfp_associativity_Q0right_index_bound. fom_gap_pfp_associativity_Q0right_index_bound + S (fom_index_pfp_associativity_Q0right) = j) -> exists fom_value_pfp_associativity_Q0right. ((((exists fom_beta_height_pfp_associativity_Q0right_entry. fom_beta_height_pfp_associativity_Q0right_entry + S (fom_value_pfp_associativity_Q0right) = S ((S (fom_index_pfp_associativity_Q0right)) * dc)) /\\ exists fom_beta_quotient_pfp_associativity_Q0right_entry. db = fom_beta_quotient_pfp_associativity_Q0right_entry * S ((S (fom_index_pfp_associativity_Q0right)) * dc) + (fom_value_pfp_associativity_Q0right))) /\\ (exists fom_gap_pfp_associativity_Q0right_value_bound. fom_gap_pfp_associativity_Q0right_value_bound + S (fom_value_pfp_associativity_Q0right) = p))) /\\ (((((((M)=0 \\/ (j)=0) /\\ (((x1)=0)))) \\/ (((~((M)=0)) /\\ (((~((j)=0)) /\\ (((M)+(j)=S (x1)))))))) /\\ ((forall pfc_index_associativity_Q0coefficients. (exists pfa_gap_associativity_Q0coefficientsbound. pfa_gap_associativity_Q0coefficientsbound + S (pfc_index_associativity_Q0coefficients) = (x1)) -> exists pfc_value_associativity_Q0coefficients. ((((exists ff_h_pfp_associativity_Q0coefficientsentry. ff_h_pfp_associativity_Q0coefficientsentry + S (pfc_value_associativity_Q0coefficients) = S ((S (pfc_index_associativity_Q0coefficients)) * q0c)) /\\ exists ff_q_pfp_associativity_Q0coefficientsentry. q0b = ff_q_pfp_associativity_Q0coefficientsentry * S ((S (pfc_index_associativity_Q0coefficients)) * q0c) + (pfc_value_associativity_Q0coefficients))) /\\ ((exists pfc_terms_code_associativity_Q0coefficientscoefficient pfc_terms_scale_associativity_Q0coefficientscoefficient pfc_natural_sum_associativity_Q0coefficientscoefficient. ((forall pfc_index_associativity_Q0coefficientscoefficientdiagonal. (exists pfa_gap_associativity_Q0coefficientscoefficientdiagonalbound. pfa_gap_associativity_Q0coefficientscoefficientdiagonalbound + S (pfc_index_associativity_Q0coefficientscoefficientdiagonal) = (S (pfc_index_associativity_Q0coefficients))) -> exists pfc_value_associativity_Q0coefficientscoefficientdiagonal. ((((exists ff_h_pfp_associativity_Q0coefficientscoefficientdiagonalentry. ff_h_pfp_associativity_Q0coefficientscoefficientdiagonalentry + S (pfc_value_associativity_Q0coefficientscoefficientdiagonal) = S ((S (pfc_index_associativity_Q0coefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_Q0coefficientscoefficient)) /\\ exists ff_q_pfp_associativity_Q0coefficientscoefficientdiagonalentry. pfc_terms_code_associativity_Q0coefficientscoefficient = ff_q_pfp_associativity_Q0coefficientscoefficientdiagonalentry * S ((S (pfc_index_associativity_Q0coefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_Q0coefficientscoefficient) + (pfc_value_associativity_Q0coefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associativity_Q0coefficientscoefficientdiagonalterm pfc_left_associativity_Q0coefficientscoefficientdiagonalterm pfc_right_associativity_Q0coefficientscoefficientdiagonalterm. (((pfc_index_associativity_Q0coefficientscoefficientdiagonal)+pfc_complement_associativity_Q0coefficientscoefficientdiagonalterm=(pfc_index_associativity_Q0coefficients)) /\\ ((((((exists pfa_gap_associativity_Q0coefficientscoefficientdiagonaltermleftinside. pfa_gap_associativity_Q0coefficientscoefficientdiagonaltermleftinside + S (pfc_index_associativity_Q0coefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_associativity_Q0coefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associativity_Q0coefficientscoefficientdiagonaltermleftentry + S (pfc_left_associativity_Q0coefficientscoefficientdiagonalterm) = S ((S (pfc_index_associativity_Q0coefficientscoefficientdiagonal)) * bc)) /\\ exists ff_q_pfp_associativity_Q0coefficientscoefficientdiagonaltermleftentry. bb = ff_q_pfp_associativity_Q0coefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associativity_Q0coefficientscoefficientdiagonal)) * bc) + (pfc_left_associativity_Q0coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_Q0coefficientscoefficientdiagonaltermleftoutside. pfc_gap_associativity_Q0coefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_associativity_Q0coefficientscoefficientdiagonal)) /\\ (((pfc_left_associativity_Q0coefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associativity_Q0coefficientscoefficientdiagonaltermrightinside. pfa_gap_associativity_Q0coefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associativity_Q0coefficientscoefficientdiagonalterm) = (j)) /\\ ((((exists ff_h_pfp_associativity_Q0coefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associativity_Q0coefficientscoefficientdiagonaltermrightentry + S (pfc_right_associativity_Q0coefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associativity_Q0coefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_associativity_Q0coefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_associativity_Q0coefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associativity_Q0coefficientscoefficientdiagonalterm)) * dc) + (pfc_right_associativity_Q0coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_Q0coefficientscoefficientdiagonaltermrightoutside. pfc_gap_associativity_Q0coefficientscoefficientdiagonaltermrightoutside+(j)=(pfc_complement_associativity_Q0coefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associativity_Q0coefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associativity_Q0coefficientscoefficientdiagonal)=pfc_left_associativity_Q0coefficientscoefficientdiagonalterm*pfc_right_associativity_Q0coefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associativity_Q0coefficientscoefficientsum fs_v_pfc_associativity_Q0coefficientscoefficientsum. ((((exists fs_h_pfc_associativity_Q0coefficientscoefficientsum_body_start. fs_h_pfc_associativity_Q0coefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associativity_Q0coefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_Q0coefficientscoefficientsum_body_start. fs_u_pfc_associativity_Q0coefficientscoefficientsum = fs_q_pfc_associativity_Q0coefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associativity_Q0coefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associativity_Q0coefficientscoefficientsum_body_terminal. fs_h_pfc_associativity_Q0coefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associativity_Q0coefficientscoefficient) = S ((S (S (pfc_index_associativity_Q0coefficients))) * fs_v_pfc_associativity_Q0coefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_Q0coefficientscoefficientsum_body_terminal. fs_u_pfc_associativity_Q0coefficientscoefficientsum = fs_q_pfc_associativity_Q0coefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associativity_Q0coefficients))) * fs_v_pfc_associativity_Q0coefficientscoefficientsum) + (pfc_natural_sum_associativity_Q0coefficientscoefficient))) /\\ forall fs_i_pfc_associativity_Q0coefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associativity_Q0coefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associativity_Q0coefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associativity_Q0coefficientscoefficientsum_body_steps = S (pfc_index_associativity_Q0coefficients)) -> exists fs_a_pfc_associativity_Q0coefficientscoefficientsum_body_steps fs_r_pfc_associativity_Q0coefficientscoefficientsum_body_steps fs_s_pfc_associativity_Q0coefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associativity_Q0coefficientscoefficientsum_body_steps_summand. fs_h_pfc_associativity_Q0coefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associativity_Q0coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_Q0coefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_Q0coefficientscoefficient)) /\\ exists fs_q_pfc_associativity_Q0coefficientscoefficientsum_body_steps_summand. pfc_terms_code_associativity_Q0coefficientscoefficient = fs_q_pfc_associativity_Q0coefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associativity_Q0coefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_Q0coefficientscoefficient) + (fs_a_pfc_associativity_Q0coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_Q0coefficientscoefficientsum_body_steps_partial. fs_h_pfc_associativity_Q0coefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associativity_Q0coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_Q0coefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_Q0coefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_Q0coefficientscoefficientsum_body_steps_partial. fs_u_pfc_associativity_Q0coefficientscoefficientsum = fs_q_pfc_associativity_Q0coefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associativity_Q0coefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_Q0coefficientscoefficientsum) + (fs_r_pfc_associativity_Q0coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_Q0coefficientscoefficientsum_body_steps_successor. fs_h_pfc_associativity_Q0coefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associativity_Q0coefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associativity_Q0coefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_Q0coefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_Q0coefficientscoefficientsum_body_steps_successor. fs_u_pfc_associativity_Q0coefficientscoefficientsum = fs_q_pfc_associativity_Q0coefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associativity_Q0coefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_Q0coefficientscoefficientsum) + (fs_s_pfc_associativity_Q0coefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associativity_Q0coefficientscoefficientsum_body_steps = fs_r_pfc_associativity_Q0coefficientscoefficientsum_body_steps + fs_a_pfc_associativity_Q0coefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associativity_Q0coefficientscoefficientresiduebound. pfa_gap_associativity_Q0coefficientscoefficientresiduebound + S (pfc_value_associativity_Q0coefficients) = (p)) /\\ ((exists pfa_offset_left_associativity_Q0coefficientscoefficientresiduecongruence pfa_offset_right_associativity_Q0coefficientscoefficientresiduecongruence. (pfc_natural_sum_associativity_Q0coefficientscoefficient) + (p) * pfa_offset_left_associativity_Q0coefficientscoefficientresiduecongruence = (pfc_value_associativity_Q0coefficients) + (p) * pfa_offset_right_associativity_Q0coefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (bb)", "specialize prime_field_polynomial_convolution_at_length_exists (bc)", "specialize prime_field_polynomial_convolution_at_length_exists (M)", "specialize prime_field_polynomial_convolution_at_length_exists (db)", "specialize prime_field_polynomial_convolution_at_length_exists (dc)", "specialize prime_field_polynomial_convolution_at_length_exists (j)", "specialize prime_field_polynomial_convolution_at_length_exists (x1)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hABcopy_right_left", "exact hprefix_bound", "exact hQlength_witness", "cases hQ0", "cases hQ0_witness", "have hRlength : exists u0. (((((N)=0 \\/ (j)=0) /\\ (((u0)=0)))) \\/ (((~((N)=0)) /\\ (((~((j)=0)) /\\ (((N)+(j)=S (u0))))))))", "specialize polynomial_product_length_exists (N)", "specialize polynomial_product_length_exists (j)", "apply polynomial_product_length_exists", "cases hRlength", "have hR0 : exists r0b r0c. ((forall fom_index_pfp_associativity_R0left. (exists fom_gap_pfp_associativity_R0left_index_bound. fom_gap_pfp_associativity_R0left_index_bound + S (fom_index_pfp_associativity_R0left) = N) -> exists fom_value_pfp_associativity_R0left. ((((exists fom_beta_height_pfp_associativity_R0left_entry. fom_beta_height_pfp_associativity_R0left_entry + S (fom_value_pfp_associativity_R0left) = S ((S (fom_index_pfp_associativity_R0left)) * pc)) /\\ exists fom_beta_quotient_pfp_associativity_R0left_entry. pb = fom_beta_quotient_pfp_associativity_R0left_entry * S ((S (fom_index_pfp_associativity_R0left)) * pc) + (fom_value_pfp_associativity_R0left))) /\\ (exists fom_gap_pfp_associativity_R0left_value_bound. fom_gap_pfp_associativity_R0left_value_bound + S (fom_value_pfp_associativity_R0left) = p))) /\\ (((forall fom_index_pfp_associativity_R0right. (exists fom_gap_pfp_associativity_R0right_index_bound. fom_gap_pfp_associativity_R0right_index_bound + S (fom_index_pfp_associativity_R0right) = j) -> exists fom_value_pfp_associativity_R0right. ((((exists fom_beta_height_pfp_associativity_R0right_entry. fom_beta_height_pfp_associativity_R0right_entry + S (fom_value_pfp_associativity_R0right) = S ((S (fom_index_pfp_associativity_R0right)) * dc)) /\\ exists fom_beta_quotient_pfp_associativity_R0right_entry. db = fom_beta_quotient_pfp_associativity_R0right_entry * S ((S (fom_index_pfp_associativity_R0right)) * dc) + (fom_value_pfp_associativity_R0right))) /\\ (exists fom_gap_pfp_associativity_R0right_value_bound. fom_gap_pfp_associativity_R0right_value_bound + S (fom_value_pfp_associativity_R0right) = p))) /\\ (((((((N)=0 \\/ (j)=0) /\\ (((x4)=0)))) \\/ (((~((N)=0)) /\\ (((~((j)=0)) /\\ (((N)+(j)=S (x4)))))))) /\\ ((forall pfc_index_associativity_R0coefficients. (exists pfa_gap_associativity_R0coefficientsbound. pfa_gap_associativity_R0coefficientsbound + S (pfc_index_associativity_R0coefficients) = (x4)) -> exists pfc_value_associativity_R0coefficients. ((((exists ff_h_pfp_associativity_R0coefficientsentry. ff_h_pfp_associativity_R0coefficientsentry + S (pfc_value_associativity_R0coefficients) = S ((S (pfc_index_associativity_R0coefficients)) * r0c)) /\\ exists ff_q_pfp_associativity_R0coefficientsentry. r0b = ff_q_pfp_associativity_R0coefficientsentry * S ((S (pfc_index_associativity_R0coefficients)) * r0c) + (pfc_value_associativity_R0coefficients))) /\\ ((exists pfc_terms_code_associativity_R0coefficientscoefficient pfc_terms_scale_associativity_R0coefficientscoefficient pfc_natural_sum_associativity_R0coefficientscoefficient. ((forall pfc_index_associativity_R0coefficientscoefficientdiagonal. (exists pfa_gap_associativity_R0coefficientscoefficientdiagonalbound. pfa_gap_associativity_R0coefficientscoefficientdiagonalbound + S (pfc_index_associativity_R0coefficientscoefficientdiagonal) = (S (pfc_index_associativity_R0coefficients))) -> exists pfc_value_associativity_R0coefficientscoefficientdiagonal. ((((exists ff_h_pfp_associativity_R0coefficientscoefficientdiagonalentry. ff_h_pfp_associativity_R0coefficientscoefficientdiagonalentry + S (pfc_value_associativity_R0coefficientscoefficientdiagonal) = S ((S (pfc_index_associativity_R0coefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_R0coefficientscoefficient)) /\\ exists ff_q_pfp_associativity_R0coefficientscoefficientdiagonalentry. pfc_terms_code_associativity_R0coefficientscoefficient = ff_q_pfp_associativity_R0coefficientscoefficientdiagonalentry * S ((S (pfc_index_associativity_R0coefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_R0coefficientscoefficient) + (pfc_value_associativity_R0coefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associativity_R0coefficientscoefficientdiagonalterm pfc_left_associativity_R0coefficientscoefficientdiagonalterm pfc_right_associativity_R0coefficientscoefficientdiagonalterm. (((pfc_index_associativity_R0coefficientscoefficientdiagonal)+pfc_complement_associativity_R0coefficientscoefficientdiagonalterm=(pfc_index_associativity_R0coefficients)) /\\ ((((((exists pfa_gap_associativity_R0coefficientscoefficientdiagonaltermleftinside. pfa_gap_associativity_R0coefficientscoefficientdiagonaltermleftinside + S (pfc_index_associativity_R0coefficientscoefficientdiagonal) = (N)) /\\ ((((exists ff_h_pfp_associativity_R0coefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associativity_R0coefficientscoefficientdiagonaltermleftentry + S (pfc_left_associativity_R0coefficientscoefficientdiagonalterm) = S ((S (pfc_index_associativity_R0coefficientscoefficientdiagonal)) * pc)) /\\ exists ff_q_pfp_associativity_R0coefficientscoefficientdiagonaltermleftentry. pb = ff_q_pfp_associativity_R0coefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associativity_R0coefficientscoefficientdiagonal)) * pc) + (pfc_left_associativity_R0coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_R0coefficientscoefficientdiagonaltermleftoutside. pfc_gap_associativity_R0coefficientscoefficientdiagonaltermleftoutside+(N)=(pfc_index_associativity_R0coefficientscoefficientdiagonal)) /\\ (((pfc_left_associativity_R0coefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associativity_R0coefficientscoefficientdiagonaltermrightinside. pfa_gap_associativity_R0coefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associativity_R0coefficientscoefficientdiagonalterm) = (j)) /\\ ((((exists ff_h_pfp_associativity_R0coefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associativity_R0coefficientscoefficientdiagonaltermrightentry + S (pfc_right_associativity_R0coefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associativity_R0coefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_associativity_R0coefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_associativity_R0coefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associativity_R0coefficientscoefficientdiagonalterm)) * dc) + (pfc_right_associativity_R0coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_R0coefficientscoefficientdiagonaltermrightoutside. pfc_gap_associativity_R0coefficientscoefficientdiagonaltermrightoutside+(j)=(pfc_complement_associativity_R0coefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associativity_R0coefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associativity_R0coefficientscoefficientdiagonal)=pfc_left_associativity_R0coefficientscoefficientdiagonalterm*pfc_right_associativity_R0coefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associativity_R0coefficientscoefficientsum fs_v_pfc_associativity_R0coefficientscoefficientsum. ((((exists fs_h_pfc_associativity_R0coefficientscoefficientsum_body_start. fs_h_pfc_associativity_R0coefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associativity_R0coefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_R0coefficientscoefficientsum_body_start. fs_u_pfc_associativity_R0coefficientscoefficientsum = fs_q_pfc_associativity_R0coefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associativity_R0coefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associativity_R0coefficientscoefficientsum_body_terminal. fs_h_pfc_associativity_R0coefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associativity_R0coefficientscoefficient) = S ((S (S (pfc_index_associativity_R0coefficients))) * fs_v_pfc_associativity_R0coefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_R0coefficientscoefficientsum_body_terminal. fs_u_pfc_associativity_R0coefficientscoefficientsum = fs_q_pfc_associativity_R0coefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associativity_R0coefficients))) * fs_v_pfc_associativity_R0coefficientscoefficientsum) + (pfc_natural_sum_associativity_R0coefficientscoefficient))) /\\ forall fs_i_pfc_associativity_R0coefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associativity_R0coefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associativity_R0coefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associativity_R0coefficientscoefficientsum_body_steps = S (pfc_index_associativity_R0coefficients)) -> exists fs_a_pfc_associativity_R0coefficientscoefficientsum_body_steps fs_r_pfc_associativity_R0coefficientscoefficientsum_body_steps fs_s_pfc_associativity_R0coefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associativity_R0coefficientscoefficientsum_body_steps_summand. fs_h_pfc_associativity_R0coefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associativity_R0coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_R0coefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_R0coefficientscoefficient)) /\\ exists fs_q_pfc_associativity_R0coefficientscoefficientsum_body_steps_summand. pfc_terms_code_associativity_R0coefficientscoefficient = fs_q_pfc_associativity_R0coefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associativity_R0coefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_R0coefficientscoefficient) + (fs_a_pfc_associativity_R0coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_R0coefficientscoefficientsum_body_steps_partial. fs_h_pfc_associativity_R0coefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associativity_R0coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_R0coefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_R0coefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_R0coefficientscoefficientsum_body_steps_partial. fs_u_pfc_associativity_R0coefficientscoefficientsum = fs_q_pfc_associativity_R0coefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associativity_R0coefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_R0coefficientscoefficientsum) + (fs_r_pfc_associativity_R0coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_R0coefficientscoefficientsum_body_steps_successor. fs_h_pfc_associativity_R0coefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associativity_R0coefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associativity_R0coefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_R0coefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_R0coefficientscoefficientsum_body_steps_successor. fs_u_pfc_associativity_R0coefficientscoefficientsum = fs_q_pfc_associativity_R0coefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associativity_R0coefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_R0coefficientscoefficientsum) + (fs_s_pfc_associativity_R0coefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associativity_R0coefficientscoefficientsum_body_steps = fs_r_pfc_associativity_R0coefficientscoefficientsum_body_steps + fs_a_pfc_associativity_R0coefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associativity_R0coefficientscoefficientresiduebound. pfa_gap_associativity_R0coefficientscoefficientresiduebound + S (pfc_value_associativity_R0coefficients) = (p)) /\\ ((exists pfa_offset_left_associativity_R0coefficientscoefficientresiduecongruence pfa_offset_right_associativity_R0coefficientscoefficientresiduecongruence. (pfc_natural_sum_associativity_R0coefficientscoefficient) + (p) * pfa_offset_left_associativity_R0coefficientscoefficientresiduecongruence = (pfc_value_associativity_R0coefficients) + (p) * pfa_offset_right_associativity_R0coefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (pb)", "specialize prime_field_polynomial_convolution_at_length_exists (pc)", "specialize prime_field_polynomial_convolution_at_length_exists (N)", "specialize prime_field_polynomial_convolution_at_length_exists (db)", "specialize prime_field_polynomial_convolution_at_length_exists (dc)", "specialize prime_field_polynomial_convolution_at_length_exists (j)", "specialize prime_field_polynomial_convolution_at_length_exists (x4)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hPbound", "exact hprefix_bound", "exact hRlength_witness", "cases hR0", "cases hR0_witness", "have hQ0bound : forall fom_index_pfp_associativity_Q0_bound. (exists fom_gap_pfp_associativity_Q0_bound_index_bound. fom_gap_pfp_associativity_Q0_bound_index_bound + S (fom_index_pfp_associativity_Q0_bound) = x1) -> exists fom_value_pfp_associativity_Q0_bound. ((((exists fom_beta_height_pfp_associativity_Q0_bound_entry. fom_beta_height_pfp_associativity_Q0_bound_entry + S (fom_value_pfp_associativity_Q0_bound) = S ((S (fom_index_pfp_associativity_Q0_bound)) * x3)) /\\ exists fom_beta_quotient_pfp_associativity_Q0_bound_entry. x2 = fom_beta_quotient_pfp_associativity_Q0_bound_entry * S ((S (fom_index_pfp_associativity_Q0_bound)) * x3) + (fom_value_pfp_associativity_Q0_bound))) /\\ (exists fom_gap_pfp_associativity_Q0_bound_value_bound. fom_gap_pfp_associativity_Q0_bound_value_bound + S (fom_value_pfp_associativity_Q0_bound) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (bb)", "specialize prime_field_polynomial_convolution_bounded (bc)", "specialize prime_field_polynomial_convolution_bounded (M)", "specialize prime_field_polynomial_convolution_bounded (db)", "specialize prime_field_polynomial_convolution_bounded (dc)", "specialize prime_field_polynomial_convolution_bounded (j)", "specialize prime_field_polynomial_convolution_bounded (x2)", "specialize prime_field_polynomial_convolution_bounded (x3)", "specialize prime_field_polynomial_convolution_bounded (x1)", "apply prime_field_polynomial_convolution_bounded", "exact hQ0_witness_witness", "have hSlength : exists v0. (((((L)=0 \\/ (x1)=0) /\\ (((v0)=0)))) \\/ (((~((L)=0)) /\\ (((~((x1)=0)) /\\ (((L)+(x1)=S (v0))))))))", "specialize polynomial_product_length_exists (L)", "specialize polynomial_product_length_exists (x1)", "apply polynomial_product_length_exists", "cases hSlength", "have hS0 : exists s0b s0c. ((forall fom_index_pfp_associativity_S0left. (exists fom_gap_pfp_associativity_S0left_index_bound. fom_gap_pfp_associativity_S0left_index_bound + S (fom_index_pfp_associativity_S0left) = L) -> exists fom_value_pfp_associativity_S0left. ((((exists fom_beta_height_pfp_associativity_S0left_entry. fom_beta_height_pfp_associativity_S0left_entry + S (fom_value_pfp_associativity_S0left) = S ((S (fom_index_pfp_associativity_S0left)) * ac)) /\\ exists fom_beta_quotient_pfp_associativity_S0left_entry. ab = fom_beta_quotient_pfp_associativity_S0left_entry * S ((S (fom_index_pfp_associativity_S0left)) * ac) + (fom_value_pfp_associativity_S0left))) /\\ (exists fom_gap_pfp_associativity_S0left_value_bound. fom_gap_pfp_associativity_S0left_value_bound + S (fom_value_pfp_associativity_S0left) = p))) /\\ (((forall fom_index_pfp_associativity_S0right. (exists fom_gap_pfp_associativity_S0right_index_bound. fom_gap_pfp_associativity_S0right_index_bound + S (fom_index_pfp_associativity_S0right) = x1) -> exists fom_value_pfp_associativity_S0right. ((((exists fom_beta_height_pfp_associativity_S0right_entry. fom_beta_height_pfp_associativity_S0right_entry + S (fom_value_pfp_associativity_S0right) = S ((S (fom_index_pfp_associativity_S0right)) * x3)) /\\ exists fom_beta_quotient_pfp_associativity_S0right_entry. x2 = fom_beta_quotient_pfp_associativity_S0right_entry * S ((S (fom_index_pfp_associativity_S0right)) * x3) + (fom_value_pfp_associativity_S0right))) /\\ (exists fom_gap_pfp_associativity_S0right_value_bound. fom_gap_pfp_associativity_S0right_value_bound + S (fom_value_pfp_associativity_S0right) = p))) /\\ (((((((L)=0 \\/ (x1)=0) /\\ (((x7)=0)))) \\/ (((~((L)=0)) /\\ (((~((x1)=0)) /\\ (((L)+(x1)=S (x7)))))))) /\\ ((forall pfc_index_associativity_S0coefficients. (exists pfa_gap_associativity_S0coefficientsbound. pfa_gap_associativity_S0coefficientsbound + S (pfc_index_associativity_S0coefficients) = (x7)) -> exists pfc_value_associativity_S0coefficients. ((((exists ff_h_pfp_associativity_S0coefficientsentry. ff_h_pfp_associativity_S0coefficientsentry + S (pfc_value_associativity_S0coefficients) = S ((S (pfc_index_associativity_S0coefficients)) * s0c)) /\\ exists ff_q_pfp_associativity_S0coefficientsentry. s0b = ff_q_pfp_associativity_S0coefficientsentry * S ((S (pfc_index_associativity_S0coefficients)) * s0c) + (pfc_value_associativity_S0coefficients))) /\\ ((exists pfc_terms_code_associativity_S0coefficientscoefficient pfc_terms_scale_associativity_S0coefficientscoefficient pfc_natural_sum_associativity_S0coefficientscoefficient. ((forall pfc_index_associativity_S0coefficientscoefficientdiagonal. (exists pfa_gap_associativity_S0coefficientscoefficientdiagonalbound. pfa_gap_associativity_S0coefficientscoefficientdiagonalbound + S (pfc_index_associativity_S0coefficientscoefficientdiagonal) = (S (pfc_index_associativity_S0coefficients))) -> exists pfc_value_associativity_S0coefficientscoefficientdiagonal. ((((exists ff_h_pfp_associativity_S0coefficientscoefficientdiagonalentry. ff_h_pfp_associativity_S0coefficientscoefficientdiagonalentry + S (pfc_value_associativity_S0coefficientscoefficientdiagonal) = S ((S (pfc_index_associativity_S0coefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_S0coefficientscoefficient)) /\\ exists ff_q_pfp_associativity_S0coefficientscoefficientdiagonalentry. pfc_terms_code_associativity_S0coefficientscoefficient = ff_q_pfp_associativity_S0coefficientscoefficientdiagonalentry * S ((S (pfc_index_associativity_S0coefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_S0coefficientscoefficient) + (pfc_value_associativity_S0coefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associativity_S0coefficientscoefficientdiagonalterm pfc_left_associativity_S0coefficientscoefficientdiagonalterm pfc_right_associativity_S0coefficientscoefficientdiagonalterm. (((pfc_index_associativity_S0coefficientscoefficientdiagonal)+pfc_complement_associativity_S0coefficientscoefficientdiagonalterm=(pfc_index_associativity_S0coefficients)) /\\ ((((((exists pfa_gap_associativity_S0coefficientscoefficientdiagonaltermleftinside. pfa_gap_associativity_S0coefficientscoefficientdiagonaltermleftinside + S (pfc_index_associativity_S0coefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_associativity_S0coefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associativity_S0coefficientscoefficientdiagonaltermleftentry + S (pfc_left_associativity_S0coefficientscoefficientdiagonalterm) = S ((S (pfc_index_associativity_S0coefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_associativity_S0coefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_associativity_S0coefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associativity_S0coefficientscoefficientdiagonal)) * ac) + (pfc_left_associativity_S0coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_S0coefficientscoefficientdiagonaltermleftoutside. pfc_gap_associativity_S0coefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_associativity_S0coefficientscoefficientdiagonal)) /\\ (((pfc_left_associativity_S0coefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associativity_S0coefficientscoefficientdiagonaltermrightinside. pfa_gap_associativity_S0coefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associativity_S0coefficientscoefficientdiagonalterm) = (x1)) /\\ ((((exists ff_h_pfp_associativity_S0coefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associativity_S0coefficientscoefficientdiagonaltermrightentry + S (pfc_right_associativity_S0coefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associativity_S0coefficientscoefficientdiagonalterm)) * x3)) /\\ exists ff_q_pfp_associativity_S0coefficientscoefficientdiagonaltermrightentry. x2 = ff_q_pfp_associativity_S0coefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associativity_S0coefficientscoefficientdiagonalterm)) * x3) + (pfc_right_associativity_S0coefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_S0coefficientscoefficientdiagonaltermrightoutside. pfc_gap_associativity_S0coefficientscoefficientdiagonaltermrightoutside+(x1)=(pfc_complement_associativity_S0coefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associativity_S0coefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associativity_S0coefficientscoefficientdiagonal)=pfc_left_associativity_S0coefficientscoefficientdiagonalterm*pfc_right_associativity_S0coefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associativity_S0coefficientscoefficientsum fs_v_pfc_associativity_S0coefficientscoefficientsum. ((((exists fs_h_pfc_associativity_S0coefficientscoefficientsum_body_start. fs_h_pfc_associativity_S0coefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associativity_S0coefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_S0coefficientscoefficientsum_body_start. fs_u_pfc_associativity_S0coefficientscoefficientsum = fs_q_pfc_associativity_S0coefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associativity_S0coefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associativity_S0coefficientscoefficientsum_body_terminal. fs_h_pfc_associativity_S0coefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associativity_S0coefficientscoefficient) = S ((S (S (pfc_index_associativity_S0coefficients))) * fs_v_pfc_associativity_S0coefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_S0coefficientscoefficientsum_body_terminal. fs_u_pfc_associativity_S0coefficientscoefficientsum = fs_q_pfc_associativity_S0coefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associativity_S0coefficients))) * fs_v_pfc_associativity_S0coefficientscoefficientsum) + (pfc_natural_sum_associativity_S0coefficientscoefficient))) /\\ forall fs_i_pfc_associativity_S0coefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associativity_S0coefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associativity_S0coefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associativity_S0coefficientscoefficientsum_body_steps = S (pfc_index_associativity_S0coefficients)) -> exists fs_a_pfc_associativity_S0coefficientscoefficientsum_body_steps fs_r_pfc_associativity_S0coefficientscoefficientsum_body_steps fs_s_pfc_associativity_S0coefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associativity_S0coefficientscoefficientsum_body_steps_summand. fs_h_pfc_associativity_S0coefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associativity_S0coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_S0coefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_S0coefficientscoefficient)) /\\ exists fs_q_pfc_associativity_S0coefficientscoefficientsum_body_steps_summand. pfc_terms_code_associativity_S0coefficientscoefficient = fs_q_pfc_associativity_S0coefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associativity_S0coefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_S0coefficientscoefficient) + (fs_a_pfc_associativity_S0coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_S0coefficientscoefficientsum_body_steps_partial. fs_h_pfc_associativity_S0coefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associativity_S0coefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_S0coefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_S0coefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_S0coefficientscoefficientsum_body_steps_partial. fs_u_pfc_associativity_S0coefficientscoefficientsum = fs_q_pfc_associativity_S0coefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associativity_S0coefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_S0coefficientscoefficientsum) + (fs_r_pfc_associativity_S0coefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_S0coefficientscoefficientsum_body_steps_successor. fs_h_pfc_associativity_S0coefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associativity_S0coefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associativity_S0coefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_S0coefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_S0coefficientscoefficientsum_body_steps_successor. fs_u_pfc_associativity_S0coefficientscoefficientsum = fs_q_pfc_associativity_S0coefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associativity_S0coefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_S0coefficientscoefficientsum) + (fs_s_pfc_associativity_S0coefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associativity_S0coefficientscoefficientsum_body_steps = fs_r_pfc_associativity_S0coefficientscoefficientsum_body_steps + fs_a_pfc_associativity_S0coefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associativity_S0coefficientscoefficientresiduebound. pfa_gap_associativity_S0coefficientscoefficientresiduebound + S (pfc_value_associativity_S0coefficients) = (p)) /\\ ((exists pfa_offset_left_associativity_S0coefficientscoefficientresiduecongruence pfa_offset_right_associativity_S0coefficientscoefficientresiduecongruence. (pfc_natural_sum_associativity_S0coefficientscoefficient) + (p) * pfa_offset_left_associativity_S0coefficientscoefficientresiduecongruence = (pfc_value_associativity_S0coefficients) + (p) * pfa_offset_right_associativity_S0coefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (ab)", "specialize prime_field_polynomial_convolution_at_length_exists (ac)", "specialize prime_field_polynomial_convolution_at_length_exists (L)", "specialize prime_field_polynomial_convolution_at_length_exists (x2)", "specialize prime_field_polynomial_convolution_at_length_exists (x3)", "specialize prime_field_polynomial_convolution_at_length_exists (x1)", "specialize prime_field_polynomial_convolution_at_length_exists (x7)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hABcopy_left", "exact hQ0bound", "exact hSlength_witness", "cases hS0", "cases hS0_witness", "have hprevious : forall pfrep_power_associativity_actual_IH pfrep_left_associativity_actual_IH pfrep_right_associativity_actual_IH. ((exists pfrep_position_associativity_actual_IHfirst. ((pfrep_position_associativity_actual_IHfirst+S (pfrep_power_associativity_actual_IH)=(x4)) /\\ ((((exists ff_h_pfp_associativity_actual_IHfirstentry. ff_h_pfp_associativity_actual_IHfirstentry + S (pfrep_left_associativity_actual_IH) = S ((S (pfrep_position_associativity_actual_IHfirst)) * x6)) /\\ exists ff_q_pfp_associativity_actual_IHfirstentry. x5 = ff_q_pfp_associativity_actual_IHfirstentry * S ((S (pfrep_position_associativity_actual_IHfirst)) * x6) + (pfrep_left_associativity_actual_IH)))))) \\/ (((exists pfrep_gap_associativity_actual_IHfirstoutside. pfrep_gap_associativity_actual_IHfirstoutside+(x4)=(pfrep_power_associativity_actual_IH)) /\\ (((pfrep_left_associativity_actual_IH)=0))))) -> ((exists pfrep_position_associativity_actual_IHsecond. ((pfrep_position_associativity_actual_IHsecond+S (pfrep_power_associativity_actual_IH)=(x7)) /\\ ((((exists ff_h_pfp_associativity_actual_IHsecondentry. ff_h_pfp_associativity_actual_IHsecondentry + S (pfrep_right_associativity_actual_IH) = S ((S (pfrep_position_associativity_actual_IHsecond)) * x9)) /\\ exists ff_q_pfp_associativity_actual_IHsecondentry. x8 = ff_q_pfp_associativity_actual_IHsecondentry * S ((S (pfrep_position_associativity_actual_IHsecond)) * x9) + (pfrep_right_associativity_actual_IH)))))) \\/ (((exists pfrep_gap_associativity_actual_IHsecondoutside. pfrep_gap_associativity_actual_IHsecondoutside+(x7)=(pfrep_power_associativity_actual_IH)) /\\ (((pfrep_right_associativity_actual_IH)=0))))) -> pfrep_left_associativity_actual_IH=pfrep_right_associativity_actual_IH", "specialize IH (db)", "specialize IH (dc)", "specialize IH (x2)", "specialize IH (x3)", "specialize IH (x1)", "specialize IH (x5)", "specialize IH (x6)", "specialize IH (x4)", "specialize IH (x8)", "specialize IH (x9)", "specialize IH (x7)", "apply IH", "exact hQ0_witness_witness", "exact hR0_witness_witness", "exact hS0_witness_witness", "specialize prime_field_polynomial_convolution_associativity_append_step (p)", "specialize prime_field_polynomial_convolution_associativity_append_step (ab)", "specialize prime_field_polynomial_convolution_associativity_append_step (ac)", "specialize prime_field_polynomial_convolution_associativity_append_step (L)", "specialize prime_field_polynomial_convolution_associativity_append_step (bb)", "specialize prime_field_polynomial_convolution_associativity_append_step (bc)", "specialize prime_field_polynomial_convolution_associativity_append_step (M)", "specialize prime_field_polynomial_convolution_associativity_append_step (pb)", "specialize prime_field_polynomial_convolution_associativity_append_step (pc)", "specialize prime_field_polynomial_convolution_associativity_append_step (N)", "specialize prime_field_polynomial_convolution_associativity_append_step (db)", "specialize prime_field_polynomial_convolution_associativity_append_step (dc)", "specialize prime_field_polynomial_convolution_associativity_append_step (j)", "specialize prime_field_polynomial_convolution_associativity_append_step (x2)", "specialize prime_field_polynomial_convolution_associativity_append_step (x3)", "specialize prime_field_polynomial_convolution_associativity_append_step (x1)", "specialize prime_field_polynomial_convolution_associativity_append_step (x5)", "specialize prime_field_polynomial_convolution_associativity_append_step (x6)", "specialize prime_field_polynomial_convolution_associativity_append_step (x4)", "specialize prime_field_polynomial_convolution_associativity_append_step (x8)", "specialize prime_field_polynomial_convolution_associativity_append_step (x9)", "specialize prime_field_polynomial_convolution_associativity_append_step (x7)", "specialize prime_field_polynomial_convolution_associativity_append_step (x)", "specialize prime_field_polynomial_convolution_associativity_append_step (db)", "specialize prime_field_polynomial_convolution_associativity_append_step (dc)", "specialize prime_field_polynomial_convolution_associativity_append_step (qxb)", "specialize prime_field_polynomial_convolution_associativity_append_step (qxc)", "specialize prime_field_polynomial_convolution_associativity_append_step (k)", "specialize prime_field_polynomial_convolution_associativity_append_step (rxb)", "specialize prime_field_polynomial_convolution_associativity_append_step (rxc)", "specialize prime_field_polynomial_convolution_associativity_append_step (u)", "specialize prime_field_polynomial_convolution_associativity_append_step (sxb)", "specialize prime_field_polynomial_convolution_associativity_append_step (sxc)", "specialize prime_field_polynomial_convolution_associativity_append_step (v)", "apply prime_field_polynomial_convolution_associativity_append_step", "exact hp", "exact hAB", "exact hQ0_witness_witness", "exact hR0_witness_witness", "exact hS0_witness_witness", "exact hprevious", "intro i", "intro a", "intro hi", "intro ha", "exact ha", "exact hlast_witness", "exact hQ", "exact hR", "exact hS", "specialize hall (J)", "specialize hall (cb)", "specialize hall (cc)", "specialize hall (qb)", "specialize hall (qc)", "specialize hall (K)", "specialize hall (rb)", "specialize hall (rc)", "specialize hall (U)", "specialize hall (sb)", "specialize hall (sc)", "specialize hall (V)", "apply hall", "exact hBC", "exact hPC", "exact hAQ" ], "script_sha256": "7468ced32f238f954aa1611ef0f23a72a304901f0e6f7ba955d18b5bc11d3a07", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_associativity_induction_candidate.py", "sha256": "8d276a028764cd08e6eaebbf25bb4e21fcd5076a610d356a77d52ba6603ebe4c" }, "statement": "forall p ab ac L bb bc M pb pc N cb cc J qb qc K rb rc U sb sc V. (~((p) = 1) /\\ forall pfa_factor_left_associativity_prime pfa_factor_right_associativity_prime. (p) = pfa_factor_left_associativity_prime * pfa_factor_right_associativity_prime -> pfa_factor_left_associativity_prime = 1 \\/ pfa_factor_right_associativity_prime = 1) -> (((forall fom_index_pfp_associativity_ABleft. (exists fom_gap_pfp_associativity_ABleft_index_bound. fom_gap_pfp_associativity_ABleft_index_bound + S (fom_index_pfp_associativity_ABleft) = L) -> exists fom_value_pfp_associativity_ABleft. ((((exists fom_beta_height_pfp_associativity_ABleft_entry. fom_beta_height_pfp_associativity_ABleft_entry + S (fom_value_pfp_associativity_ABleft) = S ((S (fom_index_pfp_associativity_ABleft)) * ac)) /\\ exists fom_beta_quotient_pfp_associativity_ABleft_entry. ab = fom_beta_quotient_pfp_associativity_ABleft_entry * S ((S (fom_index_pfp_associativity_ABleft)) * ac) + (fom_value_pfp_associativity_ABleft))) /\\ (exists fom_gap_pfp_associativity_ABleft_value_bound. fom_gap_pfp_associativity_ABleft_value_bound + S (fom_value_pfp_associativity_ABleft) = p))) /\\ (((forall fom_index_pfp_associativity_ABright. (exists fom_gap_pfp_associativity_ABright_index_bound. fom_gap_pfp_associativity_ABright_index_bound + S (fom_index_pfp_associativity_ABright) = M) -> exists fom_value_pfp_associativity_ABright. ((((exists fom_beta_height_pfp_associativity_ABright_entry. fom_beta_height_pfp_associativity_ABright_entry + S (fom_value_pfp_associativity_ABright) = S ((S (fom_index_pfp_associativity_ABright)) * bc)) /\\ exists fom_beta_quotient_pfp_associativity_ABright_entry. bb = fom_beta_quotient_pfp_associativity_ABright_entry * S ((S (fom_index_pfp_associativity_ABright)) * bc) + (fom_value_pfp_associativity_ABright))) /\\ (exists fom_gap_pfp_associativity_ABright_value_bound. fom_gap_pfp_associativity_ABright_value_bound + S (fom_value_pfp_associativity_ABright) = p))) /\\ (((((((L)=0 \\/ (M)=0) /\\ (((N)=0)))) \\/ (((~((L)=0)) /\\ (((~((M)=0)) /\\ (((L)+(M)=S (N)))))))) /\\ ((forall pfc_index_associativity_ABcoefficients. (exists pfa_gap_associativity_ABcoefficientsbound. pfa_gap_associativity_ABcoefficientsbound + S (pfc_index_associativity_ABcoefficients) = (N)) -> exists pfc_value_associativity_ABcoefficients. ((((exists ff_h_pfp_associativity_ABcoefficientsentry. ff_h_pfp_associativity_ABcoefficientsentry + S (pfc_value_associativity_ABcoefficients) = S ((S (pfc_index_associativity_ABcoefficients)) * pc)) /\\ exists ff_q_pfp_associativity_ABcoefficientsentry. pb = ff_q_pfp_associativity_ABcoefficientsentry * S ((S (pfc_index_associativity_ABcoefficients)) * pc) + (pfc_value_associativity_ABcoefficients))) /\\ ((exists pfc_terms_code_associativity_ABcoefficientscoefficient pfc_terms_scale_associativity_ABcoefficientscoefficient pfc_natural_sum_associativity_ABcoefficientscoefficient. ((forall pfc_index_associativity_ABcoefficientscoefficientdiagonal. (exists pfa_gap_associativity_ABcoefficientscoefficientdiagonalbound. pfa_gap_associativity_ABcoefficientscoefficientdiagonalbound + S (pfc_index_associativity_ABcoefficientscoefficientdiagonal) = (S (pfc_index_associativity_ABcoefficients))) -> exists pfc_value_associativity_ABcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associativity_ABcoefficientscoefficientdiagonalentry. ff_h_pfp_associativity_ABcoefficientscoefficientdiagonalentry + S (pfc_value_associativity_ABcoefficientscoefficientdiagonal) = S ((S (pfc_index_associativity_ABcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_ABcoefficientscoefficient)) /\\ exists ff_q_pfp_associativity_ABcoefficientscoefficientdiagonalentry. pfc_terms_code_associativity_ABcoefficientscoefficient = ff_q_pfp_associativity_ABcoefficientscoefficientdiagonalentry * S ((S (pfc_index_associativity_ABcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_ABcoefficientscoefficient) + (pfc_value_associativity_ABcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associativity_ABcoefficientscoefficientdiagonalterm pfc_left_associativity_ABcoefficientscoefficientdiagonalterm pfc_right_associativity_ABcoefficientscoefficientdiagonalterm. (((pfc_index_associativity_ABcoefficientscoefficientdiagonal)+pfc_complement_associativity_ABcoefficientscoefficientdiagonalterm=(pfc_index_associativity_ABcoefficients)) /\\ ((((((exists pfa_gap_associativity_ABcoefficientscoefficientdiagonaltermleftinside. pfa_gap_associativity_ABcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associativity_ABcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_associativity_ABcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associativity_ABcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associativity_ABcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associativity_ABcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_associativity_ABcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_associativity_ABcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associativity_ABcoefficientscoefficientdiagonal)) * ac) + (pfc_left_associativity_ABcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_ABcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associativity_ABcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_associativity_ABcoefficientscoefficientdiagonal)) /\\ (((pfc_left_associativity_ABcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associativity_ABcoefficientscoefficientdiagonaltermrightinside. pfa_gap_associativity_ABcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associativity_ABcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_associativity_ABcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associativity_ABcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associativity_ABcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associativity_ABcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_associativity_ABcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_associativity_ABcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associativity_ABcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_associativity_ABcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_ABcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associativity_ABcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_associativity_ABcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associativity_ABcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associativity_ABcoefficientscoefficientdiagonal)=pfc_left_associativity_ABcoefficientscoefficientdiagonalterm*pfc_right_associativity_ABcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associativity_ABcoefficientscoefficientsum fs_v_pfc_associativity_ABcoefficientscoefficientsum. ((((exists fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_start. fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associativity_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_start. fs_u_pfc_associativity_ABcoefficientscoefficientsum = fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associativity_ABcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_terminal. fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associativity_ABcoefficientscoefficient) = S ((S (S (pfc_index_associativity_ABcoefficients))) * fs_v_pfc_associativity_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_terminal. fs_u_pfc_associativity_ABcoefficientscoefficientsum = fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associativity_ABcoefficients))) * fs_v_pfc_associativity_ABcoefficientscoefficientsum) + (pfc_natural_sum_associativity_ABcoefficientscoefficient))) /\\ forall fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associativity_ABcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associativity_ABcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps = S (pfc_index_associativity_ABcoefficients)) -> exists fs_a_pfc_associativity_ABcoefficientscoefficientsum_body_steps fs_r_pfc_associativity_ABcoefficientscoefficientsum_body_steps fs_s_pfc_associativity_ABcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associativity_ABcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_ABcoefficientscoefficient)) /\\ exists fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associativity_ABcoefficientscoefficient = fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_ABcoefficientscoefficient) + (fs_a_pfc_associativity_ABcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associativity_ABcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associativity_ABcoefficientscoefficientsum = fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_ABcoefficientscoefficientsum) + (fs_r_pfc_associativity_ABcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associativity_ABcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associativity_ABcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_ABcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associativity_ABcoefficientscoefficientsum = fs_q_pfc_associativity_ABcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associativity_ABcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_ABcoefficientscoefficientsum) + (fs_s_pfc_associativity_ABcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associativity_ABcoefficientscoefficientsum_body_steps = fs_r_pfc_associativity_ABcoefficientscoefficientsum_body_steps + fs_a_pfc_associativity_ABcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associativity_ABcoefficientscoefficientresiduebound. pfa_gap_associativity_ABcoefficientscoefficientresiduebound + S (pfc_value_associativity_ABcoefficients) = (p)) /\\ ((exists pfa_offset_left_associativity_ABcoefficientscoefficientresiduecongruence pfa_offset_right_associativity_ABcoefficientscoefficientresiduecongruence. (pfc_natural_sum_associativity_ABcoefficientscoefficient) + (p) * pfa_offset_left_associativity_ABcoefficientscoefficientresiduecongruence = (pfc_value_associativity_ABcoefficients) + (p) * pfa_offset_right_associativity_ABcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_associativity_BCleft. (exists fom_gap_pfp_associativity_BCleft_index_bound. fom_gap_pfp_associativity_BCleft_index_bound + S (fom_index_pfp_associativity_BCleft) = M) -> exists fom_value_pfp_associativity_BCleft. ((((exists fom_beta_height_pfp_associativity_BCleft_entry. fom_beta_height_pfp_associativity_BCleft_entry + S (fom_value_pfp_associativity_BCleft) = S ((S (fom_index_pfp_associativity_BCleft)) * bc)) /\\ exists fom_beta_quotient_pfp_associativity_BCleft_entry. bb = fom_beta_quotient_pfp_associativity_BCleft_entry * S ((S (fom_index_pfp_associativity_BCleft)) * bc) + (fom_value_pfp_associativity_BCleft))) /\\ (exists fom_gap_pfp_associativity_BCleft_value_bound. fom_gap_pfp_associativity_BCleft_value_bound + S (fom_value_pfp_associativity_BCleft) = p))) /\\ (((forall fom_index_pfp_associativity_BCright. (exists fom_gap_pfp_associativity_BCright_index_bound. fom_gap_pfp_associativity_BCright_index_bound + S (fom_index_pfp_associativity_BCright) = J) -> exists fom_value_pfp_associativity_BCright. ((((exists fom_beta_height_pfp_associativity_BCright_entry. fom_beta_height_pfp_associativity_BCright_entry + S (fom_value_pfp_associativity_BCright) = S ((S (fom_index_pfp_associativity_BCright)) * cc)) /\\ exists fom_beta_quotient_pfp_associativity_BCright_entry. cb = fom_beta_quotient_pfp_associativity_BCright_entry * S ((S (fom_index_pfp_associativity_BCright)) * cc) + (fom_value_pfp_associativity_BCright))) /\\ (exists fom_gap_pfp_associativity_BCright_value_bound. fom_gap_pfp_associativity_BCright_value_bound + S (fom_value_pfp_associativity_BCright) = p))) /\\ (((((((M)=0 \\/ (J)=0) /\\ (((K)=0)))) \\/ (((~((M)=0)) /\\ (((~((J)=0)) /\\ (((M)+(J)=S (K)))))))) /\\ ((forall pfc_index_associativity_BCcoefficients. (exists pfa_gap_associativity_BCcoefficientsbound. pfa_gap_associativity_BCcoefficientsbound + S (pfc_index_associativity_BCcoefficients) = (K)) -> exists pfc_value_associativity_BCcoefficients. ((((exists ff_h_pfp_associativity_BCcoefficientsentry. ff_h_pfp_associativity_BCcoefficientsentry + S (pfc_value_associativity_BCcoefficients) = S ((S (pfc_index_associativity_BCcoefficients)) * qc)) /\\ exists ff_q_pfp_associativity_BCcoefficientsentry. qb = ff_q_pfp_associativity_BCcoefficientsentry * S ((S (pfc_index_associativity_BCcoefficients)) * qc) + (pfc_value_associativity_BCcoefficients))) /\\ ((exists pfc_terms_code_associativity_BCcoefficientscoefficient pfc_terms_scale_associativity_BCcoefficientscoefficient pfc_natural_sum_associativity_BCcoefficientscoefficient. ((forall pfc_index_associativity_BCcoefficientscoefficientdiagonal. (exists pfa_gap_associativity_BCcoefficientscoefficientdiagonalbound. pfa_gap_associativity_BCcoefficientscoefficientdiagonalbound + S (pfc_index_associativity_BCcoefficientscoefficientdiagonal) = (S (pfc_index_associativity_BCcoefficients))) -> exists pfc_value_associativity_BCcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associativity_BCcoefficientscoefficientdiagonalentry. ff_h_pfp_associativity_BCcoefficientscoefficientdiagonalentry + S (pfc_value_associativity_BCcoefficientscoefficientdiagonal) = S ((S (pfc_index_associativity_BCcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_BCcoefficientscoefficient)) /\\ exists ff_q_pfp_associativity_BCcoefficientscoefficientdiagonalentry. pfc_terms_code_associativity_BCcoefficientscoefficient = ff_q_pfp_associativity_BCcoefficientscoefficientdiagonalentry * S ((S (pfc_index_associativity_BCcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_BCcoefficientscoefficient) + (pfc_value_associativity_BCcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associativity_BCcoefficientscoefficientdiagonalterm pfc_left_associativity_BCcoefficientscoefficientdiagonalterm pfc_right_associativity_BCcoefficientscoefficientdiagonalterm. (((pfc_index_associativity_BCcoefficientscoefficientdiagonal)+pfc_complement_associativity_BCcoefficientscoefficientdiagonalterm=(pfc_index_associativity_BCcoefficients)) /\\ ((((((exists pfa_gap_associativity_BCcoefficientscoefficientdiagonaltermleftinside. pfa_gap_associativity_BCcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associativity_BCcoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_associativity_BCcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associativity_BCcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associativity_BCcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associativity_BCcoefficientscoefficientdiagonal)) * bc)) /\\ exists ff_q_pfp_associativity_BCcoefficientscoefficientdiagonaltermleftentry. bb = ff_q_pfp_associativity_BCcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associativity_BCcoefficientscoefficientdiagonal)) * bc) + (pfc_left_associativity_BCcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_BCcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associativity_BCcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_associativity_BCcoefficientscoefficientdiagonal)) /\\ (((pfc_left_associativity_BCcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associativity_BCcoefficientscoefficientdiagonaltermrightinside. pfa_gap_associativity_BCcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associativity_BCcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_associativity_BCcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associativity_BCcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associativity_BCcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associativity_BCcoefficientscoefficientdiagonalterm)) * cc)) /\\ exists ff_q_pfp_associativity_BCcoefficientscoefficientdiagonaltermrightentry. cb = ff_q_pfp_associativity_BCcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associativity_BCcoefficientscoefficientdiagonalterm)) * cc) + (pfc_right_associativity_BCcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_BCcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associativity_BCcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_associativity_BCcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associativity_BCcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associativity_BCcoefficientscoefficientdiagonal)=pfc_left_associativity_BCcoefficientscoefficientdiagonalterm*pfc_right_associativity_BCcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associativity_BCcoefficientscoefficientsum fs_v_pfc_associativity_BCcoefficientscoefficientsum. ((((exists fs_h_pfc_associativity_BCcoefficientscoefficientsum_body_start. fs_h_pfc_associativity_BCcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associativity_BCcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_BCcoefficientscoefficientsum_body_start. fs_u_pfc_associativity_BCcoefficientscoefficientsum = fs_q_pfc_associativity_BCcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associativity_BCcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associativity_BCcoefficientscoefficientsum_body_terminal. fs_h_pfc_associativity_BCcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associativity_BCcoefficientscoefficient) = S ((S (S (pfc_index_associativity_BCcoefficients))) * fs_v_pfc_associativity_BCcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_BCcoefficientscoefficientsum_body_terminal. fs_u_pfc_associativity_BCcoefficientscoefficientsum = fs_q_pfc_associativity_BCcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associativity_BCcoefficients))) * fs_v_pfc_associativity_BCcoefficientscoefficientsum) + (pfc_natural_sum_associativity_BCcoefficientscoefficient))) /\\ forall fs_i_pfc_associativity_BCcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associativity_BCcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associativity_BCcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associativity_BCcoefficientscoefficientsum_body_steps = S (pfc_index_associativity_BCcoefficients)) -> exists fs_a_pfc_associativity_BCcoefficientscoefficientsum_body_steps fs_r_pfc_associativity_BCcoefficientscoefficientsum_body_steps fs_s_pfc_associativity_BCcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associativity_BCcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associativity_BCcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associativity_BCcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_BCcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_BCcoefficientscoefficient)) /\\ exists fs_q_pfc_associativity_BCcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associativity_BCcoefficientscoefficient = fs_q_pfc_associativity_BCcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associativity_BCcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_BCcoefficientscoefficient) + (fs_a_pfc_associativity_BCcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_BCcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associativity_BCcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associativity_BCcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_BCcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_BCcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_BCcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associativity_BCcoefficientscoefficientsum = fs_q_pfc_associativity_BCcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associativity_BCcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_BCcoefficientscoefficientsum) + (fs_r_pfc_associativity_BCcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_BCcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associativity_BCcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associativity_BCcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associativity_BCcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_BCcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_BCcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associativity_BCcoefficientscoefficientsum = fs_q_pfc_associativity_BCcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associativity_BCcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_BCcoefficientscoefficientsum) + (fs_s_pfc_associativity_BCcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associativity_BCcoefficientscoefficientsum_body_steps = fs_r_pfc_associativity_BCcoefficientscoefficientsum_body_steps + fs_a_pfc_associativity_BCcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associativity_BCcoefficientscoefficientresiduebound. pfa_gap_associativity_BCcoefficientscoefficientresiduebound + S (pfc_value_associativity_BCcoefficients) = (p)) /\\ ((exists pfa_offset_left_associativity_BCcoefficientscoefficientresiduecongruence pfa_offset_right_associativity_BCcoefficientscoefficientresiduecongruence. (pfc_natural_sum_associativity_BCcoefficientscoefficient) + (p) * pfa_offset_left_associativity_BCcoefficientscoefficientresiduecongruence = (pfc_value_associativity_BCcoefficients) + (p) * pfa_offset_right_associativity_BCcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_associativity_PCleft. (exists fom_gap_pfp_associativity_PCleft_index_bound. fom_gap_pfp_associativity_PCleft_index_bound + S (fom_index_pfp_associativity_PCleft) = N) -> exists fom_value_pfp_associativity_PCleft. ((((exists fom_beta_height_pfp_associativity_PCleft_entry. fom_beta_height_pfp_associativity_PCleft_entry + S (fom_value_pfp_associativity_PCleft) = S ((S (fom_index_pfp_associativity_PCleft)) * pc)) /\\ exists fom_beta_quotient_pfp_associativity_PCleft_entry. pb = fom_beta_quotient_pfp_associativity_PCleft_entry * S ((S (fom_index_pfp_associativity_PCleft)) * pc) + (fom_value_pfp_associativity_PCleft))) /\\ (exists fom_gap_pfp_associativity_PCleft_value_bound. fom_gap_pfp_associativity_PCleft_value_bound + S (fom_value_pfp_associativity_PCleft) = p))) /\\ (((forall fom_index_pfp_associativity_PCright. (exists fom_gap_pfp_associativity_PCright_index_bound. fom_gap_pfp_associativity_PCright_index_bound + S (fom_index_pfp_associativity_PCright) = J) -> exists fom_value_pfp_associativity_PCright. ((((exists fom_beta_height_pfp_associativity_PCright_entry. fom_beta_height_pfp_associativity_PCright_entry + S (fom_value_pfp_associativity_PCright) = S ((S (fom_index_pfp_associativity_PCright)) * cc)) /\\ exists fom_beta_quotient_pfp_associativity_PCright_entry. cb = fom_beta_quotient_pfp_associativity_PCright_entry * S ((S (fom_index_pfp_associativity_PCright)) * cc) + (fom_value_pfp_associativity_PCright))) /\\ (exists fom_gap_pfp_associativity_PCright_value_bound. fom_gap_pfp_associativity_PCright_value_bound + S (fom_value_pfp_associativity_PCright) = p))) /\\ (((((((N)=0 \\/ (J)=0) /\\ (((U)=0)))) \\/ (((~((N)=0)) /\\ (((~((J)=0)) /\\ (((N)+(J)=S (U)))))))) /\\ ((forall pfc_index_associativity_PCcoefficients. (exists pfa_gap_associativity_PCcoefficientsbound. pfa_gap_associativity_PCcoefficientsbound + S (pfc_index_associativity_PCcoefficients) = (U)) -> exists pfc_value_associativity_PCcoefficients. ((((exists ff_h_pfp_associativity_PCcoefficientsentry. ff_h_pfp_associativity_PCcoefficientsentry + S (pfc_value_associativity_PCcoefficients) = S ((S (pfc_index_associativity_PCcoefficients)) * rc)) /\\ exists ff_q_pfp_associativity_PCcoefficientsentry. rb = ff_q_pfp_associativity_PCcoefficientsentry * S ((S (pfc_index_associativity_PCcoefficients)) * rc) + (pfc_value_associativity_PCcoefficients))) /\\ ((exists pfc_terms_code_associativity_PCcoefficientscoefficient pfc_terms_scale_associativity_PCcoefficientscoefficient pfc_natural_sum_associativity_PCcoefficientscoefficient. ((forall pfc_index_associativity_PCcoefficientscoefficientdiagonal. (exists pfa_gap_associativity_PCcoefficientscoefficientdiagonalbound. pfa_gap_associativity_PCcoefficientscoefficientdiagonalbound + S (pfc_index_associativity_PCcoefficientscoefficientdiagonal) = (S (pfc_index_associativity_PCcoefficients))) -> exists pfc_value_associativity_PCcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associativity_PCcoefficientscoefficientdiagonalentry. ff_h_pfp_associativity_PCcoefficientscoefficientdiagonalentry + S (pfc_value_associativity_PCcoefficientscoefficientdiagonal) = S ((S (pfc_index_associativity_PCcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_PCcoefficientscoefficient)) /\\ exists ff_q_pfp_associativity_PCcoefficientscoefficientdiagonalentry. pfc_terms_code_associativity_PCcoefficientscoefficient = ff_q_pfp_associativity_PCcoefficientscoefficientdiagonalentry * S ((S (pfc_index_associativity_PCcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_PCcoefficientscoefficient) + (pfc_value_associativity_PCcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associativity_PCcoefficientscoefficientdiagonalterm pfc_left_associativity_PCcoefficientscoefficientdiagonalterm pfc_right_associativity_PCcoefficientscoefficientdiagonalterm. (((pfc_index_associativity_PCcoefficientscoefficientdiagonal)+pfc_complement_associativity_PCcoefficientscoefficientdiagonalterm=(pfc_index_associativity_PCcoefficients)) /\\ ((((((exists pfa_gap_associativity_PCcoefficientscoefficientdiagonaltermleftinside. pfa_gap_associativity_PCcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associativity_PCcoefficientscoefficientdiagonal) = (N)) /\\ ((((exists ff_h_pfp_associativity_PCcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associativity_PCcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associativity_PCcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associativity_PCcoefficientscoefficientdiagonal)) * pc)) /\\ exists ff_q_pfp_associativity_PCcoefficientscoefficientdiagonaltermleftentry. pb = ff_q_pfp_associativity_PCcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associativity_PCcoefficientscoefficientdiagonal)) * pc) + (pfc_left_associativity_PCcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_PCcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associativity_PCcoefficientscoefficientdiagonaltermleftoutside+(N)=(pfc_index_associativity_PCcoefficientscoefficientdiagonal)) /\\ (((pfc_left_associativity_PCcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associativity_PCcoefficientscoefficientdiagonaltermrightinside. pfa_gap_associativity_PCcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associativity_PCcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_associativity_PCcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associativity_PCcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associativity_PCcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associativity_PCcoefficientscoefficientdiagonalterm)) * cc)) /\\ exists ff_q_pfp_associativity_PCcoefficientscoefficientdiagonaltermrightentry. cb = ff_q_pfp_associativity_PCcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associativity_PCcoefficientscoefficientdiagonalterm)) * cc) + (pfc_right_associativity_PCcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_PCcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associativity_PCcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_associativity_PCcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associativity_PCcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associativity_PCcoefficientscoefficientdiagonal)=pfc_left_associativity_PCcoefficientscoefficientdiagonalterm*pfc_right_associativity_PCcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associativity_PCcoefficientscoefficientsum fs_v_pfc_associativity_PCcoefficientscoefficientsum. ((((exists fs_h_pfc_associativity_PCcoefficientscoefficientsum_body_start. fs_h_pfc_associativity_PCcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associativity_PCcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_PCcoefficientscoefficientsum_body_start. fs_u_pfc_associativity_PCcoefficientscoefficientsum = fs_q_pfc_associativity_PCcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associativity_PCcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associativity_PCcoefficientscoefficientsum_body_terminal. fs_h_pfc_associativity_PCcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associativity_PCcoefficientscoefficient) = S ((S (S (pfc_index_associativity_PCcoefficients))) * fs_v_pfc_associativity_PCcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_PCcoefficientscoefficientsum_body_terminal. fs_u_pfc_associativity_PCcoefficientscoefficientsum = fs_q_pfc_associativity_PCcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associativity_PCcoefficients))) * fs_v_pfc_associativity_PCcoefficientscoefficientsum) + (pfc_natural_sum_associativity_PCcoefficientscoefficient))) /\\ forall fs_i_pfc_associativity_PCcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associativity_PCcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associativity_PCcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associativity_PCcoefficientscoefficientsum_body_steps = S (pfc_index_associativity_PCcoefficients)) -> exists fs_a_pfc_associativity_PCcoefficientscoefficientsum_body_steps fs_r_pfc_associativity_PCcoefficientscoefficientsum_body_steps fs_s_pfc_associativity_PCcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associativity_PCcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associativity_PCcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associativity_PCcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_PCcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_PCcoefficientscoefficient)) /\\ exists fs_q_pfc_associativity_PCcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associativity_PCcoefficientscoefficient = fs_q_pfc_associativity_PCcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associativity_PCcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_PCcoefficientscoefficient) + (fs_a_pfc_associativity_PCcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_PCcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associativity_PCcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associativity_PCcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_PCcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_PCcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_PCcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associativity_PCcoefficientscoefficientsum = fs_q_pfc_associativity_PCcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associativity_PCcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_PCcoefficientscoefficientsum) + (fs_r_pfc_associativity_PCcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_PCcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associativity_PCcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associativity_PCcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associativity_PCcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_PCcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_PCcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associativity_PCcoefficientscoefficientsum = fs_q_pfc_associativity_PCcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associativity_PCcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_PCcoefficientscoefficientsum) + (fs_s_pfc_associativity_PCcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associativity_PCcoefficientscoefficientsum_body_steps = fs_r_pfc_associativity_PCcoefficientscoefficientsum_body_steps + fs_a_pfc_associativity_PCcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associativity_PCcoefficientscoefficientresiduebound. pfa_gap_associativity_PCcoefficientscoefficientresiduebound + S (pfc_value_associativity_PCcoefficients) = (p)) /\\ ((exists pfa_offset_left_associativity_PCcoefficientscoefficientresiduecongruence pfa_offset_right_associativity_PCcoefficientscoefficientresiduecongruence. (pfc_natural_sum_associativity_PCcoefficientscoefficient) + (p) * pfa_offset_left_associativity_PCcoefficientscoefficientresiduecongruence = (pfc_value_associativity_PCcoefficients) + (p) * pfa_offset_right_associativity_PCcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_associativity_AQleft. (exists fom_gap_pfp_associativity_AQleft_index_bound. fom_gap_pfp_associativity_AQleft_index_bound + S (fom_index_pfp_associativity_AQleft) = L) -> exists fom_value_pfp_associativity_AQleft. ((((exists fom_beta_height_pfp_associativity_AQleft_entry. fom_beta_height_pfp_associativity_AQleft_entry + S (fom_value_pfp_associativity_AQleft) = S ((S (fom_index_pfp_associativity_AQleft)) * ac)) /\\ exists fom_beta_quotient_pfp_associativity_AQleft_entry. ab = fom_beta_quotient_pfp_associativity_AQleft_entry * S ((S (fom_index_pfp_associativity_AQleft)) * ac) + (fom_value_pfp_associativity_AQleft))) /\\ (exists fom_gap_pfp_associativity_AQleft_value_bound. fom_gap_pfp_associativity_AQleft_value_bound + S (fom_value_pfp_associativity_AQleft) = p))) /\\ (((forall fom_index_pfp_associativity_AQright. (exists fom_gap_pfp_associativity_AQright_index_bound. fom_gap_pfp_associativity_AQright_index_bound + S (fom_index_pfp_associativity_AQright) = K) -> exists fom_value_pfp_associativity_AQright. ((((exists fom_beta_height_pfp_associativity_AQright_entry. fom_beta_height_pfp_associativity_AQright_entry + S (fom_value_pfp_associativity_AQright) = S ((S (fom_index_pfp_associativity_AQright)) * qc)) /\\ exists fom_beta_quotient_pfp_associativity_AQright_entry. qb = fom_beta_quotient_pfp_associativity_AQright_entry * S ((S (fom_index_pfp_associativity_AQright)) * qc) + (fom_value_pfp_associativity_AQright))) /\\ (exists fom_gap_pfp_associativity_AQright_value_bound. fom_gap_pfp_associativity_AQright_value_bound + S (fom_value_pfp_associativity_AQright) = p))) /\\ (((((((L)=0 \\/ (K)=0) /\\ (((V)=0)))) \\/ (((~((L)=0)) /\\ (((~((K)=0)) /\\ (((L)+(K)=S (V)))))))) /\\ ((forall pfc_index_associativity_AQcoefficients. (exists pfa_gap_associativity_AQcoefficientsbound. pfa_gap_associativity_AQcoefficientsbound + S (pfc_index_associativity_AQcoefficients) = (V)) -> exists pfc_value_associativity_AQcoefficients. ((((exists ff_h_pfp_associativity_AQcoefficientsentry. ff_h_pfp_associativity_AQcoefficientsentry + S (pfc_value_associativity_AQcoefficients) = S ((S (pfc_index_associativity_AQcoefficients)) * sc)) /\\ exists ff_q_pfp_associativity_AQcoefficientsentry. sb = ff_q_pfp_associativity_AQcoefficientsentry * S ((S (pfc_index_associativity_AQcoefficients)) * sc) + (pfc_value_associativity_AQcoefficients))) /\\ ((exists pfc_terms_code_associativity_AQcoefficientscoefficient pfc_terms_scale_associativity_AQcoefficientscoefficient pfc_natural_sum_associativity_AQcoefficientscoefficient. ((forall pfc_index_associativity_AQcoefficientscoefficientdiagonal. (exists pfa_gap_associativity_AQcoefficientscoefficientdiagonalbound. pfa_gap_associativity_AQcoefficientscoefficientdiagonalbound + S (pfc_index_associativity_AQcoefficientscoefficientdiagonal) = (S (pfc_index_associativity_AQcoefficients))) -> exists pfc_value_associativity_AQcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associativity_AQcoefficientscoefficientdiagonalentry. ff_h_pfp_associativity_AQcoefficientscoefficientdiagonalentry + S (pfc_value_associativity_AQcoefficientscoefficientdiagonal) = S ((S (pfc_index_associativity_AQcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_AQcoefficientscoefficient)) /\\ exists ff_q_pfp_associativity_AQcoefficientscoefficientdiagonalentry. pfc_terms_code_associativity_AQcoefficientscoefficient = ff_q_pfp_associativity_AQcoefficientscoefficientdiagonalentry * S ((S (pfc_index_associativity_AQcoefficientscoefficientdiagonal)) * pfc_terms_scale_associativity_AQcoefficientscoefficient) + (pfc_value_associativity_AQcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associativity_AQcoefficientscoefficientdiagonalterm pfc_left_associativity_AQcoefficientscoefficientdiagonalterm pfc_right_associativity_AQcoefficientscoefficientdiagonalterm. (((pfc_index_associativity_AQcoefficientscoefficientdiagonal)+pfc_complement_associativity_AQcoefficientscoefficientdiagonalterm=(pfc_index_associativity_AQcoefficients)) /\\ ((((((exists pfa_gap_associativity_AQcoefficientscoefficientdiagonaltermleftinside. pfa_gap_associativity_AQcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associativity_AQcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_associativity_AQcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associativity_AQcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associativity_AQcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associativity_AQcoefficientscoefficientdiagonal)) * ac)) /\\ exists ff_q_pfp_associativity_AQcoefficientscoefficientdiagonaltermleftentry. ab = ff_q_pfp_associativity_AQcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associativity_AQcoefficientscoefficientdiagonal)) * ac) + (pfc_left_associativity_AQcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_AQcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associativity_AQcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_associativity_AQcoefficientscoefficientdiagonal)) /\\ (((pfc_left_associativity_AQcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associativity_AQcoefficientscoefficientdiagonaltermrightinside. pfa_gap_associativity_AQcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associativity_AQcoefficientscoefficientdiagonalterm) = (K)) /\\ ((((exists ff_h_pfp_associativity_AQcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associativity_AQcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associativity_AQcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associativity_AQcoefficientscoefficientdiagonalterm)) * qc)) /\\ exists ff_q_pfp_associativity_AQcoefficientscoefficientdiagonaltermrightentry. qb = ff_q_pfp_associativity_AQcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associativity_AQcoefficientscoefficientdiagonalterm)) * qc) + (pfc_right_associativity_AQcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associativity_AQcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associativity_AQcoefficientscoefficientdiagonaltermrightoutside+(K)=(pfc_complement_associativity_AQcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associativity_AQcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associativity_AQcoefficientscoefficientdiagonal)=pfc_left_associativity_AQcoefficientscoefficientdiagonalterm*pfc_right_associativity_AQcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associativity_AQcoefficientscoefficientsum fs_v_pfc_associativity_AQcoefficientscoefficientsum. ((((exists fs_h_pfc_associativity_AQcoefficientscoefficientsum_body_start. fs_h_pfc_associativity_AQcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associativity_AQcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_AQcoefficientscoefficientsum_body_start. fs_u_pfc_associativity_AQcoefficientscoefficientsum = fs_q_pfc_associativity_AQcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associativity_AQcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associativity_AQcoefficientscoefficientsum_body_terminal. fs_h_pfc_associativity_AQcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associativity_AQcoefficientscoefficient) = S ((S (S (pfc_index_associativity_AQcoefficients))) * fs_v_pfc_associativity_AQcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_AQcoefficientscoefficientsum_body_terminal. fs_u_pfc_associativity_AQcoefficientscoefficientsum = fs_q_pfc_associativity_AQcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associativity_AQcoefficients))) * fs_v_pfc_associativity_AQcoefficientscoefficientsum) + (pfc_natural_sum_associativity_AQcoefficientscoefficient))) /\\ forall fs_i_pfc_associativity_AQcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associativity_AQcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associativity_AQcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associativity_AQcoefficientscoefficientsum_body_steps = S (pfc_index_associativity_AQcoefficients)) -> exists fs_a_pfc_associativity_AQcoefficientscoefficientsum_body_steps fs_r_pfc_associativity_AQcoefficientscoefficientsum_body_steps fs_s_pfc_associativity_AQcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associativity_AQcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associativity_AQcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associativity_AQcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_AQcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_AQcoefficientscoefficient)) /\\ exists fs_q_pfc_associativity_AQcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associativity_AQcoefficientscoefficient = fs_q_pfc_associativity_AQcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associativity_AQcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associativity_AQcoefficientscoefficient) + (fs_a_pfc_associativity_AQcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_AQcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associativity_AQcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associativity_AQcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associativity_AQcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_AQcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_AQcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associativity_AQcoefficientscoefficientsum = fs_q_pfc_associativity_AQcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associativity_AQcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_AQcoefficientscoefficientsum) + (fs_r_pfc_associativity_AQcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associativity_AQcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associativity_AQcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associativity_AQcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associativity_AQcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_AQcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associativity_AQcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associativity_AQcoefficientscoefficientsum = fs_q_pfc_associativity_AQcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associativity_AQcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associativity_AQcoefficientscoefficientsum) + (fs_s_pfc_associativity_AQcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associativity_AQcoefficientscoefficientsum_body_steps = fs_r_pfc_associativity_AQcoefficientscoefficientsum_body_steps + fs_a_pfc_associativity_AQcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associativity_AQcoefficientscoefficientresiduebound. pfa_gap_associativity_AQcoefficientscoefficientresiduebound + S (pfc_value_associativity_AQcoefficients) = (p)) /\\ ((exists pfa_offset_left_associativity_AQcoefficientscoefficientresiduecongruence pfa_offset_right_associativity_AQcoefficientscoefficientresiduecongruence. (pfc_natural_sum_associativity_AQcoefficientscoefficient) + (p) * pfa_offset_left_associativity_AQcoefficientscoefficientresiduecongruence = (pfc_value_associativity_AQcoefficients) + (p) * pfa_offset_right_associativity_AQcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_associativity_result pfrep_left_associativity_result pfrep_right_associativity_result. ((exists pfrep_position_associativity_resultfirst. ((pfrep_position_associativity_resultfirst+S (pfrep_power_associativity_result)=(U)) /\\ ((((exists ff_h_pfp_associativity_resultfirstentry. ff_h_pfp_associativity_resultfirstentry + S (pfrep_left_associativity_result) = S ((S (pfrep_position_associativity_resultfirst)) * rc)) /\\ exists ff_q_pfp_associativity_resultfirstentry. rb = ff_q_pfp_associativity_resultfirstentry * S ((S (pfrep_position_associativity_resultfirst)) * rc) + (pfrep_left_associativity_result)))))) \\/ (((exists pfrep_gap_associativity_resultfirstoutside. pfrep_gap_associativity_resultfirstoutside+(U)=(pfrep_power_associativity_result)) /\\ (((pfrep_left_associativity_result)=0))))) -> ((exists pfrep_position_associativity_resultsecond. ((pfrep_position_associativity_resultsecond+S (pfrep_power_associativity_result)=(V)) /\\ ((((exists ff_h_pfp_associativity_resultsecondentry. ff_h_pfp_associativity_resultsecondentry + S (pfrep_right_associativity_result) = S ((S (pfrep_position_associativity_resultsecond)) * sc)) /\\ exists ff_q_pfp_associativity_resultsecondentry. sb = ff_q_pfp_associativity_resultsecondentry * S ((S (pfrep_position_associativity_resultsecond)) * sc) + (pfrep_right_associativity_result)))))) \\/ (((exists pfrep_gap_associativity_resultsecondoutside. pfrep_gap_associativity_resultsecondoutside+(V)=(pfrep_power_associativity_result)) /\\ (((pfrep_right_associativity_result)=0))))) -> pfrep_left_associativity_result=pfrep_right_associativity_result)", "statement_sha256": "7b693d78212d80c6406b09f6ca5151ac88862da29d824494ed6338f601fb6912", "summary": "Draft universal rightmost-length induction for formal equivalence of actual (A*B)*C and A*(B*C). The induction predicate quantifies all rightmost codes and proper-length output triples, the successor genuinely constructs three prefix products and decodes the actual endpoint, and the empty base retains arbitrary encodings. This statement is not a successful proof observation until its original body and its exact step dependency are genuinely checked.", "summary_sha256": "f2dfe1490f1e95631eb714fed6411408f77646e8dcb29d2944b559785c650652" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "bd41fe64499cce55c30f96cfadda5d5da15326de6421464cb5a8526b5139d237", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 308, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "f544adedd3ce963e4a773e8582efcb0f91ba7491207c9792d477d452e854f2b8", "test_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85" }, "body_checked": true, "body_receipt": { "command_count": 27, "dependency_count": 0, "dne_command_count": 0, "name": "prime_field_polynomial_right_divides_from_product", "proof_depth": 25, "proof_edges": 26, "proof_nodes": 27, "proof_objects": 27, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [], "dependencies_sha256": "01ba4719c80b6fe911b091a7c05124b64eeece964e09c058ef8f9805daca546b", "empty_context_closure": { "body_proof_depth": 25, "body_proof_nodes": 27, "bundle_campaign": "polynomial-gcd-bezout", "bundle_dependency_edge_count": 1578, "bundle_node_count": 493, "bundle_node_id": 308, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "bundle_root_id": 492, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "cce9cf77a9e9bc71e96c431dbe18be513e9020461125447d9e6a50e6ad2d4377", "status": "checked" }, "enrollment_index": 4129, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "f544adedd3ce963e4a773e8582efcb0f91ba7491207c9792d477d452e854f2b8", "kind": "alpha_v34_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_divisibility_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85", "kind": "alpha_v34_frontier_executable_audit", "path": "peano-lab/py/tests/test_campaign_research_v34_closure.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "kind": "alpha_v34_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-field-polynomial-gcd-bezout-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "kind": "alpha_v34_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=308]" }, { "document_sha256": "81943a545e7b9f667ad758c24de14bb47280e8cf3fbb3ca714efec55fb096408", "kind": "alpha_v34_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "2081c85ef059f0e4936ae68416e114a0159499070768f454214e986ac92a8bb5", "membership": "alpha_only", "name": "prime_field_polynomial_right_divides_from_product", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro db", "intro dc", "intro D", "intro ab", "intro ac", "intro L", "intro qb", "intro qc", "intro Q", "intro pb", "intro pc", "intro P", "intro hA", "intro hproduct", "intro hequivalent", "split", "exact hA", "exists qb", "exists qc", "exists Q", "exists pb", "exists pc", "exists P", "split", "exact hproduct", "exact hequivalent" ], "script_sha256": "41b188991885fe4bbb9da98e622835330e45dd3bda26b8ab1abc37db7fa7de8a", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_divisibility_candidate.py", "sha256": "f544adedd3ce963e4a773e8582efcb0f91ba7491207c9792d477d452e854f2b8" }, "statement": "forall p db dc D ab ac L qb qc Q pb pc P. (forall fom_index_pfp_right_divides_intro_bound. (exists fom_gap_pfp_right_divides_intro_bound_index_bound. fom_gap_pfp_right_divides_intro_bound_index_bound + S (fom_index_pfp_right_divides_intro_bound) = L) -> exists fom_value_pfp_right_divides_intro_bound. ((((exists fom_beta_height_pfp_right_divides_intro_bound_entry. fom_beta_height_pfp_right_divides_intro_bound_entry + S (fom_value_pfp_right_divides_intro_bound) = S ((S (fom_index_pfp_right_divides_intro_bound)) * ac)) /\\ exists fom_beta_quotient_pfp_right_divides_intro_bound_entry. ab = fom_beta_quotient_pfp_right_divides_intro_bound_entry * S ((S (fom_index_pfp_right_divides_intro_bound)) * ac) + (fom_value_pfp_right_divides_intro_bound))) /\\ (exists fom_gap_pfp_right_divides_intro_bound_value_bound. fom_gap_pfp_right_divides_intro_bound_value_bound + S (fom_value_pfp_right_divides_intro_bound) = p))) -> (((forall fom_index_pfp_right_divides_intro_productleft. (exists fom_gap_pfp_right_divides_intro_productleft_index_bound. fom_gap_pfp_right_divides_intro_productleft_index_bound + S (fom_index_pfp_right_divides_intro_productleft) = Q) -> exists fom_value_pfp_right_divides_intro_productleft. ((((exists fom_beta_height_pfp_right_divides_intro_productleft_entry. fom_beta_height_pfp_right_divides_intro_productleft_entry + S (fom_value_pfp_right_divides_intro_productleft) = S ((S (fom_index_pfp_right_divides_intro_productleft)) * qc)) /\\ exists fom_beta_quotient_pfp_right_divides_intro_productleft_entry. qb = fom_beta_quotient_pfp_right_divides_intro_productleft_entry * S ((S (fom_index_pfp_right_divides_intro_productleft)) * qc) + (fom_value_pfp_right_divides_intro_productleft))) /\\ (exists fom_gap_pfp_right_divides_intro_productleft_value_bound. fom_gap_pfp_right_divides_intro_productleft_value_bound + S (fom_value_pfp_right_divides_intro_productleft) = p))) /\\ (((forall fom_index_pfp_right_divides_intro_productright. (exists fom_gap_pfp_right_divides_intro_productright_index_bound. fom_gap_pfp_right_divides_intro_productright_index_bound + S (fom_index_pfp_right_divides_intro_productright) = D) -> exists fom_value_pfp_right_divides_intro_productright. ((((exists fom_beta_height_pfp_right_divides_intro_productright_entry. fom_beta_height_pfp_right_divides_intro_productright_entry + S (fom_value_pfp_right_divides_intro_productright) = S ((S (fom_index_pfp_right_divides_intro_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_right_divides_intro_productright_entry. db = fom_beta_quotient_pfp_right_divides_intro_productright_entry * S ((S (fom_index_pfp_right_divides_intro_productright)) * dc) + (fom_value_pfp_right_divides_intro_productright))) /\\ (exists fom_gap_pfp_right_divides_intro_productright_value_bound. fom_gap_pfp_right_divides_intro_productright_value_bound + S (fom_value_pfp_right_divides_intro_productright) = p))) /\\ (((((((Q)=0 \\/ (D)=0) /\\ (((P)=0)))) \\/ (((~((Q)=0)) /\\ (((~((D)=0)) /\\ (((Q)+(D)=S (P)))))))) /\\ ((forall pfc_index_right_divides_intro_productcoefficients. (exists pfa_gap_right_divides_intro_productcoefficientsbound. pfa_gap_right_divides_intro_productcoefficientsbound + S (pfc_index_right_divides_intro_productcoefficients) = (P)) -> exists pfc_value_right_divides_intro_productcoefficients. ((((exists ff_h_pfp_right_divides_intro_productcoefficientsentry. ff_h_pfp_right_divides_intro_productcoefficientsentry + S (pfc_value_right_divides_intro_productcoefficients) = S ((S (pfc_index_right_divides_intro_productcoefficients)) * pc)) /\\ exists ff_q_pfp_right_divides_intro_productcoefficientsentry. pb = ff_q_pfp_right_divides_intro_productcoefficientsentry * S ((S (pfc_index_right_divides_intro_productcoefficients)) * pc) + (pfc_value_right_divides_intro_productcoefficients))) /\\ ((exists pfc_terms_code_right_divides_intro_productcoefficientscoefficient pfc_terms_scale_right_divides_intro_productcoefficientscoefficient pfc_natural_sum_right_divides_intro_productcoefficientscoefficient. ((forall pfc_index_right_divides_intro_productcoefficientscoefficientdiagonal. (exists pfa_gap_right_divides_intro_productcoefficientscoefficientdiagonalbound. pfa_gap_right_divides_intro_productcoefficientscoefficientdiagonalbound + S (pfc_index_right_divides_intro_productcoefficientscoefficientdiagonal) = (S (pfc_index_right_divides_intro_productcoefficients))) -> exists pfc_value_right_divides_intro_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_right_divides_intro_productcoefficientscoefficientdiagonalentry. ff_h_pfp_right_divides_intro_productcoefficientscoefficientdiagonalentry + S (pfc_value_right_divides_intro_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_right_divides_intro_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_divides_intro_productcoefficientscoefficient)) /\\ exists ff_q_pfp_right_divides_intro_productcoefficientscoefficientdiagonalentry. pfc_terms_code_right_divides_intro_productcoefficientscoefficient = ff_q_pfp_right_divides_intro_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_right_divides_intro_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_divides_intro_productcoefficientscoefficient) + (pfc_value_right_divides_intro_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_right_divides_intro_productcoefficientscoefficientdiagonalterm pfc_left_right_divides_intro_productcoefficientscoefficientdiagonalterm pfc_right_right_divides_intro_productcoefficientscoefficientdiagonalterm. (((pfc_index_right_divides_intro_productcoefficientscoefficientdiagonal)+pfc_complement_right_divides_intro_productcoefficientscoefficientdiagonalterm=(pfc_index_right_divides_intro_productcoefficients)) /\\ ((((((exists pfa_gap_right_divides_intro_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_right_divides_intro_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_right_divides_intro_productcoefficientscoefficientdiagonal) = (Q)) /\\ ((((exists ff_h_pfp_right_divides_intro_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_right_divides_intro_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_right_divides_intro_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_right_divides_intro_productcoefficientscoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_right_divides_intro_productcoefficientscoefficientdiagonaltermleftentry. qb = ff_q_pfp_right_divides_intro_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_right_divides_intro_productcoefficientscoefficientdiagonal)) * qc) + (pfc_left_right_divides_intro_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_divides_intro_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_right_divides_intro_productcoefficientscoefficientdiagonaltermleftoutside+(Q)=(pfc_index_right_divides_intro_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_right_divides_intro_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_right_divides_intro_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_right_divides_intro_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_right_divides_intro_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_right_divides_intro_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_right_divides_intro_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_right_divides_intro_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_right_divides_intro_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_right_divides_intro_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_right_divides_intro_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_right_divides_intro_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_right_divides_intro_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_divides_intro_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_right_divides_intro_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_right_divides_intro_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_right_divides_intro_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_right_divides_intro_productcoefficientscoefficientdiagonal)=pfc_left_right_divides_intro_productcoefficientscoefficientdiagonalterm*pfc_right_right_divides_intro_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_right_divides_intro_productcoefficientscoefficientsum fs_v_pfc_right_divides_intro_productcoefficientscoefficientsum. ((((exists fs_h_pfc_right_divides_intro_productcoefficientscoefficientsum_body_start. fs_h_pfc_right_divides_intro_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_right_divides_intro_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divides_intro_productcoefficientscoefficientsum_body_start. fs_u_pfc_right_divides_intro_productcoefficientscoefficientsum = fs_q_pfc_right_divides_intro_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_right_divides_intro_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_right_divides_intro_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_right_divides_intro_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_right_divides_intro_productcoefficientscoefficient) = S ((S (S (pfc_index_right_divides_intro_productcoefficients))) * fs_v_pfc_right_divides_intro_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divides_intro_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_right_divides_intro_productcoefficientscoefficientsum = fs_q_pfc_right_divides_intro_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_right_divides_intro_productcoefficients))) * fs_v_pfc_right_divides_intro_productcoefficientscoefficientsum) + (pfc_natural_sum_right_divides_intro_productcoefficientscoefficient))) /\\ forall fs_i_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps = S (pfc_index_right_divides_intro_productcoefficients)) -> exists fs_a_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps fs_r_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps fs_s_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_divides_intro_productcoefficientscoefficient)) /\\ exists fs_q_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_right_divides_intro_productcoefficientscoefficient = fs_q_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_divides_intro_productcoefficientscoefficient) + (fs_a_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divides_intro_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_right_divides_intro_productcoefficientscoefficientsum = fs_q_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divides_intro_productcoefficientscoefficientsum) + (fs_r_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divides_intro_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_right_divides_intro_productcoefficientscoefficientsum = fs_q_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divides_intro_productcoefficientscoefficientsum) + (fs_s_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps = fs_r_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps + fs_a_pfc_right_divides_intro_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_right_divides_intro_productcoefficientscoefficientresiduebound. pfa_gap_right_divides_intro_productcoefficientscoefficientresiduebound + S (pfc_value_right_divides_intro_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_right_divides_intro_productcoefficientscoefficientresiduecongruence pfa_offset_right_right_divides_intro_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_right_divides_intro_productcoefficientscoefficient) + (p) * pfa_offset_left_right_divides_intro_productcoefficientscoefficientresiduecongruence = (pfc_value_right_divides_intro_productcoefficients) + (p) * pfa_offset_right_right_divides_intro_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_right_divides_intro_equivalent pfrep_left_right_divides_intro_equivalent pfrep_right_right_divides_intro_equivalent. ((exists pfrep_position_right_divides_intro_equivalentfirst. ((pfrep_position_right_divides_intro_equivalentfirst+S (pfrep_power_right_divides_intro_equivalent)=(P)) /\\ ((((exists ff_h_pfp_right_divides_intro_equivalentfirstentry. ff_h_pfp_right_divides_intro_equivalentfirstentry + S (pfrep_left_right_divides_intro_equivalent) = S ((S (pfrep_position_right_divides_intro_equivalentfirst)) * pc)) /\\ exists ff_q_pfp_right_divides_intro_equivalentfirstentry. pb = ff_q_pfp_right_divides_intro_equivalentfirstentry * S ((S (pfrep_position_right_divides_intro_equivalentfirst)) * pc) + (pfrep_left_right_divides_intro_equivalent)))))) \\/ (((exists pfrep_gap_right_divides_intro_equivalentfirstoutside. pfrep_gap_right_divides_intro_equivalentfirstoutside+(P)=(pfrep_power_right_divides_intro_equivalent)) /\\ (((pfrep_left_right_divides_intro_equivalent)=0))))) -> ((exists pfrep_position_right_divides_intro_equivalentsecond. ((pfrep_position_right_divides_intro_equivalentsecond+S (pfrep_power_right_divides_intro_equivalent)=(L)) /\\ ((((exists ff_h_pfp_right_divides_intro_equivalentsecondentry. ff_h_pfp_right_divides_intro_equivalentsecondentry + S (pfrep_right_right_divides_intro_equivalent) = S ((S (pfrep_position_right_divides_intro_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_right_divides_intro_equivalentsecondentry. ab = ff_q_pfp_right_divides_intro_equivalentsecondentry * S ((S (pfrep_position_right_divides_intro_equivalentsecond)) * ac) + (pfrep_right_right_divides_intro_equivalent)))))) \\/ (((exists pfrep_gap_right_divides_intro_equivalentsecondoutside. pfrep_gap_right_divides_intro_equivalentsecondoutside+(L)=(pfrep_power_right_divides_intro_equivalent)) /\\ (((pfrep_right_right_divides_intro_equivalent)=0))))) -> pfrep_left_right_divides_intro_equivalent=pfrep_right_right_divides_intro_equivalent) -> (((forall fom_index_pfp_right_divides_intro_result_canonical. (exists fom_gap_pfp_right_divides_intro_result_canonical_index_bound. fom_gap_pfp_right_divides_intro_result_canonical_index_bound + S (fom_index_pfp_right_divides_intro_result_canonical) = L) -> exists fom_value_pfp_right_divides_intro_result_canonical. ((((exists fom_beta_height_pfp_right_divides_intro_result_canonical_entry. fom_beta_height_pfp_right_divides_intro_result_canonical_entry + S (fom_value_pfp_right_divides_intro_result_canonical) = S ((S (fom_index_pfp_right_divides_intro_result_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_right_divides_intro_result_canonical_entry. ab = fom_beta_quotient_pfp_right_divides_intro_result_canonical_entry * S ((S (fom_index_pfp_right_divides_intro_result_canonical)) * ac) + (fom_value_pfp_right_divides_intro_result_canonical))) /\\ (exists fom_gap_pfp_right_divides_intro_result_canonical_value_bound. fom_gap_pfp_right_divides_intro_result_canonical_value_bound + S (fom_value_pfp_right_divides_intro_result_canonical) = p))) /\\ ((exists pfrd_qb_right_divides_intro_result pfrd_qc_right_divides_intro_result pfrd_qlen_right_divides_intro_result pfrd_pb_right_divides_intro_result pfrd_pc_right_divides_intro_result pfrd_plen_right_divides_intro_result. ((((forall fom_index_pfp_right_divides_intro_result_productleft. (exists fom_gap_pfp_right_divides_intro_result_productleft_index_bound. fom_gap_pfp_right_divides_intro_result_productleft_index_bound + S (fom_index_pfp_right_divides_intro_result_productleft) = pfrd_qlen_right_divides_intro_result) -> exists fom_value_pfp_right_divides_intro_result_productleft. ((((exists fom_beta_height_pfp_right_divides_intro_result_productleft_entry. fom_beta_height_pfp_right_divides_intro_result_productleft_entry + S (fom_value_pfp_right_divides_intro_result_productleft) = S ((S (fom_index_pfp_right_divides_intro_result_productleft)) * pfrd_qc_right_divides_intro_result)) /\\ exists fom_beta_quotient_pfp_right_divides_intro_result_productleft_entry. pfrd_qb_right_divides_intro_result = fom_beta_quotient_pfp_right_divides_intro_result_productleft_entry * S ((S (fom_index_pfp_right_divides_intro_result_productleft)) * pfrd_qc_right_divides_intro_result) + (fom_value_pfp_right_divides_intro_result_productleft))) /\\ (exists fom_gap_pfp_right_divides_intro_result_productleft_value_bound. fom_gap_pfp_right_divides_intro_result_productleft_value_bound + S (fom_value_pfp_right_divides_intro_result_productleft) = p))) /\\ (((forall fom_index_pfp_right_divides_intro_result_productright. (exists fom_gap_pfp_right_divides_intro_result_productright_index_bound. fom_gap_pfp_right_divides_intro_result_productright_index_bound + S (fom_index_pfp_right_divides_intro_result_productright) = D) -> exists fom_value_pfp_right_divides_intro_result_productright. ((((exists fom_beta_height_pfp_right_divides_intro_result_productright_entry. fom_beta_height_pfp_right_divides_intro_result_productright_entry + S (fom_value_pfp_right_divides_intro_result_productright) = S ((S (fom_index_pfp_right_divides_intro_result_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_right_divides_intro_result_productright_entry. db = fom_beta_quotient_pfp_right_divides_intro_result_productright_entry * S ((S (fom_index_pfp_right_divides_intro_result_productright)) * dc) + (fom_value_pfp_right_divides_intro_result_productright))) /\\ (exists fom_gap_pfp_right_divides_intro_result_productright_value_bound. fom_gap_pfp_right_divides_intro_result_productright_value_bound + S (fom_value_pfp_right_divides_intro_result_productright) = p))) /\\ (((((((pfrd_qlen_right_divides_intro_result)=0 \\/ (D)=0) /\\ (((pfrd_plen_right_divides_intro_result)=0)))) \\/ (((~((pfrd_qlen_right_divides_intro_result)=0)) /\\ (((~((D)=0)) /\\ (((pfrd_qlen_right_divides_intro_result)+(D)=S (pfrd_plen_right_divides_intro_result)))))))) /\\ ((forall pfc_index_right_divides_intro_result_productcoefficients. (exists pfa_gap_right_divides_intro_result_productcoefficientsbound. pfa_gap_right_divides_intro_result_productcoefficientsbound + S (pfc_index_right_divides_intro_result_productcoefficients) = (pfrd_plen_right_divides_intro_result)) -> exists pfc_value_right_divides_intro_result_productcoefficients. ((((exists ff_h_pfp_right_divides_intro_result_productcoefficientsentry. ff_h_pfp_right_divides_intro_result_productcoefficientsentry + S (pfc_value_right_divides_intro_result_productcoefficients) = S ((S (pfc_index_right_divides_intro_result_productcoefficients)) * pfrd_pc_right_divides_intro_result)) /\\ exists ff_q_pfp_right_divides_intro_result_productcoefficientsentry. pfrd_pb_right_divides_intro_result = ff_q_pfp_right_divides_intro_result_productcoefficientsentry * S ((S (pfc_index_right_divides_intro_result_productcoefficients)) * pfrd_pc_right_divides_intro_result) + (pfc_value_right_divides_intro_result_productcoefficients))) /\\ ((exists pfc_terms_code_right_divides_intro_result_productcoefficientscoefficient pfc_terms_scale_right_divides_intro_result_productcoefficientscoefficient pfc_natural_sum_right_divides_intro_result_productcoefficientscoefficient. ((forall pfc_index_right_divides_intro_result_productcoefficientscoefficientdiagonal. (exists pfa_gap_right_divides_intro_result_productcoefficientscoefficientdiagonalbound. pfa_gap_right_divides_intro_result_productcoefficientscoefficientdiagonalbound + S (pfc_index_right_divides_intro_result_productcoefficientscoefficientdiagonal) = (S (pfc_index_right_divides_intro_result_productcoefficients))) -> exists pfc_value_right_divides_intro_result_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_right_divides_intro_result_productcoefficientscoefficientdiagonalentry. ff_h_pfp_right_divides_intro_result_productcoefficientscoefficientdiagonalentry + S (pfc_value_right_divides_intro_result_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_right_divides_intro_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_divides_intro_result_productcoefficientscoefficient)) /\\ exists ff_q_pfp_right_divides_intro_result_productcoefficientscoefficientdiagonalentry. pfc_terms_code_right_divides_intro_result_productcoefficientscoefficient = ff_q_pfp_right_divides_intro_result_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_right_divides_intro_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_divides_intro_result_productcoefficientscoefficient) + (pfc_value_right_divides_intro_result_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_right_divides_intro_result_productcoefficientscoefficientdiagonalterm pfc_left_right_divides_intro_result_productcoefficientscoefficientdiagonalterm pfc_right_right_divides_intro_result_productcoefficientscoefficientdiagonalterm. (((pfc_index_right_divides_intro_result_productcoefficientscoefficientdiagonal)+pfc_complement_right_divides_intro_result_productcoefficientscoefficientdiagonalterm=(pfc_index_right_divides_intro_result_productcoefficients)) /\\ ((((((exists pfa_gap_right_divides_intro_result_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_right_divides_intro_result_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_right_divides_intro_result_productcoefficientscoefficientdiagonal) = (pfrd_qlen_right_divides_intro_result)) /\\ ((((exists ff_h_pfp_right_divides_intro_result_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_right_divides_intro_result_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_right_divides_intro_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_right_divides_intro_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_divides_intro_result)) /\\ exists ff_q_pfp_right_divides_intro_result_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_right_divides_intro_result = ff_q_pfp_right_divides_intro_result_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_right_divides_intro_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_divides_intro_result) + (pfc_left_right_divides_intro_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_divides_intro_result_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_right_divides_intro_result_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_right_divides_intro_result)=(pfc_index_right_divides_intro_result_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_right_divides_intro_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_right_divides_intro_result_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_right_divides_intro_result_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_right_divides_intro_result_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_right_divides_intro_result_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_right_divides_intro_result_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_right_divides_intro_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_right_divides_intro_result_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_right_divides_intro_result_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_right_divides_intro_result_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_right_divides_intro_result_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_right_divides_intro_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_divides_intro_result_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_right_divides_intro_result_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_right_divides_intro_result_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_right_divides_intro_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_right_divides_intro_result_productcoefficientscoefficientdiagonal)=pfc_left_right_divides_intro_result_productcoefficientscoefficientdiagonalterm*pfc_right_right_divides_intro_result_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_right_divides_intro_result_productcoefficientscoefficientsum fs_v_pfc_right_divides_intro_result_productcoefficientscoefficientsum. ((((exists fs_h_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_start. fs_h_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_right_divides_intro_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_start. fs_u_pfc_right_divides_intro_result_productcoefficientscoefficientsum = fs_q_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_right_divides_intro_result_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_right_divides_intro_result_productcoefficientscoefficient) = S ((S (S (pfc_index_right_divides_intro_result_productcoefficients))) * fs_v_pfc_right_divides_intro_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_right_divides_intro_result_productcoefficientscoefficientsum = fs_q_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_right_divides_intro_result_productcoefficients))) * fs_v_pfc_right_divides_intro_result_productcoefficientscoefficientsum) + (pfc_natural_sum_right_divides_intro_result_productcoefficientscoefficient))) /\\ forall fs_i_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps = S (pfc_index_right_divides_intro_result_productcoefficients)) -> exists fs_a_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps fs_r_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps fs_s_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_divides_intro_result_productcoefficientscoefficient)) /\\ exists fs_q_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_right_divides_intro_result_productcoefficientscoefficient = fs_q_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_divides_intro_result_productcoefficientscoefficient) + (fs_a_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divides_intro_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_right_divides_intro_result_productcoefficientscoefficientsum = fs_q_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divides_intro_result_productcoefficientscoefficientsum) + (fs_r_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divides_intro_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_right_divides_intro_result_productcoefficientscoefficientsum = fs_q_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divides_intro_result_productcoefficientscoefficientsum) + (fs_s_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps = fs_r_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps + fs_a_pfc_right_divides_intro_result_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_right_divides_intro_result_productcoefficientscoefficientresiduebound. pfa_gap_right_divides_intro_result_productcoefficientscoefficientresiduebound + S (pfc_value_right_divides_intro_result_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_right_divides_intro_result_productcoefficientscoefficientresiduecongruence pfa_offset_right_right_divides_intro_result_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_right_divides_intro_result_productcoefficientscoefficient) + (p) * pfa_offset_left_right_divides_intro_result_productcoefficientscoefficientresiduecongruence = (pfc_value_right_divides_intro_result_productcoefficients) + (p) * pfa_offset_right_right_divides_intro_result_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_right_divides_intro_result_target pfrep_left_right_divides_intro_result_target pfrep_right_right_divides_intro_result_target. ((exists pfrep_position_right_divides_intro_result_targetfirst. ((pfrep_position_right_divides_intro_result_targetfirst+S (pfrep_power_right_divides_intro_result_target)=(pfrd_plen_right_divides_intro_result)) /\\ ((((exists ff_h_pfp_right_divides_intro_result_targetfirstentry. ff_h_pfp_right_divides_intro_result_targetfirstentry + S (pfrep_left_right_divides_intro_result_target) = S ((S (pfrep_position_right_divides_intro_result_targetfirst)) * pfrd_pc_right_divides_intro_result)) /\\ exists ff_q_pfp_right_divides_intro_result_targetfirstentry. pfrd_pb_right_divides_intro_result = ff_q_pfp_right_divides_intro_result_targetfirstentry * S ((S (pfrep_position_right_divides_intro_result_targetfirst)) * pfrd_pc_right_divides_intro_result) + (pfrep_left_right_divides_intro_result_target)))))) \\/ (((exists pfrep_gap_right_divides_intro_result_targetfirstoutside. pfrep_gap_right_divides_intro_result_targetfirstoutside+(pfrd_plen_right_divides_intro_result)=(pfrep_power_right_divides_intro_result_target)) /\\ (((pfrep_left_right_divides_intro_result_target)=0))))) -> ((exists pfrep_position_right_divides_intro_result_targetsecond. 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"6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "3a8d6ac5fa6ca6a68d71ed7d05e957d6ec8838161e6191143eadc98443fc2dc9", "membership": "alpha_only", "name": "prime_field_polynomial_right_divides_divisor_bounded", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro db", "intro dc", "intro D", "intro ab", "intro ac", "intro L", "intro h", "cases h", "cases h_right", "cases h_right_witness", "cases h_right_witness_witness", "cases h_right_witness_witness_witness", "cases h_right_witness_witness_witness_witness", "cases h_right_witness_witness_witness_witness_witness", "cases h_right_witness_witness_witness_witness_witness_witness", "cases h_right_witness_witness_witness_witness_witness_witness_left", "cases h_right_witness_witness_witness_witness_witness_witness_left_right", "cases h_right_witness_witness_witness_witness_witness_witness_left_right_right", "exact h_right_witness_witness_witness_witness_witness_witness_left_right_left" ], "script_sha256": "51ecd5b543f2acb17915c5f134e8cbc6de8b1ab73855759df61dbaf93745b128", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_divisibility_candidate.py", "sha256": "f544adedd3ce963e4a773e8582efcb0f91ba7491207c9792d477d452e854f2b8" }, "statement": "forall p db dc D ab ac L. (((forall fom_index_pfp_right_divisor_bound_source_canonical. (exists fom_gap_pfp_right_divisor_bound_source_canonical_index_bound. fom_gap_pfp_right_divisor_bound_source_canonical_index_bound + S (fom_index_pfp_right_divisor_bound_source_canonical) = L) -> exists fom_value_pfp_right_divisor_bound_source_canonical. ((((exists fom_beta_height_pfp_right_divisor_bound_source_canonical_entry. fom_beta_height_pfp_right_divisor_bound_source_canonical_entry + S (fom_value_pfp_right_divisor_bound_source_canonical) = S ((S (fom_index_pfp_right_divisor_bound_source_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_right_divisor_bound_source_canonical_entry. ab = fom_beta_quotient_pfp_right_divisor_bound_source_canonical_entry * S ((S (fom_index_pfp_right_divisor_bound_source_canonical)) * ac) + (fom_value_pfp_right_divisor_bound_source_canonical))) /\\ (exists fom_gap_pfp_right_divisor_bound_source_canonical_value_bound. fom_gap_pfp_right_divisor_bound_source_canonical_value_bound + S (fom_value_pfp_right_divisor_bound_source_canonical) = p))) /\\ ((exists pfrd_qb_right_divisor_bound_source pfrd_qc_right_divisor_bound_source pfrd_qlen_right_divisor_bound_source pfrd_pb_right_divisor_bound_source pfrd_pc_right_divisor_bound_source pfrd_plen_right_divisor_bound_source. ((((forall fom_index_pfp_right_divisor_bound_source_productleft. (exists fom_gap_pfp_right_divisor_bound_source_productleft_index_bound. fom_gap_pfp_right_divisor_bound_source_productleft_index_bound + S (fom_index_pfp_right_divisor_bound_source_productleft) = pfrd_qlen_right_divisor_bound_source) -> exists fom_value_pfp_right_divisor_bound_source_productleft. ((((exists fom_beta_height_pfp_right_divisor_bound_source_productleft_entry. fom_beta_height_pfp_right_divisor_bound_source_productleft_entry + S (fom_value_pfp_right_divisor_bound_source_productleft) = S ((S (fom_index_pfp_right_divisor_bound_source_productleft)) * pfrd_qc_right_divisor_bound_source)) /\\ exists fom_beta_quotient_pfp_right_divisor_bound_source_productleft_entry. pfrd_qb_right_divisor_bound_source = fom_beta_quotient_pfp_right_divisor_bound_source_productleft_entry * S ((S (fom_index_pfp_right_divisor_bound_source_productleft)) * pfrd_qc_right_divisor_bound_source) + (fom_value_pfp_right_divisor_bound_source_productleft))) /\\ (exists fom_gap_pfp_right_divisor_bound_source_productleft_value_bound. fom_gap_pfp_right_divisor_bound_source_productleft_value_bound + S (fom_value_pfp_right_divisor_bound_source_productleft) = p))) /\\ (((forall fom_index_pfp_right_divisor_bound_source_productright. (exists fom_gap_pfp_right_divisor_bound_source_productright_index_bound. fom_gap_pfp_right_divisor_bound_source_productright_index_bound + S (fom_index_pfp_right_divisor_bound_source_productright) = D) -> exists fom_value_pfp_right_divisor_bound_source_productright. ((((exists fom_beta_height_pfp_right_divisor_bound_source_productright_entry. fom_beta_height_pfp_right_divisor_bound_source_productright_entry + S (fom_value_pfp_right_divisor_bound_source_productright) = S ((S (fom_index_pfp_right_divisor_bound_source_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_right_divisor_bound_source_productright_entry. db = fom_beta_quotient_pfp_right_divisor_bound_source_productright_entry * S ((S (fom_index_pfp_right_divisor_bound_source_productright)) * dc) + (fom_value_pfp_right_divisor_bound_source_productright))) /\\ (exists fom_gap_pfp_right_divisor_bound_source_productright_value_bound. fom_gap_pfp_right_divisor_bound_source_productright_value_bound + S (fom_value_pfp_right_divisor_bound_source_productright) = p))) /\\ (((((((pfrd_qlen_right_divisor_bound_source)=0 \\/ (D)=0) /\\ (((pfrd_plen_right_divisor_bound_source)=0)))) \\/ (((~((pfrd_qlen_right_divisor_bound_source)=0)) /\\ (((~((D)=0)) /\\ (((pfrd_qlen_right_divisor_bound_source)+(D)=S (pfrd_plen_right_divisor_bound_source)))))))) /\\ ((forall pfc_index_right_divisor_bound_source_productcoefficients. (exists pfa_gap_right_divisor_bound_source_productcoefficientsbound. pfa_gap_right_divisor_bound_source_productcoefficientsbound + S (pfc_index_right_divisor_bound_source_productcoefficients) = (pfrd_plen_right_divisor_bound_source)) -> exists pfc_value_right_divisor_bound_source_productcoefficients. ((((exists ff_h_pfp_right_divisor_bound_source_productcoefficientsentry. ff_h_pfp_right_divisor_bound_source_productcoefficientsentry + S (pfc_value_right_divisor_bound_source_productcoefficients) = S ((S (pfc_index_right_divisor_bound_source_productcoefficients)) * pfrd_pc_right_divisor_bound_source)) /\\ exists ff_q_pfp_right_divisor_bound_source_productcoefficientsentry. pfrd_pb_right_divisor_bound_source = ff_q_pfp_right_divisor_bound_source_productcoefficientsentry * S ((S (pfc_index_right_divisor_bound_source_productcoefficients)) * pfrd_pc_right_divisor_bound_source) + (pfc_value_right_divisor_bound_source_productcoefficients))) /\\ ((exists pfc_terms_code_right_divisor_bound_source_productcoefficientscoefficient pfc_terms_scale_right_divisor_bound_source_productcoefficientscoefficient pfc_natural_sum_right_divisor_bound_source_productcoefficientscoefficient. ((forall pfc_index_right_divisor_bound_source_productcoefficientscoefficientdiagonal. (exists pfa_gap_right_divisor_bound_source_productcoefficientscoefficientdiagonalbound. pfa_gap_right_divisor_bound_source_productcoefficientscoefficientdiagonalbound + S (pfc_index_right_divisor_bound_source_productcoefficientscoefficientdiagonal) = (S (pfc_index_right_divisor_bound_source_productcoefficients))) -> exists pfc_value_right_divisor_bound_source_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_right_divisor_bound_source_productcoefficientscoefficientdiagonalentry. ff_h_pfp_right_divisor_bound_source_productcoefficientscoefficientdiagonalentry + S (pfc_value_right_divisor_bound_source_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_right_divisor_bound_source_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_divisor_bound_source_productcoefficientscoefficient)) /\\ exists ff_q_pfp_right_divisor_bound_source_productcoefficientscoefficientdiagonalentry. pfc_terms_code_right_divisor_bound_source_productcoefficientscoefficient = ff_q_pfp_right_divisor_bound_source_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_right_divisor_bound_source_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_divisor_bound_source_productcoefficientscoefficient) + (pfc_value_right_divisor_bound_source_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm pfc_left_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm pfc_right_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm. (((pfc_index_right_divisor_bound_source_productcoefficientscoefficientdiagonal)+pfc_complement_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm=(pfc_index_right_divisor_bound_source_productcoefficients)) /\\ ((((((exists pfa_gap_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_right_divisor_bound_source_productcoefficientscoefficientdiagonal) = (pfrd_qlen_right_divisor_bound_source)) /\\ ((((exists ff_h_pfp_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_right_divisor_bound_source_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_divisor_bound_source)) /\\ exists ff_q_pfp_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_right_divisor_bound_source = ff_q_pfp_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_right_divisor_bound_source_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_divisor_bound_source) + (pfc_left_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_right_divisor_bound_source)=(pfc_index_right_divisor_bound_source_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_right_divisor_bound_source_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_right_divisor_bound_source_productcoefficientscoefficientdiagonal)=pfc_left_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm*pfc_right_right_divisor_bound_source_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_right_divisor_bound_source_productcoefficientscoefficientsum fs_v_pfc_right_divisor_bound_source_productcoefficientscoefficientsum. ((((exists fs_h_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_start. fs_h_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_right_divisor_bound_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_start. fs_u_pfc_right_divisor_bound_source_productcoefficientscoefficientsum = fs_q_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_right_divisor_bound_source_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_right_divisor_bound_source_productcoefficientscoefficient) = S ((S (S (pfc_index_right_divisor_bound_source_productcoefficients))) * fs_v_pfc_right_divisor_bound_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_right_divisor_bound_source_productcoefficientscoefficientsum = fs_q_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_right_divisor_bound_source_productcoefficients))) * fs_v_pfc_right_divisor_bound_source_productcoefficientscoefficientsum) + (pfc_natural_sum_right_divisor_bound_source_productcoefficientscoefficient))) /\\ forall fs_i_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps = S (pfc_index_right_divisor_bound_source_productcoefficients)) -> exists fs_a_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps fs_r_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps fs_s_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_divisor_bound_source_productcoefficientscoefficient)) /\\ exists fs_q_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_right_divisor_bound_source_productcoefficientscoefficient = fs_q_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_divisor_bound_source_productcoefficientscoefficient) + (fs_a_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_bound_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_right_divisor_bound_source_productcoefficientscoefficientsum = fs_q_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_bound_source_productcoefficientscoefficientsum) + (fs_r_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_bound_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_right_divisor_bound_source_productcoefficientscoefficientsum = fs_q_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_bound_source_productcoefficientscoefficientsum) + (fs_s_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps = fs_r_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps + fs_a_pfc_right_divisor_bound_source_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_right_divisor_bound_source_productcoefficientscoefficientresiduebound. pfa_gap_right_divisor_bound_source_productcoefficientscoefficientresiduebound + S (pfc_value_right_divisor_bound_source_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_right_divisor_bound_source_productcoefficientscoefficientresiduecongruence pfa_offset_right_right_divisor_bound_source_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_right_divisor_bound_source_productcoefficientscoefficient) + (p) * pfa_offset_left_right_divisor_bound_source_productcoefficientscoefficientresiduecongruence = (pfc_value_right_divisor_bound_source_productcoefficients) + (p) * pfa_offset_right_right_divisor_bound_source_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_right_divisor_bound_source_target pfrep_left_right_divisor_bound_source_target pfrep_right_right_divisor_bound_source_target. ((exists pfrep_position_right_divisor_bound_source_targetfirst. ((pfrep_position_right_divisor_bound_source_targetfirst+S (pfrep_power_right_divisor_bound_source_target)=(pfrd_plen_right_divisor_bound_source)) /\\ ((((exists ff_h_pfp_right_divisor_bound_source_targetfirstentry. ff_h_pfp_right_divisor_bound_source_targetfirstentry + S (pfrep_left_right_divisor_bound_source_target) = S ((S (pfrep_position_right_divisor_bound_source_targetfirst)) * pfrd_pc_right_divisor_bound_source)) /\\ exists ff_q_pfp_right_divisor_bound_source_targetfirstentry. pfrd_pb_right_divisor_bound_source = ff_q_pfp_right_divisor_bound_source_targetfirstentry * S ((S (pfrep_position_right_divisor_bound_source_targetfirst)) * pfrd_pc_right_divisor_bound_source) + (pfrep_left_right_divisor_bound_source_target)))))) \\/ (((exists pfrep_gap_right_divisor_bound_source_targetfirstoutside. pfrep_gap_right_divisor_bound_source_targetfirstoutside+(pfrd_plen_right_divisor_bound_source)=(pfrep_power_right_divisor_bound_source_target)) /\\ (((pfrep_left_right_divisor_bound_source_target)=0))))) -> ((exists pfrep_position_right_divisor_bound_source_targetsecond. ((pfrep_position_right_divisor_bound_source_targetsecond+S (pfrep_power_right_divisor_bound_source_target)=(L)) /\\ ((((exists ff_h_pfp_right_divisor_bound_source_targetsecondentry. ff_h_pfp_right_divisor_bound_source_targetsecondentry + S (pfrep_right_right_divisor_bound_source_target) = S ((S (pfrep_position_right_divisor_bound_source_targetsecond)) * ac)) /\\ exists ff_q_pfp_right_divisor_bound_source_targetsecondentry. ab = ff_q_pfp_right_divisor_bound_source_targetsecondentry * S ((S (pfrep_position_right_divisor_bound_source_targetsecond)) * ac) + (pfrep_right_right_divisor_bound_source_target)))))) \\/ (((exists pfrep_gap_right_divisor_bound_source_targetsecondoutside. pfrep_gap_right_divisor_bound_source_targetsecondoutside+(L)=(pfrep_power_right_divisor_bound_source_target)) /\\ (((pfrep_right_right_divisor_bound_source_target)=0))))) -> pfrep_left_right_divisor_bound_source_target=pfrep_right_right_divisor_bound_source_target))))))) -> (forall fom_index_pfp_right_divisor_bound_result. (exists fom_gap_pfp_right_divisor_bound_result_index_bound. fom_gap_pfp_right_divisor_bound_result_index_bound + S (fom_index_pfp_right_divisor_bound_result) = D) -> exists fom_value_pfp_right_divisor_bound_result. ((((exists fom_beta_height_pfp_right_divisor_bound_result_entry. fom_beta_height_pfp_right_divisor_bound_result_entry + S (fom_value_pfp_right_divisor_bound_result) = S ((S (fom_index_pfp_right_divisor_bound_result)) * dc)) /\\ exists fom_beta_quotient_pfp_right_divisor_bound_result_entry. db = fom_beta_quotient_pfp_right_divisor_bound_result_entry * S ((S (fom_index_pfp_right_divisor_bound_result)) * dc) + (fom_value_pfp_right_divisor_bound_result))) /\\ (exists fom_gap_pfp_right_divisor_bound_result_value_bound. fom_gap_pfp_right_divisor_bound_result_value_bound + S (fom_value_pfp_right_divisor_bound_result) = p)))", "statement_sha256": "544318213b2b7d1697a9d395876f8de05096b5f5d66464fcf35d0607b2766f8d", "summary": "The actual witnessed convolution supplies a canonical divisor prefix, including zero-length divisors.", "summary_sha256": "131bdbb0c96006e6fecccf874780e105f98ebd4783be14825d68d70a72c7ce81" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": 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(((forall fom_index_pfp_right_dividend_bound_source_canonical. (exists fom_gap_pfp_right_dividend_bound_source_canonical_index_bound. fom_gap_pfp_right_dividend_bound_source_canonical_index_bound + S (fom_index_pfp_right_dividend_bound_source_canonical) = L) -> exists fom_value_pfp_right_dividend_bound_source_canonical. ((((exists fom_beta_height_pfp_right_dividend_bound_source_canonical_entry. fom_beta_height_pfp_right_dividend_bound_source_canonical_entry + S (fom_value_pfp_right_dividend_bound_source_canonical) = S ((S (fom_index_pfp_right_dividend_bound_source_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_right_dividend_bound_source_canonical_entry. ab = fom_beta_quotient_pfp_right_dividend_bound_source_canonical_entry * S ((S (fom_index_pfp_right_dividend_bound_source_canonical)) * ac) + (fom_value_pfp_right_dividend_bound_source_canonical))) /\\ (exists fom_gap_pfp_right_dividend_bound_source_canonical_value_bound. fom_gap_pfp_right_dividend_bound_source_canonical_value_bound + S (fom_value_pfp_right_dividend_bound_source_canonical) = p))) /\\ ((exists pfrd_qb_right_dividend_bound_source pfrd_qc_right_dividend_bound_source pfrd_qlen_right_dividend_bound_source pfrd_pb_right_dividend_bound_source pfrd_pc_right_dividend_bound_source pfrd_plen_right_dividend_bound_source. ((((forall fom_index_pfp_right_dividend_bound_source_productleft. (exists fom_gap_pfp_right_dividend_bound_source_productleft_index_bound. fom_gap_pfp_right_dividend_bound_source_productleft_index_bound + S (fom_index_pfp_right_dividend_bound_source_productleft) = pfrd_qlen_right_dividend_bound_source) -> exists fom_value_pfp_right_dividend_bound_source_productleft. ((((exists fom_beta_height_pfp_right_dividend_bound_source_productleft_entry. fom_beta_height_pfp_right_dividend_bound_source_productleft_entry + S (fom_value_pfp_right_dividend_bound_source_productleft) = S ((S (fom_index_pfp_right_dividend_bound_source_productleft)) * pfrd_qc_right_dividend_bound_source)) /\\ exists fom_beta_quotient_pfp_right_dividend_bound_source_productleft_entry. pfrd_qb_right_dividend_bound_source = fom_beta_quotient_pfp_right_dividend_bound_source_productleft_entry * S ((S (fom_index_pfp_right_dividend_bound_source_productleft)) * pfrd_qc_right_dividend_bound_source) + (fom_value_pfp_right_dividend_bound_source_productleft))) /\\ (exists fom_gap_pfp_right_dividend_bound_source_productleft_value_bound. fom_gap_pfp_right_dividend_bound_source_productleft_value_bound + S (fom_value_pfp_right_dividend_bound_source_productleft) = p))) /\\ (((forall fom_index_pfp_right_dividend_bound_source_productright. (exists fom_gap_pfp_right_dividend_bound_source_productright_index_bound. fom_gap_pfp_right_dividend_bound_source_productright_index_bound + S (fom_index_pfp_right_dividend_bound_source_productright) = D) -> exists fom_value_pfp_right_dividend_bound_source_productright. ((((exists fom_beta_height_pfp_right_dividend_bound_source_productright_entry. fom_beta_height_pfp_right_dividend_bound_source_productright_entry + S (fom_value_pfp_right_dividend_bound_source_productright) = S ((S (fom_index_pfp_right_dividend_bound_source_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_right_dividend_bound_source_productright_entry. db = fom_beta_quotient_pfp_right_dividend_bound_source_productright_entry * S ((S (fom_index_pfp_right_dividend_bound_source_productright)) * dc) + (fom_value_pfp_right_dividend_bound_source_productright))) /\\ (exists fom_gap_pfp_right_dividend_bound_source_productright_value_bound. fom_gap_pfp_right_dividend_bound_source_productright_value_bound + S (fom_value_pfp_right_dividend_bound_source_productright) = p))) /\\ (((((((pfrd_qlen_right_dividend_bound_source)=0 \\/ (D)=0) /\\ (((pfrd_plen_right_dividend_bound_source)=0)))) \\/ (((~((pfrd_qlen_right_dividend_bound_source)=0)) /\\ (((~((D)=0)) /\\ (((pfrd_qlen_right_dividend_bound_source)+(D)=S (pfrd_plen_right_dividend_bound_source)))))))) /\\ ((forall pfc_index_right_dividend_bound_source_productcoefficients. (exists pfa_gap_right_dividend_bound_source_productcoefficientsbound. pfa_gap_right_dividend_bound_source_productcoefficientsbound + S (pfc_index_right_dividend_bound_source_productcoefficients) = (pfrd_plen_right_dividend_bound_source)) -> exists pfc_value_right_dividend_bound_source_productcoefficients. ((((exists ff_h_pfp_right_dividend_bound_source_productcoefficientsentry. ff_h_pfp_right_dividend_bound_source_productcoefficientsentry + S (pfc_value_right_dividend_bound_source_productcoefficients) = S ((S (pfc_index_right_dividend_bound_source_productcoefficients)) * pfrd_pc_right_dividend_bound_source)) /\\ exists ff_q_pfp_right_dividend_bound_source_productcoefficientsentry. pfrd_pb_right_dividend_bound_source = ff_q_pfp_right_dividend_bound_source_productcoefficientsentry * S ((S (pfc_index_right_dividend_bound_source_productcoefficients)) * pfrd_pc_right_dividend_bound_source) + (pfc_value_right_dividend_bound_source_productcoefficients))) /\\ ((exists pfc_terms_code_right_dividend_bound_source_productcoefficientscoefficient pfc_terms_scale_right_dividend_bound_source_productcoefficientscoefficient pfc_natural_sum_right_dividend_bound_source_productcoefficientscoefficient. ((forall pfc_index_right_dividend_bound_source_productcoefficientscoefficientdiagonal. (exists pfa_gap_right_dividend_bound_source_productcoefficientscoefficientdiagonalbound. pfa_gap_right_dividend_bound_source_productcoefficientscoefficientdiagonalbound + S (pfc_index_right_dividend_bound_source_productcoefficientscoefficientdiagonal) = (S (pfc_index_right_dividend_bound_source_productcoefficients))) -> exists pfc_value_right_dividend_bound_source_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_right_dividend_bound_source_productcoefficientscoefficientdiagonalentry. ff_h_pfp_right_dividend_bound_source_productcoefficientscoefficientdiagonalentry + S (pfc_value_right_dividend_bound_source_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_right_dividend_bound_source_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_dividend_bound_source_productcoefficientscoefficient)) /\\ exists ff_q_pfp_right_dividend_bound_source_productcoefficientscoefficientdiagonalentry. pfc_terms_code_right_dividend_bound_source_productcoefficientscoefficient = ff_q_pfp_right_dividend_bound_source_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_right_dividend_bound_source_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_dividend_bound_source_productcoefficientscoefficient) + (pfc_value_right_dividend_bound_source_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm pfc_left_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm pfc_right_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm. (((pfc_index_right_dividend_bound_source_productcoefficientscoefficientdiagonal)+pfc_complement_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm=(pfc_index_right_dividend_bound_source_productcoefficients)) /\\ ((((((exists pfa_gap_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_right_dividend_bound_source_productcoefficientscoefficientdiagonal) = (pfrd_qlen_right_dividend_bound_source)) /\\ ((((exists ff_h_pfp_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_right_dividend_bound_source_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_dividend_bound_source)) /\\ exists ff_q_pfp_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_right_dividend_bound_source = ff_q_pfp_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_right_dividend_bound_source_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_dividend_bound_source) + (pfc_left_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_right_dividend_bound_source)=(pfc_index_right_dividend_bound_source_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_right_dividend_bound_source_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_right_dividend_bound_source_productcoefficientscoefficientdiagonal)=pfc_left_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm*pfc_right_right_dividend_bound_source_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_right_dividend_bound_source_productcoefficientscoefficientsum fs_v_pfc_right_dividend_bound_source_productcoefficientscoefficientsum. ((((exists fs_h_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_start. fs_h_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_right_dividend_bound_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_start. fs_u_pfc_right_dividend_bound_source_productcoefficientscoefficientsum = fs_q_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_right_dividend_bound_source_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_right_dividend_bound_source_productcoefficientscoefficient) = S ((S (S (pfc_index_right_dividend_bound_source_productcoefficients))) * fs_v_pfc_right_dividend_bound_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_right_dividend_bound_source_productcoefficientscoefficientsum = fs_q_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_right_dividend_bound_source_productcoefficients))) * fs_v_pfc_right_dividend_bound_source_productcoefficientscoefficientsum) + (pfc_natural_sum_right_dividend_bound_source_productcoefficientscoefficient))) /\\ forall fs_i_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps = S (pfc_index_right_dividend_bound_source_productcoefficients)) -> exists fs_a_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps fs_r_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps fs_s_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_dividend_bound_source_productcoefficientscoefficient)) /\\ exists fs_q_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_right_dividend_bound_source_productcoefficientscoefficient = fs_q_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_dividend_bound_source_productcoefficientscoefficient) + (fs_a_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_dividend_bound_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_right_dividend_bound_source_productcoefficientscoefficientsum = fs_q_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_dividend_bound_source_productcoefficientscoefficientsum) + (fs_r_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_dividend_bound_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_right_dividend_bound_source_productcoefficientscoefficientsum = fs_q_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_dividend_bound_source_productcoefficientscoefficientsum) + (fs_s_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps = fs_r_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps + fs_a_pfc_right_dividend_bound_source_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_right_dividend_bound_source_productcoefficientscoefficientresiduebound. pfa_gap_right_dividend_bound_source_productcoefficientscoefficientresiduebound + S (pfc_value_right_dividend_bound_source_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_right_dividend_bound_source_productcoefficientscoefficientresiduecongruence pfa_offset_right_right_dividend_bound_source_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_right_dividend_bound_source_productcoefficientscoefficient) + (p) * pfa_offset_left_right_dividend_bound_source_productcoefficientscoefficientresiduecongruence = (pfc_value_right_dividend_bound_source_productcoefficients) + (p) * pfa_offset_right_right_dividend_bound_source_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_right_dividend_bound_source_target pfrep_left_right_dividend_bound_source_target pfrep_right_right_dividend_bound_source_target. ((exists pfrep_position_right_dividend_bound_source_targetfirst. ((pfrep_position_right_dividend_bound_source_targetfirst+S (pfrep_power_right_dividend_bound_source_target)=(pfrd_plen_right_dividend_bound_source)) /\\ ((((exists ff_h_pfp_right_dividend_bound_source_targetfirstentry. ff_h_pfp_right_dividend_bound_source_targetfirstentry + S (pfrep_left_right_dividend_bound_source_target) = S ((S (pfrep_position_right_dividend_bound_source_targetfirst)) * pfrd_pc_right_dividend_bound_source)) /\\ exists ff_q_pfp_right_dividend_bound_source_targetfirstentry. pfrd_pb_right_dividend_bound_source = ff_q_pfp_right_dividend_bound_source_targetfirstentry * S ((S (pfrep_position_right_dividend_bound_source_targetfirst)) * pfrd_pc_right_dividend_bound_source) + (pfrep_left_right_dividend_bound_source_target)))))) \\/ (((exists pfrep_gap_right_dividend_bound_source_targetfirstoutside. pfrep_gap_right_dividend_bound_source_targetfirstoutside+(pfrd_plen_right_dividend_bound_source)=(pfrep_power_right_dividend_bound_source_target)) /\\ (((pfrep_left_right_dividend_bound_source_target)=0))))) -> ((exists pfrep_position_right_dividend_bound_source_targetsecond. ((pfrep_position_right_dividend_bound_source_targetsecond+S (pfrep_power_right_dividend_bound_source_target)=(L)) /\\ ((((exists ff_h_pfp_right_dividend_bound_source_targetsecondentry. ff_h_pfp_right_dividend_bound_source_targetsecondentry + S (pfrep_right_right_dividend_bound_source_target) = S ((S (pfrep_position_right_dividend_bound_source_targetsecond)) * ac)) /\\ exists ff_q_pfp_right_dividend_bound_source_targetsecondentry. ab = ff_q_pfp_right_dividend_bound_source_targetsecondentry * S ((S (pfrep_position_right_dividend_bound_source_targetsecond)) * ac) + (pfrep_right_right_dividend_bound_source_target)))))) \\/ (((exists pfrep_gap_right_dividend_bound_source_targetsecondoutside. pfrep_gap_right_dividend_bound_source_targetsecondoutside+(L)=(pfrep_power_right_dividend_bound_source_target)) /\\ (((pfrep_right_right_dividend_bound_source_target)=0))))) -> pfrep_left_right_dividend_bound_source_target=pfrep_right_right_dividend_bound_source_target))))))) -> (forall fom_index_pfp_right_dividend_bound_result. 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(forall fom_index_pfp_right_target_bound. (exists fom_gap_pfp_right_target_bound_index_bound. fom_gap_pfp_right_target_bound_index_bound + S (fom_index_pfp_right_target_bound) = M) -> exists fom_value_pfp_right_target_bound. ((((exists fom_beta_height_pfp_right_target_bound_entry. fom_beta_height_pfp_right_target_bound_entry + S (fom_value_pfp_right_target_bound) = S ((S (fom_index_pfp_right_target_bound)) * bc)) /\\ exists fom_beta_quotient_pfp_right_target_bound_entry. bb = fom_beta_quotient_pfp_right_target_bound_entry * S ((S (fom_index_pfp_right_target_bound)) * bc) + (fom_value_pfp_right_target_bound))) /\\ (exists fom_gap_pfp_right_target_bound_value_bound. fom_gap_pfp_right_target_bound_value_bound + S (fom_value_pfp_right_target_bound) = p))) -> (((forall fom_index_pfp_right_target_source_canonical. (exists fom_gap_pfp_right_target_source_canonical_index_bound. fom_gap_pfp_right_target_source_canonical_index_bound + S (fom_index_pfp_right_target_source_canonical) = L) -> exists fom_value_pfp_right_target_source_canonical. ((((exists fom_beta_height_pfp_right_target_source_canonical_entry. fom_beta_height_pfp_right_target_source_canonical_entry + S (fom_value_pfp_right_target_source_canonical) = S ((S (fom_index_pfp_right_target_source_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_right_target_source_canonical_entry. ab = fom_beta_quotient_pfp_right_target_source_canonical_entry * S ((S (fom_index_pfp_right_target_source_canonical)) * ac) + (fom_value_pfp_right_target_source_canonical))) /\\ (exists fom_gap_pfp_right_target_source_canonical_value_bound. fom_gap_pfp_right_target_source_canonical_value_bound + S (fom_value_pfp_right_target_source_canonical) = p))) /\\ ((exists pfrd_qb_right_target_source pfrd_qc_right_target_source pfrd_qlen_right_target_source pfrd_pb_right_target_source pfrd_pc_right_target_source pfrd_plen_right_target_source. ((((forall fom_index_pfp_right_target_source_productleft. (exists fom_gap_pfp_right_target_source_productleft_index_bound. fom_gap_pfp_right_target_source_productleft_index_bound + S (fom_index_pfp_right_target_source_productleft) = pfrd_qlen_right_target_source) -> exists fom_value_pfp_right_target_source_productleft. ((((exists fom_beta_height_pfp_right_target_source_productleft_entry. fom_beta_height_pfp_right_target_source_productleft_entry + S (fom_value_pfp_right_target_source_productleft) = S ((S (fom_index_pfp_right_target_source_productleft)) * pfrd_qc_right_target_source)) /\\ exists fom_beta_quotient_pfp_right_target_source_productleft_entry. pfrd_qb_right_target_source = fom_beta_quotient_pfp_right_target_source_productleft_entry * S ((S (fom_index_pfp_right_target_source_productleft)) * pfrd_qc_right_target_source) + (fom_value_pfp_right_target_source_productleft))) /\\ (exists fom_gap_pfp_right_target_source_productleft_value_bound. fom_gap_pfp_right_target_source_productleft_value_bound + S (fom_value_pfp_right_target_source_productleft) = p))) /\\ (((forall fom_index_pfp_right_target_source_productright. (exists fom_gap_pfp_right_target_source_productright_index_bound. fom_gap_pfp_right_target_source_productright_index_bound + S (fom_index_pfp_right_target_source_productright) = D) -> exists fom_value_pfp_right_target_source_productright. ((((exists fom_beta_height_pfp_right_target_source_productright_entry. fom_beta_height_pfp_right_target_source_productright_entry + S (fom_value_pfp_right_target_source_productright) = S ((S (fom_index_pfp_right_target_source_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_right_target_source_productright_entry. db = fom_beta_quotient_pfp_right_target_source_productright_entry * S ((S (fom_index_pfp_right_target_source_productright)) * dc) + (fom_value_pfp_right_target_source_productright))) /\\ (exists fom_gap_pfp_right_target_source_productright_value_bound. fom_gap_pfp_right_target_source_productright_value_bound + S (fom_value_pfp_right_target_source_productright) = p))) /\\ (((((((pfrd_qlen_right_target_source)=0 \\/ (D)=0) /\\ (((pfrd_plen_right_target_source)=0)))) \\/ (((~((pfrd_qlen_right_target_source)=0)) /\\ (((~((D)=0)) /\\ (((pfrd_qlen_right_target_source)+(D)=S (pfrd_plen_right_target_source)))))))) /\\ ((forall pfc_index_right_target_source_productcoefficients. (exists pfa_gap_right_target_source_productcoefficientsbound. pfa_gap_right_target_source_productcoefficientsbound + S (pfc_index_right_target_source_productcoefficients) = (pfrd_plen_right_target_source)) -> exists pfc_value_right_target_source_productcoefficients. ((((exists ff_h_pfp_right_target_source_productcoefficientsentry. ff_h_pfp_right_target_source_productcoefficientsentry + S (pfc_value_right_target_source_productcoefficients) = S ((S (pfc_index_right_target_source_productcoefficients)) * pfrd_pc_right_target_source)) /\\ exists ff_q_pfp_right_target_source_productcoefficientsentry. pfrd_pb_right_target_source = ff_q_pfp_right_target_source_productcoefficientsentry * S ((S (pfc_index_right_target_source_productcoefficients)) * pfrd_pc_right_target_source) + (pfc_value_right_target_source_productcoefficients))) /\\ ((exists pfc_terms_code_right_target_source_productcoefficientscoefficient pfc_terms_scale_right_target_source_productcoefficientscoefficient pfc_natural_sum_right_target_source_productcoefficientscoefficient. ((forall pfc_index_right_target_source_productcoefficientscoefficientdiagonal. (exists pfa_gap_right_target_source_productcoefficientscoefficientdiagonalbound. pfa_gap_right_target_source_productcoefficientscoefficientdiagonalbound + S (pfc_index_right_target_source_productcoefficientscoefficientdiagonal) = (S (pfc_index_right_target_source_productcoefficients))) -> exists pfc_value_right_target_source_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_right_target_source_productcoefficientscoefficientdiagonalentry. ff_h_pfp_right_target_source_productcoefficientscoefficientdiagonalentry + S (pfc_value_right_target_source_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_right_target_source_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_target_source_productcoefficientscoefficient)) /\\ exists ff_q_pfp_right_target_source_productcoefficientscoefficientdiagonalentry. pfc_terms_code_right_target_source_productcoefficientscoefficient = ff_q_pfp_right_target_source_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_right_target_source_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_target_source_productcoefficientscoefficient) + (pfc_value_right_target_source_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_right_target_source_productcoefficientscoefficientdiagonalterm pfc_left_right_target_source_productcoefficientscoefficientdiagonalterm pfc_right_right_target_source_productcoefficientscoefficientdiagonalterm. (((pfc_index_right_target_source_productcoefficientscoefficientdiagonal)+pfc_complement_right_target_source_productcoefficientscoefficientdiagonalterm=(pfc_index_right_target_source_productcoefficients)) /\\ ((((((exists pfa_gap_right_target_source_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_right_target_source_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_right_target_source_productcoefficientscoefficientdiagonal) = (pfrd_qlen_right_target_source)) /\\ ((((exists ff_h_pfp_right_target_source_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_right_target_source_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_right_target_source_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_right_target_source_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_target_source)) /\\ exists ff_q_pfp_right_target_source_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_right_target_source = ff_q_pfp_right_target_source_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_right_target_source_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_target_source) + (pfc_left_right_target_source_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_target_source_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_right_target_source_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_right_target_source)=(pfc_index_right_target_source_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_right_target_source_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_right_target_source_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_right_target_source_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_right_target_source_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_right_target_source_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_right_target_source_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_right_target_source_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_right_target_source_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_right_target_source_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_right_target_source_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_right_target_source_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_right_target_source_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_target_source_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_right_target_source_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_right_target_source_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_right_target_source_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_right_target_source_productcoefficientscoefficientdiagonal)=pfc_left_right_target_source_productcoefficientscoefficientdiagonalterm*pfc_right_right_target_source_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_right_target_source_productcoefficientscoefficientsum fs_v_pfc_right_target_source_productcoefficientscoefficientsum. ((((exists fs_h_pfc_right_target_source_productcoefficientscoefficientsum_body_start. fs_h_pfc_right_target_source_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_right_target_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_target_source_productcoefficientscoefficientsum_body_start. fs_u_pfc_right_target_source_productcoefficientscoefficientsum = fs_q_pfc_right_target_source_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_right_target_source_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_right_target_source_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_right_target_source_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_right_target_source_productcoefficientscoefficient) = S ((S (S (pfc_index_right_target_source_productcoefficients))) * fs_v_pfc_right_target_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_target_source_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_right_target_source_productcoefficientscoefficientsum = fs_q_pfc_right_target_source_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_right_target_source_productcoefficients))) * fs_v_pfc_right_target_source_productcoefficientscoefficientsum) + (pfc_natural_sum_right_target_source_productcoefficientscoefficient))) /\\ forall fs_i_pfc_right_target_source_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_right_target_source_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_right_target_source_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_right_target_source_productcoefficientscoefficientsum_body_steps = S (pfc_index_right_target_source_productcoefficients)) -> exists fs_a_pfc_right_target_source_productcoefficientscoefficientsum_body_steps fs_r_pfc_right_target_source_productcoefficientscoefficientsum_body_steps fs_s_pfc_right_target_source_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_right_target_source_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_right_target_source_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_right_target_source_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_target_source_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_target_source_productcoefficientscoefficient)) /\\ exists fs_q_pfc_right_target_source_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_right_target_source_productcoefficientscoefficient = fs_q_pfc_right_target_source_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_right_target_source_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_target_source_productcoefficientscoefficient) + (fs_a_pfc_right_target_source_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_target_source_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_right_target_source_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_right_target_source_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_target_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_target_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_target_source_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_right_target_source_productcoefficientscoefficientsum = fs_q_pfc_right_target_source_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_right_target_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_target_source_productcoefficientscoefficientsum) + (fs_r_pfc_right_target_source_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_target_source_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_right_target_source_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_right_target_source_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_right_target_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_target_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_target_source_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_right_target_source_productcoefficientscoefficientsum = fs_q_pfc_right_target_source_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_right_target_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_target_source_productcoefficientscoefficientsum) + (fs_s_pfc_right_target_source_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_right_target_source_productcoefficientscoefficientsum_body_steps = fs_r_pfc_right_target_source_productcoefficientscoefficientsum_body_steps + fs_a_pfc_right_target_source_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_right_target_source_productcoefficientscoefficientresiduebound. pfa_gap_right_target_source_productcoefficientscoefficientresiduebound + S (pfc_value_right_target_source_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_right_target_source_productcoefficientscoefficientresiduecongruence pfa_offset_right_right_target_source_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_right_target_source_productcoefficientscoefficient) + (p) * pfa_offset_left_right_target_source_productcoefficientscoefficientresiduecongruence = (pfc_value_right_target_source_productcoefficients) + (p) * pfa_offset_right_right_target_source_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_right_target_source_target pfrep_left_right_target_source_target pfrep_right_right_target_source_target. ((exists pfrep_position_right_target_source_targetfirst. ((pfrep_position_right_target_source_targetfirst+S (pfrep_power_right_target_source_target)=(pfrd_plen_right_target_source)) /\\ ((((exists ff_h_pfp_right_target_source_targetfirstentry. ff_h_pfp_right_target_source_targetfirstentry + S (pfrep_left_right_target_source_target) = S ((S (pfrep_position_right_target_source_targetfirst)) * pfrd_pc_right_target_source)) /\\ exists ff_q_pfp_right_target_source_targetfirstentry. pfrd_pb_right_target_source = ff_q_pfp_right_target_source_targetfirstentry * S ((S (pfrep_position_right_target_source_targetfirst)) * pfrd_pc_right_target_source) + (pfrep_left_right_target_source_target)))))) \\/ (((exists pfrep_gap_right_target_source_targetfirstoutside. pfrep_gap_right_target_source_targetfirstoutside+(pfrd_plen_right_target_source)=(pfrep_power_right_target_source_target)) /\\ (((pfrep_left_right_target_source_target)=0))))) -> ((exists pfrep_position_right_target_source_targetsecond. ((pfrep_position_right_target_source_targetsecond+S (pfrep_power_right_target_source_target)=(L)) /\\ ((((exists ff_h_pfp_right_target_source_targetsecondentry. ff_h_pfp_right_target_source_targetsecondentry + S (pfrep_right_right_target_source_target) = S ((S (pfrep_position_right_target_source_targetsecond)) * ac)) /\\ exists ff_q_pfp_right_target_source_targetsecondentry. ab = ff_q_pfp_right_target_source_targetsecondentry * S ((S (pfrep_position_right_target_source_targetsecond)) * ac) + (pfrep_right_right_target_source_target)))))) \\/ (((exists pfrep_gap_right_target_source_targetsecondoutside. pfrep_gap_right_target_source_targetsecondoutside+(L)=(pfrep_power_right_target_source_target)) /\\ (((pfrep_right_right_target_source_target)=0))))) -> pfrep_left_right_target_source_target=pfrep_right_right_target_source_target))))))) -> (forall pfrep_power_right_target_equivalent pfrep_left_right_target_equivalent pfrep_right_right_target_equivalent. ((exists pfrep_position_right_target_equivalentfirst. ((pfrep_position_right_target_equivalentfirst+S (pfrep_power_right_target_equivalent)=(L)) /\\ ((((exists ff_h_pfp_right_target_equivalentfirstentry. ff_h_pfp_right_target_equivalentfirstentry + S (pfrep_left_right_target_equivalent) = S ((S (pfrep_position_right_target_equivalentfirst)) * ac)) /\\ exists ff_q_pfp_right_target_equivalentfirstentry. ab = ff_q_pfp_right_target_equivalentfirstentry * S ((S (pfrep_position_right_target_equivalentfirst)) * ac) + (pfrep_left_right_target_equivalent)))))) \\/ (((exists pfrep_gap_right_target_equivalentfirstoutside. pfrep_gap_right_target_equivalentfirstoutside+(L)=(pfrep_power_right_target_equivalent)) /\\ (((pfrep_left_right_target_equivalent)=0))))) -> ((exists pfrep_position_right_target_equivalentsecond. ((pfrep_position_right_target_equivalentsecond+S (pfrep_power_right_target_equivalent)=(M)) /\\ ((((exists ff_h_pfp_right_target_equivalentsecondentry. ff_h_pfp_right_target_equivalentsecondentry + S (pfrep_right_right_target_equivalent) = S ((S (pfrep_position_right_target_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_right_target_equivalentsecondentry. bb = ff_q_pfp_right_target_equivalentsecondentry * S ((S (pfrep_position_right_target_equivalentsecond)) * bc) + (pfrep_right_right_target_equivalent)))))) \\/ (((exists pfrep_gap_right_target_equivalentsecondoutside. pfrep_gap_right_target_equivalentsecondoutside+(M)=(pfrep_power_right_target_equivalent)) /\\ (((pfrep_right_right_target_equivalent)=0))))) -> pfrep_left_right_target_equivalent=pfrep_right_right_target_equivalent) -> (((forall fom_index_pfp_right_target_result_canonical. (exists fom_gap_pfp_right_target_result_canonical_index_bound. fom_gap_pfp_right_target_result_canonical_index_bound + S (fom_index_pfp_right_target_result_canonical) = M) -> exists fom_value_pfp_right_target_result_canonical. ((((exists fom_beta_height_pfp_right_target_result_canonical_entry. fom_beta_height_pfp_right_target_result_canonical_entry + S (fom_value_pfp_right_target_result_canonical) = S ((S (fom_index_pfp_right_target_result_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_right_target_result_canonical_entry. bb = fom_beta_quotient_pfp_right_target_result_canonical_entry * S ((S (fom_index_pfp_right_target_result_canonical)) * bc) + (fom_value_pfp_right_target_result_canonical))) /\\ (exists fom_gap_pfp_right_target_result_canonical_value_bound. fom_gap_pfp_right_target_result_canonical_value_bound + S (fom_value_pfp_right_target_result_canonical) = p))) /\\ ((exists pfrd_qb_right_target_result pfrd_qc_right_target_result pfrd_qlen_right_target_result pfrd_pb_right_target_result pfrd_pc_right_target_result pfrd_plen_right_target_result. ((((forall fom_index_pfp_right_target_result_productleft. (exists fom_gap_pfp_right_target_result_productleft_index_bound. fom_gap_pfp_right_target_result_productleft_index_bound + S (fom_index_pfp_right_target_result_productleft) = pfrd_qlen_right_target_result) -> exists fom_value_pfp_right_target_result_productleft. ((((exists fom_beta_height_pfp_right_target_result_productleft_entry. fom_beta_height_pfp_right_target_result_productleft_entry + S (fom_value_pfp_right_target_result_productleft) = S ((S (fom_index_pfp_right_target_result_productleft)) * pfrd_qc_right_target_result)) /\\ exists fom_beta_quotient_pfp_right_target_result_productleft_entry. pfrd_qb_right_target_result = fom_beta_quotient_pfp_right_target_result_productleft_entry * S ((S (fom_index_pfp_right_target_result_productleft)) * pfrd_qc_right_target_result) + (fom_value_pfp_right_target_result_productleft))) /\\ (exists fom_gap_pfp_right_target_result_productleft_value_bound. fom_gap_pfp_right_target_result_productleft_value_bound + S (fom_value_pfp_right_target_result_productleft) = p))) /\\ (((forall fom_index_pfp_right_target_result_productright. (exists fom_gap_pfp_right_target_result_productright_index_bound. fom_gap_pfp_right_target_result_productright_index_bound + S (fom_index_pfp_right_target_result_productright) = D) -> exists fom_value_pfp_right_target_result_productright. ((((exists fom_beta_height_pfp_right_target_result_productright_entry. fom_beta_height_pfp_right_target_result_productright_entry + S (fom_value_pfp_right_target_result_productright) = S ((S (fom_index_pfp_right_target_result_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_right_target_result_productright_entry. db = fom_beta_quotient_pfp_right_target_result_productright_entry * S ((S (fom_index_pfp_right_target_result_productright)) * dc) + (fom_value_pfp_right_target_result_productright))) /\\ (exists fom_gap_pfp_right_target_result_productright_value_bound. fom_gap_pfp_right_target_result_productright_value_bound + S (fom_value_pfp_right_target_result_productright) = p))) /\\ (((((((pfrd_qlen_right_target_result)=0 \\/ (D)=0) /\\ (((pfrd_plen_right_target_result)=0)))) \\/ (((~((pfrd_qlen_right_target_result)=0)) /\\ (((~((D)=0)) /\\ (((pfrd_qlen_right_target_result)+(D)=S (pfrd_plen_right_target_result)))))))) /\\ ((forall pfc_index_right_target_result_productcoefficients. (exists pfa_gap_right_target_result_productcoefficientsbound. pfa_gap_right_target_result_productcoefficientsbound + S (pfc_index_right_target_result_productcoefficients) = (pfrd_plen_right_target_result)) -> exists pfc_value_right_target_result_productcoefficients. ((((exists ff_h_pfp_right_target_result_productcoefficientsentry. ff_h_pfp_right_target_result_productcoefficientsentry + S (pfc_value_right_target_result_productcoefficients) = S ((S (pfc_index_right_target_result_productcoefficients)) * pfrd_pc_right_target_result)) /\\ exists ff_q_pfp_right_target_result_productcoefficientsentry. pfrd_pb_right_target_result = ff_q_pfp_right_target_result_productcoefficientsentry * S ((S (pfc_index_right_target_result_productcoefficients)) * pfrd_pc_right_target_result) + (pfc_value_right_target_result_productcoefficients))) /\\ ((exists pfc_terms_code_right_target_result_productcoefficientscoefficient pfc_terms_scale_right_target_result_productcoefficientscoefficient pfc_natural_sum_right_target_result_productcoefficientscoefficient. ((forall pfc_index_right_target_result_productcoefficientscoefficientdiagonal. (exists pfa_gap_right_target_result_productcoefficientscoefficientdiagonalbound. pfa_gap_right_target_result_productcoefficientscoefficientdiagonalbound + S (pfc_index_right_target_result_productcoefficientscoefficientdiagonal) = (S (pfc_index_right_target_result_productcoefficients))) -> exists pfc_value_right_target_result_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_right_target_result_productcoefficientscoefficientdiagonalentry. ff_h_pfp_right_target_result_productcoefficientscoefficientdiagonalentry + S (pfc_value_right_target_result_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_right_target_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_target_result_productcoefficientscoefficient)) /\\ exists ff_q_pfp_right_target_result_productcoefficientscoefficientdiagonalentry. pfc_terms_code_right_target_result_productcoefficientscoefficient = ff_q_pfp_right_target_result_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_right_target_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_target_result_productcoefficientscoefficient) + (pfc_value_right_target_result_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_right_target_result_productcoefficientscoefficientdiagonalterm pfc_left_right_target_result_productcoefficientscoefficientdiagonalterm pfc_right_right_target_result_productcoefficientscoefficientdiagonalterm. (((pfc_index_right_target_result_productcoefficientscoefficientdiagonal)+pfc_complement_right_target_result_productcoefficientscoefficientdiagonalterm=(pfc_index_right_target_result_productcoefficients)) /\\ ((((((exists pfa_gap_right_target_result_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_right_target_result_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_right_target_result_productcoefficientscoefficientdiagonal) = (pfrd_qlen_right_target_result)) /\\ ((((exists ff_h_pfp_right_target_result_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_right_target_result_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_right_target_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_right_target_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_target_result)) /\\ exists ff_q_pfp_right_target_result_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_right_target_result = ff_q_pfp_right_target_result_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_right_target_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_target_result) + (pfc_left_right_target_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_target_result_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_right_target_result_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_right_target_result)=(pfc_index_right_target_result_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_right_target_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_right_target_result_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_right_target_result_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_right_target_result_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_right_target_result_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_right_target_result_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_right_target_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_right_target_result_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_right_target_result_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_right_target_result_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_right_target_result_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_right_target_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_target_result_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_right_target_result_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_right_target_result_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_right_target_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_right_target_result_productcoefficientscoefficientdiagonal)=pfc_left_right_target_result_productcoefficientscoefficientdiagonalterm*pfc_right_right_target_result_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_right_target_result_productcoefficientscoefficientsum fs_v_pfc_right_target_result_productcoefficientscoefficientsum. ((((exists fs_h_pfc_right_target_result_productcoefficientscoefficientsum_body_start. fs_h_pfc_right_target_result_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_right_target_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_target_result_productcoefficientscoefficientsum_body_start. fs_u_pfc_right_target_result_productcoefficientscoefficientsum = fs_q_pfc_right_target_result_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_right_target_result_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_right_target_result_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_right_target_result_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_right_target_result_productcoefficientscoefficient) = S ((S (S (pfc_index_right_target_result_productcoefficients))) * fs_v_pfc_right_target_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_target_result_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_right_target_result_productcoefficientscoefficientsum = fs_q_pfc_right_target_result_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_right_target_result_productcoefficients))) * fs_v_pfc_right_target_result_productcoefficientscoefficientsum) + (pfc_natural_sum_right_target_result_productcoefficientscoefficient))) /\\ forall fs_i_pfc_right_target_result_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_right_target_result_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_right_target_result_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_right_target_result_productcoefficientscoefficientsum_body_steps = S (pfc_index_right_target_result_productcoefficients)) -> exists fs_a_pfc_right_target_result_productcoefficientscoefficientsum_body_steps fs_r_pfc_right_target_result_productcoefficientscoefficientsum_body_steps fs_s_pfc_right_target_result_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_right_target_result_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_right_target_result_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_right_target_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_target_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_target_result_productcoefficientscoefficient)) /\\ exists fs_q_pfc_right_target_result_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_right_target_result_productcoefficientscoefficient = fs_q_pfc_right_target_result_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_right_target_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_target_result_productcoefficientscoefficient) + (fs_a_pfc_right_target_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_target_result_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_right_target_result_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_right_target_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_target_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_target_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_target_result_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_right_target_result_productcoefficientscoefficientsum = fs_q_pfc_right_target_result_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_right_target_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_target_result_productcoefficientscoefficientsum) + (fs_r_pfc_right_target_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_target_result_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_right_target_result_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_right_target_result_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_right_target_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_target_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_target_result_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_right_target_result_productcoefficientscoefficientsum = fs_q_pfc_right_target_result_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_right_target_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_target_result_productcoefficientscoefficientsum) + (fs_s_pfc_right_target_result_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_right_target_result_productcoefficientscoefficientsum_body_steps = fs_r_pfc_right_target_result_productcoefficientscoefficientsum_body_steps + fs_a_pfc_right_target_result_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_right_target_result_productcoefficientscoefficientresiduebound. pfa_gap_right_target_result_productcoefficientscoefficientresiduebound + S (pfc_value_right_target_result_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_right_target_result_productcoefficientscoefficientresiduecongruence pfa_offset_right_right_target_result_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_right_target_result_productcoefficientscoefficient) + (p) * pfa_offset_left_right_target_result_productcoefficientscoefficientresiduecongruence = (pfc_value_right_target_result_productcoefficients) + (p) * pfa_offset_right_right_target_result_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_right_target_result_target pfrep_left_right_target_result_target pfrep_right_right_target_result_target. ((exists pfrep_position_right_target_result_targetfirst. ((pfrep_position_right_target_result_targetfirst+S (pfrep_power_right_target_result_target)=(pfrd_plen_right_target_result)) /\\ ((((exists ff_h_pfp_right_target_result_targetfirstentry. ff_h_pfp_right_target_result_targetfirstentry + S (pfrep_left_right_target_result_target) = S ((S (pfrep_position_right_target_result_targetfirst)) * pfrd_pc_right_target_result)) /\\ exists ff_q_pfp_right_target_result_targetfirstentry. pfrd_pb_right_target_result = ff_q_pfp_right_target_result_targetfirstentry * S ((S (pfrep_position_right_target_result_targetfirst)) * pfrd_pc_right_target_result) + (pfrep_left_right_target_result_target)))))) \\/ (((exists pfrep_gap_right_target_result_targetfirstoutside. pfrep_gap_right_target_result_targetfirstoutside+(pfrd_plen_right_target_result)=(pfrep_power_right_target_result_target)) /\\ (((pfrep_left_right_target_result_target)=0))))) -> ((exists pfrep_position_right_target_result_targetsecond. ((pfrep_position_right_target_result_targetsecond+S (pfrep_power_right_target_result_target)=(M)) /\\ ((((exists ff_h_pfp_right_target_result_targetsecondentry. ff_h_pfp_right_target_result_targetsecondentry + S (pfrep_right_right_target_result_target) = S ((S (pfrep_position_right_target_result_targetsecond)) * bc)) /\\ exists ff_q_pfp_right_target_result_targetsecondentry. bb = ff_q_pfp_right_target_result_targetsecondentry * S ((S (pfrep_position_right_target_result_targetsecond)) * bc) + (pfrep_right_right_target_result_target)))))) \\/ (((exists pfrep_gap_right_target_result_targetsecondoutside. pfrep_gap_right_target_result_targetsecondoutside+(M)=(pfrep_power_right_target_result_target)) /\\ (((pfrep_right_right_target_result_target)=0))))) -> pfrep_left_right_target_result_target=pfrep_right_right_target_result_target)))))))", "statement_sha256": "87d9faf6a5934bdd2423446f49f7a9b58ff6b0da1c60509d4f0b9af98bb04b14", "summary": "Changing the canonical target by all-power formal 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(forall fom_index_pfp_right_empty_divisor. (exists fom_gap_pfp_right_empty_divisor_index_bound. fom_gap_pfp_right_empty_divisor_index_bound + S (fom_index_pfp_right_empty_divisor) = D) -> exists fom_value_pfp_right_empty_divisor. ((((exists fom_beta_height_pfp_right_empty_divisor_entry. fom_beta_height_pfp_right_empty_divisor_entry + S (fom_value_pfp_right_empty_divisor) = S ((S (fom_index_pfp_right_empty_divisor)) * dc)) /\\ exists fom_beta_quotient_pfp_right_empty_divisor_entry. db = fom_beta_quotient_pfp_right_empty_divisor_entry * S ((S (fom_index_pfp_right_empty_divisor)) * dc) + (fom_value_pfp_right_empty_divisor))) /\\ (exists fom_gap_pfp_right_empty_divisor_value_bound. fom_gap_pfp_right_empty_divisor_value_bound + S (fom_value_pfp_right_empty_divisor) = p))) -> (((forall fom_index_pfp_right_empty_result_canonical. (exists fom_gap_pfp_right_empty_result_canonical_index_bound. fom_gap_pfp_right_empty_result_canonical_index_bound + S (fom_index_pfp_right_empty_result_canonical) = 0) -> exists fom_value_pfp_right_empty_result_canonical. ((((exists fom_beta_height_pfp_right_empty_result_canonical_entry. fom_beta_height_pfp_right_empty_result_canonical_entry + S (fom_value_pfp_right_empty_result_canonical) = S ((S (fom_index_pfp_right_empty_result_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_right_empty_result_canonical_entry. ab = fom_beta_quotient_pfp_right_empty_result_canonical_entry * S ((S (fom_index_pfp_right_empty_result_canonical)) * ac) + (fom_value_pfp_right_empty_result_canonical))) /\\ (exists fom_gap_pfp_right_empty_result_canonical_value_bound. fom_gap_pfp_right_empty_result_canonical_value_bound + S (fom_value_pfp_right_empty_result_canonical) = p))) /\\ ((exists pfrd_qb_right_empty_result pfrd_qc_right_empty_result pfrd_qlen_right_empty_result pfrd_pb_right_empty_result pfrd_pc_right_empty_result pfrd_plen_right_empty_result. ((((forall fom_index_pfp_right_empty_result_productleft. (exists fom_gap_pfp_right_empty_result_productleft_index_bound. fom_gap_pfp_right_empty_result_productleft_index_bound + S (fom_index_pfp_right_empty_result_productleft) = pfrd_qlen_right_empty_result) -> exists fom_value_pfp_right_empty_result_productleft. ((((exists fom_beta_height_pfp_right_empty_result_productleft_entry. fom_beta_height_pfp_right_empty_result_productleft_entry + S (fom_value_pfp_right_empty_result_productleft) = S ((S (fom_index_pfp_right_empty_result_productleft)) * pfrd_qc_right_empty_result)) /\\ exists fom_beta_quotient_pfp_right_empty_result_productleft_entry. pfrd_qb_right_empty_result = fom_beta_quotient_pfp_right_empty_result_productleft_entry * S ((S (fom_index_pfp_right_empty_result_productleft)) * pfrd_qc_right_empty_result) + (fom_value_pfp_right_empty_result_productleft))) /\\ (exists fom_gap_pfp_right_empty_result_productleft_value_bound. fom_gap_pfp_right_empty_result_productleft_value_bound + S (fom_value_pfp_right_empty_result_productleft) = p))) /\\ (((forall fom_index_pfp_right_empty_result_productright. (exists fom_gap_pfp_right_empty_result_productright_index_bound. fom_gap_pfp_right_empty_result_productright_index_bound + S (fom_index_pfp_right_empty_result_productright) = D) -> exists fom_value_pfp_right_empty_result_productright. ((((exists fom_beta_height_pfp_right_empty_result_productright_entry. fom_beta_height_pfp_right_empty_result_productright_entry + S (fom_value_pfp_right_empty_result_productright) = S ((S (fom_index_pfp_right_empty_result_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_right_empty_result_productright_entry. db = fom_beta_quotient_pfp_right_empty_result_productright_entry * S ((S (fom_index_pfp_right_empty_result_productright)) * dc) + (fom_value_pfp_right_empty_result_productright))) /\\ (exists fom_gap_pfp_right_empty_result_productright_value_bound. fom_gap_pfp_right_empty_result_productright_value_bound + S (fom_value_pfp_right_empty_result_productright) = p))) /\\ (((((((pfrd_qlen_right_empty_result)=0 \\/ (D)=0) /\\ (((pfrd_plen_right_empty_result)=0)))) \\/ (((~((pfrd_qlen_right_empty_result)=0)) /\\ (((~((D)=0)) /\\ (((pfrd_qlen_right_empty_result)+(D)=S (pfrd_plen_right_empty_result)))))))) /\\ ((forall pfc_index_right_empty_result_productcoefficients. (exists pfa_gap_right_empty_result_productcoefficientsbound. pfa_gap_right_empty_result_productcoefficientsbound + S (pfc_index_right_empty_result_productcoefficients) = (pfrd_plen_right_empty_result)) -> exists pfc_value_right_empty_result_productcoefficients. ((((exists ff_h_pfp_right_empty_result_productcoefficientsentry. ff_h_pfp_right_empty_result_productcoefficientsentry + S (pfc_value_right_empty_result_productcoefficients) = S ((S (pfc_index_right_empty_result_productcoefficients)) * pfrd_pc_right_empty_result)) /\\ exists ff_q_pfp_right_empty_result_productcoefficientsentry. pfrd_pb_right_empty_result = ff_q_pfp_right_empty_result_productcoefficientsentry * S ((S (pfc_index_right_empty_result_productcoefficients)) * pfrd_pc_right_empty_result) + (pfc_value_right_empty_result_productcoefficients))) /\\ ((exists pfc_terms_code_right_empty_result_productcoefficientscoefficient pfc_terms_scale_right_empty_result_productcoefficientscoefficient pfc_natural_sum_right_empty_result_productcoefficientscoefficient. ((forall pfc_index_right_empty_result_productcoefficientscoefficientdiagonal. (exists pfa_gap_right_empty_result_productcoefficientscoefficientdiagonalbound. pfa_gap_right_empty_result_productcoefficientscoefficientdiagonalbound + S (pfc_index_right_empty_result_productcoefficientscoefficientdiagonal) = (S (pfc_index_right_empty_result_productcoefficients))) -> exists pfc_value_right_empty_result_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_right_empty_result_productcoefficientscoefficientdiagonalentry. ff_h_pfp_right_empty_result_productcoefficientscoefficientdiagonalentry + S (pfc_value_right_empty_result_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_right_empty_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_empty_result_productcoefficientscoefficient)) /\\ exists ff_q_pfp_right_empty_result_productcoefficientscoefficientdiagonalentry. pfc_terms_code_right_empty_result_productcoefficientscoefficient = ff_q_pfp_right_empty_result_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_right_empty_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_empty_result_productcoefficientscoefficient) + (pfc_value_right_empty_result_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_right_empty_result_productcoefficientscoefficientdiagonalterm pfc_left_right_empty_result_productcoefficientscoefficientdiagonalterm pfc_right_right_empty_result_productcoefficientscoefficientdiagonalterm. (((pfc_index_right_empty_result_productcoefficientscoefficientdiagonal)+pfc_complement_right_empty_result_productcoefficientscoefficientdiagonalterm=(pfc_index_right_empty_result_productcoefficients)) /\\ ((((((exists pfa_gap_right_empty_result_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_right_empty_result_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_right_empty_result_productcoefficientscoefficientdiagonal) = (pfrd_qlen_right_empty_result)) /\\ ((((exists ff_h_pfp_right_empty_result_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_right_empty_result_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_right_empty_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_right_empty_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_empty_result)) /\\ exists ff_q_pfp_right_empty_result_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_right_empty_result = ff_q_pfp_right_empty_result_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_right_empty_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_empty_result) + (pfc_left_right_empty_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_empty_result_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_right_empty_result_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_right_empty_result)=(pfc_index_right_empty_result_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_right_empty_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_right_empty_result_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_right_empty_result_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_right_empty_result_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_right_empty_result_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_right_empty_result_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_right_empty_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_right_empty_result_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_right_empty_result_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_right_empty_result_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_right_empty_result_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_right_empty_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_empty_result_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_right_empty_result_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_right_empty_result_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_right_empty_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_right_empty_result_productcoefficientscoefficientdiagonal)=pfc_left_right_empty_result_productcoefficientscoefficientdiagonalterm*pfc_right_right_empty_result_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_right_empty_result_productcoefficientscoefficientsum fs_v_pfc_right_empty_result_productcoefficientscoefficientsum. ((((exists fs_h_pfc_right_empty_result_productcoefficientscoefficientsum_body_start. fs_h_pfc_right_empty_result_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_right_empty_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_empty_result_productcoefficientscoefficientsum_body_start. fs_u_pfc_right_empty_result_productcoefficientscoefficientsum = fs_q_pfc_right_empty_result_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_right_empty_result_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_right_empty_result_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_right_empty_result_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_right_empty_result_productcoefficientscoefficient) = S ((S (S (pfc_index_right_empty_result_productcoefficients))) * fs_v_pfc_right_empty_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_empty_result_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_right_empty_result_productcoefficientscoefficientsum = fs_q_pfc_right_empty_result_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_right_empty_result_productcoefficients))) * fs_v_pfc_right_empty_result_productcoefficientscoefficientsum) + (pfc_natural_sum_right_empty_result_productcoefficientscoefficient))) /\\ forall fs_i_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps = S (pfc_index_right_empty_result_productcoefficients)) -> exists fs_a_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps fs_r_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps fs_s_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_empty_result_productcoefficientscoefficient)) /\\ exists fs_q_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_right_empty_result_productcoefficientscoefficient = fs_q_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_empty_result_productcoefficientscoefficient) + (fs_a_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_empty_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_right_empty_result_productcoefficientscoefficientsum = fs_q_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_empty_result_productcoefficientscoefficientsum) + (fs_r_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_empty_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_right_empty_result_productcoefficientscoefficientsum = fs_q_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_empty_result_productcoefficientscoefficientsum) + (fs_s_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps = fs_r_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps + fs_a_pfc_right_empty_result_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_right_empty_result_productcoefficientscoefficientresiduebound. pfa_gap_right_empty_result_productcoefficientscoefficientresiduebound + S (pfc_value_right_empty_result_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_right_empty_result_productcoefficientscoefficientresiduecongruence pfa_offset_right_right_empty_result_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_right_empty_result_productcoefficientscoefficient) + (p) * pfa_offset_left_right_empty_result_productcoefficientscoefficientresiduecongruence = (pfc_value_right_empty_result_productcoefficients) + (p) * pfa_offset_right_right_empty_result_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_right_empty_result_target pfrep_left_right_empty_result_target pfrep_right_right_empty_result_target. ((exists pfrep_position_right_empty_result_targetfirst. ((pfrep_position_right_empty_result_targetfirst+S (pfrep_power_right_empty_result_target)=(pfrd_plen_right_empty_result)) /\\ ((((exists ff_h_pfp_right_empty_result_targetfirstentry. ff_h_pfp_right_empty_result_targetfirstentry + S (pfrep_left_right_empty_result_target) = S ((S (pfrep_position_right_empty_result_targetfirst)) * pfrd_pc_right_empty_result)) /\\ exists ff_q_pfp_right_empty_result_targetfirstentry. pfrd_pb_right_empty_result = ff_q_pfp_right_empty_result_targetfirstentry * S ((S (pfrep_position_right_empty_result_targetfirst)) * pfrd_pc_right_empty_result) + (pfrep_left_right_empty_result_target)))))) \\/ (((exists pfrep_gap_right_empty_result_targetfirstoutside. pfrep_gap_right_empty_result_targetfirstoutside+(pfrd_plen_right_empty_result)=(pfrep_power_right_empty_result_target)) /\\ (((pfrep_left_right_empty_result_target)=0))))) -> ((exists pfrep_position_right_empty_result_targetsecond. ((pfrep_position_right_empty_result_targetsecond+S (pfrep_power_right_empty_result_target)=(0)) /\\ ((((exists ff_h_pfp_right_empty_result_targetsecondentry. ff_h_pfp_right_empty_result_targetsecondentry + S (pfrep_right_right_empty_result_target) = S ((S (pfrep_position_right_empty_result_targetsecond)) * ac)) /\\ exists ff_q_pfp_right_empty_result_targetsecondentry. ab = ff_q_pfp_right_empty_result_targetsecondentry * S ((S (pfrep_position_right_empty_result_targetsecond)) * ac) + (pfrep_right_right_empty_result_target)))))) \\/ (((exists pfrep_gap_right_empty_result_targetsecondoutside. pfrep_gap_right_empty_result_targetsecondoutside+(0)=(pfrep_power_right_empty_result_target)) /\\ (((pfrep_right_right_empty_result_target)=0))))) -> pfrep_left_right_empty_result_target=pfrep_right_right_empty_result_target)))))))", "statement_sha256": "3fdc5c341af454a1bf0018deaae0d0f6bdf04d524f98073f07d191911246c9de", "summary": "Every canonical divisor divides every encoding of the empty polynomial, using an actual empty quotient and actual empty product. 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hdivides_right_witness_witness_witness_witness", "cases hdivides_right_witness_witness_witness_witness_witness", "cases hdivides_right_witness_witness_witness_witness_witness_witness", "have hsource : ((forall fom_index_pfp_right_divisor_source_copyleft. (exists fom_gap_pfp_right_divisor_source_copyleft_index_bound. fom_gap_pfp_right_divisor_source_copyleft_index_bound + S (fom_index_pfp_right_divisor_source_copyleft) = x2) -> exists fom_value_pfp_right_divisor_source_copyleft. ((((exists fom_beta_height_pfp_right_divisor_source_copyleft_entry. fom_beta_height_pfp_right_divisor_source_copyleft_entry + S (fom_value_pfp_right_divisor_source_copyleft) = S ((S (fom_index_pfp_right_divisor_source_copyleft)) * x1)) /\\ exists fom_beta_quotient_pfp_right_divisor_source_copyleft_entry. x = fom_beta_quotient_pfp_right_divisor_source_copyleft_entry * S ((S (fom_index_pfp_right_divisor_source_copyleft)) * x1) + (fom_value_pfp_right_divisor_source_copyleft))) /\\ (exists fom_gap_pfp_right_divisor_source_copyleft_value_bound. fom_gap_pfp_right_divisor_source_copyleft_value_bound + S (fom_value_pfp_right_divisor_source_copyleft) = p))) /\\ (((forall fom_index_pfp_right_divisor_source_copyright. (exists fom_gap_pfp_right_divisor_source_copyright_index_bound. fom_gap_pfp_right_divisor_source_copyright_index_bound + S (fom_index_pfp_right_divisor_source_copyright) = D) -> exists fom_value_pfp_right_divisor_source_copyright. ((((exists fom_beta_height_pfp_right_divisor_source_copyright_entry. fom_beta_height_pfp_right_divisor_source_copyright_entry + S (fom_value_pfp_right_divisor_source_copyright) = S ((S (fom_index_pfp_right_divisor_source_copyright)) * dc)) /\\ exists fom_beta_quotient_pfp_right_divisor_source_copyright_entry. db = fom_beta_quotient_pfp_right_divisor_source_copyright_entry * S ((S (fom_index_pfp_right_divisor_source_copyright)) * dc) + (fom_value_pfp_right_divisor_source_copyright))) /\\ (exists fom_gap_pfp_right_divisor_source_copyright_value_bound. fom_gap_pfp_right_divisor_source_copyright_value_bound + S (fom_value_pfp_right_divisor_source_copyright) = p))) /\\ (((((((x2)=0 \\/ (D)=0) /\\ (((x5)=0)))) \\/ (((~((x2)=0)) /\\ (((~((D)=0)) /\\ (((x2)+(D)=S (x5)))))))) /\\ ((forall pfc_index_right_divisor_source_copycoefficients. (exists pfa_gap_right_divisor_source_copycoefficientsbound. pfa_gap_right_divisor_source_copycoefficientsbound + S (pfc_index_right_divisor_source_copycoefficients) = (x5)) -> exists pfc_value_right_divisor_source_copycoefficients. ((((exists ff_h_pfp_right_divisor_source_copycoefficientsentry. ff_h_pfp_right_divisor_source_copycoefficientsentry + S (pfc_value_right_divisor_source_copycoefficients) = S ((S (pfc_index_right_divisor_source_copycoefficients)) * x4)) /\\ exists ff_q_pfp_right_divisor_source_copycoefficientsentry. x3 = ff_q_pfp_right_divisor_source_copycoefficientsentry * S ((S (pfc_index_right_divisor_source_copycoefficients)) * x4) + (pfc_value_right_divisor_source_copycoefficients))) /\\ ((exists pfc_terms_code_right_divisor_source_copycoefficientscoefficient pfc_terms_scale_right_divisor_source_copycoefficientscoefficient pfc_natural_sum_right_divisor_source_copycoefficientscoefficient. ((forall pfc_index_right_divisor_source_copycoefficientscoefficientdiagonal. (exists pfa_gap_right_divisor_source_copycoefficientscoefficientdiagonalbound. pfa_gap_right_divisor_source_copycoefficientscoefficientdiagonalbound + S (pfc_index_right_divisor_source_copycoefficientscoefficientdiagonal) = (S (pfc_index_right_divisor_source_copycoefficients))) -> exists pfc_value_right_divisor_source_copycoefficientscoefficientdiagonal. ((((exists ff_h_pfp_right_divisor_source_copycoefficientscoefficientdiagonalentry. ff_h_pfp_right_divisor_source_copycoefficientscoefficientdiagonalentry + S (pfc_value_right_divisor_source_copycoefficientscoefficientdiagonal) = S ((S (pfc_index_right_divisor_source_copycoefficientscoefficientdiagonal)) * pfc_terms_scale_right_divisor_source_copycoefficientscoefficient)) /\\ exists ff_q_pfp_right_divisor_source_copycoefficientscoefficientdiagonalentry. pfc_terms_code_right_divisor_source_copycoefficientscoefficient = ff_q_pfp_right_divisor_source_copycoefficientscoefficientdiagonalentry * S ((S (pfc_index_right_divisor_source_copycoefficientscoefficientdiagonal)) * pfc_terms_scale_right_divisor_source_copycoefficientscoefficient) + (pfc_value_right_divisor_source_copycoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_right_divisor_source_copycoefficientscoefficientdiagonalterm pfc_left_right_divisor_source_copycoefficientscoefficientdiagonalterm pfc_right_right_divisor_source_copycoefficientscoefficientdiagonalterm. (((pfc_index_right_divisor_source_copycoefficientscoefficientdiagonal)+pfc_complement_right_divisor_source_copycoefficientscoefficientdiagonalterm=(pfc_index_right_divisor_source_copycoefficients)) /\\ ((((((exists pfa_gap_right_divisor_source_copycoefficientscoefficientdiagonaltermleftinside. pfa_gap_right_divisor_source_copycoefficientscoefficientdiagonaltermleftinside + S (pfc_index_right_divisor_source_copycoefficientscoefficientdiagonal) = (x2)) /\\ ((((exists ff_h_pfp_right_divisor_source_copycoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_right_divisor_source_copycoefficientscoefficientdiagonaltermleftentry + S (pfc_left_right_divisor_source_copycoefficientscoefficientdiagonalterm) = S ((S (pfc_index_right_divisor_source_copycoefficientscoefficientdiagonal)) * x1)) /\\ exists ff_q_pfp_right_divisor_source_copycoefficientscoefficientdiagonaltermleftentry. x = ff_q_pfp_right_divisor_source_copycoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_right_divisor_source_copycoefficientscoefficientdiagonal)) * x1) + (pfc_left_right_divisor_source_copycoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_divisor_source_copycoefficientscoefficientdiagonaltermleftoutside. pfc_gap_right_divisor_source_copycoefficientscoefficientdiagonaltermleftoutside+(x2)=(pfc_index_right_divisor_source_copycoefficientscoefficientdiagonal)) /\\ (((pfc_left_right_divisor_source_copycoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_right_divisor_source_copycoefficientscoefficientdiagonaltermrightinside. pfa_gap_right_divisor_source_copycoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_right_divisor_source_copycoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_right_divisor_source_copycoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_right_divisor_source_copycoefficientscoefficientdiagonaltermrightentry + S (pfc_right_right_divisor_source_copycoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_right_divisor_source_copycoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_right_divisor_source_copycoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_right_divisor_source_copycoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_right_divisor_source_copycoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_right_divisor_source_copycoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_divisor_source_copycoefficientscoefficientdiagonaltermrightoutside. pfc_gap_right_divisor_source_copycoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_right_divisor_source_copycoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_right_divisor_source_copycoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_right_divisor_source_copycoefficientscoefficientdiagonal)=pfc_left_right_divisor_source_copycoefficientscoefficientdiagonalterm*pfc_right_right_divisor_source_copycoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_right_divisor_source_copycoefficientscoefficientsum fs_v_pfc_right_divisor_source_copycoefficientscoefficientsum. ((((exists fs_h_pfc_right_divisor_source_copycoefficientscoefficientsum_body_start. fs_h_pfc_right_divisor_source_copycoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_right_divisor_source_copycoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_source_copycoefficientscoefficientsum_body_start. fs_u_pfc_right_divisor_source_copycoefficientscoefficientsum = fs_q_pfc_right_divisor_source_copycoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_right_divisor_source_copycoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_right_divisor_source_copycoefficientscoefficientsum_body_terminal. fs_h_pfc_right_divisor_source_copycoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_right_divisor_source_copycoefficientscoefficient) = S ((S (S (pfc_index_right_divisor_source_copycoefficients))) * fs_v_pfc_right_divisor_source_copycoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_source_copycoefficientscoefficientsum_body_terminal. fs_u_pfc_right_divisor_source_copycoefficientscoefficientsum = fs_q_pfc_right_divisor_source_copycoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_right_divisor_source_copycoefficients))) * fs_v_pfc_right_divisor_source_copycoefficientscoefficientsum) + (pfc_natural_sum_right_divisor_source_copycoefficientscoefficient))) /\\ forall fs_i_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps = S (pfc_index_right_divisor_source_copycoefficients)) -> exists fs_a_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps fs_r_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps fs_s_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps_summand. fs_h_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_divisor_source_copycoefficientscoefficient)) /\\ exists fs_q_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps_summand. pfc_terms_code_right_divisor_source_copycoefficientscoefficient = fs_q_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_divisor_source_copycoefficientscoefficient) + (fs_a_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps_partial. fs_h_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_source_copycoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps_partial. fs_u_pfc_right_divisor_source_copycoefficientscoefficientsum = fs_q_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_source_copycoefficientscoefficientsum) + (fs_r_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps_successor. fs_h_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_source_copycoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps_successor. fs_u_pfc_right_divisor_source_copycoefficientscoefficientsum = fs_q_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_source_copycoefficientscoefficientsum) + (fs_s_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps = fs_r_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps + fs_a_pfc_right_divisor_source_copycoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_right_divisor_source_copycoefficientscoefficientresiduebound. pfa_gap_right_divisor_source_copycoefficientscoefficientresiduebound + S (pfc_value_right_divisor_source_copycoefficients) = (p)) /\\ ((exists pfa_offset_left_right_divisor_source_copycoefficientscoefficientresiduecongruence pfa_offset_right_right_divisor_source_copycoefficientscoefficientresiduecongruence. (pfc_natural_sum_right_divisor_source_copycoefficientscoefficient) + (p) * pfa_offset_left_right_divisor_source_copycoefficientscoefficientresiduecongruence = (pfc_value_right_divisor_source_copycoefficients) + (p) * pfa_offset_right_right_divisor_source_copycoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hdivides_right_witness_witness_witness_witness_witness_witness_left", "cases hsource", "cases hsource_right", "cases hsource_right_right", "have hnew_length : exists n. (((((x2)=0 \\/ (E)=0) /\\ (((n)=0)))) \\/ (((~((x2)=0)) /\\ (((~((E)=0)) /\\ (((x2)+(E)=S (n))))))))", "specialize polynomial_product_length_exists (x2)", "specialize polynomial_product_length_exists (E)", "apply polynomial_product_length_exists", "cases hnew_length", "have hnew_product : exists b c. (((forall fom_index_pfp_hnew_graphleft. (exists fom_gap_pfp_hnew_graphleft_index_bound. fom_gap_pfp_hnew_graphleft_index_bound + S (fom_index_pfp_hnew_graphleft) = x2) -> exists fom_value_pfp_hnew_graphleft. ((((exists fom_beta_height_pfp_hnew_graphleft_entry. fom_beta_height_pfp_hnew_graphleft_entry + S (fom_value_pfp_hnew_graphleft) = S ((S (fom_index_pfp_hnew_graphleft)) * x1)) /\\ exists fom_beta_quotient_pfp_hnew_graphleft_entry. x = fom_beta_quotient_pfp_hnew_graphleft_entry * S ((S (fom_index_pfp_hnew_graphleft)) * x1) + (fom_value_pfp_hnew_graphleft))) /\\ (exists fom_gap_pfp_hnew_graphleft_value_bound. fom_gap_pfp_hnew_graphleft_value_bound + S (fom_value_pfp_hnew_graphleft) = p))) /\\ (((forall fom_index_pfp_hnew_graphright. (exists fom_gap_pfp_hnew_graphright_index_bound. fom_gap_pfp_hnew_graphright_index_bound + S (fom_index_pfp_hnew_graphright) = E) -> exists fom_value_pfp_hnew_graphright. ((((exists fom_beta_height_pfp_hnew_graphright_entry. fom_beta_height_pfp_hnew_graphright_entry + S (fom_value_pfp_hnew_graphright) = S ((S (fom_index_pfp_hnew_graphright)) * ec)) /\\ exists fom_beta_quotient_pfp_hnew_graphright_entry. eb = fom_beta_quotient_pfp_hnew_graphright_entry * S ((S (fom_index_pfp_hnew_graphright)) * ec) + (fom_value_pfp_hnew_graphright))) /\\ (exists fom_gap_pfp_hnew_graphright_value_bound. fom_gap_pfp_hnew_graphright_value_bound + S (fom_value_pfp_hnew_graphright) = p))) /\\ (((((((x2)=0 \\/ (E)=0) /\\ (((x6)=0)))) \\/ (((~((x2)=0)) /\\ (((~((E)=0)) /\\ (((x2)+(E)=S (x6)))))))) /\\ ((forall pfc_index_hnew_graphcoefficients. (exists pfa_gap_hnew_graphcoefficientsbound. pfa_gap_hnew_graphcoefficientsbound + S (pfc_index_hnew_graphcoefficients) = (x6)) -> exists pfc_value_hnew_graphcoefficients. ((((exists ff_h_pfp_hnew_graphcoefficientsentry. ff_h_pfp_hnew_graphcoefficientsentry + S (pfc_value_hnew_graphcoefficients) = S ((S (pfc_index_hnew_graphcoefficients)) * c)) /\\ exists ff_q_pfp_hnew_graphcoefficientsentry. b = ff_q_pfp_hnew_graphcoefficientsentry * S ((S (pfc_index_hnew_graphcoefficients)) * c) + (pfc_value_hnew_graphcoefficients))) /\\ ((exists pfc_terms_code_hnew_graphcoefficientscoefficient pfc_terms_scale_hnew_graphcoefficientscoefficient pfc_natural_sum_hnew_graphcoefficientscoefficient. ((forall pfc_index_hnew_graphcoefficientscoefficientdiagonal. (exists pfa_gap_hnew_graphcoefficientscoefficientdiagonalbound. pfa_gap_hnew_graphcoefficientscoefficientdiagonalbound + S (pfc_index_hnew_graphcoefficientscoefficientdiagonal) = (S (pfc_index_hnew_graphcoefficients))) -> exists pfc_value_hnew_graphcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_hnew_graphcoefficientscoefficientdiagonalentry. ff_h_pfp_hnew_graphcoefficientscoefficientdiagonalentry + S (pfc_value_hnew_graphcoefficientscoefficientdiagonal) = S ((S (pfc_index_hnew_graphcoefficientscoefficientdiagonal)) * pfc_terms_scale_hnew_graphcoefficientscoefficient)) /\\ exists ff_q_pfp_hnew_graphcoefficientscoefficientdiagonalentry. pfc_terms_code_hnew_graphcoefficientscoefficient = ff_q_pfp_hnew_graphcoefficientscoefficientdiagonalentry * S ((S (pfc_index_hnew_graphcoefficientscoefficientdiagonal)) * pfc_terms_scale_hnew_graphcoefficientscoefficient) + (pfc_value_hnew_graphcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_hnew_graphcoefficientscoefficientdiagonalterm pfc_left_hnew_graphcoefficientscoefficientdiagonalterm pfc_right_hnew_graphcoefficientscoefficientdiagonalterm. (((pfc_index_hnew_graphcoefficientscoefficientdiagonal)+pfc_complement_hnew_graphcoefficientscoefficientdiagonalterm=(pfc_index_hnew_graphcoefficients)) /\\ ((((((exists pfa_gap_hnew_graphcoefficientscoefficientdiagonaltermleftinside. pfa_gap_hnew_graphcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_hnew_graphcoefficientscoefficientdiagonal) = (x2)) /\\ ((((exists ff_h_pfp_hnew_graphcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_hnew_graphcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_hnew_graphcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_hnew_graphcoefficientscoefficientdiagonal)) * x1)) /\\ exists ff_q_pfp_hnew_graphcoefficientscoefficientdiagonaltermleftentry. x = ff_q_pfp_hnew_graphcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_hnew_graphcoefficientscoefficientdiagonal)) * x1) + (pfc_left_hnew_graphcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hnew_graphcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_hnew_graphcoefficientscoefficientdiagonaltermleftoutside+(x2)=(pfc_index_hnew_graphcoefficientscoefficientdiagonal)) /\\ (((pfc_left_hnew_graphcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_hnew_graphcoefficientscoefficientdiagonaltermrightinside. pfa_gap_hnew_graphcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_hnew_graphcoefficientscoefficientdiagonalterm) = (E)) /\\ ((((exists ff_h_pfp_hnew_graphcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_hnew_graphcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_hnew_graphcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_hnew_graphcoefficientscoefficientdiagonalterm)) * ec)) /\\ exists ff_q_pfp_hnew_graphcoefficientscoefficientdiagonaltermrightentry. eb = ff_q_pfp_hnew_graphcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_hnew_graphcoefficientscoefficientdiagonalterm)) * ec) + (pfc_right_hnew_graphcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hnew_graphcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_hnew_graphcoefficientscoefficientdiagonaltermrightoutside+(E)=(pfc_complement_hnew_graphcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_hnew_graphcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_hnew_graphcoefficientscoefficientdiagonal)=pfc_left_hnew_graphcoefficientscoefficientdiagonalterm*pfc_right_hnew_graphcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_hnew_graphcoefficientscoefficientsum fs_v_pfc_hnew_graphcoefficientscoefficientsum. ((((exists fs_h_pfc_hnew_graphcoefficientscoefficientsum_body_start. fs_h_pfc_hnew_graphcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_hnew_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnew_graphcoefficientscoefficientsum_body_start. fs_u_pfc_hnew_graphcoefficientscoefficientsum = fs_q_pfc_hnew_graphcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_hnew_graphcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_hnew_graphcoefficientscoefficientsum_body_terminal. fs_h_pfc_hnew_graphcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_hnew_graphcoefficientscoefficient) = S ((S (S (pfc_index_hnew_graphcoefficients))) * fs_v_pfc_hnew_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnew_graphcoefficientscoefficientsum_body_terminal. fs_u_pfc_hnew_graphcoefficientscoefficientsum = fs_q_pfc_hnew_graphcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_hnew_graphcoefficients))) * fs_v_pfc_hnew_graphcoefficientscoefficientsum) + (pfc_natural_sum_hnew_graphcoefficientscoefficient))) /\\ forall fs_i_pfc_hnew_graphcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_hnew_graphcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_hnew_graphcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_hnew_graphcoefficientscoefficientsum_body_steps = S (pfc_index_hnew_graphcoefficients)) -> exists fs_a_pfc_hnew_graphcoefficientscoefficientsum_body_steps fs_r_pfc_hnew_graphcoefficientscoefficientsum_body_steps fs_s_pfc_hnew_graphcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_hnew_graphcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_hnew_graphcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_hnew_graphcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hnew_graphcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hnew_graphcoefficientscoefficient)) /\\ exists fs_q_pfc_hnew_graphcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_hnew_graphcoefficientscoefficient = fs_q_pfc_hnew_graphcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_hnew_graphcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hnew_graphcoefficientscoefficient) + (fs_a_pfc_hnew_graphcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hnew_graphcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_hnew_graphcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_hnew_graphcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hnew_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnew_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnew_graphcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_hnew_graphcoefficientscoefficientsum = fs_q_pfc_hnew_graphcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_hnew_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnew_graphcoefficientscoefficientsum) + (fs_r_pfc_hnew_graphcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hnew_graphcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_hnew_graphcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_hnew_graphcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_hnew_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnew_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnew_graphcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_hnew_graphcoefficientscoefficientsum = fs_q_pfc_hnew_graphcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_hnew_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnew_graphcoefficientscoefficientsum) + (fs_s_pfc_hnew_graphcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_hnew_graphcoefficientscoefficientsum_body_steps = fs_r_pfc_hnew_graphcoefficientscoefficientsum_body_steps + fs_a_pfc_hnew_graphcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_hnew_graphcoefficientscoefficientresiduebound. pfa_gap_hnew_graphcoefficientscoefficientresiduebound + S (pfc_value_hnew_graphcoefficients) = (p)) /\\ ((exists pfa_offset_left_hnew_graphcoefficientscoefficientresiduecongruence pfa_offset_right_hnew_graphcoefficientscoefficientresiduecongruence. (pfc_natural_sum_hnew_graphcoefficientscoefficient) + (p) * pfa_offset_left_hnew_graphcoefficientscoefficientresiduecongruence = (pfc_value_hnew_graphcoefficients) + (p) * pfa_offset_right_hnew_graphcoefficientscoefficientresiduecongruence)))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (x)", "specialize prime_field_polynomial_convolution_at_length_exists (x1)", "specialize prime_field_polynomial_convolution_at_length_exists (x2)", "specialize prime_field_polynomial_convolution_at_length_exists (eb)", "specialize prime_field_polynomial_convolution_at_length_exists (ec)", "specialize prime_field_polynomial_convolution_at_length_exists (E)", "specialize prime_field_polynomial_convolution_at_length_exists (x6)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hsource_left", "exact hE", "exact hnew_length_witness", "cases hnew_product", "cases hnew_product_witness", "specialize prime_field_polynomial_right_divides_from_product (p)", "specialize prime_field_polynomial_right_divides_from_product (eb)", "specialize prime_field_polynomial_right_divides_from_product (ec)", "specialize prime_field_polynomial_right_divides_from_product (E)", "specialize prime_field_polynomial_right_divides_from_product (ab)", "specialize prime_field_polynomial_right_divides_from_product (ac)", "specialize prime_field_polynomial_right_divides_from_product (L)", "specialize prime_field_polynomial_right_divides_from_product (x)", "specialize prime_field_polynomial_right_divides_from_product (x1)", "specialize prime_field_polynomial_right_divides_from_product (x2)", "specialize prime_field_polynomial_right_divides_from_product (x7)", "specialize prime_field_polynomial_right_divides_from_product (x8)", "specialize prime_field_polynomial_right_divides_from_product (x6)", "apply prime_field_polynomial_right_divides_from_product", "exact hdivides_left", "exact hnew_product_witness_witness", "specialize prime_field_polynomial_equivalent_transitive (x7)", "specialize prime_field_polynomial_equivalent_transitive (x8)", "specialize prime_field_polynomial_equivalent_transitive (x6)", "specialize prime_field_polynomial_equivalent_transitive (x3)", "specialize prime_field_polynomial_equivalent_transitive (x4)", "specialize prime_field_polynomial_equivalent_transitive (x5)", "specialize prime_field_polynomial_equivalent_transitive (ab)", "specialize prime_field_polynomial_equivalent_transitive (ac)", "specialize prime_field_polynomial_equivalent_transitive (L)", "apply prime_field_polynomial_equivalent_transitive", "specialize prime_field_polynomial_equivalent_symmetric (x3)", "specialize prime_field_polynomial_equivalent_symmetric (x4)", "specialize prime_field_polynomial_equivalent_symmetric (x5)", "specialize prime_field_polynomial_equivalent_symmetric (x7)", "specialize prime_field_polynomial_equivalent_symmetric (x8)", "specialize prime_field_polynomial_equivalent_symmetric (x6)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (p)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x1)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x2)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (db)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (dc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (D)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x3)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x4)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x5)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (eb)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (ec)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (E)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x7)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x8)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x6)", "apply prime_field_polynomial_convolution_equivalent_congruent_right", "exact hp0", "exact hequivalent", "exact hdivides_right_witness_witness_witness_witness_witness_witness_left", "exact hnew_product_witness_witness", "exact hdivides_right_witness_witness_witness_witness_witness_witness_right" ], "script_sha256": "cf07c83682fed1045f79cb4ae1b54a04093efa1696e65a88bacc88ad22ce79c8", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_divisibility_candidate.py", "sha256": "f544adedd3ce963e4a773e8582efcb0f91ba7491207c9792d477d452e854f2b8" }, "statement": "forall p db dc D ab ac L eb ec E. (~(p=0)) -> (forall fom_index_pfp_right_divisor_replacement_bound. (exists fom_gap_pfp_right_divisor_replacement_bound_index_bound. fom_gap_pfp_right_divisor_replacement_bound_index_bound + S (fom_index_pfp_right_divisor_replacement_bound) = E) -> exists fom_value_pfp_right_divisor_replacement_bound. ((((exists fom_beta_height_pfp_right_divisor_replacement_bound_entry. fom_beta_height_pfp_right_divisor_replacement_bound_entry + S (fom_value_pfp_right_divisor_replacement_bound) = S ((S (fom_index_pfp_right_divisor_replacement_bound)) * ec)) /\\ exists fom_beta_quotient_pfp_right_divisor_replacement_bound_entry. eb = fom_beta_quotient_pfp_right_divisor_replacement_bound_entry * S ((S (fom_index_pfp_right_divisor_replacement_bound)) * ec) + (fom_value_pfp_right_divisor_replacement_bound))) /\\ (exists fom_gap_pfp_right_divisor_replacement_bound_value_bound. fom_gap_pfp_right_divisor_replacement_bound_value_bound + S (fom_value_pfp_right_divisor_replacement_bound) = p))) -> (forall pfrep_power_right_divisor_equivalent pfrep_left_right_divisor_equivalent pfrep_right_right_divisor_equivalent. ((exists pfrep_position_right_divisor_equivalentfirst. ((pfrep_position_right_divisor_equivalentfirst+S (pfrep_power_right_divisor_equivalent)=(D)) /\\ ((((exists ff_h_pfp_right_divisor_equivalentfirstentry. ff_h_pfp_right_divisor_equivalentfirstentry + S (pfrep_left_right_divisor_equivalent) = S ((S (pfrep_position_right_divisor_equivalentfirst)) * dc)) /\\ exists ff_q_pfp_right_divisor_equivalentfirstentry. db = ff_q_pfp_right_divisor_equivalentfirstentry * S ((S (pfrep_position_right_divisor_equivalentfirst)) * dc) + (pfrep_left_right_divisor_equivalent)))))) \\/ (((exists pfrep_gap_right_divisor_equivalentfirstoutside. pfrep_gap_right_divisor_equivalentfirstoutside+(D)=(pfrep_power_right_divisor_equivalent)) /\\ (((pfrep_left_right_divisor_equivalent)=0))))) -> ((exists pfrep_position_right_divisor_equivalentsecond. ((pfrep_position_right_divisor_equivalentsecond+S (pfrep_power_right_divisor_equivalent)=(E)) /\\ ((((exists ff_h_pfp_right_divisor_equivalentsecondentry. ff_h_pfp_right_divisor_equivalentsecondentry + S (pfrep_right_right_divisor_equivalent) = S ((S (pfrep_position_right_divisor_equivalentsecond)) * ec)) /\\ exists ff_q_pfp_right_divisor_equivalentsecondentry. eb = ff_q_pfp_right_divisor_equivalentsecondentry * S ((S (pfrep_position_right_divisor_equivalentsecond)) * ec) + (pfrep_right_right_divisor_equivalent)))))) \\/ (((exists pfrep_gap_right_divisor_equivalentsecondoutside. pfrep_gap_right_divisor_equivalentsecondoutside+(E)=(pfrep_power_right_divisor_equivalent)) /\\ (((pfrep_right_right_divisor_equivalent)=0))))) -> pfrep_left_right_divisor_equivalent=pfrep_right_right_divisor_equivalent) -> (((forall fom_index_pfp_right_divisor_source_canonical. (exists fom_gap_pfp_right_divisor_source_canonical_index_bound. fom_gap_pfp_right_divisor_source_canonical_index_bound + S (fom_index_pfp_right_divisor_source_canonical) = L) -> exists fom_value_pfp_right_divisor_source_canonical. ((((exists fom_beta_height_pfp_right_divisor_source_canonical_entry. fom_beta_height_pfp_right_divisor_source_canonical_entry + S (fom_value_pfp_right_divisor_source_canonical) = S ((S (fom_index_pfp_right_divisor_source_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_right_divisor_source_canonical_entry. ab = fom_beta_quotient_pfp_right_divisor_source_canonical_entry * S ((S (fom_index_pfp_right_divisor_source_canonical)) * ac) + (fom_value_pfp_right_divisor_source_canonical))) /\\ (exists fom_gap_pfp_right_divisor_source_canonical_value_bound. fom_gap_pfp_right_divisor_source_canonical_value_bound + S (fom_value_pfp_right_divisor_source_canonical) = p))) /\\ ((exists pfrd_qb_right_divisor_source pfrd_qc_right_divisor_source pfrd_qlen_right_divisor_source pfrd_pb_right_divisor_source pfrd_pc_right_divisor_source pfrd_plen_right_divisor_source. ((((forall fom_index_pfp_right_divisor_source_productleft. (exists fom_gap_pfp_right_divisor_source_productleft_index_bound. fom_gap_pfp_right_divisor_source_productleft_index_bound + S (fom_index_pfp_right_divisor_source_productleft) = pfrd_qlen_right_divisor_source) -> exists fom_value_pfp_right_divisor_source_productleft. ((((exists fom_beta_height_pfp_right_divisor_source_productleft_entry. fom_beta_height_pfp_right_divisor_source_productleft_entry + S (fom_value_pfp_right_divisor_source_productleft) = S ((S (fom_index_pfp_right_divisor_source_productleft)) * pfrd_qc_right_divisor_source)) /\\ exists fom_beta_quotient_pfp_right_divisor_source_productleft_entry. pfrd_qb_right_divisor_source = fom_beta_quotient_pfp_right_divisor_source_productleft_entry * S ((S (fom_index_pfp_right_divisor_source_productleft)) * pfrd_qc_right_divisor_source) + (fom_value_pfp_right_divisor_source_productleft))) /\\ (exists fom_gap_pfp_right_divisor_source_productleft_value_bound. fom_gap_pfp_right_divisor_source_productleft_value_bound + S (fom_value_pfp_right_divisor_source_productleft) = p))) /\\ (((forall fom_index_pfp_right_divisor_source_productright. (exists fom_gap_pfp_right_divisor_source_productright_index_bound. fom_gap_pfp_right_divisor_source_productright_index_bound + S (fom_index_pfp_right_divisor_source_productright) = D) -> exists fom_value_pfp_right_divisor_source_productright. ((((exists fom_beta_height_pfp_right_divisor_source_productright_entry. fom_beta_height_pfp_right_divisor_source_productright_entry + S (fom_value_pfp_right_divisor_source_productright) = S ((S (fom_index_pfp_right_divisor_source_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_right_divisor_source_productright_entry. db = fom_beta_quotient_pfp_right_divisor_source_productright_entry * S ((S (fom_index_pfp_right_divisor_source_productright)) * dc) + (fom_value_pfp_right_divisor_source_productright))) /\\ (exists fom_gap_pfp_right_divisor_source_productright_value_bound. fom_gap_pfp_right_divisor_source_productright_value_bound + S (fom_value_pfp_right_divisor_source_productright) = p))) /\\ (((((((pfrd_qlen_right_divisor_source)=0 \\/ (D)=0) /\\ (((pfrd_plen_right_divisor_source)=0)))) \\/ (((~((pfrd_qlen_right_divisor_source)=0)) /\\ (((~((D)=0)) /\\ (((pfrd_qlen_right_divisor_source)+(D)=S (pfrd_plen_right_divisor_source)))))))) /\\ ((forall pfc_index_right_divisor_source_productcoefficients. (exists pfa_gap_right_divisor_source_productcoefficientsbound. pfa_gap_right_divisor_source_productcoefficientsbound + S (pfc_index_right_divisor_source_productcoefficients) = (pfrd_plen_right_divisor_source)) -> exists pfc_value_right_divisor_source_productcoefficients. ((((exists ff_h_pfp_right_divisor_source_productcoefficientsentry. ff_h_pfp_right_divisor_source_productcoefficientsentry + S (pfc_value_right_divisor_source_productcoefficients) = S ((S (pfc_index_right_divisor_source_productcoefficients)) * pfrd_pc_right_divisor_source)) /\\ exists ff_q_pfp_right_divisor_source_productcoefficientsentry. pfrd_pb_right_divisor_source = ff_q_pfp_right_divisor_source_productcoefficientsentry * S ((S (pfc_index_right_divisor_source_productcoefficients)) * pfrd_pc_right_divisor_source) + (pfc_value_right_divisor_source_productcoefficients))) /\\ ((exists pfc_terms_code_right_divisor_source_productcoefficientscoefficient pfc_terms_scale_right_divisor_source_productcoefficientscoefficient pfc_natural_sum_right_divisor_source_productcoefficientscoefficient. ((forall pfc_index_right_divisor_source_productcoefficientscoefficientdiagonal. (exists pfa_gap_right_divisor_source_productcoefficientscoefficientdiagonalbound. pfa_gap_right_divisor_source_productcoefficientscoefficientdiagonalbound + S (pfc_index_right_divisor_source_productcoefficientscoefficientdiagonal) = (S (pfc_index_right_divisor_source_productcoefficients))) -> exists pfc_value_right_divisor_source_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_right_divisor_source_productcoefficientscoefficientdiagonalentry. ff_h_pfp_right_divisor_source_productcoefficientscoefficientdiagonalentry + S (pfc_value_right_divisor_source_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_right_divisor_source_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_divisor_source_productcoefficientscoefficient)) /\\ exists ff_q_pfp_right_divisor_source_productcoefficientscoefficientdiagonalentry. pfc_terms_code_right_divisor_source_productcoefficientscoefficient = ff_q_pfp_right_divisor_source_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_right_divisor_source_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_divisor_source_productcoefficientscoefficient) + (pfc_value_right_divisor_source_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_right_divisor_source_productcoefficientscoefficientdiagonalterm pfc_left_right_divisor_source_productcoefficientscoefficientdiagonalterm pfc_right_right_divisor_source_productcoefficientscoefficientdiagonalterm. (((pfc_index_right_divisor_source_productcoefficientscoefficientdiagonal)+pfc_complement_right_divisor_source_productcoefficientscoefficientdiagonalterm=(pfc_index_right_divisor_source_productcoefficients)) /\\ ((((((exists pfa_gap_right_divisor_source_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_right_divisor_source_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_right_divisor_source_productcoefficientscoefficientdiagonal) = (pfrd_qlen_right_divisor_source)) /\\ ((((exists ff_h_pfp_right_divisor_source_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_right_divisor_source_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_right_divisor_source_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_right_divisor_source_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_divisor_source)) /\\ exists ff_q_pfp_right_divisor_source_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_right_divisor_source = ff_q_pfp_right_divisor_source_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_right_divisor_source_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_divisor_source) + (pfc_left_right_divisor_source_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_divisor_source_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_right_divisor_source_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_right_divisor_source)=(pfc_index_right_divisor_source_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_right_divisor_source_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_right_divisor_source_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_right_divisor_source_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_right_divisor_source_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_right_divisor_source_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_right_divisor_source_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_right_divisor_source_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_right_divisor_source_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_right_divisor_source_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_right_divisor_source_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_right_divisor_source_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_right_divisor_source_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_divisor_source_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_right_divisor_source_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_right_divisor_source_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_right_divisor_source_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_right_divisor_source_productcoefficientscoefficientdiagonal)=pfc_left_right_divisor_source_productcoefficientscoefficientdiagonalterm*pfc_right_right_divisor_source_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_right_divisor_source_productcoefficientscoefficientsum fs_v_pfc_right_divisor_source_productcoefficientscoefficientsum. ((((exists fs_h_pfc_right_divisor_source_productcoefficientscoefficientsum_body_start. fs_h_pfc_right_divisor_source_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_right_divisor_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_source_productcoefficientscoefficientsum_body_start. fs_u_pfc_right_divisor_source_productcoefficientscoefficientsum = fs_q_pfc_right_divisor_source_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_right_divisor_source_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_right_divisor_source_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_right_divisor_source_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_right_divisor_source_productcoefficientscoefficient) = S ((S (S (pfc_index_right_divisor_source_productcoefficients))) * fs_v_pfc_right_divisor_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_source_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_right_divisor_source_productcoefficientscoefficientsum = fs_q_pfc_right_divisor_source_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_right_divisor_source_productcoefficients))) * fs_v_pfc_right_divisor_source_productcoefficientscoefficientsum) + (pfc_natural_sum_right_divisor_source_productcoefficientscoefficient))) /\\ forall fs_i_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps = S (pfc_index_right_divisor_source_productcoefficients)) -> exists fs_a_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps fs_r_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps fs_s_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_divisor_source_productcoefficientscoefficient)) /\\ exists fs_q_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_right_divisor_source_productcoefficientscoefficient = fs_q_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_divisor_source_productcoefficientscoefficient) + (fs_a_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_right_divisor_source_productcoefficientscoefficientsum = fs_q_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_source_productcoefficientscoefficientsum) + (fs_r_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_source_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_right_divisor_source_productcoefficientscoefficientsum = fs_q_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_source_productcoefficientscoefficientsum) + (fs_s_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps = fs_r_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps + fs_a_pfc_right_divisor_source_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_right_divisor_source_productcoefficientscoefficientresiduebound. pfa_gap_right_divisor_source_productcoefficientscoefficientresiduebound + S (pfc_value_right_divisor_source_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_right_divisor_source_productcoefficientscoefficientresiduecongruence pfa_offset_right_right_divisor_source_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_right_divisor_source_productcoefficientscoefficient) + (p) * pfa_offset_left_right_divisor_source_productcoefficientscoefficientresiduecongruence = (pfc_value_right_divisor_source_productcoefficients) + (p) * pfa_offset_right_right_divisor_source_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_right_divisor_source_target pfrep_left_right_divisor_source_target pfrep_right_right_divisor_source_target. ((exists pfrep_position_right_divisor_source_targetfirst. ((pfrep_position_right_divisor_source_targetfirst+S (pfrep_power_right_divisor_source_target)=(pfrd_plen_right_divisor_source)) /\\ ((((exists ff_h_pfp_right_divisor_source_targetfirstentry. ff_h_pfp_right_divisor_source_targetfirstentry + S (pfrep_left_right_divisor_source_target) = S ((S (pfrep_position_right_divisor_source_targetfirst)) * pfrd_pc_right_divisor_source)) /\\ exists ff_q_pfp_right_divisor_source_targetfirstentry. pfrd_pb_right_divisor_source = ff_q_pfp_right_divisor_source_targetfirstentry * S ((S (pfrep_position_right_divisor_source_targetfirst)) * pfrd_pc_right_divisor_source) + (pfrep_left_right_divisor_source_target)))))) \\/ (((exists pfrep_gap_right_divisor_source_targetfirstoutside. pfrep_gap_right_divisor_source_targetfirstoutside+(pfrd_plen_right_divisor_source)=(pfrep_power_right_divisor_source_target)) /\\ (((pfrep_left_right_divisor_source_target)=0))))) -> ((exists pfrep_position_right_divisor_source_targetsecond. ((pfrep_position_right_divisor_source_targetsecond+S (pfrep_power_right_divisor_source_target)=(L)) /\\ ((((exists ff_h_pfp_right_divisor_source_targetsecondentry. ff_h_pfp_right_divisor_source_targetsecondentry + S (pfrep_right_right_divisor_source_target) = S ((S (pfrep_position_right_divisor_source_targetsecond)) * ac)) /\\ exists ff_q_pfp_right_divisor_source_targetsecondentry. ab = ff_q_pfp_right_divisor_source_targetsecondentry * S ((S (pfrep_position_right_divisor_source_targetsecond)) * ac) + (pfrep_right_right_divisor_source_target)))))) \\/ (((exists pfrep_gap_right_divisor_source_targetsecondoutside. pfrep_gap_right_divisor_source_targetsecondoutside+(L)=(pfrep_power_right_divisor_source_target)) /\\ (((pfrep_right_right_divisor_source_target)=0))))) -> pfrep_left_right_divisor_source_target=pfrep_right_right_divisor_source_target))))))) -> (((forall fom_index_pfp_right_divisor_result_canonical. (exists fom_gap_pfp_right_divisor_result_canonical_index_bound. fom_gap_pfp_right_divisor_result_canonical_index_bound + S (fom_index_pfp_right_divisor_result_canonical) = L) -> exists fom_value_pfp_right_divisor_result_canonical. ((((exists fom_beta_height_pfp_right_divisor_result_canonical_entry. fom_beta_height_pfp_right_divisor_result_canonical_entry + S (fom_value_pfp_right_divisor_result_canonical) = S ((S (fom_index_pfp_right_divisor_result_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_right_divisor_result_canonical_entry. ab = fom_beta_quotient_pfp_right_divisor_result_canonical_entry * S ((S (fom_index_pfp_right_divisor_result_canonical)) * ac) + (fom_value_pfp_right_divisor_result_canonical))) /\\ (exists fom_gap_pfp_right_divisor_result_canonical_value_bound. fom_gap_pfp_right_divisor_result_canonical_value_bound + S (fom_value_pfp_right_divisor_result_canonical) = p))) /\\ ((exists pfrd_qb_right_divisor_result pfrd_qc_right_divisor_result pfrd_qlen_right_divisor_result pfrd_pb_right_divisor_result pfrd_pc_right_divisor_result pfrd_plen_right_divisor_result. ((((forall fom_index_pfp_right_divisor_result_productleft. (exists fom_gap_pfp_right_divisor_result_productleft_index_bound. fom_gap_pfp_right_divisor_result_productleft_index_bound + S (fom_index_pfp_right_divisor_result_productleft) = pfrd_qlen_right_divisor_result) -> exists fom_value_pfp_right_divisor_result_productleft. ((((exists fom_beta_height_pfp_right_divisor_result_productleft_entry. fom_beta_height_pfp_right_divisor_result_productleft_entry + S (fom_value_pfp_right_divisor_result_productleft) = S ((S (fom_index_pfp_right_divisor_result_productleft)) * pfrd_qc_right_divisor_result)) /\\ exists fom_beta_quotient_pfp_right_divisor_result_productleft_entry. pfrd_qb_right_divisor_result = fom_beta_quotient_pfp_right_divisor_result_productleft_entry * S ((S (fom_index_pfp_right_divisor_result_productleft)) * pfrd_qc_right_divisor_result) + (fom_value_pfp_right_divisor_result_productleft))) /\\ (exists fom_gap_pfp_right_divisor_result_productleft_value_bound. fom_gap_pfp_right_divisor_result_productleft_value_bound + S (fom_value_pfp_right_divisor_result_productleft) = p))) /\\ (((forall fom_index_pfp_right_divisor_result_productright. (exists fom_gap_pfp_right_divisor_result_productright_index_bound. fom_gap_pfp_right_divisor_result_productright_index_bound + S (fom_index_pfp_right_divisor_result_productright) = E) -> exists fom_value_pfp_right_divisor_result_productright. ((((exists fom_beta_height_pfp_right_divisor_result_productright_entry. fom_beta_height_pfp_right_divisor_result_productright_entry + S (fom_value_pfp_right_divisor_result_productright) = S ((S (fom_index_pfp_right_divisor_result_productright)) * ec)) /\\ exists fom_beta_quotient_pfp_right_divisor_result_productright_entry. eb = fom_beta_quotient_pfp_right_divisor_result_productright_entry * S ((S (fom_index_pfp_right_divisor_result_productright)) * ec) + (fom_value_pfp_right_divisor_result_productright))) /\\ (exists fom_gap_pfp_right_divisor_result_productright_value_bound. fom_gap_pfp_right_divisor_result_productright_value_bound + S (fom_value_pfp_right_divisor_result_productright) = p))) /\\ (((((((pfrd_qlen_right_divisor_result)=0 \\/ (E)=0) /\\ (((pfrd_plen_right_divisor_result)=0)))) \\/ (((~((pfrd_qlen_right_divisor_result)=0)) /\\ (((~((E)=0)) /\\ (((pfrd_qlen_right_divisor_result)+(E)=S (pfrd_plen_right_divisor_result)))))))) /\\ ((forall pfc_index_right_divisor_result_productcoefficients. (exists pfa_gap_right_divisor_result_productcoefficientsbound. pfa_gap_right_divisor_result_productcoefficientsbound + S (pfc_index_right_divisor_result_productcoefficients) = (pfrd_plen_right_divisor_result)) -> exists pfc_value_right_divisor_result_productcoefficients. ((((exists ff_h_pfp_right_divisor_result_productcoefficientsentry. ff_h_pfp_right_divisor_result_productcoefficientsentry + S (pfc_value_right_divisor_result_productcoefficients) = S ((S (pfc_index_right_divisor_result_productcoefficients)) * pfrd_pc_right_divisor_result)) /\\ exists ff_q_pfp_right_divisor_result_productcoefficientsentry. pfrd_pb_right_divisor_result = ff_q_pfp_right_divisor_result_productcoefficientsentry * S ((S (pfc_index_right_divisor_result_productcoefficients)) * pfrd_pc_right_divisor_result) + (pfc_value_right_divisor_result_productcoefficients))) /\\ ((exists pfc_terms_code_right_divisor_result_productcoefficientscoefficient pfc_terms_scale_right_divisor_result_productcoefficientscoefficient pfc_natural_sum_right_divisor_result_productcoefficientscoefficient. ((forall pfc_index_right_divisor_result_productcoefficientscoefficientdiagonal. (exists pfa_gap_right_divisor_result_productcoefficientscoefficientdiagonalbound. pfa_gap_right_divisor_result_productcoefficientscoefficientdiagonalbound + S (pfc_index_right_divisor_result_productcoefficientscoefficientdiagonal) = (S (pfc_index_right_divisor_result_productcoefficients))) -> exists pfc_value_right_divisor_result_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_right_divisor_result_productcoefficientscoefficientdiagonalentry. ff_h_pfp_right_divisor_result_productcoefficientscoefficientdiagonalentry + S (pfc_value_right_divisor_result_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_right_divisor_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_divisor_result_productcoefficientscoefficient)) /\\ exists ff_q_pfp_right_divisor_result_productcoefficientscoefficientdiagonalentry. pfc_terms_code_right_divisor_result_productcoefficientscoefficient = ff_q_pfp_right_divisor_result_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_right_divisor_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_divisor_result_productcoefficientscoefficient) + (pfc_value_right_divisor_result_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_right_divisor_result_productcoefficientscoefficientdiagonalterm pfc_left_right_divisor_result_productcoefficientscoefficientdiagonalterm pfc_right_right_divisor_result_productcoefficientscoefficientdiagonalterm. (((pfc_index_right_divisor_result_productcoefficientscoefficientdiagonal)+pfc_complement_right_divisor_result_productcoefficientscoefficientdiagonalterm=(pfc_index_right_divisor_result_productcoefficients)) /\\ ((((((exists pfa_gap_right_divisor_result_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_right_divisor_result_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_right_divisor_result_productcoefficientscoefficientdiagonal) = (pfrd_qlen_right_divisor_result)) /\\ ((((exists ff_h_pfp_right_divisor_result_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_right_divisor_result_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_right_divisor_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_right_divisor_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_divisor_result)) /\\ exists ff_q_pfp_right_divisor_result_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_right_divisor_result = ff_q_pfp_right_divisor_result_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_right_divisor_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_divisor_result) + (pfc_left_right_divisor_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_divisor_result_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_right_divisor_result_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_right_divisor_result)=(pfc_index_right_divisor_result_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_right_divisor_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_right_divisor_result_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_right_divisor_result_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_right_divisor_result_productcoefficientscoefficientdiagonalterm) = (E)) /\\ ((((exists ff_h_pfp_right_divisor_result_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_right_divisor_result_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_right_divisor_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_right_divisor_result_productcoefficientscoefficientdiagonalterm)) * ec)) /\\ exists ff_q_pfp_right_divisor_result_productcoefficientscoefficientdiagonaltermrightentry. eb = ff_q_pfp_right_divisor_result_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_right_divisor_result_productcoefficientscoefficientdiagonalterm)) * ec) + (pfc_right_right_divisor_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_divisor_result_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_right_divisor_result_productcoefficientscoefficientdiagonaltermrightoutside+(E)=(pfc_complement_right_divisor_result_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_right_divisor_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_right_divisor_result_productcoefficientscoefficientdiagonal)=pfc_left_right_divisor_result_productcoefficientscoefficientdiagonalterm*pfc_right_right_divisor_result_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_right_divisor_result_productcoefficientscoefficientsum fs_v_pfc_right_divisor_result_productcoefficientscoefficientsum. ((((exists fs_h_pfc_right_divisor_result_productcoefficientscoefficientsum_body_start. fs_h_pfc_right_divisor_result_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_right_divisor_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_result_productcoefficientscoefficientsum_body_start. fs_u_pfc_right_divisor_result_productcoefficientscoefficientsum = fs_q_pfc_right_divisor_result_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_right_divisor_result_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_right_divisor_result_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_right_divisor_result_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_right_divisor_result_productcoefficientscoefficient) = S ((S (S (pfc_index_right_divisor_result_productcoefficients))) * fs_v_pfc_right_divisor_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_result_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_right_divisor_result_productcoefficientscoefficientsum = fs_q_pfc_right_divisor_result_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_right_divisor_result_productcoefficients))) * fs_v_pfc_right_divisor_result_productcoefficientscoefficientsum) + (pfc_natural_sum_right_divisor_result_productcoefficientscoefficient))) /\\ forall fs_i_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps = S (pfc_index_right_divisor_result_productcoefficients)) -> exists fs_a_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps fs_r_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps fs_s_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_divisor_result_productcoefficientscoefficient)) /\\ exists fs_q_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_right_divisor_result_productcoefficientscoefficient = fs_q_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_divisor_result_productcoefficientscoefficient) + (fs_a_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_right_divisor_result_productcoefficientscoefficientsum = fs_q_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_result_productcoefficientscoefficientsum) + (fs_r_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_right_divisor_result_productcoefficientscoefficientsum = fs_q_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_divisor_result_productcoefficientscoefficientsum) + (fs_s_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps = fs_r_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps + fs_a_pfc_right_divisor_result_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_right_divisor_result_productcoefficientscoefficientresiduebound. pfa_gap_right_divisor_result_productcoefficientscoefficientresiduebound + S (pfc_value_right_divisor_result_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_right_divisor_result_productcoefficientscoefficientresiduecongruence pfa_offset_right_right_divisor_result_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_right_divisor_result_productcoefficientscoefficient) + (p) * pfa_offset_left_right_divisor_result_productcoefficientscoefficientresiduecongruence = (pfc_value_right_divisor_result_productcoefficients) + (p) * pfa_offset_right_right_divisor_result_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_right_divisor_result_target pfrep_left_right_divisor_result_target pfrep_right_right_divisor_result_target. ((exists pfrep_position_right_divisor_result_targetfirst. ((pfrep_position_right_divisor_result_targetfirst+S (pfrep_power_right_divisor_result_target)=(pfrd_plen_right_divisor_result)) /\\ ((((exists ff_h_pfp_right_divisor_result_targetfirstentry. ff_h_pfp_right_divisor_result_targetfirstentry + S (pfrep_left_right_divisor_result_target) = S ((S (pfrep_position_right_divisor_result_targetfirst)) * pfrd_pc_right_divisor_result)) /\\ exists ff_q_pfp_right_divisor_result_targetfirstentry. pfrd_pb_right_divisor_result = ff_q_pfp_right_divisor_result_targetfirstentry * S ((S (pfrep_position_right_divisor_result_targetfirst)) * pfrd_pc_right_divisor_result) + (pfrep_left_right_divisor_result_target)))))) \\/ (((exists pfrep_gap_right_divisor_result_targetfirstoutside. pfrep_gap_right_divisor_result_targetfirstoutside+(pfrd_plen_right_divisor_result)=(pfrep_power_right_divisor_result_target)) /\\ (((pfrep_left_right_divisor_result_target)=0))))) -> ((exists pfrep_position_right_divisor_result_targetsecond. ((pfrep_position_right_divisor_result_targetsecond+S (pfrep_power_right_divisor_result_target)=(L)) /\\ ((((exists ff_h_pfp_right_divisor_result_targetsecondentry. ff_h_pfp_right_divisor_result_targetsecondentry + S (pfrep_right_right_divisor_result_target) = S ((S (pfrep_position_right_divisor_result_targetsecond)) * ac)) /\\ exists ff_q_pfp_right_divisor_result_targetsecondentry. ab = ff_q_pfp_right_divisor_result_targetsecondentry * S ((S (pfrep_position_right_divisor_result_targetsecond)) * ac) + (pfrep_right_right_divisor_result_target)))))) \\/ (((exists pfrep_gap_right_divisor_result_targetsecondoutside. pfrep_gap_right_divisor_result_targetsecondoutside+(L)=(pfrep_power_right_divisor_result_target)) /\\ (((pfrep_right_right_divisor_result_target)=0))))) -> pfrep_left_right_divisor_result_target=pfrep_right_right_divisor_result_target)))))))", "statement_sha256": "608a6407b35414fe4335c37afae0bbd3de15caa82ee1d3c67f25ef9cf9b76ccb", "summary": "An equivalent canonical right divisor preserves divisibility by constructing the quotient times the replacement at its own actual proper length. 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"research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "821a43b9741cdd9740ca823ab72d578896e292349d039cce04211a171416d3ac", "membership": "alpha_only", "name": "prime_field_polynomial_right_divides_transitive", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro db", "intro dc", "intro D", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro M", "intro hp", "intro hDA", "intro hAB", "have hp0 : ~(p=0)", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hz", "cases hDA", "cases hDA_right", "cases hDA_right_witness", "cases hDA_right_witness_witness", "cases hDA_right_witness_witness_witness", "cases hDA_right_witness_witness_witness_witness", "cases hDA_right_witness_witness_witness_witness_witness", "cases hDA_right_witness_witness_witness_witness_witness_witness", "cases hAB", "cases hAB_right", "cases hAB_right_witness", "cases hAB_right_witness_witness", "cases hAB_right_witness_witness_witness", "cases hAB_right_witness_witness_witness_witness", "cases hAB_right_witness_witness_witness_witness_witness", "cases hAB_right_witness_witness_witness_witness_witness_witness", "have hfirst : ((forall fom_index_pfp_right_transitive_hfirstleft. (exists fom_gap_pfp_right_transitive_hfirstleft_index_bound. fom_gap_pfp_right_transitive_hfirstleft_index_bound + S (fom_index_pfp_right_transitive_hfirstleft) = x2) -> exists fom_value_pfp_right_transitive_hfirstleft. ((((exists fom_beta_height_pfp_right_transitive_hfirstleft_entry. fom_beta_height_pfp_right_transitive_hfirstleft_entry + S (fom_value_pfp_right_transitive_hfirstleft) = S ((S (fom_index_pfp_right_transitive_hfirstleft)) * x1)) /\\ exists fom_beta_quotient_pfp_right_transitive_hfirstleft_entry. x = fom_beta_quotient_pfp_right_transitive_hfirstleft_entry * S ((S (fom_index_pfp_right_transitive_hfirstleft)) * x1) + (fom_value_pfp_right_transitive_hfirstleft))) /\\ (exists fom_gap_pfp_right_transitive_hfirstleft_value_bound. fom_gap_pfp_right_transitive_hfirstleft_value_bound + S (fom_value_pfp_right_transitive_hfirstleft) = p))) /\\ (((forall fom_index_pfp_right_transitive_hfirstright. (exists fom_gap_pfp_right_transitive_hfirstright_index_bound. fom_gap_pfp_right_transitive_hfirstright_index_bound + S (fom_index_pfp_right_transitive_hfirstright) = D) -> exists fom_value_pfp_right_transitive_hfirstright. ((((exists fom_beta_height_pfp_right_transitive_hfirstright_entry. fom_beta_height_pfp_right_transitive_hfirstright_entry + S (fom_value_pfp_right_transitive_hfirstright) = S ((S (fom_index_pfp_right_transitive_hfirstright)) * dc)) /\\ exists fom_beta_quotient_pfp_right_transitive_hfirstright_entry. db = fom_beta_quotient_pfp_right_transitive_hfirstright_entry * S ((S (fom_index_pfp_right_transitive_hfirstright)) * dc) + (fom_value_pfp_right_transitive_hfirstright))) /\\ (exists fom_gap_pfp_right_transitive_hfirstright_value_bound. fom_gap_pfp_right_transitive_hfirstright_value_bound + S (fom_value_pfp_right_transitive_hfirstright) = p))) /\\ (((((((x2)=0 \\/ (D)=0) /\\ (((x5)=0)))) \\/ (((~((x2)=0)) /\\ (((~((D)=0)) /\\ (((x2)+(D)=S (x5)))))))) /\\ ((forall pfc_index_right_transitive_hfirstcoefficients. (exists pfa_gap_right_transitive_hfirstcoefficientsbound. pfa_gap_right_transitive_hfirstcoefficientsbound + S (pfc_index_right_transitive_hfirstcoefficients) = (x5)) -> exists pfc_value_right_transitive_hfirstcoefficients. ((((exists ff_h_pfp_right_transitive_hfirstcoefficientsentry. ff_h_pfp_right_transitive_hfirstcoefficientsentry + S (pfc_value_right_transitive_hfirstcoefficients) = S ((S (pfc_index_right_transitive_hfirstcoefficients)) * x4)) /\\ exists ff_q_pfp_right_transitive_hfirstcoefficientsentry. x3 = ff_q_pfp_right_transitive_hfirstcoefficientsentry * S ((S (pfc_index_right_transitive_hfirstcoefficients)) * x4) + (pfc_value_right_transitive_hfirstcoefficients))) /\\ ((exists pfc_terms_code_right_transitive_hfirstcoefficientscoefficient pfc_terms_scale_right_transitive_hfirstcoefficientscoefficient pfc_natural_sum_right_transitive_hfirstcoefficientscoefficient. ((forall pfc_index_right_transitive_hfirstcoefficientscoefficientdiagonal. (exists pfa_gap_right_transitive_hfirstcoefficientscoefficientdiagonalbound. pfa_gap_right_transitive_hfirstcoefficientscoefficientdiagonalbound + S (pfc_index_right_transitive_hfirstcoefficientscoefficientdiagonal) = (S (pfc_index_right_transitive_hfirstcoefficients))) -> exists pfc_value_right_transitive_hfirstcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_right_transitive_hfirstcoefficientscoefficientdiagonalentry. ff_h_pfp_right_transitive_hfirstcoefficientscoefficientdiagonalentry + S (pfc_value_right_transitive_hfirstcoefficientscoefficientdiagonal) = S ((S (pfc_index_right_transitive_hfirstcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_transitive_hfirstcoefficientscoefficient)) /\\ exists ff_q_pfp_right_transitive_hfirstcoefficientscoefficientdiagonalentry. pfc_terms_code_right_transitive_hfirstcoefficientscoefficient = ff_q_pfp_right_transitive_hfirstcoefficientscoefficientdiagonalentry * S ((S (pfc_index_right_transitive_hfirstcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_transitive_hfirstcoefficientscoefficient) + (pfc_value_right_transitive_hfirstcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_right_transitive_hfirstcoefficientscoefficientdiagonalterm pfc_left_right_transitive_hfirstcoefficientscoefficientdiagonalterm pfc_right_right_transitive_hfirstcoefficientscoefficientdiagonalterm. (((pfc_index_right_transitive_hfirstcoefficientscoefficientdiagonal)+pfc_complement_right_transitive_hfirstcoefficientscoefficientdiagonalterm=(pfc_index_right_transitive_hfirstcoefficients)) /\\ ((((((exists pfa_gap_right_transitive_hfirstcoefficientscoefficientdiagonaltermleftinside. pfa_gap_right_transitive_hfirstcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_right_transitive_hfirstcoefficientscoefficientdiagonal) = (x2)) /\\ ((((exists ff_h_pfp_right_transitive_hfirstcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_right_transitive_hfirstcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_right_transitive_hfirstcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_right_transitive_hfirstcoefficientscoefficientdiagonal)) * x1)) /\\ exists ff_q_pfp_right_transitive_hfirstcoefficientscoefficientdiagonaltermleftentry. x = ff_q_pfp_right_transitive_hfirstcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_right_transitive_hfirstcoefficientscoefficientdiagonal)) * x1) + (pfc_left_right_transitive_hfirstcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_transitive_hfirstcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_right_transitive_hfirstcoefficientscoefficientdiagonaltermleftoutside+(x2)=(pfc_index_right_transitive_hfirstcoefficientscoefficientdiagonal)) /\\ (((pfc_left_right_transitive_hfirstcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_right_transitive_hfirstcoefficientscoefficientdiagonaltermrightinside. pfa_gap_right_transitive_hfirstcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_right_transitive_hfirstcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_right_transitive_hfirstcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_right_transitive_hfirstcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_right_transitive_hfirstcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_right_transitive_hfirstcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_right_transitive_hfirstcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_right_transitive_hfirstcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_right_transitive_hfirstcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_right_transitive_hfirstcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_transitive_hfirstcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_right_transitive_hfirstcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_right_transitive_hfirstcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_right_transitive_hfirstcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_right_transitive_hfirstcoefficientscoefficientdiagonal)=pfc_left_right_transitive_hfirstcoefficientscoefficientdiagonalterm*pfc_right_right_transitive_hfirstcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_right_transitive_hfirstcoefficientscoefficientsum fs_v_pfc_right_transitive_hfirstcoefficientscoefficientsum. ((((exists fs_h_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_start. fs_h_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_right_transitive_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_start. fs_u_pfc_right_transitive_hfirstcoefficientscoefficientsum = fs_q_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_right_transitive_hfirstcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_terminal. fs_h_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_right_transitive_hfirstcoefficientscoefficient) = S ((S (S (pfc_index_right_transitive_hfirstcoefficients))) * fs_v_pfc_right_transitive_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_terminal. fs_u_pfc_right_transitive_hfirstcoefficientscoefficientsum = fs_q_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_right_transitive_hfirstcoefficients))) * fs_v_pfc_right_transitive_hfirstcoefficientscoefficientsum) + (pfc_natural_sum_right_transitive_hfirstcoefficientscoefficient))) /\\ forall fs_i_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps = S (pfc_index_right_transitive_hfirstcoefficients)) -> exists fs_a_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps fs_r_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps fs_s_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_transitive_hfirstcoefficientscoefficient)) /\\ exists fs_q_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_right_transitive_hfirstcoefficientscoefficient = fs_q_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_transitive_hfirstcoefficientscoefficient) + (fs_a_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_right_transitive_hfirstcoefficientscoefficientsum = fs_q_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_hfirstcoefficientscoefficientsum) + (fs_r_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_right_transitive_hfirstcoefficientscoefficientsum = fs_q_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_hfirstcoefficientscoefficientsum) + (fs_s_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps = fs_r_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps + fs_a_pfc_right_transitive_hfirstcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_right_transitive_hfirstcoefficientscoefficientresiduebound. pfa_gap_right_transitive_hfirstcoefficientscoefficientresiduebound + S (pfc_value_right_transitive_hfirstcoefficients) = (p)) /\\ ((exists pfa_offset_left_right_transitive_hfirstcoefficientscoefficientresiduecongruence pfa_offset_right_right_transitive_hfirstcoefficientscoefficientresiduecongruence. (pfc_natural_sum_right_transitive_hfirstcoefficientscoefficient) + (p) * pfa_offset_left_right_transitive_hfirstcoefficientscoefficientresiduecongruence = (pfc_value_right_transitive_hfirstcoefficients) + (p) * pfa_offset_right_right_transitive_hfirstcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hDA_right_witness_witness_witness_witness_witness_witness_left", "cases hfirst", "cases hfirst_right", "cases hfirst_right_right", "have hsecond : ((forall fom_index_pfp_right_transitive_hsecondleft. (exists fom_gap_pfp_right_transitive_hsecondleft_index_bound. fom_gap_pfp_right_transitive_hsecondleft_index_bound + S (fom_index_pfp_right_transitive_hsecondleft) = x8) -> exists fom_value_pfp_right_transitive_hsecondleft. ((((exists fom_beta_height_pfp_right_transitive_hsecondleft_entry. fom_beta_height_pfp_right_transitive_hsecondleft_entry + S (fom_value_pfp_right_transitive_hsecondleft) = S ((S (fom_index_pfp_right_transitive_hsecondleft)) * x7)) /\\ exists fom_beta_quotient_pfp_right_transitive_hsecondleft_entry. x6 = fom_beta_quotient_pfp_right_transitive_hsecondleft_entry * S ((S (fom_index_pfp_right_transitive_hsecondleft)) * x7) + (fom_value_pfp_right_transitive_hsecondleft))) /\\ (exists fom_gap_pfp_right_transitive_hsecondleft_value_bound. fom_gap_pfp_right_transitive_hsecondleft_value_bound + S (fom_value_pfp_right_transitive_hsecondleft) = p))) /\\ (((forall fom_index_pfp_right_transitive_hsecondright. (exists fom_gap_pfp_right_transitive_hsecondright_index_bound. fom_gap_pfp_right_transitive_hsecondright_index_bound + S (fom_index_pfp_right_transitive_hsecondright) = L) -> exists fom_value_pfp_right_transitive_hsecondright. ((((exists fom_beta_height_pfp_right_transitive_hsecondright_entry. fom_beta_height_pfp_right_transitive_hsecondright_entry + S (fom_value_pfp_right_transitive_hsecondright) = S ((S (fom_index_pfp_right_transitive_hsecondright)) * ac)) /\\ exists fom_beta_quotient_pfp_right_transitive_hsecondright_entry. ab = fom_beta_quotient_pfp_right_transitive_hsecondright_entry * S ((S (fom_index_pfp_right_transitive_hsecondright)) * ac) + (fom_value_pfp_right_transitive_hsecondright))) /\\ (exists fom_gap_pfp_right_transitive_hsecondright_value_bound. fom_gap_pfp_right_transitive_hsecondright_value_bound + S (fom_value_pfp_right_transitive_hsecondright) = p))) /\\ (((((((x8)=0 \\/ (L)=0) /\\ (((x11)=0)))) \\/ (((~((x8)=0)) /\\ (((~((L)=0)) /\\ (((x8)+(L)=S (x11)))))))) /\\ ((forall pfc_index_right_transitive_hsecondcoefficients. (exists pfa_gap_right_transitive_hsecondcoefficientsbound. pfa_gap_right_transitive_hsecondcoefficientsbound + S (pfc_index_right_transitive_hsecondcoefficients) = (x11)) -> exists pfc_value_right_transitive_hsecondcoefficients. ((((exists ff_h_pfp_right_transitive_hsecondcoefficientsentry. ff_h_pfp_right_transitive_hsecondcoefficientsentry + S (pfc_value_right_transitive_hsecondcoefficients) = S ((S (pfc_index_right_transitive_hsecondcoefficients)) * x10)) /\\ exists ff_q_pfp_right_transitive_hsecondcoefficientsentry. x9 = ff_q_pfp_right_transitive_hsecondcoefficientsentry * S ((S (pfc_index_right_transitive_hsecondcoefficients)) * x10) + (pfc_value_right_transitive_hsecondcoefficients))) /\\ ((exists pfc_terms_code_right_transitive_hsecondcoefficientscoefficient pfc_terms_scale_right_transitive_hsecondcoefficientscoefficient pfc_natural_sum_right_transitive_hsecondcoefficientscoefficient. ((forall pfc_index_right_transitive_hsecondcoefficientscoefficientdiagonal. (exists pfa_gap_right_transitive_hsecondcoefficientscoefficientdiagonalbound. pfa_gap_right_transitive_hsecondcoefficientscoefficientdiagonalbound + S (pfc_index_right_transitive_hsecondcoefficientscoefficientdiagonal) = (S (pfc_index_right_transitive_hsecondcoefficients))) -> exists pfc_value_right_transitive_hsecondcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_right_transitive_hsecondcoefficientscoefficientdiagonalentry. ff_h_pfp_right_transitive_hsecondcoefficientscoefficientdiagonalentry + S (pfc_value_right_transitive_hsecondcoefficientscoefficientdiagonal) = S ((S (pfc_index_right_transitive_hsecondcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_transitive_hsecondcoefficientscoefficient)) /\\ exists ff_q_pfp_right_transitive_hsecondcoefficientscoefficientdiagonalentry. pfc_terms_code_right_transitive_hsecondcoefficientscoefficient = ff_q_pfp_right_transitive_hsecondcoefficientscoefficientdiagonalentry * S ((S (pfc_index_right_transitive_hsecondcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_transitive_hsecondcoefficientscoefficient) + (pfc_value_right_transitive_hsecondcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_right_transitive_hsecondcoefficientscoefficientdiagonalterm pfc_left_right_transitive_hsecondcoefficientscoefficientdiagonalterm pfc_right_right_transitive_hsecondcoefficientscoefficientdiagonalterm. (((pfc_index_right_transitive_hsecondcoefficientscoefficientdiagonal)+pfc_complement_right_transitive_hsecondcoefficientscoefficientdiagonalterm=(pfc_index_right_transitive_hsecondcoefficients)) /\\ ((((((exists pfa_gap_right_transitive_hsecondcoefficientscoefficientdiagonaltermleftinside. pfa_gap_right_transitive_hsecondcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_right_transitive_hsecondcoefficientscoefficientdiagonal) = (x8)) /\\ ((((exists ff_h_pfp_right_transitive_hsecondcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_right_transitive_hsecondcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_right_transitive_hsecondcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_right_transitive_hsecondcoefficientscoefficientdiagonal)) * x7)) /\\ exists ff_q_pfp_right_transitive_hsecondcoefficientscoefficientdiagonaltermleftentry. x6 = ff_q_pfp_right_transitive_hsecondcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_right_transitive_hsecondcoefficientscoefficientdiagonal)) * x7) + (pfc_left_right_transitive_hsecondcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_transitive_hsecondcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_right_transitive_hsecondcoefficientscoefficientdiagonaltermleftoutside+(x8)=(pfc_index_right_transitive_hsecondcoefficientscoefficientdiagonal)) /\\ (((pfc_left_right_transitive_hsecondcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_right_transitive_hsecondcoefficientscoefficientdiagonaltermrightinside. pfa_gap_right_transitive_hsecondcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_right_transitive_hsecondcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_right_transitive_hsecondcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_right_transitive_hsecondcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_right_transitive_hsecondcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_right_transitive_hsecondcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_right_transitive_hsecondcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_right_transitive_hsecondcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_right_transitive_hsecondcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_right_transitive_hsecondcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_transitive_hsecondcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_right_transitive_hsecondcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_right_transitive_hsecondcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_right_transitive_hsecondcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_right_transitive_hsecondcoefficientscoefficientdiagonal)=pfc_left_right_transitive_hsecondcoefficientscoefficientdiagonalterm*pfc_right_right_transitive_hsecondcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_right_transitive_hsecondcoefficientscoefficientsum fs_v_pfc_right_transitive_hsecondcoefficientscoefficientsum. ((((exists fs_h_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_start. fs_h_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_right_transitive_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_start. fs_u_pfc_right_transitive_hsecondcoefficientscoefficientsum = fs_q_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_right_transitive_hsecondcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_terminal. fs_h_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_right_transitive_hsecondcoefficientscoefficient) = S ((S (S (pfc_index_right_transitive_hsecondcoefficients))) * fs_v_pfc_right_transitive_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_terminal. fs_u_pfc_right_transitive_hsecondcoefficientscoefficientsum = fs_q_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_right_transitive_hsecondcoefficients))) * fs_v_pfc_right_transitive_hsecondcoefficientscoefficientsum) + (pfc_natural_sum_right_transitive_hsecondcoefficientscoefficient))) /\\ forall fs_i_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps = S (pfc_index_right_transitive_hsecondcoefficients)) -> exists fs_a_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps fs_r_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps fs_s_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_transitive_hsecondcoefficientscoefficient)) /\\ exists fs_q_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_right_transitive_hsecondcoefficientscoefficient = fs_q_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_transitive_hsecondcoefficientscoefficient) + (fs_a_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_right_transitive_hsecondcoefficientscoefficientsum = fs_q_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_hsecondcoefficientscoefficientsum) + (fs_r_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_right_transitive_hsecondcoefficientscoefficientsum = fs_q_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_hsecondcoefficientscoefficientsum) + (fs_s_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps = fs_r_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps + fs_a_pfc_right_transitive_hsecondcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_right_transitive_hsecondcoefficientscoefficientresiduebound. pfa_gap_right_transitive_hsecondcoefficientscoefficientresiduebound + S (pfc_value_right_transitive_hsecondcoefficients) = (p)) /\\ ((exists pfa_offset_left_right_transitive_hsecondcoefficientscoefficientresiduecongruence pfa_offset_right_right_transitive_hsecondcoefficientscoefficientresiduecongruence. (pfc_natural_sum_right_transitive_hsecondcoefficientscoefficient) + (p) * pfa_offset_left_right_transitive_hsecondcoefficientscoefficientresiduecongruence = (pfc_value_right_transitive_hsecondcoefficients) + (p) * pfa_offset_right_right_transitive_hsecondcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hAB_right_witness_witness_witness_witness_witness_witness_left", "cases hsecond", "cases hsecond_right", "cases hsecond_right_right", "have hPbound : forall fom_index_pfp_right_transitive_P_bound. (exists fom_gap_pfp_right_transitive_P_bound_index_bound. fom_gap_pfp_right_transitive_P_bound_index_bound + S (fom_index_pfp_right_transitive_P_bound) = x5) -> exists fom_value_pfp_right_transitive_P_bound. ((((exists fom_beta_height_pfp_right_transitive_P_bound_entry. fom_beta_height_pfp_right_transitive_P_bound_entry + S (fom_value_pfp_right_transitive_P_bound) = S ((S (fom_index_pfp_right_transitive_P_bound)) * x4)) /\\ exists fom_beta_quotient_pfp_right_transitive_P_bound_entry. x3 = fom_beta_quotient_pfp_right_transitive_P_bound_entry * S ((S (fom_index_pfp_right_transitive_P_bound)) * x4) + (fom_value_pfp_right_transitive_P_bound))) /\\ (exists fom_gap_pfp_right_transitive_P_bound_value_bound. fom_gap_pfp_right_transitive_P_bound_value_bound + S (fom_value_pfp_right_transitive_P_bound) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (x)", "specialize prime_field_polynomial_convolution_bounded (x1)", "specialize prime_field_polynomial_convolution_bounded (x2)", "specialize prime_field_polynomial_convolution_bounded (db)", "specialize prime_field_polynomial_convolution_bounded (dc)", "specialize prime_field_polynomial_convolution_bounded (D)", "specialize prime_field_polynomial_convolution_bounded (x3)", "specialize prime_field_polynomial_convolution_bounded (x4)", "specialize prime_field_polynomial_convolution_bounded (x5)", "apply prime_field_polynomial_convolution_bounded", "exact hDA_right_witness_witness_witness_witness_witness_witness_left", "have hcomposite_length : exists n. (((((x8)=0 \\/ (x2)=0) /\\ (((n)=0)))) \\/ (((~((x8)=0)) /\\ (((~((x2)=0)) /\\ (((x8)+(x2)=S (n))))))))", "specialize polynomial_product_length_exists (x8)", "specialize polynomial_product_length_exists (x2)", "apply polynomial_product_length_exists", "cases hcomposite_length", "have hcomposite_product : exists b c. (((forall fom_index_pfp_hcomposite_graphleft. (exists fom_gap_pfp_hcomposite_graphleft_index_bound. fom_gap_pfp_hcomposite_graphleft_index_bound + S (fom_index_pfp_hcomposite_graphleft) = x8) -> exists fom_value_pfp_hcomposite_graphleft. ((((exists fom_beta_height_pfp_hcomposite_graphleft_entry. fom_beta_height_pfp_hcomposite_graphleft_entry + S (fom_value_pfp_hcomposite_graphleft) = S ((S (fom_index_pfp_hcomposite_graphleft)) * x7)) /\\ exists fom_beta_quotient_pfp_hcomposite_graphleft_entry. x6 = fom_beta_quotient_pfp_hcomposite_graphleft_entry * S ((S (fom_index_pfp_hcomposite_graphleft)) * x7) + (fom_value_pfp_hcomposite_graphleft))) /\\ (exists fom_gap_pfp_hcomposite_graphleft_value_bound. fom_gap_pfp_hcomposite_graphleft_value_bound + S (fom_value_pfp_hcomposite_graphleft) = p))) /\\ (((forall fom_index_pfp_hcomposite_graphright. (exists fom_gap_pfp_hcomposite_graphright_index_bound. fom_gap_pfp_hcomposite_graphright_index_bound + S (fom_index_pfp_hcomposite_graphright) = x2) -> exists fom_value_pfp_hcomposite_graphright. ((((exists fom_beta_height_pfp_hcomposite_graphright_entry. fom_beta_height_pfp_hcomposite_graphright_entry + S (fom_value_pfp_hcomposite_graphright) = S ((S (fom_index_pfp_hcomposite_graphright)) * x1)) /\\ exists fom_beta_quotient_pfp_hcomposite_graphright_entry. x = fom_beta_quotient_pfp_hcomposite_graphright_entry * S ((S (fom_index_pfp_hcomposite_graphright)) * x1) + (fom_value_pfp_hcomposite_graphright))) /\\ (exists fom_gap_pfp_hcomposite_graphright_value_bound. fom_gap_pfp_hcomposite_graphright_value_bound + S (fom_value_pfp_hcomposite_graphright) = p))) /\\ (((((((x8)=0 \\/ (x2)=0) /\\ (((x12)=0)))) \\/ (((~((x8)=0)) /\\ (((~((x2)=0)) /\\ (((x8)+(x2)=S (x12)))))))) /\\ ((forall pfc_index_hcomposite_graphcoefficients. (exists pfa_gap_hcomposite_graphcoefficientsbound. pfa_gap_hcomposite_graphcoefficientsbound + S (pfc_index_hcomposite_graphcoefficients) = (x12)) -> exists pfc_value_hcomposite_graphcoefficients. ((((exists ff_h_pfp_hcomposite_graphcoefficientsentry. ff_h_pfp_hcomposite_graphcoefficientsentry + S (pfc_value_hcomposite_graphcoefficients) = S ((S (pfc_index_hcomposite_graphcoefficients)) * c)) /\\ exists ff_q_pfp_hcomposite_graphcoefficientsentry. b = ff_q_pfp_hcomposite_graphcoefficientsentry * S ((S (pfc_index_hcomposite_graphcoefficients)) * c) + (pfc_value_hcomposite_graphcoefficients))) /\\ ((exists pfc_terms_code_hcomposite_graphcoefficientscoefficient pfc_terms_scale_hcomposite_graphcoefficientscoefficient pfc_natural_sum_hcomposite_graphcoefficientscoefficient. ((forall pfc_index_hcomposite_graphcoefficientscoefficientdiagonal. (exists pfa_gap_hcomposite_graphcoefficientscoefficientdiagonalbound. pfa_gap_hcomposite_graphcoefficientscoefficientdiagonalbound + S (pfc_index_hcomposite_graphcoefficientscoefficientdiagonal) = (S (pfc_index_hcomposite_graphcoefficients))) -> exists pfc_value_hcomposite_graphcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_hcomposite_graphcoefficientscoefficientdiagonalentry. ff_h_pfp_hcomposite_graphcoefficientscoefficientdiagonalentry + S (pfc_value_hcomposite_graphcoefficientscoefficientdiagonal) = S ((S (pfc_index_hcomposite_graphcoefficientscoefficientdiagonal)) * pfc_terms_scale_hcomposite_graphcoefficientscoefficient)) /\\ exists ff_q_pfp_hcomposite_graphcoefficientscoefficientdiagonalentry. pfc_terms_code_hcomposite_graphcoefficientscoefficient = ff_q_pfp_hcomposite_graphcoefficientscoefficientdiagonalentry * S ((S (pfc_index_hcomposite_graphcoefficientscoefficientdiagonal)) * pfc_terms_scale_hcomposite_graphcoefficientscoefficient) + (pfc_value_hcomposite_graphcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_hcomposite_graphcoefficientscoefficientdiagonalterm pfc_left_hcomposite_graphcoefficientscoefficientdiagonalterm pfc_right_hcomposite_graphcoefficientscoefficientdiagonalterm. (((pfc_index_hcomposite_graphcoefficientscoefficientdiagonal)+pfc_complement_hcomposite_graphcoefficientscoefficientdiagonalterm=(pfc_index_hcomposite_graphcoefficients)) /\\ ((((((exists pfa_gap_hcomposite_graphcoefficientscoefficientdiagonaltermleftinside. pfa_gap_hcomposite_graphcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_hcomposite_graphcoefficientscoefficientdiagonal) = (x8)) /\\ ((((exists ff_h_pfp_hcomposite_graphcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_hcomposite_graphcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_hcomposite_graphcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_hcomposite_graphcoefficientscoefficientdiagonal)) * x7)) /\\ exists ff_q_pfp_hcomposite_graphcoefficientscoefficientdiagonaltermleftentry. x6 = ff_q_pfp_hcomposite_graphcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_hcomposite_graphcoefficientscoefficientdiagonal)) * x7) + (pfc_left_hcomposite_graphcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hcomposite_graphcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_hcomposite_graphcoefficientscoefficientdiagonaltermleftoutside+(x8)=(pfc_index_hcomposite_graphcoefficientscoefficientdiagonal)) /\\ (((pfc_left_hcomposite_graphcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_hcomposite_graphcoefficientscoefficientdiagonaltermrightinside. pfa_gap_hcomposite_graphcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_hcomposite_graphcoefficientscoefficientdiagonalterm) = (x2)) /\\ ((((exists ff_h_pfp_hcomposite_graphcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_hcomposite_graphcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_hcomposite_graphcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_hcomposite_graphcoefficientscoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_hcomposite_graphcoefficientscoefficientdiagonaltermrightentry. x = ff_q_pfp_hcomposite_graphcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_hcomposite_graphcoefficientscoefficientdiagonalterm)) * x1) + (pfc_right_hcomposite_graphcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hcomposite_graphcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_hcomposite_graphcoefficientscoefficientdiagonaltermrightoutside+(x2)=(pfc_complement_hcomposite_graphcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_hcomposite_graphcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_hcomposite_graphcoefficientscoefficientdiagonal)=pfc_left_hcomposite_graphcoefficientscoefficientdiagonalterm*pfc_right_hcomposite_graphcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_hcomposite_graphcoefficientscoefficientsum fs_v_pfc_hcomposite_graphcoefficientscoefficientsum. ((((exists fs_h_pfc_hcomposite_graphcoefficientscoefficientsum_body_start. fs_h_pfc_hcomposite_graphcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_hcomposite_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hcomposite_graphcoefficientscoefficientsum_body_start. fs_u_pfc_hcomposite_graphcoefficientscoefficientsum = fs_q_pfc_hcomposite_graphcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_hcomposite_graphcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_hcomposite_graphcoefficientscoefficientsum_body_terminal. fs_h_pfc_hcomposite_graphcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_hcomposite_graphcoefficientscoefficient) = S ((S (S (pfc_index_hcomposite_graphcoefficients))) * fs_v_pfc_hcomposite_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hcomposite_graphcoefficientscoefficientsum_body_terminal. fs_u_pfc_hcomposite_graphcoefficientscoefficientsum = fs_q_pfc_hcomposite_graphcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_hcomposite_graphcoefficients))) * fs_v_pfc_hcomposite_graphcoefficientscoefficientsum) + (pfc_natural_sum_hcomposite_graphcoefficientscoefficient))) /\\ forall fs_i_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps = S (pfc_index_hcomposite_graphcoefficients)) -> exists fs_a_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps fs_r_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps fs_s_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hcomposite_graphcoefficientscoefficient)) /\\ exists fs_q_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_hcomposite_graphcoefficientscoefficient = fs_q_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hcomposite_graphcoefficientscoefficient) + (fs_a_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hcomposite_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_hcomposite_graphcoefficientscoefficientsum = fs_q_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hcomposite_graphcoefficientscoefficientsum) + (fs_r_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hcomposite_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_hcomposite_graphcoefficientscoefficientsum = fs_q_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hcomposite_graphcoefficientscoefficientsum) + (fs_s_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps = fs_r_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps + fs_a_pfc_hcomposite_graphcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_hcomposite_graphcoefficientscoefficientresiduebound. pfa_gap_hcomposite_graphcoefficientscoefficientresiduebound + S (pfc_value_hcomposite_graphcoefficients) = (p)) /\\ ((exists pfa_offset_left_hcomposite_graphcoefficientscoefficientresiduecongruence pfa_offset_right_hcomposite_graphcoefficientscoefficientresiduecongruence. (pfc_natural_sum_hcomposite_graphcoefficientscoefficient) + (p) * pfa_offset_left_hcomposite_graphcoefficientscoefficientresiduecongruence = (pfc_value_hcomposite_graphcoefficients) + (p) * pfa_offset_right_hcomposite_graphcoefficientscoefficientresiduecongruence)))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (x6)", "specialize prime_field_polynomial_convolution_at_length_exists (x7)", "specialize prime_field_polynomial_convolution_at_length_exists (x8)", "specialize prime_field_polynomial_convolution_at_length_exists (x)", "specialize prime_field_polynomial_convolution_at_length_exists (x1)", "specialize prime_field_polynomial_convolution_at_length_exists (x2)", "specialize prime_field_polynomial_convolution_at_length_exists (x12)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hsecond_left", "exact hfirst_left", "exact hcomposite_length_witness", "cases hcomposite_product", "cases hcomposite_product_witness", "have hQbound : forall fom_index_pfp_right_transitive_Q_bound. (exists fom_gap_pfp_right_transitive_Q_bound_index_bound. fom_gap_pfp_right_transitive_Q_bound_index_bound + S (fom_index_pfp_right_transitive_Q_bound) = x12) -> exists fom_value_pfp_right_transitive_Q_bound. ((((exists fom_beta_height_pfp_right_transitive_Q_bound_entry. fom_beta_height_pfp_right_transitive_Q_bound_entry + S (fom_value_pfp_right_transitive_Q_bound) = S ((S (fom_index_pfp_right_transitive_Q_bound)) * x14)) /\\ exists fom_beta_quotient_pfp_right_transitive_Q_bound_entry. x13 = fom_beta_quotient_pfp_right_transitive_Q_bound_entry * S ((S (fom_index_pfp_right_transitive_Q_bound)) * x14) + (fom_value_pfp_right_transitive_Q_bound))) /\\ (exists fom_gap_pfp_right_transitive_Q_bound_value_bound. fom_gap_pfp_right_transitive_Q_bound_value_bound + S (fom_value_pfp_right_transitive_Q_bound) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (x6)", "specialize prime_field_polynomial_convolution_bounded (x7)", "specialize prime_field_polynomial_convolution_bounded (x8)", "specialize prime_field_polynomial_convolution_bounded (x)", "specialize prime_field_polynomial_convolution_bounded (x1)", "specialize prime_field_polynomial_convolution_bounded (x2)", "specialize prime_field_polynomial_convolution_bounded (x13)", "specialize prime_field_polynomial_convolution_bounded (x14)", "specialize prime_field_polynomial_convolution_bounded (x12)", "apply prime_field_polynomial_convolution_bounded", "exact hcomposite_product_witness_witness", "have hresult_length : exists n. (((((x12)=0 \\/ (D)=0) /\\ (((n)=0)))) \\/ (((~((x12)=0)) /\\ (((~((D)=0)) /\\ (((x12)+(D)=S (n))))))))", "specialize polynomial_product_length_exists (x12)", "specialize polynomial_product_length_exists (D)", "apply polynomial_product_length_exists", "cases hresult_length", "have hresult_product : exists b c. (((forall fom_index_pfp_hresult_graphleft. (exists fom_gap_pfp_hresult_graphleft_index_bound. fom_gap_pfp_hresult_graphleft_index_bound + S (fom_index_pfp_hresult_graphleft) = x12) -> exists fom_value_pfp_hresult_graphleft. ((((exists fom_beta_height_pfp_hresult_graphleft_entry. fom_beta_height_pfp_hresult_graphleft_entry + S (fom_value_pfp_hresult_graphleft) = S ((S (fom_index_pfp_hresult_graphleft)) * x14)) /\\ exists fom_beta_quotient_pfp_hresult_graphleft_entry. x13 = fom_beta_quotient_pfp_hresult_graphleft_entry * S ((S (fom_index_pfp_hresult_graphleft)) * x14) + (fom_value_pfp_hresult_graphleft))) /\\ (exists fom_gap_pfp_hresult_graphleft_value_bound. fom_gap_pfp_hresult_graphleft_value_bound + S (fom_value_pfp_hresult_graphleft) = p))) /\\ (((forall fom_index_pfp_hresult_graphright. (exists fom_gap_pfp_hresult_graphright_index_bound. fom_gap_pfp_hresult_graphright_index_bound + S (fom_index_pfp_hresult_graphright) = D) -> exists fom_value_pfp_hresult_graphright. ((((exists fom_beta_height_pfp_hresult_graphright_entry. fom_beta_height_pfp_hresult_graphright_entry + S (fom_value_pfp_hresult_graphright) = S ((S (fom_index_pfp_hresult_graphright)) * dc)) /\\ exists fom_beta_quotient_pfp_hresult_graphright_entry. db = fom_beta_quotient_pfp_hresult_graphright_entry * S ((S (fom_index_pfp_hresult_graphright)) * dc) + (fom_value_pfp_hresult_graphright))) /\\ (exists fom_gap_pfp_hresult_graphright_value_bound. fom_gap_pfp_hresult_graphright_value_bound + S (fom_value_pfp_hresult_graphright) = p))) /\\ (((((((x12)=0 \\/ (D)=0) /\\ (((x15)=0)))) \\/ (((~((x12)=0)) /\\ (((~((D)=0)) /\\ (((x12)+(D)=S (x15)))))))) /\\ ((forall pfc_index_hresult_graphcoefficients. (exists pfa_gap_hresult_graphcoefficientsbound. pfa_gap_hresult_graphcoefficientsbound + S (pfc_index_hresult_graphcoefficients) = (x15)) -> exists pfc_value_hresult_graphcoefficients. ((((exists ff_h_pfp_hresult_graphcoefficientsentry. ff_h_pfp_hresult_graphcoefficientsentry + S (pfc_value_hresult_graphcoefficients) = S ((S (pfc_index_hresult_graphcoefficients)) * c)) /\\ exists ff_q_pfp_hresult_graphcoefficientsentry. b = ff_q_pfp_hresult_graphcoefficientsentry * S ((S (pfc_index_hresult_graphcoefficients)) * c) + (pfc_value_hresult_graphcoefficients))) /\\ ((exists pfc_terms_code_hresult_graphcoefficientscoefficient pfc_terms_scale_hresult_graphcoefficientscoefficient pfc_natural_sum_hresult_graphcoefficientscoefficient. ((forall pfc_index_hresult_graphcoefficientscoefficientdiagonal. (exists pfa_gap_hresult_graphcoefficientscoefficientdiagonalbound. pfa_gap_hresult_graphcoefficientscoefficientdiagonalbound + S (pfc_index_hresult_graphcoefficientscoefficientdiagonal) = (S (pfc_index_hresult_graphcoefficients))) -> exists pfc_value_hresult_graphcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_hresult_graphcoefficientscoefficientdiagonalentry. ff_h_pfp_hresult_graphcoefficientscoefficientdiagonalentry + S (pfc_value_hresult_graphcoefficientscoefficientdiagonal) = S ((S (pfc_index_hresult_graphcoefficientscoefficientdiagonal)) * pfc_terms_scale_hresult_graphcoefficientscoefficient)) /\\ exists ff_q_pfp_hresult_graphcoefficientscoefficientdiagonalentry. pfc_terms_code_hresult_graphcoefficientscoefficient = ff_q_pfp_hresult_graphcoefficientscoefficientdiagonalentry * S ((S (pfc_index_hresult_graphcoefficientscoefficientdiagonal)) * pfc_terms_scale_hresult_graphcoefficientscoefficient) + (pfc_value_hresult_graphcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_hresult_graphcoefficientscoefficientdiagonalterm pfc_left_hresult_graphcoefficientscoefficientdiagonalterm pfc_right_hresult_graphcoefficientscoefficientdiagonalterm. (((pfc_index_hresult_graphcoefficientscoefficientdiagonal)+pfc_complement_hresult_graphcoefficientscoefficientdiagonalterm=(pfc_index_hresult_graphcoefficients)) /\\ ((((((exists pfa_gap_hresult_graphcoefficientscoefficientdiagonaltermleftinside. pfa_gap_hresult_graphcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_hresult_graphcoefficientscoefficientdiagonal) = (x12)) /\\ ((((exists ff_h_pfp_hresult_graphcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_hresult_graphcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_hresult_graphcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_hresult_graphcoefficientscoefficientdiagonal)) * x14)) /\\ exists ff_q_pfp_hresult_graphcoefficientscoefficientdiagonaltermleftentry. x13 = ff_q_pfp_hresult_graphcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_hresult_graphcoefficientscoefficientdiagonal)) * x14) + (pfc_left_hresult_graphcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hresult_graphcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_hresult_graphcoefficientscoefficientdiagonaltermleftoutside+(x12)=(pfc_index_hresult_graphcoefficientscoefficientdiagonal)) /\\ (((pfc_left_hresult_graphcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_hresult_graphcoefficientscoefficientdiagonaltermrightinside. pfa_gap_hresult_graphcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_hresult_graphcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_hresult_graphcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_hresult_graphcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_hresult_graphcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_hresult_graphcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_hresult_graphcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_hresult_graphcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_hresult_graphcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_hresult_graphcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hresult_graphcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_hresult_graphcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_hresult_graphcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_hresult_graphcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_hresult_graphcoefficientscoefficientdiagonal)=pfc_left_hresult_graphcoefficientscoefficientdiagonalterm*pfc_right_hresult_graphcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_hresult_graphcoefficientscoefficientsum fs_v_pfc_hresult_graphcoefficientscoefficientsum. ((((exists fs_h_pfc_hresult_graphcoefficientscoefficientsum_body_start. fs_h_pfc_hresult_graphcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_hresult_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hresult_graphcoefficientscoefficientsum_body_start. fs_u_pfc_hresult_graphcoefficientscoefficientsum = fs_q_pfc_hresult_graphcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_hresult_graphcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_hresult_graphcoefficientscoefficientsum_body_terminal. fs_h_pfc_hresult_graphcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_hresult_graphcoefficientscoefficient) = S ((S (S (pfc_index_hresult_graphcoefficients))) * fs_v_pfc_hresult_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hresult_graphcoefficientscoefficientsum_body_terminal. fs_u_pfc_hresult_graphcoefficientscoefficientsum = fs_q_pfc_hresult_graphcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_hresult_graphcoefficients))) * fs_v_pfc_hresult_graphcoefficientscoefficientsum) + (pfc_natural_sum_hresult_graphcoefficientscoefficient))) /\\ forall fs_i_pfc_hresult_graphcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_hresult_graphcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_hresult_graphcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_hresult_graphcoefficientscoefficientsum_body_steps = S (pfc_index_hresult_graphcoefficients)) -> exists fs_a_pfc_hresult_graphcoefficientscoefficientsum_body_steps fs_r_pfc_hresult_graphcoefficientscoefficientsum_body_steps fs_s_pfc_hresult_graphcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_hresult_graphcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_hresult_graphcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_hresult_graphcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hresult_graphcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hresult_graphcoefficientscoefficient)) /\\ exists fs_q_pfc_hresult_graphcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_hresult_graphcoefficientscoefficient = fs_q_pfc_hresult_graphcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_hresult_graphcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hresult_graphcoefficientscoefficient) + (fs_a_pfc_hresult_graphcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hresult_graphcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_hresult_graphcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_hresult_graphcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hresult_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hresult_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hresult_graphcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_hresult_graphcoefficientscoefficientsum = fs_q_pfc_hresult_graphcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_hresult_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hresult_graphcoefficientscoefficientsum) + (fs_r_pfc_hresult_graphcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hresult_graphcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_hresult_graphcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_hresult_graphcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_hresult_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hresult_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hresult_graphcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_hresult_graphcoefficientscoefficientsum = fs_q_pfc_hresult_graphcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_hresult_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hresult_graphcoefficientscoefficientsum) + (fs_s_pfc_hresult_graphcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_hresult_graphcoefficientscoefficientsum_body_steps = fs_r_pfc_hresult_graphcoefficientscoefficientsum_body_steps + fs_a_pfc_hresult_graphcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_hresult_graphcoefficientscoefficientresiduebound. pfa_gap_hresult_graphcoefficientscoefficientresiduebound + S (pfc_value_hresult_graphcoefficients) = (p)) /\\ ((exists pfa_offset_left_hresult_graphcoefficientscoefficientresiduecongruence pfa_offset_right_hresult_graphcoefficientscoefficientresiduecongruence. (pfc_natural_sum_hresult_graphcoefficientscoefficient) + (p) * pfa_offset_left_hresult_graphcoefficientscoefficientresiduecongruence = (pfc_value_hresult_graphcoefficients) + (p) * pfa_offset_right_hresult_graphcoefficientscoefficientresiduecongruence)))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (x13)", "specialize prime_field_polynomial_convolution_at_length_exists (x14)", "specialize prime_field_polynomial_convolution_at_length_exists (x12)", "specialize prime_field_polynomial_convolution_at_length_exists (db)", "specialize prime_field_polynomial_convolution_at_length_exists (dc)", "specialize prime_field_polynomial_convolution_at_length_exists (D)", "specialize prime_field_polynomial_convolution_at_length_exists (x15)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hQbound", "exact hfirst_right_left", "exact hresult_length_witness", "cases hresult_product", "cases hresult_product_witness", "have hmixed_length : exists n. (((((x8)=0 \\/ (x5)=0) /\\ (((n)=0)))) \\/ (((~((x8)=0)) /\\ (((~((x5)=0)) /\\ (((x8)+(x5)=S (n))))))))", "specialize polynomial_product_length_exists (x8)", "specialize polynomial_product_length_exists (x5)", "apply polynomial_product_length_exists", "cases hmixed_length", "have hmixed_product : exists b c. (((forall fom_index_pfp_hmixed_graphleft. (exists fom_gap_pfp_hmixed_graphleft_index_bound. fom_gap_pfp_hmixed_graphleft_index_bound + S (fom_index_pfp_hmixed_graphleft) = x8) -> exists fom_value_pfp_hmixed_graphleft. ((((exists fom_beta_height_pfp_hmixed_graphleft_entry. fom_beta_height_pfp_hmixed_graphleft_entry + S (fom_value_pfp_hmixed_graphleft) = S ((S (fom_index_pfp_hmixed_graphleft)) * x7)) /\\ exists fom_beta_quotient_pfp_hmixed_graphleft_entry. x6 = fom_beta_quotient_pfp_hmixed_graphleft_entry * S ((S (fom_index_pfp_hmixed_graphleft)) * x7) + (fom_value_pfp_hmixed_graphleft))) /\\ (exists fom_gap_pfp_hmixed_graphleft_value_bound. fom_gap_pfp_hmixed_graphleft_value_bound + S (fom_value_pfp_hmixed_graphleft) = p))) /\\ (((forall fom_index_pfp_hmixed_graphright. (exists fom_gap_pfp_hmixed_graphright_index_bound. fom_gap_pfp_hmixed_graphright_index_bound + S (fom_index_pfp_hmixed_graphright) = x5) -> exists fom_value_pfp_hmixed_graphright. ((((exists fom_beta_height_pfp_hmixed_graphright_entry. fom_beta_height_pfp_hmixed_graphright_entry + S (fom_value_pfp_hmixed_graphright) = S ((S (fom_index_pfp_hmixed_graphright)) * x4)) /\\ exists fom_beta_quotient_pfp_hmixed_graphright_entry. x3 = fom_beta_quotient_pfp_hmixed_graphright_entry * S ((S (fom_index_pfp_hmixed_graphright)) * x4) + (fom_value_pfp_hmixed_graphright))) /\\ (exists fom_gap_pfp_hmixed_graphright_value_bound. fom_gap_pfp_hmixed_graphright_value_bound + S (fom_value_pfp_hmixed_graphright) = p))) /\\ (((((((x8)=0 \\/ (x5)=0) /\\ (((x18)=0)))) \\/ (((~((x8)=0)) /\\ (((~((x5)=0)) /\\ (((x8)+(x5)=S (x18)))))))) /\\ ((forall pfc_index_hmixed_graphcoefficients. (exists pfa_gap_hmixed_graphcoefficientsbound. pfa_gap_hmixed_graphcoefficientsbound + S (pfc_index_hmixed_graphcoefficients) = (x18)) -> exists pfc_value_hmixed_graphcoefficients. ((((exists ff_h_pfp_hmixed_graphcoefficientsentry. ff_h_pfp_hmixed_graphcoefficientsentry + S (pfc_value_hmixed_graphcoefficients) = S ((S (pfc_index_hmixed_graphcoefficients)) * c)) /\\ exists ff_q_pfp_hmixed_graphcoefficientsentry. b = ff_q_pfp_hmixed_graphcoefficientsentry * S ((S (pfc_index_hmixed_graphcoefficients)) * c) + (pfc_value_hmixed_graphcoefficients))) /\\ ((exists pfc_terms_code_hmixed_graphcoefficientscoefficient pfc_terms_scale_hmixed_graphcoefficientscoefficient pfc_natural_sum_hmixed_graphcoefficientscoefficient. ((forall pfc_index_hmixed_graphcoefficientscoefficientdiagonal. (exists pfa_gap_hmixed_graphcoefficientscoefficientdiagonalbound. pfa_gap_hmixed_graphcoefficientscoefficientdiagonalbound + S (pfc_index_hmixed_graphcoefficientscoefficientdiagonal) = (S (pfc_index_hmixed_graphcoefficients))) -> exists pfc_value_hmixed_graphcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_hmixed_graphcoefficientscoefficientdiagonalentry. ff_h_pfp_hmixed_graphcoefficientscoefficientdiagonalentry + S (pfc_value_hmixed_graphcoefficientscoefficientdiagonal) = S ((S (pfc_index_hmixed_graphcoefficientscoefficientdiagonal)) * pfc_terms_scale_hmixed_graphcoefficientscoefficient)) /\\ exists ff_q_pfp_hmixed_graphcoefficientscoefficientdiagonalentry. pfc_terms_code_hmixed_graphcoefficientscoefficient = ff_q_pfp_hmixed_graphcoefficientscoefficientdiagonalentry * S ((S (pfc_index_hmixed_graphcoefficientscoefficientdiagonal)) * pfc_terms_scale_hmixed_graphcoefficientscoefficient) + (pfc_value_hmixed_graphcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_hmixed_graphcoefficientscoefficientdiagonalterm pfc_left_hmixed_graphcoefficientscoefficientdiagonalterm pfc_right_hmixed_graphcoefficientscoefficientdiagonalterm. (((pfc_index_hmixed_graphcoefficientscoefficientdiagonal)+pfc_complement_hmixed_graphcoefficientscoefficientdiagonalterm=(pfc_index_hmixed_graphcoefficients)) /\\ ((((((exists pfa_gap_hmixed_graphcoefficientscoefficientdiagonaltermleftinside. pfa_gap_hmixed_graphcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_hmixed_graphcoefficientscoefficientdiagonal) = (x8)) /\\ ((((exists ff_h_pfp_hmixed_graphcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_hmixed_graphcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_hmixed_graphcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_hmixed_graphcoefficientscoefficientdiagonal)) * x7)) /\\ exists ff_q_pfp_hmixed_graphcoefficientscoefficientdiagonaltermleftentry. x6 = ff_q_pfp_hmixed_graphcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_hmixed_graphcoefficientscoefficientdiagonal)) * x7) + (pfc_left_hmixed_graphcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hmixed_graphcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_hmixed_graphcoefficientscoefficientdiagonaltermleftoutside+(x8)=(pfc_index_hmixed_graphcoefficientscoefficientdiagonal)) /\\ (((pfc_left_hmixed_graphcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_hmixed_graphcoefficientscoefficientdiagonaltermrightinside. pfa_gap_hmixed_graphcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_hmixed_graphcoefficientscoefficientdiagonalterm) = (x5)) /\\ ((((exists ff_h_pfp_hmixed_graphcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_hmixed_graphcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_hmixed_graphcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_hmixed_graphcoefficientscoefficientdiagonalterm)) * x4)) /\\ exists ff_q_pfp_hmixed_graphcoefficientscoefficientdiagonaltermrightentry. x3 = ff_q_pfp_hmixed_graphcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_hmixed_graphcoefficientscoefficientdiagonalterm)) * x4) + (pfc_right_hmixed_graphcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hmixed_graphcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_hmixed_graphcoefficientscoefficientdiagonaltermrightoutside+(x5)=(pfc_complement_hmixed_graphcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_hmixed_graphcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_hmixed_graphcoefficientscoefficientdiagonal)=pfc_left_hmixed_graphcoefficientscoefficientdiagonalterm*pfc_right_hmixed_graphcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_hmixed_graphcoefficientscoefficientsum fs_v_pfc_hmixed_graphcoefficientscoefficientsum. ((((exists fs_h_pfc_hmixed_graphcoefficientscoefficientsum_body_start. fs_h_pfc_hmixed_graphcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_hmixed_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hmixed_graphcoefficientscoefficientsum_body_start. fs_u_pfc_hmixed_graphcoefficientscoefficientsum = fs_q_pfc_hmixed_graphcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_hmixed_graphcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_hmixed_graphcoefficientscoefficientsum_body_terminal. fs_h_pfc_hmixed_graphcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_hmixed_graphcoefficientscoefficient) = S ((S (S (pfc_index_hmixed_graphcoefficients))) * fs_v_pfc_hmixed_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hmixed_graphcoefficientscoefficientsum_body_terminal. fs_u_pfc_hmixed_graphcoefficientscoefficientsum = fs_q_pfc_hmixed_graphcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_hmixed_graphcoefficients))) * fs_v_pfc_hmixed_graphcoefficientscoefficientsum) + (pfc_natural_sum_hmixed_graphcoefficientscoefficient))) /\\ forall fs_i_pfc_hmixed_graphcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_hmixed_graphcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_hmixed_graphcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_hmixed_graphcoefficientscoefficientsum_body_steps = S (pfc_index_hmixed_graphcoefficients)) -> exists fs_a_pfc_hmixed_graphcoefficientscoefficientsum_body_steps fs_r_pfc_hmixed_graphcoefficientscoefficientsum_body_steps fs_s_pfc_hmixed_graphcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_hmixed_graphcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_hmixed_graphcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_hmixed_graphcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hmixed_graphcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hmixed_graphcoefficientscoefficient)) /\\ exists fs_q_pfc_hmixed_graphcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_hmixed_graphcoefficientscoefficient = fs_q_pfc_hmixed_graphcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_hmixed_graphcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hmixed_graphcoefficientscoefficient) + (fs_a_pfc_hmixed_graphcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hmixed_graphcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_hmixed_graphcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_hmixed_graphcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hmixed_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hmixed_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hmixed_graphcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_hmixed_graphcoefficientscoefficientsum = fs_q_pfc_hmixed_graphcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_hmixed_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hmixed_graphcoefficientscoefficientsum) + (fs_r_pfc_hmixed_graphcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hmixed_graphcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_hmixed_graphcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_hmixed_graphcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_hmixed_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hmixed_graphcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hmixed_graphcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_hmixed_graphcoefficientscoefficientsum = fs_q_pfc_hmixed_graphcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_hmixed_graphcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hmixed_graphcoefficientscoefficientsum) + (fs_s_pfc_hmixed_graphcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_hmixed_graphcoefficientscoefficientsum_body_steps = fs_r_pfc_hmixed_graphcoefficientscoefficientsum_body_steps + fs_a_pfc_hmixed_graphcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_hmixed_graphcoefficientscoefficientresiduebound. pfa_gap_hmixed_graphcoefficientscoefficientresiduebound + S (pfc_value_hmixed_graphcoefficients) = (p)) /\\ ((exists pfa_offset_left_hmixed_graphcoefficientscoefficientresiduecongruence pfa_offset_right_hmixed_graphcoefficientscoefficientresiduecongruence. (pfc_natural_sum_hmixed_graphcoefficientscoefficient) + (p) * pfa_offset_left_hmixed_graphcoefficientscoefficientresiduecongruence = (pfc_value_hmixed_graphcoefficients) + (p) * pfa_offset_right_hmixed_graphcoefficientscoefficientresiduecongruence)))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (x6)", "specialize prime_field_polynomial_convolution_at_length_exists (x7)", "specialize prime_field_polynomial_convolution_at_length_exists (x8)", "specialize prime_field_polynomial_convolution_at_length_exists (x3)", "specialize prime_field_polynomial_convolution_at_length_exists (x4)", "specialize prime_field_polynomial_convolution_at_length_exists (x5)", "specialize prime_field_polynomial_convolution_at_length_exists (x18)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hsecond_left", "exact hPbound", "exact hmixed_length_witness", "cases hmixed_product", "cases hmixed_product_witness", "have htarget_equivalent : forall pfrep_power_right_transitive_target pfrep_left_right_transitive_target pfrep_right_right_transitive_target. ((exists pfrep_position_right_transitive_targetfirst. ((pfrep_position_right_transitive_targetfirst+S (pfrep_power_right_transitive_target)=(x18)) /\\ ((((exists ff_h_pfp_right_transitive_targetfirstentry. ff_h_pfp_right_transitive_targetfirstentry + S (pfrep_left_right_transitive_target) = S ((S (pfrep_position_right_transitive_targetfirst)) * x20)) /\\ exists ff_q_pfp_right_transitive_targetfirstentry. x19 = ff_q_pfp_right_transitive_targetfirstentry * S ((S (pfrep_position_right_transitive_targetfirst)) * x20) + (pfrep_left_right_transitive_target)))))) \\/ (((exists pfrep_gap_right_transitive_targetfirstoutside. pfrep_gap_right_transitive_targetfirstoutside+(x18)=(pfrep_power_right_transitive_target)) /\\ (((pfrep_left_right_transitive_target)=0))))) -> ((exists pfrep_position_right_transitive_targetsecond. ((pfrep_position_right_transitive_targetsecond+S (pfrep_power_right_transitive_target)=(M)) /\\ ((((exists ff_h_pfp_right_transitive_targetsecondentry. ff_h_pfp_right_transitive_targetsecondentry + S (pfrep_right_right_transitive_target) = S ((S (pfrep_position_right_transitive_targetsecond)) * bc)) /\\ exists ff_q_pfp_right_transitive_targetsecondentry. bb = ff_q_pfp_right_transitive_targetsecondentry * S ((S (pfrep_position_right_transitive_targetsecond)) * bc) + (pfrep_right_right_transitive_target)))))) \\/ (((exists pfrep_gap_right_transitive_targetsecondoutside. pfrep_gap_right_transitive_targetsecondoutside+(M)=(pfrep_power_right_transitive_target)) /\\ (((pfrep_right_right_transitive_target)=0))))) -> pfrep_left_right_transitive_target=pfrep_right_right_transitive_target", "specialize prime_field_polynomial_equivalent_transitive (x19)", "specialize prime_field_polynomial_equivalent_transitive (x20)", "specialize prime_field_polynomial_equivalent_transitive (x18)", "specialize prime_field_polynomial_equivalent_transitive (x9)", "specialize prime_field_polynomial_equivalent_transitive (x10)", "specialize prime_field_polynomial_equivalent_transitive (x11)", "specialize prime_field_polynomial_equivalent_transitive (bb)", "specialize prime_field_polynomial_equivalent_transitive (bc)", "specialize prime_field_polynomial_equivalent_transitive (M)", "apply prime_field_polynomial_equivalent_transitive", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (p)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x6)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x7)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x8)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x3)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x4)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x5)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x19)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x20)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x18)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (ab)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (ac)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (L)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x9)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x10)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x11)", "apply prime_field_polynomial_convolution_equivalent_congruent_right", "exact hp0", "exact hDA_right_witness_witness_witness_witness_witness_witness_right", "exact hmixed_product_witness_witness", "exact hAB_right_witness_witness_witness_witness_witness_witness_left", "exact hAB_right_witness_witness_witness_witness_witness_witness_right", "specialize prime_field_polynomial_right_divides_from_product (p)", "specialize prime_field_polynomial_right_divides_from_product (db)", "specialize prime_field_polynomial_right_divides_from_product (dc)", "specialize prime_field_polynomial_right_divides_from_product (D)", "specialize prime_field_polynomial_right_divides_from_product (bb)", "specialize prime_field_polynomial_right_divides_from_product (bc)", "specialize prime_field_polynomial_right_divides_from_product (M)", "specialize prime_field_polynomial_right_divides_from_product (x13)", "specialize prime_field_polynomial_right_divides_from_product (x14)", "specialize prime_field_polynomial_right_divides_from_product (x12)", "specialize prime_field_polynomial_right_divides_from_product (x16)", "specialize prime_field_polynomial_right_divides_from_product (x17)", "specialize prime_field_polynomial_right_divides_from_product (x15)", "apply prime_field_polynomial_right_divides_from_product", "exact hAB_left", "exact hresult_product_witness_witness", "specialize prime_field_polynomial_equivalent_transitive (x16)", "specialize prime_field_polynomial_equivalent_transitive (x17)", "specialize prime_field_polynomial_equivalent_transitive (x15)", "specialize prime_field_polynomial_equivalent_transitive (x19)", "specialize prime_field_polynomial_equivalent_transitive (x20)", "specialize prime_field_polynomial_equivalent_transitive (x18)", "specialize prime_field_polynomial_equivalent_transitive (bb)", "specialize prime_field_polynomial_equivalent_transitive (bc)", "specialize prime_field_polynomial_equivalent_transitive (M)", "apply prime_field_polynomial_equivalent_transitive", "specialize prime_field_polynomial_convolution_associative_equivalent (p)", "specialize prime_field_polynomial_convolution_associative_equivalent (x6)", "specialize prime_field_polynomial_convolution_associative_equivalent (x7)", "specialize prime_field_polynomial_convolution_associative_equivalent (x8)", "specialize prime_field_polynomial_convolution_associative_equivalent (x)", "specialize prime_field_polynomial_convolution_associative_equivalent (x1)", "specialize prime_field_polynomial_convolution_associative_equivalent (x2)", "specialize prime_field_polynomial_convolution_associative_equivalent (x13)", "specialize prime_field_polynomial_convolution_associative_equivalent (x14)", "specialize prime_field_polynomial_convolution_associative_equivalent (x12)", "specialize prime_field_polynomial_convolution_associative_equivalent (db)", "specialize prime_field_polynomial_convolution_associative_equivalent (dc)", "specialize prime_field_polynomial_convolution_associative_equivalent (D)", "specialize prime_field_polynomial_convolution_associative_equivalent (x3)", "specialize prime_field_polynomial_convolution_associative_equivalent (x4)", "specialize prime_field_polynomial_convolution_associative_equivalent (x5)", "specialize prime_field_polynomial_convolution_associative_equivalent (x16)", "specialize prime_field_polynomial_convolution_associative_equivalent (x17)", "specialize prime_field_polynomial_convolution_associative_equivalent (x15)", "specialize prime_field_polynomial_convolution_associative_equivalent (x19)", "specialize prime_field_polynomial_convolution_associative_equivalent (x20)", "specialize prime_field_polynomial_convolution_associative_equivalent (x18)", "apply prime_field_polynomial_convolution_associative_equivalent", "exact hp", "exact hcomposite_product_witness_witness", "exact hDA_right_witness_witness_witness_witness_witness_witness_left", "exact hresult_product_witness_witness", "exact hmixed_product_witness_witness", "exact htarget_equivalent" ], "script_sha256": "5e0b80f4d1d01560067b69a0b833b1a06e449baf5404136d5645daed25a9d1ee", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_divisibility_candidate.py", "sha256": "f544adedd3ce963e4a773e8582efcb0f91ba7491207c9792d477d452e854f2b8" }, "statement": "forall p db dc D ab ac L bb bc M. (~((p) = 1) /\\ forall pfa_factor_left_right_transitive_prime pfa_factor_right_right_transitive_prime. (p) = pfa_factor_left_right_transitive_prime * pfa_factor_right_right_transitive_prime -> pfa_factor_left_right_transitive_prime = 1 \\/ pfa_factor_right_right_transitive_prime = 1) -> (((forall fom_index_pfp_right_transitive_first_canonical. (exists fom_gap_pfp_right_transitive_first_canonical_index_bound. fom_gap_pfp_right_transitive_first_canonical_index_bound + S (fom_index_pfp_right_transitive_first_canonical) = L) -> exists fom_value_pfp_right_transitive_first_canonical. ((((exists fom_beta_height_pfp_right_transitive_first_canonical_entry. fom_beta_height_pfp_right_transitive_first_canonical_entry + S (fom_value_pfp_right_transitive_first_canonical) = S ((S (fom_index_pfp_right_transitive_first_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_right_transitive_first_canonical_entry. ab = fom_beta_quotient_pfp_right_transitive_first_canonical_entry * S ((S (fom_index_pfp_right_transitive_first_canonical)) * ac) + (fom_value_pfp_right_transitive_first_canonical))) /\\ (exists fom_gap_pfp_right_transitive_first_canonical_value_bound. fom_gap_pfp_right_transitive_first_canonical_value_bound + S (fom_value_pfp_right_transitive_first_canonical) = p))) /\\ ((exists pfrd_qb_right_transitive_first pfrd_qc_right_transitive_first pfrd_qlen_right_transitive_first pfrd_pb_right_transitive_first pfrd_pc_right_transitive_first pfrd_plen_right_transitive_first. ((((forall fom_index_pfp_right_transitive_first_productleft. (exists fom_gap_pfp_right_transitive_first_productleft_index_bound. fom_gap_pfp_right_transitive_first_productleft_index_bound + S (fom_index_pfp_right_transitive_first_productleft) = pfrd_qlen_right_transitive_first) -> exists fom_value_pfp_right_transitive_first_productleft. ((((exists fom_beta_height_pfp_right_transitive_first_productleft_entry. fom_beta_height_pfp_right_transitive_first_productleft_entry + S (fom_value_pfp_right_transitive_first_productleft) = S ((S (fom_index_pfp_right_transitive_first_productleft)) * pfrd_qc_right_transitive_first)) /\\ exists fom_beta_quotient_pfp_right_transitive_first_productleft_entry. pfrd_qb_right_transitive_first = fom_beta_quotient_pfp_right_transitive_first_productleft_entry * S ((S (fom_index_pfp_right_transitive_first_productleft)) * pfrd_qc_right_transitive_first) + (fom_value_pfp_right_transitive_first_productleft))) /\\ (exists fom_gap_pfp_right_transitive_first_productleft_value_bound. fom_gap_pfp_right_transitive_first_productleft_value_bound + S (fom_value_pfp_right_transitive_first_productleft) = p))) /\\ (((forall fom_index_pfp_right_transitive_first_productright. (exists fom_gap_pfp_right_transitive_first_productright_index_bound. fom_gap_pfp_right_transitive_first_productright_index_bound + S (fom_index_pfp_right_transitive_first_productright) = D) -> exists fom_value_pfp_right_transitive_first_productright. ((((exists fom_beta_height_pfp_right_transitive_first_productright_entry. fom_beta_height_pfp_right_transitive_first_productright_entry + S (fom_value_pfp_right_transitive_first_productright) = S ((S (fom_index_pfp_right_transitive_first_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_right_transitive_first_productright_entry. db = fom_beta_quotient_pfp_right_transitive_first_productright_entry * S ((S (fom_index_pfp_right_transitive_first_productright)) * dc) + (fom_value_pfp_right_transitive_first_productright))) /\\ (exists fom_gap_pfp_right_transitive_first_productright_value_bound. fom_gap_pfp_right_transitive_first_productright_value_bound + S (fom_value_pfp_right_transitive_first_productright) = p))) /\\ (((((((pfrd_qlen_right_transitive_first)=0 \\/ (D)=0) /\\ (((pfrd_plen_right_transitive_first)=0)))) \\/ (((~((pfrd_qlen_right_transitive_first)=0)) /\\ (((~((D)=0)) /\\ (((pfrd_qlen_right_transitive_first)+(D)=S (pfrd_plen_right_transitive_first)))))))) /\\ ((forall pfc_index_right_transitive_first_productcoefficients. (exists pfa_gap_right_transitive_first_productcoefficientsbound. pfa_gap_right_transitive_first_productcoefficientsbound + S (pfc_index_right_transitive_first_productcoefficients) = (pfrd_plen_right_transitive_first)) -> exists pfc_value_right_transitive_first_productcoefficients. ((((exists ff_h_pfp_right_transitive_first_productcoefficientsentry. ff_h_pfp_right_transitive_first_productcoefficientsentry + S (pfc_value_right_transitive_first_productcoefficients) = S ((S (pfc_index_right_transitive_first_productcoefficients)) * pfrd_pc_right_transitive_first)) /\\ exists ff_q_pfp_right_transitive_first_productcoefficientsentry. pfrd_pb_right_transitive_first = ff_q_pfp_right_transitive_first_productcoefficientsentry * S ((S (pfc_index_right_transitive_first_productcoefficients)) * pfrd_pc_right_transitive_first) + (pfc_value_right_transitive_first_productcoefficients))) /\\ ((exists pfc_terms_code_right_transitive_first_productcoefficientscoefficient pfc_terms_scale_right_transitive_first_productcoefficientscoefficient pfc_natural_sum_right_transitive_first_productcoefficientscoefficient. ((forall pfc_index_right_transitive_first_productcoefficientscoefficientdiagonal. (exists pfa_gap_right_transitive_first_productcoefficientscoefficientdiagonalbound. pfa_gap_right_transitive_first_productcoefficientscoefficientdiagonalbound + S (pfc_index_right_transitive_first_productcoefficientscoefficientdiagonal) = (S (pfc_index_right_transitive_first_productcoefficients))) -> exists pfc_value_right_transitive_first_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_right_transitive_first_productcoefficientscoefficientdiagonalentry. ff_h_pfp_right_transitive_first_productcoefficientscoefficientdiagonalentry + S (pfc_value_right_transitive_first_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_right_transitive_first_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_transitive_first_productcoefficientscoefficient)) /\\ exists ff_q_pfp_right_transitive_first_productcoefficientscoefficientdiagonalentry. pfc_terms_code_right_transitive_first_productcoefficientscoefficient = ff_q_pfp_right_transitive_first_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_right_transitive_first_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_transitive_first_productcoefficientscoefficient) + (pfc_value_right_transitive_first_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_right_transitive_first_productcoefficientscoefficientdiagonalterm pfc_left_right_transitive_first_productcoefficientscoefficientdiagonalterm pfc_right_right_transitive_first_productcoefficientscoefficientdiagonalterm. (((pfc_index_right_transitive_first_productcoefficientscoefficientdiagonal)+pfc_complement_right_transitive_first_productcoefficientscoefficientdiagonalterm=(pfc_index_right_transitive_first_productcoefficients)) /\\ ((((((exists pfa_gap_right_transitive_first_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_right_transitive_first_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_right_transitive_first_productcoefficientscoefficientdiagonal) = (pfrd_qlen_right_transitive_first)) /\\ ((((exists ff_h_pfp_right_transitive_first_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_right_transitive_first_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_right_transitive_first_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_right_transitive_first_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_transitive_first)) /\\ exists ff_q_pfp_right_transitive_first_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_right_transitive_first = ff_q_pfp_right_transitive_first_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_right_transitive_first_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_transitive_first) + (pfc_left_right_transitive_first_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_transitive_first_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_right_transitive_first_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_right_transitive_first)=(pfc_index_right_transitive_first_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_right_transitive_first_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_right_transitive_first_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_right_transitive_first_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_right_transitive_first_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_right_transitive_first_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_right_transitive_first_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_right_transitive_first_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_right_transitive_first_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_right_transitive_first_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_right_transitive_first_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_right_transitive_first_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_right_transitive_first_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_transitive_first_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_right_transitive_first_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_right_transitive_first_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_right_transitive_first_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_right_transitive_first_productcoefficientscoefficientdiagonal)=pfc_left_right_transitive_first_productcoefficientscoefficientdiagonalterm*pfc_right_right_transitive_first_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_right_transitive_first_productcoefficientscoefficientsum fs_v_pfc_right_transitive_first_productcoefficientscoefficientsum. ((((exists fs_h_pfc_right_transitive_first_productcoefficientscoefficientsum_body_start. fs_h_pfc_right_transitive_first_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_right_transitive_first_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_first_productcoefficientscoefficientsum_body_start. fs_u_pfc_right_transitive_first_productcoefficientscoefficientsum = fs_q_pfc_right_transitive_first_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_right_transitive_first_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_right_transitive_first_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_right_transitive_first_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_right_transitive_first_productcoefficientscoefficient) = S ((S (S (pfc_index_right_transitive_first_productcoefficients))) * fs_v_pfc_right_transitive_first_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_first_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_right_transitive_first_productcoefficientscoefficientsum = fs_q_pfc_right_transitive_first_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_right_transitive_first_productcoefficients))) * fs_v_pfc_right_transitive_first_productcoefficientscoefficientsum) + (pfc_natural_sum_right_transitive_first_productcoefficientscoefficient))) /\\ forall fs_i_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps = S (pfc_index_right_transitive_first_productcoefficients)) -> exists fs_a_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps fs_r_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps fs_s_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_transitive_first_productcoefficientscoefficient)) /\\ exists fs_q_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_right_transitive_first_productcoefficientscoefficient = fs_q_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_transitive_first_productcoefficientscoefficient) + (fs_a_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_first_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_right_transitive_first_productcoefficientscoefficientsum = fs_q_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_first_productcoefficientscoefficientsum) + (fs_r_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_first_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_right_transitive_first_productcoefficientscoefficientsum = fs_q_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_first_productcoefficientscoefficientsum) + (fs_s_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps = fs_r_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps + fs_a_pfc_right_transitive_first_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_right_transitive_first_productcoefficientscoefficientresiduebound. pfa_gap_right_transitive_first_productcoefficientscoefficientresiduebound + S (pfc_value_right_transitive_first_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_right_transitive_first_productcoefficientscoefficientresiduecongruence pfa_offset_right_right_transitive_first_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_right_transitive_first_productcoefficientscoefficient) + (p) * pfa_offset_left_right_transitive_first_productcoefficientscoefficientresiduecongruence = (pfc_value_right_transitive_first_productcoefficients) + (p) * pfa_offset_right_right_transitive_first_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_right_transitive_first_target pfrep_left_right_transitive_first_target pfrep_right_right_transitive_first_target. ((exists pfrep_position_right_transitive_first_targetfirst. ((pfrep_position_right_transitive_first_targetfirst+S (pfrep_power_right_transitive_first_target)=(pfrd_plen_right_transitive_first)) /\\ ((((exists ff_h_pfp_right_transitive_first_targetfirstentry. ff_h_pfp_right_transitive_first_targetfirstentry + S (pfrep_left_right_transitive_first_target) = S ((S (pfrep_position_right_transitive_first_targetfirst)) * pfrd_pc_right_transitive_first)) /\\ exists ff_q_pfp_right_transitive_first_targetfirstentry. pfrd_pb_right_transitive_first = ff_q_pfp_right_transitive_first_targetfirstentry * S ((S (pfrep_position_right_transitive_first_targetfirst)) * pfrd_pc_right_transitive_first) + (pfrep_left_right_transitive_first_target)))))) \\/ (((exists pfrep_gap_right_transitive_first_targetfirstoutside. pfrep_gap_right_transitive_first_targetfirstoutside+(pfrd_plen_right_transitive_first)=(pfrep_power_right_transitive_first_target)) /\\ (((pfrep_left_right_transitive_first_target)=0))))) -> ((exists pfrep_position_right_transitive_first_targetsecond. ((pfrep_position_right_transitive_first_targetsecond+S (pfrep_power_right_transitive_first_target)=(L)) /\\ ((((exists ff_h_pfp_right_transitive_first_targetsecondentry. ff_h_pfp_right_transitive_first_targetsecondentry + S (pfrep_right_right_transitive_first_target) = S ((S (pfrep_position_right_transitive_first_targetsecond)) * ac)) /\\ exists ff_q_pfp_right_transitive_first_targetsecondentry. ab = ff_q_pfp_right_transitive_first_targetsecondentry * S ((S (pfrep_position_right_transitive_first_targetsecond)) * ac) + (pfrep_right_right_transitive_first_target)))))) \\/ (((exists pfrep_gap_right_transitive_first_targetsecondoutside. pfrep_gap_right_transitive_first_targetsecondoutside+(L)=(pfrep_power_right_transitive_first_target)) /\\ (((pfrep_right_right_transitive_first_target)=0))))) -> pfrep_left_right_transitive_first_target=pfrep_right_right_transitive_first_target))))))) -> (((forall fom_index_pfp_right_transitive_second_canonical. (exists fom_gap_pfp_right_transitive_second_canonical_index_bound. fom_gap_pfp_right_transitive_second_canonical_index_bound + S (fom_index_pfp_right_transitive_second_canonical) = M) -> exists fom_value_pfp_right_transitive_second_canonical. ((((exists fom_beta_height_pfp_right_transitive_second_canonical_entry. fom_beta_height_pfp_right_transitive_second_canonical_entry + S (fom_value_pfp_right_transitive_second_canonical) = S ((S (fom_index_pfp_right_transitive_second_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_right_transitive_second_canonical_entry. bb = fom_beta_quotient_pfp_right_transitive_second_canonical_entry * S ((S (fom_index_pfp_right_transitive_second_canonical)) * bc) + (fom_value_pfp_right_transitive_second_canonical))) /\\ (exists fom_gap_pfp_right_transitive_second_canonical_value_bound. fom_gap_pfp_right_transitive_second_canonical_value_bound + S (fom_value_pfp_right_transitive_second_canonical) = p))) /\\ ((exists pfrd_qb_right_transitive_second pfrd_qc_right_transitive_second pfrd_qlen_right_transitive_second pfrd_pb_right_transitive_second pfrd_pc_right_transitive_second pfrd_plen_right_transitive_second. ((((forall fom_index_pfp_right_transitive_second_productleft. (exists fom_gap_pfp_right_transitive_second_productleft_index_bound. fom_gap_pfp_right_transitive_second_productleft_index_bound + S (fom_index_pfp_right_transitive_second_productleft) = pfrd_qlen_right_transitive_second) -> exists fom_value_pfp_right_transitive_second_productleft. ((((exists fom_beta_height_pfp_right_transitive_second_productleft_entry. fom_beta_height_pfp_right_transitive_second_productleft_entry + S (fom_value_pfp_right_transitive_second_productleft) = S ((S (fom_index_pfp_right_transitive_second_productleft)) * pfrd_qc_right_transitive_second)) /\\ exists fom_beta_quotient_pfp_right_transitive_second_productleft_entry. pfrd_qb_right_transitive_second = fom_beta_quotient_pfp_right_transitive_second_productleft_entry * S ((S (fom_index_pfp_right_transitive_second_productleft)) * pfrd_qc_right_transitive_second) + (fom_value_pfp_right_transitive_second_productleft))) /\\ (exists fom_gap_pfp_right_transitive_second_productleft_value_bound. fom_gap_pfp_right_transitive_second_productleft_value_bound + S (fom_value_pfp_right_transitive_second_productleft) = p))) /\\ (((forall fom_index_pfp_right_transitive_second_productright. (exists fom_gap_pfp_right_transitive_second_productright_index_bound. fom_gap_pfp_right_transitive_second_productright_index_bound + S (fom_index_pfp_right_transitive_second_productright) = L) -> exists fom_value_pfp_right_transitive_second_productright. ((((exists fom_beta_height_pfp_right_transitive_second_productright_entry. fom_beta_height_pfp_right_transitive_second_productright_entry + S (fom_value_pfp_right_transitive_second_productright) = S ((S (fom_index_pfp_right_transitive_second_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_right_transitive_second_productright_entry. ab = fom_beta_quotient_pfp_right_transitive_second_productright_entry * S ((S (fom_index_pfp_right_transitive_second_productright)) * ac) + (fom_value_pfp_right_transitive_second_productright))) /\\ (exists fom_gap_pfp_right_transitive_second_productright_value_bound. fom_gap_pfp_right_transitive_second_productright_value_bound + S (fom_value_pfp_right_transitive_second_productright) = p))) /\\ (((((((pfrd_qlen_right_transitive_second)=0 \\/ (L)=0) /\\ (((pfrd_plen_right_transitive_second)=0)))) \\/ (((~((pfrd_qlen_right_transitive_second)=0)) /\\ (((~((L)=0)) /\\ (((pfrd_qlen_right_transitive_second)+(L)=S (pfrd_plen_right_transitive_second)))))))) /\\ ((forall pfc_index_right_transitive_second_productcoefficients. (exists pfa_gap_right_transitive_second_productcoefficientsbound. pfa_gap_right_transitive_second_productcoefficientsbound + S (pfc_index_right_transitive_second_productcoefficients) = (pfrd_plen_right_transitive_second)) -> exists pfc_value_right_transitive_second_productcoefficients. ((((exists ff_h_pfp_right_transitive_second_productcoefficientsentry. ff_h_pfp_right_transitive_second_productcoefficientsentry + S (pfc_value_right_transitive_second_productcoefficients) = S ((S (pfc_index_right_transitive_second_productcoefficients)) * pfrd_pc_right_transitive_second)) /\\ exists ff_q_pfp_right_transitive_second_productcoefficientsentry. pfrd_pb_right_transitive_second = ff_q_pfp_right_transitive_second_productcoefficientsentry * S ((S (pfc_index_right_transitive_second_productcoefficients)) * pfrd_pc_right_transitive_second) + (pfc_value_right_transitive_second_productcoefficients))) /\\ ((exists pfc_terms_code_right_transitive_second_productcoefficientscoefficient pfc_terms_scale_right_transitive_second_productcoefficientscoefficient pfc_natural_sum_right_transitive_second_productcoefficientscoefficient. ((forall pfc_index_right_transitive_second_productcoefficientscoefficientdiagonal. (exists pfa_gap_right_transitive_second_productcoefficientscoefficientdiagonalbound. pfa_gap_right_transitive_second_productcoefficientscoefficientdiagonalbound + S (pfc_index_right_transitive_second_productcoefficientscoefficientdiagonal) = (S (pfc_index_right_transitive_second_productcoefficients))) -> exists pfc_value_right_transitive_second_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_right_transitive_second_productcoefficientscoefficientdiagonalentry. ff_h_pfp_right_transitive_second_productcoefficientscoefficientdiagonalentry + S (pfc_value_right_transitive_second_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_right_transitive_second_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_transitive_second_productcoefficientscoefficient)) /\\ exists ff_q_pfp_right_transitive_second_productcoefficientscoefficientdiagonalentry. pfc_terms_code_right_transitive_second_productcoefficientscoefficient = ff_q_pfp_right_transitive_second_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_right_transitive_second_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_transitive_second_productcoefficientscoefficient) + (pfc_value_right_transitive_second_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_right_transitive_second_productcoefficientscoefficientdiagonalterm pfc_left_right_transitive_second_productcoefficientscoefficientdiagonalterm pfc_right_right_transitive_second_productcoefficientscoefficientdiagonalterm. (((pfc_index_right_transitive_second_productcoefficientscoefficientdiagonal)+pfc_complement_right_transitive_second_productcoefficientscoefficientdiagonalterm=(pfc_index_right_transitive_second_productcoefficients)) /\\ ((((((exists pfa_gap_right_transitive_second_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_right_transitive_second_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_right_transitive_second_productcoefficientscoefficientdiagonal) = (pfrd_qlen_right_transitive_second)) /\\ ((((exists ff_h_pfp_right_transitive_second_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_right_transitive_second_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_right_transitive_second_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_right_transitive_second_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_transitive_second)) /\\ exists ff_q_pfp_right_transitive_second_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_right_transitive_second = ff_q_pfp_right_transitive_second_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_right_transitive_second_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_transitive_second) + (pfc_left_right_transitive_second_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_transitive_second_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_right_transitive_second_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_right_transitive_second)=(pfc_index_right_transitive_second_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_right_transitive_second_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_right_transitive_second_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_right_transitive_second_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_right_transitive_second_productcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_right_transitive_second_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_right_transitive_second_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_right_transitive_second_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_right_transitive_second_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_right_transitive_second_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_right_transitive_second_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_right_transitive_second_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_right_transitive_second_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_transitive_second_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_right_transitive_second_productcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_right_transitive_second_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_right_transitive_second_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_right_transitive_second_productcoefficientscoefficientdiagonal)=pfc_left_right_transitive_second_productcoefficientscoefficientdiagonalterm*pfc_right_right_transitive_second_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_right_transitive_second_productcoefficientscoefficientsum fs_v_pfc_right_transitive_second_productcoefficientscoefficientsum. ((((exists fs_h_pfc_right_transitive_second_productcoefficientscoefficientsum_body_start. fs_h_pfc_right_transitive_second_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_right_transitive_second_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_second_productcoefficientscoefficientsum_body_start. fs_u_pfc_right_transitive_second_productcoefficientscoefficientsum = fs_q_pfc_right_transitive_second_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_right_transitive_second_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_right_transitive_second_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_right_transitive_second_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_right_transitive_second_productcoefficientscoefficient) = S ((S (S (pfc_index_right_transitive_second_productcoefficients))) * fs_v_pfc_right_transitive_second_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_second_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_right_transitive_second_productcoefficientscoefficientsum = fs_q_pfc_right_transitive_second_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_right_transitive_second_productcoefficients))) * fs_v_pfc_right_transitive_second_productcoefficientscoefficientsum) + (pfc_natural_sum_right_transitive_second_productcoefficientscoefficient))) /\\ forall fs_i_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps = S (pfc_index_right_transitive_second_productcoefficients)) -> exists fs_a_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps fs_r_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps fs_s_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_transitive_second_productcoefficientscoefficient)) /\\ exists fs_q_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_right_transitive_second_productcoefficientscoefficient = fs_q_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_transitive_second_productcoefficientscoefficient) + (fs_a_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_second_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_right_transitive_second_productcoefficientscoefficientsum = fs_q_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_second_productcoefficientscoefficientsum) + (fs_r_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_second_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_right_transitive_second_productcoefficientscoefficientsum = fs_q_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_second_productcoefficientscoefficientsum) + (fs_s_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps = fs_r_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps + fs_a_pfc_right_transitive_second_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_right_transitive_second_productcoefficientscoefficientresiduebound. pfa_gap_right_transitive_second_productcoefficientscoefficientresiduebound + S (pfc_value_right_transitive_second_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_right_transitive_second_productcoefficientscoefficientresiduecongruence pfa_offset_right_right_transitive_second_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_right_transitive_second_productcoefficientscoefficient) + (p) * pfa_offset_left_right_transitive_second_productcoefficientscoefficientresiduecongruence = (pfc_value_right_transitive_second_productcoefficients) + (p) * pfa_offset_right_right_transitive_second_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_right_transitive_second_target pfrep_left_right_transitive_second_target pfrep_right_right_transitive_second_target. ((exists pfrep_position_right_transitive_second_targetfirst. ((pfrep_position_right_transitive_second_targetfirst+S (pfrep_power_right_transitive_second_target)=(pfrd_plen_right_transitive_second)) /\\ ((((exists ff_h_pfp_right_transitive_second_targetfirstentry. ff_h_pfp_right_transitive_second_targetfirstentry + S (pfrep_left_right_transitive_second_target) = S ((S (pfrep_position_right_transitive_second_targetfirst)) * pfrd_pc_right_transitive_second)) /\\ exists ff_q_pfp_right_transitive_second_targetfirstentry. pfrd_pb_right_transitive_second = ff_q_pfp_right_transitive_second_targetfirstentry * S ((S (pfrep_position_right_transitive_second_targetfirst)) * pfrd_pc_right_transitive_second) + (pfrep_left_right_transitive_second_target)))))) \\/ (((exists pfrep_gap_right_transitive_second_targetfirstoutside. pfrep_gap_right_transitive_second_targetfirstoutside+(pfrd_plen_right_transitive_second)=(pfrep_power_right_transitive_second_target)) /\\ (((pfrep_left_right_transitive_second_target)=0))))) -> ((exists pfrep_position_right_transitive_second_targetsecond. ((pfrep_position_right_transitive_second_targetsecond+S (pfrep_power_right_transitive_second_target)=(M)) /\\ ((((exists ff_h_pfp_right_transitive_second_targetsecondentry. ff_h_pfp_right_transitive_second_targetsecondentry + S (pfrep_right_right_transitive_second_target) = S ((S (pfrep_position_right_transitive_second_targetsecond)) * bc)) /\\ exists ff_q_pfp_right_transitive_second_targetsecondentry. bb = ff_q_pfp_right_transitive_second_targetsecondentry * S ((S (pfrep_position_right_transitive_second_targetsecond)) * bc) + (pfrep_right_right_transitive_second_target)))))) \\/ (((exists pfrep_gap_right_transitive_second_targetsecondoutside. pfrep_gap_right_transitive_second_targetsecondoutside+(M)=(pfrep_power_right_transitive_second_target)) /\\ (((pfrep_right_right_transitive_second_target)=0))))) -> pfrep_left_right_transitive_second_target=pfrep_right_right_transitive_second_target))))))) -> (((forall fom_index_pfp_right_transitive_result_canonical. (exists fom_gap_pfp_right_transitive_result_canonical_index_bound. fom_gap_pfp_right_transitive_result_canonical_index_bound + S (fom_index_pfp_right_transitive_result_canonical) = M) -> exists fom_value_pfp_right_transitive_result_canonical. ((((exists fom_beta_height_pfp_right_transitive_result_canonical_entry. fom_beta_height_pfp_right_transitive_result_canonical_entry + S (fom_value_pfp_right_transitive_result_canonical) = S ((S (fom_index_pfp_right_transitive_result_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_right_transitive_result_canonical_entry. bb = fom_beta_quotient_pfp_right_transitive_result_canonical_entry * S ((S (fom_index_pfp_right_transitive_result_canonical)) * bc) + (fom_value_pfp_right_transitive_result_canonical))) /\\ (exists fom_gap_pfp_right_transitive_result_canonical_value_bound. fom_gap_pfp_right_transitive_result_canonical_value_bound + S (fom_value_pfp_right_transitive_result_canonical) = p))) /\\ ((exists pfrd_qb_right_transitive_result pfrd_qc_right_transitive_result pfrd_qlen_right_transitive_result pfrd_pb_right_transitive_result pfrd_pc_right_transitive_result pfrd_plen_right_transitive_result. ((((forall fom_index_pfp_right_transitive_result_productleft. (exists fom_gap_pfp_right_transitive_result_productleft_index_bound. fom_gap_pfp_right_transitive_result_productleft_index_bound + S (fom_index_pfp_right_transitive_result_productleft) = pfrd_qlen_right_transitive_result) -> exists fom_value_pfp_right_transitive_result_productleft. ((((exists fom_beta_height_pfp_right_transitive_result_productleft_entry. fom_beta_height_pfp_right_transitive_result_productleft_entry + S (fom_value_pfp_right_transitive_result_productleft) = S ((S (fom_index_pfp_right_transitive_result_productleft)) * pfrd_qc_right_transitive_result)) /\\ exists fom_beta_quotient_pfp_right_transitive_result_productleft_entry. pfrd_qb_right_transitive_result = fom_beta_quotient_pfp_right_transitive_result_productleft_entry * S ((S (fom_index_pfp_right_transitive_result_productleft)) * pfrd_qc_right_transitive_result) + (fom_value_pfp_right_transitive_result_productleft))) /\\ (exists fom_gap_pfp_right_transitive_result_productleft_value_bound. fom_gap_pfp_right_transitive_result_productleft_value_bound + S (fom_value_pfp_right_transitive_result_productleft) = p))) /\\ (((forall fom_index_pfp_right_transitive_result_productright. (exists fom_gap_pfp_right_transitive_result_productright_index_bound. fom_gap_pfp_right_transitive_result_productright_index_bound + S (fom_index_pfp_right_transitive_result_productright) = D) -> exists fom_value_pfp_right_transitive_result_productright. ((((exists fom_beta_height_pfp_right_transitive_result_productright_entry. fom_beta_height_pfp_right_transitive_result_productright_entry + S (fom_value_pfp_right_transitive_result_productright) = S ((S (fom_index_pfp_right_transitive_result_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_right_transitive_result_productright_entry. db = fom_beta_quotient_pfp_right_transitive_result_productright_entry * S ((S (fom_index_pfp_right_transitive_result_productright)) * dc) + (fom_value_pfp_right_transitive_result_productright))) /\\ (exists fom_gap_pfp_right_transitive_result_productright_value_bound. fom_gap_pfp_right_transitive_result_productright_value_bound + S (fom_value_pfp_right_transitive_result_productright) = p))) /\\ (((((((pfrd_qlen_right_transitive_result)=0 \\/ (D)=0) /\\ (((pfrd_plen_right_transitive_result)=0)))) \\/ (((~((pfrd_qlen_right_transitive_result)=0)) /\\ (((~((D)=0)) /\\ (((pfrd_qlen_right_transitive_result)+(D)=S (pfrd_plen_right_transitive_result)))))))) /\\ ((forall pfc_index_right_transitive_result_productcoefficients. (exists pfa_gap_right_transitive_result_productcoefficientsbound. pfa_gap_right_transitive_result_productcoefficientsbound + S (pfc_index_right_transitive_result_productcoefficients) = (pfrd_plen_right_transitive_result)) -> exists pfc_value_right_transitive_result_productcoefficients. ((((exists ff_h_pfp_right_transitive_result_productcoefficientsentry. ff_h_pfp_right_transitive_result_productcoefficientsentry + S (pfc_value_right_transitive_result_productcoefficients) = S ((S (pfc_index_right_transitive_result_productcoefficients)) * pfrd_pc_right_transitive_result)) /\\ exists ff_q_pfp_right_transitive_result_productcoefficientsentry. pfrd_pb_right_transitive_result = ff_q_pfp_right_transitive_result_productcoefficientsentry * S ((S (pfc_index_right_transitive_result_productcoefficients)) * pfrd_pc_right_transitive_result) + (pfc_value_right_transitive_result_productcoefficients))) /\\ ((exists pfc_terms_code_right_transitive_result_productcoefficientscoefficient pfc_terms_scale_right_transitive_result_productcoefficientscoefficient pfc_natural_sum_right_transitive_result_productcoefficientscoefficient. ((forall pfc_index_right_transitive_result_productcoefficientscoefficientdiagonal. (exists pfa_gap_right_transitive_result_productcoefficientscoefficientdiagonalbound. pfa_gap_right_transitive_result_productcoefficientscoefficientdiagonalbound + S (pfc_index_right_transitive_result_productcoefficientscoefficientdiagonal) = (S (pfc_index_right_transitive_result_productcoefficients))) -> exists pfc_value_right_transitive_result_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_right_transitive_result_productcoefficientscoefficientdiagonalentry. ff_h_pfp_right_transitive_result_productcoefficientscoefficientdiagonalentry + S (pfc_value_right_transitive_result_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_right_transitive_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_transitive_result_productcoefficientscoefficient)) /\\ exists ff_q_pfp_right_transitive_result_productcoefficientscoefficientdiagonalentry. pfc_terms_code_right_transitive_result_productcoefficientscoefficient = ff_q_pfp_right_transitive_result_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_right_transitive_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_right_transitive_result_productcoefficientscoefficient) + (pfc_value_right_transitive_result_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_right_transitive_result_productcoefficientscoefficientdiagonalterm pfc_left_right_transitive_result_productcoefficientscoefficientdiagonalterm pfc_right_right_transitive_result_productcoefficientscoefficientdiagonalterm. (((pfc_index_right_transitive_result_productcoefficientscoefficientdiagonal)+pfc_complement_right_transitive_result_productcoefficientscoefficientdiagonalterm=(pfc_index_right_transitive_result_productcoefficients)) /\\ ((((((exists pfa_gap_right_transitive_result_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_right_transitive_result_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_right_transitive_result_productcoefficientscoefficientdiagonal) = (pfrd_qlen_right_transitive_result)) /\\ ((((exists ff_h_pfp_right_transitive_result_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_right_transitive_result_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_right_transitive_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_right_transitive_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_transitive_result)) /\\ exists ff_q_pfp_right_transitive_result_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_right_transitive_result = ff_q_pfp_right_transitive_result_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_right_transitive_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_right_transitive_result) + (pfc_left_right_transitive_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_transitive_result_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_right_transitive_result_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_right_transitive_result)=(pfc_index_right_transitive_result_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_right_transitive_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_right_transitive_result_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_right_transitive_result_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_right_transitive_result_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_right_transitive_result_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_right_transitive_result_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_right_transitive_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_right_transitive_result_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_right_transitive_result_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_right_transitive_result_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_right_transitive_result_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_right_transitive_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_right_transitive_result_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_right_transitive_result_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_right_transitive_result_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_right_transitive_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_right_transitive_result_productcoefficientscoefficientdiagonal)=pfc_left_right_transitive_result_productcoefficientscoefficientdiagonalterm*pfc_right_right_transitive_result_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_right_transitive_result_productcoefficientscoefficientsum fs_v_pfc_right_transitive_result_productcoefficientscoefficientsum. ((((exists fs_h_pfc_right_transitive_result_productcoefficientscoefficientsum_body_start. fs_h_pfc_right_transitive_result_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_right_transitive_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_result_productcoefficientscoefficientsum_body_start. fs_u_pfc_right_transitive_result_productcoefficientscoefficientsum = fs_q_pfc_right_transitive_result_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_right_transitive_result_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_right_transitive_result_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_right_transitive_result_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_right_transitive_result_productcoefficientscoefficient) = S ((S (S (pfc_index_right_transitive_result_productcoefficients))) * fs_v_pfc_right_transitive_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_result_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_right_transitive_result_productcoefficientscoefficientsum = fs_q_pfc_right_transitive_result_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_right_transitive_result_productcoefficients))) * fs_v_pfc_right_transitive_result_productcoefficientscoefficientsum) + (pfc_natural_sum_right_transitive_result_productcoefficientscoefficient))) /\\ forall fs_i_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps = S (pfc_index_right_transitive_result_productcoefficients)) -> exists fs_a_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps fs_r_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps fs_s_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_transitive_result_productcoefficientscoefficient)) /\\ exists fs_q_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_right_transitive_result_productcoefficientscoefficient = fs_q_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_right_transitive_result_productcoefficientscoefficient) + (fs_a_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_right_transitive_result_productcoefficientscoefficientsum = fs_q_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_result_productcoefficientscoefficientsum) + (fs_r_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_right_transitive_result_productcoefficientscoefficientsum = fs_q_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_right_transitive_result_productcoefficientscoefficientsum) + (fs_s_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps = fs_r_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps + fs_a_pfc_right_transitive_result_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_right_transitive_result_productcoefficientscoefficientresiduebound. pfa_gap_right_transitive_result_productcoefficientscoefficientresiduebound + S (pfc_value_right_transitive_result_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_right_transitive_result_productcoefficientscoefficientresiduecongruence pfa_offset_right_right_transitive_result_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_right_transitive_result_productcoefficientscoefficient) + (p) * pfa_offset_left_right_transitive_result_productcoefficientscoefficientresiduecongruence = (pfc_value_right_transitive_result_productcoefficients) + (p) * pfa_offset_right_right_transitive_result_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_right_transitive_result_target pfrep_left_right_transitive_result_target pfrep_right_right_transitive_result_target. ((exists pfrep_position_right_transitive_result_targetfirst. ((pfrep_position_right_transitive_result_targetfirst+S (pfrep_power_right_transitive_result_target)=(pfrd_plen_right_transitive_result)) /\\ ((((exists ff_h_pfp_right_transitive_result_targetfirstentry. ff_h_pfp_right_transitive_result_targetfirstentry + S (pfrep_left_right_transitive_result_target) = S ((S (pfrep_position_right_transitive_result_targetfirst)) * pfrd_pc_right_transitive_result)) /\\ exists ff_q_pfp_right_transitive_result_targetfirstentry. pfrd_pb_right_transitive_result = ff_q_pfp_right_transitive_result_targetfirstentry * S ((S (pfrep_position_right_transitive_result_targetfirst)) * pfrd_pc_right_transitive_result) + (pfrep_left_right_transitive_result_target)))))) \\/ (((exists pfrep_gap_right_transitive_result_targetfirstoutside. pfrep_gap_right_transitive_result_targetfirstoutside+(pfrd_plen_right_transitive_result)=(pfrep_power_right_transitive_result_target)) /\\ (((pfrep_left_right_transitive_result_target)=0))))) -> ((exists pfrep_position_right_transitive_result_targetsecond. ((pfrep_position_right_transitive_result_targetsecond+S (pfrep_power_right_transitive_result_target)=(M)) /\\ ((((exists ff_h_pfp_right_transitive_result_targetsecondentry. ff_h_pfp_right_transitive_result_targetsecondentry + S (pfrep_right_right_transitive_result_target) = S ((S (pfrep_position_right_transitive_result_targetsecond)) * bc)) /\\ exists ff_q_pfp_right_transitive_result_targetsecondentry. bb = ff_q_pfp_right_transitive_result_targetsecondentry * S ((S (pfrep_position_right_transitive_result_targetsecond)) * bc) + (pfrep_right_right_transitive_result_target)))))) \\/ (((exists pfrep_gap_right_transitive_result_targetsecondoutside. pfrep_gap_right_transitive_result_targetsecondoutside+(M)=(pfrep_power_right_transitive_result_target)) /\\ (((pfrep_right_right_transitive_result_target)=0))))) -> pfrep_left_right_transitive_result_target=pfrep_right_right_transitive_result_target)))))))", "statement_sha256": "581d2fad3dc745194f2297dad6cff729117d75f5b686c87dd8716374d0a98d10", "summary": "Actual Q1*D equivalent to A and Q2*A equivalent to B give the actual composite quotient Q2*Q1. Three genuine intermediate products, formal associativity and right-input congruence prove its product with D equivalent to B. No commutativity, fixed representation lengths or raw-code identities are assumed.", "summary_sha256": "f8491ba0c585c654eebb55ef36a99a21d639a2fc16d9e1dc995aeeb82e59d0ac" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "58b36f0d0bedb52917aea7672041702d95f0ea9645cab09548ea5ab006ad3760", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 316, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "dbb8debb4716b6bb9b246700f7e93865c8a6c1b12a3b65c0ffbb62206a890ba6", "test_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85" }, "body_checked": true, "body_receipt": { "command_count": 42, "dependency_count": 3, "dne_command_count": 0, "name": 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"catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "ceb7a2b8803caa9d936798059704a83c787eb42e96f9a7c3c505b99674999114", "membership": "alpha_only", "name": "polynomial_diagonal_left_unit_first_term", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro ub", "intro uc", "intro ab", "intro ac", "intro L", "intro i", "intro a", "intro t", "intro hu", "intro hi", "intro ha", "intro ht", "specialize polynomial_diagonal_term_functional (ub)", "specialize polynomial_diagonal_term_functional (uc)", "specialize polynomial_diagonal_term_functional (1)", "specialize polynomial_diagonal_term_functional (ab)", "specialize polynomial_diagonal_term_functional (ac)", "specialize polynomial_diagonal_term_functional (L)", "specialize polynomial_diagonal_term_functional (i)", "specialize polynomial_diagonal_term_functional (0)", "specialize polynomial_diagonal_term_functional (t)", "specialize polynomial_diagonal_term_functional (a)", "apply polynomial_diagonal_term_functional", "exact ht", "exists i", "exists 1", "exists a", "split", "apply zero_add", "split", "left", "split", "exists 0", "apply zero_add", "exact hu", "split", "left", "split", "exact hi", "exact ha", "symm", "apply one_mul" ], "script_sha256": "3b303f48ac2dd5f3509dd74ec948741d12616b9e30b3ceac618d6a8f68990dbe", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_left_unit_candidate.py", "sha256": "dbb8debb4716b6bb9b246700f7e93865c8a6c1b12a3b65c0ffbb62206a890ba6" }, "statement": "forall ub uc ab ac L i a t. (((exists ff_h_pfp_unit_first_entry. ff_h_pfp_unit_first_entry + S (1) = S ((S (0)) * uc)) /\\ exists ff_q_pfp_unit_first_entry. ub = ff_q_pfp_unit_first_entry * S ((S (0)) * uc) + (1))) -> (exists pfa_gap_unit_first_index. pfa_gap_unit_first_index + S (i) = (L)) -> (((exists ff_h_pfp_unit_first_A. ff_h_pfp_unit_first_A + S (a) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_unit_first_A. ab = ff_q_pfp_unit_first_A * S ((S (i)) * ac) + (a))) -> (exists pfc_complement_unit_first_actual pfc_left_unit_first_actual pfc_right_unit_first_actual. (((0)+pfc_complement_unit_first_actual=(i)) /\\ ((((((exists pfa_gap_unit_first_actualleftinside. pfa_gap_unit_first_actualleftinside + S (0) = (1)) /\\ ((((exists ff_h_pfp_unit_first_actualleftentry. ff_h_pfp_unit_first_actualleftentry + S (pfc_left_unit_first_actual) = S ((S (0)) * uc)) /\\ exists ff_q_pfp_unit_first_actualleftentry. ub = ff_q_pfp_unit_first_actualleftentry * S ((S (0)) * uc) + (pfc_left_unit_first_actual)))))) \\/ (((exists pfc_gap_unit_first_actualleftoutside. pfc_gap_unit_first_actualleftoutside+(1)=(0)) /\\ (((pfc_left_unit_first_actual)=0))))) /\\ ((((((exists pfa_gap_unit_first_actualrightinside. pfa_gap_unit_first_actualrightinside + S (pfc_complement_unit_first_actual) = (L)) /\\ ((((exists ff_h_pfp_unit_first_actualrightentry. ff_h_pfp_unit_first_actualrightentry + S (pfc_right_unit_first_actual) = S ((S (pfc_complement_unit_first_actual)) * ac)) /\\ exists ff_q_pfp_unit_first_actualrightentry. ab = ff_q_pfp_unit_first_actualrightentry * S ((S (pfc_complement_unit_first_actual)) * ac) + (pfc_right_unit_first_actual)))))) \\/ (((exists pfc_gap_unit_first_actualrightoutside. pfc_gap_unit_first_actualrightoutside+(L)=(pfc_complement_unit_first_actual)) /\\ (((pfc_right_unit_first_actual)=0))))) /\\ (((t)=pfc_left_unit_first_actual*pfc_right_unit_first_actual)))))))) -> (t=a)", "statement_sha256": "7a6f01d0db546e919bcae4e6e48a3206b3a00faa0c24d327c789ee55a1a9309c", "summary": "The first actual antidiagonal term of a length-one left unit is the chosen right-input coefficient, without any primality or commutativity assumption.", "summary_sha256": "a91482f26234164770a236b54c12cd26273452a01d21a8e89940f81bbd7f7c8b" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "5508b37613d93d27ce615b15fb175d4b87efb437df6926f3e1d735fb8c30fb8f", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 317, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", 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"cases ht_witness", "cases ht_witness_witness", "cases ht_witness_witness_witness", "cases ht_witness_witness_witness_right", "cases ht_witness_witness_witness_right_right", "have hz : x1=0", "specialize polynomial_zero_extended_entry_functional (ub)", "specialize polynomial_zero_extended_entry_functional (uc)", "specialize polynomial_zero_extended_entry_functional (1)", "specialize polynomial_zero_extended_entry_functional (j)", "specialize polynomial_zero_extended_entry_functional (x1)", "specialize polynomial_zero_extended_entry_functional (0)", "apply polynomial_zero_extended_entry_functional", "exact ht_witness_witness_witness_right_left", "right", "split", "exact hj", "refl", "trans x1*x2", "exact ht_witness_witness_witness_right_right_right", "rewrite hz", "apply mul_zero_left" ], "script_sha256": "86a3de08512536cbb56a577ddcd1d0d76a959eb6ab430d869161166c9cbda571", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_left_unit_candidate.py", "sha256": "dbb8debb4716b6bb9b246700f7e93865c8a6c1b12a3b65c0ffbb62206a890ba6" }, "statement": "forall ub uc ab ac L i j t. (exists pfc_gap_unit_tail_index. pfc_gap_unit_tail_index+(1)=(j)) -> (exists pfc_complement_unit_tail_actual pfc_left_unit_tail_actual pfc_right_unit_tail_actual. (((j)+pfc_complement_unit_tail_actual=(i)) /\\ ((((((exists pfa_gap_unit_tail_actualleftinside. pfa_gap_unit_tail_actualleftinside + S (j) = (1)) /\\ ((((exists ff_h_pfp_unit_tail_actualleftentry. ff_h_pfp_unit_tail_actualleftentry + S (pfc_left_unit_tail_actual) = S ((S (j)) * uc)) /\\ exists ff_q_pfp_unit_tail_actualleftentry. ub = ff_q_pfp_unit_tail_actualleftentry * S ((S (j)) * uc) + (pfc_left_unit_tail_actual)))))) \\/ (((exists pfc_gap_unit_tail_actualleftoutside. pfc_gap_unit_tail_actualleftoutside+(1)=(j)) /\\ (((pfc_left_unit_tail_actual)=0))))) /\\ ((((((exists pfa_gap_unit_tail_actualrightinside. pfa_gap_unit_tail_actualrightinside + S (pfc_complement_unit_tail_actual) = (L)) /\\ ((((exists ff_h_pfp_unit_tail_actualrightentry. ff_h_pfp_unit_tail_actualrightentry + S (pfc_right_unit_tail_actual) = S ((S (pfc_complement_unit_tail_actual)) * ac)) /\\ exists ff_q_pfp_unit_tail_actualrightentry. ab = ff_q_pfp_unit_tail_actualrightentry * S ((S (pfc_complement_unit_tail_actual)) * ac) + (pfc_right_unit_tail_actual)))))) \\/ (((exists pfc_gap_unit_tail_actualrightoutside. pfc_gap_unit_tail_actualrightoutside+(L)=(pfc_complement_unit_tail_actual)) /\\ (((pfc_right_unit_tail_actual)=0))))) /\\ (((t)=pfc_left_unit_tail_actual*pfc_right_unit_tail_actual)))))))) -> (t=0)", "statement_sha256": "e6f20add80e772621b1c88d09eeb0829d63438fa5566e95a823022a7d8d96604", "summary": "Every later term is zero because its left input index is outside a genuine length-one prefix. 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((((exists ff_h_pfp_unit_sum_first_entry. ff_h_pfp_unit_sum_first_entry + S (t) = S ((S (0)) * dc)) /\\ exists ff_q_pfp_unit_sum_first_entry. db = ff_q_pfp_unit_sum_first_entry * S ((S (0)) * dc) + (t))) /\\ ((exists pfc_complement_unit_sum_first_term pfc_left_unit_sum_first_term pfc_right_unit_sum_first_term. (((0)+pfc_complement_unit_sum_first_term=(i)) /\\ ((((((exists pfa_gap_unit_sum_first_termleftinside. pfa_gap_unit_sum_first_termleftinside + S (0) = (1)) /\\ ((((exists ff_h_pfp_unit_sum_first_termleftentry. ff_h_pfp_unit_sum_first_termleftentry + S (pfc_left_unit_sum_first_term) = S ((S (0)) * uc)) /\\ exists ff_q_pfp_unit_sum_first_termleftentry. ub = ff_q_pfp_unit_sum_first_termleftentry * S ((S (0)) * uc) + (pfc_left_unit_sum_first_term)))))) \\/ (((exists pfc_gap_unit_sum_first_termleftoutside. pfc_gap_unit_sum_first_termleftoutside+(1)=(0)) /\\ (((pfc_left_unit_sum_first_term)=0))))) /\\ ((((((exists pfa_gap_unit_sum_first_termrightinside. pfa_gap_unit_sum_first_termrightinside + S (pfc_complement_unit_sum_first_term) = (L)) /\\ ((((exists ff_h_pfp_unit_sum_first_termrightentry. ff_h_pfp_unit_sum_first_termrightentry + S (pfc_right_unit_sum_first_term) = S ((S (pfc_complement_unit_sum_first_term)) * ac)) /\\ exists ff_q_pfp_unit_sum_first_termrightentry. ab = ff_q_pfp_unit_sum_first_termrightentry * S ((S (pfc_complement_unit_sum_first_term)) * ac) + (pfc_right_unit_sum_first_term)))))) \\/ (((exists pfc_gap_unit_sum_first_termrightoutside. pfc_gap_unit_sum_first_termrightoutside+(L)=(pfc_complement_unit_sum_first_term)) /\\ (((pfc_right_unit_sum_first_term)=0))))) /\\ (((t)=pfc_left_unit_sum_first_term*pfc_right_unit_sum_first_term))))))))))", "specialize hd (0)", "apply hd", "exists i", "simp", "cases hv", "cases hv_witness", "have heq : x=a", "specialize polynomial_diagonal_left_unit_first_term (ub)", "specialize polynomial_diagonal_left_unit_first_term (uc)", "specialize polynomial_diagonal_left_unit_first_term (ab)", "specialize polynomial_diagonal_left_unit_first_term (ac)", "specialize polynomial_diagonal_left_unit_first_term (L)", "specialize polynomial_diagonal_left_unit_first_term (i)", "specialize polynomial_diagonal_left_unit_first_term (a)", "specialize polynomial_diagonal_left_unit_first_term (x)", "apply polynomial_diagonal_left_unit_first_term", "exact hu", "exact hi", "exact ha", "exact hv_witness_right", "rewrite heq at hv_witness_left", "rewrite heq at hv_witness_left", "exact hv_witness_left", "have htail : forall pfpad_tail_index_unit_sum_tail. (exists pfa_gap_unit_sum_tailbound. pfa_gap_unit_sum_tailbound + S (pfpad_tail_index_unit_sum_tail) = (i)) -> (((exists ff_h_pfp_unit_sum_tailzero. ff_h_pfp_unit_sum_tailzero + S (0) = S ((S ((1)+pfpad_tail_index_unit_sum_tail)) * dc)) /\\ exists ff_q_pfp_unit_sum_tailzero. db = ff_q_pfp_unit_sum_tailzero * S ((S ((1)+pfpad_tail_index_unit_sum_tail)) * dc) + (0)))", "intro j", "intro hj", "have hv : exists t. ((((exists ff_h_pfp_unit_sum_tail_entry. ff_h_pfp_unit_sum_tail_entry + S (t) = S ((S (1+j)) * dc)) /\\ exists ff_q_pfp_unit_sum_tail_entry. db = ff_q_pfp_unit_sum_tail_entry * S ((S (1+j)) * dc) + (t))) /\\ ((exists pfc_complement_unit_sum_tail_term pfc_left_unit_sum_tail_term pfc_right_unit_sum_tail_term. (((1+j)+pfc_complement_unit_sum_tail_term=(i)) /\\ ((((((exists pfa_gap_unit_sum_tail_termleftinside. pfa_gap_unit_sum_tail_termleftinside + S (1+j) = (1)) /\\ ((((exists ff_h_pfp_unit_sum_tail_termleftentry. ff_h_pfp_unit_sum_tail_termleftentry + S (pfc_left_unit_sum_tail_term) = S ((S (1+j)) * uc)) /\\ exists ff_q_pfp_unit_sum_tail_termleftentry. ub = ff_q_pfp_unit_sum_tail_termleftentry * S ((S (1+j)) * uc) + (pfc_left_unit_sum_tail_term)))))) \\/ (((exists pfc_gap_unit_sum_tail_termleftoutside. pfc_gap_unit_sum_tail_termleftoutside+(1)=(1+j)) /\\ (((pfc_left_unit_sum_tail_term)=0))))) /\\ ((((((exists pfa_gap_unit_sum_tail_termrightinside. pfa_gap_unit_sum_tail_termrightinside + S (pfc_complement_unit_sum_tail_term) = (L)) /\\ ((((exists ff_h_pfp_unit_sum_tail_termrightentry. ff_h_pfp_unit_sum_tail_termrightentry + S (pfc_right_unit_sum_tail_term) = S ((S (pfc_complement_unit_sum_tail_term)) * ac)) /\\ exists ff_q_pfp_unit_sum_tail_termrightentry. ab = ff_q_pfp_unit_sum_tail_termrightentry * S ((S (pfc_complement_unit_sum_tail_term)) * ac) + (pfc_right_unit_sum_tail_term)))))) \\/ (((exists pfc_gap_unit_sum_tail_termrightoutside. pfc_gap_unit_sum_tail_termrightoutside+(L)=(pfc_complement_unit_sum_tail_term)) /\\ (((pfc_right_unit_sum_tail_term)=0))))) /\\ (((t)=pfc_left_unit_sum_tail_term*pfc_right_unit_sum_tail_term))))))))))", "specialize hd (1+j)", "apply hd", "have hindex : 1+j=S j", "simp [add_succ_left,zero_add]", "rewrite hindex", "specialize succ_le_succ (S j)", "specialize succ_le_succ (i)", "apply succ_le_succ", "exact hj", "cases hv", "cases hv_witness", "have heq : x=0", "specialize polynomial_diagonal_left_unit_tail_term (ub)", "specialize polynomial_diagonal_left_unit_tail_term (uc)", "specialize polynomial_diagonal_left_unit_tail_term (ab)", "specialize polynomial_diagonal_left_unit_tail_term (ac)", "specialize polynomial_diagonal_left_unit_tail_term (L)", "specialize polynomial_diagonal_left_unit_tail_term (i)", "specialize polynomial_diagonal_left_unit_tail_term (1+j)", "specialize polynomial_diagonal_left_unit_tail_term (x)", "apply polynomial_diagonal_left_unit_tail_term", "specialize le_add_right (1)", "specialize le_add_right (j)", "apply le_add_right", "exact hv_witness_right", "rewrite heq at hv_witness_left", "rewrite heq at hv_witness_left", "exact hv_witness_left", "have hsingle : exists m. (exists fs_u_pfc_unit_single_sum fs_v_pfc_unit_single_sum. ((((exists fs_h_pfc_unit_single_sum_body_start. fs_h_pfc_unit_single_sum_body_start + S (0) = S ((S (0)) * fs_v_pfc_unit_single_sum)) /\\ exists fs_q_pfc_unit_single_sum_body_start. fs_u_pfc_unit_single_sum = fs_q_pfc_unit_single_sum_body_start * S ((S (0)) * fs_v_pfc_unit_single_sum) + (0))) /\\ ((((exists fs_h_pfc_unit_single_sum_body_terminal. fs_h_pfc_unit_single_sum_body_terminal + S (m) = S ((S (1)) * fs_v_pfc_unit_single_sum)) /\\ exists fs_q_pfc_unit_single_sum_body_terminal. fs_u_pfc_unit_single_sum = fs_q_pfc_unit_single_sum_body_terminal * S ((S (1)) * fs_v_pfc_unit_single_sum) + (m))) /\\ forall fs_i_pfc_unit_single_sum_body_steps. (exists fs_lt_pfc_unit_single_sum_body_steps_bound. fs_lt_pfc_unit_single_sum_body_steps_bound + S fs_i_pfc_unit_single_sum_body_steps = 1) -> exists fs_a_pfc_unit_single_sum_body_steps fs_r_pfc_unit_single_sum_body_steps fs_s_pfc_unit_single_sum_body_steps. ((((exists fs_h_pfc_unit_single_sum_body_steps_summand. fs_h_pfc_unit_single_sum_body_steps_summand + S (fs_a_pfc_unit_single_sum_body_steps) = S ((S (fs_i_pfc_unit_single_sum_body_steps)) * dc)) /\\ exists fs_q_pfc_unit_single_sum_body_steps_summand. db = fs_q_pfc_unit_single_sum_body_steps_summand * S ((S (fs_i_pfc_unit_single_sum_body_steps)) * dc) + (fs_a_pfc_unit_single_sum_body_steps))) /\\ ((((exists fs_h_pfc_unit_single_sum_body_steps_partial. fs_h_pfc_unit_single_sum_body_steps_partial + S (fs_r_pfc_unit_single_sum_body_steps) = S ((S (fs_i_pfc_unit_single_sum_body_steps)) * fs_v_pfc_unit_single_sum)) /\\ exists fs_q_pfc_unit_single_sum_body_steps_partial. fs_u_pfc_unit_single_sum = fs_q_pfc_unit_single_sum_body_steps_partial * S ((S (fs_i_pfc_unit_single_sum_body_steps)) * fs_v_pfc_unit_single_sum) + (fs_r_pfc_unit_single_sum_body_steps))) /\\ ((((exists fs_h_pfc_unit_single_sum_body_steps_successor. fs_h_pfc_unit_single_sum_body_steps_successor + S (fs_s_pfc_unit_single_sum_body_steps) = S ((S (S fs_i_pfc_unit_single_sum_body_steps)) * fs_v_pfc_unit_single_sum)) /\\ exists fs_q_pfc_unit_single_sum_body_steps_successor. fs_u_pfc_unit_single_sum = fs_q_pfc_unit_single_sum_body_steps_successor * S ((S (S fs_i_pfc_unit_single_sum_body_steps)) * fs_v_pfc_unit_single_sum) + (fs_s_pfc_unit_single_sum_body_steps))) /\\ fs_s_pfc_unit_single_sum_body_steps = fs_r_pfc_unit_single_sum_body_steps + fs_a_pfc_unit_single_sum_body_steps))))))", "specialize beta_sum_exists (db)", "specialize beta_sum_exists (dc)", "specialize beta_sum_exists (1)", "apply beta_sum_exists", "cases hsingle", "have hdecomp : exists t s. ((((exists ff_h_pfp_unit_single_entry. ff_h_pfp_unit_single_entry + S (t) = S ((S (0)) * dc)) /\\ exists ff_q_pfp_unit_single_entry. db = ff_q_pfp_unit_single_entry * S ((S (0)) * dc) + (t))) /\\ (((exists fs_u_pfc_unit_single_empty fs_v_pfc_unit_single_empty. ((((exists fs_h_pfc_unit_single_empty_body_start. fs_h_pfc_unit_single_empty_body_start + S (0) = S ((S (0)) * fs_v_pfc_unit_single_empty)) /\\ exists fs_q_pfc_unit_single_empty_body_start. fs_u_pfc_unit_single_empty = fs_q_pfc_unit_single_empty_body_start * S ((S (0)) * fs_v_pfc_unit_single_empty) + (0))) /\\ ((((exists fs_h_pfc_unit_single_empty_body_terminal. fs_h_pfc_unit_single_empty_body_terminal + S (s) = S ((S (0)) * fs_v_pfc_unit_single_empty)) /\\ exists fs_q_pfc_unit_single_empty_body_terminal. fs_u_pfc_unit_single_empty = fs_q_pfc_unit_single_empty_body_terminal * S ((S (0)) * fs_v_pfc_unit_single_empty) + (s))) /\\ forall fs_i_pfc_unit_single_empty_body_steps. (exists fs_lt_pfc_unit_single_empty_body_steps_bound. fs_lt_pfc_unit_single_empty_body_steps_bound + S fs_i_pfc_unit_single_empty_body_steps = 0) -> exists fs_a_pfc_unit_single_empty_body_steps fs_r_pfc_unit_single_empty_body_steps fs_s_pfc_unit_single_empty_body_steps. ((((exists fs_h_pfc_unit_single_empty_body_steps_summand. fs_h_pfc_unit_single_empty_body_steps_summand + S (fs_a_pfc_unit_single_empty_body_steps) = S ((S (fs_i_pfc_unit_single_empty_body_steps)) * dc)) /\\ exists fs_q_pfc_unit_single_empty_body_steps_summand. db = fs_q_pfc_unit_single_empty_body_steps_summand * S ((S (fs_i_pfc_unit_single_empty_body_steps)) * dc) + (fs_a_pfc_unit_single_empty_body_steps))) /\\ ((((exists fs_h_pfc_unit_single_empty_body_steps_partial. fs_h_pfc_unit_single_empty_body_steps_partial + S (fs_r_pfc_unit_single_empty_body_steps) = S ((S (fs_i_pfc_unit_single_empty_body_steps)) * fs_v_pfc_unit_single_empty)) /\\ exists fs_q_pfc_unit_single_empty_body_steps_partial. fs_u_pfc_unit_single_empty = fs_q_pfc_unit_single_empty_body_steps_partial * S ((S (fs_i_pfc_unit_single_empty_body_steps)) * fs_v_pfc_unit_single_empty) + (fs_r_pfc_unit_single_empty_body_steps))) /\\ ((((exists fs_h_pfc_unit_single_empty_body_steps_successor. fs_h_pfc_unit_single_empty_body_steps_successor + S (fs_s_pfc_unit_single_empty_body_steps) = S ((S (S fs_i_pfc_unit_single_empty_body_steps)) * fs_v_pfc_unit_single_empty)) /\\ exists fs_q_pfc_unit_single_empty_body_steps_successor. fs_u_pfc_unit_single_empty = fs_q_pfc_unit_single_empty_body_steps_successor * S ((S (S fs_i_pfc_unit_single_empty_body_steps)) * fs_v_pfc_unit_single_empty) + (fs_s_pfc_unit_single_empty_body_steps))) /\\ fs_s_pfc_unit_single_empty_body_steps = fs_r_pfc_unit_single_empty_body_steps + fs_a_pfc_unit_single_empty_body_steps)))))) /\\ ((x=s+t)))))", "specialize beta_sum_succ_decompose (db)", "specialize beta_sum_succ_decompose (dc)", "specialize beta_sum_succ_decompose (0)", "specialize beta_sum_succ_decompose (x)", "apply beta_sum_succ_decompose", "exact hsingle_witness", "cases hdecomp", "cases hdecomp_witness", "cases hdecomp_witness_witness", "cases hdecomp_witness_witness_right", "have hzero : x2=0", "specialize beta_sum_zero (db)", "specialize beta_sum_zero (dc)", "specialize beta_sum_zero (x2)", "apply beta_sum_zero", "exact hdecomp_witness_witness_right_left", "have hentry : x1=a", "specialize beta_at_unique (db)", "specialize beta_at_unique (dc)", "specialize beta_at_unique (0)", "specialize beta_at_unique (x1)", "specialize beta_at_unique (a)", "apply beta_at_unique", "exact hdecomp_witness_witness_left", "exact hhead", "have hvalue : x=a", "trans x2+x1", "exact hdecomp_witness_witness_right_right", "rewrite hzero", "trans x1", "apply zero_add", "exact hentry", "trans x", "specialize polynomial_zero_tail_natural_sum_invariant (db)", "specialize polynomial_zero_tail_natural_sum_invariant (dc)", "specialize polynomial_zero_tail_natural_sum_invariant (db)", "specialize polynomial_zero_tail_natural_sum_invariant (dc)", "specialize polynomial_zero_tail_natural_sum_invariant (1)", "specialize polynomial_zero_tail_natural_sum_invariant (i)", "specialize polynomial_zero_tail_natural_sum_invariant (x)", "specialize polynomial_zero_tail_natural_sum_invariant (n)", "apply polynomial_zero_tail_natural_sum_invariant", "intro k", "intro v", "intro hk", "intro hv", "exact hv", "exact htail", "exact hsingle_witness", "have hlength : 1+i=S i", "simp [add_succ_left,zero_add]", "rewrite hlength", "rewrite hlength", "rewrite hlength", "exact hs", "exact hvalue" ], "script_sha256": "d59b48ed73faeb57243620706eacc28d2aa23240c68e40409a99c387465318d1", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_left_unit_candidate.py", "sha256": "dbb8debb4716b6bb9b246700f7e93865c8a6c1b12a3b65c0ffbb62206a890ba6" }, "statement": "forall ub uc ab ac L i a db dc n. (((exists ff_h_pfp_unit_sum_unit. ff_h_pfp_unit_sum_unit + S (1) = S ((S (0)) * uc)) /\\ exists ff_q_pfp_unit_sum_unit. ub = ff_q_pfp_unit_sum_unit * S ((S (0)) * uc) + (1))) -> (exists pfa_gap_unit_sum_index. pfa_gap_unit_sum_index + S (i) = (L)) -> (((exists ff_h_pfp_unit_sum_A. ff_h_pfp_unit_sum_A + S (a) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_unit_sum_A. ab = ff_q_pfp_unit_sum_A * S ((S (i)) * ac) + (a))) -> (forall pfc_index_unit_sum_diagonal. (exists pfa_gap_unit_sum_diagonalbound. pfa_gap_unit_sum_diagonalbound + S (pfc_index_unit_sum_diagonal) = (S i)) -> exists pfc_value_unit_sum_diagonal. ((((exists ff_h_pfp_unit_sum_diagonalentry. ff_h_pfp_unit_sum_diagonalentry + S (pfc_value_unit_sum_diagonal) = S ((S (pfc_index_unit_sum_diagonal)) * dc)) /\\ exists ff_q_pfp_unit_sum_diagonalentry. db = ff_q_pfp_unit_sum_diagonalentry * S ((S (pfc_index_unit_sum_diagonal)) * dc) + (pfc_value_unit_sum_diagonal))) /\\ ((exists pfc_complement_unit_sum_diagonalterm pfc_left_unit_sum_diagonalterm pfc_right_unit_sum_diagonalterm. (((pfc_index_unit_sum_diagonal)+pfc_complement_unit_sum_diagonalterm=(i)) /\\ ((((((exists pfa_gap_unit_sum_diagonaltermleftinside. pfa_gap_unit_sum_diagonaltermleftinside + S (pfc_index_unit_sum_diagonal) = (1)) /\\ ((((exists ff_h_pfp_unit_sum_diagonaltermleftentry. ff_h_pfp_unit_sum_diagonaltermleftentry + S (pfc_left_unit_sum_diagonalterm) = S ((S (pfc_index_unit_sum_diagonal)) * uc)) /\\ exists ff_q_pfp_unit_sum_diagonaltermleftentry. ub = ff_q_pfp_unit_sum_diagonaltermleftentry * S ((S (pfc_index_unit_sum_diagonal)) * uc) + (pfc_left_unit_sum_diagonalterm)))))) \\/ (((exists pfc_gap_unit_sum_diagonaltermleftoutside. pfc_gap_unit_sum_diagonaltermleftoutside+(1)=(pfc_index_unit_sum_diagonal)) /\\ (((pfc_left_unit_sum_diagonalterm)=0))))) /\\ ((((((exists pfa_gap_unit_sum_diagonaltermrightinside. pfa_gap_unit_sum_diagonaltermrightinside + S (pfc_complement_unit_sum_diagonalterm) = (L)) /\\ ((((exists ff_h_pfp_unit_sum_diagonaltermrightentry. ff_h_pfp_unit_sum_diagonaltermrightentry + S (pfc_right_unit_sum_diagonalterm) = S ((S (pfc_complement_unit_sum_diagonalterm)) * ac)) /\\ exists ff_q_pfp_unit_sum_diagonaltermrightentry. ab = ff_q_pfp_unit_sum_diagonaltermrightentry * S ((S (pfc_complement_unit_sum_diagonalterm)) * ac) + (pfc_right_unit_sum_diagonalterm)))))) \\/ (((exists pfc_gap_unit_sum_diagonaltermrightoutside. pfc_gap_unit_sum_diagonaltermrightoutside+(L)=(pfc_complement_unit_sum_diagonalterm)) /\\ (((pfc_right_unit_sum_diagonalterm)=0))))) /\\ (((pfc_value_unit_sum_diagonal)=pfc_left_unit_sum_diagonalterm*pfc_right_unit_sum_diagonalterm))))))))))) -> (exists fs_u_pfc_unit_sum_actual fs_v_pfc_unit_sum_actual. ((((exists fs_h_pfc_unit_sum_actual_body_start. fs_h_pfc_unit_sum_actual_body_start + S (0) = S ((S (0)) * fs_v_pfc_unit_sum_actual)) /\\ exists fs_q_pfc_unit_sum_actual_body_start. fs_u_pfc_unit_sum_actual = fs_q_pfc_unit_sum_actual_body_start * S ((S (0)) * fs_v_pfc_unit_sum_actual) + (0))) /\\ ((((exists fs_h_pfc_unit_sum_actual_body_terminal. fs_h_pfc_unit_sum_actual_body_terminal + S (n) = S ((S (S i)) * fs_v_pfc_unit_sum_actual)) /\\ exists fs_q_pfc_unit_sum_actual_body_terminal. fs_u_pfc_unit_sum_actual = fs_q_pfc_unit_sum_actual_body_terminal * S ((S (S i)) * fs_v_pfc_unit_sum_actual) + (n))) /\\ forall fs_i_pfc_unit_sum_actual_body_steps. (exists fs_lt_pfc_unit_sum_actual_body_steps_bound. fs_lt_pfc_unit_sum_actual_body_steps_bound + S fs_i_pfc_unit_sum_actual_body_steps = S i) -> exists fs_a_pfc_unit_sum_actual_body_steps fs_r_pfc_unit_sum_actual_body_steps fs_s_pfc_unit_sum_actual_body_steps. ((((exists fs_h_pfc_unit_sum_actual_body_steps_summand. fs_h_pfc_unit_sum_actual_body_steps_summand + S (fs_a_pfc_unit_sum_actual_body_steps) = S ((S (fs_i_pfc_unit_sum_actual_body_steps)) * dc)) /\\ exists fs_q_pfc_unit_sum_actual_body_steps_summand. db = fs_q_pfc_unit_sum_actual_body_steps_summand * S ((S (fs_i_pfc_unit_sum_actual_body_steps)) * dc) + (fs_a_pfc_unit_sum_actual_body_steps))) /\\ ((((exists fs_h_pfc_unit_sum_actual_body_steps_partial. fs_h_pfc_unit_sum_actual_body_steps_partial + S (fs_r_pfc_unit_sum_actual_body_steps) = S ((S (fs_i_pfc_unit_sum_actual_body_steps)) * fs_v_pfc_unit_sum_actual)) /\\ exists fs_q_pfc_unit_sum_actual_body_steps_partial. fs_u_pfc_unit_sum_actual = fs_q_pfc_unit_sum_actual_body_steps_partial * S ((S (fs_i_pfc_unit_sum_actual_body_steps)) * fs_v_pfc_unit_sum_actual) + (fs_r_pfc_unit_sum_actual_body_steps))) /\\ ((((exists fs_h_pfc_unit_sum_actual_body_steps_successor. fs_h_pfc_unit_sum_actual_body_steps_successor + S (fs_s_pfc_unit_sum_actual_body_steps) = S ((S (S fs_i_pfc_unit_sum_actual_body_steps)) * fs_v_pfc_unit_sum_actual)) /\\ exists fs_q_pfc_unit_sum_actual_body_steps_successor. fs_u_pfc_unit_sum_actual = fs_q_pfc_unit_sum_actual_body_steps_successor * S ((S (S fs_i_pfc_unit_sum_actual_body_steps)) * fs_v_pfc_unit_sum_actual) + (fs_s_pfc_unit_sum_actual_body_steps))) /\\ fs_s_pfc_unit_sum_actual_body_steps = fs_r_pfc_unit_sum_actual_body_steps + fs_a_pfc_unit_sum_actual_body_steps)))))) -> (n=a)", "statement_sha256": "c30de45412742e6dfdd00a7094334ce45b06b7364b5a71c58e2f806882941579", "summary": "An actual unit-left antidiagonal sum equals its first coefficient: construct the one-term sum and use the proved zero-tail invariant on all remaining actual summands.", "summary_sha256": "85638921534a920c99d04c6c7a50f4fb07c8ab8e2e8b3d7f8571bac8c1e3e072" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "de4ee6b404e0b6acb409fda40b6046a7e4c1e2e85f1117941287df40ac78514d", 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(((exists ff_h_pfp_unit_coefficient_unit. ff_h_pfp_unit_coefficient_unit + S (1) = S ((S (0)) * uc)) /\\ exists ff_q_pfp_unit_coefficient_unit. ub = ff_q_pfp_unit_coefficient_unit * S ((S (0)) * uc) + (1))) -> (exists pfa_gap_unit_coefficient_index. pfa_gap_unit_coefficient_index + S (i) = (L)) -> (((exists ff_h_pfp_unit_coefficient_A. ff_h_pfp_unit_coefficient_A + S (a) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_unit_coefficient_A. ab = ff_q_pfp_unit_coefficient_A * S ((S (i)) * ac) + (a))) -> (exists pfa_gap_unit_coefficient_bound. pfa_gap_unit_coefficient_bound + S (a) = (p)) -> (exists pfc_terms_code_unit_coefficient_actual pfc_terms_scale_unit_coefficient_actual pfc_natural_sum_unit_coefficient_actual. ((forall pfc_index_unit_coefficient_actualdiagonal. (exists pfa_gap_unit_coefficient_actualdiagonalbound. pfa_gap_unit_coefficient_actualdiagonalbound + S (pfc_index_unit_coefficient_actualdiagonal) = (S (i))) -> exists pfc_value_unit_coefficient_actualdiagonal. ((((exists ff_h_pfp_unit_coefficient_actualdiagonalentry. ff_h_pfp_unit_coefficient_actualdiagonalentry + S (pfc_value_unit_coefficient_actualdiagonal) = S ((S (pfc_index_unit_coefficient_actualdiagonal)) * pfc_terms_scale_unit_coefficient_actual)) /\\ exists ff_q_pfp_unit_coefficient_actualdiagonalentry. pfc_terms_code_unit_coefficient_actual = ff_q_pfp_unit_coefficient_actualdiagonalentry * S ((S (pfc_index_unit_coefficient_actualdiagonal)) * pfc_terms_scale_unit_coefficient_actual) + (pfc_value_unit_coefficient_actualdiagonal))) /\\ ((exists pfc_complement_unit_coefficient_actualdiagonalterm pfc_left_unit_coefficient_actualdiagonalterm pfc_right_unit_coefficient_actualdiagonalterm. (((pfc_index_unit_coefficient_actualdiagonal)+pfc_complement_unit_coefficient_actualdiagonalterm=(i)) /\\ ((((((exists pfa_gap_unit_coefficient_actualdiagonaltermleftinside. pfa_gap_unit_coefficient_actualdiagonaltermleftinside + S (pfc_index_unit_coefficient_actualdiagonal) = (1)) /\\ ((((exists ff_h_pfp_unit_coefficient_actualdiagonaltermleftentry. ff_h_pfp_unit_coefficient_actualdiagonaltermleftentry + S (pfc_left_unit_coefficient_actualdiagonalterm) = S ((S (pfc_index_unit_coefficient_actualdiagonal)) * uc)) /\\ exists ff_q_pfp_unit_coefficient_actualdiagonaltermleftentry. ub = ff_q_pfp_unit_coefficient_actualdiagonaltermleftentry * S ((S (pfc_index_unit_coefficient_actualdiagonal)) * uc) + (pfc_left_unit_coefficient_actualdiagonalterm)))))) \\/ (((exists pfc_gap_unit_coefficient_actualdiagonaltermleftoutside. pfc_gap_unit_coefficient_actualdiagonaltermleftoutside+(1)=(pfc_index_unit_coefficient_actualdiagonal)) /\\ (((pfc_left_unit_coefficient_actualdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_unit_coefficient_actualdiagonaltermrightinside. pfa_gap_unit_coefficient_actualdiagonaltermrightinside + S (pfc_complement_unit_coefficient_actualdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_unit_coefficient_actualdiagonaltermrightentry. ff_h_pfp_unit_coefficient_actualdiagonaltermrightentry + S (pfc_right_unit_coefficient_actualdiagonalterm) = S ((S (pfc_complement_unit_coefficient_actualdiagonalterm)) * ac)) /\\ exists ff_q_pfp_unit_coefficient_actualdiagonaltermrightentry. ab = ff_q_pfp_unit_coefficient_actualdiagonaltermrightentry * S ((S (pfc_complement_unit_coefficient_actualdiagonalterm)) * ac) + (pfc_right_unit_coefficient_actualdiagonalterm)))))) \\/ (((exists pfc_gap_unit_coefficient_actualdiagonaltermrightoutside. pfc_gap_unit_coefficient_actualdiagonaltermrightoutside+(L)=(pfc_complement_unit_coefficient_actualdiagonalterm)) /\\ (((pfc_right_unit_coefficient_actualdiagonalterm)=0))))) /\\ (((pfc_value_unit_coefficient_actualdiagonal)=pfc_left_unit_coefficient_actualdiagonalterm*pfc_right_unit_coefficient_actualdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_unit_coefficient_actualsum fs_v_pfc_unit_coefficient_actualsum. ((((exists fs_h_pfc_unit_coefficient_actualsum_body_start. fs_h_pfc_unit_coefficient_actualsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_unit_coefficient_actualsum)) /\\ exists fs_q_pfc_unit_coefficient_actualsum_body_start. fs_u_pfc_unit_coefficient_actualsum = fs_q_pfc_unit_coefficient_actualsum_body_start * S ((S (0)) * fs_v_pfc_unit_coefficient_actualsum) + (0))) /\\ ((((exists fs_h_pfc_unit_coefficient_actualsum_body_terminal. fs_h_pfc_unit_coefficient_actualsum_body_terminal + S (pfc_natural_sum_unit_coefficient_actual) = S ((S (S (i))) * fs_v_pfc_unit_coefficient_actualsum)) /\\ exists fs_q_pfc_unit_coefficient_actualsum_body_terminal. fs_u_pfc_unit_coefficient_actualsum = fs_q_pfc_unit_coefficient_actualsum_body_terminal * S ((S (S (i))) * fs_v_pfc_unit_coefficient_actualsum) + (pfc_natural_sum_unit_coefficient_actual))) /\\ forall fs_i_pfc_unit_coefficient_actualsum_body_steps. (exists fs_lt_pfc_unit_coefficient_actualsum_body_steps_bound. fs_lt_pfc_unit_coefficient_actualsum_body_steps_bound + S fs_i_pfc_unit_coefficient_actualsum_body_steps = S (i)) -> exists fs_a_pfc_unit_coefficient_actualsum_body_steps fs_r_pfc_unit_coefficient_actualsum_body_steps fs_s_pfc_unit_coefficient_actualsum_body_steps. ((((exists fs_h_pfc_unit_coefficient_actualsum_body_steps_summand. fs_h_pfc_unit_coefficient_actualsum_body_steps_summand + S (fs_a_pfc_unit_coefficient_actualsum_body_steps) = S ((S (fs_i_pfc_unit_coefficient_actualsum_body_steps)) * pfc_terms_scale_unit_coefficient_actual)) /\\ exists fs_q_pfc_unit_coefficient_actualsum_body_steps_summand. pfc_terms_code_unit_coefficient_actual = fs_q_pfc_unit_coefficient_actualsum_body_steps_summand * S ((S (fs_i_pfc_unit_coefficient_actualsum_body_steps)) * pfc_terms_scale_unit_coefficient_actual) + (fs_a_pfc_unit_coefficient_actualsum_body_steps))) /\\ ((((exists fs_h_pfc_unit_coefficient_actualsum_body_steps_partial. fs_h_pfc_unit_coefficient_actualsum_body_steps_partial + S (fs_r_pfc_unit_coefficient_actualsum_body_steps) = S ((S (fs_i_pfc_unit_coefficient_actualsum_body_steps)) * fs_v_pfc_unit_coefficient_actualsum)) /\\ exists fs_q_pfc_unit_coefficient_actualsum_body_steps_partial. fs_u_pfc_unit_coefficient_actualsum = fs_q_pfc_unit_coefficient_actualsum_body_steps_partial * S ((S (fs_i_pfc_unit_coefficient_actualsum_body_steps)) * fs_v_pfc_unit_coefficient_actualsum) + (fs_r_pfc_unit_coefficient_actualsum_body_steps))) /\\ ((((exists fs_h_pfc_unit_coefficient_actualsum_body_steps_successor. fs_h_pfc_unit_coefficient_actualsum_body_steps_successor + S (fs_s_pfc_unit_coefficient_actualsum_body_steps) = S ((S (S fs_i_pfc_unit_coefficient_actualsum_body_steps)) * fs_v_pfc_unit_coefficient_actualsum)) /\\ exists fs_q_pfc_unit_coefficient_actualsum_body_steps_successor. fs_u_pfc_unit_coefficient_actualsum = fs_q_pfc_unit_coefficient_actualsum_body_steps_successor * S ((S (S fs_i_pfc_unit_coefficient_actualsum_body_steps)) * fs_v_pfc_unit_coefficient_actualsum) + (fs_s_pfc_unit_coefficient_actualsum_body_steps))) /\\ fs_s_pfc_unit_coefficient_actualsum_body_steps = fs_r_pfc_unit_coefficient_actualsum_body_steps + fs_a_pfc_unit_coefficient_actualsum_body_steps)))))) /\\ ((((exists pfa_gap_unit_coefficient_actualresiduebound. pfa_gap_unit_coefficient_actualresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_unit_coefficient_actualresiduecongruence pfa_offset_right_unit_coefficient_actualresiduecongruence. (pfc_natural_sum_unit_coefficient_actual) + (p) * pfa_offset_left_unit_coefficient_actualresiduecongruence = (r) + (p) * pfa_offset_right_unit_coefficient_actualresiduecongruence))))))))) -> (r=a)", "statement_sha256": "80480e6637752b1bd651633f253ff4d17a5e0f9fb98367c77cd0e3d3da8ea3ca", "summary": "The actual residue of the unit-left natural sum is its already bounded right-input value. 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(exists fom_gap_pfp_unit_equal_A_index_bound. fom_gap_pfp_unit_equal_A_index_bound + S (fom_index_pfp_unit_equal_A) = L) -> exists fom_value_pfp_unit_equal_A. ((((exists fom_beta_height_pfp_unit_equal_A_entry. fom_beta_height_pfp_unit_equal_A_entry + S (fom_value_pfp_unit_equal_A) = S ((S (fom_index_pfp_unit_equal_A)) * ac)) /\\ exists fom_beta_quotient_pfp_unit_equal_A_entry. ab = fom_beta_quotient_pfp_unit_equal_A_entry * S ((S (fom_index_pfp_unit_equal_A)) * ac) + (fom_value_pfp_unit_equal_A))) /\\ (exists fom_gap_pfp_unit_equal_A_value_bound. fom_gap_pfp_unit_equal_A_value_bound + S (fom_value_pfp_unit_equal_A) = p))", "cases hc", "cases hc_right", "exact hc_right_left", "intro i", "intro r", "intro hi", "intro hr", "have ha : exists a. (((exists ff_h_pfp_unit_equal_entry. ff_h_pfp_unit_equal_entry + S (a) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_unit_equal_entry. ab = ff_q_pfp_unit_equal_entry * S ((S (i)) * ac) + (a)))", "specialize beta_at_exists (ab)", "specialize beta_at_exists (ac)", "specialize beta_at_exists (i)", "apply beta_at_exists", "cases ha", "have heq : r=x", "specialize prime_field_convolution_coefficient_left_unit (p)", "specialize prime_field_convolution_coefficient_left_unit (ub)", "specialize prime_field_convolution_coefficient_left_unit (uc)", "specialize prime_field_convolution_coefficient_left_unit (ab)", "specialize prime_field_convolution_coefficient_left_unit (ac)", "specialize prime_field_convolution_coefficient_left_unit (L)", "specialize prime_field_convolution_coefficient_left_unit (i)", "specialize prime_field_convolution_coefficient_left_unit (x)", "specialize prime_field_convolution_coefficient_left_unit (r)", "apply prime_field_convolution_coefficient_left_unit", "exact hu", "exact hi", "exact ha_witness", "specialize matrix_rank_bounded_prefix_value (ab)", "specialize matrix_rank_bounded_prefix_value (ac)", "specialize matrix_rank_bounded_prefix_value (L)", "specialize matrix_rank_bounded_prefix_value (p)", "specialize matrix_rank_bounded_prefix_value (i)", "specialize matrix_rank_bounded_prefix_value (x)", "apply matrix_rank_bounded_prefix_value", "exact hA", "exact hi", "exact ha_witness", "specialize prime_field_polynomial_convolution_entry (p)", "specialize prime_field_polynomial_convolution_entry (ub)", "specialize prime_field_polynomial_convolution_entry (uc)", "specialize prime_field_polynomial_convolution_entry (1)", "specialize prime_field_polynomial_convolution_entry (ab)", "specialize prime_field_polynomial_convolution_entry (ac)", "specialize prime_field_polynomial_convolution_entry (L)", "specialize prime_field_polynomial_convolution_entry (cb)", "specialize prime_field_polynomial_convolution_entry (cc)", "specialize prime_field_polynomial_convolution_entry (L)", "specialize prime_field_polynomial_convolution_entry (i)", "specialize prime_field_polynomial_convolution_entry (r)", "apply prime_field_polynomial_convolution_entry", "exact hc", "exact hi", "exact hr", "rewrite heq", "rewrite heq", "exact ha_witness" ], "script_sha256": "413915d346b8ec93db7c7cb71810820c1ff1b381abab2563c5db63ce4ace32bb", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_left_unit_candidate.py", "sha256": "dbb8debb4716b6bb9b246700f7e93865c8a6c1b12a3b65c0ffbb62206a890ba6" }, "statement": "forall p ub uc ab ac L cb cc. (((exists ff_h_pfp_unit_equal_unit. ff_h_pfp_unit_equal_unit + S (1) = S ((S (0)) * uc)) /\\ exists ff_q_pfp_unit_equal_unit. ub = ff_q_pfp_unit_equal_unit * S ((S (0)) * uc) + (1))) -> (((forall fom_index_pfp_unit_equal_actualleft. (exists fom_gap_pfp_unit_equal_actualleft_index_bound. fom_gap_pfp_unit_equal_actualleft_index_bound + S (fom_index_pfp_unit_equal_actualleft) = 1) -> exists fom_value_pfp_unit_equal_actualleft. ((((exists fom_beta_height_pfp_unit_equal_actualleft_entry. fom_beta_height_pfp_unit_equal_actualleft_entry + S (fom_value_pfp_unit_equal_actualleft) = S ((S (fom_index_pfp_unit_equal_actualleft)) * uc)) /\\ exists fom_beta_quotient_pfp_unit_equal_actualleft_entry. ub = fom_beta_quotient_pfp_unit_equal_actualleft_entry * S ((S (fom_index_pfp_unit_equal_actualleft)) * uc) + (fom_value_pfp_unit_equal_actualleft))) /\\ (exists fom_gap_pfp_unit_equal_actualleft_value_bound. fom_gap_pfp_unit_equal_actualleft_value_bound + S (fom_value_pfp_unit_equal_actualleft) = p))) /\\ (((forall fom_index_pfp_unit_equal_actualright. (exists fom_gap_pfp_unit_equal_actualright_index_bound. fom_gap_pfp_unit_equal_actualright_index_bound + S (fom_index_pfp_unit_equal_actualright) = L) -> exists fom_value_pfp_unit_equal_actualright. ((((exists fom_beta_height_pfp_unit_equal_actualright_entry. fom_beta_height_pfp_unit_equal_actualright_entry + S (fom_value_pfp_unit_equal_actualright) = S ((S (fom_index_pfp_unit_equal_actualright)) * ac)) /\\ exists fom_beta_quotient_pfp_unit_equal_actualright_entry. ab = fom_beta_quotient_pfp_unit_equal_actualright_entry * S ((S (fom_index_pfp_unit_equal_actualright)) * ac) + (fom_value_pfp_unit_equal_actualright))) /\\ (exists fom_gap_pfp_unit_equal_actualright_value_bound. fom_gap_pfp_unit_equal_actualright_value_bound + S (fom_value_pfp_unit_equal_actualright) = p))) /\\ (((((((1)=0 \\/ (L)=0) /\\ (((L)=0)))) \\/ (((~((1)=0)) /\\ (((~((L)=0)) /\\ (((1)+(L)=S (L)))))))) /\\ ((forall pfc_index_unit_equal_actualcoefficients. (exists pfa_gap_unit_equal_actualcoefficientsbound. pfa_gap_unit_equal_actualcoefficientsbound + S (pfc_index_unit_equal_actualcoefficients) = (L)) -> exists pfc_value_unit_equal_actualcoefficients. ((((exists ff_h_pfp_unit_equal_actualcoefficientsentry. ff_h_pfp_unit_equal_actualcoefficientsentry + S (pfc_value_unit_equal_actualcoefficients) = S ((S (pfc_index_unit_equal_actualcoefficients)) * cc)) /\\ exists ff_q_pfp_unit_equal_actualcoefficientsentry. cb = ff_q_pfp_unit_equal_actualcoefficientsentry * S ((S (pfc_index_unit_equal_actualcoefficients)) * cc) + (pfc_value_unit_equal_actualcoefficients))) /\\ ((exists pfc_terms_code_unit_equal_actualcoefficientscoefficient pfc_terms_scale_unit_equal_actualcoefficientscoefficient pfc_natural_sum_unit_equal_actualcoefficientscoefficient. ((forall pfc_index_unit_equal_actualcoefficientscoefficientdiagonal. (exists pfa_gap_unit_equal_actualcoefficientscoefficientdiagonalbound. pfa_gap_unit_equal_actualcoefficientscoefficientdiagonalbound + S (pfc_index_unit_equal_actualcoefficientscoefficientdiagonal) = (S (pfc_index_unit_equal_actualcoefficients))) -> exists pfc_value_unit_equal_actualcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_unit_equal_actualcoefficientscoefficientdiagonalentry. ff_h_pfp_unit_equal_actualcoefficientscoefficientdiagonalentry + S (pfc_value_unit_equal_actualcoefficientscoefficientdiagonal) = S ((S (pfc_index_unit_equal_actualcoefficientscoefficientdiagonal)) * pfc_terms_scale_unit_equal_actualcoefficientscoefficient)) /\\ exists ff_q_pfp_unit_equal_actualcoefficientscoefficientdiagonalentry. pfc_terms_code_unit_equal_actualcoefficientscoefficient = ff_q_pfp_unit_equal_actualcoefficientscoefficientdiagonalentry * S ((S (pfc_index_unit_equal_actualcoefficientscoefficientdiagonal)) * pfc_terms_scale_unit_equal_actualcoefficientscoefficient) + (pfc_value_unit_equal_actualcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_unit_equal_actualcoefficientscoefficientdiagonalterm pfc_left_unit_equal_actualcoefficientscoefficientdiagonalterm pfc_right_unit_equal_actualcoefficientscoefficientdiagonalterm. (((pfc_index_unit_equal_actualcoefficientscoefficientdiagonal)+pfc_complement_unit_equal_actualcoefficientscoefficientdiagonalterm=(pfc_index_unit_equal_actualcoefficients)) /\\ ((((((exists pfa_gap_unit_equal_actualcoefficientscoefficientdiagonaltermleftinside. pfa_gap_unit_equal_actualcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_unit_equal_actualcoefficientscoefficientdiagonal) = (1)) /\\ ((((exists ff_h_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_unit_equal_actualcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_unit_equal_actualcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_unit_equal_actualcoefficientscoefficientdiagonal)) * uc) + (pfc_left_unit_equal_actualcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_unit_equal_actualcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_unit_equal_actualcoefficientscoefficientdiagonaltermleftoutside+(1)=(pfc_index_unit_equal_actualcoefficientscoefficientdiagonal)) /\\ (((pfc_left_unit_equal_actualcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_unit_equal_actualcoefficientscoefficientdiagonaltermrightinside. pfa_gap_unit_equal_actualcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_unit_equal_actualcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_unit_equal_actualcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_unit_equal_actualcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_unit_equal_actualcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_unit_equal_actualcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_unit_equal_actualcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_unit_equal_actualcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_unit_equal_actualcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_unit_equal_actualcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_unit_equal_actualcoefficientscoefficientdiagonal)=pfc_left_unit_equal_actualcoefficientscoefficientdiagonalterm*pfc_right_unit_equal_actualcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_unit_equal_actualcoefficientscoefficientsum fs_v_pfc_unit_equal_actualcoefficientscoefficientsum. ((((exists fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_start. fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_start. fs_u_pfc_unit_equal_actualcoefficientscoefficientsum = fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_terminal. fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_unit_equal_actualcoefficientscoefficient) = S ((S (S (pfc_index_unit_equal_actualcoefficients))) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_terminal. fs_u_pfc_unit_equal_actualcoefficientscoefficientsum = fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_unit_equal_actualcoefficients))) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum) + (pfc_natural_sum_unit_equal_actualcoefficientscoefficient))) /\\ forall fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps = S (pfc_index_unit_equal_actualcoefficients)) -> exists fs_a_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps fs_r_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps fs_s_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_unit_equal_actualcoefficientscoefficient)) /\\ exists fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_unit_equal_actualcoefficientscoefficient = fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_unit_equal_actualcoefficientscoefficient) + (fs_a_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_unit_equal_actualcoefficientscoefficientsum = fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum) + (fs_r_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_unit_equal_actualcoefficientscoefficientsum = fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum) + (fs_s_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps = fs_r_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps + fs_a_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_unit_equal_actualcoefficientscoefficientresiduebound. pfa_gap_unit_equal_actualcoefficientscoefficientresiduebound + S (pfc_value_unit_equal_actualcoefficients) = (p)) /\\ ((exists pfa_offset_left_unit_equal_actualcoefficientscoefficientresiduecongruence pfa_offset_right_unit_equal_actualcoefficientscoefficientresiduecongruence. (pfc_natural_sum_unit_equal_actualcoefficientscoefficient) + (p) * pfa_offset_left_unit_equal_actualcoefficientscoefficientresiduecongruence = (pfc_value_unit_equal_actualcoefficients) + (p) * pfa_offset_right_unit_equal_actualcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall mdr_i_pfp_unit_equal_result mdr_a_pfp_unit_equal_result. 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(((exists ff_h_pfp_unit_equal_unit. ff_h_pfp_unit_equal_unit + S (1) = S ((S (0)) * uc)) /\\ exists ff_q_pfp_unit_equal_unit. ub = ff_q_pfp_unit_equal_unit * S ((S (0)) * uc) + (1))) -> (((forall fom_index_pfp_unit_equal_actualleft. (exists fom_gap_pfp_unit_equal_actualleft_index_bound. fom_gap_pfp_unit_equal_actualleft_index_bound + S (fom_index_pfp_unit_equal_actualleft) = 1) -> exists fom_value_pfp_unit_equal_actualleft. ((((exists fom_beta_height_pfp_unit_equal_actualleft_entry. fom_beta_height_pfp_unit_equal_actualleft_entry + S (fom_value_pfp_unit_equal_actualleft) = S ((S (fom_index_pfp_unit_equal_actualleft)) * uc)) /\\ exists fom_beta_quotient_pfp_unit_equal_actualleft_entry. ub = fom_beta_quotient_pfp_unit_equal_actualleft_entry * S ((S (fom_index_pfp_unit_equal_actualleft)) * uc) + (fom_value_pfp_unit_equal_actualleft))) /\\ (exists fom_gap_pfp_unit_equal_actualleft_value_bound. fom_gap_pfp_unit_equal_actualleft_value_bound + S (fom_value_pfp_unit_equal_actualleft) = p))) /\\ (((forall fom_index_pfp_unit_equal_actualright. (exists fom_gap_pfp_unit_equal_actualright_index_bound. fom_gap_pfp_unit_equal_actualright_index_bound + S (fom_index_pfp_unit_equal_actualright) = L) -> exists fom_value_pfp_unit_equal_actualright. ((((exists fom_beta_height_pfp_unit_equal_actualright_entry. fom_beta_height_pfp_unit_equal_actualright_entry + S (fom_value_pfp_unit_equal_actualright) = S ((S (fom_index_pfp_unit_equal_actualright)) * ac)) /\\ exists fom_beta_quotient_pfp_unit_equal_actualright_entry. ab = fom_beta_quotient_pfp_unit_equal_actualright_entry * S ((S (fom_index_pfp_unit_equal_actualright)) * ac) + (fom_value_pfp_unit_equal_actualright))) /\\ (exists fom_gap_pfp_unit_equal_actualright_value_bound. fom_gap_pfp_unit_equal_actualright_value_bound + S (fom_value_pfp_unit_equal_actualright) = p))) /\\ (((((((1)=0 \\/ (L)=0) /\\ (((L)=0)))) \\/ (((~((1)=0)) /\\ (((~((L)=0)) /\\ (((1)+(L)=S (L)))))))) /\\ ((forall pfc_index_unit_equal_actualcoefficients. (exists pfa_gap_unit_equal_actualcoefficientsbound. pfa_gap_unit_equal_actualcoefficientsbound + S (pfc_index_unit_equal_actualcoefficients) = (L)) -> exists pfc_value_unit_equal_actualcoefficients. ((((exists ff_h_pfp_unit_equal_actualcoefficientsentry. ff_h_pfp_unit_equal_actualcoefficientsentry + S (pfc_value_unit_equal_actualcoefficients) = S ((S (pfc_index_unit_equal_actualcoefficients)) * cc)) /\\ exists ff_q_pfp_unit_equal_actualcoefficientsentry. cb = ff_q_pfp_unit_equal_actualcoefficientsentry * S ((S (pfc_index_unit_equal_actualcoefficients)) * cc) + (pfc_value_unit_equal_actualcoefficients))) /\\ ((exists pfc_terms_code_unit_equal_actualcoefficientscoefficient pfc_terms_scale_unit_equal_actualcoefficientscoefficient pfc_natural_sum_unit_equal_actualcoefficientscoefficient. ((forall pfc_index_unit_equal_actualcoefficientscoefficientdiagonal. (exists pfa_gap_unit_equal_actualcoefficientscoefficientdiagonalbound. pfa_gap_unit_equal_actualcoefficientscoefficientdiagonalbound + S (pfc_index_unit_equal_actualcoefficientscoefficientdiagonal) = (S (pfc_index_unit_equal_actualcoefficients))) -> exists pfc_value_unit_equal_actualcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_unit_equal_actualcoefficientscoefficientdiagonalentry. ff_h_pfp_unit_equal_actualcoefficientscoefficientdiagonalentry + S (pfc_value_unit_equal_actualcoefficientscoefficientdiagonal) = S ((S (pfc_index_unit_equal_actualcoefficientscoefficientdiagonal)) * pfc_terms_scale_unit_equal_actualcoefficientscoefficient)) /\\ exists ff_q_pfp_unit_equal_actualcoefficientscoefficientdiagonalentry. pfc_terms_code_unit_equal_actualcoefficientscoefficient = ff_q_pfp_unit_equal_actualcoefficientscoefficientdiagonalentry * S ((S (pfc_index_unit_equal_actualcoefficientscoefficientdiagonal)) * pfc_terms_scale_unit_equal_actualcoefficientscoefficient) + (pfc_value_unit_equal_actualcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_unit_equal_actualcoefficientscoefficientdiagonalterm pfc_left_unit_equal_actualcoefficientscoefficientdiagonalterm pfc_right_unit_equal_actualcoefficientscoefficientdiagonalterm. (((pfc_index_unit_equal_actualcoefficientscoefficientdiagonal)+pfc_complement_unit_equal_actualcoefficientscoefficientdiagonalterm=(pfc_index_unit_equal_actualcoefficients)) /\\ ((((((exists pfa_gap_unit_equal_actualcoefficientscoefficientdiagonaltermleftinside. pfa_gap_unit_equal_actualcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_unit_equal_actualcoefficientscoefficientdiagonal) = (1)) /\\ ((((exists ff_h_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_unit_equal_actualcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_unit_equal_actualcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_unit_equal_actualcoefficientscoefficientdiagonal)) * uc) + (pfc_left_unit_equal_actualcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_unit_equal_actualcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_unit_equal_actualcoefficientscoefficientdiagonaltermleftoutside+(1)=(pfc_index_unit_equal_actualcoefficientscoefficientdiagonal)) /\\ (((pfc_left_unit_equal_actualcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_unit_equal_actualcoefficientscoefficientdiagonaltermrightinside. pfa_gap_unit_equal_actualcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_unit_equal_actualcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_unit_equal_actualcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_unit_equal_actualcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_unit_equal_actualcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_unit_equal_actualcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_unit_equal_actualcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_unit_equal_actualcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_unit_equal_actualcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_unit_equal_actualcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_unit_equal_actualcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_unit_equal_actualcoefficientscoefficientdiagonal)=pfc_left_unit_equal_actualcoefficientscoefficientdiagonalterm*pfc_right_unit_equal_actualcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_unit_equal_actualcoefficientscoefficientsum fs_v_pfc_unit_equal_actualcoefficientscoefficientsum. ((((exists fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_start. fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_start. fs_u_pfc_unit_equal_actualcoefficientscoefficientsum = fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_terminal. fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_unit_equal_actualcoefficientscoefficient) = S ((S (S (pfc_index_unit_equal_actualcoefficients))) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_terminal. fs_u_pfc_unit_equal_actualcoefficientscoefficientsum = fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_unit_equal_actualcoefficients))) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum) + (pfc_natural_sum_unit_equal_actualcoefficientscoefficient))) /\\ forall fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps = S (pfc_index_unit_equal_actualcoefficients)) -> exists fs_a_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps fs_r_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps fs_s_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_unit_equal_actualcoefficientscoefficient)) /\\ exists fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_unit_equal_actualcoefficientscoefficient = fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_unit_equal_actualcoefficientscoefficient) + (fs_a_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_unit_equal_actualcoefficientscoefficientsum = fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum) + (fs_r_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_unit_equal_actualcoefficientscoefficientsum = fs_q_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_equal_actualcoefficientscoefficientsum) + (fs_s_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps = fs_r_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps + fs_a_pfc_unit_equal_actualcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_unit_equal_actualcoefficientscoefficientresiduebound. pfa_gap_unit_equal_actualcoefficientscoefficientresiduebound + S (pfc_value_unit_equal_actualcoefficients) = (p)) /\\ ((exists pfa_offset_left_unit_equal_actualcoefficientscoefficientresiduecongruence pfa_offset_right_unit_equal_actualcoefficientscoefficientresiduecongruence. (pfc_natural_sum_unit_equal_actualcoefficientscoefficient) + (p) * pfa_offset_left_unit_equal_actualcoefficientscoefficientresiduecongruence = (pfc_value_unit_equal_actualcoefficients) + (p) * pfa_offset_right_unit_equal_actualcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_unit_equivalent_result pfrep_left_unit_equivalent_result pfrep_right_unit_equivalent_result. ((exists pfrep_position_unit_equivalent_resultfirst. ((pfrep_position_unit_equivalent_resultfirst+S (pfrep_power_unit_equivalent_result)=(L)) /\\ ((((exists ff_h_pfp_unit_equivalent_resultfirstentry. ff_h_pfp_unit_equivalent_resultfirstentry + S (pfrep_left_unit_equivalent_result) = S ((S (pfrep_position_unit_equivalent_resultfirst)) * cc)) /\\ exists ff_q_pfp_unit_equivalent_resultfirstentry. cb = ff_q_pfp_unit_equivalent_resultfirstentry * S ((S (pfrep_position_unit_equivalent_resultfirst)) * cc) + (pfrep_left_unit_equivalent_result)))))) \\/ (((exists pfrep_gap_unit_equivalent_resultfirstoutside. pfrep_gap_unit_equivalent_resultfirstoutside+(L)=(pfrep_power_unit_equivalent_result)) /\\ (((pfrep_left_unit_equivalent_result)=0))))) -> ((exists pfrep_position_unit_equivalent_resultsecond. ((pfrep_position_unit_equivalent_resultsecond+S (pfrep_power_unit_equivalent_result)=(L)) /\\ ((((exists ff_h_pfp_unit_equivalent_resultsecondentry. ff_h_pfp_unit_equivalent_resultsecondentry + S (pfrep_right_unit_equivalent_result) = S ((S (pfrep_position_unit_equivalent_resultsecond)) * ac)) /\\ exists ff_q_pfp_unit_equivalent_resultsecondentry. ab = ff_q_pfp_unit_equivalent_resultsecondentry * S ((S (pfrep_position_unit_equivalent_resultsecond)) * ac) + (pfrep_right_unit_equivalent_result)))))) \\/ (((exists pfrep_gap_unit_equivalent_resultsecondoutside. pfrep_gap_unit_equivalent_resultsecondoutside+(L)=(pfrep_power_unit_equivalent_result)) /\\ (((pfrep_right_unit_equivalent_result)=0))))) -> pfrep_left_unit_equivalent_result=pfrep_right_unit_equivalent_result)", "statement_sha256": "3feadd7bc4a60b0ff0ac6e067b5d124e6155d16d8a88d90f0ce78a519896eebb", "summary": "Actual left multiplication by a length-one unit preserves formal coefficients, not just evaluations or a selected 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((forall fom_index_pfp_unit_exists_bound. (exists fom_gap_pfp_unit_exists_bound_index_bound. fom_gap_pfp_unit_exists_bound_index_bound + S (fom_index_pfp_unit_exists_bound) = 1) -> exists fom_value_pfp_unit_exists_bound. ((((exists fom_beta_height_pfp_unit_exists_bound_entry. fom_beta_height_pfp_unit_exists_bound_entry + S (fom_value_pfp_unit_exists_bound) = S ((S (fom_index_pfp_unit_exists_bound)) * uc)) /\\ exists fom_beta_quotient_pfp_unit_exists_bound_entry. ub = fom_beta_quotient_pfp_unit_exists_bound_entry * S ((S (fom_index_pfp_unit_exists_bound)) * uc) + (fom_value_pfp_unit_exists_bound))) /\\ (exists fom_gap_pfp_unit_exists_bound_value_bound. fom_gap_pfp_unit_exists_bound_value_bound + S (fom_value_pfp_unit_exists_bound) = p))) /\\ ((forall pfp_repeat_index_unit_exists_repeat. (exists pfa_gap_unit_exists_repeatindex. pfa_gap_unit_exists_repeatindex + S (pfp_repeat_index_unit_exists_repeat) = (1)) -> (((exists ff_h_pfp_unit_exists_repeatentry. ff_h_pfp_unit_exists_repeatentry + S (1) = S ((S (pfp_repeat_index_unit_exists_repeat)) * uc)) /\\ exists ff_q_pfp_unit_exists_repeatentry. ub = ff_q_pfp_unit_exists_repeatentry * S ((S (pfp_repeat_index_unit_exists_repeat)) * uc) + (1))))))", "specialize prime_field_polynomial_repeat_exists (p)", "specialize prime_field_polynomial_repeat_exists (1)", "specialize prime_field_polynomial_repeat_exists (1)", "apply prime_field_polynomial_repeat_exists", "specialize prime_two_le (p)", "apply prime_two_le", "exact hp", "cases hu", "cases hu_witness", "cases hu_witness_witness", "have hone : ((exists ff_h_pfp_unit_exists_value. ff_h_pfp_unit_exists_value + S (1) = S ((S (0)) * x1)) /\\ exists ff_q_pfp_unit_exists_value. x = ff_q_pfp_unit_exists_value * S ((S (0)) * x1) + (1))", "specialize hu_witness_witness_right (0)", "apply hu_witness_witness_right", "exists 0", "apply zero_add", "have hc : exists cb cc. (((forall fom_index_pfp_unit_exists_chosenleft. (exists fom_gap_pfp_unit_exists_chosenleft_index_bound. fom_gap_pfp_unit_exists_chosenleft_index_bound + S (fom_index_pfp_unit_exists_chosenleft) = 1) -> exists fom_value_pfp_unit_exists_chosenleft. ((((exists fom_beta_height_pfp_unit_exists_chosenleft_entry. fom_beta_height_pfp_unit_exists_chosenleft_entry + S (fom_value_pfp_unit_exists_chosenleft) = S ((S (fom_index_pfp_unit_exists_chosenleft)) * x1)) /\\ exists fom_beta_quotient_pfp_unit_exists_chosenleft_entry. x = fom_beta_quotient_pfp_unit_exists_chosenleft_entry * S ((S (fom_index_pfp_unit_exists_chosenleft)) * x1) + (fom_value_pfp_unit_exists_chosenleft))) /\\ (exists fom_gap_pfp_unit_exists_chosenleft_value_bound. fom_gap_pfp_unit_exists_chosenleft_value_bound + S (fom_value_pfp_unit_exists_chosenleft) = p))) /\\ (((forall fom_index_pfp_unit_exists_chosenright. (exists fom_gap_pfp_unit_exists_chosenright_index_bound. fom_gap_pfp_unit_exists_chosenright_index_bound + S (fom_index_pfp_unit_exists_chosenright) = L) -> exists fom_value_pfp_unit_exists_chosenright. ((((exists fom_beta_height_pfp_unit_exists_chosenright_entry. fom_beta_height_pfp_unit_exists_chosenright_entry + S (fom_value_pfp_unit_exists_chosenright) = S ((S (fom_index_pfp_unit_exists_chosenright)) * ac)) /\\ exists fom_beta_quotient_pfp_unit_exists_chosenright_entry. ab = fom_beta_quotient_pfp_unit_exists_chosenright_entry * S ((S (fom_index_pfp_unit_exists_chosenright)) * ac) + (fom_value_pfp_unit_exists_chosenright))) /\\ (exists fom_gap_pfp_unit_exists_chosenright_value_bound. fom_gap_pfp_unit_exists_chosenright_value_bound + S (fom_value_pfp_unit_exists_chosenright) = p))) /\\ (((((((1)=0 \\/ (L)=0) /\\ (((L)=0)))) \\/ (((~((1)=0)) /\\ (((~((L)=0)) /\\ (((1)+(L)=S (L)))))))) /\\ ((forall pfc_index_unit_exists_chosencoefficients. (exists pfa_gap_unit_exists_chosencoefficientsbound. pfa_gap_unit_exists_chosencoefficientsbound + S (pfc_index_unit_exists_chosencoefficients) = (L)) -> exists pfc_value_unit_exists_chosencoefficients. ((((exists ff_h_pfp_unit_exists_chosencoefficientsentry. ff_h_pfp_unit_exists_chosencoefficientsentry + S (pfc_value_unit_exists_chosencoefficients) = S ((S (pfc_index_unit_exists_chosencoefficients)) * cc)) /\\ exists ff_q_pfp_unit_exists_chosencoefficientsentry. cb = ff_q_pfp_unit_exists_chosencoefficientsentry * S ((S (pfc_index_unit_exists_chosencoefficients)) * cc) + (pfc_value_unit_exists_chosencoefficients))) /\\ ((exists pfc_terms_code_unit_exists_chosencoefficientscoefficient pfc_terms_scale_unit_exists_chosencoefficientscoefficient pfc_natural_sum_unit_exists_chosencoefficientscoefficient. ((forall pfc_index_unit_exists_chosencoefficientscoefficientdiagonal. (exists pfa_gap_unit_exists_chosencoefficientscoefficientdiagonalbound. pfa_gap_unit_exists_chosencoefficientscoefficientdiagonalbound + S (pfc_index_unit_exists_chosencoefficientscoefficientdiagonal) = (S (pfc_index_unit_exists_chosencoefficients))) -> exists pfc_value_unit_exists_chosencoefficientscoefficientdiagonal. ((((exists ff_h_pfp_unit_exists_chosencoefficientscoefficientdiagonalentry. ff_h_pfp_unit_exists_chosencoefficientscoefficientdiagonalentry + S (pfc_value_unit_exists_chosencoefficientscoefficientdiagonal) = S ((S (pfc_index_unit_exists_chosencoefficientscoefficientdiagonal)) * pfc_terms_scale_unit_exists_chosencoefficientscoefficient)) /\\ exists ff_q_pfp_unit_exists_chosencoefficientscoefficientdiagonalentry. pfc_terms_code_unit_exists_chosencoefficientscoefficient = ff_q_pfp_unit_exists_chosencoefficientscoefficientdiagonalentry * S ((S (pfc_index_unit_exists_chosencoefficientscoefficientdiagonal)) * pfc_terms_scale_unit_exists_chosencoefficientscoefficient) + (pfc_value_unit_exists_chosencoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_unit_exists_chosencoefficientscoefficientdiagonalterm pfc_left_unit_exists_chosencoefficientscoefficientdiagonalterm pfc_right_unit_exists_chosencoefficientscoefficientdiagonalterm. (((pfc_index_unit_exists_chosencoefficientscoefficientdiagonal)+pfc_complement_unit_exists_chosencoefficientscoefficientdiagonalterm=(pfc_index_unit_exists_chosencoefficients)) /\\ ((((((exists pfa_gap_unit_exists_chosencoefficientscoefficientdiagonaltermleftinside. pfa_gap_unit_exists_chosencoefficientscoefficientdiagonaltermleftinside + S (pfc_index_unit_exists_chosencoefficientscoefficientdiagonal) = (1)) /\\ ((((exists ff_h_pfp_unit_exists_chosencoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_unit_exists_chosencoefficientscoefficientdiagonaltermleftentry + S (pfc_left_unit_exists_chosencoefficientscoefficientdiagonalterm) = S ((S (pfc_index_unit_exists_chosencoefficientscoefficientdiagonal)) * x1)) /\\ exists ff_q_pfp_unit_exists_chosencoefficientscoefficientdiagonaltermleftentry. x = ff_q_pfp_unit_exists_chosencoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_unit_exists_chosencoefficientscoefficientdiagonal)) * x1) + (pfc_left_unit_exists_chosencoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_unit_exists_chosencoefficientscoefficientdiagonaltermleftoutside. pfc_gap_unit_exists_chosencoefficientscoefficientdiagonaltermleftoutside+(1)=(pfc_index_unit_exists_chosencoefficientscoefficientdiagonal)) /\\ (((pfc_left_unit_exists_chosencoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_unit_exists_chosencoefficientscoefficientdiagonaltermrightinside. pfa_gap_unit_exists_chosencoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_unit_exists_chosencoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_unit_exists_chosencoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_unit_exists_chosencoefficientscoefficientdiagonaltermrightentry + S (pfc_right_unit_exists_chosencoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_unit_exists_chosencoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_unit_exists_chosencoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_unit_exists_chosencoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_unit_exists_chosencoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_unit_exists_chosencoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_unit_exists_chosencoefficientscoefficientdiagonaltermrightoutside. pfc_gap_unit_exists_chosencoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_unit_exists_chosencoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_unit_exists_chosencoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_unit_exists_chosencoefficientscoefficientdiagonal)=pfc_left_unit_exists_chosencoefficientscoefficientdiagonalterm*pfc_right_unit_exists_chosencoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_unit_exists_chosencoefficientscoefficientsum fs_v_pfc_unit_exists_chosencoefficientscoefficientsum. ((((exists fs_h_pfc_unit_exists_chosencoefficientscoefficientsum_body_start. fs_h_pfc_unit_exists_chosencoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_unit_exists_chosencoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_exists_chosencoefficientscoefficientsum_body_start. fs_u_pfc_unit_exists_chosencoefficientscoefficientsum = fs_q_pfc_unit_exists_chosencoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_unit_exists_chosencoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_unit_exists_chosencoefficientscoefficientsum_body_terminal. fs_h_pfc_unit_exists_chosencoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_unit_exists_chosencoefficientscoefficient) = S ((S (S (pfc_index_unit_exists_chosencoefficients))) * fs_v_pfc_unit_exists_chosencoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_exists_chosencoefficientscoefficientsum_body_terminal. fs_u_pfc_unit_exists_chosencoefficientscoefficientsum = fs_q_pfc_unit_exists_chosencoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_unit_exists_chosencoefficients))) * fs_v_pfc_unit_exists_chosencoefficientscoefficientsum) + (pfc_natural_sum_unit_exists_chosencoefficientscoefficient))) /\\ forall fs_i_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps = S (pfc_index_unit_exists_chosencoefficients)) -> exists fs_a_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps fs_r_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps fs_s_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps_summand. fs_h_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps)) * pfc_terms_scale_unit_exists_chosencoefficientscoefficient)) /\\ exists fs_q_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps_summand. pfc_terms_code_unit_exists_chosencoefficientscoefficient = fs_q_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps)) * pfc_terms_scale_unit_exists_chosencoefficientscoefficient) + (fs_a_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps_partial. fs_h_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_exists_chosencoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps_partial. fs_u_pfc_unit_exists_chosencoefficientscoefficientsum = fs_q_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_exists_chosencoefficientscoefficientsum) + (fs_r_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps_successor. fs_h_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_exists_chosencoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps_successor. fs_u_pfc_unit_exists_chosencoefficientscoefficientsum = fs_q_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_exists_chosencoefficientscoefficientsum) + (fs_s_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps = fs_r_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps + fs_a_pfc_unit_exists_chosencoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_unit_exists_chosencoefficientscoefficientresiduebound. pfa_gap_unit_exists_chosencoefficientscoefficientresiduebound + S (pfc_value_unit_exists_chosencoefficients) = (p)) /\\ ((exists pfa_offset_left_unit_exists_chosencoefficientscoefficientresiduecongruence pfa_offset_right_unit_exists_chosencoefficientscoefficientresiduecongruence. (pfc_natural_sum_unit_exists_chosencoefficientscoefficient) + (p) * pfa_offset_left_unit_exists_chosencoefficientscoefficientresiduecongruence = (pfc_value_unit_exists_chosencoefficients) + (p) * pfa_offset_right_unit_exists_chosencoefficientscoefficientresiduecongruence)))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (x)", "specialize prime_field_polynomial_convolution_at_length_exists (x1)", "specialize prime_field_polynomial_convolution_at_length_exists (1)", "specialize prime_field_polynomial_convolution_at_length_exists (ab)", "specialize prime_field_polynomial_convolution_at_length_exists (ac)", "specialize prime_field_polynomial_convolution_at_length_exists (L)", "specialize prime_field_polynomial_convolution_at_length_exists (L)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hu_witness_witness_left", "exact hA", "have hzero : L=0 \\/ ~(L=0)", "specialize eq_decidable (L)", "specialize eq_decidable (0)", "apply eq_decidable", "cases hzero", "left", "split", "right", "exact hzero_left", "exact hzero_left", "right", "split", "specialize succ_ne_zero (0)", "apply succ_ne_zero", "split", "exact hzero_right", "simp [add_succ_left,zero_add]", "cases hc", "cases hc_witness", "exists x", "exists x1", "exists x2", "exists x3", "split", "exact hu_witness_witness_left", "split", "exact hone", "split", "exact hc_witness_witness", "specialize prime_field_polynomial_convolution_left_unit_equivalent (p)", "specialize prime_field_polynomial_convolution_left_unit_equivalent (x)", "specialize prime_field_polynomial_convolution_left_unit_equivalent (x1)", "specialize prime_field_polynomial_convolution_left_unit_equivalent (ab)", "specialize prime_field_polynomial_convolution_left_unit_equivalent (ac)", "specialize prime_field_polynomial_convolution_left_unit_equivalent (L)", "specialize prime_field_polynomial_convolution_left_unit_equivalent (x2)", "specialize prime_field_polynomial_convolution_left_unit_equivalent (x3)", "apply prime_field_polynomial_convolution_left_unit_equivalent", "exact hone", "exact hc_witness_witness" ], "script_sha256": "cb736259422e26b2e34db01b8a94edb3854ea05b51659fba03b9910d32bea5c3", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_left_unit_candidate.py", "sha256": "dbb8debb4716b6bb9b246700f7e93865c8a6c1b12a3b65c0ffbb62206a890ba6" }, "statement": "forall p ab ac L. (~((p) = 1) /\\ forall pfa_factor_left_unit_exists_prime pfa_factor_right_unit_exists_prime. (p) = pfa_factor_left_unit_exists_prime * pfa_factor_right_unit_exists_prime -> pfa_factor_left_unit_exists_prime = 1 \\/ pfa_factor_right_unit_exists_prime = 1) -> (forall fom_index_pfp_unit_exists_A. (exists fom_gap_pfp_unit_exists_A_index_bound. fom_gap_pfp_unit_exists_A_index_bound + S (fom_index_pfp_unit_exists_A) = L) -> exists fom_value_pfp_unit_exists_A. ((((exists fom_beta_height_pfp_unit_exists_A_entry. fom_beta_height_pfp_unit_exists_A_entry + S (fom_value_pfp_unit_exists_A) = S ((S (fom_index_pfp_unit_exists_A)) * ac)) /\\ exists fom_beta_quotient_pfp_unit_exists_A_entry. ab = fom_beta_quotient_pfp_unit_exists_A_entry * S ((S (fom_index_pfp_unit_exists_A)) * ac) + (fom_value_pfp_unit_exists_A))) /\\ (exists fom_gap_pfp_unit_exists_A_value_bound. fom_gap_pfp_unit_exists_A_value_bound + S (fom_value_pfp_unit_exists_A) = p))) -> (exists ub uc cb cc. ((forall fom_index_pfp_unit_exists_result_unit_bound. (exists fom_gap_pfp_unit_exists_result_unit_bound_index_bound. fom_gap_pfp_unit_exists_result_unit_bound_index_bound + S (fom_index_pfp_unit_exists_result_unit_bound) = 1) -> exists fom_value_pfp_unit_exists_result_unit_bound. ((((exists fom_beta_height_pfp_unit_exists_result_unit_bound_entry. fom_beta_height_pfp_unit_exists_result_unit_bound_entry + S (fom_value_pfp_unit_exists_result_unit_bound) = S ((S (fom_index_pfp_unit_exists_result_unit_bound)) * uc)) /\\ exists fom_beta_quotient_pfp_unit_exists_result_unit_bound_entry. ub = fom_beta_quotient_pfp_unit_exists_result_unit_bound_entry * S ((S (fom_index_pfp_unit_exists_result_unit_bound)) * uc) + (fom_value_pfp_unit_exists_result_unit_bound))) /\\ (exists fom_gap_pfp_unit_exists_result_unit_bound_value_bound. fom_gap_pfp_unit_exists_result_unit_bound_value_bound + S (fom_value_pfp_unit_exists_result_unit_bound) = p))) /\\ (((((exists ff_h_pfp_unit_exists_result_unit_value. ff_h_pfp_unit_exists_result_unit_value + S (1) = S ((S (0)) * uc)) /\\ exists ff_q_pfp_unit_exists_result_unit_value. ub = ff_q_pfp_unit_exists_result_unit_value * S ((S (0)) * uc) + (1))) /\\ (((((forall fom_index_pfp_unit_exists_result_productleft. (exists fom_gap_pfp_unit_exists_result_productleft_index_bound. fom_gap_pfp_unit_exists_result_productleft_index_bound + S (fom_index_pfp_unit_exists_result_productleft) = 1) -> exists fom_value_pfp_unit_exists_result_productleft. ((((exists fom_beta_height_pfp_unit_exists_result_productleft_entry. fom_beta_height_pfp_unit_exists_result_productleft_entry + S (fom_value_pfp_unit_exists_result_productleft) = S ((S (fom_index_pfp_unit_exists_result_productleft)) * uc)) /\\ exists fom_beta_quotient_pfp_unit_exists_result_productleft_entry. ub = fom_beta_quotient_pfp_unit_exists_result_productleft_entry * S ((S (fom_index_pfp_unit_exists_result_productleft)) * uc) + (fom_value_pfp_unit_exists_result_productleft))) /\\ (exists fom_gap_pfp_unit_exists_result_productleft_value_bound. fom_gap_pfp_unit_exists_result_productleft_value_bound + S (fom_value_pfp_unit_exists_result_productleft) = p))) /\\ (((forall fom_index_pfp_unit_exists_result_productright. (exists fom_gap_pfp_unit_exists_result_productright_index_bound. fom_gap_pfp_unit_exists_result_productright_index_bound + S (fom_index_pfp_unit_exists_result_productright) = L) -> exists fom_value_pfp_unit_exists_result_productright. ((((exists fom_beta_height_pfp_unit_exists_result_productright_entry. fom_beta_height_pfp_unit_exists_result_productright_entry + S (fom_value_pfp_unit_exists_result_productright) = S ((S (fom_index_pfp_unit_exists_result_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_unit_exists_result_productright_entry. ab = fom_beta_quotient_pfp_unit_exists_result_productright_entry * S ((S (fom_index_pfp_unit_exists_result_productright)) * ac) + (fom_value_pfp_unit_exists_result_productright))) /\\ (exists fom_gap_pfp_unit_exists_result_productright_value_bound. fom_gap_pfp_unit_exists_result_productright_value_bound + S (fom_value_pfp_unit_exists_result_productright) = p))) /\\ (((((((1)=0 \\/ (L)=0) /\\ (((L)=0)))) \\/ (((~((1)=0)) /\\ (((~((L)=0)) /\\ (((1)+(L)=S (L)))))))) /\\ ((forall pfc_index_unit_exists_result_productcoefficients. (exists pfa_gap_unit_exists_result_productcoefficientsbound. pfa_gap_unit_exists_result_productcoefficientsbound + S (pfc_index_unit_exists_result_productcoefficients) = (L)) -> exists pfc_value_unit_exists_result_productcoefficients. ((((exists ff_h_pfp_unit_exists_result_productcoefficientsentry. ff_h_pfp_unit_exists_result_productcoefficientsentry + S (pfc_value_unit_exists_result_productcoefficients) = S ((S (pfc_index_unit_exists_result_productcoefficients)) * cc)) /\\ exists ff_q_pfp_unit_exists_result_productcoefficientsentry. cb = ff_q_pfp_unit_exists_result_productcoefficientsentry * S ((S (pfc_index_unit_exists_result_productcoefficients)) * cc) + (pfc_value_unit_exists_result_productcoefficients))) /\\ ((exists pfc_terms_code_unit_exists_result_productcoefficientscoefficient pfc_terms_scale_unit_exists_result_productcoefficientscoefficient pfc_natural_sum_unit_exists_result_productcoefficientscoefficient. ((forall pfc_index_unit_exists_result_productcoefficientscoefficientdiagonal. (exists pfa_gap_unit_exists_result_productcoefficientscoefficientdiagonalbound. pfa_gap_unit_exists_result_productcoefficientscoefficientdiagonalbound + S (pfc_index_unit_exists_result_productcoefficientscoefficientdiagonal) = (S (pfc_index_unit_exists_result_productcoefficients))) -> exists pfc_value_unit_exists_result_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_unit_exists_result_productcoefficientscoefficientdiagonalentry. ff_h_pfp_unit_exists_result_productcoefficientscoefficientdiagonalentry + S (pfc_value_unit_exists_result_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_unit_exists_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_unit_exists_result_productcoefficientscoefficient)) /\\ exists ff_q_pfp_unit_exists_result_productcoefficientscoefficientdiagonalentry. pfc_terms_code_unit_exists_result_productcoefficientscoefficient = ff_q_pfp_unit_exists_result_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_unit_exists_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_unit_exists_result_productcoefficientscoefficient) + (pfc_value_unit_exists_result_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_unit_exists_result_productcoefficientscoefficientdiagonalterm pfc_left_unit_exists_result_productcoefficientscoefficientdiagonalterm pfc_right_unit_exists_result_productcoefficientscoefficientdiagonalterm. (((pfc_index_unit_exists_result_productcoefficientscoefficientdiagonal)+pfc_complement_unit_exists_result_productcoefficientscoefficientdiagonalterm=(pfc_index_unit_exists_result_productcoefficients)) /\\ ((((((exists pfa_gap_unit_exists_result_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_unit_exists_result_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_unit_exists_result_productcoefficientscoefficientdiagonal) = (1)) /\\ ((((exists ff_h_pfp_unit_exists_result_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_unit_exists_result_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_unit_exists_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_unit_exists_result_productcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_unit_exists_result_productcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_unit_exists_result_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_unit_exists_result_productcoefficientscoefficientdiagonal)) * uc) + (pfc_left_unit_exists_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_unit_exists_result_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_unit_exists_result_productcoefficientscoefficientdiagonaltermleftoutside+(1)=(pfc_index_unit_exists_result_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_unit_exists_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_unit_exists_result_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_unit_exists_result_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_unit_exists_result_productcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_unit_exists_result_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_unit_exists_result_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_unit_exists_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_unit_exists_result_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_unit_exists_result_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_unit_exists_result_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_unit_exists_result_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_unit_exists_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_unit_exists_result_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_unit_exists_result_productcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_unit_exists_result_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_unit_exists_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_unit_exists_result_productcoefficientscoefficientdiagonal)=pfc_left_unit_exists_result_productcoefficientscoefficientdiagonalterm*pfc_right_unit_exists_result_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_unit_exists_result_productcoefficientscoefficientsum fs_v_pfc_unit_exists_result_productcoefficientscoefficientsum. ((((exists fs_h_pfc_unit_exists_result_productcoefficientscoefficientsum_body_start. fs_h_pfc_unit_exists_result_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_unit_exists_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_exists_result_productcoefficientscoefficientsum_body_start. fs_u_pfc_unit_exists_result_productcoefficientscoefficientsum = fs_q_pfc_unit_exists_result_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_unit_exists_result_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_unit_exists_result_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_unit_exists_result_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_unit_exists_result_productcoefficientscoefficient) = S ((S (S (pfc_index_unit_exists_result_productcoefficients))) * fs_v_pfc_unit_exists_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_exists_result_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_unit_exists_result_productcoefficientscoefficientsum = fs_q_pfc_unit_exists_result_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_unit_exists_result_productcoefficients))) * fs_v_pfc_unit_exists_result_productcoefficientscoefficientsum) + (pfc_natural_sum_unit_exists_result_productcoefficientscoefficient))) /\\ forall fs_i_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps = S (pfc_index_unit_exists_result_productcoefficients)) -> exists fs_a_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps fs_r_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps fs_s_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_unit_exists_result_productcoefficientscoefficient)) /\\ exists fs_q_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_unit_exists_result_productcoefficientscoefficient = fs_q_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_unit_exists_result_productcoefficientscoefficient) + (fs_a_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_exists_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_unit_exists_result_productcoefficientscoefficientsum = fs_q_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_exists_result_productcoefficientscoefficientsum) + (fs_r_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_exists_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_unit_exists_result_productcoefficientscoefficientsum = fs_q_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_exists_result_productcoefficientscoefficientsum) + (fs_s_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps = fs_r_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps + fs_a_pfc_unit_exists_result_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_unit_exists_result_productcoefficientscoefficientresiduebound. pfa_gap_unit_exists_result_productcoefficientscoefficientresiduebound + S (pfc_value_unit_exists_result_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_unit_exists_result_productcoefficientscoefficientresiduecongruence pfa_offset_right_unit_exists_result_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_unit_exists_result_productcoefficientscoefficient) + (p) * pfa_offset_left_unit_exists_result_productcoefficientscoefficientresiduecongruence = (pfc_value_unit_exists_result_productcoefficients) + (p) * pfa_offset_right_unit_exists_result_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_unit_exists_result_equivalent pfrep_left_unit_exists_result_equivalent pfrep_right_unit_exists_result_equivalent. ((exists pfrep_position_unit_exists_result_equivalentfirst. ((pfrep_position_unit_exists_result_equivalentfirst+S (pfrep_power_unit_exists_result_equivalent)=(L)) /\\ ((((exists ff_h_pfp_unit_exists_result_equivalentfirstentry. ff_h_pfp_unit_exists_result_equivalentfirstentry + S (pfrep_left_unit_exists_result_equivalent) = S ((S (pfrep_position_unit_exists_result_equivalentfirst)) * cc)) /\\ exists ff_q_pfp_unit_exists_result_equivalentfirstentry. cb = ff_q_pfp_unit_exists_result_equivalentfirstentry * S ((S (pfrep_position_unit_exists_result_equivalentfirst)) * cc) + (pfrep_left_unit_exists_result_equivalent)))))) \\/ (((exists pfrep_gap_unit_exists_result_equivalentfirstoutside. pfrep_gap_unit_exists_result_equivalentfirstoutside+(L)=(pfrep_power_unit_exists_result_equivalent)) /\\ (((pfrep_left_unit_exists_result_equivalent)=0))))) -> ((exists pfrep_position_unit_exists_result_equivalentsecond. ((pfrep_position_unit_exists_result_equivalentsecond+S (pfrep_power_unit_exists_result_equivalent)=(L)) /\\ ((((exists ff_h_pfp_unit_exists_result_equivalentsecondentry. ff_h_pfp_unit_exists_result_equivalentsecondentry + S (pfrep_right_unit_exists_result_equivalent) = S ((S (pfrep_position_unit_exists_result_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_unit_exists_result_equivalentsecondentry. ab = ff_q_pfp_unit_exists_result_equivalentsecondentry * S ((S (pfrep_position_unit_exists_result_equivalentsecond)) * ac) + (pfrep_right_unit_exists_result_equivalent)))))) \\/ (((exists pfrep_gap_unit_exists_result_equivalentsecondoutside. pfrep_gap_unit_exists_result_equivalentsecondoutside+(L)=(pfrep_power_unit_exists_result_equivalent)) /\\ (((pfrep_right_unit_exists_result_equivalent)=0))))) -> pfrep_left_unit_exists_result_equivalent=pfrep_right_unit_exists_result_equivalent))))))))", "statement_sha256": "963a274110f5998e09e17f6031760ed4662f9bee1c452fee0388dd3d450bde13", "summary": "Construct an actual canonical length-one unit and its actual length-L left product, formally equal to A. 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((forall fom_index_pfp_unit_divides_witness_unit_bound. (exists fom_gap_pfp_unit_divides_witness_unit_bound_index_bound. fom_gap_pfp_unit_divides_witness_unit_bound_index_bound + S (fom_index_pfp_unit_divides_witness_unit_bound) = 1) -> exists fom_value_pfp_unit_divides_witness_unit_bound. ((((exists fom_beta_height_pfp_unit_divides_witness_unit_bound_entry. fom_beta_height_pfp_unit_divides_witness_unit_bound_entry + S (fom_value_pfp_unit_divides_witness_unit_bound) = S ((S (fom_index_pfp_unit_divides_witness_unit_bound)) * uc)) /\\ exists fom_beta_quotient_pfp_unit_divides_witness_unit_bound_entry. ub = fom_beta_quotient_pfp_unit_divides_witness_unit_bound_entry * S ((S (fom_index_pfp_unit_divides_witness_unit_bound)) * uc) + (fom_value_pfp_unit_divides_witness_unit_bound))) /\\ (exists fom_gap_pfp_unit_divides_witness_unit_bound_value_bound. fom_gap_pfp_unit_divides_witness_unit_bound_value_bound + S (fom_value_pfp_unit_divides_witness_unit_bound) = p))) /\\ (((((exists ff_h_pfp_unit_divides_witness_unit_value. ff_h_pfp_unit_divides_witness_unit_value + S (1) = S ((S (0)) * uc)) /\\ exists ff_q_pfp_unit_divides_witness_unit_value. ub = ff_q_pfp_unit_divides_witness_unit_value * S ((S (0)) * uc) + (1))) /\\ (((((forall fom_index_pfp_unit_divides_witness_productleft. (exists fom_gap_pfp_unit_divides_witness_productleft_index_bound. fom_gap_pfp_unit_divides_witness_productleft_index_bound + S (fom_index_pfp_unit_divides_witness_productleft) = 1) -> exists fom_value_pfp_unit_divides_witness_productleft. ((((exists fom_beta_height_pfp_unit_divides_witness_productleft_entry. fom_beta_height_pfp_unit_divides_witness_productleft_entry + S (fom_value_pfp_unit_divides_witness_productleft) = S ((S (fom_index_pfp_unit_divides_witness_productleft)) * uc)) /\\ exists fom_beta_quotient_pfp_unit_divides_witness_productleft_entry. ub = fom_beta_quotient_pfp_unit_divides_witness_productleft_entry * S ((S (fom_index_pfp_unit_divides_witness_productleft)) * uc) + (fom_value_pfp_unit_divides_witness_productleft))) /\\ (exists fom_gap_pfp_unit_divides_witness_productleft_value_bound. fom_gap_pfp_unit_divides_witness_productleft_value_bound + S (fom_value_pfp_unit_divides_witness_productleft) = p))) /\\ (((forall fom_index_pfp_unit_divides_witness_productright. (exists fom_gap_pfp_unit_divides_witness_productright_index_bound. fom_gap_pfp_unit_divides_witness_productright_index_bound + S (fom_index_pfp_unit_divides_witness_productright) = L) -> exists fom_value_pfp_unit_divides_witness_productright. ((((exists fom_beta_height_pfp_unit_divides_witness_productright_entry. fom_beta_height_pfp_unit_divides_witness_productright_entry + S (fom_value_pfp_unit_divides_witness_productright) = S ((S (fom_index_pfp_unit_divides_witness_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_unit_divides_witness_productright_entry. ab = fom_beta_quotient_pfp_unit_divides_witness_productright_entry * S ((S (fom_index_pfp_unit_divides_witness_productright)) * ac) + (fom_value_pfp_unit_divides_witness_productright))) /\\ (exists fom_gap_pfp_unit_divides_witness_productright_value_bound. fom_gap_pfp_unit_divides_witness_productright_value_bound + S (fom_value_pfp_unit_divides_witness_productright) = p))) /\\ (((((((1)=0 \\/ (L)=0) /\\ (((L)=0)))) \\/ (((~((1)=0)) /\\ (((~((L)=0)) /\\ (((1)+(L)=S (L)))))))) /\\ ((forall pfc_index_unit_divides_witness_productcoefficients. (exists pfa_gap_unit_divides_witness_productcoefficientsbound. pfa_gap_unit_divides_witness_productcoefficientsbound + S (pfc_index_unit_divides_witness_productcoefficients) = (L)) -> exists pfc_value_unit_divides_witness_productcoefficients. ((((exists ff_h_pfp_unit_divides_witness_productcoefficientsentry. ff_h_pfp_unit_divides_witness_productcoefficientsentry + S (pfc_value_unit_divides_witness_productcoefficients) = S ((S (pfc_index_unit_divides_witness_productcoefficients)) * cc)) /\\ exists ff_q_pfp_unit_divides_witness_productcoefficientsentry. cb = ff_q_pfp_unit_divides_witness_productcoefficientsentry * S ((S (pfc_index_unit_divides_witness_productcoefficients)) * cc) + (pfc_value_unit_divides_witness_productcoefficients))) /\\ ((exists pfc_terms_code_unit_divides_witness_productcoefficientscoefficient pfc_terms_scale_unit_divides_witness_productcoefficientscoefficient pfc_natural_sum_unit_divides_witness_productcoefficientscoefficient. ((forall pfc_index_unit_divides_witness_productcoefficientscoefficientdiagonal. (exists pfa_gap_unit_divides_witness_productcoefficientscoefficientdiagonalbound. pfa_gap_unit_divides_witness_productcoefficientscoefficientdiagonalbound + S (pfc_index_unit_divides_witness_productcoefficientscoefficientdiagonal) = (S (pfc_index_unit_divides_witness_productcoefficients))) -> exists pfc_value_unit_divides_witness_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_unit_divides_witness_productcoefficientscoefficientdiagonalentry. ff_h_pfp_unit_divides_witness_productcoefficientscoefficientdiagonalentry + S (pfc_value_unit_divides_witness_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_unit_divides_witness_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_unit_divides_witness_productcoefficientscoefficient)) /\\ exists ff_q_pfp_unit_divides_witness_productcoefficientscoefficientdiagonalentry. pfc_terms_code_unit_divides_witness_productcoefficientscoefficient = ff_q_pfp_unit_divides_witness_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_unit_divides_witness_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_unit_divides_witness_productcoefficientscoefficient) + (pfc_value_unit_divides_witness_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_unit_divides_witness_productcoefficientscoefficientdiagonalterm pfc_left_unit_divides_witness_productcoefficientscoefficientdiagonalterm pfc_right_unit_divides_witness_productcoefficientscoefficientdiagonalterm. (((pfc_index_unit_divides_witness_productcoefficientscoefficientdiagonal)+pfc_complement_unit_divides_witness_productcoefficientscoefficientdiagonalterm=(pfc_index_unit_divides_witness_productcoefficients)) /\\ ((((((exists pfa_gap_unit_divides_witness_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_unit_divides_witness_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_unit_divides_witness_productcoefficientscoefficientdiagonal) = (1)) /\\ ((((exists ff_h_pfp_unit_divides_witness_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_unit_divides_witness_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_unit_divides_witness_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_unit_divides_witness_productcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_unit_divides_witness_productcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_unit_divides_witness_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_unit_divides_witness_productcoefficientscoefficientdiagonal)) * uc) + (pfc_left_unit_divides_witness_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_unit_divides_witness_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_unit_divides_witness_productcoefficientscoefficientdiagonaltermleftoutside+(1)=(pfc_index_unit_divides_witness_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_unit_divides_witness_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_unit_divides_witness_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_unit_divides_witness_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_unit_divides_witness_productcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_unit_divides_witness_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_unit_divides_witness_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_unit_divides_witness_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_unit_divides_witness_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_unit_divides_witness_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_unit_divides_witness_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_unit_divides_witness_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_unit_divides_witness_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_unit_divides_witness_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_unit_divides_witness_productcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_unit_divides_witness_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_unit_divides_witness_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_unit_divides_witness_productcoefficientscoefficientdiagonal)=pfc_left_unit_divides_witness_productcoefficientscoefficientdiagonalterm*pfc_right_unit_divides_witness_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_unit_divides_witness_productcoefficientscoefficientsum fs_v_pfc_unit_divides_witness_productcoefficientscoefficientsum. ((((exists fs_h_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_start. fs_h_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_unit_divides_witness_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_start. fs_u_pfc_unit_divides_witness_productcoefficientscoefficientsum = fs_q_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_unit_divides_witness_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_unit_divides_witness_productcoefficientscoefficient) = S ((S (S (pfc_index_unit_divides_witness_productcoefficients))) * fs_v_pfc_unit_divides_witness_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_unit_divides_witness_productcoefficientscoefficientsum = fs_q_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_unit_divides_witness_productcoefficients))) * fs_v_pfc_unit_divides_witness_productcoefficientscoefficientsum) + (pfc_natural_sum_unit_divides_witness_productcoefficientscoefficient))) /\\ forall fs_i_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps = S (pfc_index_unit_divides_witness_productcoefficients)) -> exists fs_a_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps fs_r_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps fs_s_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_unit_divides_witness_productcoefficientscoefficient)) /\\ exists fs_q_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_unit_divides_witness_productcoefficientscoefficient = fs_q_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_unit_divides_witness_productcoefficientscoefficient) + (fs_a_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_divides_witness_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_unit_divides_witness_productcoefficientscoefficientsum = fs_q_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_divides_witness_productcoefficientscoefficientsum) + (fs_r_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_divides_witness_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_unit_divides_witness_productcoefficientscoefficientsum = fs_q_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_divides_witness_productcoefficientscoefficientsum) + (fs_s_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps = fs_r_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps + fs_a_pfc_unit_divides_witness_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_unit_divides_witness_productcoefficientscoefficientresiduebound. pfa_gap_unit_divides_witness_productcoefficientscoefficientresiduebound + S (pfc_value_unit_divides_witness_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_unit_divides_witness_productcoefficientscoefficientresiduecongruence pfa_offset_right_unit_divides_witness_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_unit_divides_witness_productcoefficientscoefficient) + (p) * pfa_offset_left_unit_divides_witness_productcoefficientscoefficientresiduecongruence = (pfc_value_unit_divides_witness_productcoefficients) + (p) * pfa_offset_right_unit_divides_witness_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_unit_divides_witness_equivalent pfrep_left_unit_divides_witness_equivalent pfrep_right_unit_divides_witness_equivalent. ((exists pfrep_position_unit_divides_witness_equivalentfirst. ((pfrep_position_unit_divides_witness_equivalentfirst+S (pfrep_power_unit_divides_witness_equivalent)=(L)) /\\ ((((exists ff_h_pfp_unit_divides_witness_equivalentfirstentry. ff_h_pfp_unit_divides_witness_equivalentfirstentry + S (pfrep_left_unit_divides_witness_equivalent) = S ((S (pfrep_position_unit_divides_witness_equivalentfirst)) * cc)) /\\ exists ff_q_pfp_unit_divides_witness_equivalentfirstentry. cb = ff_q_pfp_unit_divides_witness_equivalentfirstentry * S ((S (pfrep_position_unit_divides_witness_equivalentfirst)) * cc) + (pfrep_left_unit_divides_witness_equivalent)))))) \\/ (((exists pfrep_gap_unit_divides_witness_equivalentfirstoutside. pfrep_gap_unit_divides_witness_equivalentfirstoutside+(L)=(pfrep_power_unit_divides_witness_equivalent)) /\\ (((pfrep_left_unit_divides_witness_equivalent)=0))))) -> ((exists pfrep_position_unit_divides_witness_equivalentsecond. ((pfrep_position_unit_divides_witness_equivalentsecond+S (pfrep_power_unit_divides_witness_equivalent)=(L)) /\\ ((((exists ff_h_pfp_unit_divides_witness_equivalentsecondentry. ff_h_pfp_unit_divides_witness_equivalentsecondentry + S (pfrep_right_unit_divides_witness_equivalent) = S ((S (pfrep_position_unit_divides_witness_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_unit_divides_witness_equivalentsecondentry. ab = ff_q_pfp_unit_divides_witness_equivalentsecondentry * S ((S (pfrep_position_unit_divides_witness_equivalentsecond)) * ac) + (pfrep_right_unit_divides_witness_equivalent)))))) \\/ (((exists pfrep_gap_unit_divides_witness_equivalentsecondoutside. pfrep_gap_unit_divides_witness_equivalentsecondoutside+(L)=(pfrep_power_unit_divides_witness_equivalent)) /\\ (((pfrep_right_unit_divides_witness_equivalent)=0))))) -> pfrep_left_unit_divides_witness_equivalent=pfrep_right_unit_divides_witness_equivalent)))))))", "specialize prime_field_polynomial_convolution_left_unit_exists (p)", "specialize prime_field_polynomial_convolution_left_unit_exists (ab)", "specialize prime_field_polynomial_convolution_left_unit_exists (ac)", "specialize prime_field_polynomial_convolution_left_unit_exists (L)", "apply prime_field_polynomial_convolution_left_unit_exists", "exact hp", "exact hA", "cases hu", "cases hu_witness", "cases hu_witness_witness", "cases hu_witness_witness_witness", "cases hu_witness_witness_witness_witness", "cases hu_witness_witness_witness_witness_right", "cases hu_witness_witness_witness_witness_right_right", "specialize prime_field_polynomial_right_divides_from_product (p)", "specialize prime_field_polynomial_right_divides_from_product (ab)", "specialize prime_field_polynomial_right_divides_from_product (ac)", "specialize prime_field_polynomial_right_divides_from_product (L)", "specialize prime_field_polynomial_right_divides_from_product (ab)", "specialize prime_field_polynomial_right_divides_from_product (ac)", "specialize prime_field_polynomial_right_divides_from_product (L)", "specialize prime_field_polynomial_right_divides_from_product (x)", "specialize prime_field_polynomial_right_divides_from_product (x1)", "specialize prime_field_polynomial_right_divides_from_product (1)", "specialize prime_field_polynomial_right_divides_from_product (x2)", "specialize prime_field_polynomial_right_divides_from_product (x3)", "specialize prime_field_polynomial_right_divides_from_product (L)", "apply prime_field_polynomial_right_divides_from_product", "exact hA", "exact hu_witness_witness_witness_witness_right_right_left", "exact hu_witness_witness_witness_witness_right_right_right" ], "script_sha256": "cb1863f2247124a6d52501ad093a9f13a05e7b3042e170630762a7079ec97acd", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_left_unit_candidate.py", "sha256": "dbb8debb4716b6bb9b246700f7e93865c8a6c1b12a3b65c0ffbb62206a890ba6" }, "statement": "forall p ab ac L. (~((p) = 1) /\\ forall pfa_factor_left_unit_divides_prime pfa_factor_right_unit_divides_prime. (p) = pfa_factor_left_unit_divides_prime * pfa_factor_right_unit_divides_prime -> pfa_factor_left_unit_divides_prime = 1 \\/ pfa_factor_right_unit_divides_prime = 1) -> (forall fom_index_pfp_unit_divides_A. (exists fom_gap_pfp_unit_divides_A_index_bound. fom_gap_pfp_unit_divides_A_index_bound + S (fom_index_pfp_unit_divides_A) = L) -> exists fom_value_pfp_unit_divides_A. ((((exists fom_beta_height_pfp_unit_divides_A_entry. fom_beta_height_pfp_unit_divides_A_entry + S (fom_value_pfp_unit_divides_A) = S ((S (fom_index_pfp_unit_divides_A)) * ac)) /\\ exists fom_beta_quotient_pfp_unit_divides_A_entry. ab = fom_beta_quotient_pfp_unit_divides_A_entry * S ((S (fom_index_pfp_unit_divides_A)) * ac) + (fom_value_pfp_unit_divides_A))) /\\ (exists fom_gap_pfp_unit_divides_A_value_bound. fom_gap_pfp_unit_divides_A_value_bound + S (fom_value_pfp_unit_divides_A) = p))) -> (((forall fom_index_pfp_unit_divides_bound. (exists fom_gap_pfp_unit_divides_bound_index_bound. fom_gap_pfp_unit_divides_bound_index_bound + S (fom_index_pfp_unit_divides_bound) = L) -> exists fom_value_pfp_unit_divides_bound. ((((exists fom_beta_height_pfp_unit_divides_bound_entry. fom_beta_height_pfp_unit_divides_bound_entry + S (fom_value_pfp_unit_divides_bound) = S ((S (fom_index_pfp_unit_divides_bound)) * ac)) /\\ exists fom_beta_quotient_pfp_unit_divides_bound_entry. ab = fom_beta_quotient_pfp_unit_divides_bound_entry * S ((S (fom_index_pfp_unit_divides_bound)) * ac) + (fom_value_pfp_unit_divides_bound))) /\\ (exists fom_gap_pfp_unit_divides_bound_value_bound. fom_gap_pfp_unit_divides_bound_value_bound + S (fom_value_pfp_unit_divides_bound) = p))) /\\ ((exists qb qc Q pb pc P. ((((forall fom_index_pfp_unit_divides_productleft. (exists fom_gap_pfp_unit_divides_productleft_index_bound. fom_gap_pfp_unit_divides_productleft_index_bound + S (fom_index_pfp_unit_divides_productleft) = Q) -> exists fom_value_pfp_unit_divides_productleft. ((((exists fom_beta_height_pfp_unit_divides_productleft_entry. fom_beta_height_pfp_unit_divides_productleft_entry + S (fom_value_pfp_unit_divides_productleft) = S ((S (fom_index_pfp_unit_divides_productleft)) * qc)) /\\ exists fom_beta_quotient_pfp_unit_divides_productleft_entry. qb = fom_beta_quotient_pfp_unit_divides_productleft_entry * S ((S (fom_index_pfp_unit_divides_productleft)) * qc) + (fom_value_pfp_unit_divides_productleft))) /\\ (exists fom_gap_pfp_unit_divides_productleft_value_bound. fom_gap_pfp_unit_divides_productleft_value_bound + S (fom_value_pfp_unit_divides_productleft) = p))) /\\ (((forall fom_index_pfp_unit_divides_productright. (exists fom_gap_pfp_unit_divides_productright_index_bound. fom_gap_pfp_unit_divides_productright_index_bound + S (fom_index_pfp_unit_divides_productright) = L) -> exists fom_value_pfp_unit_divides_productright. ((((exists fom_beta_height_pfp_unit_divides_productright_entry. fom_beta_height_pfp_unit_divides_productright_entry + S (fom_value_pfp_unit_divides_productright) = S ((S (fom_index_pfp_unit_divides_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_unit_divides_productright_entry. ab = fom_beta_quotient_pfp_unit_divides_productright_entry * S ((S (fom_index_pfp_unit_divides_productright)) * ac) + (fom_value_pfp_unit_divides_productright))) /\\ (exists fom_gap_pfp_unit_divides_productright_value_bound. fom_gap_pfp_unit_divides_productright_value_bound + S (fom_value_pfp_unit_divides_productright) = p))) /\\ (((((((Q)=0 \\/ (L)=0) /\\ (((P)=0)))) \\/ (((~((Q)=0)) /\\ (((~((L)=0)) /\\ (((Q)+(L)=S (P)))))))) /\\ ((forall pfc_index_unit_divides_productcoefficients. (exists pfa_gap_unit_divides_productcoefficientsbound. pfa_gap_unit_divides_productcoefficientsbound + S (pfc_index_unit_divides_productcoefficients) = (P)) -> exists pfc_value_unit_divides_productcoefficients. ((((exists ff_h_pfp_unit_divides_productcoefficientsentry. ff_h_pfp_unit_divides_productcoefficientsentry + S (pfc_value_unit_divides_productcoefficients) = S ((S (pfc_index_unit_divides_productcoefficients)) * pc)) /\\ exists ff_q_pfp_unit_divides_productcoefficientsentry. pb = ff_q_pfp_unit_divides_productcoefficientsentry * S ((S (pfc_index_unit_divides_productcoefficients)) * pc) + (pfc_value_unit_divides_productcoefficients))) /\\ ((exists pfc_terms_code_unit_divides_productcoefficientscoefficient pfc_terms_scale_unit_divides_productcoefficientscoefficient pfc_natural_sum_unit_divides_productcoefficientscoefficient. ((forall pfc_index_unit_divides_productcoefficientscoefficientdiagonal. (exists pfa_gap_unit_divides_productcoefficientscoefficientdiagonalbound. pfa_gap_unit_divides_productcoefficientscoefficientdiagonalbound + S (pfc_index_unit_divides_productcoefficientscoefficientdiagonal) = (S (pfc_index_unit_divides_productcoefficients))) -> exists pfc_value_unit_divides_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_unit_divides_productcoefficientscoefficientdiagonalentry. ff_h_pfp_unit_divides_productcoefficientscoefficientdiagonalentry + S (pfc_value_unit_divides_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_unit_divides_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_unit_divides_productcoefficientscoefficient)) /\\ exists ff_q_pfp_unit_divides_productcoefficientscoefficientdiagonalentry. pfc_terms_code_unit_divides_productcoefficientscoefficient = ff_q_pfp_unit_divides_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_unit_divides_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_unit_divides_productcoefficientscoefficient) + (pfc_value_unit_divides_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_unit_divides_productcoefficientscoefficientdiagonalterm pfc_left_unit_divides_productcoefficientscoefficientdiagonalterm pfc_right_unit_divides_productcoefficientscoefficientdiagonalterm. (((pfc_index_unit_divides_productcoefficientscoefficientdiagonal)+pfc_complement_unit_divides_productcoefficientscoefficientdiagonalterm=(pfc_index_unit_divides_productcoefficients)) /\\ ((((((exists pfa_gap_unit_divides_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_unit_divides_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_unit_divides_productcoefficientscoefficientdiagonal) = (Q)) /\\ ((((exists ff_h_pfp_unit_divides_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_unit_divides_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_unit_divides_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_unit_divides_productcoefficientscoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_unit_divides_productcoefficientscoefficientdiagonaltermleftentry. qb = ff_q_pfp_unit_divides_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_unit_divides_productcoefficientscoefficientdiagonal)) * qc) + (pfc_left_unit_divides_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_unit_divides_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_unit_divides_productcoefficientscoefficientdiagonaltermleftoutside+(Q)=(pfc_index_unit_divides_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_unit_divides_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_unit_divides_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_unit_divides_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_unit_divides_productcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_unit_divides_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_unit_divides_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_unit_divides_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_unit_divides_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_unit_divides_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_unit_divides_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_unit_divides_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_unit_divides_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_unit_divides_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_unit_divides_productcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_unit_divides_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_unit_divides_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_unit_divides_productcoefficientscoefficientdiagonal)=pfc_left_unit_divides_productcoefficientscoefficientdiagonalterm*pfc_right_unit_divides_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_unit_divides_productcoefficientscoefficientsum fs_v_pfc_unit_divides_productcoefficientscoefficientsum. ((((exists fs_h_pfc_unit_divides_productcoefficientscoefficientsum_body_start. fs_h_pfc_unit_divides_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_unit_divides_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_divides_productcoefficientscoefficientsum_body_start. fs_u_pfc_unit_divides_productcoefficientscoefficientsum = fs_q_pfc_unit_divides_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_unit_divides_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_unit_divides_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_unit_divides_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_unit_divides_productcoefficientscoefficient) = S ((S (S (pfc_index_unit_divides_productcoefficients))) * fs_v_pfc_unit_divides_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_divides_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_unit_divides_productcoefficientscoefficientsum = fs_q_pfc_unit_divides_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_unit_divides_productcoefficients))) * fs_v_pfc_unit_divides_productcoefficientscoefficientsum) + (pfc_natural_sum_unit_divides_productcoefficientscoefficient))) /\\ forall fs_i_pfc_unit_divides_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_unit_divides_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_unit_divides_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_unit_divides_productcoefficientscoefficientsum_body_steps = S (pfc_index_unit_divides_productcoefficients)) -> exists fs_a_pfc_unit_divides_productcoefficientscoefficientsum_body_steps fs_r_pfc_unit_divides_productcoefficientscoefficientsum_body_steps fs_s_pfc_unit_divides_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_unit_divides_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_unit_divides_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_unit_divides_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_unit_divides_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_unit_divides_productcoefficientscoefficient)) /\\ exists fs_q_pfc_unit_divides_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_unit_divides_productcoefficientscoefficient = fs_q_pfc_unit_divides_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_unit_divides_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_unit_divides_productcoefficientscoefficient) + (fs_a_pfc_unit_divides_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_unit_divides_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_unit_divides_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_unit_divides_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_unit_divides_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_divides_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_divides_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_unit_divides_productcoefficientscoefficientsum = fs_q_pfc_unit_divides_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_unit_divides_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_divides_productcoefficientscoefficientsum) + (fs_r_pfc_unit_divides_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_unit_divides_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_unit_divides_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_unit_divides_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_unit_divides_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_divides_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_unit_divides_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_unit_divides_productcoefficientscoefficientsum = fs_q_pfc_unit_divides_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_unit_divides_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_unit_divides_productcoefficientscoefficientsum) + (fs_s_pfc_unit_divides_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_unit_divides_productcoefficientscoefficientsum_body_steps = fs_r_pfc_unit_divides_productcoefficientscoefficientsum_body_steps + fs_a_pfc_unit_divides_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_unit_divides_productcoefficientscoefficientresiduebound. pfa_gap_unit_divides_productcoefficientscoefficientresiduebound + S (pfc_value_unit_divides_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_unit_divides_productcoefficientscoefficientresiduecongruence pfa_offset_right_unit_divides_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_unit_divides_productcoefficientscoefficient) + (p) * pfa_offset_left_unit_divides_productcoefficientscoefficientresiduecongruence = (pfc_value_unit_divides_productcoefficients) + (p) * pfa_offset_right_unit_divides_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_unit_divides_equivalent pfrep_left_unit_divides_equivalent pfrep_right_unit_divides_equivalent. ((exists pfrep_position_unit_divides_equivalentfirst. ((pfrep_position_unit_divides_equivalentfirst+S (pfrep_power_unit_divides_equivalent)=(P)) /\\ ((((exists ff_h_pfp_unit_divides_equivalentfirstentry. ff_h_pfp_unit_divides_equivalentfirstentry + S (pfrep_left_unit_divides_equivalent) = S ((S (pfrep_position_unit_divides_equivalentfirst)) * pc)) /\\ exists ff_q_pfp_unit_divides_equivalentfirstentry. pb = ff_q_pfp_unit_divides_equivalentfirstentry * S ((S (pfrep_position_unit_divides_equivalentfirst)) * pc) + (pfrep_left_unit_divides_equivalent)))))) \\/ (((exists pfrep_gap_unit_divides_equivalentfirstoutside. pfrep_gap_unit_divides_equivalentfirstoutside+(P)=(pfrep_power_unit_divides_equivalent)) /\\ (((pfrep_left_unit_divides_equivalent)=0))))) -> ((exists pfrep_position_unit_divides_equivalentsecond. ((pfrep_position_unit_divides_equivalentsecond+S (pfrep_power_unit_divides_equivalent)=(L)) /\\ ((((exists ff_h_pfp_unit_divides_equivalentsecondentry. ff_h_pfp_unit_divides_equivalentsecondentry + S (pfrep_right_unit_divides_equivalent) = S ((S (pfrep_position_unit_divides_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_unit_divides_equivalentsecondentry. ab = ff_q_pfp_unit_divides_equivalentsecondentry * S ((S (pfrep_position_unit_divides_equivalentsecond)) * ac) + (pfrep_right_unit_divides_equivalent)))))) \\/ (((exists pfrep_gap_unit_divides_equivalentsecondoutside. pfrep_gap_unit_divides_equivalentsecondoutside+(L)=(pfrep_power_unit_divides_equivalent)) /\\ (((pfrep_right_unit_divides_equivalent)=0))))) -> pfrep_left_unit_divides_equivalent=pfrep_right_unit_divides_equivalent)))))))", "statement_sha256": "d8f3531eb2f6d2fb37e8ee936807a66a7dc1e49b71c95c7c7023c7964fc03852", "summary": "Every canonical polynomial right-divides itself using a constructed left unit and actual product. 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(((forall pfp_repeat_index_alignment_constructed_padzeros. (exists pfa_gap_alignment_constructed_padzerosindex. pfa_gap_alignment_constructed_padzerosindex + S (pfp_repeat_index_alignment_constructed_padzeros) = (x)) -> (((exists ff_h_pfp_alignment_constructed_padzerosentry. ff_h_pfp_alignment_constructed_padzerosentry + S (0) = S ((S (pfp_repeat_index_alignment_constructed_padzeros)) * uc)) /\\ exists ff_q_pfp_alignment_constructed_padzerosentry. ub = ff_q_pfp_alignment_constructed_padzerosentry * S ((S (pfp_repeat_index_alignment_constructed_padzeros)) * uc) + (0)))) /\\ ((forall pfrep_index_alignment_constructed_pad pfrep_value_alignment_constructed_pad. (exists pfa_gap_alignment_constructed_padbound. pfa_gap_alignment_constructed_padbound + S (pfrep_index_alignment_constructed_pad) = (L)) -> (((exists ff_h_pfp_alignment_constructed_padinput. ff_h_pfp_alignment_constructed_padinput + S (pfrep_value_alignment_constructed_pad) = S ((S (pfrep_index_alignment_constructed_pad)) * ac)) /\\ exists ff_q_pfp_alignment_constructed_padinput. ab = ff_q_pfp_alignment_constructed_padinput * S ((S (pfrep_index_alignment_constructed_pad)) * ac) + (pfrep_value_alignment_constructed_pad))) -> (((exists ff_h_pfp_alignment_constructed_padoutput. ff_h_pfp_alignment_constructed_padoutput + S (pfrep_value_alignment_constructed_pad) = S ((S ((x)+pfrep_index_alignment_constructed_pad)) * uc)) /\\ exists ff_q_pfp_alignment_constructed_padoutput. ub = ff_q_pfp_alignment_constructed_padoutput * S ((S ((x)+pfrep_index_alignment_constructed_pad)) * uc) + (pfrep_value_alignment_constructed_pad)))))))", "specialize prime_field_polynomial_left_pad_exists (ab)", "specialize prime_field_polynomial_left_pad_exists (ac)", "specialize prime_field_polynomial_left_pad_exists (x)", "specialize prime_field_polynomial_left_pad_exists (L)", "apply prime_field_polynomial_left_pad_exists", "cases hpad", "cases hpad_witness", "exists x1", "exists x2", "split", "rewrite <- hlen_witness", "specialize prime_field_polynomial_left_pad_bounded (p)", "specialize prime_field_polynomial_left_pad_bounded (ab)", "specialize prime_field_polynomial_left_pad_bounded (ac)", "specialize prime_field_polynomial_left_pad_bounded (L)", "specialize prime_field_polynomial_left_pad_bounded (x)", "specialize prime_field_polynomial_left_pad_bounded (x1)", "specialize prime_field_polynomial_left_pad_bounded (x2)", "apply prime_field_polynomial_left_pad_bounded", "exact hp", "exact ha", "exact hpad_witness_witness", "rewrite <- hlen_witness", "rewrite <- hlen_witness", "specialize prime_field_polynomial_left_pad_equivalent (ab)", "specialize prime_field_polynomial_left_pad_equivalent (ac)", "specialize prime_field_polynomial_left_pad_equivalent (L)", "specialize prime_field_polynomial_left_pad_equivalent (x)", "specialize prime_field_polynomial_left_pad_equivalent (x1)", "specialize prime_field_polynomial_left_pad_equivalent (x2)", "apply prime_field_polynomial_left_pad_equivalent", "exact hpad_witness_witness" ], "script_sha256": "91adaa814dd1dc8a96b0776e91924fa25f9dabf0ac9c49806d9eb0d9dff3d7dc", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_alignment_candidate.py", "sha256": "eb16e2eb02dbd66a7706e616388182992b8cf2e0715818dc1f7748938e7d798e" }, "statement": "forall p ab ac L K. (~((p) = 1) /\\ forall pfa_factor_left_alignment_prime pfa_factor_right_alignment_prime. (p) = pfa_factor_left_alignment_prime * pfa_factor_right_alignment_prime -> pfa_factor_left_alignment_prime = 1 \\/ pfa_factor_right_alignment_prime = 1) -> (forall fom_index_pfp_alignment_input. (exists fom_gap_pfp_alignment_input_index_bound. fom_gap_pfp_alignment_input_index_bound + S (fom_index_pfp_alignment_input) = L) -> exists fom_value_pfp_alignment_input. ((((exists fom_beta_height_pfp_alignment_input_entry. fom_beta_height_pfp_alignment_input_entry + S (fom_value_pfp_alignment_input) = S ((S (fom_index_pfp_alignment_input)) * ac)) /\\ exists fom_beta_quotient_pfp_alignment_input_entry. ab = fom_beta_quotient_pfp_alignment_input_entry * S ((S (fom_index_pfp_alignment_input)) * ac) + (fom_value_pfp_alignment_input))) /\\ (exists fom_gap_pfp_alignment_input_value_bound. fom_gap_pfp_alignment_input_value_bound + S (fom_value_pfp_alignment_input) = p))) -> (exists pfrep_gap_alignment_length_bound. pfrep_gap_alignment_length_bound+(L)=(K)) -> (exists ub uc. ((forall fom_index_pfp_alignment_output_bounded. (exists fom_gap_pfp_alignment_output_bounded_index_bound. fom_gap_pfp_alignment_output_bounded_index_bound + S (fom_index_pfp_alignment_output_bounded) = K) -> exists fom_value_pfp_alignment_output_bounded. ((((exists fom_beta_height_pfp_alignment_output_bounded_entry. fom_beta_height_pfp_alignment_output_bounded_entry + S (fom_value_pfp_alignment_output_bounded) = S ((S (fom_index_pfp_alignment_output_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_alignment_output_bounded_entry. ub = fom_beta_quotient_pfp_alignment_output_bounded_entry * S ((S (fom_index_pfp_alignment_output_bounded)) * uc) + (fom_value_pfp_alignment_output_bounded))) /\\ (exists fom_gap_pfp_alignment_output_bounded_value_bound. fom_gap_pfp_alignment_output_bounded_value_bound + S (fom_value_pfp_alignment_output_bounded) = p))) /\\ ((forall pfrep_power_alignment_output_equivalent pfrep_left_alignment_output_equivalent pfrep_right_alignment_output_equivalent. ((exists pfrep_position_alignment_output_equivalentfirst. ((pfrep_position_alignment_output_equivalentfirst+S (pfrep_power_alignment_output_equivalent)=(L)) /\\ ((((exists ff_h_pfp_alignment_output_equivalentfirstentry. ff_h_pfp_alignment_output_equivalentfirstentry + S (pfrep_left_alignment_output_equivalent) = S ((S (pfrep_position_alignment_output_equivalentfirst)) * ac)) /\\ exists ff_q_pfp_alignment_output_equivalentfirstentry. ab = ff_q_pfp_alignment_output_equivalentfirstentry * S ((S (pfrep_position_alignment_output_equivalentfirst)) * ac) + (pfrep_left_alignment_output_equivalent)))))) \\/ (((exists pfrep_gap_alignment_output_equivalentfirstoutside. pfrep_gap_alignment_output_equivalentfirstoutside+(L)=(pfrep_power_alignment_output_equivalent)) /\\ (((pfrep_left_alignment_output_equivalent)=0))))) -> ((exists pfrep_position_alignment_output_equivalentsecond. ((pfrep_position_alignment_output_equivalentsecond+S (pfrep_power_alignment_output_equivalent)=(K)) /\\ ((((exists ff_h_pfp_alignment_output_equivalentsecondentry. ff_h_pfp_alignment_output_equivalentsecondentry + S (pfrep_right_alignment_output_equivalent) = S ((S (pfrep_position_alignment_output_equivalentsecond)) * uc)) /\\ exists ff_q_pfp_alignment_output_equivalentsecondentry. ub = ff_q_pfp_alignment_output_equivalentsecondentry * S ((S (pfrep_position_alignment_output_equivalentsecond)) * uc) + (pfrep_right_alignment_output_equivalent)))))) \\/ (((exists pfrep_gap_alignment_output_equivalentsecondoutside. pfrep_gap_alignment_output_equivalentsecondoutside+(K)=(pfrep_power_alignment_output_equivalent)) /\\ (((pfrep_right_alignment_output_equivalent)=0))))) -> pfrep_left_alignment_output_equivalent=pfrep_right_alignment_output_equivalent))))", "statement_sha256": "51508b1ddc9693ff10ff32e2685de2491020cdd1c32753425cf2881f02c37b70", "summary": "From L<=K construct a genuine 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(K)", "apply prime_field_polynomial_power_coefficient_functional" ], "script_sha256": "1e7c1ed39a4bf1ffc16c29ef600eb36724b528f4149296655ec3d4dfcf7b54cd", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_alignment_candidate.py", "sha256": "eb16e2eb02dbd66a7706e616388182992b8cf2e0715818dc1f7748938e7d798e" }, "statement": "forall ab ac bb bc K. (((forall pfrep_power_common_self_left pfrep_left_common_self_left pfrep_right_common_self_left. ((exists pfrep_position_common_self_leftfirst. ((pfrep_position_common_self_leftfirst+S (pfrep_power_common_self_left)=(K)) /\\ ((((exists ff_h_pfp_common_self_leftfirstentry. ff_h_pfp_common_self_leftfirstentry + S (pfrep_left_common_self_left) = S ((S (pfrep_position_common_self_leftfirst)) * ac)) /\\ exists ff_q_pfp_common_self_leftfirstentry. ab = ff_q_pfp_common_self_leftfirstentry * S ((S (pfrep_position_common_self_leftfirst)) * ac) + (pfrep_left_common_self_left)))))) \\/ (((exists pfrep_gap_common_self_leftfirstoutside. pfrep_gap_common_self_leftfirstoutside+(K)=(pfrep_power_common_self_left)) /\\ (((pfrep_left_common_self_left)=0))))) -> ((exists pfrep_position_common_self_leftsecond. ((pfrep_position_common_self_leftsecond+S (pfrep_power_common_self_left)=(K)) /\\ ((((exists ff_h_pfp_common_self_leftsecondentry. ff_h_pfp_common_self_leftsecondentry + S (pfrep_right_common_self_left) = S ((S (pfrep_position_common_self_leftsecond)) * ac)) /\\ exists ff_q_pfp_common_self_leftsecondentry. ab = ff_q_pfp_common_self_leftsecondentry * S ((S (pfrep_position_common_self_leftsecond)) * ac) + (pfrep_right_common_self_left)))))) \\/ (((exists pfrep_gap_common_self_leftsecondoutside. pfrep_gap_common_self_leftsecondoutside+(K)=(pfrep_power_common_self_left)) /\\ (((pfrep_right_common_self_left)=0))))) -> pfrep_left_common_self_left=pfrep_right_common_self_left) /\\ ((forall pfrep_power_common_self_right pfrep_left_common_self_right pfrep_right_common_self_right. ((exists pfrep_position_common_self_rightfirst. ((pfrep_position_common_self_rightfirst+S (pfrep_power_common_self_right)=(K)) /\\ ((((exists ff_h_pfp_common_self_rightfirstentry. ff_h_pfp_common_self_rightfirstentry + S (pfrep_left_common_self_right) = S ((S (pfrep_position_common_self_rightfirst)) * bc)) /\\ exists ff_q_pfp_common_self_rightfirstentry. bb = ff_q_pfp_common_self_rightfirstentry * S ((S (pfrep_position_common_self_rightfirst)) * bc) + (pfrep_left_common_self_right)))))) \\/ (((exists pfrep_gap_common_self_rightfirstoutside. pfrep_gap_common_self_rightfirstoutside+(K)=(pfrep_power_common_self_right)) /\\ (((pfrep_left_common_self_right)=0))))) -> ((exists pfrep_position_common_self_rightsecond. ((pfrep_position_common_self_rightsecond+S (pfrep_power_common_self_right)=(K)) /\\ ((((exists ff_h_pfp_common_self_rightsecondentry. ff_h_pfp_common_self_rightsecondentry + S (pfrep_right_common_self_right) = S ((S (pfrep_position_common_self_rightsecond)) * bc)) /\\ exists ff_q_pfp_common_self_rightsecondentry. bb = ff_q_pfp_common_self_rightsecondentry * S ((S (pfrep_position_common_self_rightsecond)) * bc) + (pfrep_right_common_self_right)))))) \\/ (((exists pfrep_gap_common_self_rightsecondoutside. pfrep_gap_common_self_rightsecondoutside+(K)=(pfrep_power_common_self_right)) /\\ (((pfrep_right_common_self_right)=0))))) -> pfrep_left_common_self_right=pfrep_right_common_self_right))))", "statement_sha256": "f48c856d15638825baf1f0bffe4c71b47c60dbdd27265b1f2eccb0670f5d6196", "summary": "Two prefixes already at one length serve as their actual common representatives; no new beta encoding or prime premise is needed.", "summary_sha256": 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(uc)", "specialize prime_field_polynomial_equivalent_transitive (K)", "apply prime_field_polynomial_equivalent_transitive", "exact ha", "exact h_left", "specialize prime_field_polynomial_equivalent_transitive (eb)", "specialize prime_field_polynomial_equivalent_transitive (ec)", "specialize prime_field_polynomial_equivalent_transitive (N)", "specialize prime_field_polynomial_equivalent_transitive (bb)", "specialize prime_field_polynomial_equivalent_transitive (bc)", "specialize prime_field_polynomial_equivalent_transitive (M)", "specialize prime_field_polynomial_equivalent_transitive (vb)", "specialize prime_field_polynomial_equivalent_transitive (vc)", "specialize prime_field_polynomial_equivalent_transitive (K)", "apply prime_field_polynomial_equivalent_transitive", "exact hb", "exact h_right" ], "script_sha256": "63c650dbcf7b793f058537432040cb5f3067a0a5ae3ce535fffa8e2aea0894a5", "source": { "kind": "candidate_module", "path": 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(forall pfrep_power_common_transport_left pfrep_left_common_transport_left pfrep_right_common_transport_left. ((exists pfrep_position_common_transport_leftfirst. ((pfrep_position_common_transport_leftfirst+S (pfrep_power_common_transport_left)=(J)) /\\ ((((exists ff_h_pfp_common_transport_leftfirstentry. ff_h_pfp_common_transport_leftfirstentry + S (pfrep_left_common_transport_left) = S ((S (pfrep_position_common_transport_leftfirst)) * dc)) /\\ exists ff_q_pfp_common_transport_leftfirstentry. db = ff_q_pfp_common_transport_leftfirstentry * S ((S (pfrep_position_common_transport_leftfirst)) * dc) + (pfrep_left_common_transport_left)))))) \\/ (((exists pfrep_gap_common_transport_leftfirstoutside. pfrep_gap_common_transport_leftfirstoutside+(J)=(pfrep_power_common_transport_left)) /\\ (((pfrep_left_common_transport_left)=0))))) -> ((exists pfrep_position_common_transport_leftsecond. ((pfrep_position_common_transport_leftsecond+S (pfrep_power_common_transport_left)=(L)) /\\ ((((exists ff_h_pfp_common_transport_leftsecondentry. ff_h_pfp_common_transport_leftsecondentry + S (pfrep_right_common_transport_left) = S ((S (pfrep_position_common_transport_leftsecond)) * ac)) /\\ exists ff_q_pfp_common_transport_leftsecondentry. ab = ff_q_pfp_common_transport_leftsecondentry * S ((S (pfrep_position_common_transport_leftsecond)) * ac) + (pfrep_right_common_transport_left)))))) \\/ (((exists pfrep_gap_common_transport_leftsecondoutside. pfrep_gap_common_transport_leftsecondoutside+(L)=(pfrep_power_common_transport_left)) /\\ (((pfrep_right_common_transport_left)=0))))) -> pfrep_left_common_transport_left=pfrep_right_common_transport_left) -> (forall pfrep_power_common_transport_right pfrep_left_common_transport_right pfrep_right_common_transport_right. ((exists pfrep_position_common_transport_rightfirst. ((pfrep_position_common_transport_rightfirst+S (pfrep_power_common_transport_right)=(N)) /\\ ((((exists ff_h_pfp_common_transport_rightfirstentry. ff_h_pfp_common_transport_rightfirstentry + S (pfrep_left_common_transport_right) = S ((S (pfrep_position_common_transport_rightfirst)) * ec)) /\\ exists ff_q_pfp_common_transport_rightfirstentry. eb = ff_q_pfp_common_transport_rightfirstentry * S ((S (pfrep_position_common_transport_rightfirst)) * ec) + (pfrep_left_common_transport_right)))))) \\/ (((exists pfrep_gap_common_transport_rightfirstoutside. pfrep_gap_common_transport_rightfirstoutside+(N)=(pfrep_power_common_transport_right)) /\\ (((pfrep_left_common_transport_right)=0))))) -> ((exists pfrep_position_common_transport_rightsecond. ((pfrep_position_common_transport_rightsecond+S (pfrep_power_common_transport_right)=(M)) /\\ ((((exists ff_h_pfp_common_transport_rightsecondentry. ff_h_pfp_common_transport_rightsecondentry + S (pfrep_right_common_transport_right) = S ((S (pfrep_position_common_transport_rightsecond)) * bc)) /\\ exists ff_q_pfp_common_transport_rightsecondentry. bb = ff_q_pfp_common_transport_rightsecondentry * S ((S (pfrep_position_common_transport_rightsecond)) * bc) + (pfrep_right_common_transport_right)))))) \\/ (((exists pfrep_gap_common_transport_rightsecondoutside. pfrep_gap_common_transport_rightsecondoutside+(M)=(pfrep_power_common_transport_right)) /\\ (((pfrep_right_common_transport_right)=0))))) -> pfrep_left_common_transport_right=pfrep_right_common_transport_right) -> (((forall pfrep_power_common_transport_old_left pfrep_left_common_transport_old_left pfrep_right_common_transport_old_left. ((exists pfrep_position_common_transport_old_leftfirst. ((pfrep_position_common_transport_old_leftfirst+S (pfrep_power_common_transport_old_left)=(L)) /\\ ((((exists ff_h_pfp_common_transport_old_leftfirstentry. ff_h_pfp_common_transport_old_leftfirstentry + S (pfrep_left_common_transport_old_left) = S ((S (pfrep_position_common_transport_old_leftfirst)) * ac)) /\\ exists ff_q_pfp_common_transport_old_leftfirstentry. ab = ff_q_pfp_common_transport_old_leftfirstentry * S ((S (pfrep_position_common_transport_old_leftfirst)) * ac) + (pfrep_left_common_transport_old_left)))))) \\/ (((exists pfrep_gap_common_transport_old_leftfirstoutside. pfrep_gap_common_transport_old_leftfirstoutside+(L)=(pfrep_power_common_transport_old_left)) /\\ (((pfrep_left_common_transport_old_left)=0))))) -> ((exists pfrep_position_common_transport_old_leftsecond. ((pfrep_position_common_transport_old_leftsecond+S (pfrep_power_common_transport_old_left)=(K)) /\\ ((((exists ff_h_pfp_common_transport_old_leftsecondentry. ff_h_pfp_common_transport_old_leftsecondentry + S (pfrep_right_common_transport_old_left) = S ((S (pfrep_position_common_transport_old_leftsecond)) * uc)) /\\ exists ff_q_pfp_common_transport_old_leftsecondentry. ub = ff_q_pfp_common_transport_old_leftsecondentry * S ((S (pfrep_position_common_transport_old_leftsecond)) * uc) + (pfrep_right_common_transport_old_left)))))) \\/ (((exists pfrep_gap_common_transport_old_leftsecondoutside. pfrep_gap_common_transport_old_leftsecondoutside+(K)=(pfrep_power_common_transport_old_left)) /\\ (((pfrep_right_common_transport_old_left)=0))))) -> pfrep_left_common_transport_old_left=pfrep_right_common_transport_old_left) /\\ ((forall pfrep_power_common_transport_old_right pfrep_left_common_transport_old_right pfrep_right_common_transport_old_right. ((exists pfrep_position_common_transport_old_rightfirst. ((pfrep_position_common_transport_old_rightfirst+S (pfrep_power_common_transport_old_right)=(M)) /\\ ((((exists ff_h_pfp_common_transport_old_rightfirstentry. ff_h_pfp_common_transport_old_rightfirstentry + S (pfrep_left_common_transport_old_right) = S ((S (pfrep_position_common_transport_old_rightfirst)) * bc)) /\\ exists ff_q_pfp_common_transport_old_rightfirstentry. bb = ff_q_pfp_common_transport_old_rightfirstentry * S ((S (pfrep_position_common_transport_old_rightfirst)) * bc) + (pfrep_left_common_transport_old_right)))))) \\/ (((exists pfrep_gap_common_transport_old_rightfirstoutside. pfrep_gap_common_transport_old_rightfirstoutside+(M)=(pfrep_power_common_transport_old_right)) /\\ (((pfrep_left_common_transport_old_right)=0))))) -> ((exists pfrep_position_common_transport_old_rightsecond. ((pfrep_position_common_transport_old_rightsecond+S (pfrep_power_common_transport_old_right)=(K)) /\\ ((((exists ff_h_pfp_common_transport_old_rightsecondentry. ff_h_pfp_common_transport_old_rightsecondentry + S (pfrep_right_common_transport_old_right) = S ((S (pfrep_position_common_transport_old_rightsecond)) * vc)) /\\ exists ff_q_pfp_common_transport_old_rightsecondentry. vb = ff_q_pfp_common_transport_old_rightsecondentry * S ((S (pfrep_position_common_transport_old_rightsecond)) * vc) + (pfrep_right_common_transport_old_right)))))) \\/ (((exists pfrep_gap_common_transport_old_rightsecondoutside. pfrep_gap_common_transport_old_rightsecondoutside+(K)=(pfrep_power_common_transport_old_right)) /\\ (((pfrep_right_common_transport_old_right)=0))))) -> pfrep_left_common_transport_old_right=pfrep_right_common_transport_old_right)))) -> (((forall pfrep_power_common_transport_new_left pfrep_left_common_transport_new_left pfrep_right_common_transport_new_left. ((exists pfrep_position_common_transport_new_leftfirst. ((pfrep_position_common_transport_new_leftfirst+S (pfrep_power_common_transport_new_left)=(J)) /\\ ((((exists ff_h_pfp_common_transport_new_leftfirstentry. ff_h_pfp_common_transport_new_leftfirstentry + S (pfrep_left_common_transport_new_left) = S ((S (pfrep_position_common_transport_new_leftfirst)) * dc)) /\\ exists ff_q_pfp_common_transport_new_leftfirstentry. db = ff_q_pfp_common_transport_new_leftfirstentry * S ((S (pfrep_position_common_transport_new_leftfirst)) * dc) + (pfrep_left_common_transport_new_left)))))) \\/ (((exists pfrep_gap_common_transport_new_leftfirstoutside. pfrep_gap_common_transport_new_leftfirstoutside+(J)=(pfrep_power_common_transport_new_left)) /\\ (((pfrep_left_common_transport_new_left)=0))))) -> ((exists pfrep_position_common_transport_new_leftsecond. ((pfrep_position_common_transport_new_leftsecond+S (pfrep_power_common_transport_new_left)=(K)) /\\ ((((exists ff_h_pfp_common_transport_new_leftsecondentry. ff_h_pfp_common_transport_new_leftsecondentry + S (pfrep_right_common_transport_new_left) = S ((S (pfrep_position_common_transport_new_leftsecond)) * uc)) /\\ exists ff_q_pfp_common_transport_new_leftsecondentry. ub = ff_q_pfp_common_transport_new_leftsecondentry * S ((S (pfrep_position_common_transport_new_leftsecond)) * uc) + (pfrep_right_common_transport_new_left)))))) \\/ (((exists pfrep_gap_common_transport_new_leftsecondoutside. pfrep_gap_common_transport_new_leftsecondoutside+(K)=(pfrep_power_common_transport_new_left)) /\\ (((pfrep_right_common_transport_new_left)=0))))) -> pfrep_left_common_transport_new_left=pfrep_right_common_transport_new_left) /\\ ((forall pfrep_power_common_transport_new_right pfrep_left_common_transport_new_right pfrep_right_common_transport_new_right. ((exists pfrep_position_common_transport_new_rightfirst. ((pfrep_position_common_transport_new_rightfirst+S (pfrep_power_common_transport_new_right)=(N)) /\\ ((((exists ff_h_pfp_common_transport_new_rightfirstentry. ff_h_pfp_common_transport_new_rightfirstentry + S (pfrep_left_common_transport_new_right) = S ((S (pfrep_position_common_transport_new_rightfirst)) * ec)) /\\ exists ff_q_pfp_common_transport_new_rightfirstentry. eb = ff_q_pfp_common_transport_new_rightfirstentry * S ((S (pfrep_position_common_transport_new_rightfirst)) * ec) + (pfrep_left_common_transport_new_right)))))) \\/ (((exists pfrep_gap_common_transport_new_rightfirstoutside. pfrep_gap_common_transport_new_rightfirstoutside+(N)=(pfrep_power_common_transport_new_right)) /\\ (((pfrep_left_common_transport_new_right)=0))))) -> ((exists pfrep_position_common_transport_new_rightsecond. ((pfrep_position_common_transport_new_rightsecond+S (pfrep_power_common_transport_new_right)=(K)) /\\ ((((exists ff_h_pfp_common_transport_new_rightsecondentry. ff_h_pfp_common_transport_new_rightsecondentry + S (pfrep_right_common_transport_new_right) = S ((S (pfrep_position_common_transport_new_rightsecond)) * vc)) /\\ exists ff_q_pfp_common_transport_new_rightsecondentry. vb = ff_q_pfp_common_transport_new_rightsecondentry * S ((S (pfrep_position_common_transport_new_rightsecond)) * vc) + (pfrep_right_common_transport_new_right)))))) \\/ (((exists pfrep_gap_common_transport_new_rightsecondoutside. pfrep_gap_common_transport_new_rightsecondoutside+(K)=(pfrep_power_common_transport_new_right)) /\\ (((pfrep_right_common_transport_new_right)=0))))) -> pfrep_left_common_transport_new_right=pfrep_right_common_transport_new_right))))", "statement_sha256": "48e378c2913bbb6bb5a3344207742da611b205e7cb25ff604f712c3f6e463124", "summary": "Independent formal recodings of the original inputs 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((forall fom_index_pfp_common_construct_left_bounded. (exists fom_gap_pfp_common_construct_left_bounded_index_bound. fom_gap_pfp_common_construct_left_bounded_index_bound + S (fom_index_pfp_common_construct_left_bounded) = K) -> exists fom_value_pfp_common_construct_left_bounded. ((((exists fom_beta_height_pfp_common_construct_left_bounded_entry. fom_beta_height_pfp_common_construct_left_bounded_entry + S (fom_value_pfp_common_construct_left_bounded) = S ((S (fom_index_pfp_common_construct_left_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_common_construct_left_bounded_entry. ub = fom_beta_quotient_pfp_common_construct_left_bounded_entry * S ((S (fom_index_pfp_common_construct_left_bounded)) * uc) + (fom_value_pfp_common_construct_left_bounded))) /\\ (exists fom_gap_pfp_common_construct_left_bounded_value_bound. fom_gap_pfp_common_construct_left_bounded_value_bound + S (fom_value_pfp_common_construct_left_bounded) = p))) /\\ ((forall pfrep_power_common_construct_left_equivalent pfrep_left_common_construct_left_equivalent pfrep_right_common_construct_left_equivalent. ((exists pfrep_position_common_construct_left_equivalentfirst. ((pfrep_position_common_construct_left_equivalentfirst+S (pfrep_power_common_construct_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_common_construct_left_equivalentfirstentry. ff_h_pfp_common_construct_left_equivalentfirstentry + S (pfrep_left_common_construct_left_equivalent) = S ((S (pfrep_position_common_construct_left_equivalentfirst)) * ac)) /\\ exists ff_q_pfp_common_construct_left_equivalentfirstentry. ab = ff_q_pfp_common_construct_left_equivalentfirstentry * S ((S (pfrep_position_common_construct_left_equivalentfirst)) * ac) + (pfrep_left_common_construct_left_equivalent)))))) \\/ (((exists pfrep_gap_common_construct_left_equivalentfirstoutside. pfrep_gap_common_construct_left_equivalentfirstoutside+(L)=(pfrep_power_common_construct_left_equivalent)) /\\ (((pfrep_left_common_construct_left_equivalent)=0))))) -> ((exists pfrep_position_common_construct_left_equivalentsecond. ((pfrep_position_common_construct_left_equivalentsecond+S (pfrep_power_common_construct_left_equivalent)=(K)) /\\ ((((exists ff_h_pfp_common_construct_left_equivalentsecondentry. ff_h_pfp_common_construct_left_equivalentsecondentry + S (pfrep_right_common_construct_left_equivalent) = S ((S (pfrep_position_common_construct_left_equivalentsecond)) * uc)) /\\ exists ff_q_pfp_common_construct_left_equivalentsecondentry. ub = ff_q_pfp_common_construct_left_equivalentsecondentry * S ((S (pfrep_position_common_construct_left_equivalentsecond)) * uc) + (pfrep_right_common_construct_left_equivalent)))))) \\/ (((exists pfrep_gap_common_construct_left_equivalentsecondoutside. pfrep_gap_common_construct_left_equivalentsecondoutside+(K)=(pfrep_power_common_construct_left_equivalent)) /\\ (((pfrep_right_common_construct_left_equivalent)=0))))) -> pfrep_left_common_construct_left_equivalent=pfrep_right_common_construct_left_equivalent)))", "specialize prime_field_polynomial_bounded_representative_at_length_exists (p)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (ab)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (ac)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (L)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (K)", "apply prime_field_polynomial_bounded_representative_at_length_exists", "exact hp", "exact ha", "exact hL", "cases hu", "cases hu_witness", "cases hu_witness_witness", "have hv : exists vb vc. ((forall fom_index_pfp_common_construct_right_bounded. (exists fom_gap_pfp_common_construct_right_bounded_index_bound. fom_gap_pfp_common_construct_right_bounded_index_bound + S (fom_index_pfp_common_construct_right_bounded) = K) -> exists fom_value_pfp_common_construct_right_bounded. ((((exists fom_beta_height_pfp_common_construct_right_bounded_entry. fom_beta_height_pfp_common_construct_right_bounded_entry + S (fom_value_pfp_common_construct_right_bounded) = S ((S (fom_index_pfp_common_construct_right_bounded)) * vc)) /\\ exists fom_beta_quotient_pfp_common_construct_right_bounded_entry. vb = fom_beta_quotient_pfp_common_construct_right_bounded_entry * S ((S (fom_index_pfp_common_construct_right_bounded)) * vc) + (fom_value_pfp_common_construct_right_bounded))) /\\ (exists fom_gap_pfp_common_construct_right_bounded_value_bound. fom_gap_pfp_common_construct_right_bounded_value_bound + S (fom_value_pfp_common_construct_right_bounded) = p))) /\\ ((forall pfrep_power_common_construct_right_equivalent pfrep_left_common_construct_right_equivalent pfrep_right_common_construct_right_equivalent. ((exists pfrep_position_common_construct_right_equivalentfirst. ((pfrep_position_common_construct_right_equivalentfirst+S (pfrep_power_common_construct_right_equivalent)=(M)) /\\ ((((exists ff_h_pfp_common_construct_right_equivalentfirstentry. ff_h_pfp_common_construct_right_equivalentfirstentry + S (pfrep_left_common_construct_right_equivalent) = S ((S (pfrep_position_common_construct_right_equivalentfirst)) * bc)) /\\ exists ff_q_pfp_common_construct_right_equivalentfirstentry. bb = ff_q_pfp_common_construct_right_equivalentfirstentry * S ((S (pfrep_position_common_construct_right_equivalentfirst)) * bc) + (pfrep_left_common_construct_right_equivalent)))))) \\/ (((exists pfrep_gap_common_construct_right_equivalentfirstoutside. pfrep_gap_common_construct_right_equivalentfirstoutside+(M)=(pfrep_power_common_construct_right_equivalent)) /\\ (((pfrep_left_common_construct_right_equivalent)=0))))) -> ((exists pfrep_position_common_construct_right_equivalentsecond. ((pfrep_position_common_construct_right_equivalentsecond+S (pfrep_power_common_construct_right_equivalent)=(K)) /\\ ((((exists ff_h_pfp_common_construct_right_equivalentsecondentry. ff_h_pfp_common_construct_right_equivalentsecondentry + S (pfrep_right_common_construct_right_equivalent) = S ((S (pfrep_position_common_construct_right_equivalentsecond)) * vc)) /\\ exists ff_q_pfp_common_construct_right_equivalentsecondentry. vb = ff_q_pfp_common_construct_right_equivalentsecondentry * S ((S (pfrep_position_common_construct_right_equivalentsecond)) * vc) + (pfrep_right_common_construct_right_equivalent)))))) \\/ (((exists pfrep_gap_common_construct_right_equivalentsecondoutside. pfrep_gap_common_construct_right_equivalentsecondoutside+(K)=(pfrep_power_common_construct_right_equivalent)) /\\ (((pfrep_right_common_construct_right_equivalent)=0))))) -> pfrep_left_common_construct_right_equivalent=pfrep_right_common_construct_right_equivalent)))", "specialize prime_field_polynomial_bounded_representative_at_length_exists (p)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (bb)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (bc)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (M)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (K)", "apply prime_field_polynomial_bounded_representative_at_length_exists", "exact hp", "exact hb", "exact hM", "cases hv", "cases hv_witness", "cases hv_witness_witness", "exists x", "exists x1", "exists x2", "exists x3", "split", "exact hu_witness_witness_left", "split", "exact hv_witness_witness_left", "split", "exact hu_witness_witness_right", "exact hv_witness_witness_right" ], "script_sha256": "09dd04aa58ff54e5950dccf78213aa1f4c9ceef50184596b3800ea53ba07597d", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_alignment_candidate.py", "sha256": "eb16e2eb02dbd66a7706e616388182992b8cf2e0715818dc1f7748938e7d798e" }, "statement": "forall p ab ac L bb bc M K. (~((p) = 1) /\\ forall pfa_factor_left_common_at_prime pfa_factor_right_common_at_prime. (p) = pfa_factor_left_common_at_prime * pfa_factor_right_common_at_prime -> pfa_factor_left_common_at_prime = 1 \\/ pfa_factor_right_common_at_prime = 1) -> (forall fom_index_pfp_common_at_A. (exists fom_gap_pfp_common_at_A_index_bound. fom_gap_pfp_common_at_A_index_bound + S (fom_index_pfp_common_at_A) = L) -> exists fom_value_pfp_common_at_A. ((((exists fom_beta_height_pfp_common_at_A_entry. fom_beta_height_pfp_common_at_A_entry + S (fom_value_pfp_common_at_A) = S ((S (fom_index_pfp_common_at_A)) * ac)) /\\ exists fom_beta_quotient_pfp_common_at_A_entry. ab = fom_beta_quotient_pfp_common_at_A_entry * S ((S (fom_index_pfp_common_at_A)) * ac) + (fom_value_pfp_common_at_A))) /\\ (exists fom_gap_pfp_common_at_A_value_bound. fom_gap_pfp_common_at_A_value_bound + S (fom_value_pfp_common_at_A) = p))) -> (forall fom_index_pfp_common_at_B. (exists fom_gap_pfp_common_at_B_index_bound. fom_gap_pfp_common_at_B_index_bound + S (fom_index_pfp_common_at_B) = M) -> exists fom_value_pfp_common_at_B. ((((exists fom_beta_height_pfp_common_at_B_entry. fom_beta_height_pfp_common_at_B_entry + S (fom_value_pfp_common_at_B) = S ((S (fom_index_pfp_common_at_B)) * bc)) /\\ exists fom_beta_quotient_pfp_common_at_B_entry. bb = fom_beta_quotient_pfp_common_at_B_entry * S ((S (fom_index_pfp_common_at_B)) * bc) + (fom_value_pfp_common_at_B))) /\\ (exists fom_gap_pfp_common_at_B_value_bound. fom_gap_pfp_common_at_B_value_bound + S (fom_value_pfp_common_at_B) = p))) -> (exists pfrep_gap_common_at_L. pfrep_gap_common_at_L+(L)=(K)) -> (exists pfrep_gap_common_at_M. pfrep_gap_common_at_M+(M)=(K)) -> (exists ub uc vb vc. ((forall fom_index_pfp_common_at_output_left_bounded. (exists fom_gap_pfp_common_at_output_left_bounded_index_bound. fom_gap_pfp_common_at_output_left_bounded_index_bound + S (fom_index_pfp_common_at_output_left_bounded) = K) -> exists fom_value_pfp_common_at_output_left_bounded. ((((exists fom_beta_height_pfp_common_at_output_left_bounded_entry. fom_beta_height_pfp_common_at_output_left_bounded_entry + S (fom_value_pfp_common_at_output_left_bounded) = S ((S (fom_index_pfp_common_at_output_left_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_common_at_output_left_bounded_entry. ub = fom_beta_quotient_pfp_common_at_output_left_bounded_entry * S ((S (fom_index_pfp_common_at_output_left_bounded)) * uc) + (fom_value_pfp_common_at_output_left_bounded))) /\\ (exists fom_gap_pfp_common_at_output_left_bounded_value_bound. fom_gap_pfp_common_at_output_left_bounded_value_bound + S (fom_value_pfp_common_at_output_left_bounded) = p))) /\\ (((forall fom_index_pfp_common_at_output_right_bounded. (exists fom_gap_pfp_common_at_output_right_bounded_index_bound. fom_gap_pfp_common_at_output_right_bounded_index_bound + S (fom_index_pfp_common_at_output_right_bounded) = K) -> exists fom_value_pfp_common_at_output_right_bounded. ((((exists fom_beta_height_pfp_common_at_output_right_bounded_entry. fom_beta_height_pfp_common_at_output_right_bounded_entry + S (fom_value_pfp_common_at_output_right_bounded) = S ((S (fom_index_pfp_common_at_output_right_bounded)) * vc)) /\\ exists fom_beta_quotient_pfp_common_at_output_right_bounded_entry. vb = fom_beta_quotient_pfp_common_at_output_right_bounded_entry * S ((S (fom_index_pfp_common_at_output_right_bounded)) * vc) + (fom_value_pfp_common_at_output_right_bounded))) /\\ (exists fom_gap_pfp_common_at_output_right_bounded_value_bound. fom_gap_pfp_common_at_output_right_bounded_value_bound + S (fom_value_pfp_common_at_output_right_bounded) = p))) /\\ ((((forall pfrep_power_common_at_output_common_left pfrep_left_common_at_output_common_left pfrep_right_common_at_output_common_left. ((exists pfrep_position_common_at_output_common_leftfirst. ((pfrep_position_common_at_output_common_leftfirst+S (pfrep_power_common_at_output_common_left)=(L)) /\\ ((((exists ff_h_pfp_common_at_output_common_leftfirstentry. ff_h_pfp_common_at_output_common_leftfirstentry + S (pfrep_left_common_at_output_common_left) = S ((S (pfrep_position_common_at_output_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_common_at_output_common_leftfirstentry. ab = ff_q_pfp_common_at_output_common_leftfirstentry * S ((S (pfrep_position_common_at_output_common_leftfirst)) * ac) + (pfrep_left_common_at_output_common_left)))))) \\/ (((exists pfrep_gap_common_at_output_common_leftfirstoutside. pfrep_gap_common_at_output_common_leftfirstoutside+(L)=(pfrep_power_common_at_output_common_left)) /\\ (((pfrep_left_common_at_output_common_left)=0))))) -> ((exists pfrep_position_common_at_output_common_leftsecond. ((pfrep_position_common_at_output_common_leftsecond+S (pfrep_power_common_at_output_common_left)=(K)) /\\ ((((exists ff_h_pfp_common_at_output_common_leftsecondentry. ff_h_pfp_common_at_output_common_leftsecondentry + S (pfrep_right_common_at_output_common_left) = S ((S (pfrep_position_common_at_output_common_leftsecond)) * uc)) /\\ exists ff_q_pfp_common_at_output_common_leftsecondentry. ub = ff_q_pfp_common_at_output_common_leftsecondentry * S ((S (pfrep_position_common_at_output_common_leftsecond)) * uc) + (pfrep_right_common_at_output_common_left)))))) \\/ (((exists pfrep_gap_common_at_output_common_leftsecondoutside. pfrep_gap_common_at_output_common_leftsecondoutside+(K)=(pfrep_power_common_at_output_common_left)) /\\ (((pfrep_right_common_at_output_common_left)=0))))) -> pfrep_left_common_at_output_common_left=pfrep_right_common_at_output_common_left) /\\ ((forall pfrep_power_common_at_output_common_right pfrep_left_common_at_output_common_right pfrep_right_common_at_output_common_right. ((exists pfrep_position_common_at_output_common_rightfirst. ((pfrep_position_common_at_output_common_rightfirst+S (pfrep_power_common_at_output_common_right)=(M)) /\\ ((((exists ff_h_pfp_common_at_output_common_rightfirstentry. ff_h_pfp_common_at_output_common_rightfirstentry + S (pfrep_left_common_at_output_common_right) = S ((S (pfrep_position_common_at_output_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_common_at_output_common_rightfirstentry. bb = ff_q_pfp_common_at_output_common_rightfirstentry * S ((S (pfrep_position_common_at_output_common_rightfirst)) * bc) + (pfrep_left_common_at_output_common_right)))))) \\/ (((exists pfrep_gap_common_at_output_common_rightfirstoutside. pfrep_gap_common_at_output_common_rightfirstoutside+(M)=(pfrep_power_common_at_output_common_right)) /\\ (((pfrep_left_common_at_output_common_right)=0))))) -> ((exists pfrep_position_common_at_output_common_rightsecond. ((pfrep_position_common_at_output_common_rightsecond+S (pfrep_power_common_at_output_common_right)=(K)) /\\ ((((exists ff_h_pfp_common_at_output_common_rightsecondentry. ff_h_pfp_common_at_output_common_rightsecondentry + S (pfrep_right_common_at_output_common_right) = S ((S (pfrep_position_common_at_output_common_rightsecond)) * vc)) /\\ exists ff_q_pfp_common_at_output_common_rightsecondentry. vb = ff_q_pfp_common_at_output_common_rightsecondentry * S ((S (pfrep_position_common_at_output_common_rightsecond)) * vc) + (pfrep_right_common_at_output_common_right)))))) \\/ (((exists pfrep_gap_common_at_output_common_rightsecondoutside. pfrep_gap_common_at_output_common_rightsecondoutside+(K)=(pfrep_power_common_at_output_common_right)) /\\ (((pfrep_right_common_at_output_common_right)=0))))) -> pfrep_left_common_at_output_common_right=pfrep_right_common_at_output_common_right)))))))))", "statement_sha256": "4057b87d5e33934e9642b0d8cd835a12d78c8ccf9fe9ac1a64135c44727f7d7f", "summary": 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(~((p) = 1) /\\ forall pfa_factor_left_common_exists_prime pfa_factor_right_common_exists_prime. (p) = pfa_factor_left_common_exists_prime * pfa_factor_right_common_exists_prime -> pfa_factor_left_common_exists_prime = 1 \\/ pfa_factor_right_common_exists_prime = 1) -> (forall fom_index_pfp_common_exists_A. (exists fom_gap_pfp_common_exists_A_index_bound. fom_gap_pfp_common_exists_A_index_bound + S (fom_index_pfp_common_exists_A) = L) -> exists fom_value_pfp_common_exists_A. ((((exists fom_beta_height_pfp_common_exists_A_entry. fom_beta_height_pfp_common_exists_A_entry + S (fom_value_pfp_common_exists_A) = S ((S (fom_index_pfp_common_exists_A)) * ac)) /\\ exists fom_beta_quotient_pfp_common_exists_A_entry. ab = fom_beta_quotient_pfp_common_exists_A_entry * S ((S (fom_index_pfp_common_exists_A)) * ac) + (fom_value_pfp_common_exists_A))) /\\ (exists fom_gap_pfp_common_exists_A_value_bound. fom_gap_pfp_common_exists_A_value_bound + S (fom_value_pfp_common_exists_A) = p))) -> (forall fom_index_pfp_common_exists_B. (exists fom_gap_pfp_common_exists_B_index_bound. fom_gap_pfp_common_exists_B_index_bound + S (fom_index_pfp_common_exists_B) = M) -> exists fom_value_pfp_common_exists_B. ((((exists fom_beta_height_pfp_common_exists_B_entry. fom_beta_height_pfp_common_exists_B_entry + S (fom_value_pfp_common_exists_B) = S ((S (fom_index_pfp_common_exists_B)) * bc)) /\\ exists fom_beta_quotient_pfp_common_exists_B_entry. bb = fom_beta_quotient_pfp_common_exists_B_entry * S ((S (fom_index_pfp_common_exists_B)) * bc) + (fom_value_pfp_common_exists_B))) /\\ (exists fom_gap_pfp_common_exists_B_value_bound. fom_gap_pfp_common_exists_B_value_bound + S (fom_value_pfp_common_exists_B) = p))) -> (exists ub uc vb vc. ((forall fom_index_pfp_common_exists_output_left_bounded. (exists fom_gap_pfp_common_exists_output_left_bounded_index_bound. fom_gap_pfp_common_exists_output_left_bounded_index_bound + S (fom_index_pfp_common_exists_output_left_bounded) = L+M) -> exists fom_value_pfp_common_exists_output_left_bounded. ((((exists fom_beta_height_pfp_common_exists_output_left_bounded_entry. fom_beta_height_pfp_common_exists_output_left_bounded_entry + S (fom_value_pfp_common_exists_output_left_bounded) = S ((S (fom_index_pfp_common_exists_output_left_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_common_exists_output_left_bounded_entry. ub = fom_beta_quotient_pfp_common_exists_output_left_bounded_entry * S ((S (fom_index_pfp_common_exists_output_left_bounded)) * uc) + (fom_value_pfp_common_exists_output_left_bounded))) /\\ (exists fom_gap_pfp_common_exists_output_left_bounded_value_bound. fom_gap_pfp_common_exists_output_left_bounded_value_bound + S (fom_value_pfp_common_exists_output_left_bounded) = p))) /\\ (((forall fom_index_pfp_common_exists_output_right_bounded. (exists fom_gap_pfp_common_exists_output_right_bounded_index_bound. fom_gap_pfp_common_exists_output_right_bounded_index_bound + S (fom_index_pfp_common_exists_output_right_bounded) = L+M) -> exists fom_value_pfp_common_exists_output_right_bounded. ((((exists fom_beta_height_pfp_common_exists_output_right_bounded_entry. fom_beta_height_pfp_common_exists_output_right_bounded_entry + S (fom_value_pfp_common_exists_output_right_bounded) = S ((S (fom_index_pfp_common_exists_output_right_bounded)) * vc)) /\\ exists fom_beta_quotient_pfp_common_exists_output_right_bounded_entry. vb = fom_beta_quotient_pfp_common_exists_output_right_bounded_entry * S ((S (fom_index_pfp_common_exists_output_right_bounded)) * vc) + (fom_value_pfp_common_exists_output_right_bounded))) /\\ (exists fom_gap_pfp_common_exists_output_right_bounded_value_bound. fom_gap_pfp_common_exists_output_right_bounded_value_bound + S (fom_value_pfp_common_exists_output_right_bounded) = p))) /\\ ((((forall pfrep_power_common_exists_output_common_left pfrep_left_common_exists_output_common_left pfrep_right_common_exists_output_common_left. ((exists pfrep_position_common_exists_output_common_leftfirst. ((pfrep_position_common_exists_output_common_leftfirst+S (pfrep_power_common_exists_output_common_left)=(L)) /\\ ((((exists ff_h_pfp_common_exists_output_common_leftfirstentry. ff_h_pfp_common_exists_output_common_leftfirstentry + S (pfrep_left_common_exists_output_common_left) = S ((S (pfrep_position_common_exists_output_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_common_exists_output_common_leftfirstentry. ab = ff_q_pfp_common_exists_output_common_leftfirstentry * S ((S (pfrep_position_common_exists_output_common_leftfirst)) * ac) + (pfrep_left_common_exists_output_common_left)))))) \\/ (((exists pfrep_gap_common_exists_output_common_leftfirstoutside. pfrep_gap_common_exists_output_common_leftfirstoutside+(L)=(pfrep_power_common_exists_output_common_left)) /\\ (((pfrep_left_common_exists_output_common_left)=0))))) -> ((exists pfrep_position_common_exists_output_common_leftsecond. ((pfrep_position_common_exists_output_common_leftsecond+S (pfrep_power_common_exists_output_common_left)=(L+M)) /\\ ((((exists ff_h_pfp_common_exists_output_common_leftsecondentry. ff_h_pfp_common_exists_output_common_leftsecondentry + S (pfrep_right_common_exists_output_common_left) = S ((S (pfrep_position_common_exists_output_common_leftsecond)) * uc)) /\\ exists ff_q_pfp_common_exists_output_common_leftsecondentry. ub = ff_q_pfp_common_exists_output_common_leftsecondentry * S ((S (pfrep_position_common_exists_output_common_leftsecond)) * uc) + (pfrep_right_common_exists_output_common_left)))))) \\/ (((exists pfrep_gap_common_exists_output_common_leftsecondoutside. pfrep_gap_common_exists_output_common_leftsecondoutside+(L+M)=(pfrep_power_common_exists_output_common_left)) /\\ (((pfrep_right_common_exists_output_common_left)=0))))) -> pfrep_left_common_exists_output_common_left=pfrep_right_common_exists_output_common_left) /\\ ((forall pfrep_power_common_exists_output_common_right pfrep_left_common_exists_output_common_right pfrep_right_common_exists_output_common_right. ((exists pfrep_position_common_exists_output_common_rightfirst. ((pfrep_position_common_exists_output_common_rightfirst+S (pfrep_power_common_exists_output_common_right)=(M)) /\\ ((((exists ff_h_pfp_common_exists_output_common_rightfirstentry. ff_h_pfp_common_exists_output_common_rightfirstentry + S (pfrep_left_common_exists_output_common_right) = S ((S (pfrep_position_common_exists_output_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_common_exists_output_common_rightfirstentry. bb = ff_q_pfp_common_exists_output_common_rightfirstentry * S ((S (pfrep_position_common_exists_output_common_rightfirst)) * bc) + (pfrep_left_common_exists_output_common_right)))))) \\/ (((exists pfrep_gap_common_exists_output_common_rightfirstoutside. pfrep_gap_common_exists_output_common_rightfirstoutside+(M)=(pfrep_power_common_exists_output_common_right)) /\\ (((pfrep_left_common_exists_output_common_right)=0))))) -> ((exists pfrep_position_common_exists_output_common_rightsecond. ((pfrep_position_common_exists_output_common_rightsecond+S (pfrep_power_common_exists_output_common_right)=(L+M)) /\\ ((((exists ff_h_pfp_common_exists_output_common_rightsecondentry. ff_h_pfp_common_exists_output_common_rightsecondentry + S (pfrep_right_common_exists_output_common_right) = S ((S (pfrep_position_common_exists_output_common_rightsecond)) * vc)) /\\ exists ff_q_pfp_common_exists_output_common_rightsecondentry. vb = ff_q_pfp_common_exists_output_common_rightsecondentry * S ((S (pfrep_position_common_exists_output_common_rightsecond)) * vc) + (pfrep_right_common_exists_output_common_right)))))) \\/ (((exists pfrep_gap_common_exists_output_common_rightsecondoutside. pfrep_gap_common_exists_output_common_rightsecondoutside+(L+M)=(pfrep_power_common_exists_output_common_right)) /\\ (((pfrep_right_common_exists_output_common_right)=0))))) -> pfrep_left_common_exists_output_common_right=pfrep_right_common_exists_output_common_right)))))))))", "statement_sha256": 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((exists pfrep_position_common_function_left_reversefirst. ((pfrep_position_common_function_left_reversefirst+S (pfrep_power_common_function_left_reverse)=(K)) /\\ ((((exists ff_h_pfp_common_function_left_reversefirstentry. ff_h_pfp_common_function_left_reversefirstentry + S (pfrep_left_common_function_left_reverse) = S ((S (pfrep_position_common_function_left_reversefirst)) * uc)) /\\ exists ff_q_pfp_common_function_left_reversefirstentry. ub = ff_q_pfp_common_function_left_reversefirstentry * S ((S (pfrep_position_common_function_left_reversefirst)) * uc) + (pfrep_left_common_function_left_reverse)))))) \\/ (((exists pfrep_gap_common_function_left_reversefirstoutside. pfrep_gap_common_function_left_reversefirstoutside+(K)=(pfrep_power_common_function_left_reverse)) /\\ (((pfrep_left_common_function_left_reverse)=0))))) -> ((exists pfrep_position_common_function_left_reversesecond. ((pfrep_position_common_function_left_reversesecond+S (pfrep_power_common_function_left_reverse)=(L)) /\\ ((((exists ff_h_pfp_common_function_left_reversesecondentry. ff_h_pfp_common_function_left_reversesecondentry + S (pfrep_right_common_function_left_reverse) = S ((S (pfrep_position_common_function_left_reversesecond)) * ac)) /\\ exists ff_q_pfp_common_function_left_reversesecondentry. ab = ff_q_pfp_common_function_left_reversesecondentry * S ((S (pfrep_position_common_function_left_reversesecond)) * ac) + (pfrep_right_common_function_left_reverse)))))) \\/ (((exists pfrep_gap_common_function_left_reversesecondoutside. pfrep_gap_common_function_left_reversesecondoutside+(L)=(pfrep_power_common_function_left_reverse)) /\\ (((pfrep_right_common_function_left_reverse)=0))))) -> pfrep_left_common_function_left_reverse=pfrep_right_common_function_left_reverse", "specialize prime_field_polynomial_equivalent_symmetric (ab)", "specialize prime_field_polynomial_equivalent_symmetric (ac)", "specialize prime_field_polynomial_equivalent_symmetric (L)", "specialize prime_field_polynomial_equivalent_symmetric (ub)", "specialize prime_field_polynomial_equivalent_symmetric (uc)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "apply prime_field_polynomial_equivalent_symmetric", "exact h_left", "specialize prime_field_polynomial_equivalent_transitive (ub)", "specialize prime_field_polynomial_equivalent_transitive (uc)", "specialize prime_field_polynomial_equivalent_transitive (K)", "specialize prime_field_polynomial_equivalent_transitive (ab)", "specialize prime_field_polynomial_equivalent_transitive (ac)", "specialize prime_field_polynomial_equivalent_transitive (L)", "specialize prime_field_polynomial_equivalent_transitive (db)", "specialize prime_field_polynomial_equivalent_transitive (dc)", "specialize prime_field_polynomial_equivalent_transitive (J)", "apply prime_field_polynomial_equivalent_transitive", "exact hr", "exact hnew_left", "have hr : forall pfrep_power_common_function_right_reverse pfrep_left_common_function_right_reverse pfrep_right_common_function_right_reverse. ((exists pfrep_position_common_function_right_reversefirst. ((pfrep_position_common_function_right_reversefirst+S (pfrep_power_common_function_right_reverse)=(K)) /\\ ((((exists ff_h_pfp_common_function_right_reversefirstentry. ff_h_pfp_common_function_right_reversefirstentry + S (pfrep_left_common_function_right_reverse) = S ((S (pfrep_position_common_function_right_reversefirst)) * vc)) /\\ exists ff_q_pfp_common_function_right_reversefirstentry. vb = ff_q_pfp_common_function_right_reversefirstentry * S ((S (pfrep_position_common_function_right_reversefirst)) * vc) + (pfrep_left_common_function_right_reverse)))))) \\/ (((exists pfrep_gap_common_function_right_reversefirstoutside. pfrep_gap_common_function_right_reversefirstoutside+(K)=(pfrep_power_common_function_right_reverse)) /\\ (((pfrep_left_common_function_right_reverse)=0))))) -> ((exists pfrep_position_common_function_right_reversesecond. ((pfrep_position_common_function_right_reversesecond+S (pfrep_power_common_function_right_reverse)=(M)) /\\ ((((exists ff_h_pfp_common_function_right_reversesecondentry. ff_h_pfp_common_function_right_reversesecondentry + S (pfrep_right_common_function_right_reverse) = S ((S (pfrep_position_common_function_right_reversesecond)) * bc)) /\\ exists ff_q_pfp_common_function_right_reversesecondentry. bb = ff_q_pfp_common_function_right_reversesecondentry * S ((S (pfrep_position_common_function_right_reversesecond)) * bc) + (pfrep_right_common_function_right_reverse)))))) \\/ (((exists pfrep_gap_common_function_right_reversesecondoutside. pfrep_gap_common_function_right_reversesecondoutside+(M)=(pfrep_power_common_function_right_reverse)) /\\ (((pfrep_right_common_function_right_reverse)=0))))) -> pfrep_left_common_function_right_reverse=pfrep_right_common_function_right_reverse", "specialize prime_field_polynomial_equivalent_symmetric (bb)", "specialize prime_field_polynomial_equivalent_symmetric (bc)", "specialize prime_field_polynomial_equivalent_symmetric (M)", "specialize prime_field_polynomial_equivalent_symmetric (vb)", "specialize prime_field_polynomial_equivalent_symmetric (vc)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "apply prime_field_polynomial_equivalent_symmetric", "exact h_right", "specialize prime_field_polynomial_equivalent_transitive (vb)", "specialize prime_field_polynomial_equivalent_transitive (vc)", "specialize prime_field_polynomial_equivalent_transitive (K)", "specialize prime_field_polynomial_equivalent_transitive (bb)", "specialize prime_field_polynomial_equivalent_transitive (bc)", "specialize prime_field_polynomial_equivalent_transitive (M)", "specialize prime_field_polynomial_equivalent_transitive (eb)", "specialize prime_field_polynomial_equivalent_transitive (ec)", "specialize prime_field_polynomial_equivalent_transitive (J)", "apply prime_field_polynomial_equivalent_transitive", "exact hr", "exact hnew_right" ], "script_sha256": "7d9685e1babfc6121356a1cf2155935481106b180a3ee372c234ffce09cd980e", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_alignment_candidate.py", "sha256": "eb16e2eb02dbd66a7706e616388182992b8cf2e0715818dc1f7748938e7d798e" }, "statement": "forall ab ac L bb bc M ub uc vb vc K db dc eb ec J. (((forall pfrep_power_common_function_first_left pfrep_left_common_function_first_left pfrep_right_common_function_first_left. ((exists pfrep_position_common_function_first_leftfirst. ((pfrep_position_common_function_first_leftfirst+S (pfrep_power_common_function_first_left)=(L)) /\\ ((((exists ff_h_pfp_common_function_first_leftfirstentry. ff_h_pfp_common_function_first_leftfirstentry + S (pfrep_left_common_function_first_left) = S ((S (pfrep_position_common_function_first_leftfirst)) * ac)) /\\ exists ff_q_pfp_common_function_first_leftfirstentry. ab = ff_q_pfp_common_function_first_leftfirstentry * S ((S (pfrep_position_common_function_first_leftfirst)) * ac) + (pfrep_left_common_function_first_left)))))) \\/ (((exists pfrep_gap_common_function_first_leftfirstoutside. pfrep_gap_common_function_first_leftfirstoutside+(L)=(pfrep_power_common_function_first_left)) /\\ (((pfrep_left_common_function_first_left)=0))))) -> ((exists pfrep_position_common_function_first_leftsecond. ((pfrep_position_common_function_first_leftsecond+S (pfrep_power_common_function_first_left)=(K)) /\\ ((((exists ff_h_pfp_common_function_first_leftsecondentry. ff_h_pfp_common_function_first_leftsecondentry + S (pfrep_right_common_function_first_left) = S ((S (pfrep_position_common_function_first_leftsecond)) * uc)) /\\ exists ff_q_pfp_common_function_first_leftsecondentry. ub = ff_q_pfp_common_function_first_leftsecondentry * S ((S (pfrep_position_common_function_first_leftsecond)) * uc) + (pfrep_right_common_function_first_left)))))) \\/ (((exists pfrep_gap_common_function_first_leftsecondoutside. pfrep_gap_common_function_first_leftsecondoutside+(K)=(pfrep_power_common_function_first_left)) /\\ (((pfrep_right_common_function_first_left)=0))))) -> pfrep_left_common_function_first_left=pfrep_right_common_function_first_left) /\\ ((forall pfrep_power_common_function_first_right pfrep_left_common_function_first_right pfrep_right_common_function_first_right. ((exists pfrep_position_common_function_first_rightfirst. ((pfrep_position_common_function_first_rightfirst+S (pfrep_power_common_function_first_right)=(M)) /\\ ((((exists ff_h_pfp_common_function_first_rightfirstentry. ff_h_pfp_common_function_first_rightfirstentry + S (pfrep_left_common_function_first_right) = S ((S (pfrep_position_common_function_first_rightfirst)) * bc)) /\\ exists ff_q_pfp_common_function_first_rightfirstentry. bb = ff_q_pfp_common_function_first_rightfirstentry * S ((S (pfrep_position_common_function_first_rightfirst)) * bc) + (pfrep_left_common_function_first_right)))))) \\/ (((exists pfrep_gap_common_function_first_rightfirstoutside. pfrep_gap_common_function_first_rightfirstoutside+(M)=(pfrep_power_common_function_first_right)) /\\ (((pfrep_left_common_function_first_right)=0))))) -> ((exists pfrep_position_common_function_first_rightsecond. ((pfrep_position_common_function_first_rightsecond+S (pfrep_power_common_function_first_right)=(K)) /\\ ((((exists ff_h_pfp_common_function_first_rightsecondentry. ff_h_pfp_common_function_first_rightsecondentry + S (pfrep_right_common_function_first_right) = S ((S (pfrep_position_common_function_first_rightsecond)) * vc)) /\\ exists ff_q_pfp_common_function_first_rightsecondentry. vb = ff_q_pfp_common_function_first_rightsecondentry * S ((S (pfrep_position_common_function_first_rightsecond)) * vc) + (pfrep_right_common_function_first_right)))))) \\/ (((exists pfrep_gap_common_function_first_rightsecondoutside. pfrep_gap_common_function_first_rightsecondoutside+(K)=(pfrep_power_common_function_first_right)) /\\ (((pfrep_right_common_function_first_right)=0))))) -> pfrep_left_common_function_first_right=pfrep_right_common_function_first_right)))) -> (((forall pfrep_power_common_function_second_left pfrep_left_common_function_second_left pfrep_right_common_function_second_left. ((exists pfrep_position_common_function_second_leftfirst. ((pfrep_position_common_function_second_leftfirst+S (pfrep_power_common_function_second_left)=(L)) /\\ ((((exists ff_h_pfp_common_function_second_leftfirstentry. ff_h_pfp_common_function_second_leftfirstentry + S (pfrep_left_common_function_second_left) = S ((S (pfrep_position_common_function_second_leftfirst)) * ac)) /\\ exists ff_q_pfp_common_function_second_leftfirstentry. ab = ff_q_pfp_common_function_second_leftfirstentry * S ((S (pfrep_position_common_function_second_leftfirst)) * ac) + (pfrep_left_common_function_second_left)))))) \\/ (((exists pfrep_gap_common_function_second_leftfirstoutside. pfrep_gap_common_function_second_leftfirstoutside+(L)=(pfrep_power_common_function_second_left)) /\\ (((pfrep_left_common_function_second_left)=0))))) -> ((exists pfrep_position_common_function_second_leftsecond. ((pfrep_position_common_function_second_leftsecond+S (pfrep_power_common_function_second_left)=(J)) /\\ ((((exists ff_h_pfp_common_function_second_leftsecondentry. ff_h_pfp_common_function_second_leftsecondentry + S (pfrep_right_common_function_second_left) = S ((S (pfrep_position_common_function_second_leftsecond)) * dc)) /\\ exists ff_q_pfp_common_function_second_leftsecondentry. db = ff_q_pfp_common_function_second_leftsecondentry * S ((S (pfrep_position_common_function_second_leftsecond)) * dc) + (pfrep_right_common_function_second_left)))))) \\/ (((exists pfrep_gap_common_function_second_leftsecondoutside. pfrep_gap_common_function_second_leftsecondoutside+(J)=(pfrep_power_common_function_second_left)) /\\ (((pfrep_right_common_function_second_left)=0))))) -> pfrep_left_common_function_second_left=pfrep_right_common_function_second_left) /\\ ((forall pfrep_power_common_function_second_right pfrep_left_common_function_second_right pfrep_right_common_function_second_right. ((exists pfrep_position_common_function_second_rightfirst. ((pfrep_position_common_function_second_rightfirst+S (pfrep_power_common_function_second_right)=(M)) /\\ ((((exists ff_h_pfp_common_function_second_rightfirstentry. ff_h_pfp_common_function_second_rightfirstentry + S (pfrep_left_common_function_second_right) = S ((S (pfrep_position_common_function_second_rightfirst)) * bc)) /\\ exists ff_q_pfp_common_function_second_rightfirstentry. bb = ff_q_pfp_common_function_second_rightfirstentry * S ((S (pfrep_position_common_function_second_rightfirst)) * bc) + (pfrep_left_common_function_second_right)))))) \\/ (((exists pfrep_gap_common_function_second_rightfirstoutside. pfrep_gap_common_function_second_rightfirstoutside+(M)=(pfrep_power_common_function_second_right)) /\\ (((pfrep_left_common_function_second_right)=0))))) -> ((exists pfrep_position_common_function_second_rightsecond. ((pfrep_position_common_function_second_rightsecond+S (pfrep_power_common_function_second_right)=(J)) /\\ ((((exists ff_h_pfp_common_function_second_rightsecondentry. ff_h_pfp_common_function_second_rightsecondentry + S (pfrep_right_common_function_second_right) = S ((S (pfrep_position_common_function_second_rightsecond)) * ec)) /\\ exists ff_q_pfp_common_function_second_rightsecondentry. eb = ff_q_pfp_common_function_second_rightsecondentry * S ((S (pfrep_position_common_function_second_rightsecond)) * ec) + (pfrep_right_common_function_second_right)))))) \\/ (((exists pfrep_gap_common_function_second_rightsecondoutside. pfrep_gap_common_function_second_rightsecondoutside+(J)=(pfrep_power_common_function_second_right)) /\\ (((pfrep_right_common_function_second_right)=0))))) -> pfrep_left_common_function_second_right=pfrep_right_common_function_second_right)))) -> (((forall pfrep_power_common_function_result_left pfrep_left_common_function_result_left pfrep_right_common_function_result_left. ((exists pfrep_position_common_function_result_leftfirst. ((pfrep_position_common_function_result_leftfirst+S (pfrep_power_common_function_result_left)=(K)) /\\ ((((exists ff_h_pfp_common_function_result_leftfirstentry. ff_h_pfp_common_function_result_leftfirstentry + S (pfrep_left_common_function_result_left) = S ((S (pfrep_position_common_function_result_leftfirst)) * uc)) /\\ exists ff_q_pfp_common_function_result_leftfirstentry. ub = ff_q_pfp_common_function_result_leftfirstentry * S ((S (pfrep_position_common_function_result_leftfirst)) * uc) + (pfrep_left_common_function_result_left)))))) \\/ (((exists pfrep_gap_common_function_result_leftfirstoutside. pfrep_gap_common_function_result_leftfirstoutside+(K)=(pfrep_power_common_function_result_left)) /\\ (((pfrep_left_common_function_result_left)=0))))) -> ((exists pfrep_position_common_function_result_leftsecond. ((pfrep_position_common_function_result_leftsecond+S (pfrep_power_common_function_result_left)=(J)) /\\ ((((exists ff_h_pfp_common_function_result_leftsecondentry. ff_h_pfp_common_function_result_leftsecondentry + S (pfrep_right_common_function_result_left) = S ((S (pfrep_position_common_function_result_leftsecond)) * dc)) /\\ exists ff_q_pfp_common_function_result_leftsecondentry. db = ff_q_pfp_common_function_result_leftsecondentry * S ((S (pfrep_position_common_function_result_leftsecond)) * dc) + (pfrep_right_common_function_result_left)))))) \\/ (((exists pfrep_gap_common_function_result_leftsecondoutside. pfrep_gap_common_function_result_leftsecondoutside+(J)=(pfrep_power_common_function_result_left)) /\\ (((pfrep_right_common_function_result_left)=0))))) -> pfrep_left_common_function_result_left=pfrep_right_common_function_result_left) /\\ ((forall pfrep_power_common_function_result_right pfrep_left_common_function_result_right pfrep_right_common_function_result_right. ((exists pfrep_position_common_function_result_rightfirst. ((pfrep_position_common_function_result_rightfirst+S (pfrep_power_common_function_result_right)=(K)) /\\ ((((exists ff_h_pfp_common_function_result_rightfirstentry. ff_h_pfp_common_function_result_rightfirstentry + S (pfrep_left_common_function_result_right) = S ((S (pfrep_position_common_function_result_rightfirst)) * vc)) /\\ exists ff_q_pfp_common_function_result_rightfirstentry. vb = ff_q_pfp_common_function_result_rightfirstentry * S ((S (pfrep_position_common_function_result_rightfirst)) * vc) + (pfrep_left_common_function_result_right)))))) \\/ (((exists pfrep_gap_common_function_result_rightfirstoutside. pfrep_gap_common_function_result_rightfirstoutside+(K)=(pfrep_power_common_function_result_right)) /\\ (((pfrep_left_common_function_result_right)=0))))) -> ((exists pfrep_position_common_function_result_rightsecond. ((pfrep_position_common_function_result_rightsecond+S (pfrep_power_common_function_result_right)=(J)) /\\ ((((exists ff_h_pfp_common_function_result_rightsecondentry. ff_h_pfp_common_function_result_rightsecondentry + S (pfrep_right_common_function_result_right) = S ((S (pfrep_position_common_function_result_rightsecond)) * ec)) /\\ exists ff_q_pfp_common_function_result_rightsecondentry. eb = ff_q_pfp_common_function_result_rightsecondentry * S ((S (pfrep_position_common_function_result_rightsecond)) * ec) + (pfrep_right_common_function_result_right)))))) \\/ (((exists pfrep_gap_common_function_result_rightsecondoutside. pfrep_gap_common_function_result_rightsecondoutside+(J)=(pfrep_power_common_function_result_right)) /\\ (((pfrep_right_common_function_result_right)=0))))) -> pfrep_left_common_function_result_right=pfrep_right_common_function_result_right))))", "statement_sha256": "5b2fa84d8801413b35903b1fd1f886223da5aa6ef0f0e6ee1e7b53fb9977105d", "summary": "Any two choices of common representatives are pairwise formally equivalent, even at different common lengths; 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(((forall pfrep_power_common_symmetric_old_left pfrep_left_common_symmetric_old_left pfrep_right_common_symmetric_old_left. ((exists pfrep_position_common_symmetric_old_leftfirst. ((pfrep_position_common_symmetric_old_leftfirst+S (pfrep_power_common_symmetric_old_left)=(L)) /\\ ((((exists ff_h_pfp_common_symmetric_old_leftfirstentry. ff_h_pfp_common_symmetric_old_leftfirstentry + S (pfrep_left_common_symmetric_old_left) = S ((S (pfrep_position_common_symmetric_old_leftfirst)) * ac)) /\\ exists ff_q_pfp_common_symmetric_old_leftfirstentry. ab = ff_q_pfp_common_symmetric_old_leftfirstentry * S ((S (pfrep_position_common_symmetric_old_leftfirst)) * ac) + (pfrep_left_common_symmetric_old_left)))))) \\/ (((exists pfrep_gap_common_symmetric_old_leftfirstoutside. pfrep_gap_common_symmetric_old_leftfirstoutside+(L)=(pfrep_power_common_symmetric_old_left)) /\\ (((pfrep_left_common_symmetric_old_left)=0))))) -> ((exists pfrep_position_common_symmetric_old_leftsecond. ((pfrep_position_common_symmetric_old_leftsecond+S (pfrep_power_common_symmetric_old_left)=(K)) /\\ ((((exists ff_h_pfp_common_symmetric_old_leftsecondentry. ff_h_pfp_common_symmetric_old_leftsecondentry + S (pfrep_right_common_symmetric_old_left) = S ((S (pfrep_position_common_symmetric_old_leftsecond)) * uc)) /\\ exists ff_q_pfp_common_symmetric_old_leftsecondentry. ub = ff_q_pfp_common_symmetric_old_leftsecondentry * S ((S (pfrep_position_common_symmetric_old_leftsecond)) * uc) + (pfrep_right_common_symmetric_old_left)))))) \\/ (((exists pfrep_gap_common_symmetric_old_leftsecondoutside. pfrep_gap_common_symmetric_old_leftsecondoutside+(K)=(pfrep_power_common_symmetric_old_left)) /\\ (((pfrep_right_common_symmetric_old_left)=0))))) -> pfrep_left_common_symmetric_old_left=pfrep_right_common_symmetric_old_left) /\\ ((forall pfrep_power_common_symmetric_old_right pfrep_left_common_symmetric_old_right pfrep_right_common_symmetric_old_right. ((exists pfrep_position_common_symmetric_old_rightfirst. ((pfrep_position_common_symmetric_old_rightfirst+S (pfrep_power_common_symmetric_old_right)=(M)) /\\ ((((exists ff_h_pfp_common_symmetric_old_rightfirstentry. ff_h_pfp_common_symmetric_old_rightfirstentry + S (pfrep_left_common_symmetric_old_right) = S ((S (pfrep_position_common_symmetric_old_rightfirst)) * bc)) /\\ exists ff_q_pfp_common_symmetric_old_rightfirstentry. bb = ff_q_pfp_common_symmetric_old_rightfirstentry * S ((S (pfrep_position_common_symmetric_old_rightfirst)) * bc) + (pfrep_left_common_symmetric_old_right)))))) \\/ (((exists pfrep_gap_common_symmetric_old_rightfirstoutside. pfrep_gap_common_symmetric_old_rightfirstoutside+(M)=(pfrep_power_common_symmetric_old_right)) /\\ (((pfrep_left_common_symmetric_old_right)=0))))) -> ((exists pfrep_position_common_symmetric_old_rightsecond. ((pfrep_position_common_symmetric_old_rightsecond+S (pfrep_power_common_symmetric_old_right)=(K)) /\\ ((((exists ff_h_pfp_common_symmetric_old_rightsecondentry. ff_h_pfp_common_symmetric_old_rightsecondentry + S (pfrep_right_common_symmetric_old_right) = S ((S (pfrep_position_common_symmetric_old_rightsecond)) * vc)) /\\ exists ff_q_pfp_common_symmetric_old_rightsecondentry. vb = ff_q_pfp_common_symmetric_old_rightsecondentry * S ((S (pfrep_position_common_symmetric_old_rightsecond)) * vc) + (pfrep_right_common_symmetric_old_right)))))) \\/ (((exists pfrep_gap_common_symmetric_old_rightsecondoutside. pfrep_gap_common_symmetric_old_rightsecondoutside+(K)=(pfrep_power_common_symmetric_old_right)) /\\ (((pfrep_right_common_symmetric_old_right)=0))))) -> pfrep_left_common_symmetric_old_right=pfrep_right_common_symmetric_old_right)))) -> (((forall pfrep_power_common_symmetric_new_left pfrep_left_common_symmetric_new_left pfrep_right_common_symmetric_new_left. ((exists pfrep_position_common_symmetric_new_leftfirst. ((pfrep_position_common_symmetric_new_leftfirst+S (pfrep_power_common_symmetric_new_left)=(M)) /\\ ((((exists ff_h_pfp_common_symmetric_new_leftfirstentry. ff_h_pfp_common_symmetric_new_leftfirstentry + S (pfrep_left_common_symmetric_new_left) = S ((S (pfrep_position_common_symmetric_new_leftfirst)) * bc)) /\\ exists ff_q_pfp_common_symmetric_new_leftfirstentry. bb = ff_q_pfp_common_symmetric_new_leftfirstentry * S ((S (pfrep_position_common_symmetric_new_leftfirst)) * bc) + (pfrep_left_common_symmetric_new_left)))))) \\/ (((exists pfrep_gap_common_symmetric_new_leftfirstoutside. pfrep_gap_common_symmetric_new_leftfirstoutside+(M)=(pfrep_power_common_symmetric_new_left)) /\\ (((pfrep_left_common_symmetric_new_left)=0))))) -> ((exists pfrep_position_common_symmetric_new_leftsecond. ((pfrep_position_common_symmetric_new_leftsecond+S (pfrep_power_common_symmetric_new_left)=(K)) /\\ ((((exists ff_h_pfp_common_symmetric_new_leftsecondentry. ff_h_pfp_common_symmetric_new_leftsecondentry + S (pfrep_right_common_symmetric_new_left) = S ((S (pfrep_position_common_symmetric_new_leftsecond)) * vc)) /\\ exists ff_q_pfp_common_symmetric_new_leftsecondentry. vb = ff_q_pfp_common_symmetric_new_leftsecondentry * S ((S (pfrep_position_common_symmetric_new_leftsecond)) * vc) + (pfrep_right_common_symmetric_new_left)))))) \\/ (((exists pfrep_gap_common_symmetric_new_leftsecondoutside. pfrep_gap_common_symmetric_new_leftsecondoutside+(K)=(pfrep_power_common_symmetric_new_left)) /\\ (((pfrep_right_common_symmetric_new_left)=0))))) -> pfrep_left_common_symmetric_new_left=pfrep_right_common_symmetric_new_left) /\\ ((forall pfrep_power_common_symmetric_new_right pfrep_left_common_symmetric_new_right pfrep_right_common_symmetric_new_right. ((exists pfrep_position_common_symmetric_new_rightfirst. ((pfrep_position_common_symmetric_new_rightfirst+S (pfrep_power_common_symmetric_new_right)=(L)) /\\ ((((exists ff_h_pfp_common_symmetric_new_rightfirstentry. ff_h_pfp_common_symmetric_new_rightfirstentry + S (pfrep_left_common_symmetric_new_right) = S ((S (pfrep_position_common_symmetric_new_rightfirst)) * ac)) /\\ exists ff_q_pfp_common_symmetric_new_rightfirstentry. ab = ff_q_pfp_common_symmetric_new_rightfirstentry * S ((S (pfrep_position_common_symmetric_new_rightfirst)) * ac) + (pfrep_left_common_symmetric_new_right)))))) \\/ (((exists pfrep_gap_common_symmetric_new_rightfirstoutside. pfrep_gap_common_symmetric_new_rightfirstoutside+(L)=(pfrep_power_common_symmetric_new_right)) /\\ (((pfrep_left_common_symmetric_new_right)=0))))) -> ((exists pfrep_position_common_symmetric_new_rightsecond. ((pfrep_position_common_symmetric_new_rightsecond+S (pfrep_power_common_symmetric_new_right)=(K)) /\\ ((((exists ff_h_pfp_common_symmetric_new_rightsecondentry. ff_h_pfp_common_symmetric_new_rightsecondentry + S (pfrep_right_common_symmetric_new_right) = S ((S (pfrep_position_common_symmetric_new_rightsecond)) * uc)) /\\ exists ff_q_pfp_common_symmetric_new_rightsecondentry. ub = ff_q_pfp_common_symmetric_new_rightsecondentry * S ((S (pfrep_position_common_symmetric_new_rightsecond)) * uc) + (pfrep_right_common_symmetric_new_right)))))) \\/ (((exists pfrep_gap_common_symmetric_new_rightsecondoutside. pfrep_gap_common_symmetric_new_rightsecondoutside+(K)=(pfrep_power_common_symmetric_new_right)) /\\ (((pfrep_right_common_symmetric_new_right)=0))))) -> pfrep_left_common_symmetric_new_right=pfrep_right_common_symmetric_new_right))))", "statement_sha256": "b61f777d6eb54e37ba25ea76bc1eb4cc37fdf97ec323f727f3ba1aa4a2665886", "summary": "Swapping the two original inputs and their actual 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"evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "7638b077a19c65d8b933cab1f71fd0473a91c891a0dd24140399c82f9510fcf6", "membership": "alpha_only", "name": "prime_field_polynomial_aligned_add_from_common", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro M", "intro rb", "intro rc", "intro N", "intro ub", "intro uc", "intro vb", "intro vc", "intro tb", "intro tc", "intro K", "intro ha", "intro hb", "intro hr", "intro hc", "intro hs", "intro he", "split", "exact ha", "split", "exact hb", "split", "exact hr", "exists ub", "exists uc", "exists vb", "exists vc", "exists tb", "exists tc", "exists K", "split", "exact hc", "split", "exact hs", "exact he" ], "script_sha256": "98322118a803b17f367c070c1b6ba9660f870e668b7b1da1672b83b73b831223", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_aligned_add_candidate.py", "sha256": "a05bb4f5c4230ca05f51690d3ab82e33ff4596af65176874e25fbe38cf87a0db" }, "statement": "forall p ab ac L bb bc M rb rc N ub uc vb vc tb tc K. (forall fom_index_pfp_from_common_0. (exists fom_gap_pfp_from_common_0_index_bound. fom_gap_pfp_from_common_0_index_bound + S (fom_index_pfp_from_common_0) = L) -> exists fom_value_pfp_from_common_0. ((((exists fom_beta_height_pfp_from_common_0_entry. fom_beta_height_pfp_from_common_0_entry + S (fom_value_pfp_from_common_0) = S ((S (fom_index_pfp_from_common_0)) * ac)) /\\ exists fom_beta_quotient_pfp_from_common_0_entry. ab = fom_beta_quotient_pfp_from_common_0_entry * S ((S (fom_index_pfp_from_common_0)) * ac) + (fom_value_pfp_from_common_0))) /\\ (exists fom_gap_pfp_from_common_0_value_bound. fom_gap_pfp_from_common_0_value_bound + S (fom_value_pfp_from_common_0) = p))) -> (forall fom_index_pfp_from_common_1. (exists fom_gap_pfp_from_common_1_index_bound. fom_gap_pfp_from_common_1_index_bound + S (fom_index_pfp_from_common_1) = M) -> exists fom_value_pfp_from_common_1. ((((exists fom_beta_height_pfp_from_common_1_entry. fom_beta_height_pfp_from_common_1_entry + S (fom_value_pfp_from_common_1) = S ((S (fom_index_pfp_from_common_1)) * bc)) /\\ exists fom_beta_quotient_pfp_from_common_1_entry. bb = fom_beta_quotient_pfp_from_common_1_entry * S ((S (fom_index_pfp_from_common_1)) * bc) + (fom_value_pfp_from_common_1))) /\\ (exists fom_gap_pfp_from_common_1_value_bound. fom_gap_pfp_from_common_1_value_bound + S (fom_value_pfp_from_common_1) = p))) -> (forall fom_index_pfp_from_common_2. (exists fom_gap_pfp_from_common_2_index_bound. fom_gap_pfp_from_common_2_index_bound + S (fom_index_pfp_from_common_2) = N) -> exists fom_value_pfp_from_common_2. ((((exists fom_beta_height_pfp_from_common_2_entry. fom_beta_height_pfp_from_common_2_entry + S (fom_value_pfp_from_common_2) = S ((S (fom_index_pfp_from_common_2)) * rc)) /\\ exists fom_beta_quotient_pfp_from_common_2_entry. rb = fom_beta_quotient_pfp_from_common_2_entry * S ((S (fom_index_pfp_from_common_2)) * rc) + (fom_value_pfp_from_common_2))) /\\ (exists fom_gap_pfp_from_common_2_value_bound. fom_gap_pfp_from_common_2_value_bound + S (fom_value_pfp_from_common_2) = p))) -> (((forall pfrep_power_from_common_reps_left pfrep_left_from_common_reps_left pfrep_right_from_common_reps_left. ((exists pfrep_position_from_common_reps_leftfirst. ((pfrep_position_from_common_reps_leftfirst+S (pfrep_power_from_common_reps_left)=(L)) /\\ ((((exists ff_h_pfp_from_common_reps_leftfirstentry. ff_h_pfp_from_common_reps_leftfirstentry + S (pfrep_left_from_common_reps_left) = S ((S (pfrep_position_from_common_reps_leftfirst)) * ac)) /\\ exists ff_q_pfp_from_common_reps_leftfirstentry. ab = ff_q_pfp_from_common_reps_leftfirstentry * S ((S (pfrep_position_from_common_reps_leftfirst)) * ac) + (pfrep_left_from_common_reps_left)))))) \\/ (((exists pfrep_gap_from_common_reps_leftfirstoutside. pfrep_gap_from_common_reps_leftfirstoutside+(L)=(pfrep_power_from_common_reps_left)) /\\ (((pfrep_left_from_common_reps_left)=0))))) -> ((exists pfrep_position_from_common_reps_leftsecond. ((pfrep_position_from_common_reps_leftsecond+S (pfrep_power_from_common_reps_left)=(K)) /\\ ((((exists ff_h_pfp_from_common_reps_leftsecondentry. ff_h_pfp_from_common_reps_leftsecondentry + S (pfrep_right_from_common_reps_left) = S ((S (pfrep_position_from_common_reps_leftsecond)) * uc)) /\\ exists ff_q_pfp_from_common_reps_leftsecondentry. ub = ff_q_pfp_from_common_reps_leftsecondentry * S ((S (pfrep_position_from_common_reps_leftsecond)) * uc) + (pfrep_right_from_common_reps_left)))))) \\/ (((exists pfrep_gap_from_common_reps_leftsecondoutside. pfrep_gap_from_common_reps_leftsecondoutside+(K)=(pfrep_power_from_common_reps_left)) /\\ (((pfrep_right_from_common_reps_left)=0))))) -> pfrep_left_from_common_reps_left=pfrep_right_from_common_reps_left) /\\ ((forall pfrep_power_from_common_reps_right pfrep_left_from_common_reps_right pfrep_right_from_common_reps_right. ((exists pfrep_position_from_common_reps_rightfirst. ((pfrep_position_from_common_reps_rightfirst+S (pfrep_power_from_common_reps_right)=(M)) /\\ ((((exists ff_h_pfp_from_common_reps_rightfirstentry. ff_h_pfp_from_common_reps_rightfirstentry + S (pfrep_left_from_common_reps_right) = S ((S (pfrep_position_from_common_reps_rightfirst)) * bc)) /\\ exists ff_q_pfp_from_common_reps_rightfirstentry. bb = ff_q_pfp_from_common_reps_rightfirstentry * S ((S (pfrep_position_from_common_reps_rightfirst)) * bc) + (pfrep_left_from_common_reps_right)))))) \\/ (((exists pfrep_gap_from_common_reps_rightfirstoutside. pfrep_gap_from_common_reps_rightfirstoutside+(M)=(pfrep_power_from_common_reps_right)) /\\ (((pfrep_left_from_common_reps_right)=0))))) -> ((exists pfrep_position_from_common_reps_rightsecond. ((pfrep_position_from_common_reps_rightsecond+S (pfrep_power_from_common_reps_right)=(K)) /\\ ((((exists ff_h_pfp_from_common_reps_rightsecondentry. ff_h_pfp_from_common_reps_rightsecondentry + S (pfrep_right_from_common_reps_right) = S ((S (pfrep_position_from_common_reps_rightsecond)) * vc)) /\\ exists ff_q_pfp_from_common_reps_rightsecondentry. vb = ff_q_pfp_from_common_reps_rightsecondentry * S ((S (pfrep_position_from_common_reps_rightsecond)) * vc) + (pfrep_right_from_common_reps_right)))))) \\/ (((exists pfrep_gap_from_common_reps_rightsecondoutside. pfrep_gap_from_common_reps_rightsecondoutside+(K)=(pfrep_power_from_common_reps_right)) /\\ (((pfrep_right_from_common_reps_right)=0))))) -> pfrep_left_from_common_reps_right=pfrep_right_from_common_reps_right)))) -> (forall pfp_index_from_common_sum. (exists pfa_gap_from_common_sumindex. pfa_gap_from_common_sumindex + S (pfp_index_from_common_sum) = (K)) -> exists pfp_left_from_common_sum pfp_right_from_common_sum pfp_value_from_common_sum. ((((exists ff_h_pfp_from_common_sumleft. ff_h_pfp_from_common_sumleft + S (pfp_left_from_common_sum) = S ((S (pfp_index_from_common_sum)) * uc)) /\\ exists ff_q_pfp_from_common_sumleft. ub = ff_q_pfp_from_common_sumleft * S ((S (pfp_index_from_common_sum)) * uc) + (pfp_left_from_common_sum))) /\\ (((((exists ff_h_pfp_from_common_sumright. ff_h_pfp_from_common_sumright + S (pfp_right_from_common_sum) = S ((S (pfp_index_from_common_sum)) * vc)) /\\ exists ff_q_pfp_from_common_sumright. vb = ff_q_pfp_from_common_sumright * S ((S (pfp_index_from_common_sum)) * vc) + (pfp_right_from_common_sum))) /\\ (((((exists ff_h_pfp_from_common_sumtarget. ff_h_pfp_from_common_sumtarget + S (pfp_value_from_common_sum) = S ((S (pfp_index_from_common_sum)) * tc)) /\\ exists ff_q_pfp_from_common_sumtarget. tb = ff_q_pfp_from_common_sumtarget * S ((S (pfp_index_from_common_sum)) * tc) + (pfp_value_from_common_sum))) /\\ ((((exists pfa_gap_from_common_sumoperationleft. pfa_gap_from_common_sumoperationleft + S (pfp_left_from_common_sum) = (p)) /\\ (((exists pfa_gap_from_common_sumoperationright. pfa_gap_from_common_sumoperationright + S (pfp_right_from_common_sum) = (p)) /\\ ((((exists pfa_gap_from_common_sumoperationresultbound. pfa_gap_from_common_sumoperationresultbound + S (pfp_value_from_common_sum) = (p)) /\\ ((exists pfa_offset_left_from_common_sumoperationresultcongruence pfa_offset_right_from_common_sumoperationresultcongruence. ((pfp_left_from_common_sum) + (pfp_right_from_common_sum)) + (p) * pfa_offset_left_from_common_sumoperationresultcongruence = (pfp_value_from_common_sum) + (p) * pfa_offset_right_from_common_sumoperationresultcongruence)))))))))))))))) -> (forall pfrep_power_from_common_output pfrep_left_from_common_output pfrep_right_from_common_output. ((exists pfrep_position_from_common_outputfirst. ((pfrep_position_from_common_outputfirst+S (pfrep_power_from_common_output)=(K)) /\\ ((((exists ff_h_pfp_from_common_outputfirstentry. ff_h_pfp_from_common_outputfirstentry + S (pfrep_left_from_common_output) = S ((S (pfrep_position_from_common_outputfirst)) * tc)) /\\ exists ff_q_pfp_from_common_outputfirstentry. tb = ff_q_pfp_from_common_outputfirstentry * S ((S (pfrep_position_from_common_outputfirst)) * tc) + (pfrep_left_from_common_output)))))) \\/ (((exists pfrep_gap_from_common_outputfirstoutside. pfrep_gap_from_common_outputfirstoutside+(K)=(pfrep_power_from_common_output)) /\\ (((pfrep_left_from_common_output)=0))))) -> ((exists pfrep_position_from_common_outputsecond. ((pfrep_position_from_common_outputsecond+S (pfrep_power_from_common_output)=(N)) /\\ ((((exists ff_h_pfp_from_common_outputsecondentry. ff_h_pfp_from_common_outputsecondentry + S (pfrep_right_from_common_output) = S ((S (pfrep_position_from_common_outputsecond)) * rc)) /\\ exists ff_q_pfp_from_common_outputsecondentry. rb = ff_q_pfp_from_common_outputsecondentry * S ((S (pfrep_position_from_common_outputsecond)) * rc) + (pfrep_right_from_common_output)))))) \\/ (((exists pfrep_gap_from_common_outputsecondoutside. pfrep_gap_from_common_outputsecondoutside+(N)=(pfrep_power_from_common_output)) /\\ (((pfrep_right_from_common_output)=0))))) -> pfrep_left_from_common_output=pfrep_right_from_common_output) -> (((forall fom_index_pfp_from_common_result_left_bounded. (exists fom_gap_pfp_from_common_result_left_bounded_index_bound. fom_gap_pfp_from_common_result_left_bounded_index_bound + S (fom_index_pfp_from_common_result_left_bounded) = L) -> exists fom_value_pfp_from_common_result_left_bounded. ((((exists fom_beta_height_pfp_from_common_result_left_bounded_entry. fom_beta_height_pfp_from_common_result_left_bounded_entry + S (fom_value_pfp_from_common_result_left_bounded) = S ((S (fom_index_pfp_from_common_result_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_from_common_result_left_bounded_entry. ab = fom_beta_quotient_pfp_from_common_result_left_bounded_entry * S ((S (fom_index_pfp_from_common_result_left_bounded)) * ac) + (fom_value_pfp_from_common_result_left_bounded))) /\\ (exists fom_gap_pfp_from_common_result_left_bounded_value_bound. fom_gap_pfp_from_common_result_left_bounded_value_bound + S (fom_value_pfp_from_common_result_left_bounded) = p))) /\\ (((forall fom_index_pfp_from_common_result_right_bounded. (exists fom_gap_pfp_from_common_result_right_bounded_index_bound. fom_gap_pfp_from_common_result_right_bounded_index_bound + S (fom_index_pfp_from_common_result_right_bounded) = M) -> exists fom_value_pfp_from_common_result_right_bounded. ((((exists fom_beta_height_pfp_from_common_result_right_bounded_entry. fom_beta_height_pfp_from_common_result_right_bounded_entry + S (fom_value_pfp_from_common_result_right_bounded) = S ((S (fom_index_pfp_from_common_result_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_from_common_result_right_bounded_entry. bb = fom_beta_quotient_pfp_from_common_result_right_bounded_entry * S ((S (fom_index_pfp_from_common_result_right_bounded)) * bc) + (fom_value_pfp_from_common_result_right_bounded))) /\\ (exists fom_gap_pfp_from_common_result_right_bounded_value_bound. fom_gap_pfp_from_common_result_right_bounded_value_bound + S (fom_value_pfp_from_common_result_right_bounded) = p))) /\\ (((forall fom_index_pfp_from_common_result_result_bounded. (exists fom_gap_pfp_from_common_result_result_bounded_index_bound. fom_gap_pfp_from_common_result_result_bounded_index_bound + S (fom_index_pfp_from_common_result_result_bounded) = N) -> exists fom_value_pfp_from_common_result_result_bounded. ((((exists fom_beta_height_pfp_from_common_result_result_bounded_entry. fom_beta_height_pfp_from_common_result_result_bounded_entry + S (fom_value_pfp_from_common_result_result_bounded) = S ((S (fom_index_pfp_from_common_result_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_from_common_result_result_bounded_entry. rb = fom_beta_quotient_pfp_from_common_result_result_bounded_entry * S ((S (fom_index_pfp_from_common_result_result_bounded)) * rc) + (fom_value_pfp_from_common_result_result_bounded))) /\\ (exists fom_gap_pfp_from_common_result_result_bounded_value_bound. fom_gap_pfp_from_common_result_result_bounded_value_bound + S (fom_value_pfp_from_common_result_result_bounded) = p))) /\\ ((exists pfaa_left_b_from_common_result pfaa_left_c_from_common_result pfaa_right_b_from_common_result pfaa_right_c_from_common_result pfaa_sum_b_from_common_result pfaa_sum_c_from_common_result pfaa_length_from_common_result. ((((forall pfrep_power_from_common_result_witness_common_left pfrep_left_from_common_result_witness_common_left pfrep_right_from_common_result_witness_common_left. ((exists pfrep_position_from_common_result_witness_common_leftfirst. ((pfrep_position_from_common_result_witness_common_leftfirst+S (pfrep_power_from_common_result_witness_common_left)=(L)) /\\ ((((exists ff_h_pfp_from_common_result_witness_common_leftfirstentry. ff_h_pfp_from_common_result_witness_common_leftfirstentry + S (pfrep_left_from_common_result_witness_common_left) = S ((S (pfrep_position_from_common_result_witness_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_from_common_result_witness_common_leftfirstentry. ab = ff_q_pfp_from_common_result_witness_common_leftfirstentry * S ((S (pfrep_position_from_common_result_witness_common_leftfirst)) * ac) + (pfrep_left_from_common_result_witness_common_left)))))) \\/ (((exists pfrep_gap_from_common_result_witness_common_leftfirstoutside. pfrep_gap_from_common_result_witness_common_leftfirstoutside+(L)=(pfrep_power_from_common_result_witness_common_left)) /\\ (((pfrep_left_from_common_result_witness_common_left)=0))))) -> ((exists pfrep_position_from_common_result_witness_common_leftsecond. ((pfrep_position_from_common_result_witness_common_leftsecond+S (pfrep_power_from_common_result_witness_common_left)=(pfaa_length_from_common_result)) /\\ ((((exists ff_h_pfp_from_common_result_witness_common_leftsecondentry. ff_h_pfp_from_common_result_witness_common_leftsecondentry + S (pfrep_right_from_common_result_witness_common_left) = S ((S (pfrep_position_from_common_result_witness_common_leftsecond)) * pfaa_left_c_from_common_result)) /\\ exists ff_q_pfp_from_common_result_witness_common_leftsecondentry. pfaa_left_b_from_common_result = ff_q_pfp_from_common_result_witness_common_leftsecondentry * S ((S (pfrep_position_from_common_result_witness_common_leftsecond)) * pfaa_left_c_from_common_result) + (pfrep_right_from_common_result_witness_common_left)))))) \\/ (((exists pfrep_gap_from_common_result_witness_common_leftsecondoutside. pfrep_gap_from_common_result_witness_common_leftsecondoutside+(pfaa_length_from_common_result)=(pfrep_power_from_common_result_witness_common_left)) /\\ (((pfrep_right_from_common_result_witness_common_left)=0))))) -> pfrep_left_from_common_result_witness_common_left=pfrep_right_from_common_result_witness_common_left) /\\ ((forall pfrep_power_from_common_result_witness_common_right pfrep_left_from_common_result_witness_common_right pfrep_right_from_common_result_witness_common_right. ((exists pfrep_position_from_common_result_witness_common_rightfirst. ((pfrep_position_from_common_result_witness_common_rightfirst+S (pfrep_power_from_common_result_witness_common_right)=(M)) /\\ ((((exists ff_h_pfp_from_common_result_witness_common_rightfirstentry. ff_h_pfp_from_common_result_witness_common_rightfirstentry + S (pfrep_left_from_common_result_witness_common_right) = S ((S (pfrep_position_from_common_result_witness_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_from_common_result_witness_common_rightfirstentry. bb = ff_q_pfp_from_common_result_witness_common_rightfirstentry * S ((S (pfrep_position_from_common_result_witness_common_rightfirst)) * bc) + (pfrep_left_from_common_result_witness_common_right)))))) \\/ (((exists pfrep_gap_from_common_result_witness_common_rightfirstoutside. pfrep_gap_from_common_result_witness_common_rightfirstoutside+(M)=(pfrep_power_from_common_result_witness_common_right)) /\\ (((pfrep_left_from_common_result_witness_common_right)=0))))) -> ((exists pfrep_position_from_common_result_witness_common_rightsecond. ((pfrep_position_from_common_result_witness_common_rightsecond+S (pfrep_power_from_common_result_witness_common_right)=(pfaa_length_from_common_result)) /\\ ((((exists ff_h_pfp_from_common_result_witness_common_rightsecondentry. ff_h_pfp_from_common_result_witness_common_rightsecondentry + S (pfrep_right_from_common_result_witness_common_right) = S ((S (pfrep_position_from_common_result_witness_common_rightsecond)) * pfaa_right_c_from_common_result)) /\\ exists ff_q_pfp_from_common_result_witness_common_rightsecondentry. pfaa_right_b_from_common_result = ff_q_pfp_from_common_result_witness_common_rightsecondentry * S ((S (pfrep_position_from_common_result_witness_common_rightsecond)) * pfaa_right_c_from_common_result) + (pfrep_right_from_common_result_witness_common_right)))))) \\/ (((exists pfrep_gap_from_common_result_witness_common_rightsecondoutside. pfrep_gap_from_common_result_witness_common_rightsecondoutside+(pfaa_length_from_common_result)=(pfrep_power_from_common_result_witness_common_right)) /\\ (((pfrep_right_from_common_result_witness_common_right)=0))))) -> pfrep_left_from_common_result_witness_common_right=pfrep_right_from_common_result_witness_common_right)))) /\\ (((forall pfp_index_from_common_result_witness_operation. (exists pfa_gap_from_common_result_witness_operationindex. pfa_gap_from_common_result_witness_operationindex + S (pfp_index_from_common_result_witness_operation) = (pfaa_length_from_common_result)) -> exists pfp_left_from_common_result_witness_operation pfp_right_from_common_result_witness_operation pfp_value_from_common_result_witness_operation. ((((exists ff_h_pfp_from_common_result_witness_operationleft. ff_h_pfp_from_common_result_witness_operationleft + S (pfp_left_from_common_result_witness_operation) = S ((S (pfp_index_from_common_result_witness_operation)) * pfaa_left_c_from_common_result)) /\\ exists ff_q_pfp_from_common_result_witness_operationleft. pfaa_left_b_from_common_result = ff_q_pfp_from_common_result_witness_operationleft * S ((S (pfp_index_from_common_result_witness_operation)) * pfaa_left_c_from_common_result) + (pfp_left_from_common_result_witness_operation))) /\\ (((((exists ff_h_pfp_from_common_result_witness_operationright. ff_h_pfp_from_common_result_witness_operationright + S (pfp_right_from_common_result_witness_operation) = S ((S (pfp_index_from_common_result_witness_operation)) * pfaa_right_c_from_common_result)) /\\ exists ff_q_pfp_from_common_result_witness_operationright. pfaa_right_b_from_common_result = ff_q_pfp_from_common_result_witness_operationright * S ((S (pfp_index_from_common_result_witness_operation)) * pfaa_right_c_from_common_result) + (pfp_right_from_common_result_witness_operation))) /\\ (((((exists ff_h_pfp_from_common_result_witness_operationtarget. ff_h_pfp_from_common_result_witness_operationtarget + S (pfp_value_from_common_result_witness_operation) = S ((S (pfp_index_from_common_result_witness_operation)) * pfaa_sum_c_from_common_result)) /\\ exists ff_q_pfp_from_common_result_witness_operationtarget. pfaa_sum_b_from_common_result = ff_q_pfp_from_common_result_witness_operationtarget * S ((S (pfp_index_from_common_result_witness_operation)) * pfaa_sum_c_from_common_result) + (pfp_value_from_common_result_witness_operation))) /\\ ((((exists pfa_gap_from_common_result_witness_operationoperationleft. pfa_gap_from_common_result_witness_operationoperationleft + S (pfp_left_from_common_result_witness_operation) = (p)) /\\ (((exists pfa_gap_from_common_result_witness_operationoperationright. pfa_gap_from_common_result_witness_operationoperationright + S (pfp_right_from_common_result_witness_operation) = (p)) /\\ ((((exists pfa_gap_from_common_result_witness_operationoperationresultbound. pfa_gap_from_common_result_witness_operationoperationresultbound + S (pfp_value_from_common_result_witness_operation) = (p)) /\\ ((exists pfa_offset_left_from_common_result_witness_operationoperationresultcongruence pfa_offset_right_from_common_result_witness_operationoperationresultcongruence. ((pfp_left_from_common_result_witness_operation) + (pfp_right_from_common_result_witness_operation)) + (p) * pfa_offset_left_from_common_result_witness_operationoperationresultcongruence = (pfp_value_from_common_result_witness_operation) + (p) * pfa_offset_right_from_common_result_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_from_common_result_witness_output pfrep_left_from_common_result_witness_output pfrep_right_from_common_result_witness_output. ((exists pfrep_position_from_common_result_witness_outputfirst. ((pfrep_position_from_common_result_witness_outputfirst+S (pfrep_power_from_common_result_witness_output)=(pfaa_length_from_common_result)) /\\ ((((exists ff_h_pfp_from_common_result_witness_outputfirstentry. ff_h_pfp_from_common_result_witness_outputfirstentry + S (pfrep_left_from_common_result_witness_output) = S ((S (pfrep_position_from_common_result_witness_outputfirst)) * pfaa_sum_c_from_common_result)) /\\ exists ff_q_pfp_from_common_result_witness_outputfirstentry. pfaa_sum_b_from_common_result = ff_q_pfp_from_common_result_witness_outputfirstentry * S ((S (pfrep_position_from_common_result_witness_outputfirst)) * pfaa_sum_c_from_common_result) + (pfrep_left_from_common_result_witness_output)))))) \\/ (((exists pfrep_gap_from_common_result_witness_outputfirstoutside. pfrep_gap_from_common_result_witness_outputfirstoutside+(pfaa_length_from_common_result)=(pfrep_power_from_common_result_witness_output)) /\\ (((pfrep_left_from_common_result_witness_output)=0))))) -> ((exists pfrep_position_from_common_result_witness_outputsecond. ((pfrep_position_from_common_result_witness_outputsecond+S (pfrep_power_from_common_result_witness_output)=(N)) /\\ ((((exists ff_h_pfp_from_common_result_witness_outputsecondentry. ff_h_pfp_from_common_result_witness_outputsecondentry + S (pfrep_right_from_common_result_witness_output) = S ((S (pfrep_position_from_common_result_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_from_common_result_witness_outputsecondentry. rb = ff_q_pfp_from_common_result_witness_outputsecondentry * S ((S (pfrep_position_from_common_result_witness_outputsecond)) * rc) + (pfrep_right_from_common_result_witness_output)))))) \\/ (((exists pfrep_gap_from_common_result_witness_outputsecondoutside. pfrep_gap_from_common_result_witness_outputsecondoutside+(N)=(pfrep_power_from_common_result_witness_output)) /\\ (((pfrep_right_from_common_result_witness_output)=0))))) -> pfrep_left_from_common_result_witness_output=pfrep_right_from_common_result_witness_output)))))))))))))", "statement_sha256": "bd82fb68cb62222615655c29679c320e4196c46971c97786152fcda9d2551105", "summary": "Package actual common representatives, an actual coefficient sum, and formal output equivalence while retaining all three original canonical coefficient guards.", "summary_sha256": "07d14a2b6e5031ddc8817e933c7bda7a2fb0e70a72ef470f0b1ea73d3c338952" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "2ff484eea1b0b476bfcdb90699f87263a3d685119734acd9899fe470364d7142", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 335, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "a05bb4f5c4230ca05f51690d3ab82e33ff4596af65176874e25fbe38cf87a0db", "test_sha256": 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"statement": "forall p ab ac L bb bc M rb rc N. (((forall fom_index_pfp_aligned_bound_input_left_bounded. (exists fom_gap_pfp_aligned_bound_input_left_bounded_index_bound. fom_gap_pfp_aligned_bound_input_left_bounded_index_bound + S (fom_index_pfp_aligned_bound_input_left_bounded) = L) -> exists fom_value_pfp_aligned_bound_input_left_bounded. ((((exists fom_beta_height_pfp_aligned_bound_input_left_bounded_entry. fom_beta_height_pfp_aligned_bound_input_left_bounded_entry + S (fom_value_pfp_aligned_bound_input_left_bounded) = S ((S (fom_index_pfp_aligned_bound_input_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_aligned_bound_input_left_bounded_entry. ab = fom_beta_quotient_pfp_aligned_bound_input_left_bounded_entry * S ((S (fom_index_pfp_aligned_bound_input_left_bounded)) * ac) + (fom_value_pfp_aligned_bound_input_left_bounded))) /\\ (exists fom_gap_pfp_aligned_bound_input_left_bounded_value_bound. fom_gap_pfp_aligned_bound_input_left_bounded_value_bound + S (fom_value_pfp_aligned_bound_input_left_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_bound_input_right_bounded. (exists fom_gap_pfp_aligned_bound_input_right_bounded_index_bound. fom_gap_pfp_aligned_bound_input_right_bounded_index_bound + S (fom_index_pfp_aligned_bound_input_right_bounded) = M) -> exists fom_value_pfp_aligned_bound_input_right_bounded. ((((exists fom_beta_height_pfp_aligned_bound_input_right_bounded_entry. fom_beta_height_pfp_aligned_bound_input_right_bounded_entry + S (fom_value_pfp_aligned_bound_input_right_bounded) = S ((S (fom_index_pfp_aligned_bound_input_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_aligned_bound_input_right_bounded_entry. bb = fom_beta_quotient_pfp_aligned_bound_input_right_bounded_entry * S ((S (fom_index_pfp_aligned_bound_input_right_bounded)) * bc) + (fom_value_pfp_aligned_bound_input_right_bounded))) /\\ (exists fom_gap_pfp_aligned_bound_input_right_bounded_value_bound. fom_gap_pfp_aligned_bound_input_right_bounded_value_bound + S (fom_value_pfp_aligned_bound_input_right_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_bound_input_result_bounded. (exists fom_gap_pfp_aligned_bound_input_result_bounded_index_bound. fom_gap_pfp_aligned_bound_input_result_bounded_index_bound + S (fom_index_pfp_aligned_bound_input_result_bounded) = N) -> exists fom_value_pfp_aligned_bound_input_result_bounded. ((((exists fom_beta_height_pfp_aligned_bound_input_result_bounded_entry. fom_beta_height_pfp_aligned_bound_input_result_bounded_entry + S (fom_value_pfp_aligned_bound_input_result_bounded) = S ((S (fom_index_pfp_aligned_bound_input_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_aligned_bound_input_result_bounded_entry. rb = fom_beta_quotient_pfp_aligned_bound_input_result_bounded_entry * S ((S (fom_index_pfp_aligned_bound_input_result_bounded)) * rc) + (fom_value_pfp_aligned_bound_input_result_bounded))) /\\ (exists fom_gap_pfp_aligned_bound_input_result_bounded_value_bound. fom_gap_pfp_aligned_bound_input_result_bounded_value_bound + S (fom_value_pfp_aligned_bound_input_result_bounded) = p))) /\\ ((exists pfaa_left_b_aligned_bound_input pfaa_left_c_aligned_bound_input pfaa_right_b_aligned_bound_input pfaa_right_c_aligned_bound_input pfaa_sum_b_aligned_bound_input pfaa_sum_c_aligned_bound_input pfaa_length_aligned_bound_input. ((((forall pfrep_power_aligned_bound_input_witness_common_left pfrep_left_aligned_bound_input_witness_common_left pfrep_right_aligned_bound_input_witness_common_left. ((exists pfrep_position_aligned_bound_input_witness_common_leftfirst. ((pfrep_position_aligned_bound_input_witness_common_leftfirst+S (pfrep_power_aligned_bound_input_witness_common_left)=(L)) /\\ ((((exists ff_h_pfp_aligned_bound_input_witness_common_leftfirstentry. ff_h_pfp_aligned_bound_input_witness_common_leftfirstentry + S (pfrep_left_aligned_bound_input_witness_common_left) = S ((S (pfrep_position_aligned_bound_input_witness_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_aligned_bound_input_witness_common_leftfirstentry. ab = ff_q_pfp_aligned_bound_input_witness_common_leftfirstentry * S ((S (pfrep_position_aligned_bound_input_witness_common_leftfirst)) * ac) + (pfrep_left_aligned_bound_input_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_bound_input_witness_common_leftfirstoutside. pfrep_gap_aligned_bound_input_witness_common_leftfirstoutside+(L)=(pfrep_power_aligned_bound_input_witness_common_left)) /\\ (((pfrep_left_aligned_bound_input_witness_common_left)=0))))) -> ((exists pfrep_position_aligned_bound_input_witness_common_leftsecond. ((pfrep_position_aligned_bound_input_witness_common_leftsecond+S (pfrep_power_aligned_bound_input_witness_common_left)=(pfaa_length_aligned_bound_input)) /\\ ((((exists ff_h_pfp_aligned_bound_input_witness_common_leftsecondentry. ff_h_pfp_aligned_bound_input_witness_common_leftsecondentry + S (pfrep_right_aligned_bound_input_witness_common_left) = S ((S (pfrep_position_aligned_bound_input_witness_common_leftsecond)) * pfaa_left_c_aligned_bound_input)) /\\ exists ff_q_pfp_aligned_bound_input_witness_common_leftsecondentry. pfaa_left_b_aligned_bound_input = ff_q_pfp_aligned_bound_input_witness_common_leftsecondentry * S ((S (pfrep_position_aligned_bound_input_witness_common_leftsecond)) * pfaa_left_c_aligned_bound_input) + (pfrep_right_aligned_bound_input_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_bound_input_witness_common_leftsecondoutside. pfrep_gap_aligned_bound_input_witness_common_leftsecondoutside+(pfaa_length_aligned_bound_input)=(pfrep_power_aligned_bound_input_witness_common_left)) /\\ (((pfrep_right_aligned_bound_input_witness_common_left)=0))))) -> pfrep_left_aligned_bound_input_witness_common_left=pfrep_right_aligned_bound_input_witness_common_left) /\\ ((forall pfrep_power_aligned_bound_input_witness_common_right pfrep_left_aligned_bound_input_witness_common_right pfrep_right_aligned_bound_input_witness_common_right. ((exists pfrep_position_aligned_bound_input_witness_common_rightfirst. ((pfrep_position_aligned_bound_input_witness_common_rightfirst+S (pfrep_power_aligned_bound_input_witness_common_right)=(M)) /\\ ((((exists ff_h_pfp_aligned_bound_input_witness_common_rightfirstentry. ff_h_pfp_aligned_bound_input_witness_common_rightfirstentry + S (pfrep_left_aligned_bound_input_witness_common_right) = S ((S (pfrep_position_aligned_bound_input_witness_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_aligned_bound_input_witness_common_rightfirstentry. bb = ff_q_pfp_aligned_bound_input_witness_common_rightfirstentry * S ((S (pfrep_position_aligned_bound_input_witness_common_rightfirst)) * bc) + (pfrep_left_aligned_bound_input_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_bound_input_witness_common_rightfirstoutside. pfrep_gap_aligned_bound_input_witness_common_rightfirstoutside+(M)=(pfrep_power_aligned_bound_input_witness_common_right)) /\\ (((pfrep_left_aligned_bound_input_witness_common_right)=0))))) -> ((exists pfrep_position_aligned_bound_input_witness_common_rightsecond. ((pfrep_position_aligned_bound_input_witness_common_rightsecond+S (pfrep_power_aligned_bound_input_witness_common_right)=(pfaa_length_aligned_bound_input)) /\\ ((((exists ff_h_pfp_aligned_bound_input_witness_common_rightsecondentry. ff_h_pfp_aligned_bound_input_witness_common_rightsecondentry + S (pfrep_right_aligned_bound_input_witness_common_right) = S ((S (pfrep_position_aligned_bound_input_witness_common_rightsecond)) * pfaa_right_c_aligned_bound_input)) /\\ exists ff_q_pfp_aligned_bound_input_witness_common_rightsecondentry. pfaa_right_b_aligned_bound_input = ff_q_pfp_aligned_bound_input_witness_common_rightsecondentry * S ((S (pfrep_position_aligned_bound_input_witness_common_rightsecond)) * pfaa_right_c_aligned_bound_input) + (pfrep_right_aligned_bound_input_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_bound_input_witness_common_rightsecondoutside. pfrep_gap_aligned_bound_input_witness_common_rightsecondoutside+(pfaa_length_aligned_bound_input)=(pfrep_power_aligned_bound_input_witness_common_right)) /\\ (((pfrep_right_aligned_bound_input_witness_common_right)=0))))) -> pfrep_left_aligned_bound_input_witness_common_right=pfrep_right_aligned_bound_input_witness_common_right)))) /\\ (((forall pfp_index_aligned_bound_input_witness_operation. (exists pfa_gap_aligned_bound_input_witness_operationindex. pfa_gap_aligned_bound_input_witness_operationindex + S (pfp_index_aligned_bound_input_witness_operation) = (pfaa_length_aligned_bound_input)) -> exists pfp_left_aligned_bound_input_witness_operation pfp_right_aligned_bound_input_witness_operation pfp_value_aligned_bound_input_witness_operation. ((((exists ff_h_pfp_aligned_bound_input_witness_operationleft. ff_h_pfp_aligned_bound_input_witness_operationleft + S (pfp_left_aligned_bound_input_witness_operation) = S ((S (pfp_index_aligned_bound_input_witness_operation)) * pfaa_left_c_aligned_bound_input)) /\\ exists ff_q_pfp_aligned_bound_input_witness_operationleft. pfaa_left_b_aligned_bound_input = ff_q_pfp_aligned_bound_input_witness_operationleft * S ((S (pfp_index_aligned_bound_input_witness_operation)) * pfaa_left_c_aligned_bound_input) + (pfp_left_aligned_bound_input_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_bound_input_witness_operationright. ff_h_pfp_aligned_bound_input_witness_operationright + S (pfp_right_aligned_bound_input_witness_operation) = S ((S (pfp_index_aligned_bound_input_witness_operation)) * pfaa_right_c_aligned_bound_input)) /\\ exists ff_q_pfp_aligned_bound_input_witness_operationright. pfaa_right_b_aligned_bound_input = ff_q_pfp_aligned_bound_input_witness_operationright * S ((S (pfp_index_aligned_bound_input_witness_operation)) * pfaa_right_c_aligned_bound_input) + (pfp_right_aligned_bound_input_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_bound_input_witness_operationtarget. ff_h_pfp_aligned_bound_input_witness_operationtarget + S (pfp_value_aligned_bound_input_witness_operation) = S ((S (pfp_index_aligned_bound_input_witness_operation)) * pfaa_sum_c_aligned_bound_input)) /\\ exists ff_q_pfp_aligned_bound_input_witness_operationtarget. pfaa_sum_b_aligned_bound_input = ff_q_pfp_aligned_bound_input_witness_operationtarget * S ((S (pfp_index_aligned_bound_input_witness_operation)) * pfaa_sum_c_aligned_bound_input) + (pfp_value_aligned_bound_input_witness_operation))) /\\ ((((exists pfa_gap_aligned_bound_input_witness_operationoperationleft. pfa_gap_aligned_bound_input_witness_operationoperationleft + S (pfp_left_aligned_bound_input_witness_operation) = (p)) /\\ (((exists pfa_gap_aligned_bound_input_witness_operationoperationright. pfa_gap_aligned_bound_input_witness_operationoperationright + S (pfp_right_aligned_bound_input_witness_operation) = (p)) /\\ ((((exists pfa_gap_aligned_bound_input_witness_operationoperationresultbound. pfa_gap_aligned_bound_input_witness_operationoperationresultbound + S (pfp_value_aligned_bound_input_witness_operation) = (p)) /\\ ((exists pfa_offset_left_aligned_bound_input_witness_operationoperationresultcongruence pfa_offset_right_aligned_bound_input_witness_operationoperationresultcongruence. ((pfp_left_aligned_bound_input_witness_operation) + (pfp_right_aligned_bound_input_witness_operation)) + (p) * pfa_offset_left_aligned_bound_input_witness_operationoperationresultcongruence = (pfp_value_aligned_bound_input_witness_operation) + (p) * pfa_offset_right_aligned_bound_input_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_aligned_bound_input_witness_output pfrep_left_aligned_bound_input_witness_output pfrep_right_aligned_bound_input_witness_output. ((exists pfrep_position_aligned_bound_input_witness_outputfirst. ((pfrep_position_aligned_bound_input_witness_outputfirst+S (pfrep_power_aligned_bound_input_witness_output)=(pfaa_length_aligned_bound_input)) /\\ ((((exists ff_h_pfp_aligned_bound_input_witness_outputfirstentry. ff_h_pfp_aligned_bound_input_witness_outputfirstentry + S (pfrep_left_aligned_bound_input_witness_output) = S ((S (pfrep_position_aligned_bound_input_witness_outputfirst)) * pfaa_sum_c_aligned_bound_input)) /\\ exists ff_q_pfp_aligned_bound_input_witness_outputfirstentry. pfaa_sum_b_aligned_bound_input = ff_q_pfp_aligned_bound_input_witness_outputfirstentry * S ((S (pfrep_position_aligned_bound_input_witness_outputfirst)) * pfaa_sum_c_aligned_bound_input) + (pfrep_left_aligned_bound_input_witness_output)))))) \\/ (((exists pfrep_gap_aligned_bound_input_witness_outputfirstoutside. pfrep_gap_aligned_bound_input_witness_outputfirstoutside+(pfaa_length_aligned_bound_input)=(pfrep_power_aligned_bound_input_witness_output)) /\\ (((pfrep_left_aligned_bound_input_witness_output)=0))))) -> ((exists pfrep_position_aligned_bound_input_witness_outputsecond. ((pfrep_position_aligned_bound_input_witness_outputsecond+S (pfrep_power_aligned_bound_input_witness_output)=(N)) /\\ ((((exists ff_h_pfp_aligned_bound_input_witness_outputsecondentry. ff_h_pfp_aligned_bound_input_witness_outputsecondentry + S (pfrep_right_aligned_bound_input_witness_output) = S ((S (pfrep_position_aligned_bound_input_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_aligned_bound_input_witness_outputsecondentry. rb = ff_q_pfp_aligned_bound_input_witness_outputsecondentry * S ((S (pfrep_position_aligned_bound_input_witness_outputsecond)) * rc) + (pfrep_right_aligned_bound_input_witness_output)))))) \\/ (((exists pfrep_gap_aligned_bound_input_witness_outputsecondoutside. pfrep_gap_aligned_bound_input_witness_outputsecondoutside+(N)=(pfrep_power_aligned_bound_input_witness_output)) /\\ (((pfrep_right_aligned_bound_input_witness_output)=0))))) -> pfrep_left_aligned_bound_input_witness_output=pfrep_right_aligned_bound_input_witness_output))))))))))))) -> (((forall fom_index_pfp_aligned_bound_0. (exists fom_gap_pfp_aligned_bound_0_index_bound. fom_gap_pfp_aligned_bound_0_index_bound + S (fom_index_pfp_aligned_bound_0) = L) -> exists fom_value_pfp_aligned_bound_0. ((((exists fom_beta_height_pfp_aligned_bound_0_entry. fom_beta_height_pfp_aligned_bound_0_entry + S (fom_value_pfp_aligned_bound_0) = S ((S (fom_index_pfp_aligned_bound_0)) * ac)) /\\ exists fom_beta_quotient_pfp_aligned_bound_0_entry. ab = fom_beta_quotient_pfp_aligned_bound_0_entry * S ((S (fom_index_pfp_aligned_bound_0)) * ac) + (fom_value_pfp_aligned_bound_0))) /\\ (exists fom_gap_pfp_aligned_bound_0_value_bound. fom_gap_pfp_aligned_bound_0_value_bound + S (fom_value_pfp_aligned_bound_0) = p))) /\\ (((forall fom_index_pfp_aligned_bound_1. (exists fom_gap_pfp_aligned_bound_1_index_bound. fom_gap_pfp_aligned_bound_1_index_bound + S (fom_index_pfp_aligned_bound_1) = M) -> exists fom_value_pfp_aligned_bound_1. ((((exists fom_beta_height_pfp_aligned_bound_1_entry. fom_beta_height_pfp_aligned_bound_1_entry + S (fom_value_pfp_aligned_bound_1) = S ((S (fom_index_pfp_aligned_bound_1)) * bc)) /\\ exists fom_beta_quotient_pfp_aligned_bound_1_entry. bb = fom_beta_quotient_pfp_aligned_bound_1_entry * S ((S (fom_index_pfp_aligned_bound_1)) * bc) + (fom_value_pfp_aligned_bound_1))) /\\ (exists fom_gap_pfp_aligned_bound_1_value_bound. fom_gap_pfp_aligned_bound_1_value_bound + S (fom_value_pfp_aligned_bound_1) = p))) /\\ ((forall fom_index_pfp_aligned_bound_2. (exists fom_gap_pfp_aligned_bound_2_index_bound. fom_gap_pfp_aligned_bound_2_index_bound + S (fom_index_pfp_aligned_bound_2) = N) -> exists fom_value_pfp_aligned_bound_2. ((((exists fom_beta_height_pfp_aligned_bound_2_entry. fom_beta_height_pfp_aligned_bound_2_entry + S (fom_value_pfp_aligned_bound_2) = S ((S (fom_index_pfp_aligned_bound_2)) * rc)) /\\ exists fom_beta_quotient_pfp_aligned_bound_2_entry. rb = fom_beta_quotient_pfp_aligned_bound_2_entry * S ((S (fom_index_pfp_aligned_bound_2)) * rc) + (fom_value_pfp_aligned_bound_2))) /\\ (exists fom_gap_pfp_aligned_bound_2_value_bound. fom_gap_pfp_aligned_bound_2_value_bound + S (fom_value_pfp_aligned_bound_2) = p))))))))", "statement_sha256": "a7ad10b1c37620d45990d130df7c576667b905c3fd51cf603229b39fb6ca5e7e", "summary": "Aligned addition includes canonical coefficients for the actual originals and output, not merely for equivalent witnesses.", "summary_sha256": "03c3649b1b71a7d5ae3d4d5d2800cff693dd4b60256d13397a82167b13a8c799" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "f0b54ff717dfb27e2e571bf28540a383460b7e15d85d140db2c8d846e75186d9", "bundle_campaign": "polynomial-gcd-bezout", 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"3577ec6d847575fa3b5dfd8448eacdd4461891849cd4175ff92de19f60ed2221", "membership": "alpha_only", "name": "prime_field_polynomial_aligned_add_from_fixed", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro bb", "intro bc", "intro rb", "intro rc", "intro K", "intro h", "have hb : ((forall fom_index_pfp_fixed_bound_ab. (exists fom_gap_pfp_fixed_bound_ab_index_bound. fom_gap_pfp_fixed_bound_ab_index_bound + S (fom_index_pfp_fixed_bound_ab) = K) -> exists fom_value_pfp_fixed_bound_ab. ((((exists fom_beta_height_pfp_fixed_bound_ab_entry. fom_beta_height_pfp_fixed_bound_ab_entry + S (fom_value_pfp_fixed_bound_ab) = S ((S (fom_index_pfp_fixed_bound_ab)) * ac)) /\\ exists fom_beta_quotient_pfp_fixed_bound_ab_entry. ab = fom_beta_quotient_pfp_fixed_bound_ab_entry * S ((S (fom_index_pfp_fixed_bound_ab)) * ac) + (fom_value_pfp_fixed_bound_ab))) /\\ (exists fom_gap_pfp_fixed_bound_ab_value_bound. fom_gap_pfp_fixed_bound_ab_value_bound + S (fom_value_pfp_fixed_bound_ab) = p))) /\\ (((forall fom_index_pfp_fixed_bound_bb. (exists fom_gap_pfp_fixed_bound_bb_index_bound. fom_gap_pfp_fixed_bound_bb_index_bound + S (fom_index_pfp_fixed_bound_bb) = K) -> exists fom_value_pfp_fixed_bound_bb. ((((exists fom_beta_height_pfp_fixed_bound_bb_entry. fom_beta_height_pfp_fixed_bound_bb_entry + S (fom_value_pfp_fixed_bound_bb) = S ((S (fom_index_pfp_fixed_bound_bb)) * bc)) /\\ exists fom_beta_quotient_pfp_fixed_bound_bb_entry. bb = fom_beta_quotient_pfp_fixed_bound_bb_entry * S ((S (fom_index_pfp_fixed_bound_bb)) * bc) + (fom_value_pfp_fixed_bound_bb))) /\\ (exists fom_gap_pfp_fixed_bound_bb_value_bound. fom_gap_pfp_fixed_bound_bb_value_bound + S (fom_value_pfp_fixed_bound_bb) = p))) /\\ ((forall fom_index_pfp_fixed_bound_rb. (exists fom_gap_pfp_fixed_bound_rb_index_bound. fom_gap_pfp_fixed_bound_rb_index_bound + S (fom_index_pfp_fixed_bound_rb) = K) -> exists fom_value_pfp_fixed_bound_rb. ((((exists fom_beta_height_pfp_fixed_bound_rb_entry. fom_beta_height_pfp_fixed_bound_rb_entry + S (fom_value_pfp_fixed_bound_rb) = S ((S (fom_index_pfp_fixed_bound_rb)) * rc)) /\\ exists fom_beta_quotient_pfp_fixed_bound_rb_entry. rb = fom_beta_quotient_pfp_fixed_bound_rb_entry * S ((S (fom_index_pfp_fixed_bound_rb)) * rc) + (fom_value_pfp_fixed_bound_rb))) /\\ (exists fom_gap_pfp_fixed_bound_rb_value_bound. fom_gap_pfp_fixed_bound_rb_value_bound + S (fom_value_pfp_fixed_bound_rb) = p)))))))", "specialize prime_field_polynomial_add_bounded (p)", "specialize prime_field_polynomial_add_bounded (ab)", "specialize prime_field_polynomial_add_bounded (ac)", "specialize prime_field_polynomial_add_bounded (bb)", "specialize prime_field_polynomial_add_bounded (bc)", "specialize prime_field_polynomial_add_bounded (rb)", "specialize prime_field_polynomial_add_bounded (rc)", "specialize prime_field_polynomial_add_bounded (K)", "apply prime_field_polynomial_add_bounded", "exact h", "cases hb", "cases hb_right", "specialize prime_field_polynomial_aligned_add_from_common (p)", "specialize prime_field_polynomial_aligned_add_from_common (ab)", "specialize prime_field_polynomial_aligned_add_from_common (ac)", "specialize prime_field_polynomial_aligned_add_from_common (K)", "specialize prime_field_polynomial_aligned_add_from_common (bb)", "specialize prime_field_polynomial_aligned_add_from_common (bc)", "specialize prime_field_polynomial_aligned_add_from_common (K)", "specialize prime_field_polynomial_aligned_add_from_common (rb)", "specialize prime_field_polynomial_aligned_add_from_common (rc)", "specialize prime_field_polynomial_aligned_add_from_common (K)", "specialize prime_field_polynomial_aligned_add_from_common (ab)", "specialize prime_field_polynomial_aligned_add_from_common (ac)", "specialize prime_field_polynomial_aligned_add_from_common (bb)", "specialize prime_field_polynomial_aligned_add_from_common (bc)", "specialize prime_field_polynomial_aligned_add_from_common (rb)", "specialize prime_field_polynomial_aligned_add_from_common (rc)", "specialize prime_field_polynomial_aligned_add_from_common (K)", "apply prime_field_polynomial_aligned_add_from_common", "exact hb_left", "exact hb_right_left", "exact hb_right_right", "specialize prime_field_polynomial_common_representatives_same_length (ab)", "specialize prime_field_polynomial_common_representatives_same_length (ac)", "specialize prime_field_polynomial_common_representatives_same_length (bb)", "specialize prime_field_polynomial_common_representatives_same_length (bc)", "specialize prime_field_polynomial_common_representatives_same_length (K)", "apply prime_field_polynomial_common_representatives_same_length", "exact h", "specialize prime_field_polynomial_power_coefficient_functional (rb)", "specialize prime_field_polynomial_power_coefficient_functional (rc)", "specialize prime_field_polynomial_power_coefficient_functional (K)", "apply prime_field_polynomial_power_coefficient_functional" ], "script_sha256": "46e1bc28bc6572385c2795814f35d2fe486c86b9a363809cf20137a1c84e4e04", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_aligned_add_candidate.py", "sha256": "a05bb4f5c4230ca05f51690d3ab82e33ff4596af65176874e25fbe38cf87a0db" }, "statement": "forall p ab ac bb bc rb rc K. (forall pfp_index_fixed_input. (exists pfa_gap_fixed_inputindex. pfa_gap_fixed_inputindex + S (pfp_index_fixed_input) = (K)) -> exists pfp_left_fixed_input pfp_right_fixed_input pfp_value_fixed_input. ((((exists ff_h_pfp_fixed_inputleft. ff_h_pfp_fixed_inputleft + S (pfp_left_fixed_input) = S ((S (pfp_index_fixed_input)) * ac)) /\\ exists ff_q_pfp_fixed_inputleft. ab = ff_q_pfp_fixed_inputleft * S ((S (pfp_index_fixed_input)) * ac) + (pfp_left_fixed_input))) /\\ (((((exists ff_h_pfp_fixed_inputright. ff_h_pfp_fixed_inputright + S (pfp_right_fixed_input) = S ((S (pfp_index_fixed_input)) * bc)) /\\ exists ff_q_pfp_fixed_inputright. bb = ff_q_pfp_fixed_inputright * S ((S (pfp_index_fixed_input)) * bc) + (pfp_right_fixed_input))) /\\ (((((exists ff_h_pfp_fixed_inputtarget. ff_h_pfp_fixed_inputtarget + S (pfp_value_fixed_input) = S ((S (pfp_index_fixed_input)) * rc)) /\\ exists ff_q_pfp_fixed_inputtarget. rb = ff_q_pfp_fixed_inputtarget * S ((S (pfp_index_fixed_input)) * rc) + (pfp_value_fixed_input))) /\\ ((((exists pfa_gap_fixed_inputoperationleft. pfa_gap_fixed_inputoperationleft + S (pfp_left_fixed_input) = (p)) /\\ (((exists pfa_gap_fixed_inputoperationright. pfa_gap_fixed_inputoperationright + S (pfp_right_fixed_input) = (p)) /\\ ((((exists pfa_gap_fixed_inputoperationresultbound. pfa_gap_fixed_inputoperationresultbound + S (pfp_value_fixed_input) = (p)) /\\ ((exists pfa_offset_left_fixed_inputoperationresultcongruence pfa_offset_right_fixed_inputoperationresultcongruence. ((pfp_left_fixed_input) + (pfp_right_fixed_input)) + (p) * pfa_offset_left_fixed_inputoperationresultcongruence = (pfp_value_fixed_input) + (p) * pfa_offset_right_fixed_inputoperationresultcongruence)))))))))))))))) -> (((forall fom_index_pfp_fixed_result_left_bounded. (exists fom_gap_pfp_fixed_result_left_bounded_index_bound. fom_gap_pfp_fixed_result_left_bounded_index_bound + S (fom_index_pfp_fixed_result_left_bounded) = K) -> exists fom_value_pfp_fixed_result_left_bounded. ((((exists fom_beta_height_pfp_fixed_result_left_bounded_entry. fom_beta_height_pfp_fixed_result_left_bounded_entry + S (fom_value_pfp_fixed_result_left_bounded) = S ((S (fom_index_pfp_fixed_result_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_fixed_result_left_bounded_entry. ab = fom_beta_quotient_pfp_fixed_result_left_bounded_entry * S ((S (fom_index_pfp_fixed_result_left_bounded)) * ac) + (fom_value_pfp_fixed_result_left_bounded))) /\\ (exists fom_gap_pfp_fixed_result_left_bounded_value_bound. fom_gap_pfp_fixed_result_left_bounded_value_bound + S (fom_value_pfp_fixed_result_left_bounded) = p))) /\\ (((forall fom_index_pfp_fixed_result_right_bounded. (exists fom_gap_pfp_fixed_result_right_bounded_index_bound. fom_gap_pfp_fixed_result_right_bounded_index_bound + S (fom_index_pfp_fixed_result_right_bounded) = K) -> exists fom_value_pfp_fixed_result_right_bounded. ((((exists fom_beta_height_pfp_fixed_result_right_bounded_entry. fom_beta_height_pfp_fixed_result_right_bounded_entry + S (fom_value_pfp_fixed_result_right_bounded) = S ((S (fom_index_pfp_fixed_result_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_fixed_result_right_bounded_entry. bb = fom_beta_quotient_pfp_fixed_result_right_bounded_entry * S ((S (fom_index_pfp_fixed_result_right_bounded)) * bc) + (fom_value_pfp_fixed_result_right_bounded))) /\\ (exists fom_gap_pfp_fixed_result_right_bounded_value_bound. fom_gap_pfp_fixed_result_right_bounded_value_bound + S (fom_value_pfp_fixed_result_right_bounded) = p))) /\\ (((forall fom_index_pfp_fixed_result_result_bounded. (exists fom_gap_pfp_fixed_result_result_bounded_index_bound. fom_gap_pfp_fixed_result_result_bounded_index_bound + S (fom_index_pfp_fixed_result_result_bounded) = K) -> exists fom_value_pfp_fixed_result_result_bounded. ((((exists fom_beta_height_pfp_fixed_result_result_bounded_entry. fom_beta_height_pfp_fixed_result_result_bounded_entry + S (fom_value_pfp_fixed_result_result_bounded) = S ((S (fom_index_pfp_fixed_result_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_fixed_result_result_bounded_entry. rb = fom_beta_quotient_pfp_fixed_result_result_bounded_entry * S ((S (fom_index_pfp_fixed_result_result_bounded)) * rc) + (fom_value_pfp_fixed_result_result_bounded))) /\\ (exists fom_gap_pfp_fixed_result_result_bounded_value_bound. fom_gap_pfp_fixed_result_result_bounded_value_bound + S (fom_value_pfp_fixed_result_result_bounded) = p))) /\\ ((exists pfaa_left_b_fixed_result pfaa_left_c_fixed_result pfaa_right_b_fixed_result pfaa_right_c_fixed_result pfaa_sum_b_fixed_result pfaa_sum_c_fixed_result pfaa_length_fixed_result. ((((forall pfrep_power_fixed_result_witness_common_left pfrep_left_fixed_result_witness_common_left pfrep_right_fixed_result_witness_common_left. ((exists pfrep_position_fixed_result_witness_common_leftfirst. ((pfrep_position_fixed_result_witness_common_leftfirst+S (pfrep_power_fixed_result_witness_common_left)=(K)) /\\ ((((exists ff_h_pfp_fixed_result_witness_common_leftfirstentry. ff_h_pfp_fixed_result_witness_common_leftfirstentry + S (pfrep_left_fixed_result_witness_common_left) = S ((S (pfrep_position_fixed_result_witness_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_fixed_result_witness_common_leftfirstentry. ab = ff_q_pfp_fixed_result_witness_common_leftfirstentry * S ((S (pfrep_position_fixed_result_witness_common_leftfirst)) * ac) + (pfrep_left_fixed_result_witness_common_left)))))) \\/ (((exists pfrep_gap_fixed_result_witness_common_leftfirstoutside. pfrep_gap_fixed_result_witness_common_leftfirstoutside+(K)=(pfrep_power_fixed_result_witness_common_left)) /\\ (((pfrep_left_fixed_result_witness_common_left)=0))))) -> ((exists pfrep_position_fixed_result_witness_common_leftsecond. ((pfrep_position_fixed_result_witness_common_leftsecond+S (pfrep_power_fixed_result_witness_common_left)=(pfaa_length_fixed_result)) /\\ ((((exists ff_h_pfp_fixed_result_witness_common_leftsecondentry. ff_h_pfp_fixed_result_witness_common_leftsecondentry + S (pfrep_right_fixed_result_witness_common_left) = S ((S (pfrep_position_fixed_result_witness_common_leftsecond)) * pfaa_left_c_fixed_result)) /\\ exists ff_q_pfp_fixed_result_witness_common_leftsecondentry. pfaa_left_b_fixed_result = ff_q_pfp_fixed_result_witness_common_leftsecondentry * S ((S (pfrep_position_fixed_result_witness_common_leftsecond)) * pfaa_left_c_fixed_result) + (pfrep_right_fixed_result_witness_common_left)))))) \\/ (((exists pfrep_gap_fixed_result_witness_common_leftsecondoutside. pfrep_gap_fixed_result_witness_common_leftsecondoutside+(pfaa_length_fixed_result)=(pfrep_power_fixed_result_witness_common_left)) /\\ (((pfrep_right_fixed_result_witness_common_left)=0))))) -> pfrep_left_fixed_result_witness_common_left=pfrep_right_fixed_result_witness_common_left) /\\ ((forall pfrep_power_fixed_result_witness_common_right pfrep_left_fixed_result_witness_common_right pfrep_right_fixed_result_witness_common_right. ((exists pfrep_position_fixed_result_witness_common_rightfirst. ((pfrep_position_fixed_result_witness_common_rightfirst+S (pfrep_power_fixed_result_witness_common_right)=(K)) /\\ ((((exists ff_h_pfp_fixed_result_witness_common_rightfirstentry. ff_h_pfp_fixed_result_witness_common_rightfirstentry + S (pfrep_left_fixed_result_witness_common_right) = S ((S (pfrep_position_fixed_result_witness_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_fixed_result_witness_common_rightfirstentry. bb = ff_q_pfp_fixed_result_witness_common_rightfirstentry * S ((S (pfrep_position_fixed_result_witness_common_rightfirst)) * bc) + (pfrep_left_fixed_result_witness_common_right)))))) \\/ (((exists pfrep_gap_fixed_result_witness_common_rightfirstoutside. pfrep_gap_fixed_result_witness_common_rightfirstoutside+(K)=(pfrep_power_fixed_result_witness_common_right)) /\\ (((pfrep_left_fixed_result_witness_common_right)=0))))) -> ((exists pfrep_position_fixed_result_witness_common_rightsecond. ((pfrep_position_fixed_result_witness_common_rightsecond+S (pfrep_power_fixed_result_witness_common_right)=(pfaa_length_fixed_result)) /\\ ((((exists ff_h_pfp_fixed_result_witness_common_rightsecondentry. ff_h_pfp_fixed_result_witness_common_rightsecondentry + S (pfrep_right_fixed_result_witness_common_right) = S ((S (pfrep_position_fixed_result_witness_common_rightsecond)) * pfaa_right_c_fixed_result)) /\\ exists ff_q_pfp_fixed_result_witness_common_rightsecondentry. pfaa_right_b_fixed_result = ff_q_pfp_fixed_result_witness_common_rightsecondentry * S ((S (pfrep_position_fixed_result_witness_common_rightsecond)) * pfaa_right_c_fixed_result) + (pfrep_right_fixed_result_witness_common_right)))))) \\/ (((exists pfrep_gap_fixed_result_witness_common_rightsecondoutside. pfrep_gap_fixed_result_witness_common_rightsecondoutside+(pfaa_length_fixed_result)=(pfrep_power_fixed_result_witness_common_right)) /\\ (((pfrep_right_fixed_result_witness_common_right)=0))))) -> pfrep_left_fixed_result_witness_common_right=pfrep_right_fixed_result_witness_common_right)))) /\\ (((forall pfp_index_fixed_result_witness_operation. (exists pfa_gap_fixed_result_witness_operationindex. pfa_gap_fixed_result_witness_operationindex + S (pfp_index_fixed_result_witness_operation) = (pfaa_length_fixed_result)) -> exists pfp_left_fixed_result_witness_operation pfp_right_fixed_result_witness_operation pfp_value_fixed_result_witness_operation. ((((exists ff_h_pfp_fixed_result_witness_operationleft. ff_h_pfp_fixed_result_witness_operationleft + S (pfp_left_fixed_result_witness_operation) = S ((S (pfp_index_fixed_result_witness_operation)) * pfaa_left_c_fixed_result)) /\\ exists ff_q_pfp_fixed_result_witness_operationleft. pfaa_left_b_fixed_result = ff_q_pfp_fixed_result_witness_operationleft * S ((S (pfp_index_fixed_result_witness_operation)) * pfaa_left_c_fixed_result) + (pfp_left_fixed_result_witness_operation))) /\\ (((((exists ff_h_pfp_fixed_result_witness_operationright. ff_h_pfp_fixed_result_witness_operationright + S (pfp_right_fixed_result_witness_operation) = S ((S (pfp_index_fixed_result_witness_operation)) * pfaa_right_c_fixed_result)) /\\ exists ff_q_pfp_fixed_result_witness_operationright. pfaa_right_b_fixed_result = ff_q_pfp_fixed_result_witness_operationright * S ((S (pfp_index_fixed_result_witness_operation)) * pfaa_right_c_fixed_result) + (pfp_right_fixed_result_witness_operation))) /\\ (((((exists ff_h_pfp_fixed_result_witness_operationtarget. ff_h_pfp_fixed_result_witness_operationtarget + S (pfp_value_fixed_result_witness_operation) = S ((S (pfp_index_fixed_result_witness_operation)) * pfaa_sum_c_fixed_result)) /\\ exists ff_q_pfp_fixed_result_witness_operationtarget. pfaa_sum_b_fixed_result = ff_q_pfp_fixed_result_witness_operationtarget * S ((S (pfp_index_fixed_result_witness_operation)) * pfaa_sum_c_fixed_result) + (pfp_value_fixed_result_witness_operation))) /\\ ((((exists pfa_gap_fixed_result_witness_operationoperationleft. pfa_gap_fixed_result_witness_operationoperationleft + S (pfp_left_fixed_result_witness_operation) = (p)) /\\ (((exists pfa_gap_fixed_result_witness_operationoperationright. pfa_gap_fixed_result_witness_operationoperationright + S (pfp_right_fixed_result_witness_operation) = (p)) /\\ ((((exists pfa_gap_fixed_result_witness_operationoperationresultbound. pfa_gap_fixed_result_witness_operationoperationresultbound + S (pfp_value_fixed_result_witness_operation) = (p)) /\\ ((exists pfa_offset_left_fixed_result_witness_operationoperationresultcongruence pfa_offset_right_fixed_result_witness_operationoperationresultcongruence. ((pfp_left_fixed_result_witness_operation) + (pfp_right_fixed_result_witness_operation)) + (p) * pfa_offset_left_fixed_result_witness_operationoperationresultcongruence = (pfp_value_fixed_result_witness_operation) + (p) * pfa_offset_right_fixed_result_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_fixed_result_witness_output pfrep_left_fixed_result_witness_output pfrep_right_fixed_result_witness_output. ((exists pfrep_position_fixed_result_witness_outputfirst. ((pfrep_position_fixed_result_witness_outputfirst+S (pfrep_power_fixed_result_witness_output)=(pfaa_length_fixed_result)) /\\ ((((exists ff_h_pfp_fixed_result_witness_outputfirstentry. ff_h_pfp_fixed_result_witness_outputfirstentry + S (pfrep_left_fixed_result_witness_output) = S ((S (pfrep_position_fixed_result_witness_outputfirst)) * pfaa_sum_c_fixed_result)) /\\ exists ff_q_pfp_fixed_result_witness_outputfirstentry. pfaa_sum_b_fixed_result = ff_q_pfp_fixed_result_witness_outputfirstentry * S ((S (pfrep_position_fixed_result_witness_outputfirst)) * pfaa_sum_c_fixed_result) + (pfrep_left_fixed_result_witness_output)))))) \\/ (((exists pfrep_gap_fixed_result_witness_outputfirstoutside. pfrep_gap_fixed_result_witness_outputfirstoutside+(pfaa_length_fixed_result)=(pfrep_power_fixed_result_witness_output)) /\\ (((pfrep_left_fixed_result_witness_output)=0))))) -> ((exists pfrep_position_fixed_result_witness_outputsecond. ((pfrep_position_fixed_result_witness_outputsecond+S (pfrep_power_fixed_result_witness_output)=(K)) /\\ ((((exists ff_h_pfp_fixed_result_witness_outputsecondentry. ff_h_pfp_fixed_result_witness_outputsecondentry + S (pfrep_right_fixed_result_witness_output) = S ((S (pfrep_position_fixed_result_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_fixed_result_witness_outputsecondentry. rb = ff_q_pfp_fixed_result_witness_outputsecondentry * S ((S (pfrep_position_fixed_result_witness_outputsecond)) * rc) + (pfrep_right_fixed_result_witness_output)))))) \\/ (((exists pfrep_gap_fixed_result_witness_outputsecondoutside. pfrep_gap_fixed_result_witness_outputsecondoutside+(K)=(pfrep_power_fixed_result_witness_output)) /\\ (((pfrep_right_fixed_result_witness_output)=0))))) -> pfrep_left_fixed_result_witness_output=pfrep_right_fixed_result_witness_output)))))))))))))", "statement_sha256": "3836f4f98fcb1ec1cc3c50e14433f814e910bca6e85f50759dc034d6a2aeaad1", "summary": "Every genuine fixed-length 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"exact hrf" ], "script_sha256": "3a189797e7a27154f9603b4ad8ed4905e9ef51a4f9e82d936643ce2f386ba219", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_aligned_add_candidate.py", "sha256": "a05bb4f5c4230ca05f51690d3ab82e33ff4596af65176874e25fbe38cf87a0db" }, "statement": "forall p ab ac L bb bc M rb rc N db dc J eb ec H fb fc I. (forall fom_index_pfp_aligned_transport_0. (exists fom_gap_pfp_aligned_transport_0_index_bound. fom_gap_pfp_aligned_transport_0_index_bound + S (fom_index_pfp_aligned_transport_0) = J) -> exists fom_value_pfp_aligned_transport_0. ((((exists fom_beta_height_pfp_aligned_transport_0_entry. fom_beta_height_pfp_aligned_transport_0_entry + S (fom_value_pfp_aligned_transport_0) = S ((S (fom_index_pfp_aligned_transport_0)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_transport_0_entry. db = fom_beta_quotient_pfp_aligned_transport_0_entry * S ((S (fom_index_pfp_aligned_transport_0)) * dc) + (fom_value_pfp_aligned_transport_0))) /\\ (exists fom_gap_pfp_aligned_transport_0_value_bound. fom_gap_pfp_aligned_transport_0_value_bound + S (fom_value_pfp_aligned_transport_0) = p))) -> (forall fom_index_pfp_aligned_transport_1. (exists fom_gap_pfp_aligned_transport_1_index_bound. fom_gap_pfp_aligned_transport_1_index_bound + S (fom_index_pfp_aligned_transport_1) = H) -> exists fom_value_pfp_aligned_transport_1. ((((exists fom_beta_height_pfp_aligned_transport_1_entry. fom_beta_height_pfp_aligned_transport_1_entry + S (fom_value_pfp_aligned_transport_1) = S ((S (fom_index_pfp_aligned_transport_1)) * ec)) /\\ exists fom_beta_quotient_pfp_aligned_transport_1_entry. eb = fom_beta_quotient_pfp_aligned_transport_1_entry * S ((S (fom_index_pfp_aligned_transport_1)) * ec) + (fom_value_pfp_aligned_transport_1))) /\\ (exists fom_gap_pfp_aligned_transport_1_value_bound. fom_gap_pfp_aligned_transport_1_value_bound + S (fom_value_pfp_aligned_transport_1) = p))) -> (forall fom_index_pfp_aligned_transport_2. (exists fom_gap_pfp_aligned_transport_2_index_bound. fom_gap_pfp_aligned_transport_2_index_bound + S (fom_index_pfp_aligned_transport_2) = I) -> exists fom_value_pfp_aligned_transport_2. ((((exists fom_beta_height_pfp_aligned_transport_2_entry. fom_beta_height_pfp_aligned_transport_2_entry + S (fom_value_pfp_aligned_transport_2) = S ((S (fom_index_pfp_aligned_transport_2)) * fc)) /\\ exists fom_beta_quotient_pfp_aligned_transport_2_entry. fb = fom_beta_quotient_pfp_aligned_transport_2_entry * S ((S (fom_index_pfp_aligned_transport_2)) * fc) + (fom_value_pfp_aligned_transport_2))) /\\ (exists fom_gap_pfp_aligned_transport_2_value_bound. fom_gap_pfp_aligned_transport_2_value_bound + S (fom_value_pfp_aligned_transport_2) = p))) -> (forall pfrep_power_aligned_transport_left pfrep_left_aligned_transport_left pfrep_right_aligned_transport_left. ((exists pfrep_position_aligned_transport_leftfirst. ((pfrep_position_aligned_transport_leftfirst+S (pfrep_power_aligned_transport_left)=(J)) /\\ ((((exists ff_h_pfp_aligned_transport_leftfirstentry. ff_h_pfp_aligned_transport_leftfirstentry + S (pfrep_left_aligned_transport_left) = S ((S (pfrep_position_aligned_transport_leftfirst)) * dc)) /\\ exists ff_q_pfp_aligned_transport_leftfirstentry. db = ff_q_pfp_aligned_transport_leftfirstentry * S ((S (pfrep_position_aligned_transport_leftfirst)) * dc) + (pfrep_left_aligned_transport_left)))))) \\/ (((exists pfrep_gap_aligned_transport_leftfirstoutside. pfrep_gap_aligned_transport_leftfirstoutside+(J)=(pfrep_power_aligned_transport_left)) /\\ (((pfrep_left_aligned_transport_left)=0))))) -> ((exists pfrep_position_aligned_transport_leftsecond. ((pfrep_position_aligned_transport_leftsecond+S (pfrep_power_aligned_transport_left)=(L)) /\\ ((((exists ff_h_pfp_aligned_transport_leftsecondentry. ff_h_pfp_aligned_transport_leftsecondentry + S (pfrep_right_aligned_transport_left) = S ((S (pfrep_position_aligned_transport_leftsecond)) * ac)) /\\ exists ff_q_pfp_aligned_transport_leftsecondentry. ab = ff_q_pfp_aligned_transport_leftsecondentry * S ((S (pfrep_position_aligned_transport_leftsecond)) * ac) + (pfrep_right_aligned_transport_left)))))) \\/ (((exists pfrep_gap_aligned_transport_leftsecondoutside. pfrep_gap_aligned_transport_leftsecondoutside+(L)=(pfrep_power_aligned_transport_left)) /\\ (((pfrep_right_aligned_transport_left)=0))))) -> pfrep_left_aligned_transport_left=pfrep_right_aligned_transport_left) -> (forall pfrep_power_aligned_transport_right pfrep_left_aligned_transport_right pfrep_right_aligned_transport_right. ((exists pfrep_position_aligned_transport_rightfirst. ((pfrep_position_aligned_transport_rightfirst+S (pfrep_power_aligned_transport_right)=(H)) /\\ ((((exists ff_h_pfp_aligned_transport_rightfirstentry. ff_h_pfp_aligned_transport_rightfirstentry + S (pfrep_left_aligned_transport_right) = S ((S (pfrep_position_aligned_transport_rightfirst)) * ec)) /\\ exists ff_q_pfp_aligned_transport_rightfirstentry. eb = ff_q_pfp_aligned_transport_rightfirstentry * S ((S (pfrep_position_aligned_transport_rightfirst)) * ec) + (pfrep_left_aligned_transport_right)))))) \\/ (((exists pfrep_gap_aligned_transport_rightfirstoutside. pfrep_gap_aligned_transport_rightfirstoutside+(H)=(pfrep_power_aligned_transport_right)) /\\ (((pfrep_left_aligned_transport_right)=0))))) -> ((exists pfrep_position_aligned_transport_rightsecond. ((pfrep_position_aligned_transport_rightsecond+S (pfrep_power_aligned_transport_right)=(M)) /\\ ((((exists ff_h_pfp_aligned_transport_rightsecondentry. ff_h_pfp_aligned_transport_rightsecondentry + S (pfrep_right_aligned_transport_right) = S ((S (pfrep_position_aligned_transport_rightsecond)) * bc)) /\\ exists ff_q_pfp_aligned_transport_rightsecondentry. bb = ff_q_pfp_aligned_transport_rightsecondentry * S ((S (pfrep_position_aligned_transport_rightsecond)) * bc) + (pfrep_right_aligned_transport_right)))))) \\/ (((exists pfrep_gap_aligned_transport_rightsecondoutside. pfrep_gap_aligned_transport_rightsecondoutside+(M)=(pfrep_power_aligned_transport_right)) /\\ (((pfrep_right_aligned_transport_right)=0))))) -> pfrep_left_aligned_transport_right=pfrep_right_aligned_transport_right) -> (forall pfrep_power_aligned_transport_output pfrep_left_aligned_transport_output pfrep_right_aligned_transport_output. ((exists pfrep_position_aligned_transport_outputfirst. ((pfrep_position_aligned_transport_outputfirst+S (pfrep_power_aligned_transport_output)=(N)) /\\ ((((exists ff_h_pfp_aligned_transport_outputfirstentry. ff_h_pfp_aligned_transport_outputfirstentry + S (pfrep_left_aligned_transport_output) = S ((S (pfrep_position_aligned_transport_outputfirst)) * rc)) /\\ exists ff_q_pfp_aligned_transport_outputfirstentry. rb = ff_q_pfp_aligned_transport_outputfirstentry * S ((S (pfrep_position_aligned_transport_outputfirst)) * rc) + (pfrep_left_aligned_transport_output)))))) \\/ (((exists pfrep_gap_aligned_transport_outputfirstoutside. pfrep_gap_aligned_transport_outputfirstoutside+(N)=(pfrep_power_aligned_transport_output)) /\\ (((pfrep_left_aligned_transport_output)=0))))) -> ((exists pfrep_position_aligned_transport_outputsecond. ((pfrep_position_aligned_transport_outputsecond+S (pfrep_power_aligned_transport_output)=(I)) /\\ ((((exists ff_h_pfp_aligned_transport_outputsecondentry. ff_h_pfp_aligned_transport_outputsecondentry + S (pfrep_right_aligned_transport_output) = S ((S (pfrep_position_aligned_transport_outputsecond)) * fc)) /\\ exists ff_q_pfp_aligned_transport_outputsecondentry. fb = ff_q_pfp_aligned_transport_outputsecondentry * S ((S (pfrep_position_aligned_transport_outputsecond)) * fc) + (pfrep_right_aligned_transport_output)))))) \\/ (((exists pfrep_gap_aligned_transport_outputsecondoutside. pfrep_gap_aligned_transport_outputsecondoutside+(I)=(pfrep_power_aligned_transport_output)) /\\ (((pfrep_right_aligned_transport_output)=0))))) -> pfrep_left_aligned_transport_output=pfrep_right_aligned_transport_output) -> (((forall fom_index_pfp_aligned_transport_old_left_bounded. (exists fom_gap_pfp_aligned_transport_old_left_bounded_index_bound. fom_gap_pfp_aligned_transport_old_left_bounded_index_bound + S (fom_index_pfp_aligned_transport_old_left_bounded) = L) -> exists fom_value_pfp_aligned_transport_old_left_bounded. ((((exists fom_beta_height_pfp_aligned_transport_old_left_bounded_entry. fom_beta_height_pfp_aligned_transport_old_left_bounded_entry + S (fom_value_pfp_aligned_transport_old_left_bounded) = S ((S (fom_index_pfp_aligned_transport_old_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_aligned_transport_old_left_bounded_entry. ab = fom_beta_quotient_pfp_aligned_transport_old_left_bounded_entry * S ((S (fom_index_pfp_aligned_transport_old_left_bounded)) * ac) + (fom_value_pfp_aligned_transport_old_left_bounded))) /\\ (exists fom_gap_pfp_aligned_transport_old_left_bounded_value_bound. fom_gap_pfp_aligned_transport_old_left_bounded_value_bound + S (fom_value_pfp_aligned_transport_old_left_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_transport_old_right_bounded. (exists fom_gap_pfp_aligned_transport_old_right_bounded_index_bound. fom_gap_pfp_aligned_transport_old_right_bounded_index_bound + S (fom_index_pfp_aligned_transport_old_right_bounded) = M) -> exists fom_value_pfp_aligned_transport_old_right_bounded. ((((exists fom_beta_height_pfp_aligned_transport_old_right_bounded_entry. fom_beta_height_pfp_aligned_transport_old_right_bounded_entry + S (fom_value_pfp_aligned_transport_old_right_bounded) = S ((S (fom_index_pfp_aligned_transport_old_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_aligned_transport_old_right_bounded_entry. bb = fom_beta_quotient_pfp_aligned_transport_old_right_bounded_entry * S ((S (fom_index_pfp_aligned_transport_old_right_bounded)) * bc) + (fom_value_pfp_aligned_transport_old_right_bounded))) /\\ (exists fom_gap_pfp_aligned_transport_old_right_bounded_value_bound. fom_gap_pfp_aligned_transport_old_right_bounded_value_bound + S (fom_value_pfp_aligned_transport_old_right_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_transport_old_result_bounded. (exists fom_gap_pfp_aligned_transport_old_result_bounded_index_bound. fom_gap_pfp_aligned_transport_old_result_bounded_index_bound + S (fom_index_pfp_aligned_transport_old_result_bounded) = N) -> exists fom_value_pfp_aligned_transport_old_result_bounded. ((((exists fom_beta_height_pfp_aligned_transport_old_result_bounded_entry. fom_beta_height_pfp_aligned_transport_old_result_bounded_entry + S (fom_value_pfp_aligned_transport_old_result_bounded) = S ((S (fom_index_pfp_aligned_transport_old_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_aligned_transport_old_result_bounded_entry. rb = fom_beta_quotient_pfp_aligned_transport_old_result_bounded_entry * S ((S (fom_index_pfp_aligned_transport_old_result_bounded)) * rc) + (fom_value_pfp_aligned_transport_old_result_bounded))) /\\ (exists fom_gap_pfp_aligned_transport_old_result_bounded_value_bound. fom_gap_pfp_aligned_transport_old_result_bounded_value_bound + S (fom_value_pfp_aligned_transport_old_result_bounded) = p))) /\\ ((exists pfaa_left_b_aligned_transport_old pfaa_left_c_aligned_transport_old pfaa_right_b_aligned_transport_old pfaa_right_c_aligned_transport_old pfaa_sum_b_aligned_transport_old pfaa_sum_c_aligned_transport_old pfaa_length_aligned_transport_old. ((((forall pfrep_power_aligned_transport_old_witness_common_left pfrep_left_aligned_transport_old_witness_common_left pfrep_right_aligned_transport_old_witness_common_left. ((exists pfrep_position_aligned_transport_old_witness_common_leftfirst. ((pfrep_position_aligned_transport_old_witness_common_leftfirst+S (pfrep_power_aligned_transport_old_witness_common_left)=(L)) /\\ ((((exists ff_h_pfp_aligned_transport_old_witness_common_leftfirstentry. ff_h_pfp_aligned_transport_old_witness_common_leftfirstentry + S (pfrep_left_aligned_transport_old_witness_common_left) = S ((S (pfrep_position_aligned_transport_old_witness_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_aligned_transport_old_witness_common_leftfirstentry. ab = ff_q_pfp_aligned_transport_old_witness_common_leftfirstentry * S ((S (pfrep_position_aligned_transport_old_witness_common_leftfirst)) * ac) + (pfrep_left_aligned_transport_old_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_transport_old_witness_common_leftfirstoutside. pfrep_gap_aligned_transport_old_witness_common_leftfirstoutside+(L)=(pfrep_power_aligned_transport_old_witness_common_left)) /\\ (((pfrep_left_aligned_transport_old_witness_common_left)=0))))) -> ((exists pfrep_position_aligned_transport_old_witness_common_leftsecond. ((pfrep_position_aligned_transport_old_witness_common_leftsecond+S (pfrep_power_aligned_transport_old_witness_common_left)=(pfaa_length_aligned_transport_old)) /\\ ((((exists ff_h_pfp_aligned_transport_old_witness_common_leftsecondentry. ff_h_pfp_aligned_transport_old_witness_common_leftsecondentry + S (pfrep_right_aligned_transport_old_witness_common_left) = S ((S (pfrep_position_aligned_transport_old_witness_common_leftsecond)) * pfaa_left_c_aligned_transport_old)) /\\ exists ff_q_pfp_aligned_transport_old_witness_common_leftsecondentry. pfaa_left_b_aligned_transport_old = ff_q_pfp_aligned_transport_old_witness_common_leftsecondentry * S ((S (pfrep_position_aligned_transport_old_witness_common_leftsecond)) * pfaa_left_c_aligned_transport_old) + (pfrep_right_aligned_transport_old_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_transport_old_witness_common_leftsecondoutside. pfrep_gap_aligned_transport_old_witness_common_leftsecondoutside+(pfaa_length_aligned_transport_old)=(pfrep_power_aligned_transport_old_witness_common_left)) /\\ (((pfrep_right_aligned_transport_old_witness_common_left)=0))))) -> pfrep_left_aligned_transport_old_witness_common_left=pfrep_right_aligned_transport_old_witness_common_left) /\\ ((forall pfrep_power_aligned_transport_old_witness_common_right pfrep_left_aligned_transport_old_witness_common_right pfrep_right_aligned_transport_old_witness_common_right. ((exists pfrep_position_aligned_transport_old_witness_common_rightfirst. ((pfrep_position_aligned_transport_old_witness_common_rightfirst+S (pfrep_power_aligned_transport_old_witness_common_right)=(M)) /\\ ((((exists ff_h_pfp_aligned_transport_old_witness_common_rightfirstentry. ff_h_pfp_aligned_transport_old_witness_common_rightfirstentry + S (pfrep_left_aligned_transport_old_witness_common_right) = S ((S (pfrep_position_aligned_transport_old_witness_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_aligned_transport_old_witness_common_rightfirstentry. bb = ff_q_pfp_aligned_transport_old_witness_common_rightfirstentry * S ((S (pfrep_position_aligned_transport_old_witness_common_rightfirst)) * bc) + (pfrep_left_aligned_transport_old_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_transport_old_witness_common_rightfirstoutside. pfrep_gap_aligned_transport_old_witness_common_rightfirstoutside+(M)=(pfrep_power_aligned_transport_old_witness_common_right)) /\\ (((pfrep_left_aligned_transport_old_witness_common_right)=0))))) -> ((exists pfrep_position_aligned_transport_old_witness_common_rightsecond. ((pfrep_position_aligned_transport_old_witness_common_rightsecond+S (pfrep_power_aligned_transport_old_witness_common_right)=(pfaa_length_aligned_transport_old)) /\\ ((((exists ff_h_pfp_aligned_transport_old_witness_common_rightsecondentry. ff_h_pfp_aligned_transport_old_witness_common_rightsecondentry + S (pfrep_right_aligned_transport_old_witness_common_right) = S ((S (pfrep_position_aligned_transport_old_witness_common_rightsecond)) * pfaa_right_c_aligned_transport_old)) /\\ exists ff_q_pfp_aligned_transport_old_witness_common_rightsecondentry. pfaa_right_b_aligned_transport_old = ff_q_pfp_aligned_transport_old_witness_common_rightsecondentry * S ((S (pfrep_position_aligned_transport_old_witness_common_rightsecond)) * pfaa_right_c_aligned_transport_old) + (pfrep_right_aligned_transport_old_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_transport_old_witness_common_rightsecondoutside. pfrep_gap_aligned_transport_old_witness_common_rightsecondoutside+(pfaa_length_aligned_transport_old)=(pfrep_power_aligned_transport_old_witness_common_right)) /\\ (((pfrep_right_aligned_transport_old_witness_common_right)=0))))) -> pfrep_left_aligned_transport_old_witness_common_right=pfrep_right_aligned_transport_old_witness_common_right)))) /\\ (((forall pfp_index_aligned_transport_old_witness_operation. (exists pfa_gap_aligned_transport_old_witness_operationindex. pfa_gap_aligned_transport_old_witness_operationindex + S (pfp_index_aligned_transport_old_witness_operation) = (pfaa_length_aligned_transport_old)) -> exists pfp_left_aligned_transport_old_witness_operation pfp_right_aligned_transport_old_witness_operation pfp_value_aligned_transport_old_witness_operation. ((((exists ff_h_pfp_aligned_transport_old_witness_operationleft. ff_h_pfp_aligned_transport_old_witness_operationleft + S (pfp_left_aligned_transport_old_witness_operation) = S ((S (pfp_index_aligned_transport_old_witness_operation)) * pfaa_left_c_aligned_transport_old)) /\\ exists ff_q_pfp_aligned_transport_old_witness_operationleft. pfaa_left_b_aligned_transport_old = ff_q_pfp_aligned_transport_old_witness_operationleft * S ((S (pfp_index_aligned_transport_old_witness_operation)) * pfaa_left_c_aligned_transport_old) + (pfp_left_aligned_transport_old_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_transport_old_witness_operationright. ff_h_pfp_aligned_transport_old_witness_operationright + S (pfp_right_aligned_transport_old_witness_operation) = S ((S (pfp_index_aligned_transport_old_witness_operation)) * pfaa_right_c_aligned_transport_old)) /\\ exists ff_q_pfp_aligned_transport_old_witness_operationright. pfaa_right_b_aligned_transport_old = ff_q_pfp_aligned_transport_old_witness_operationright * S ((S (pfp_index_aligned_transport_old_witness_operation)) * pfaa_right_c_aligned_transport_old) + (pfp_right_aligned_transport_old_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_transport_old_witness_operationtarget. ff_h_pfp_aligned_transport_old_witness_operationtarget + S (pfp_value_aligned_transport_old_witness_operation) = S ((S (pfp_index_aligned_transport_old_witness_operation)) * pfaa_sum_c_aligned_transport_old)) /\\ exists ff_q_pfp_aligned_transport_old_witness_operationtarget. pfaa_sum_b_aligned_transport_old = ff_q_pfp_aligned_transport_old_witness_operationtarget * S ((S (pfp_index_aligned_transport_old_witness_operation)) * pfaa_sum_c_aligned_transport_old) + (pfp_value_aligned_transport_old_witness_operation))) /\\ ((((exists pfa_gap_aligned_transport_old_witness_operationoperationleft. pfa_gap_aligned_transport_old_witness_operationoperationleft + S (pfp_left_aligned_transport_old_witness_operation) = (p)) /\\ (((exists pfa_gap_aligned_transport_old_witness_operationoperationright. pfa_gap_aligned_transport_old_witness_operationoperationright + S (pfp_right_aligned_transport_old_witness_operation) = (p)) /\\ ((((exists pfa_gap_aligned_transport_old_witness_operationoperationresultbound. pfa_gap_aligned_transport_old_witness_operationoperationresultbound + S (pfp_value_aligned_transport_old_witness_operation) = (p)) /\\ ((exists pfa_offset_left_aligned_transport_old_witness_operationoperationresultcongruence pfa_offset_right_aligned_transport_old_witness_operationoperationresultcongruence. ((pfp_left_aligned_transport_old_witness_operation) + (pfp_right_aligned_transport_old_witness_operation)) + (p) * pfa_offset_left_aligned_transport_old_witness_operationoperationresultcongruence = (pfp_value_aligned_transport_old_witness_operation) + (p) * pfa_offset_right_aligned_transport_old_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_aligned_transport_old_witness_output pfrep_left_aligned_transport_old_witness_output pfrep_right_aligned_transport_old_witness_output. ((exists pfrep_position_aligned_transport_old_witness_outputfirst. ((pfrep_position_aligned_transport_old_witness_outputfirst+S (pfrep_power_aligned_transport_old_witness_output)=(pfaa_length_aligned_transport_old)) /\\ ((((exists ff_h_pfp_aligned_transport_old_witness_outputfirstentry. ff_h_pfp_aligned_transport_old_witness_outputfirstentry + S (pfrep_left_aligned_transport_old_witness_output) = S ((S (pfrep_position_aligned_transport_old_witness_outputfirst)) * pfaa_sum_c_aligned_transport_old)) /\\ exists ff_q_pfp_aligned_transport_old_witness_outputfirstentry. pfaa_sum_b_aligned_transport_old = ff_q_pfp_aligned_transport_old_witness_outputfirstentry * S ((S (pfrep_position_aligned_transport_old_witness_outputfirst)) * pfaa_sum_c_aligned_transport_old) + (pfrep_left_aligned_transport_old_witness_output)))))) \\/ (((exists pfrep_gap_aligned_transport_old_witness_outputfirstoutside. pfrep_gap_aligned_transport_old_witness_outputfirstoutside+(pfaa_length_aligned_transport_old)=(pfrep_power_aligned_transport_old_witness_output)) /\\ (((pfrep_left_aligned_transport_old_witness_output)=0))))) -> ((exists pfrep_position_aligned_transport_old_witness_outputsecond. ((pfrep_position_aligned_transport_old_witness_outputsecond+S (pfrep_power_aligned_transport_old_witness_output)=(N)) /\\ ((((exists ff_h_pfp_aligned_transport_old_witness_outputsecondentry. ff_h_pfp_aligned_transport_old_witness_outputsecondentry + S (pfrep_right_aligned_transport_old_witness_output) = S ((S (pfrep_position_aligned_transport_old_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_aligned_transport_old_witness_outputsecondentry. rb = ff_q_pfp_aligned_transport_old_witness_outputsecondentry * S ((S (pfrep_position_aligned_transport_old_witness_outputsecond)) * rc) + (pfrep_right_aligned_transport_old_witness_output)))))) \\/ (((exists pfrep_gap_aligned_transport_old_witness_outputsecondoutside. pfrep_gap_aligned_transport_old_witness_outputsecondoutside+(N)=(pfrep_power_aligned_transport_old_witness_output)) /\\ (((pfrep_right_aligned_transport_old_witness_output)=0))))) -> pfrep_left_aligned_transport_old_witness_output=pfrep_right_aligned_transport_old_witness_output))))))))))))) -> (((forall fom_index_pfp_aligned_transport_new_left_bounded. (exists fom_gap_pfp_aligned_transport_new_left_bounded_index_bound. fom_gap_pfp_aligned_transport_new_left_bounded_index_bound + S (fom_index_pfp_aligned_transport_new_left_bounded) = J) -> exists fom_value_pfp_aligned_transport_new_left_bounded. ((((exists fom_beta_height_pfp_aligned_transport_new_left_bounded_entry. fom_beta_height_pfp_aligned_transport_new_left_bounded_entry + S (fom_value_pfp_aligned_transport_new_left_bounded) = S ((S (fom_index_pfp_aligned_transport_new_left_bounded)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_transport_new_left_bounded_entry. db = fom_beta_quotient_pfp_aligned_transport_new_left_bounded_entry * S ((S (fom_index_pfp_aligned_transport_new_left_bounded)) * dc) + (fom_value_pfp_aligned_transport_new_left_bounded))) /\\ (exists fom_gap_pfp_aligned_transport_new_left_bounded_value_bound. fom_gap_pfp_aligned_transport_new_left_bounded_value_bound + S (fom_value_pfp_aligned_transport_new_left_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_transport_new_right_bounded. (exists fom_gap_pfp_aligned_transport_new_right_bounded_index_bound. fom_gap_pfp_aligned_transport_new_right_bounded_index_bound + S (fom_index_pfp_aligned_transport_new_right_bounded) = H) -> exists fom_value_pfp_aligned_transport_new_right_bounded. ((((exists fom_beta_height_pfp_aligned_transport_new_right_bounded_entry. fom_beta_height_pfp_aligned_transport_new_right_bounded_entry + S (fom_value_pfp_aligned_transport_new_right_bounded) = S ((S (fom_index_pfp_aligned_transport_new_right_bounded)) * ec)) /\\ exists fom_beta_quotient_pfp_aligned_transport_new_right_bounded_entry. eb = fom_beta_quotient_pfp_aligned_transport_new_right_bounded_entry * S ((S (fom_index_pfp_aligned_transport_new_right_bounded)) * ec) + (fom_value_pfp_aligned_transport_new_right_bounded))) /\\ (exists fom_gap_pfp_aligned_transport_new_right_bounded_value_bound. fom_gap_pfp_aligned_transport_new_right_bounded_value_bound + S (fom_value_pfp_aligned_transport_new_right_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_transport_new_result_bounded. (exists fom_gap_pfp_aligned_transport_new_result_bounded_index_bound. fom_gap_pfp_aligned_transport_new_result_bounded_index_bound + S (fom_index_pfp_aligned_transport_new_result_bounded) = I) -> exists fom_value_pfp_aligned_transport_new_result_bounded. ((((exists fom_beta_height_pfp_aligned_transport_new_result_bounded_entry. fom_beta_height_pfp_aligned_transport_new_result_bounded_entry + S (fom_value_pfp_aligned_transport_new_result_bounded) = S ((S (fom_index_pfp_aligned_transport_new_result_bounded)) * fc)) /\\ exists fom_beta_quotient_pfp_aligned_transport_new_result_bounded_entry. fb = fom_beta_quotient_pfp_aligned_transport_new_result_bounded_entry * S ((S (fom_index_pfp_aligned_transport_new_result_bounded)) * fc) + (fom_value_pfp_aligned_transport_new_result_bounded))) /\\ (exists fom_gap_pfp_aligned_transport_new_result_bounded_value_bound. fom_gap_pfp_aligned_transport_new_result_bounded_value_bound + S (fom_value_pfp_aligned_transport_new_result_bounded) = p))) /\\ ((exists pfaa_left_b_aligned_transport_new pfaa_left_c_aligned_transport_new pfaa_right_b_aligned_transport_new pfaa_right_c_aligned_transport_new pfaa_sum_b_aligned_transport_new pfaa_sum_c_aligned_transport_new pfaa_length_aligned_transport_new. ((((forall pfrep_power_aligned_transport_new_witness_common_left pfrep_left_aligned_transport_new_witness_common_left pfrep_right_aligned_transport_new_witness_common_left. ((exists pfrep_position_aligned_transport_new_witness_common_leftfirst. ((pfrep_position_aligned_transport_new_witness_common_leftfirst+S (pfrep_power_aligned_transport_new_witness_common_left)=(J)) /\\ ((((exists ff_h_pfp_aligned_transport_new_witness_common_leftfirstentry. ff_h_pfp_aligned_transport_new_witness_common_leftfirstentry + S (pfrep_left_aligned_transport_new_witness_common_left) = S ((S (pfrep_position_aligned_transport_new_witness_common_leftfirst)) * dc)) /\\ exists ff_q_pfp_aligned_transport_new_witness_common_leftfirstentry. db = ff_q_pfp_aligned_transport_new_witness_common_leftfirstentry * S ((S (pfrep_position_aligned_transport_new_witness_common_leftfirst)) * dc) + (pfrep_left_aligned_transport_new_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_transport_new_witness_common_leftfirstoutside. pfrep_gap_aligned_transport_new_witness_common_leftfirstoutside+(J)=(pfrep_power_aligned_transport_new_witness_common_left)) /\\ (((pfrep_left_aligned_transport_new_witness_common_left)=0))))) -> ((exists pfrep_position_aligned_transport_new_witness_common_leftsecond. ((pfrep_position_aligned_transport_new_witness_common_leftsecond+S (pfrep_power_aligned_transport_new_witness_common_left)=(pfaa_length_aligned_transport_new)) /\\ ((((exists ff_h_pfp_aligned_transport_new_witness_common_leftsecondentry. ff_h_pfp_aligned_transport_new_witness_common_leftsecondentry + S (pfrep_right_aligned_transport_new_witness_common_left) = S ((S (pfrep_position_aligned_transport_new_witness_common_leftsecond)) * pfaa_left_c_aligned_transport_new)) /\\ exists ff_q_pfp_aligned_transport_new_witness_common_leftsecondentry. pfaa_left_b_aligned_transport_new = ff_q_pfp_aligned_transport_new_witness_common_leftsecondentry * S ((S (pfrep_position_aligned_transport_new_witness_common_leftsecond)) * pfaa_left_c_aligned_transport_new) + (pfrep_right_aligned_transport_new_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_transport_new_witness_common_leftsecondoutside. pfrep_gap_aligned_transport_new_witness_common_leftsecondoutside+(pfaa_length_aligned_transport_new)=(pfrep_power_aligned_transport_new_witness_common_left)) /\\ (((pfrep_right_aligned_transport_new_witness_common_left)=0))))) -> pfrep_left_aligned_transport_new_witness_common_left=pfrep_right_aligned_transport_new_witness_common_left) /\\ ((forall pfrep_power_aligned_transport_new_witness_common_right pfrep_left_aligned_transport_new_witness_common_right pfrep_right_aligned_transport_new_witness_common_right. ((exists pfrep_position_aligned_transport_new_witness_common_rightfirst. ((pfrep_position_aligned_transport_new_witness_common_rightfirst+S (pfrep_power_aligned_transport_new_witness_common_right)=(H)) /\\ ((((exists ff_h_pfp_aligned_transport_new_witness_common_rightfirstentry. ff_h_pfp_aligned_transport_new_witness_common_rightfirstentry + S (pfrep_left_aligned_transport_new_witness_common_right) = S ((S (pfrep_position_aligned_transport_new_witness_common_rightfirst)) * ec)) /\\ exists ff_q_pfp_aligned_transport_new_witness_common_rightfirstentry. eb = ff_q_pfp_aligned_transport_new_witness_common_rightfirstentry * S ((S (pfrep_position_aligned_transport_new_witness_common_rightfirst)) * ec) + (pfrep_left_aligned_transport_new_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_transport_new_witness_common_rightfirstoutside. pfrep_gap_aligned_transport_new_witness_common_rightfirstoutside+(H)=(pfrep_power_aligned_transport_new_witness_common_right)) /\\ (((pfrep_left_aligned_transport_new_witness_common_right)=0))))) -> ((exists pfrep_position_aligned_transport_new_witness_common_rightsecond. ((pfrep_position_aligned_transport_new_witness_common_rightsecond+S (pfrep_power_aligned_transport_new_witness_common_right)=(pfaa_length_aligned_transport_new)) /\\ ((((exists ff_h_pfp_aligned_transport_new_witness_common_rightsecondentry. ff_h_pfp_aligned_transport_new_witness_common_rightsecondentry + S (pfrep_right_aligned_transport_new_witness_common_right) = S ((S (pfrep_position_aligned_transport_new_witness_common_rightsecond)) * pfaa_right_c_aligned_transport_new)) /\\ exists ff_q_pfp_aligned_transport_new_witness_common_rightsecondentry. pfaa_right_b_aligned_transport_new = ff_q_pfp_aligned_transport_new_witness_common_rightsecondentry * S ((S (pfrep_position_aligned_transport_new_witness_common_rightsecond)) * pfaa_right_c_aligned_transport_new) + (pfrep_right_aligned_transport_new_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_transport_new_witness_common_rightsecondoutside. pfrep_gap_aligned_transport_new_witness_common_rightsecondoutside+(pfaa_length_aligned_transport_new)=(pfrep_power_aligned_transport_new_witness_common_right)) /\\ (((pfrep_right_aligned_transport_new_witness_common_right)=0))))) -> pfrep_left_aligned_transport_new_witness_common_right=pfrep_right_aligned_transport_new_witness_common_right)))) /\\ (((forall pfp_index_aligned_transport_new_witness_operation. (exists pfa_gap_aligned_transport_new_witness_operationindex. pfa_gap_aligned_transport_new_witness_operationindex + S (pfp_index_aligned_transport_new_witness_operation) = (pfaa_length_aligned_transport_new)) -> exists pfp_left_aligned_transport_new_witness_operation pfp_right_aligned_transport_new_witness_operation pfp_value_aligned_transport_new_witness_operation. ((((exists ff_h_pfp_aligned_transport_new_witness_operationleft. ff_h_pfp_aligned_transport_new_witness_operationleft + S (pfp_left_aligned_transport_new_witness_operation) = S ((S (pfp_index_aligned_transport_new_witness_operation)) * pfaa_left_c_aligned_transport_new)) /\\ exists ff_q_pfp_aligned_transport_new_witness_operationleft. pfaa_left_b_aligned_transport_new = ff_q_pfp_aligned_transport_new_witness_operationleft * S ((S (pfp_index_aligned_transport_new_witness_operation)) * pfaa_left_c_aligned_transport_new) + (pfp_left_aligned_transport_new_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_transport_new_witness_operationright. ff_h_pfp_aligned_transport_new_witness_operationright + S (pfp_right_aligned_transport_new_witness_operation) = S ((S (pfp_index_aligned_transport_new_witness_operation)) * pfaa_right_c_aligned_transport_new)) /\\ exists ff_q_pfp_aligned_transport_new_witness_operationright. pfaa_right_b_aligned_transport_new = ff_q_pfp_aligned_transport_new_witness_operationright * S ((S (pfp_index_aligned_transport_new_witness_operation)) * pfaa_right_c_aligned_transport_new) + (pfp_right_aligned_transport_new_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_transport_new_witness_operationtarget. ff_h_pfp_aligned_transport_new_witness_operationtarget + S (pfp_value_aligned_transport_new_witness_operation) = S ((S (pfp_index_aligned_transport_new_witness_operation)) * pfaa_sum_c_aligned_transport_new)) /\\ exists ff_q_pfp_aligned_transport_new_witness_operationtarget. pfaa_sum_b_aligned_transport_new = ff_q_pfp_aligned_transport_new_witness_operationtarget * S ((S (pfp_index_aligned_transport_new_witness_operation)) * pfaa_sum_c_aligned_transport_new) + (pfp_value_aligned_transport_new_witness_operation))) /\\ ((((exists pfa_gap_aligned_transport_new_witness_operationoperationleft. pfa_gap_aligned_transport_new_witness_operationoperationleft + S (pfp_left_aligned_transport_new_witness_operation) = (p)) /\\ (((exists pfa_gap_aligned_transport_new_witness_operationoperationright. pfa_gap_aligned_transport_new_witness_operationoperationright + S (pfp_right_aligned_transport_new_witness_operation) = (p)) /\\ ((((exists pfa_gap_aligned_transport_new_witness_operationoperationresultbound. pfa_gap_aligned_transport_new_witness_operationoperationresultbound + S (pfp_value_aligned_transport_new_witness_operation) = (p)) /\\ ((exists pfa_offset_left_aligned_transport_new_witness_operationoperationresultcongruence pfa_offset_right_aligned_transport_new_witness_operationoperationresultcongruence. ((pfp_left_aligned_transport_new_witness_operation) + (pfp_right_aligned_transport_new_witness_operation)) + (p) * pfa_offset_left_aligned_transport_new_witness_operationoperationresultcongruence = (pfp_value_aligned_transport_new_witness_operation) + (p) * pfa_offset_right_aligned_transport_new_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_aligned_transport_new_witness_output pfrep_left_aligned_transport_new_witness_output pfrep_right_aligned_transport_new_witness_output. ((exists pfrep_position_aligned_transport_new_witness_outputfirst. ((pfrep_position_aligned_transport_new_witness_outputfirst+S (pfrep_power_aligned_transport_new_witness_output)=(pfaa_length_aligned_transport_new)) /\\ ((((exists ff_h_pfp_aligned_transport_new_witness_outputfirstentry. ff_h_pfp_aligned_transport_new_witness_outputfirstentry + S (pfrep_left_aligned_transport_new_witness_output) = S ((S (pfrep_position_aligned_transport_new_witness_outputfirst)) * pfaa_sum_c_aligned_transport_new)) /\\ exists ff_q_pfp_aligned_transport_new_witness_outputfirstentry. pfaa_sum_b_aligned_transport_new = ff_q_pfp_aligned_transport_new_witness_outputfirstentry * S ((S (pfrep_position_aligned_transport_new_witness_outputfirst)) * pfaa_sum_c_aligned_transport_new) + (pfrep_left_aligned_transport_new_witness_output)))))) \\/ (((exists pfrep_gap_aligned_transport_new_witness_outputfirstoutside. pfrep_gap_aligned_transport_new_witness_outputfirstoutside+(pfaa_length_aligned_transport_new)=(pfrep_power_aligned_transport_new_witness_output)) /\\ (((pfrep_left_aligned_transport_new_witness_output)=0))))) -> ((exists pfrep_position_aligned_transport_new_witness_outputsecond. 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prime_field_polynomial_add_commutative (x3)", "specialize prime_field_polynomial_add_commutative (x4)", "specialize prime_field_polynomial_add_commutative (x5)", "specialize prime_field_polynomial_add_commutative (x6)", "apply prime_field_polynomial_add_commutative", "exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_left", "exact h_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right" ], "script_sha256": "2c80278bc611576e81446d93e4cb43cf49da741a21974c1d330ae47d17add585", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_aligned_add_candidate.py", "sha256": "a05bb4f5c4230ca05f51690d3ab82e33ff4596af65176874e25fbe38cf87a0db" }, "statement": "forall p ab ac L bb bc M rb rc N. (((forall fom_index_pfp_aligned_comm_old_left_bounded. (exists fom_gap_pfp_aligned_comm_old_left_bounded_index_bound. fom_gap_pfp_aligned_comm_old_left_bounded_index_bound + S (fom_index_pfp_aligned_comm_old_left_bounded) = L) -> exists fom_value_pfp_aligned_comm_old_left_bounded. ((((exists fom_beta_height_pfp_aligned_comm_old_left_bounded_entry. fom_beta_height_pfp_aligned_comm_old_left_bounded_entry + S (fom_value_pfp_aligned_comm_old_left_bounded) = S ((S (fom_index_pfp_aligned_comm_old_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_aligned_comm_old_left_bounded_entry. ab = fom_beta_quotient_pfp_aligned_comm_old_left_bounded_entry * S ((S (fom_index_pfp_aligned_comm_old_left_bounded)) * ac) + (fom_value_pfp_aligned_comm_old_left_bounded))) /\\ (exists fom_gap_pfp_aligned_comm_old_left_bounded_value_bound. fom_gap_pfp_aligned_comm_old_left_bounded_value_bound + S (fom_value_pfp_aligned_comm_old_left_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_comm_old_right_bounded. (exists fom_gap_pfp_aligned_comm_old_right_bounded_index_bound. fom_gap_pfp_aligned_comm_old_right_bounded_index_bound + S (fom_index_pfp_aligned_comm_old_right_bounded) = M) -> exists fom_value_pfp_aligned_comm_old_right_bounded. ((((exists fom_beta_height_pfp_aligned_comm_old_right_bounded_entry. fom_beta_height_pfp_aligned_comm_old_right_bounded_entry + S (fom_value_pfp_aligned_comm_old_right_bounded) = S ((S (fom_index_pfp_aligned_comm_old_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_aligned_comm_old_right_bounded_entry. bb = fom_beta_quotient_pfp_aligned_comm_old_right_bounded_entry * S ((S (fom_index_pfp_aligned_comm_old_right_bounded)) * bc) + (fom_value_pfp_aligned_comm_old_right_bounded))) /\\ (exists fom_gap_pfp_aligned_comm_old_right_bounded_value_bound. fom_gap_pfp_aligned_comm_old_right_bounded_value_bound + S (fom_value_pfp_aligned_comm_old_right_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_comm_old_result_bounded. (exists fom_gap_pfp_aligned_comm_old_result_bounded_index_bound. fom_gap_pfp_aligned_comm_old_result_bounded_index_bound + S (fom_index_pfp_aligned_comm_old_result_bounded) = N) -> exists fom_value_pfp_aligned_comm_old_result_bounded. ((((exists fom_beta_height_pfp_aligned_comm_old_result_bounded_entry. fom_beta_height_pfp_aligned_comm_old_result_bounded_entry + S (fom_value_pfp_aligned_comm_old_result_bounded) = S ((S (fom_index_pfp_aligned_comm_old_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_aligned_comm_old_result_bounded_entry. rb = fom_beta_quotient_pfp_aligned_comm_old_result_bounded_entry * S ((S (fom_index_pfp_aligned_comm_old_result_bounded)) * rc) + (fom_value_pfp_aligned_comm_old_result_bounded))) /\\ (exists fom_gap_pfp_aligned_comm_old_result_bounded_value_bound. fom_gap_pfp_aligned_comm_old_result_bounded_value_bound + S (fom_value_pfp_aligned_comm_old_result_bounded) = p))) /\\ ((exists pfaa_left_b_aligned_comm_old pfaa_left_c_aligned_comm_old pfaa_right_b_aligned_comm_old pfaa_right_c_aligned_comm_old pfaa_sum_b_aligned_comm_old pfaa_sum_c_aligned_comm_old pfaa_length_aligned_comm_old. ((((forall pfrep_power_aligned_comm_old_witness_common_left pfrep_left_aligned_comm_old_witness_common_left pfrep_right_aligned_comm_old_witness_common_left. ((exists pfrep_position_aligned_comm_old_witness_common_leftfirst. ((pfrep_position_aligned_comm_old_witness_common_leftfirst+S (pfrep_power_aligned_comm_old_witness_common_left)=(L)) /\\ ((((exists ff_h_pfp_aligned_comm_old_witness_common_leftfirstentry. ff_h_pfp_aligned_comm_old_witness_common_leftfirstentry + S (pfrep_left_aligned_comm_old_witness_common_left) = S ((S (pfrep_position_aligned_comm_old_witness_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_aligned_comm_old_witness_common_leftfirstentry. ab = ff_q_pfp_aligned_comm_old_witness_common_leftfirstentry * S ((S (pfrep_position_aligned_comm_old_witness_common_leftfirst)) * ac) + (pfrep_left_aligned_comm_old_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_comm_old_witness_common_leftfirstoutside. pfrep_gap_aligned_comm_old_witness_common_leftfirstoutside+(L)=(pfrep_power_aligned_comm_old_witness_common_left)) /\\ (((pfrep_left_aligned_comm_old_witness_common_left)=0))))) -> ((exists pfrep_position_aligned_comm_old_witness_common_leftsecond. ((pfrep_position_aligned_comm_old_witness_common_leftsecond+S (pfrep_power_aligned_comm_old_witness_common_left)=(pfaa_length_aligned_comm_old)) /\\ ((((exists ff_h_pfp_aligned_comm_old_witness_common_leftsecondentry. ff_h_pfp_aligned_comm_old_witness_common_leftsecondentry + S (pfrep_right_aligned_comm_old_witness_common_left) = S ((S (pfrep_position_aligned_comm_old_witness_common_leftsecond)) * pfaa_left_c_aligned_comm_old)) /\\ exists ff_q_pfp_aligned_comm_old_witness_common_leftsecondentry. pfaa_left_b_aligned_comm_old = ff_q_pfp_aligned_comm_old_witness_common_leftsecondentry * S ((S (pfrep_position_aligned_comm_old_witness_common_leftsecond)) * pfaa_left_c_aligned_comm_old) + (pfrep_right_aligned_comm_old_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_comm_old_witness_common_leftsecondoutside. pfrep_gap_aligned_comm_old_witness_common_leftsecondoutside+(pfaa_length_aligned_comm_old)=(pfrep_power_aligned_comm_old_witness_common_left)) /\\ (((pfrep_right_aligned_comm_old_witness_common_left)=0))))) -> pfrep_left_aligned_comm_old_witness_common_left=pfrep_right_aligned_comm_old_witness_common_left) /\\ ((forall pfrep_power_aligned_comm_old_witness_common_right pfrep_left_aligned_comm_old_witness_common_right pfrep_right_aligned_comm_old_witness_common_right. ((exists pfrep_position_aligned_comm_old_witness_common_rightfirst. ((pfrep_position_aligned_comm_old_witness_common_rightfirst+S (pfrep_power_aligned_comm_old_witness_common_right)=(M)) /\\ ((((exists ff_h_pfp_aligned_comm_old_witness_common_rightfirstentry. ff_h_pfp_aligned_comm_old_witness_common_rightfirstentry + S (pfrep_left_aligned_comm_old_witness_common_right) = S ((S (pfrep_position_aligned_comm_old_witness_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_aligned_comm_old_witness_common_rightfirstentry. bb = ff_q_pfp_aligned_comm_old_witness_common_rightfirstentry * S ((S (pfrep_position_aligned_comm_old_witness_common_rightfirst)) * bc) + (pfrep_left_aligned_comm_old_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_comm_old_witness_common_rightfirstoutside. pfrep_gap_aligned_comm_old_witness_common_rightfirstoutside+(M)=(pfrep_power_aligned_comm_old_witness_common_right)) /\\ (((pfrep_left_aligned_comm_old_witness_common_right)=0))))) -> ((exists pfrep_position_aligned_comm_old_witness_common_rightsecond. ((pfrep_position_aligned_comm_old_witness_common_rightsecond+S (pfrep_power_aligned_comm_old_witness_common_right)=(pfaa_length_aligned_comm_old)) /\\ ((((exists ff_h_pfp_aligned_comm_old_witness_common_rightsecondentry. ff_h_pfp_aligned_comm_old_witness_common_rightsecondentry + S (pfrep_right_aligned_comm_old_witness_common_right) = S ((S (pfrep_position_aligned_comm_old_witness_common_rightsecond)) * pfaa_right_c_aligned_comm_old)) /\\ exists ff_q_pfp_aligned_comm_old_witness_common_rightsecondentry. pfaa_right_b_aligned_comm_old = ff_q_pfp_aligned_comm_old_witness_common_rightsecondentry * S ((S (pfrep_position_aligned_comm_old_witness_common_rightsecond)) * pfaa_right_c_aligned_comm_old) + (pfrep_right_aligned_comm_old_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_comm_old_witness_common_rightsecondoutside. pfrep_gap_aligned_comm_old_witness_common_rightsecondoutside+(pfaa_length_aligned_comm_old)=(pfrep_power_aligned_comm_old_witness_common_right)) /\\ (((pfrep_right_aligned_comm_old_witness_common_right)=0))))) -> pfrep_left_aligned_comm_old_witness_common_right=pfrep_right_aligned_comm_old_witness_common_right)))) /\\ (((forall pfp_index_aligned_comm_old_witness_operation. (exists pfa_gap_aligned_comm_old_witness_operationindex. pfa_gap_aligned_comm_old_witness_operationindex + S (pfp_index_aligned_comm_old_witness_operation) = (pfaa_length_aligned_comm_old)) -> exists pfp_left_aligned_comm_old_witness_operation pfp_right_aligned_comm_old_witness_operation pfp_value_aligned_comm_old_witness_operation. ((((exists ff_h_pfp_aligned_comm_old_witness_operationleft. ff_h_pfp_aligned_comm_old_witness_operationleft + S (pfp_left_aligned_comm_old_witness_operation) = S ((S (pfp_index_aligned_comm_old_witness_operation)) * pfaa_left_c_aligned_comm_old)) /\\ exists ff_q_pfp_aligned_comm_old_witness_operationleft. pfaa_left_b_aligned_comm_old = ff_q_pfp_aligned_comm_old_witness_operationleft * S ((S (pfp_index_aligned_comm_old_witness_operation)) * pfaa_left_c_aligned_comm_old) + (pfp_left_aligned_comm_old_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_comm_old_witness_operationright. ff_h_pfp_aligned_comm_old_witness_operationright + S (pfp_right_aligned_comm_old_witness_operation) = S ((S (pfp_index_aligned_comm_old_witness_operation)) * pfaa_right_c_aligned_comm_old)) /\\ exists ff_q_pfp_aligned_comm_old_witness_operationright. pfaa_right_b_aligned_comm_old = ff_q_pfp_aligned_comm_old_witness_operationright * S ((S (pfp_index_aligned_comm_old_witness_operation)) * pfaa_right_c_aligned_comm_old) + (pfp_right_aligned_comm_old_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_comm_old_witness_operationtarget. ff_h_pfp_aligned_comm_old_witness_operationtarget + S (pfp_value_aligned_comm_old_witness_operation) = S ((S (pfp_index_aligned_comm_old_witness_operation)) * pfaa_sum_c_aligned_comm_old)) /\\ exists ff_q_pfp_aligned_comm_old_witness_operationtarget. pfaa_sum_b_aligned_comm_old = ff_q_pfp_aligned_comm_old_witness_operationtarget * S ((S (pfp_index_aligned_comm_old_witness_operation)) * pfaa_sum_c_aligned_comm_old) + (pfp_value_aligned_comm_old_witness_operation))) /\\ ((((exists pfa_gap_aligned_comm_old_witness_operationoperationleft. pfa_gap_aligned_comm_old_witness_operationoperationleft + S (pfp_left_aligned_comm_old_witness_operation) = (p)) /\\ (((exists pfa_gap_aligned_comm_old_witness_operationoperationright. pfa_gap_aligned_comm_old_witness_operationoperationright + S (pfp_right_aligned_comm_old_witness_operation) = (p)) /\\ ((((exists pfa_gap_aligned_comm_old_witness_operationoperationresultbound. pfa_gap_aligned_comm_old_witness_operationoperationresultbound + S (pfp_value_aligned_comm_old_witness_operation) = (p)) /\\ ((exists pfa_offset_left_aligned_comm_old_witness_operationoperationresultcongruence pfa_offset_right_aligned_comm_old_witness_operationoperationresultcongruence. ((pfp_left_aligned_comm_old_witness_operation) + (pfp_right_aligned_comm_old_witness_operation)) + (p) * pfa_offset_left_aligned_comm_old_witness_operationoperationresultcongruence = (pfp_value_aligned_comm_old_witness_operation) + (p) * pfa_offset_right_aligned_comm_old_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_aligned_comm_old_witness_output pfrep_left_aligned_comm_old_witness_output pfrep_right_aligned_comm_old_witness_output. ((exists pfrep_position_aligned_comm_old_witness_outputfirst. ((pfrep_position_aligned_comm_old_witness_outputfirst+S (pfrep_power_aligned_comm_old_witness_output)=(pfaa_length_aligned_comm_old)) /\\ ((((exists ff_h_pfp_aligned_comm_old_witness_outputfirstentry. ff_h_pfp_aligned_comm_old_witness_outputfirstentry + S (pfrep_left_aligned_comm_old_witness_output) = S ((S (pfrep_position_aligned_comm_old_witness_outputfirst)) * pfaa_sum_c_aligned_comm_old)) /\\ exists ff_q_pfp_aligned_comm_old_witness_outputfirstentry. pfaa_sum_b_aligned_comm_old = ff_q_pfp_aligned_comm_old_witness_outputfirstentry * S ((S (pfrep_position_aligned_comm_old_witness_outputfirst)) * pfaa_sum_c_aligned_comm_old) + (pfrep_left_aligned_comm_old_witness_output)))))) \\/ (((exists pfrep_gap_aligned_comm_old_witness_outputfirstoutside. pfrep_gap_aligned_comm_old_witness_outputfirstoutside+(pfaa_length_aligned_comm_old)=(pfrep_power_aligned_comm_old_witness_output)) /\\ (((pfrep_left_aligned_comm_old_witness_output)=0))))) -> ((exists pfrep_position_aligned_comm_old_witness_outputsecond. ((pfrep_position_aligned_comm_old_witness_outputsecond+S (pfrep_power_aligned_comm_old_witness_output)=(N)) /\\ ((((exists ff_h_pfp_aligned_comm_old_witness_outputsecondentry. ff_h_pfp_aligned_comm_old_witness_outputsecondentry + S (pfrep_right_aligned_comm_old_witness_output) = S ((S (pfrep_position_aligned_comm_old_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_aligned_comm_old_witness_outputsecondentry. rb = ff_q_pfp_aligned_comm_old_witness_outputsecondentry * S ((S (pfrep_position_aligned_comm_old_witness_outputsecond)) * rc) + (pfrep_right_aligned_comm_old_witness_output)))))) \\/ (((exists pfrep_gap_aligned_comm_old_witness_outputsecondoutside. pfrep_gap_aligned_comm_old_witness_outputsecondoutside+(N)=(pfrep_power_aligned_comm_old_witness_output)) /\\ (((pfrep_right_aligned_comm_old_witness_output)=0))))) -> pfrep_left_aligned_comm_old_witness_output=pfrep_right_aligned_comm_old_witness_output))))))))))))) -> (((forall fom_index_pfp_aligned_comm_new_left_bounded. (exists fom_gap_pfp_aligned_comm_new_left_bounded_index_bound. fom_gap_pfp_aligned_comm_new_left_bounded_index_bound + S (fom_index_pfp_aligned_comm_new_left_bounded) = M) -> exists fom_value_pfp_aligned_comm_new_left_bounded. ((((exists fom_beta_height_pfp_aligned_comm_new_left_bounded_entry. fom_beta_height_pfp_aligned_comm_new_left_bounded_entry + S (fom_value_pfp_aligned_comm_new_left_bounded) = S ((S (fom_index_pfp_aligned_comm_new_left_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_aligned_comm_new_left_bounded_entry. bb = fom_beta_quotient_pfp_aligned_comm_new_left_bounded_entry * S ((S (fom_index_pfp_aligned_comm_new_left_bounded)) * bc) + (fom_value_pfp_aligned_comm_new_left_bounded))) /\\ (exists fom_gap_pfp_aligned_comm_new_left_bounded_value_bound. fom_gap_pfp_aligned_comm_new_left_bounded_value_bound + S (fom_value_pfp_aligned_comm_new_left_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_comm_new_right_bounded. (exists fom_gap_pfp_aligned_comm_new_right_bounded_index_bound. fom_gap_pfp_aligned_comm_new_right_bounded_index_bound + S (fom_index_pfp_aligned_comm_new_right_bounded) = L) -> exists fom_value_pfp_aligned_comm_new_right_bounded. ((((exists fom_beta_height_pfp_aligned_comm_new_right_bounded_entry. fom_beta_height_pfp_aligned_comm_new_right_bounded_entry + S (fom_value_pfp_aligned_comm_new_right_bounded) = S ((S (fom_index_pfp_aligned_comm_new_right_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_aligned_comm_new_right_bounded_entry. ab = fom_beta_quotient_pfp_aligned_comm_new_right_bounded_entry * S ((S (fom_index_pfp_aligned_comm_new_right_bounded)) * ac) + (fom_value_pfp_aligned_comm_new_right_bounded))) /\\ (exists fom_gap_pfp_aligned_comm_new_right_bounded_value_bound. fom_gap_pfp_aligned_comm_new_right_bounded_value_bound + S (fom_value_pfp_aligned_comm_new_right_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_comm_new_result_bounded. (exists fom_gap_pfp_aligned_comm_new_result_bounded_index_bound. fom_gap_pfp_aligned_comm_new_result_bounded_index_bound + S (fom_index_pfp_aligned_comm_new_result_bounded) = N) -> exists fom_value_pfp_aligned_comm_new_result_bounded. ((((exists fom_beta_height_pfp_aligned_comm_new_result_bounded_entry. fom_beta_height_pfp_aligned_comm_new_result_bounded_entry + S (fom_value_pfp_aligned_comm_new_result_bounded) = S ((S (fom_index_pfp_aligned_comm_new_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_aligned_comm_new_result_bounded_entry. rb = fom_beta_quotient_pfp_aligned_comm_new_result_bounded_entry * S ((S (fom_index_pfp_aligned_comm_new_result_bounded)) * rc) + (fom_value_pfp_aligned_comm_new_result_bounded))) /\\ (exists fom_gap_pfp_aligned_comm_new_result_bounded_value_bound. fom_gap_pfp_aligned_comm_new_result_bounded_value_bound + S (fom_value_pfp_aligned_comm_new_result_bounded) = p))) /\\ ((exists pfaa_left_b_aligned_comm_new pfaa_left_c_aligned_comm_new pfaa_right_b_aligned_comm_new pfaa_right_c_aligned_comm_new pfaa_sum_b_aligned_comm_new pfaa_sum_c_aligned_comm_new pfaa_length_aligned_comm_new. ((((forall pfrep_power_aligned_comm_new_witness_common_left pfrep_left_aligned_comm_new_witness_common_left pfrep_right_aligned_comm_new_witness_common_left. ((exists pfrep_position_aligned_comm_new_witness_common_leftfirst. ((pfrep_position_aligned_comm_new_witness_common_leftfirst+S (pfrep_power_aligned_comm_new_witness_common_left)=(M)) /\\ ((((exists ff_h_pfp_aligned_comm_new_witness_common_leftfirstentry. ff_h_pfp_aligned_comm_new_witness_common_leftfirstentry + S (pfrep_left_aligned_comm_new_witness_common_left) = S ((S (pfrep_position_aligned_comm_new_witness_common_leftfirst)) * bc)) /\\ exists ff_q_pfp_aligned_comm_new_witness_common_leftfirstentry. bb = ff_q_pfp_aligned_comm_new_witness_common_leftfirstentry * S ((S (pfrep_position_aligned_comm_new_witness_common_leftfirst)) * bc) + (pfrep_left_aligned_comm_new_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_comm_new_witness_common_leftfirstoutside. pfrep_gap_aligned_comm_new_witness_common_leftfirstoutside+(M)=(pfrep_power_aligned_comm_new_witness_common_left)) /\\ (((pfrep_left_aligned_comm_new_witness_common_left)=0))))) -> ((exists pfrep_position_aligned_comm_new_witness_common_leftsecond. ((pfrep_position_aligned_comm_new_witness_common_leftsecond+S (pfrep_power_aligned_comm_new_witness_common_left)=(pfaa_length_aligned_comm_new)) /\\ ((((exists ff_h_pfp_aligned_comm_new_witness_common_leftsecondentry. ff_h_pfp_aligned_comm_new_witness_common_leftsecondentry + S (pfrep_right_aligned_comm_new_witness_common_left) = S ((S (pfrep_position_aligned_comm_new_witness_common_leftsecond)) * pfaa_left_c_aligned_comm_new)) /\\ exists ff_q_pfp_aligned_comm_new_witness_common_leftsecondentry. pfaa_left_b_aligned_comm_new = ff_q_pfp_aligned_comm_new_witness_common_leftsecondentry * S ((S (pfrep_position_aligned_comm_new_witness_common_leftsecond)) * pfaa_left_c_aligned_comm_new) + (pfrep_right_aligned_comm_new_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_comm_new_witness_common_leftsecondoutside. pfrep_gap_aligned_comm_new_witness_common_leftsecondoutside+(pfaa_length_aligned_comm_new)=(pfrep_power_aligned_comm_new_witness_common_left)) /\\ (((pfrep_right_aligned_comm_new_witness_common_left)=0))))) -> pfrep_left_aligned_comm_new_witness_common_left=pfrep_right_aligned_comm_new_witness_common_left) /\\ ((forall pfrep_power_aligned_comm_new_witness_common_right pfrep_left_aligned_comm_new_witness_common_right pfrep_right_aligned_comm_new_witness_common_right. ((exists pfrep_position_aligned_comm_new_witness_common_rightfirst. ((pfrep_position_aligned_comm_new_witness_common_rightfirst+S (pfrep_power_aligned_comm_new_witness_common_right)=(L)) /\\ ((((exists ff_h_pfp_aligned_comm_new_witness_common_rightfirstentry. ff_h_pfp_aligned_comm_new_witness_common_rightfirstentry + S (pfrep_left_aligned_comm_new_witness_common_right) = S ((S (pfrep_position_aligned_comm_new_witness_common_rightfirst)) * ac)) /\\ exists ff_q_pfp_aligned_comm_new_witness_common_rightfirstentry. ab = ff_q_pfp_aligned_comm_new_witness_common_rightfirstentry * S ((S (pfrep_position_aligned_comm_new_witness_common_rightfirst)) * ac) + (pfrep_left_aligned_comm_new_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_comm_new_witness_common_rightfirstoutside. pfrep_gap_aligned_comm_new_witness_common_rightfirstoutside+(L)=(pfrep_power_aligned_comm_new_witness_common_right)) /\\ (((pfrep_left_aligned_comm_new_witness_common_right)=0))))) -> ((exists pfrep_position_aligned_comm_new_witness_common_rightsecond. ((pfrep_position_aligned_comm_new_witness_common_rightsecond+S (pfrep_power_aligned_comm_new_witness_common_right)=(pfaa_length_aligned_comm_new)) /\\ ((((exists ff_h_pfp_aligned_comm_new_witness_common_rightsecondentry. ff_h_pfp_aligned_comm_new_witness_common_rightsecondentry + S (pfrep_right_aligned_comm_new_witness_common_right) = S ((S (pfrep_position_aligned_comm_new_witness_common_rightsecond)) * pfaa_right_c_aligned_comm_new)) /\\ exists ff_q_pfp_aligned_comm_new_witness_common_rightsecondentry. pfaa_right_b_aligned_comm_new = ff_q_pfp_aligned_comm_new_witness_common_rightsecondentry * S ((S (pfrep_position_aligned_comm_new_witness_common_rightsecond)) * pfaa_right_c_aligned_comm_new) + (pfrep_right_aligned_comm_new_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_comm_new_witness_common_rightsecondoutside. pfrep_gap_aligned_comm_new_witness_common_rightsecondoutside+(pfaa_length_aligned_comm_new)=(pfrep_power_aligned_comm_new_witness_common_right)) /\\ (((pfrep_right_aligned_comm_new_witness_common_right)=0))))) -> pfrep_left_aligned_comm_new_witness_common_right=pfrep_right_aligned_comm_new_witness_common_right)))) /\\ (((forall pfp_index_aligned_comm_new_witness_operation. (exists pfa_gap_aligned_comm_new_witness_operationindex. pfa_gap_aligned_comm_new_witness_operationindex + S (pfp_index_aligned_comm_new_witness_operation) = (pfaa_length_aligned_comm_new)) -> exists pfp_left_aligned_comm_new_witness_operation pfp_right_aligned_comm_new_witness_operation pfp_value_aligned_comm_new_witness_operation. ((((exists ff_h_pfp_aligned_comm_new_witness_operationleft. ff_h_pfp_aligned_comm_new_witness_operationleft + S (pfp_left_aligned_comm_new_witness_operation) = S ((S (pfp_index_aligned_comm_new_witness_operation)) * pfaa_left_c_aligned_comm_new)) /\\ exists ff_q_pfp_aligned_comm_new_witness_operationleft. pfaa_left_b_aligned_comm_new = ff_q_pfp_aligned_comm_new_witness_operationleft * S ((S (pfp_index_aligned_comm_new_witness_operation)) * pfaa_left_c_aligned_comm_new) + (pfp_left_aligned_comm_new_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_comm_new_witness_operationright. ff_h_pfp_aligned_comm_new_witness_operationright + S (pfp_right_aligned_comm_new_witness_operation) = S ((S (pfp_index_aligned_comm_new_witness_operation)) * pfaa_right_c_aligned_comm_new)) /\\ exists ff_q_pfp_aligned_comm_new_witness_operationright. pfaa_right_b_aligned_comm_new = ff_q_pfp_aligned_comm_new_witness_operationright * S ((S (pfp_index_aligned_comm_new_witness_operation)) * pfaa_right_c_aligned_comm_new) + (pfp_right_aligned_comm_new_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_comm_new_witness_operationtarget. ff_h_pfp_aligned_comm_new_witness_operationtarget + S (pfp_value_aligned_comm_new_witness_operation) = S ((S (pfp_index_aligned_comm_new_witness_operation)) * pfaa_sum_c_aligned_comm_new)) /\\ exists ff_q_pfp_aligned_comm_new_witness_operationtarget. pfaa_sum_b_aligned_comm_new = ff_q_pfp_aligned_comm_new_witness_operationtarget * S ((S (pfp_index_aligned_comm_new_witness_operation)) * pfaa_sum_c_aligned_comm_new) + (pfp_value_aligned_comm_new_witness_operation))) /\\ ((((exists pfa_gap_aligned_comm_new_witness_operationoperationleft. pfa_gap_aligned_comm_new_witness_operationoperationleft + S (pfp_left_aligned_comm_new_witness_operation) = (p)) /\\ (((exists pfa_gap_aligned_comm_new_witness_operationoperationright. pfa_gap_aligned_comm_new_witness_operationoperationright + S (pfp_right_aligned_comm_new_witness_operation) = (p)) /\\ ((((exists pfa_gap_aligned_comm_new_witness_operationoperationresultbound. pfa_gap_aligned_comm_new_witness_operationoperationresultbound + S (pfp_value_aligned_comm_new_witness_operation) = (p)) /\\ ((exists pfa_offset_left_aligned_comm_new_witness_operationoperationresultcongruence pfa_offset_right_aligned_comm_new_witness_operationoperationresultcongruence. ((pfp_left_aligned_comm_new_witness_operation) + (pfp_right_aligned_comm_new_witness_operation)) + (p) * pfa_offset_left_aligned_comm_new_witness_operationoperationresultcongruence = (pfp_value_aligned_comm_new_witness_operation) + (p) * pfa_offset_right_aligned_comm_new_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_aligned_comm_new_witness_output pfrep_left_aligned_comm_new_witness_output pfrep_right_aligned_comm_new_witness_output. ((exists pfrep_position_aligned_comm_new_witness_outputfirst. ((pfrep_position_aligned_comm_new_witness_outputfirst+S (pfrep_power_aligned_comm_new_witness_output)=(pfaa_length_aligned_comm_new)) /\\ ((((exists ff_h_pfp_aligned_comm_new_witness_outputfirstentry. ff_h_pfp_aligned_comm_new_witness_outputfirstentry + S (pfrep_left_aligned_comm_new_witness_output) = S ((S (pfrep_position_aligned_comm_new_witness_outputfirst)) * pfaa_sum_c_aligned_comm_new)) /\\ exists ff_q_pfp_aligned_comm_new_witness_outputfirstentry. pfaa_sum_b_aligned_comm_new = ff_q_pfp_aligned_comm_new_witness_outputfirstentry * S ((S (pfrep_position_aligned_comm_new_witness_outputfirst)) * pfaa_sum_c_aligned_comm_new) + (pfrep_left_aligned_comm_new_witness_output)))))) \\/ (((exists pfrep_gap_aligned_comm_new_witness_outputfirstoutside. pfrep_gap_aligned_comm_new_witness_outputfirstoutside+(pfaa_length_aligned_comm_new)=(pfrep_power_aligned_comm_new_witness_output)) /\\ (((pfrep_left_aligned_comm_new_witness_output)=0))))) -> ((exists pfrep_position_aligned_comm_new_witness_outputsecond. ((pfrep_position_aligned_comm_new_witness_outputsecond+S (pfrep_power_aligned_comm_new_witness_output)=(N)) /\\ ((((exists ff_h_pfp_aligned_comm_new_witness_outputsecondentry. ff_h_pfp_aligned_comm_new_witness_outputsecondentry + S (pfrep_right_aligned_comm_new_witness_output) = S ((S (pfrep_position_aligned_comm_new_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_aligned_comm_new_witness_outputsecondentry. rb = ff_q_pfp_aligned_comm_new_witness_outputsecondentry * S ((S (pfrep_position_aligned_comm_new_witness_outputsecond)) * rc) + (pfrep_right_aligned_comm_new_witness_output)))))) \\/ (((exists pfrep_gap_aligned_comm_new_witness_outputsecondoutside. pfrep_gap_aligned_comm_new_witness_outputsecondoutside+(N)=(pfrep_power_aligned_comm_new_witness_output)) /\\ (((pfrep_right_aligned_comm_new_witness_output)=0))))) -> pfrep_left_aligned_comm_new_witness_output=pfrep_right_aligned_comm_new_witness_output)))))))))))))", "statement_sha256": 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hr_right_right_right_witness_witness_witness", "cases hr_right_right_right_witness_witness_witness_witness", "cases hr_right_right_right_witness_witness_witness_witness_witness", "cases hr_right_right_right_witness_witness_witness_witness_witness_witness", "cases hr_right_right_right_witness_witness_witness_witness_witness_witness_witness", "cases hr_right_right_right_witness_witness_witness_witness_witness_witness_witness_right", "cases hs", "cases hs_right", "cases hs_right_right", "cases hs_right_right_right", "cases hs_right_right_right_witness", "cases hs_right_right_right_witness_witness", "cases hs_right_right_right_witness_witness_witness", "cases hs_right_right_right_witness_witness_witness_witness", "cases hs_right_right_right_witness_witness_witness_witness_witness", "cases hs_right_right_right_witness_witness_witness_witness_witness_witness", "cases hs_right_right_right_witness_witness_witness_witness_witness_witness_witness", "cases hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_right", "have hc : ((forall pfrep_power_aligned_function_left pfrep_left_aligned_function_left pfrep_right_aligned_function_left. ((exists pfrep_position_aligned_function_leftfirst. ((pfrep_position_aligned_function_leftfirst+S (pfrep_power_aligned_function_left)=(x6)) /\\ ((((exists ff_h_pfp_aligned_function_leftfirstentry. ff_h_pfp_aligned_function_leftfirstentry + S (pfrep_left_aligned_function_left) = S ((S (pfrep_position_aligned_function_leftfirst)) * x1)) /\\ exists ff_q_pfp_aligned_function_leftfirstentry. x = ff_q_pfp_aligned_function_leftfirstentry * S ((S (pfrep_position_aligned_function_leftfirst)) * x1) + (pfrep_left_aligned_function_left)))))) \\/ (((exists pfrep_gap_aligned_function_leftfirstoutside. pfrep_gap_aligned_function_leftfirstoutside+(x6)=(pfrep_power_aligned_function_left)) /\\ (((pfrep_left_aligned_function_left)=0))))) -> ((exists pfrep_position_aligned_function_leftsecond. ((pfrep_position_aligned_function_leftsecond+S (pfrep_power_aligned_function_left)=(x13)) /\\ ((((exists ff_h_pfp_aligned_function_leftsecondentry. ff_h_pfp_aligned_function_leftsecondentry + S (pfrep_right_aligned_function_left) = S ((S (pfrep_position_aligned_function_leftsecond)) * x8)) /\\ exists ff_q_pfp_aligned_function_leftsecondentry. x7 = ff_q_pfp_aligned_function_leftsecondentry * S ((S (pfrep_position_aligned_function_leftsecond)) * x8) + (pfrep_right_aligned_function_left)))))) \\/ (((exists pfrep_gap_aligned_function_leftsecondoutside. pfrep_gap_aligned_function_leftsecondoutside+(x13)=(pfrep_power_aligned_function_left)) /\\ (((pfrep_right_aligned_function_left)=0))))) -> pfrep_left_aligned_function_left=pfrep_right_aligned_function_left) /\\ ((forall pfrep_power_aligned_function_right pfrep_left_aligned_function_right pfrep_right_aligned_function_right. ((exists pfrep_position_aligned_function_rightfirst. ((pfrep_position_aligned_function_rightfirst+S (pfrep_power_aligned_function_right)=(x6)) /\\ ((((exists ff_h_pfp_aligned_function_rightfirstentry. ff_h_pfp_aligned_function_rightfirstentry + S (pfrep_left_aligned_function_right) = S ((S (pfrep_position_aligned_function_rightfirst)) * x3)) /\\ exists ff_q_pfp_aligned_function_rightfirstentry. x2 = ff_q_pfp_aligned_function_rightfirstentry * S ((S (pfrep_position_aligned_function_rightfirst)) * x3) + (pfrep_left_aligned_function_right)))))) \\/ (((exists pfrep_gap_aligned_function_rightfirstoutside. pfrep_gap_aligned_function_rightfirstoutside+(x6)=(pfrep_power_aligned_function_right)) /\\ (((pfrep_left_aligned_function_right)=0))))) -> ((exists pfrep_position_aligned_function_rightsecond. ((pfrep_position_aligned_function_rightsecond+S (pfrep_power_aligned_function_right)=(x13)) /\\ ((((exists ff_h_pfp_aligned_function_rightsecondentry. ff_h_pfp_aligned_function_rightsecondentry + S (pfrep_right_aligned_function_right) = S ((S (pfrep_position_aligned_function_rightsecond)) * x10)) /\\ exists ff_q_pfp_aligned_function_rightsecondentry. x9 = ff_q_pfp_aligned_function_rightsecondentry * S ((S (pfrep_position_aligned_function_rightsecond)) * x10) + (pfrep_right_aligned_function_right)))))) \\/ (((exists pfrep_gap_aligned_function_rightsecondoutside. pfrep_gap_aligned_function_rightsecondoutside+(x13)=(pfrep_power_aligned_function_right)) /\\ (((pfrep_right_aligned_function_right)=0))))) -> pfrep_left_aligned_function_right=pfrep_right_aligned_function_right)))", "specialize prime_field_polynomial_common_representatives_functional (ab)", "specialize prime_field_polynomial_common_representatives_functional (ac)", "specialize prime_field_polynomial_common_representatives_functional (L)", "specialize prime_field_polynomial_common_representatives_functional (bb)", "specialize prime_field_polynomial_common_representatives_functional (bc)", "specialize prime_field_polynomial_common_representatives_functional (M)", "specialize prime_field_polynomial_common_representatives_functional (x)", "specialize prime_field_polynomial_common_representatives_functional (x1)", "specialize prime_field_polynomial_common_representatives_functional (x2)", "specialize prime_field_polynomial_common_representatives_functional (x3)", "specialize prime_field_polynomial_common_representatives_functional (x6)", "specialize prime_field_polynomial_common_representatives_functional (x7)", "specialize prime_field_polynomial_common_representatives_functional (x8)", "specialize prime_field_polynomial_common_representatives_functional (x9)", "specialize prime_field_polynomial_common_representatives_functional (x10)", "specialize prime_field_polynomial_common_representatives_functional (x13)", "apply prime_field_polynomial_common_representatives_functional", "exact hr_right_right_right_witness_witness_witness_witness_witness_witness_witness_left", "exact hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_left", "cases hc", "have hm : forall pfrep_power_aligned_function_sums pfrep_left_aligned_function_sums pfrep_right_aligned_function_sums. ((exists pfrep_position_aligned_function_sumsfirst. ((pfrep_position_aligned_function_sumsfirst+S (pfrep_power_aligned_function_sums)=(x6)) /\\ ((((exists ff_h_pfp_aligned_function_sumsfirstentry. ff_h_pfp_aligned_function_sumsfirstentry + S (pfrep_left_aligned_function_sums) = S ((S (pfrep_position_aligned_function_sumsfirst)) * x5)) /\\ exists ff_q_pfp_aligned_function_sumsfirstentry. x4 = ff_q_pfp_aligned_function_sumsfirstentry * S ((S (pfrep_position_aligned_function_sumsfirst)) * x5) + (pfrep_left_aligned_function_sums)))))) \\/ (((exists pfrep_gap_aligned_function_sumsfirstoutside. pfrep_gap_aligned_function_sumsfirstoutside+(x6)=(pfrep_power_aligned_function_sums)) /\\ (((pfrep_left_aligned_function_sums)=0))))) -> ((exists pfrep_position_aligned_function_sumssecond. ((pfrep_position_aligned_function_sumssecond+S (pfrep_power_aligned_function_sums)=(x13)) /\\ ((((exists ff_h_pfp_aligned_function_sumssecondentry. ff_h_pfp_aligned_function_sumssecondentry + S (pfrep_right_aligned_function_sums) = S ((S (pfrep_position_aligned_function_sumssecond)) * x12)) /\\ exists ff_q_pfp_aligned_function_sumssecondentry. x11 = ff_q_pfp_aligned_function_sumssecondentry * S ((S (pfrep_position_aligned_function_sumssecond)) * x12) + (pfrep_right_aligned_function_sums)))))) \\/ (((exists pfrep_gap_aligned_function_sumssecondoutside. pfrep_gap_aligned_function_sumssecondoutside+(x13)=(pfrep_power_aligned_function_sums)) /\\ (((pfrep_right_aligned_function_sums)=0))))) -> pfrep_left_aligned_function_sums=pfrep_right_aligned_function_sums", "specialize prime_field_polynomial_add_equivalent_congruent (p)", "specialize prime_field_polynomial_add_equivalent_congruent (x)", "specialize prime_field_polynomial_add_equivalent_congruent (x1)", "specialize prime_field_polynomial_add_equivalent_congruent (x2)", "specialize prime_field_polynomial_add_equivalent_congruent (x3)", "specialize prime_field_polynomial_add_equivalent_congruent (x4)", "specialize prime_field_polynomial_add_equivalent_congruent (x5)", "specialize prime_field_polynomial_add_equivalent_congruent (x6)", "specialize prime_field_polynomial_add_equivalent_congruent (x7)", "specialize prime_field_polynomial_add_equivalent_congruent (x8)", "specialize prime_field_polynomial_add_equivalent_congruent (x9)", "specialize prime_field_polynomial_add_equivalent_congruent (x10)", "specialize prime_field_polynomial_add_equivalent_congruent (x11)", "specialize prime_field_polynomial_add_equivalent_congruent (x12)", "specialize prime_field_polynomial_add_equivalent_congruent (x13)", "apply prime_field_polynomial_add_equivalent_congruent", "exact hp", "exact hc_left", "exact hc_right", "exact hr_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_left", "exact hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_left", "have hreverse : forall pfrep_power_aligned_function_reverse pfrep_left_aligned_function_reverse pfrep_right_aligned_function_reverse. ((exists pfrep_position_aligned_function_reversefirst. ((pfrep_position_aligned_function_reversefirst+S (pfrep_power_aligned_function_reverse)=(N)) /\\ ((((exists ff_h_pfp_aligned_function_reversefirstentry. ff_h_pfp_aligned_function_reversefirstentry + S (pfrep_left_aligned_function_reverse) = S ((S (pfrep_position_aligned_function_reversefirst)) * rc)) /\\ exists ff_q_pfp_aligned_function_reversefirstentry. rb = ff_q_pfp_aligned_function_reversefirstentry * S ((S (pfrep_position_aligned_function_reversefirst)) * rc) + (pfrep_left_aligned_function_reverse)))))) \\/ (((exists pfrep_gap_aligned_function_reversefirstoutside. pfrep_gap_aligned_function_reversefirstoutside+(N)=(pfrep_power_aligned_function_reverse)) /\\ (((pfrep_left_aligned_function_reverse)=0))))) -> ((exists pfrep_position_aligned_function_reversesecond. ((pfrep_position_aligned_function_reversesecond+S (pfrep_power_aligned_function_reverse)=(x6)) /\\ ((((exists ff_h_pfp_aligned_function_reversesecondentry. ff_h_pfp_aligned_function_reversesecondentry + S (pfrep_right_aligned_function_reverse) = S ((S (pfrep_position_aligned_function_reversesecond)) * x5)) /\\ exists ff_q_pfp_aligned_function_reversesecondentry. x4 = ff_q_pfp_aligned_function_reversesecondentry * S ((S (pfrep_position_aligned_function_reversesecond)) * x5) + (pfrep_right_aligned_function_reverse)))))) \\/ (((exists pfrep_gap_aligned_function_reversesecondoutside. pfrep_gap_aligned_function_reversesecondoutside+(x6)=(pfrep_power_aligned_function_reverse)) /\\ (((pfrep_right_aligned_function_reverse)=0))))) -> pfrep_left_aligned_function_reverse=pfrep_right_aligned_function_reverse", "specialize prime_field_polynomial_equivalent_symmetric (x4)", "specialize prime_field_polynomial_equivalent_symmetric (x5)", "specialize prime_field_polynomial_equivalent_symmetric (x6)", "specialize prime_field_polynomial_equivalent_symmetric (rb)", "specialize prime_field_polynomial_equivalent_symmetric (rc)", "specialize prime_field_polynomial_equivalent_symmetric (N)", "apply prime_field_polynomial_equivalent_symmetric", "exact hr_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right", "have hmiddle : forall pfrep_power_aligned_function_middle pfrep_left_aligned_function_middle pfrep_right_aligned_function_middle. ((exists pfrep_position_aligned_function_middlefirst. ((pfrep_position_aligned_function_middlefirst+S (pfrep_power_aligned_function_middle)=(N)) /\\ ((((exists ff_h_pfp_aligned_function_middlefirstentry. ff_h_pfp_aligned_function_middlefirstentry + S (pfrep_left_aligned_function_middle) = S ((S (pfrep_position_aligned_function_middlefirst)) * rc)) /\\ exists ff_q_pfp_aligned_function_middlefirstentry. rb = ff_q_pfp_aligned_function_middlefirstentry * S ((S (pfrep_position_aligned_function_middlefirst)) * rc) + (pfrep_left_aligned_function_middle)))))) \\/ (((exists pfrep_gap_aligned_function_middlefirstoutside. pfrep_gap_aligned_function_middlefirstoutside+(N)=(pfrep_power_aligned_function_middle)) /\\ (((pfrep_left_aligned_function_middle)=0))))) -> ((exists pfrep_position_aligned_function_middlesecond. ((pfrep_position_aligned_function_middlesecond+S (pfrep_power_aligned_function_middle)=(x13)) /\\ ((((exists ff_h_pfp_aligned_function_middlesecondentry. ff_h_pfp_aligned_function_middlesecondentry + S (pfrep_right_aligned_function_middle) = S ((S (pfrep_position_aligned_function_middlesecond)) * x12)) /\\ exists ff_q_pfp_aligned_function_middlesecondentry. x11 = ff_q_pfp_aligned_function_middlesecondentry * S ((S (pfrep_position_aligned_function_middlesecond)) * x12) + (pfrep_right_aligned_function_middle)))))) \\/ (((exists pfrep_gap_aligned_function_middlesecondoutside. pfrep_gap_aligned_function_middlesecondoutside+(x13)=(pfrep_power_aligned_function_middle)) /\\ (((pfrep_right_aligned_function_middle)=0))))) -> pfrep_left_aligned_function_middle=pfrep_right_aligned_function_middle", "specialize prime_field_polynomial_equivalent_transitive (rb)", "specialize prime_field_polynomial_equivalent_transitive (rc)", "specialize prime_field_polynomial_equivalent_transitive (N)", "specialize prime_field_polynomial_equivalent_transitive (x4)", "specialize prime_field_polynomial_equivalent_transitive (x5)", "specialize prime_field_polynomial_equivalent_transitive (x6)", "specialize prime_field_polynomial_equivalent_transitive (x11)", "specialize prime_field_polynomial_equivalent_transitive (x12)", "specialize prime_field_polynomial_equivalent_transitive (x13)", "apply prime_field_polynomial_equivalent_transitive", "exact hreverse", "exact hm", "specialize prime_field_polynomial_equivalent_transitive (rb)", "specialize prime_field_polynomial_equivalent_transitive (rc)", "specialize prime_field_polynomial_equivalent_transitive (N)", "specialize prime_field_polynomial_equivalent_transitive (x11)", "specialize prime_field_polynomial_equivalent_transitive (x12)", "specialize prime_field_polynomial_equivalent_transitive (x13)", "specialize prime_field_polynomial_equivalent_transitive (sb)", "specialize prime_field_polynomial_equivalent_transitive (sc)", "specialize prime_field_polynomial_equivalent_transitive (J)", "apply prime_field_polynomial_equivalent_transitive", "exact hmiddle", "exact hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right" ], "script_sha256": "99e72bd84b3117d195db095d6d4f701d7d3cc20dd96eab8247c960f0127c2917", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_aligned_add_candidate.py", "sha256": "a05bb4f5c4230ca05f51690d3ab82e33ff4596af65176874e25fbe38cf87a0db" }, "statement": "forall p ab ac L bb bc M rb rc N sb sc J. (~((p) = 1) /\\ forall pfa_factor_left_aligned_function_prime pfa_factor_right_aligned_function_prime. (p) = pfa_factor_left_aligned_function_prime * pfa_factor_right_aligned_function_prime -> pfa_factor_left_aligned_function_prime = 1 \\/ pfa_factor_right_aligned_function_prime = 1) -> (((forall fom_index_pfp_aligned_function_first_left_bounded. (exists fom_gap_pfp_aligned_function_first_left_bounded_index_bound. fom_gap_pfp_aligned_function_first_left_bounded_index_bound + S (fom_index_pfp_aligned_function_first_left_bounded) = L) -> exists fom_value_pfp_aligned_function_first_left_bounded. ((((exists fom_beta_height_pfp_aligned_function_first_left_bounded_entry. fom_beta_height_pfp_aligned_function_first_left_bounded_entry + S (fom_value_pfp_aligned_function_first_left_bounded) = S ((S (fom_index_pfp_aligned_function_first_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_aligned_function_first_left_bounded_entry. ab = fom_beta_quotient_pfp_aligned_function_first_left_bounded_entry * S ((S (fom_index_pfp_aligned_function_first_left_bounded)) * ac) + (fom_value_pfp_aligned_function_first_left_bounded))) /\\ (exists fom_gap_pfp_aligned_function_first_left_bounded_value_bound. fom_gap_pfp_aligned_function_first_left_bounded_value_bound + S (fom_value_pfp_aligned_function_first_left_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_function_first_right_bounded. (exists fom_gap_pfp_aligned_function_first_right_bounded_index_bound. fom_gap_pfp_aligned_function_first_right_bounded_index_bound + S (fom_index_pfp_aligned_function_first_right_bounded) = M) -> exists fom_value_pfp_aligned_function_first_right_bounded. ((((exists fom_beta_height_pfp_aligned_function_first_right_bounded_entry. fom_beta_height_pfp_aligned_function_first_right_bounded_entry + S (fom_value_pfp_aligned_function_first_right_bounded) = S ((S (fom_index_pfp_aligned_function_first_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_aligned_function_first_right_bounded_entry. bb = fom_beta_quotient_pfp_aligned_function_first_right_bounded_entry * S ((S (fom_index_pfp_aligned_function_first_right_bounded)) * bc) + (fom_value_pfp_aligned_function_first_right_bounded))) /\\ (exists fom_gap_pfp_aligned_function_first_right_bounded_value_bound. fom_gap_pfp_aligned_function_first_right_bounded_value_bound + S (fom_value_pfp_aligned_function_first_right_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_function_first_result_bounded. (exists fom_gap_pfp_aligned_function_first_result_bounded_index_bound. fom_gap_pfp_aligned_function_first_result_bounded_index_bound + S (fom_index_pfp_aligned_function_first_result_bounded) = N) -> exists fom_value_pfp_aligned_function_first_result_bounded. ((((exists fom_beta_height_pfp_aligned_function_first_result_bounded_entry. fom_beta_height_pfp_aligned_function_first_result_bounded_entry + S (fom_value_pfp_aligned_function_first_result_bounded) = S ((S (fom_index_pfp_aligned_function_first_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_aligned_function_first_result_bounded_entry. rb = fom_beta_quotient_pfp_aligned_function_first_result_bounded_entry * S ((S (fom_index_pfp_aligned_function_first_result_bounded)) * rc) + (fom_value_pfp_aligned_function_first_result_bounded))) /\\ (exists fom_gap_pfp_aligned_function_first_result_bounded_value_bound. fom_gap_pfp_aligned_function_first_result_bounded_value_bound + S (fom_value_pfp_aligned_function_first_result_bounded) = p))) /\\ ((exists pfaa_left_b_aligned_function_first pfaa_left_c_aligned_function_first pfaa_right_b_aligned_function_first pfaa_right_c_aligned_function_first pfaa_sum_b_aligned_function_first pfaa_sum_c_aligned_function_first pfaa_length_aligned_function_first. ((((forall pfrep_power_aligned_function_first_witness_common_left pfrep_left_aligned_function_first_witness_common_left pfrep_right_aligned_function_first_witness_common_left. ((exists pfrep_position_aligned_function_first_witness_common_leftfirst. ((pfrep_position_aligned_function_first_witness_common_leftfirst+S (pfrep_power_aligned_function_first_witness_common_left)=(L)) /\\ ((((exists ff_h_pfp_aligned_function_first_witness_common_leftfirstentry. ff_h_pfp_aligned_function_first_witness_common_leftfirstentry + S (pfrep_left_aligned_function_first_witness_common_left) = S ((S (pfrep_position_aligned_function_first_witness_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_aligned_function_first_witness_common_leftfirstentry. ab = ff_q_pfp_aligned_function_first_witness_common_leftfirstentry * S ((S (pfrep_position_aligned_function_first_witness_common_leftfirst)) * ac) + (pfrep_left_aligned_function_first_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_function_first_witness_common_leftfirstoutside. pfrep_gap_aligned_function_first_witness_common_leftfirstoutside+(L)=(pfrep_power_aligned_function_first_witness_common_left)) /\\ (((pfrep_left_aligned_function_first_witness_common_left)=0))))) -> ((exists pfrep_position_aligned_function_first_witness_common_leftsecond. ((pfrep_position_aligned_function_first_witness_common_leftsecond+S (pfrep_power_aligned_function_first_witness_common_left)=(pfaa_length_aligned_function_first)) /\\ ((((exists ff_h_pfp_aligned_function_first_witness_common_leftsecondentry. ff_h_pfp_aligned_function_first_witness_common_leftsecondentry + S (pfrep_right_aligned_function_first_witness_common_left) = S ((S (pfrep_position_aligned_function_first_witness_common_leftsecond)) * pfaa_left_c_aligned_function_first)) /\\ exists ff_q_pfp_aligned_function_first_witness_common_leftsecondentry. pfaa_left_b_aligned_function_first = ff_q_pfp_aligned_function_first_witness_common_leftsecondentry * S ((S (pfrep_position_aligned_function_first_witness_common_leftsecond)) * pfaa_left_c_aligned_function_first) + (pfrep_right_aligned_function_first_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_function_first_witness_common_leftsecondoutside. pfrep_gap_aligned_function_first_witness_common_leftsecondoutside+(pfaa_length_aligned_function_first)=(pfrep_power_aligned_function_first_witness_common_left)) /\\ (((pfrep_right_aligned_function_first_witness_common_left)=0))))) -> pfrep_left_aligned_function_first_witness_common_left=pfrep_right_aligned_function_first_witness_common_left) /\\ ((forall pfrep_power_aligned_function_first_witness_common_right pfrep_left_aligned_function_first_witness_common_right pfrep_right_aligned_function_first_witness_common_right. ((exists pfrep_position_aligned_function_first_witness_common_rightfirst. ((pfrep_position_aligned_function_first_witness_common_rightfirst+S (pfrep_power_aligned_function_first_witness_common_right)=(M)) /\\ ((((exists ff_h_pfp_aligned_function_first_witness_common_rightfirstentry. ff_h_pfp_aligned_function_first_witness_common_rightfirstentry + S (pfrep_left_aligned_function_first_witness_common_right) = S ((S (pfrep_position_aligned_function_first_witness_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_aligned_function_first_witness_common_rightfirstentry. bb = ff_q_pfp_aligned_function_first_witness_common_rightfirstentry * S ((S (pfrep_position_aligned_function_first_witness_common_rightfirst)) * bc) + (pfrep_left_aligned_function_first_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_function_first_witness_common_rightfirstoutside. pfrep_gap_aligned_function_first_witness_common_rightfirstoutside+(M)=(pfrep_power_aligned_function_first_witness_common_right)) /\\ (((pfrep_left_aligned_function_first_witness_common_right)=0))))) -> ((exists pfrep_position_aligned_function_first_witness_common_rightsecond. ((pfrep_position_aligned_function_first_witness_common_rightsecond+S (pfrep_power_aligned_function_first_witness_common_right)=(pfaa_length_aligned_function_first)) /\\ ((((exists ff_h_pfp_aligned_function_first_witness_common_rightsecondentry. ff_h_pfp_aligned_function_first_witness_common_rightsecondentry + S (pfrep_right_aligned_function_first_witness_common_right) = S ((S (pfrep_position_aligned_function_first_witness_common_rightsecond)) * pfaa_right_c_aligned_function_first)) /\\ exists ff_q_pfp_aligned_function_first_witness_common_rightsecondentry. pfaa_right_b_aligned_function_first = ff_q_pfp_aligned_function_first_witness_common_rightsecondentry * S ((S (pfrep_position_aligned_function_first_witness_common_rightsecond)) * pfaa_right_c_aligned_function_first) + (pfrep_right_aligned_function_first_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_function_first_witness_common_rightsecondoutside. pfrep_gap_aligned_function_first_witness_common_rightsecondoutside+(pfaa_length_aligned_function_first)=(pfrep_power_aligned_function_first_witness_common_right)) /\\ (((pfrep_right_aligned_function_first_witness_common_right)=0))))) -> pfrep_left_aligned_function_first_witness_common_right=pfrep_right_aligned_function_first_witness_common_right)))) /\\ (((forall pfp_index_aligned_function_first_witness_operation. (exists pfa_gap_aligned_function_first_witness_operationindex. pfa_gap_aligned_function_first_witness_operationindex + S (pfp_index_aligned_function_first_witness_operation) = (pfaa_length_aligned_function_first)) -> exists pfp_left_aligned_function_first_witness_operation pfp_right_aligned_function_first_witness_operation pfp_value_aligned_function_first_witness_operation. ((((exists ff_h_pfp_aligned_function_first_witness_operationleft. ff_h_pfp_aligned_function_first_witness_operationleft + S (pfp_left_aligned_function_first_witness_operation) = S ((S (pfp_index_aligned_function_first_witness_operation)) * pfaa_left_c_aligned_function_first)) /\\ exists ff_q_pfp_aligned_function_first_witness_operationleft. pfaa_left_b_aligned_function_first = ff_q_pfp_aligned_function_first_witness_operationleft * S ((S (pfp_index_aligned_function_first_witness_operation)) * pfaa_left_c_aligned_function_first) + (pfp_left_aligned_function_first_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_function_first_witness_operationright. ff_h_pfp_aligned_function_first_witness_operationright + S (pfp_right_aligned_function_first_witness_operation) = S ((S (pfp_index_aligned_function_first_witness_operation)) * pfaa_right_c_aligned_function_first)) /\\ exists ff_q_pfp_aligned_function_first_witness_operationright. pfaa_right_b_aligned_function_first = ff_q_pfp_aligned_function_first_witness_operationright * S ((S (pfp_index_aligned_function_first_witness_operation)) * pfaa_right_c_aligned_function_first) + (pfp_right_aligned_function_first_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_function_first_witness_operationtarget. ff_h_pfp_aligned_function_first_witness_operationtarget + S (pfp_value_aligned_function_first_witness_operation) = S ((S (pfp_index_aligned_function_first_witness_operation)) * pfaa_sum_c_aligned_function_first)) /\\ exists ff_q_pfp_aligned_function_first_witness_operationtarget. pfaa_sum_b_aligned_function_first = ff_q_pfp_aligned_function_first_witness_operationtarget * S ((S (pfp_index_aligned_function_first_witness_operation)) * pfaa_sum_c_aligned_function_first) + (pfp_value_aligned_function_first_witness_operation))) /\\ ((((exists pfa_gap_aligned_function_first_witness_operationoperationleft. pfa_gap_aligned_function_first_witness_operationoperationleft + S (pfp_left_aligned_function_first_witness_operation) = (p)) /\\ (((exists pfa_gap_aligned_function_first_witness_operationoperationright. pfa_gap_aligned_function_first_witness_operationoperationright + S (pfp_right_aligned_function_first_witness_operation) = (p)) /\\ ((((exists pfa_gap_aligned_function_first_witness_operationoperationresultbound. pfa_gap_aligned_function_first_witness_operationoperationresultbound + S (pfp_value_aligned_function_first_witness_operation) = (p)) /\\ ((exists pfa_offset_left_aligned_function_first_witness_operationoperationresultcongruence pfa_offset_right_aligned_function_first_witness_operationoperationresultcongruence. ((pfp_left_aligned_function_first_witness_operation) + (pfp_right_aligned_function_first_witness_operation)) + (p) * pfa_offset_left_aligned_function_first_witness_operationoperationresultcongruence = (pfp_value_aligned_function_first_witness_operation) + (p) * pfa_offset_right_aligned_function_first_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_aligned_function_first_witness_output pfrep_left_aligned_function_first_witness_output pfrep_right_aligned_function_first_witness_output. ((exists pfrep_position_aligned_function_first_witness_outputfirst. ((pfrep_position_aligned_function_first_witness_outputfirst+S (pfrep_power_aligned_function_first_witness_output)=(pfaa_length_aligned_function_first)) /\\ ((((exists ff_h_pfp_aligned_function_first_witness_outputfirstentry. ff_h_pfp_aligned_function_first_witness_outputfirstentry + S (pfrep_left_aligned_function_first_witness_output) = S ((S (pfrep_position_aligned_function_first_witness_outputfirst)) * pfaa_sum_c_aligned_function_first)) /\\ exists ff_q_pfp_aligned_function_first_witness_outputfirstentry. pfaa_sum_b_aligned_function_first = ff_q_pfp_aligned_function_first_witness_outputfirstentry * S ((S (pfrep_position_aligned_function_first_witness_outputfirst)) * pfaa_sum_c_aligned_function_first) + (pfrep_left_aligned_function_first_witness_output)))))) \\/ (((exists pfrep_gap_aligned_function_first_witness_outputfirstoutside. pfrep_gap_aligned_function_first_witness_outputfirstoutside+(pfaa_length_aligned_function_first)=(pfrep_power_aligned_function_first_witness_output)) /\\ (((pfrep_left_aligned_function_first_witness_output)=0))))) -> ((exists pfrep_position_aligned_function_first_witness_outputsecond. ((pfrep_position_aligned_function_first_witness_outputsecond+S (pfrep_power_aligned_function_first_witness_output)=(N)) /\\ ((((exists ff_h_pfp_aligned_function_first_witness_outputsecondentry. ff_h_pfp_aligned_function_first_witness_outputsecondentry + S (pfrep_right_aligned_function_first_witness_output) = S ((S (pfrep_position_aligned_function_first_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_aligned_function_first_witness_outputsecondentry. rb = ff_q_pfp_aligned_function_first_witness_outputsecondentry * S ((S (pfrep_position_aligned_function_first_witness_outputsecond)) * rc) + (pfrep_right_aligned_function_first_witness_output)))))) \\/ (((exists pfrep_gap_aligned_function_first_witness_outputsecondoutside. pfrep_gap_aligned_function_first_witness_outputsecondoutside+(N)=(pfrep_power_aligned_function_first_witness_output)) /\\ (((pfrep_right_aligned_function_first_witness_output)=0))))) -> pfrep_left_aligned_function_first_witness_output=pfrep_right_aligned_function_first_witness_output))))))))))))) -> (((forall fom_index_pfp_aligned_function_second_left_bounded. (exists fom_gap_pfp_aligned_function_second_left_bounded_index_bound. fom_gap_pfp_aligned_function_second_left_bounded_index_bound + S (fom_index_pfp_aligned_function_second_left_bounded) = L) -> exists fom_value_pfp_aligned_function_second_left_bounded. ((((exists fom_beta_height_pfp_aligned_function_second_left_bounded_entry. fom_beta_height_pfp_aligned_function_second_left_bounded_entry + S (fom_value_pfp_aligned_function_second_left_bounded) = S ((S (fom_index_pfp_aligned_function_second_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_aligned_function_second_left_bounded_entry. ab = fom_beta_quotient_pfp_aligned_function_second_left_bounded_entry * S ((S (fom_index_pfp_aligned_function_second_left_bounded)) * ac) + (fom_value_pfp_aligned_function_second_left_bounded))) /\\ (exists fom_gap_pfp_aligned_function_second_left_bounded_value_bound. fom_gap_pfp_aligned_function_second_left_bounded_value_bound + S (fom_value_pfp_aligned_function_second_left_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_function_second_right_bounded. (exists fom_gap_pfp_aligned_function_second_right_bounded_index_bound. fom_gap_pfp_aligned_function_second_right_bounded_index_bound + S (fom_index_pfp_aligned_function_second_right_bounded) = M) -> exists fom_value_pfp_aligned_function_second_right_bounded. ((((exists fom_beta_height_pfp_aligned_function_second_right_bounded_entry. fom_beta_height_pfp_aligned_function_second_right_bounded_entry + S (fom_value_pfp_aligned_function_second_right_bounded) = S ((S (fom_index_pfp_aligned_function_second_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_aligned_function_second_right_bounded_entry. bb = fom_beta_quotient_pfp_aligned_function_second_right_bounded_entry * S ((S (fom_index_pfp_aligned_function_second_right_bounded)) * bc) + (fom_value_pfp_aligned_function_second_right_bounded))) /\\ (exists fom_gap_pfp_aligned_function_second_right_bounded_value_bound. fom_gap_pfp_aligned_function_second_right_bounded_value_bound + S (fom_value_pfp_aligned_function_second_right_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_function_second_result_bounded. (exists fom_gap_pfp_aligned_function_second_result_bounded_index_bound. fom_gap_pfp_aligned_function_second_result_bounded_index_bound + S (fom_index_pfp_aligned_function_second_result_bounded) = J) -> exists fom_value_pfp_aligned_function_second_result_bounded. ((((exists fom_beta_height_pfp_aligned_function_second_result_bounded_entry. fom_beta_height_pfp_aligned_function_second_result_bounded_entry + S (fom_value_pfp_aligned_function_second_result_bounded) = S ((S (fom_index_pfp_aligned_function_second_result_bounded)) * sc)) /\\ exists fom_beta_quotient_pfp_aligned_function_second_result_bounded_entry. sb = fom_beta_quotient_pfp_aligned_function_second_result_bounded_entry * S ((S (fom_index_pfp_aligned_function_second_result_bounded)) * sc) + (fom_value_pfp_aligned_function_second_result_bounded))) /\\ (exists fom_gap_pfp_aligned_function_second_result_bounded_value_bound. fom_gap_pfp_aligned_function_second_result_bounded_value_bound + S (fom_value_pfp_aligned_function_second_result_bounded) = p))) /\\ ((exists pfaa_left_b_aligned_function_second pfaa_left_c_aligned_function_second pfaa_right_b_aligned_function_second pfaa_right_c_aligned_function_second pfaa_sum_b_aligned_function_second pfaa_sum_c_aligned_function_second pfaa_length_aligned_function_second. ((((forall pfrep_power_aligned_function_second_witness_common_left pfrep_left_aligned_function_second_witness_common_left pfrep_right_aligned_function_second_witness_common_left. ((exists pfrep_position_aligned_function_second_witness_common_leftfirst. ((pfrep_position_aligned_function_second_witness_common_leftfirst+S (pfrep_power_aligned_function_second_witness_common_left)=(L)) /\\ ((((exists ff_h_pfp_aligned_function_second_witness_common_leftfirstentry. ff_h_pfp_aligned_function_second_witness_common_leftfirstentry + S (pfrep_left_aligned_function_second_witness_common_left) = S ((S (pfrep_position_aligned_function_second_witness_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_aligned_function_second_witness_common_leftfirstentry. ab = ff_q_pfp_aligned_function_second_witness_common_leftfirstentry * S ((S (pfrep_position_aligned_function_second_witness_common_leftfirst)) * ac) + (pfrep_left_aligned_function_second_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_function_second_witness_common_leftfirstoutside. pfrep_gap_aligned_function_second_witness_common_leftfirstoutside+(L)=(pfrep_power_aligned_function_second_witness_common_left)) /\\ (((pfrep_left_aligned_function_second_witness_common_left)=0))))) -> ((exists pfrep_position_aligned_function_second_witness_common_leftsecond. ((pfrep_position_aligned_function_second_witness_common_leftsecond+S (pfrep_power_aligned_function_second_witness_common_left)=(pfaa_length_aligned_function_second)) /\\ ((((exists ff_h_pfp_aligned_function_second_witness_common_leftsecondentry. ff_h_pfp_aligned_function_second_witness_common_leftsecondentry + S (pfrep_right_aligned_function_second_witness_common_left) = S ((S (pfrep_position_aligned_function_second_witness_common_leftsecond)) * pfaa_left_c_aligned_function_second)) /\\ exists ff_q_pfp_aligned_function_second_witness_common_leftsecondentry. pfaa_left_b_aligned_function_second = ff_q_pfp_aligned_function_second_witness_common_leftsecondentry * S ((S (pfrep_position_aligned_function_second_witness_common_leftsecond)) * pfaa_left_c_aligned_function_second) + (pfrep_right_aligned_function_second_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_function_second_witness_common_leftsecondoutside. pfrep_gap_aligned_function_second_witness_common_leftsecondoutside+(pfaa_length_aligned_function_second)=(pfrep_power_aligned_function_second_witness_common_left)) /\\ (((pfrep_right_aligned_function_second_witness_common_left)=0))))) -> pfrep_left_aligned_function_second_witness_common_left=pfrep_right_aligned_function_second_witness_common_left) /\\ ((forall pfrep_power_aligned_function_second_witness_common_right pfrep_left_aligned_function_second_witness_common_right pfrep_right_aligned_function_second_witness_common_right. ((exists pfrep_position_aligned_function_second_witness_common_rightfirst. ((pfrep_position_aligned_function_second_witness_common_rightfirst+S (pfrep_power_aligned_function_second_witness_common_right)=(M)) /\\ ((((exists ff_h_pfp_aligned_function_second_witness_common_rightfirstentry. ff_h_pfp_aligned_function_second_witness_common_rightfirstentry + S (pfrep_left_aligned_function_second_witness_common_right) = S ((S (pfrep_position_aligned_function_second_witness_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_aligned_function_second_witness_common_rightfirstentry. bb = ff_q_pfp_aligned_function_second_witness_common_rightfirstentry * S ((S (pfrep_position_aligned_function_second_witness_common_rightfirst)) * bc) + (pfrep_left_aligned_function_second_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_function_second_witness_common_rightfirstoutside. pfrep_gap_aligned_function_second_witness_common_rightfirstoutside+(M)=(pfrep_power_aligned_function_second_witness_common_right)) /\\ (((pfrep_left_aligned_function_second_witness_common_right)=0))))) -> ((exists pfrep_position_aligned_function_second_witness_common_rightsecond. ((pfrep_position_aligned_function_second_witness_common_rightsecond+S (pfrep_power_aligned_function_second_witness_common_right)=(pfaa_length_aligned_function_second)) /\\ ((((exists ff_h_pfp_aligned_function_second_witness_common_rightsecondentry. ff_h_pfp_aligned_function_second_witness_common_rightsecondentry + S (pfrep_right_aligned_function_second_witness_common_right) = S ((S (pfrep_position_aligned_function_second_witness_common_rightsecond)) * pfaa_right_c_aligned_function_second)) /\\ exists ff_q_pfp_aligned_function_second_witness_common_rightsecondentry. pfaa_right_b_aligned_function_second = ff_q_pfp_aligned_function_second_witness_common_rightsecondentry * S ((S (pfrep_position_aligned_function_second_witness_common_rightsecond)) * pfaa_right_c_aligned_function_second) + (pfrep_right_aligned_function_second_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_function_second_witness_common_rightsecondoutside. pfrep_gap_aligned_function_second_witness_common_rightsecondoutside+(pfaa_length_aligned_function_second)=(pfrep_power_aligned_function_second_witness_common_right)) /\\ (((pfrep_right_aligned_function_second_witness_common_right)=0))))) -> pfrep_left_aligned_function_second_witness_common_right=pfrep_right_aligned_function_second_witness_common_right)))) /\\ (((forall pfp_index_aligned_function_second_witness_operation. (exists pfa_gap_aligned_function_second_witness_operationindex. pfa_gap_aligned_function_second_witness_operationindex + S (pfp_index_aligned_function_second_witness_operation) = (pfaa_length_aligned_function_second)) -> exists pfp_left_aligned_function_second_witness_operation pfp_right_aligned_function_second_witness_operation pfp_value_aligned_function_second_witness_operation. ((((exists ff_h_pfp_aligned_function_second_witness_operationleft. ff_h_pfp_aligned_function_second_witness_operationleft + S (pfp_left_aligned_function_second_witness_operation) = S ((S (pfp_index_aligned_function_second_witness_operation)) * pfaa_left_c_aligned_function_second)) /\\ exists ff_q_pfp_aligned_function_second_witness_operationleft. pfaa_left_b_aligned_function_second = ff_q_pfp_aligned_function_second_witness_operationleft * S ((S (pfp_index_aligned_function_second_witness_operation)) * pfaa_left_c_aligned_function_second) + (pfp_left_aligned_function_second_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_function_second_witness_operationright. ff_h_pfp_aligned_function_second_witness_operationright + S (pfp_right_aligned_function_second_witness_operation) = S ((S (pfp_index_aligned_function_second_witness_operation)) * pfaa_right_c_aligned_function_second)) /\\ exists ff_q_pfp_aligned_function_second_witness_operationright. pfaa_right_b_aligned_function_second = ff_q_pfp_aligned_function_second_witness_operationright * S ((S (pfp_index_aligned_function_second_witness_operation)) * pfaa_right_c_aligned_function_second) + (pfp_right_aligned_function_second_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_function_second_witness_operationtarget. ff_h_pfp_aligned_function_second_witness_operationtarget + S (pfp_value_aligned_function_second_witness_operation) = S ((S (pfp_index_aligned_function_second_witness_operation)) * pfaa_sum_c_aligned_function_second)) /\\ exists ff_q_pfp_aligned_function_second_witness_operationtarget. pfaa_sum_b_aligned_function_second = ff_q_pfp_aligned_function_second_witness_operationtarget * S ((S (pfp_index_aligned_function_second_witness_operation)) * pfaa_sum_c_aligned_function_second) + (pfp_value_aligned_function_second_witness_operation))) /\\ ((((exists pfa_gap_aligned_function_second_witness_operationoperationleft. pfa_gap_aligned_function_second_witness_operationoperationleft + S (pfp_left_aligned_function_second_witness_operation) = (p)) /\\ (((exists pfa_gap_aligned_function_second_witness_operationoperationright. pfa_gap_aligned_function_second_witness_operationoperationright + S (pfp_right_aligned_function_second_witness_operation) = (p)) /\\ ((((exists pfa_gap_aligned_function_second_witness_operationoperationresultbound. pfa_gap_aligned_function_second_witness_operationoperationresultbound + S (pfp_value_aligned_function_second_witness_operation) = (p)) /\\ ((exists pfa_offset_left_aligned_function_second_witness_operationoperationresultcongruence pfa_offset_right_aligned_function_second_witness_operationoperationresultcongruence. ((pfp_left_aligned_function_second_witness_operation) + (pfp_right_aligned_function_second_witness_operation)) + (p) * pfa_offset_left_aligned_function_second_witness_operationoperationresultcongruence = (pfp_value_aligned_function_second_witness_operation) + (p) * pfa_offset_right_aligned_function_second_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_aligned_function_second_witness_output pfrep_left_aligned_function_second_witness_output pfrep_right_aligned_function_second_witness_output. ((exists pfrep_position_aligned_function_second_witness_outputfirst. ((pfrep_position_aligned_function_second_witness_outputfirst+S (pfrep_power_aligned_function_second_witness_output)=(pfaa_length_aligned_function_second)) /\\ ((((exists ff_h_pfp_aligned_function_second_witness_outputfirstentry. ff_h_pfp_aligned_function_second_witness_outputfirstentry + S (pfrep_left_aligned_function_second_witness_output) = S ((S (pfrep_position_aligned_function_second_witness_outputfirst)) * pfaa_sum_c_aligned_function_second)) /\\ exists ff_q_pfp_aligned_function_second_witness_outputfirstentry. pfaa_sum_b_aligned_function_second = ff_q_pfp_aligned_function_second_witness_outputfirstentry * S ((S (pfrep_position_aligned_function_second_witness_outputfirst)) * pfaa_sum_c_aligned_function_second) + (pfrep_left_aligned_function_second_witness_output)))))) \\/ (((exists pfrep_gap_aligned_function_second_witness_outputfirstoutside. pfrep_gap_aligned_function_second_witness_outputfirstoutside+(pfaa_length_aligned_function_second)=(pfrep_power_aligned_function_second_witness_output)) /\\ (((pfrep_left_aligned_function_second_witness_output)=0))))) -> ((exists pfrep_position_aligned_function_second_witness_outputsecond. ((pfrep_position_aligned_function_second_witness_outputsecond+S (pfrep_power_aligned_function_second_witness_output)=(J)) /\\ ((((exists ff_h_pfp_aligned_function_second_witness_outputsecondentry. ff_h_pfp_aligned_function_second_witness_outputsecondentry + S (pfrep_right_aligned_function_second_witness_output) = S ((S (pfrep_position_aligned_function_second_witness_outputsecond)) * sc)) /\\ exists ff_q_pfp_aligned_function_second_witness_outputsecondentry. sb = ff_q_pfp_aligned_function_second_witness_outputsecondentry * S ((S (pfrep_position_aligned_function_second_witness_outputsecond)) * sc) + (pfrep_right_aligned_function_second_witness_output)))))) \\/ (((exists pfrep_gap_aligned_function_second_witness_outputsecondoutside. pfrep_gap_aligned_function_second_witness_outputsecondoutside+(J)=(pfrep_power_aligned_function_second_witness_output)) /\\ (((pfrep_right_aligned_function_second_witness_output)=0))))) -> pfrep_left_aligned_function_second_witness_output=pfrep_right_aligned_function_second_witness_output))))))))))))) -> (forall pfrep_power_aligned_function_result pfrep_left_aligned_function_result pfrep_right_aligned_function_result. ((exists pfrep_position_aligned_function_resultfirst. ((pfrep_position_aligned_function_resultfirst+S (pfrep_power_aligned_function_result)=(N)) /\\ ((((exists ff_h_pfp_aligned_function_resultfirstentry. ff_h_pfp_aligned_function_resultfirstentry + S (pfrep_left_aligned_function_result) = S ((S (pfrep_position_aligned_function_resultfirst)) * rc)) /\\ exists ff_q_pfp_aligned_function_resultfirstentry. rb = ff_q_pfp_aligned_function_resultfirstentry * S ((S (pfrep_position_aligned_function_resultfirst)) * rc) + (pfrep_left_aligned_function_result)))))) \\/ (((exists pfrep_gap_aligned_function_resultfirstoutside. pfrep_gap_aligned_function_resultfirstoutside+(N)=(pfrep_power_aligned_function_result)) /\\ (((pfrep_left_aligned_function_result)=0))))) -> ((exists pfrep_position_aligned_function_resultsecond. ((pfrep_position_aligned_function_resultsecond+S (pfrep_power_aligned_function_result)=(J)) /\\ ((((exists ff_h_pfp_aligned_function_resultsecondentry. ff_h_pfp_aligned_function_resultsecondentry + S (pfrep_right_aligned_function_result) = S ((S (pfrep_position_aligned_function_resultsecond)) * sc)) /\\ exists ff_q_pfp_aligned_function_resultsecondentry. sb = ff_q_pfp_aligned_function_resultsecondentry * S ((S (pfrep_position_aligned_function_resultsecond)) * sc) + (pfrep_right_aligned_function_result)))))) \\/ (((exists pfrep_gap_aligned_function_resultsecondoutside. pfrep_gap_aligned_function_resultsecondoutside+(J)=(pfrep_power_aligned_function_result)) /\\ (((pfrep_right_aligned_function_result)=0))))) -> pfrep_left_aligned_function_result=pfrep_right_aligned_function_result)", "statement_sha256": "b775ae9e728b06fec963dcd8635abaca96e1624d1f3baf358e84d786bc353d62", "summary": "Two actual aligned sums represent the same formal polynomial even when their witnesses, original output 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((forall fom_index_pfp_aligned_exists_left_bound. (exists fom_gap_pfp_aligned_exists_left_bound_index_bound. fom_gap_pfp_aligned_exists_left_bound_index_bound + S (fom_index_pfp_aligned_exists_left_bound) = L+M) -> exists fom_value_pfp_aligned_exists_left_bound. ((((exists fom_beta_height_pfp_aligned_exists_left_bound_entry. fom_beta_height_pfp_aligned_exists_left_bound_entry + S (fom_value_pfp_aligned_exists_left_bound) = S ((S (fom_index_pfp_aligned_exists_left_bound)) * uc)) /\\ exists fom_beta_quotient_pfp_aligned_exists_left_bound_entry. ub = fom_beta_quotient_pfp_aligned_exists_left_bound_entry * S ((S (fom_index_pfp_aligned_exists_left_bound)) * uc) + (fom_value_pfp_aligned_exists_left_bound))) /\\ (exists fom_gap_pfp_aligned_exists_left_bound_value_bound. fom_gap_pfp_aligned_exists_left_bound_value_bound + S (fom_value_pfp_aligned_exists_left_bound) = p))) /\\ (((forall fom_index_pfp_aligned_exists_right_bound. (exists fom_gap_pfp_aligned_exists_right_bound_index_bound. fom_gap_pfp_aligned_exists_right_bound_index_bound + S (fom_index_pfp_aligned_exists_right_bound) = L+M) -> exists fom_value_pfp_aligned_exists_right_bound. ((((exists fom_beta_height_pfp_aligned_exists_right_bound_entry. fom_beta_height_pfp_aligned_exists_right_bound_entry + S (fom_value_pfp_aligned_exists_right_bound) = S ((S (fom_index_pfp_aligned_exists_right_bound)) * vc)) /\\ exists fom_beta_quotient_pfp_aligned_exists_right_bound_entry. vb = fom_beta_quotient_pfp_aligned_exists_right_bound_entry * S ((S (fom_index_pfp_aligned_exists_right_bound)) * vc) + (fom_value_pfp_aligned_exists_right_bound))) /\\ (exists fom_gap_pfp_aligned_exists_right_bound_value_bound. fom_gap_pfp_aligned_exists_right_bound_value_bound + S (fom_value_pfp_aligned_exists_right_bound) = p))) /\\ ((((forall pfrep_power_aligned_exists_common_left pfrep_left_aligned_exists_common_left pfrep_right_aligned_exists_common_left. ((exists pfrep_position_aligned_exists_common_leftfirst. ((pfrep_position_aligned_exists_common_leftfirst+S (pfrep_power_aligned_exists_common_left)=(L)) /\\ ((((exists ff_h_pfp_aligned_exists_common_leftfirstentry. ff_h_pfp_aligned_exists_common_leftfirstentry + S (pfrep_left_aligned_exists_common_left) = S ((S (pfrep_position_aligned_exists_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_aligned_exists_common_leftfirstentry. ab = ff_q_pfp_aligned_exists_common_leftfirstentry * S ((S (pfrep_position_aligned_exists_common_leftfirst)) * ac) + (pfrep_left_aligned_exists_common_left)))))) \\/ (((exists pfrep_gap_aligned_exists_common_leftfirstoutside. pfrep_gap_aligned_exists_common_leftfirstoutside+(L)=(pfrep_power_aligned_exists_common_left)) /\\ (((pfrep_left_aligned_exists_common_left)=0))))) -> ((exists pfrep_position_aligned_exists_common_leftsecond. ((pfrep_position_aligned_exists_common_leftsecond+S (pfrep_power_aligned_exists_common_left)=(L+M)) /\\ ((((exists ff_h_pfp_aligned_exists_common_leftsecondentry. ff_h_pfp_aligned_exists_common_leftsecondentry + S (pfrep_right_aligned_exists_common_left) = S ((S (pfrep_position_aligned_exists_common_leftsecond)) * uc)) /\\ exists ff_q_pfp_aligned_exists_common_leftsecondentry. ub = ff_q_pfp_aligned_exists_common_leftsecondentry * S ((S (pfrep_position_aligned_exists_common_leftsecond)) * uc) + (pfrep_right_aligned_exists_common_left)))))) \\/ (((exists pfrep_gap_aligned_exists_common_leftsecondoutside. pfrep_gap_aligned_exists_common_leftsecondoutside+(L+M)=(pfrep_power_aligned_exists_common_left)) /\\ (((pfrep_right_aligned_exists_common_left)=0))))) -> pfrep_left_aligned_exists_common_left=pfrep_right_aligned_exists_common_left) /\\ ((forall pfrep_power_aligned_exists_common_right pfrep_left_aligned_exists_common_right pfrep_right_aligned_exists_common_right. ((exists pfrep_position_aligned_exists_common_rightfirst. ((pfrep_position_aligned_exists_common_rightfirst+S (pfrep_power_aligned_exists_common_right)=(M)) /\\ ((((exists ff_h_pfp_aligned_exists_common_rightfirstentry. ff_h_pfp_aligned_exists_common_rightfirstentry + S (pfrep_left_aligned_exists_common_right) = S ((S (pfrep_position_aligned_exists_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_aligned_exists_common_rightfirstentry. bb = ff_q_pfp_aligned_exists_common_rightfirstentry * S ((S (pfrep_position_aligned_exists_common_rightfirst)) * bc) + (pfrep_left_aligned_exists_common_right)))))) \\/ (((exists pfrep_gap_aligned_exists_common_rightfirstoutside. pfrep_gap_aligned_exists_common_rightfirstoutside+(M)=(pfrep_power_aligned_exists_common_right)) /\\ (((pfrep_left_aligned_exists_common_right)=0))))) -> ((exists pfrep_position_aligned_exists_common_rightsecond. ((pfrep_position_aligned_exists_common_rightsecond+S (pfrep_power_aligned_exists_common_right)=(L+M)) /\\ ((((exists ff_h_pfp_aligned_exists_common_rightsecondentry. ff_h_pfp_aligned_exists_common_rightsecondentry + S (pfrep_right_aligned_exists_common_right) = S ((S (pfrep_position_aligned_exists_common_rightsecond)) * vc)) /\\ exists ff_q_pfp_aligned_exists_common_rightsecondentry. vb = ff_q_pfp_aligned_exists_common_rightsecondentry * S ((S (pfrep_position_aligned_exists_common_rightsecond)) * vc) + (pfrep_right_aligned_exists_common_right)))))) \\/ (((exists pfrep_gap_aligned_exists_common_rightsecondoutside. pfrep_gap_aligned_exists_common_rightsecondoutside+(L+M)=(pfrep_power_aligned_exists_common_right)) /\\ (((pfrep_right_aligned_exists_common_right)=0))))) -> pfrep_left_aligned_exists_common_right=pfrep_right_aligned_exists_common_right))))))))", "specialize prime_field_polynomial_common_representatives_exists (p)", "specialize prime_field_polynomial_common_representatives_exists (ab)", "specialize prime_field_polynomial_common_representatives_exists (ac)", "specialize prime_field_polynomial_common_representatives_exists (L)", "specialize prime_field_polynomial_common_representatives_exists (bb)", "specialize prime_field_polynomial_common_representatives_exists (bc)", "specialize prime_field_polynomial_common_representatives_exists (M)", "apply prime_field_polynomial_common_representatives_exists", "exact hp", "exact ha", "exact hb", "cases hc", "cases hc_witness", "cases hc_witness_witness", "cases hc_witness_witness_witness", "cases hc_witness_witness_witness_witness", "cases hc_witness_witness_witness_witness_right", "have hs : exists rb rc. forall pfp_index_aligned_exists_sum. (exists pfa_gap_aligned_exists_sumindex. pfa_gap_aligned_exists_sumindex + S (pfp_index_aligned_exists_sum) = (L+M)) -> exists pfp_left_aligned_exists_sum pfp_right_aligned_exists_sum pfp_value_aligned_exists_sum. ((((exists ff_h_pfp_aligned_exists_sumleft. ff_h_pfp_aligned_exists_sumleft + S (pfp_left_aligned_exists_sum) = S ((S (pfp_index_aligned_exists_sum)) * x1)) /\\ exists ff_q_pfp_aligned_exists_sumleft. x = ff_q_pfp_aligned_exists_sumleft * S ((S (pfp_index_aligned_exists_sum)) * x1) + (pfp_left_aligned_exists_sum))) /\\ (((((exists ff_h_pfp_aligned_exists_sumright. ff_h_pfp_aligned_exists_sumright + S (pfp_right_aligned_exists_sum) = S ((S (pfp_index_aligned_exists_sum)) * x3)) /\\ exists ff_q_pfp_aligned_exists_sumright. x2 = ff_q_pfp_aligned_exists_sumright * S ((S (pfp_index_aligned_exists_sum)) * x3) + (pfp_right_aligned_exists_sum))) /\\ (((((exists ff_h_pfp_aligned_exists_sumtarget. ff_h_pfp_aligned_exists_sumtarget + S (pfp_value_aligned_exists_sum) = S ((S (pfp_index_aligned_exists_sum)) * rc)) /\\ exists ff_q_pfp_aligned_exists_sumtarget. rb = ff_q_pfp_aligned_exists_sumtarget * S ((S (pfp_index_aligned_exists_sum)) * rc) + (pfp_value_aligned_exists_sum))) /\\ ((((exists pfa_gap_aligned_exists_sumoperationleft. pfa_gap_aligned_exists_sumoperationleft + S (pfp_left_aligned_exists_sum) = (p)) /\\ (((exists pfa_gap_aligned_exists_sumoperationright. pfa_gap_aligned_exists_sumoperationright + S (pfp_right_aligned_exists_sum) = (p)) /\\ ((((exists pfa_gap_aligned_exists_sumoperationresultbound. pfa_gap_aligned_exists_sumoperationresultbound + S (pfp_value_aligned_exists_sum) = (p)) /\\ ((exists pfa_offset_left_aligned_exists_sumoperationresultcongruence pfa_offset_right_aligned_exists_sumoperationresultcongruence. ((pfp_left_aligned_exists_sum) + (pfp_right_aligned_exists_sum)) + (p) * pfa_offset_left_aligned_exists_sumoperationresultcongruence = (pfp_value_aligned_exists_sum) + (p) * pfa_offset_right_aligned_exists_sumoperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_add_exists (p)", "specialize prime_field_polynomial_add_exists (x)", "specialize prime_field_polynomial_add_exists (x1)", "specialize prime_field_polynomial_add_exists (x2)", "specialize prime_field_polynomial_add_exists (x3)", "specialize prime_field_polynomial_add_exists (L+M)", "apply prime_field_polynomial_add_exists", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hz", "exact hc_witness_witness_witness_witness_left", "exact hc_witness_witness_witness_witness_right_left", "cases hs", "cases hs_witness", "have hsbound : ((forall fom_index_pfp_aligned_exists_sum_left. (exists fom_gap_pfp_aligned_exists_sum_left_index_bound. fom_gap_pfp_aligned_exists_sum_left_index_bound + S (fom_index_pfp_aligned_exists_sum_left) = L+M) -> exists fom_value_pfp_aligned_exists_sum_left. ((((exists fom_beta_height_pfp_aligned_exists_sum_left_entry. fom_beta_height_pfp_aligned_exists_sum_left_entry + S (fom_value_pfp_aligned_exists_sum_left) = S ((S (fom_index_pfp_aligned_exists_sum_left)) * x1)) /\\ exists fom_beta_quotient_pfp_aligned_exists_sum_left_entry. x = fom_beta_quotient_pfp_aligned_exists_sum_left_entry * S ((S (fom_index_pfp_aligned_exists_sum_left)) * x1) + (fom_value_pfp_aligned_exists_sum_left))) /\\ (exists fom_gap_pfp_aligned_exists_sum_left_value_bound. fom_gap_pfp_aligned_exists_sum_left_value_bound + S (fom_value_pfp_aligned_exists_sum_left) = p))) /\\ (((forall fom_index_pfp_aligned_exists_sum_right. (exists fom_gap_pfp_aligned_exists_sum_right_index_bound. fom_gap_pfp_aligned_exists_sum_right_index_bound + S (fom_index_pfp_aligned_exists_sum_right) = L+M) -> exists fom_value_pfp_aligned_exists_sum_right. ((((exists fom_beta_height_pfp_aligned_exists_sum_right_entry. fom_beta_height_pfp_aligned_exists_sum_right_entry + S (fom_value_pfp_aligned_exists_sum_right) = S ((S (fom_index_pfp_aligned_exists_sum_right)) * x3)) /\\ exists fom_beta_quotient_pfp_aligned_exists_sum_right_entry. x2 = fom_beta_quotient_pfp_aligned_exists_sum_right_entry * S ((S (fom_index_pfp_aligned_exists_sum_right)) * x3) + (fom_value_pfp_aligned_exists_sum_right))) /\\ (exists fom_gap_pfp_aligned_exists_sum_right_value_bound. fom_gap_pfp_aligned_exists_sum_right_value_bound + S (fom_value_pfp_aligned_exists_sum_right) = p))) /\\ ((forall fom_index_pfp_aligned_exists_sum_output. (exists fom_gap_pfp_aligned_exists_sum_output_index_bound. fom_gap_pfp_aligned_exists_sum_output_index_bound + S (fom_index_pfp_aligned_exists_sum_output) = L+M) -> exists fom_value_pfp_aligned_exists_sum_output. ((((exists fom_beta_height_pfp_aligned_exists_sum_output_entry. fom_beta_height_pfp_aligned_exists_sum_output_entry + S (fom_value_pfp_aligned_exists_sum_output) = S ((S (fom_index_pfp_aligned_exists_sum_output)) * x5)) /\\ exists fom_beta_quotient_pfp_aligned_exists_sum_output_entry. x4 = fom_beta_quotient_pfp_aligned_exists_sum_output_entry * S ((S (fom_index_pfp_aligned_exists_sum_output)) * x5) + (fom_value_pfp_aligned_exists_sum_output))) /\\ (exists fom_gap_pfp_aligned_exists_sum_output_value_bound. fom_gap_pfp_aligned_exists_sum_output_value_bound + S (fom_value_pfp_aligned_exists_sum_output) = p)))))))", "specialize prime_field_polynomial_add_bounded (p)", "specialize prime_field_polynomial_add_bounded (x)", "specialize prime_field_polynomial_add_bounded (x1)", "specialize prime_field_polynomial_add_bounded (x2)", "specialize prime_field_polynomial_add_bounded (x3)", "specialize prime_field_polynomial_add_bounded (x4)", "specialize prime_field_polynomial_add_bounded (x5)", "specialize prime_field_polynomial_add_bounded (L+M)", "apply prime_field_polynomial_add_bounded", "exact hs_witness_witness", "cases hsbound", "cases hsbound_right", "exists x4", "exists x5", "specialize prime_field_polynomial_aligned_add_from_common (p)", "specialize prime_field_polynomial_aligned_add_from_common (ab)", "specialize prime_field_polynomial_aligned_add_from_common (ac)", "specialize prime_field_polynomial_aligned_add_from_common (L)", "specialize prime_field_polynomial_aligned_add_from_common (bb)", "specialize prime_field_polynomial_aligned_add_from_common (bc)", "specialize prime_field_polynomial_aligned_add_from_common (M)", "specialize prime_field_polynomial_aligned_add_from_common (x4)", "specialize prime_field_polynomial_aligned_add_from_common (x5)", "specialize prime_field_polynomial_aligned_add_from_common (L+M)", "specialize prime_field_polynomial_aligned_add_from_common (x)", "specialize prime_field_polynomial_aligned_add_from_common (x1)", "specialize prime_field_polynomial_aligned_add_from_common (x2)", "specialize prime_field_polynomial_aligned_add_from_common (x3)", "specialize prime_field_polynomial_aligned_add_from_common (x4)", "specialize prime_field_polynomial_aligned_add_from_common (x5)", "specialize prime_field_polynomial_aligned_add_from_common (L+M)", "apply prime_field_polynomial_aligned_add_from_common", "exact ha", "exact hb", "exact hsbound_right_right", "exact hc_witness_witness_witness_witness_right_right", "exact hs_witness_witness", "specialize prime_field_polynomial_power_coefficient_functional (x4)", "specialize prime_field_polynomial_power_coefficient_functional (x5)", "specialize prime_field_polynomial_power_coefficient_functional (L+M)", "apply prime_field_polynomial_power_coefficient_functional" ], "script_sha256": "ff1a2dd62254f548b85e7f5a3c7a3cf7ad8cd4dfb64cd8a5c4f9cde268d9c86a", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_aligned_add_candidate.py", "sha256": "a05bb4f5c4230ca05f51690d3ab82e33ff4596af65176874e25fbe38cf87a0db" }, "statement": "forall p ab ac L bb bc M. (~((p) = 1) /\\ forall pfa_factor_left_aligned_exists_prime pfa_factor_right_aligned_exists_prime. (p) = pfa_factor_left_aligned_exists_prime * pfa_factor_right_aligned_exists_prime -> pfa_factor_left_aligned_exists_prime = 1 \\/ pfa_factor_right_aligned_exists_prime = 1) -> (forall fom_index_pfp_aligned_exists_A. (exists fom_gap_pfp_aligned_exists_A_index_bound. fom_gap_pfp_aligned_exists_A_index_bound + S (fom_index_pfp_aligned_exists_A) = L) -> exists fom_value_pfp_aligned_exists_A. ((((exists fom_beta_height_pfp_aligned_exists_A_entry. fom_beta_height_pfp_aligned_exists_A_entry + S (fom_value_pfp_aligned_exists_A) = S ((S (fom_index_pfp_aligned_exists_A)) * ac)) /\\ exists fom_beta_quotient_pfp_aligned_exists_A_entry. ab = fom_beta_quotient_pfp_aligned_exists_A_entry * S ((S (fom_index_pfp_aligned_exists_A)) * ac) + (fom_value_pfp_aligned_exists_A))) /\\ (exists fom_gap_pfp_aligned_exists_A_value_bound. fom_gap_pfp_aligned_exists_A_value_bound + S (fom_value_pfp_aligned_exists_A) = p))) -> (forall fom_index_pfp_aligned_exists_B. (exists fom_gap_pfp_aligned_exists_B_index_bound. fom_gap_pfp_aligned_exists_B_index_bound + S (fom_index_pfp_aligned_exists_B) = M) -> exists fom_value_pfp_aligned_exists_B. ((((exists fom_beta_height_pfp_aligned_exists_B_entry. fom_beta_height_pfp_aligned_exists_B_entry + S (fom_value_pfp_aligned_exists_B) = S ((S (fom_index_pfp_aligned_exists_B)) * bc)) /\\ exists fom_beta_quotient_pfp_aligned_exists_B_entry. bb = fom_beta_quotient_pfp_aligned_exists_B_entry * S ((S (fom_index_pfp_aligned_exists_B)) * bc) + (fom_value_pfp_aligned_exists_B))) /\\ (exists fom_gap_pfp_aligned_exists_B_value_bound. fom_gap_pfp_aligned_exists_B_value_bound + S (fom_value_pfp_aligned_exists_B) = p))) -> (exists rb rc. ((forall fom_index_pfp_aligned_exists_output_left_bounded. (exists fom_gap_pfp_aligned_exists_output_left_bounded_index_bound. fom_gap_pfp_aligned_exists_output_left_bounded_index_bound + S (fom_index_pfp_aligned_exists_output_left_bounded) = L) -> exists fom_value_pfp_aligned_exists_output_left_bounded. ((((exists fom_beta_height_pfp_aligned_exists_output_left_bounded_entry. fom_beta_height_pfp_aligned_exists_output_left_bounded_entry + S (fom_value_pfp_aligned_exists_output_left_bounded) = S ((S (fom_index_pfp_aligned_exists_output_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_aligned_exists_output_left_bounded_entry. ab = fom_beta_quotient_pfp_aligned_exists_output_left_bounded_entry * S ((S (fom_index_pfp_aligned_exists_output_left_bounded)) * ac) + (fom_value_pfp_aligned_exists_output_left_bounded))) /\\ (exists fom_gap_pfp_aligned_exists_output_left_bounded_value_bound. fom_gap_pfp_aligned_exists_output_left_bounded_value_bound + S (fom_value_pfp_aligned_exists_output_left_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_exists_output_right_bounded. (exists fom_gap_pfp_aligned_exists_output_right_bounded_index_bound. fom_gap_pfp_aligned_exists_output_right_bounded_index_bound + S (fom_index_pfp_aligned_exists_output_right_bounded) = M) -> exists fom_value_pfp_aligned_exists_output_right_bounded. ((((exists fom_beta_height_pfp_aligned_exists_output_right_bounded_entry. fom_beta_height_pfp_aligned_exists_output_right_bounded_entry + S (fom_value_pfp_aligned_exists_output_right_bounded) = S ((S (fom_index_pfp_aligned_exists_output_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_aligned_exists_output_right_bounded_entry. bb = fom_beta_quotient_pfp_aligned_exists_output_right_bounded_entry * S ((S (fom_index_pfp_aligned_exists_output_right_bounded)) * bc) + (fom_value_pfp_aligned_exists_output_right_bounded))) /\\ (exists fom_gap_pfp_aligned_exists_output_right_bounded_value_bound. fom_gap_pfp_aligned_exists_output_right_bounded_value_bound + S (fom_value_pfp_aligned_exists_output_right_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_exists_output_result_bounded. (exists fom_gap_pfp_aligned_exists_output_result_bounded_index_bound. fom_gap_pfp_aligned_exists_output_result_bounded_index_bound + S (fom_index_pfp_aligned_exists_output_result_bounded) = L+M) -> exists fom_value_pfp_aligned_exists_output_result_bounded. ((((exists fom_beta_height_pfp_aligned_exists_output_result_bounded_entry. fom_beta_height_pfp_aligned_exists_output_result_bounded_entry + S (fom_value_pfp_aligned_exists_output_result_bounded) = S ((S (fom_index_pfp_aligned_exists_output_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_aligned_exists_output_result_bounded_entry. rb = fom_beta_quotient_pfp_aligned_exists_output_result_bounded_entry * S ((S (fom_index_pfp_aligned_exists_output_result_bounded)) * rc) + (fom_value_pfp_aligned_exists_output_result_bounded))) /\\ (exists fom_gap_pfp_aligned_exists_output_result_bounded_value_bound. fom_gap_pfp_aligned_exists_output_result_bounded_value_bound + S (fom_value_pfp_aligned_exists_output_result_bounded) = p))) /\\ ((exists pfaa_left_b_aligned_exists_output pfaa_left_c_aligned_exists_output pfaa_right_b_aligned_exists_output pfaa_right_c_aligned_exists_output pfaa_sum_b_aligned_exists_output pfaa_sum_c_aligned_exists_output pfaa_length_aligned_exists_output. ((((forall pfrep_power_aligned_exists_output_witness_common_left pfrep_left_aligned_exists_output_witness_common_left pfrep_right_aligned_exists_output_witness_common_left. ((exists pfrep_position_aligned_exists_output_witness_common_leftfirst. ((pfrep_position_aligned_exists_output_witness_common_leftfirst+S (pfrep_power_aligned_exists_output_witness_common_left)=(L)) /\\ ((((exists ff_h_pfp_aligned_exists_output_witness_common_leftfirstentry. ff_h_pfp_aligned_exists_output_witness_common_leftfirstentry + S (pfrep_left_aligned_exists_output_witness_common_left) = S ((S (pfrep_position_aligned_exists_output_witness_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_aligned_exists_output_witness_common_leftfirstentry. ab = ff_q_pfp_aligned_exists_output_witness_common_leftfirstentry * S ((S (pfrep_position_aligned_exists_output_witness_common_leftfirst)) * ac) + (pfrep_left_aligned_exists_output_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_exists_output_witness_common_leftfirstoutside. pfrep_gap_aligned_exists_output_witness_common_leftfirstoutside+(L)=(pfrep_power_aligned_exists_output_witness_common_left)) /\\ (((pfrep_left_aligned_exists_output_witness_common_left)=0))))) -> ((exists pfrep_position_aligned_exists_output_witness_common_leftsecond. ((pfrep_position_aligned_exists_output_witness_common_leftsecond+S (pfrep_power_aligned_exists_output_witness_common_left)=(pfaa_length_aligned_exists_output)) /\\ ((((exists ff_h_pfp_aligned_exists_output_witness_common_leftsecondentry. ff_h_pfp_aligned_exists_output_witness_common_leftsecondentry + S (pfrep_right_aligned_exists_output_witness_common_left) = S ((S (pfrep_position_aligned_exists_output_witness_common_leftsecond)) * pfaa_left_c_aligned_exists_output)) /\\ exists ff_q_pfp_aligned_exists_output_witness_common_leftsecondentry. pfaa_left_b_aligned_exists_output = ff_q_pfp_aligned_exists_output_witness_common_leftsecondentry * S ((S (pfrep_position_aligned_exists_output_witness_common_leftsecond)) * pfaa_left_c_aligned_exists_output) + (pfrep_right_aligned_exists_output_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_exists_output_witness_common_leftsecondoutside. pfrep_gap_aligned_exists_output_witness_common_leftsecondoutside+(pfaa_length_aligned_exists_output)=(pfrep_power_aligned_exists_output_witness_common_left)) /\\ (((pfrep_right_aligned_exists_output_witness_common_left)=0))))) -> pfrep_left_aligned_exists_output_witness_common_left=pfrep_right_aligned_exists_output_witness_common_left) /\\ ((forall pfrep_power_aligned_exists_output_witness_common_right pfrep_left_aligned_exists_output_witness_common_right pfrep_right_aligned_exists_output_witness_common_right. ((exists pfrep_position_aligned_exists_output_witness_common_rightfirst. ((pfrep_position_aligned_exists_output_witness_common_rightfirst+S (pfrep_power_aligned_exists_output_witness_common_right)=(M)) /\\ ((((exists ff_h_pfp_aligned_exists_output_witness_common_rightfirstentry. ff_h_pfp_aligned_exists_output_witness_common_rightfirstentry + S (pfrep_left_aligned_exists_output_witness_common_right) = S ((S (pfrep_position_aligned_exists_output_witness_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_aligned_exists_output_witness_common_rightfirstentry. bb = ff_q_pfp_aligned_exists_output_witness_common_rightfirstentry * S ((S (pfrep_position_aligned_exists_output_witness_common_rightfirst)) * bc) + (pfrep_left_aligned_exists_output_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_exists_output_witness_common_rightfirstoutside. pfrep_gap_aligned_exists_output_witness_common_rightfirstoutside+(M)=(pfrep_power_aligned_exists_output_witness_common_right)) /\\ (((pfrep_left_aligned_exists_output_witness_common_right)=0))))) -> ((exists pfrep_position_aligned_exists_output_witness_common_rightsecond. ((pfrep_position_aligned_exists_output_witness_common_rightsecond+S (pfrep_power_aligned_exists_output_witness_common_right)=(pfaa_length_aligned_exists_output)) /\\ ((((exists ff_h_pfp_aligned_exists_output_witness_common_rightsecondentry. ff_h_pfp_aligned_exists_output_witness_common_rightsecondentry + S (pfrep_right_aligned_exists_output_witness_common_right) = S ((S (pfrep_position_aligned_exists_output_witness_common_rightsecond)) * pfaa_right_c_aligned_exists_output)) /\\ exists ff_q_pfp_aligned_exists_output_witness_common_rightsecondentry. pfaa_right_b_aligned_exists_output = ff_q_pfp_aligned_exists_output_witness_common_rightsecondentry * S ((S (pfrep_position_aligned_exists_output_witness_common_rightsecond)) * pfaa_right_c_aligned_exists_output) + (pfrep_right_aligned_exists_output_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_exists_output_witness_common_rightsecondoutside. pfrep_gap_aligned_exists_output_witness_common_rightsecondoutside+(pfaa_length_aligned_exists_output)=(pfrep_power_aligned_exists_output_witness_common_right)) /\\ (((pfrep_right_aligned_exists_output_witness_common_right)=0))))) -> pfrep_left_aligned_exists_output_witness_common_right=pfrep_right_aligned_exists_output_witness_common_right)))) /\\ (((forall pfp_index_aligned_exists_output_witness_operation. (exists pfa_gap_aligned_exists_output_witness_operationindex. pfa_gap_aligned_exists_output_witness_operationindex + S (pfp_index_aligned_exists_output_witness_operation) = (pfaa_length_aligned_exists_output)) -> exists pfp_left_aligned_exists_output_witness_operation pfp_right_aligned_exists_output_witness_operation pfp_value_aligned_exists_output_witness_operation. ((((exists ff_h_pfp_aligned_exists_output_witness_operationleft. ff_h_pfp_aligned_exists_output_witness_operationleft + S (pfp_left_aligned_exists_output_witness_operation) = S ((S (pfp_index_aligned_exists_output_witness_operation)) * pfaa_left_c_aligned_exists_output)) /\\ exists ff_q_pfp_aligned_exists_output_witness_operationleft. pfaa_left_b_aligned_exists_output = ff_q_pfp_aligned_exists_output_witness_operationleft * S ((S (pfp_index_aligned_exists_output_witness_operation)) * pfaa_left_c_aligned_exists_output) + (pfp_left_aligned_exists_output_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_exists_output_witness_operationright. ff_h_pfp_aligned_exists_output_witness_operationright + S (pfp_right_aligned_exists_output_witness_operation) = S ((S (pfp_index_aligned_exists_output_witness_operation)) * pfaa_right_c_aligned_exists_output)) /\\ exists ff_q_pfp_aligned_exists_output_witness_operationright. pfaa_right_b_aligned_exists_output = ff_q_pfp_aligned_exists_output_witness_operationright * S ((S (pfp_index_aligned_exists_output_witness_operation)) * pfaa_right_c_aligned_exists_output) + (pfp_right_aligned_exists_output_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_exists_output_witness_operationtarget. ff_h_pfp_aligned_exists_output_witness_operationtarget + S (pfp_value_aligned_exists_output_witness_operation) = S ((S (pfp_index_aligned_exists_output_witness_operation)) * pfaa_sum_c_aligned_exists_output)) /\\ exists ff_q_pfp_aligned_exists_output_witness_operationtarget. pfaa_sum_b_aligned_exists_output = ff_q_pfp_aligned_exists_output_witness_operationtarget * S ((S (pfp_index_aligned_exists_output_witness_operation)) * pfaa_sum_c_aligned_exists_output) + (pfp_value_aligned_exists_output_witness_operation))) /\\ ((((exists pfa_gap_aligned_exists_output_witness_operationoperationleft. pfa_gap_aligned_exists_output_witness_operationoperationleft + S (pfp_left_aligned_exists_output_witness_operation) = (p)) /\\ (((exists pfa_gap_aligned_exists_output_witness_operationoperationright. pfa_gap_aligned_exists_output_witness_operationoperationright + S (pfp_right_aligned_exists_output_witness_operation) = (p)) /\\ ((((exists pfa_gap_aligned_exists_output_witness_operationoperationresultbound. pfa_gap_aligned_exists_output_witness_operationoperationresultbound + S (pfp_value_aligned_exists_output_witness_operation) = (p)) /\\ ((exists pfa_offset_left_aligned_exists_output_witness_operationoperationresultcongruence pfa_offset_right_aligned_exists_output_witness_operationoperationresultcongruence. ((pfp_left_aligned_exists_output_witness_operation) + (pfp_right_aligned_exists_output_witness_operation)) + (p) * pfa_offset_left_aligned_exists_output_witness_operationoperationresultcongruence = (pfp_value_aligned_exists_output_witness_operation) + (p) * pfa_offset_right_aligned_exists_output_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_aligned_exists_output_witness_output pfrep_left_aligned_exists_output_witness_output pfrep_right_aligned_exists_output_witness_output. ((exists pfrep_position_aligned_exists_output_witness_outputfirst. ((pfrep_position_aligned_exists_output_witness_outputfirst+S (pfrep_power_aligned_exists_output_witness_output)=(pfaa_length_aligned_exists_output)) /\\ ((((exists ff_h_pfp_aligned_exists_output_witness_outputfirstentry. ff_h_pfp_aligned_exists_output_witness_outputfirstentry + S (pfrep_left_aligned_exists_output_witness_output) = S ((S (pfrep_position_aligned_exists_output_witness_outputfirst)) * pfaa_sum_c_aligned_exists_output)) /\\ exists ff_q_pfp_aligned_exists_output_witness_outputfirstentry. pfaa_sum_b_aligned_exists_output = ff_q_pfp_aligned_exists_output_witness_outputfirstentry * S ((S (pfrep_position_aligned_exists_output_witness_outputfirst)) * pfaa_sum_c_aligned_exists_output) + (pfrep_left_aligned_exists_output_witness_output)))))) \\/ (((exists pfrep_gap_aligned_exists_output_witness_outputfirstoutside. pfrep_gap_aligned_exists_output_witness_outputfirstoutside+(pfaa_length_aligned_exists_output)=(pfrep_power_aligned_exists_output_witness_output)) /\\ (((pfrep_left_aligned_exists_output_witness_output)=0))))) -> ((exists pfrep_position_aligned_exists_output_witness_outputsecond. ((pfrep_position_aligned_exists_output_witness_outputsecond+S (pfrep_power_aligned_exists_output_witness_output)=(L+M)) /\\ ((((exists ff_h_pfp_aligned_exists_output_witness_outputsecondentry. ff_h_pfp_aligned_exists_output_witness_outputsecondentry + S (pfrep_right_aligned_exists_output_witness_output) = S ((S (pfrep_position_aligned_exists_output_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_aligned_exists_output_witness_outputsecondentry. rb = ff_q_pfp_aligned_exists_output_witness_outputsecondentry * S ((S (pfrep_position_aligned_exists_output_witness_outputsecond)) * rc) + (pfrep_right_aligned_exists_output_witness_output)))))) \\/ (((exists pfrep_gap_aligned_exists_output_witness_outputsecondoutside. pfrep_gap_aligned_exists_output_witness_outputsecondoutside+(L+M)=(pfrep_power_aligned_exists_output_witness_output)) /\\ (((pfrep_right_aligned_exists_output_witness_output)=0))))) -> pfrep_left_aligned_exists_output_witness_output=pfrep_right_aligned_exists_output_witness_output)))))))))))))", "statement_sha256": "6377d2df388a803a311124a8c19ae1997f6deff0a7ea7ab664ce5736f401efce", "summary": "Construct a genuine canonical aligned sum at the explicit length L+M from any two canonical inputs, rather than assuming operation witnesses.", "summary_sha256": "13f952b9cab29b335fe8f47951a54456c54afe892f0f41352db26c9be134c505" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "914622c0d301f230ead6e5f2c00040fe0bf4c9c55072dafeeb626fe8666a1587", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 343, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": 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"intro N", "intro ub", "intro uc", "intro vb", "intro vc", "intro tb", "intro tc", "intro K", "intro hp", "intro h", "intro hu", "intro hv", "intro ht", "intro hc", "intro hr", "cases hc", "have hz : exists zb zc. forall pfp_index_realize_constructed_sum. (exists pfa_gap_realize_constructed_sumindex. pfa_gap_realize_constructed_sumindex + S (pfp_index_realize_constructed_sum) = (K)) -> exists pfp_left_realize_constructed_sum pfp_right_realize_constructed_sum pfp_value_realize_constructed_sum. ((((exists ff_h_pfp_realize_constructed_sumleft. ff_h_pfp_realize_constructed_sumleft + S (pfp_left_realize_constructed_sum) = S ((S (pfp_index_realize_constructed_sum)) * uc)) /\\ exists ff_q_pfp_realize_constructed_sumleft. ub = ff_q_pfp_realize_constructed_sumleft * S ((S (pfp_index_realize_constructed_sum)) * uc) + (pfp_left_realize_constructed_sum))) /\\ (((((exists ff_h_pfp_realize_constructed_sumright. ff_h_pfp_realize_constructed_sumright + S (pfp_right_realize_constructed_sum) = S ((S (pfp_index_realize_constructed_sum)) * vc)) /\\ exists ff_q_pfp_realize_constructed_sumright. vb = ff_q_pfp_realize_constructed_sumright * S ((S (pfp_index_realize_constructed_sum)) * vc) + (pfp_right_realize_constructed_sum))) /\\ (((((exists ff_h_pfp_realize_constructed_sumtarget. ff_h_pfp_realize_constructed_sumtarget + S (pfp_value_realize_constructed_sum) = S ((S (pfp_index_realize_constructed_sum)) * zc)) /\\ exists ff_q_pfp_realize_constructed_sumtarget. zb = ff_q_pfp_realize_constructed_sumtarget * S ((S (pfp_index_realize_constructed_sum)) * zc) + (pfp_value_realize_constructed_sum))) /\\ ((((exists pfa_gap_realize_constructed_sumoperationleft. pfa_gap_realize_constructed_sumoperationleft + S (pfp_left_realize_constructed_sum) = (p)) /\\ (((exists pfa_gap_realize_constructed_sumoperationright. pfa_gap_realize_constructed_sumoperationright + S (pfp_right_realize_constructed_sum) = (p)) /\\ ((((exists pfa_gap_realize_constructed_sumoperationresultbound. pfa_gap_realize_constructed_sumoperationresultbound + S (pfp_value_realize_constructed_sum) = (p)) /\\ ((exists pfa_offset_left_realize_constructed_sumoperationresultcongruence pfa_offset_right_realize_constructed_sumoperationresultcongruence. ((pfp_left_realize_constructed_sum) + (pfp_right_realize_constructed_sum)) + (p) * pfa_offset_left_realize_constructed_sumoperationresultcongruence = (pfp_value_realize_constructed_sum) + (p) * pfa_offset_right_realize_constructed_sumoperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_add_exists (p)", "specialize prime_field_polynomial_add_exists (ub)", "specialize prime_field_polynomial_add_exists (uc)", "specialize prime_field_polynomial_add_exists (vb)", "specialize prime_field_polynomial_add_exists (vc)", "specialize prime_field_polynomial_add_exists (K)", "apply prime_field_polynomial_add_exists", "intro hpzero", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hpzero", "exact hu", "exact hv", "cases hz", "cases hz_witness", "have hzgraph : ((forall fom_index_pfp_realize_constructed_graph_left_bounded. (exists fom_gap_pfp_realize_constructed_graph_left_bounded_index_bound. fom_gap_pfp_realize_constructed_graph_left_bounded_index_bound + S (fom_index_pfp_realize_constructed_graph_left_bounded) = K) -> exists fom_value_pfp_realize_constructed_graph_left_bounded. ((((exists fom_beta_height_pfp_realize_constructed_graph_left_bounded_entry. fom_beta_height_pfp_realize_constructed_graph_left_bounded_entry + S (fom_value_pfp_realize_constructed_graph_left_bounded) = S ((S (fom_index_pfp_realize_constructed_graph_left_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_realize_constructed_graph_left_bounded_entry. ub = fom_beta_quotient_pfp_realize_constructed_graph_left_bounded_entry * S ((S (fom_index_pfp_realize_constructed_graph_left_bounded)) * uc) + (fom_value_pfp_realize_constructed_graph_left_bounded))) /\\ (exists fom_gap_pfp_realize_constructed_graph_left_bounded_value_bound. fom_gap_pfp_realize_constructed_graph_left_bounded_value_bound + S (fom_value_pfp_realize_constructed_graph_left_bounded) = p))) /\\ (((forall fom_index_pfp_realize_constructed_graph_right_bounded. (exists fom_gap_pfp_realize_constructed_graph_right_bounded_index_bound. fom_gap_pfp_realize_constructed_graph_right_bounded_index_bound + S (fom_index_pfp_realize_constructed_graph_right_bounded) = K) -> exists fom_value_pfp_realize_constructed_graph_right_bounded. ((((exists fom_beta_height_pfp_realize_constructed_graph_right_bounded_entry. fom_beta_height_pfp_realize_constructed_graph_right_bounded_entry + S (fom_value_pfp_realize_constructed_graph_right_bounded) = S ((S (fom_index_pfp_realize_constructed_graph_right_bounded)) * vc)) /\\ exists fom_beta_quotient_pfp_realize_constructed_graph_right_bounded_entry. vb = fom_beta_quotient_pfp_realize_constructed_graph_right_bounded_entry * S ((S (fom_index_pfp_realize_constructed_graph_right_bounded)) * vc) + (fom_value_pfp_realize_constructed_graph_right_bounded))) /\\ (exists fom_gap_pfp_realize_constructed_graph_right_bounded_value_bound. fom_gap_pfp_realize_constructed_graph_right_bounded_value_bound + S (fom_value_pfp_realize_constructed_graph_right_bounded) = p))) /\\ (((forall fom_index_pfp_realize_constructed_graph_result_bounded. (exists fom_gap_pfp_realize_constructed_graph_result_bounded_index_bound. fom_gap_pfp_realize_constructed_graph_result_bounded_index_bound + S (fom_index_pfp_realize_constructed_graph_result_bounded) = K) -> exists fom_value_pfp_realize_constructed_graph_result_bounded. ((((exists fom_beta_height_pfp_realize_constructed_graph_result_bounded_entry. fom_beta_height_pfp_realize_constructed_graph_result_bounded_entry + S (fom_value_pfp_realize_constructed_graph_result_bounded) = S ((S (fom_index_pfp_realize_constructed_graph_result_bounded)) * x1)) /\\ exists fom_beta_quotient_pfp_realize_constructed_graph_result_bounded_entry. x = fom_beta_quotient_pfp_realize_constructed_graph_result_bounded_entry * S ((S (fom_index_pfp_realize_constructed_graph_result_bounded)) * x1) + (fom_value_pfp_realize_constructed_graph_result_bounded))) /\\ (exists fom_gap_pfp_realize_constructed_graph_result_bounded_value_bound. fom_gap_pfp_realize_constructed_graph_result_bounded_value_bound + S (fom_value_pfp_realize_constructed_graph_result_bounded) = p))) /\\ ((exists pfaa_left_b_realize_constructed_graph pfaa_left_c_realize_constructed_graph pfaa_right_b_realize_constructed_graph pfaa_right_c_realize_constructed_graph pfaa_sum_b_realize_constructed_graph pfaa_sum_c_realize_constructed_graph pfaa_length_realize_constructed_graph. ((((forall pfrep_power_realize_constructed_graph_witness_common_left pfrep_left_realize_constructed_graph_witness_common_left pfrep_right_realize_constructed_graph_witness_common_left. ((exists pfrep_position_realize_constructed_graph_witness_common_leftfirst. ((pfrep_position_realize_constructed_graph_witness_common_leftfirst+S (pfrep_power_realize_constructed_graph_witness_common_left)=(K)) /\\ ((((exists ff_h_pfp_realize_constructed_graph_witness_common_leftfirstentry. ff_h_pfp_realize_constructed_graph_witness_common_leftfirstentry + S (pfrep_left_realize_constructed_graph_witness_common_left) = S ((S (pfrep_position_realize_constructed_graph_witness_common_leftfirst)) * uc)) /\\ exists ff_q_pfp_realize_constructed_graph_witness_common_leftfirstentry. ub = ff_q_pfp_realize_constructed_graph_witness_common_leftfirstentry * S ((S (pfrep_position_realize_constructed_graph_witness_common_leftfirst)) * uc) + (pfrep_left_realize_constructed_graph_witness_common_left)))))) \\/ (((exists pfrep_gap_realize_constructed_graph_witness_common_leftfirstoutside. pfrep_gap_realize_constructed_graph_witness_common_leftfirstoutside+(K)=(pfrep_power_realize_constructed_graph_witness_common_left)) /\\ (((pfrep_left_realize_constructed_graph_witness_common_left)=0))))) -> ((exists pfrep_position_realize_constructed_graph_witness_common_leftsecond. ((pfrep_position_realize_constructed_graph_witness_common_leftsecond+S (pfrep_power_realize_constructed_graph_witness_common_left)=(pfaa_length_realize_constructed_graph)) /\\ ((((exists ff_h_pfp_realize_constructed_graph_witness_common_leftsecondentry. ff_h_pfp_realize_constructed_graph_witness_common_leftsecondentry + S (pfrep_right_realize_constructed_graph_witness_common_left) = S ((S (pfrep_position_realize_constructed_graph_witness_common_leftsecond)) * pfaa_left_c_realize_constructed_graph)) /\\ exists ff_q_pfp_realize_constructed_graph_witness_common_leftsecondentry. pfaa_left_b_realize_constructed_graph = ff_q_pfp_realize_constructed_graph_witness_common_leftsecondentry * S ((S (pfrep_position_realize_constructed_graph_witness_common_leftsecond)) * pfaa_left_c_realize_constructed_graph) + (pfrep_right_realize_constructed_graph_witness_common_left)))))) \\/ (((exists pfrep_gap_realize_constructed_graph_witness_common_leftsecondoutside. pfrep_gap_realize_constructed_graph_witness_common_leftsecondoutside+(pfaa_length_realize_constructed_graph)=(pfrep_power_realize_constructed_graph_witness_common_left)) /\\ (((pfrep_right_realize_constructed_graph_witness_common_left)=0))))) -> pfrep_left_realize_constructed_graph_witness_common_left=pfrep_right_realize_constructed_graph_witness_common_left) /\\ ((forall pfrep_power_realize_constructed_graph_witness_common_right pfrep_left_realize_constructed_graph_witness_common_right pfrep_right_realize_constructed_graph_witness_common_right. ((exists pfrep_position_realize_constructed_graph_witness_common_rightfirst. ((pfrep_position_realize_constructed_graph_witness_common_rightfirst+S (pfrep_power_realize_constructed_graph_witness_common_right)=(K)) /\\ ((((exists ff_h_pfp_realize_constructed_graph_witness_common_rightfirstentry. ff_h_pfp_realize_constructed_graph_witness_common_rightfirstentry + S (pfrep_left_realize_constructed_graph_witness_common_right) = S ((S (pfrep_position_realize_constructed_graph_witness_common_rightfirst)) * vc)) /\\ exists ff_q_pfp_realize_constructed_graph_witness_common_rightfirstentry. vb = ff_q_pfp_realize_constructed_graph_witness_common_rightfirstentry * S ((S (pfrep_position_realize_constructed_graph_witness_common_rightfirst)) * vc) + (pfrep_left_realize_constructed_graph_witness_common_right)))))) \\/ (((exists pfrep_gap_realize_constructed_graph_witness_common_rightfirstoutside. pfrep_gap_realize_constructed_graph_witness_common_rightfirstoutside+(K)=(pfrep_power_realize_constructed_graph_witness_common_right)) /\\ (((pfrep_left_realize_constructed_graph_witness_common_right)=0))))) -> ((exists pfrep_position_realize_constructed_graph_witness_common_rightsecond. ((pfrep_position_realize_constructed_graph_witness_common_rightsecond+S (pfrep_power_realize_constructed_graph_witness_common_right)=(pfaa_length_realize_constructed_graph)) /\\ ((((exists ff_h_pfp_realize_constructed_graph_witness_common_rightsecondentry. ff_h_pfp_realize_constructed_graph_witness_common_rightsecondentry + S (pfrep_right_realize_constructed_graph_witness_common_right) = S ((S (pfrep_position_realize_constructed_graph_witness_common_rightsecond)) * pfaa_right_c_realize_constructed_graph)) /\\ exists ff_q_pfp_realize_constructed_graph_witness_common_rightsecondentry. pfaa_right_b_realize_constructed_graph = ff_q_pfp_realize_constructed_graph_witness_common_rightsecondentry * S ((S (pfrep_position_realize_constructed_graph_witness_common_rightsecond)) * pfaa_right_c_realize_constructed_graph) + (pfrep_right_realize_constructed_graph_witness_common_right)))))) \\/ (((exists pfrep_gap_realize_constructed_graph_witness_common_rightsecondoutside. pfrep_gap_realize_constructed_graph_witness_common_rightsecondoutside+(pfaa_length_realize_constructed_graph)=(pfrep_power_realize_constructed_graph_witness_common_right)) /\\ (((pfrep_right_realize_constructed_graph_witness_common_right)=0))))) -> pfrep_left_realize_constructed_graph_witness_common_right=pfrep_right_realize_constructed_graph_witness_common_right)))) /\\ (((forall pfp_index_realize_constructed_graph_witness_operation. (exists pfa_gap_realize_constructed_graph_witness_operationindex. pfa_gap_realize_constructed_graph_witness_operationindex + S (pfp_index_realize_constructed_graph_witness_operation) = (pfaa_length_realize_constructed_graph)) -> exists pfp_left_realize_constructed_graph_witness_operation pfp_right_realize_constructed_graph_witness_operation pfp_value_realize_constructed_graph_witness_operation. ((((exists ff_h_pfp_realize_constructed_graph_witness_operationleft. ff_h_pfp_realize_constructed_graph_witness_operationleft + S (pfp_left_realize_constructed_graph_witness_operation) = S ((S (pfp_index_realize_constructed_graph_witness_operation)) * pfaa_left_c_realize_constructed_graph)) /\\ exists ff_q_pfp_realize_constructed_graph_witness_operationleft. pfaa_left_b_realize_constructed_graph = ff_q_pfp_realize_constructed_graph_witness_operationleft * S ((S (pfp_index_realize_constructed_graph_witness_operation)) * pfaa_left_c_realize_constructed_graph) + (pfp_left_realize_constructed_graph_witness_operation))) /\\ (((((exists ff_h_pfp_realize_constructed_graph_witness_operationright. ff_h_pfp_realize_constructed_graph_witness_operationright + S (pfp_right_realize_constructed_graph_witness_operation) = S ((S (pfp_index_realize_constructed_graph_witness_operation)) * pfaa_right_c_realize_constructed_graph)) /\\ exists ff_q_pfp_realize_constructed_graph_witness_operationright. pfaa_right_b_realize_constructed_graph = ff_q_pfp_realize_constructed_graph_witness_operationright * S ((S (pfp_index_realize_constructed_graph_witness_operation)) * pfaa_right_c_realize_constructed_graph) + (pfp_right_realize_constructed_graph_witness_operation))) /\\ (((((exists ff_h_pfp_realize_constructed_graph_witness_operationtarget. ff_h_pfp_realize_constructed_graph_witness_operationtarget + S (pfp_value_realize_constructed_graph_witness_operation) = S ((S (pfp_index_realize_constructed_graph_witness_operation)) * pfaa_sum_c_realize_constructed_graph)) /\\ exists ff_q_pfp_realize_constructed_graph_witness_operationtarget. pfaa_sum_b_realize_constructed_graph = ff_q_pfp_realize_constructed_graph_witness_operationtarget * S ((S (pfp_index_realize_constructed_graph_witness_operation)) * pfaa_sum_c_realize_constructed_graph) + (pfp_value_realize_constructed_graph_witness_operation))) /\\ ((((exists pfa_gap_realize_constructed_graph_witness_operationoperationleft. pfa_gap_realize_constructed_graph_witness_operationoperationleft + S (pfp_left_realize_constructed_graph_witness_operation) = (p)) /\\ (((exists pfa_gap_realize_constructed_graph_witness_operationoperationright. pfa_gap_realize_constructed_graph_witness_operationoperationright + S (pfp_right_realize_constructed_graph_witness_operation) = (p)) /\\ ((((exists pfa_gap_realize_constructed_graph_witness_operationoperationresultbound. pfa_gap_realize_constructed_graph_witness_operationoperationresultbound + S (pfp_value_realize_constructed_graph_witness_operation) = (p)) /\\ ((exists pfa_offset_left_realize_constructed_graph_witness_operationoperationresultcongruence pfa_offset_right_realize_constructed_graph_witness_operationoperationresultcongruence. ((pfp_left_realize_constructed_graph_witness_operation) + (pfp_right_realize_constructed_graph_witness_operation)) + (p) * pfa_offset_left_realize_constructed_graph_witness_operationoperationresultcongruence = (pfp_value_realize_constructed_graph_witness_operation) + (p) * pfa_offset_right_realize_constructed_graph_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_realize_constructed_graph_witness_output pfrep_left_realize_constructed_graph_witness_output pfrep_right_realize_constructed_graph_witness_output. ((exists pfrep_position_realize_constructed_graph_witness_outputfirst. ((pfrep_position_realize_constructed_graph_witness_outputfirst+S (pfrep_power_realize_constructed_graph_witness_output)=(pfaa_length_realize_constructed_graph)) /\\ ((((exists ff_h_pfp_realize_constructed_graph_witness_outputfirstentry. ff_h_pfp_realize_constructed_graph_witness_outputfirstentry + S (pfrep_left_realize_constructed_graph_witness_output) = S ((S (pfrep_position_realize_constructed_graph_witness_outputfirst)) * pfaa_sum_c_realize_constructed_graph)) /\\ exists ff_q_pfp_realize_constructed_graph_witness_outputfirstentry. pfaa_sum_b_realize_constructed_graph = ff_q_pfp_realize_constructed_graph_witness_outputfirstentry * S ((S (pfrep_position_realize_constructed_graph_witness_outputfirst)) * pfaa_sum_c_realize_constructed_graph) + (pfrep_left_realize_constructed_graph_witness_output)))))) \\/ (((exists pfrep_gap_realize_constructed_graph_witness_outputfirstoutside. pfrep_gap_realize_constructed_graph_witness_outputfirstoutside+(pfaa_length_realize_constructed_graph)=(pfrep_power_realize_constructed_graph_witness_output)) /\\ (((pfrep_left_realize_constructed_graph_witness_output)=0))))) -> ((exists pfrep_position_realize_constructed_graph_witness_outputsecond. ((pfrep_position_realize_constructed_graph_witness_outputsecond+S (pfrep_power_realize_constructed_graph_witness_output)=(K)) /\\ ((((exists ff_h_pfp_realize_constructed_graph_witness_outputsecondentry. ff_h_pfp_realize_constructed_graph_witness_outputsecondentry + S (pfrep_right_realize_constructed_graph_witness_output) = S ((S (pfrep_position_realize_constructed_graph_witness_outputsecond)) * x1)) /\\ exists ff_q_pfp_realize_constructed_graph_witness_outputsecondentry. x = ff_q_pfp_realize_constructed_graph_witness_outputsecondentry * S ((S (pfrep_position_realize_constructed_graph_witness_outputsecond)) * x1) + (pfrep_right_realize_constructed_graph_witness_output)))))) \\/ (((exists pfrep_gap_realize_constructed_graph_witness_outputsecondoutside. pfrep_gap_realize_constructed_graph_witness_outputsecondoutside+(K)=(pfrep_power_realize_constructed_graph_witness_output)) /\\ (((pfrep_right_realize_constructed_graph_witness_output)=0))))) -> pfrep_left_realize_constructed_graph_witness_output=pfrep_right_realize_constructed_graph_witness_output))))))))))))", "specialize prime_field_polynomial_aligned_add_from_fixed (p)", "specialize prime_field_polynomial_aligned_add_from_fixed (ub)", "specialize prime_field_polynomial_aligned_add_from_fixed (uc)", "specialize prime_field_polynomial_aligned_add_from_fixed (vb)", "specialize prime_field_polynomial_aligned_add_from_fixed (vc)", "specialize prime_field_polynomial_aligned_add_from_fixed (x)", "specialize prime_field_polynomial_aligned_add_from_fixed (x1)", "specialize prime_field_polynomial_aligned_add_from_fixed (K)", "apply prime_field_polynomial_aligned_add_from_fixed", "exact hz_witness_witness", "have htgraph : ((forall fom_index_pfp_realize_target_graph_left_bounded. (exists fom_gap_pfp_realize_target_graph_left_bounded_index_bound. fom_gap_pfp_realize_target_graph_left_bounded_index_bound + S (fom_index_pfp_realize_target_graph_left_bounded) = K) -> exists fom_value_pfp_realize_target_graph_left_bounded. ((((exists fom_beta_height_pfp_realize_target_graph_left_bounded_entry. fom_beta_height_pfp_realize_target_graph_left_bounded_entry + S (fom_value_pfp_realize_target_graph_left_bounded) = S ((S (fom_index_pfp_realize_target_graph_left_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_realize_target_graph_left_bounded_entry. ub = fom_beta_quotient_pfp_realize_target_graph_left_bounded_entry * S ((S (fom_index_pfp_realize_target_graph_left_bounded)) * uc) + (fom_value_pfp_realize_target_graph_left_bounded))) /\\ (exists fom_gap_pfp_realize_target_graph_left_bounded_value_bound. fom_gap_pfp_realize_target_graph_left_bounded_value_bound + S (fom_value_pfp_realize_target_graph_left_bounded) = p))) /\\ (((forall fom_index_pfp_realize_target_graph_right_bounded. (exists fom_gap_pfp_realize_target_graph_right_bounded_index_bound. fom_gap_pfp_realize_target_graph_right_bounded_index_bound + S (fom_index_pfp_realize_target_graph_right_bounded) = K) -> exists fom_value_pfp_realize_target_graph_right_bounded. ((((exists fom_beta_height_pfp_realize_target_graph_right_bounded_entry. fom_beta_height_pfp_realize_target_graph_right_bounded_entry + S (fom_value_pfp_realize_target_graph_right_bounded) = S ((S (fom_index_pfp_realize_target_graph_right_bounded)) * vc)) /\\ exists fom_beta_quotient_pfp_realize_target_graph_right_bounded_entry. vb = fom_beta_quotient_pfp_realize_target_graph_right_bounded_entry * S ((S (fom_index_pfp_realize_target_graph_right_bounded)) * vc) + (fom_value_pfp_realize_target_graph_right_bounded))) /\\ (exists fom_gap_pfp_realize_target_graph_right_bounded_value_bound. fom_gap_pfp_realize_target_graph_right_bounded_value_bound + S (fom_value_pfp_realize_target_graph_right_bounded) = p))) /\\ (((forall fom_index_pfp_realize_target_graph_result_bounded. (exists fom_gap_pfp_realize_target_graph_result_bounded_index_bound. fom_gap_pfp_realize_target_graph_result_bounded_index_bound + S (fom_index_pfp_realize_target_graph_result_bounded) = K) -> exists fom_value_pfp_realize_target_graph_result_bounded. ((((exists fom_beta_height_pfp_realize_target_graph_result_bounded_entry. fom_beta_height_pfp_realize_target_graph_result_bounded_entry + S (fom_value_pfp_realize_target_graph_result_bounded) = S ((S (fom_index_pfp_realize_target_graph_result_bounded)) * tc)) /\\ exists fom_beta_quotient_pfp_realize_target_graph_result_bounded_entry. tb = fom_beta_quotient_pfp_realize_target_graph_result_bounded_entry * S ((S (fom_index_pfp_realize_target_graph_result_bounded)) * tc) + (fom_value_pfp_realize_target_graph_result_bounded))) /\\ (exists fom_gap_pfp_realize_target_graph_result_bounded_value_bound. fom_gap_pfp_realize_target_graph_result_bounded_value_bound + S (fom_value_pfp_realize_target_graph_result_bounded) = p))) /\\ ((exists pfaa_left_b_realize_target_graph pfaa_left_c_realize_target_graph pfaa_right_b_realize_target_graph pfaa_right_c_realize_target_graph pfaa_sum_b_realize_target_graph pfaa_sum_c_realize_target_graph pfaa_length_realize_target_graph. ((((forall pfrep_power_realize_target_graph_witness_common_left pfrep_left_realize_target_graph_witness_common_left pfrep_right_realize_target_graph_witness_common_left. ((exists pfrep_position_realize_target_graph_witness_common_leftfirst. ((pfrep_position_realize_target_graph_witness_common_leftfirst+S (pfrep_power_realize_target_graph_witness_common_left)=(K)) /\\ ((((exists ff_h_pfp_realize_target_graph_witness_common_leftfirstentry. ff_h_pfp_realize_target_graph_witness_common_leftfirstentry + S (pfrep_left_realize_target_graph_witness_common_left) = S ((S (pfrep_position_realize_target_graph_witness_common_leftfirst)) * uc)) /\\ exists ff_q_pfp_realize_target_graph_witness_common_leftfirstentry. ub = ff_q_pfp_realize_target_graph_witness_common_leftfirstentry * S ((S (pfrep_position_realize_target_graph_witness_common_leftfirst)) * uc) + (pfrep_left_realize_target_graph_witness_common_left)))))) \\/ (((exists pfrep_gap_realize_target_graph_witness_common_leftfirstoutside. pfrep_gap_realize_target_graph_witness_common_leftfirstoutside+(K)=(pfrep_power_realize_target_graph_witness_common_left)) /\\ (((pfrep_left_realize_target_graph_witness_common_left)=0))))) -> ((exists pfrep_position_realize_target_graph_witness_common_leftsecond. ((pfrep_position_realize_target_graph_witness_common_leftsecond+S (pfrep_power_realize_target_graph_witness_common_left)=(pfaa_length_realize_target_graph)) /\\ ((((exists ff_h_pfp_realize_target_graph_witness_common_leftsecondentry. ff_h_pfp_realize_target_graph_witness_common_leftsecondentry + S (pfrep_right_realize_target_graph_witness_common_left) = S ((S (pfrep_position_realize_target_graph_witness_common_leftsecond)) * pfaa_left_c_realize_target_graph)) /\\ exists ff_q_pfp_realize_target_graph_witness_common_leftsecondentry. pfaa_left_b_realize_target_graph = ff_q_pfp_realize_target_graph_witness_common_leftsecondentry * S ((S (pfrep_position_realize_target_graph_witness_common_leftsecond)) * pfaa_left_c_realize_target_graph) + (pfrep_right_realize_target_graph_witness_common_left)))))) \\/ (((exists pfrep_gap_realize_target_graph_witness_common_leftsecondoutside. pfrep_gap_realize_target_graph_witness_common_leftsecondoutside+(pfaa_length_realize_target_graph)=(pfrep_power_realize_target_graph_witness_common_left)) /\\ (((pfrep_right_realize_target_graph_witness_common_left)=0))))) -> pfrep_left_realize_target_graph_witness_common_left=pfrep_right_realize_target_graph_witness_common_left) /\\ ((forall pfrep_power_realize_target_graph_witness_common_right pfrep_left_realize_target_graph_witness_common_right pfrep_right_realize_target_graph_witness_common_right. ((exists pfrep_position_realize_target_graph_witness_common_rightfirst. ((pfrep_position_realize_target_graph_witness_common_rightfirst+S (pfrep_power_realize_target_graph_witness_common_right)=(K)) /\\ ((((exists ff_h_pfp_realize_target_graph_witness_common_rightfirstentry. ff_h_pfp_realize_target_graph_witness_common_rightfirstentry + S (pfrep_left_realize_target_graph_witness_common_right) = S ((S (pfrep_position_realize_target_graph_witness_common_rightfirst)) * vc)) /\\ exists ff_q_pfp_realize_target_graph_witness_common_rightfirstentry. vb = ff_q_pfp_realize_target_graph_witness_common_rightfirstentry * S ((S (pfrep_position_realize_target_graph_witness_common_rightfirst)) * vc) + (pfrep_left_realize_target_graph_witness_common_right)))))) \\/ (((exists pfrep_gap_realize_target_graph_witness_common_rightfirstoutside. pfrep_gap_realize_target_graph_witness_common_rightfirstoutside+(K)=(pfrep_power_realize_target_graph_witness_common_right)) /\\ (((pfrep_left_realize_target_graph_witness_common_right)=0))))) -> ((exists pfrep_position_realize_target_graph_witness_common_rightsecond. ((pfrep_position_realize_target_graph_witness_common_rightsecond+S (pfrep_power_realize_target_graph_witness_common_right)=(pfaa_length_realize_target_graph)) /\\ ((((exists ff_h_pfp_realize_target_graph_witness_common_rightsecondentry. ff_h_pfp_realize_target_graph_witness_common_rightsecondentry + S (pfrep_right_realize_target_graph_witness_common_right) = S ((S (pfrep_position_realize_target_graph_witness_common_rightsecond)) * pfaa_right_c_realize_target_graph)) /\\ exists ff_q_pfp_realize_target_graph_witness_common_rightsecondentry. pfaa_right_b_realize_target_graph = ff_q_pfp_realize_target_graph_witness_common_rightsecondentry * S ((S (pfrep_position_realize_target_graph_witness_common_rightsecond)) * pfaa_right_c_realize_target_graph) + (pfrep_right_realize_target_graph_witness_common_right)))))) \\/ (((exists pfrep_gap_realize_target_graph_witness_common_rightsecondoutside. pfrep_gap_realize_target_graph_witness_common_rightsecondoutside+(pfaa_length_realize_target_graph)=(pfrep_power_realize_target_graph_witness_common_right)) /\\ (((pfrep_right_realize_target_graph_witness_common_right)=0))))) -> pfrep_left_realize_target_graph_witness_common_right=pfrep_right_realize_target_graph_witness_common_right)))) /\\ (((forall pfp_index_realize_target_graph_witness_operation. (exists pfa_gap_realize_target_graph_witness_operationindex. pfa_gap_realize_target_graph_witness_operationindex + S (pfp_index_realize_target_graph_witness_operation) = (pfaa_length_realize_target_graph)) -> exists pfp_left_realize_target_graph_witness_operation pfp_right_realize_target_graph_witness_operation pfp_value_realize_target_graph_witness_operation. ((((exists ff_h_pfp_realize_target_graph_witness_operationleft. ff_h_pfp_realize_target_graph_witness_operationleft + S (pfp_left_realize_target_graph_witness_operation) = S ((S (pfp_index_realize_target_graph_witness_operation)) * pfaa_left_c_realize_target_graph)) /\\ exists ff_q_pfp_realize_target_graph_witness_operationleft. pfaa_left_b_realize_target_graph = ff_q_pfp_realize_target_graph_witness_operationleft * S ((S (pfp_index_realize_target_graph_witness_operation)) * pfaa_left_c_realize_target_graph) + (pfp_left_realize_target_graph_witness_operation))) /\\ (((((exists ff_h_pfp_realize_target_graph_witness_operationright. ff_h_pfp_realize_target_graph_witness_operationright + S (pfp_right_realize_target_graph_witness_operation) = S ((S (pfp_index_realize_target_graph_witness_operation)) * pfaa_right_c_realize_target_graph)) /\\ exists ff_q_pfp_realize_target_graph_witness_operationright. pfaa_right_b_realize_target_graph = ff_q_pfp_realize_target_graph_witness_operationright * S ((S (pfp_index_realize_target_graph_witness_operation)) * pfaa_right_c_realize_target_graph) + (pfp_right_realize_target_graph_witness_operation))) /\\ (((((exists ff_h_pfp_realize_target_graph_witness_operationtarget. ff_h_pfp_realize_target_graph_witness_operationtarget + S (pfp_value_realize_target_graph_witness_operation) = S ((S (pfp_index_realize_target_graph_witness_operation)) * pfaa_sum_c_realize_target_graph)) /\\ exists ff_q_pfp_realize_target_graph_witness_operationtarget. pfaa_sum_b_realize_target_graph = ff_q_pfp_realize_target_graph_witness_operationtarget * S ((S (pfp_index_realize_target_graph_witness_operation)) * pfaa_sum_c_realize_target_graph) + (pfp_value_realize_target_graph_witness_operation))) /\\ ((((exists pfa_gap_realize_target_graph_witness_operationoperationleft. pfa_gap_realize_target_graph_witness_operationoperationleft + S (pfp_left_realize_target_graph_witness_operation) = (p)) /\\ (((exists pfa_gap_realize_target_graph_witness_operationoperationright. pfa_gap_realize_target_graph_witness_operationoperationright + S (pfp_right_realize_target_graph_witness_operation) = (p)) /\\ ((((exists pfa_gap_realize_target_graph_witness_operationoperationresultbound. pfa_gap_realize_target_graph_witness_operationoperationresultbound + S (pfp_value_realize_target_graph_witness_operation) = (p)) /\\ ((exists pfa_offset_left_realize_target_graph_witness_operationoperationresultcongruence pfa_offset_right_realize_target_graph_witness_operationoperationresultcongruence. ((pfp_left_realize_target_graph_witness_operation) + (pfp_right_realize_target_graph_witness_operation)) + (p) * pfa_offset_left_realize_target_graph_witness_operationoperationresultcongruence = (pfp_value_realize_target_graph_witness_operation) + (p) * pfa_offset_right_realize_target_graph_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_realize_target_graph_witness_output pfrep_left_realize_target_graph_witness_output pfrep_right_realize_target_graph_witness_output. ((exists pfrep_position_realize_target_graph_witness_outputfirst. ((pfrep_position_realize_target_graph_witness_outputfirst+S (pfrep_power_realize_target_graph_witness_output)=(pfaa_length_realize_target_graph)) /\\ ((((exists ff_h_pfp_realize_target_graph_witness_outputfirstentry. ff_h_pfp_realize_target_graph_witness_outputfirstentry + S (pfrep_left_realize_target_graph_witness_output) = S ((S (pfrep_position_realize_target_graph_witness_outputfirst)) * pfaa_sum_c_realize_target_graph)) /\\ exists ff_q_pfp_realize_target_graph_witness_outputfirstentry. pfaa_sum_b_realize_target_graph = ff_q_pfp_realize_target_graph_witness_outputfirstentry * S ((S (pfrep_position_realize_target_graph_witness_outputfirst)) * pfaa_sum_c_realize_target_graph) + (pfrep_left_realize_target_graph_witness_output)))))) \\/ (((exists pfrep_gap_realize_target_graph_witness_outputfirstoutside. pfrep_gap_realize_target_graph_witness_outputfirstoutside+(pfaa_length_realize_target_graph)=(pfrep_power_realize_target_graph_witness_output)) /\\ (((pfrep_left_realize_target_graph_witness_output)=0))))) -> ((exists pfrep_position_realize_target_graph_witness_outputsecond. ((pfrep_position_realize_target_graph_witness_outputsecond+S (pfrep_power_realize_target_graph_witness_output)=(K)) /\\ ((((exists ff_h_pfp_realize_target_graph_witness_outputsecondentry. ff_h_pfp_realize_target_graph_witness_outputsecondentry + S (pfrep_right_realize_target_graph_witness_output) = S ((S (pfrep_position_realize_target_graph_witness_outputsecond)) * tc)) /\\ exists ff_q_pfp_realize_target_graph_witness_outputsecondentry. tb = ff_q_pfp_realize_target_graph_witness_outputsecondentry * S ((S (pfrep_position_realize_target_graph_witness_outputsecond)) * tc) + (pfrep_right_realize_target_graph_witness_output)))))) \\/ (((exists pfrep_gap_realize_target_graph_witness_outputsecondoutside. pfrep_gap_realize_target_graph_witness_outputsecondoutside+(K)=(pfrep_power_realize_target_graph_witness_output)) /\\ (((pfrep_right_realize_target_graph_witness_output)=0))))) -> pfrep_left_realize_target_graph_witness_output=pfrep_right_realize_target_graph_witness_output))))))))))))", "specialize prime_field_polynomial_aligned_add_transport (p)", "specialize prime_field_polynomial_aligned_add_transport (ab)", "specialize prime_field_polynomial_aligned_add_transport (ac)", "specialize prime_field_polynomial_aligned_add_transport (L)", "specialize prime_field_polynomial_aligned_add_transport (bb)", "specialize prime_field_polynomial_aligned_add_transport (bc)", "specialize prime_field_polynomial_aligned_add_transport (M)", "specialize prime_field_polynomial_aligned_add_transport (rb)", "specialize prime_field_polynomial_aligned_add_transport (rc)", "specialize prime_field_polynomial_aligned_add_transport (N)", "specialize prime_field_polynomial_aligned_add_transport (ub)", "specialize prime_field_polynomial_aligned_add_transport (uc)", "specialize prime_field_polynomial_aligned_add_transport (K)", "specialize prime_field_polynomial_aligned_add_transport (vb)", "specialize prime_field_polynomial_aligned_add_transport (vc)", "specialize prime_field_polynomial_aligned_add_transport (K)", "specialize prime_field_polynomial_aligned_add_transport (tb)", "specialize prime_field_polynomial_aligned_add_transport (tc)", "specialize prime_field_polynomial_aligned_add_transport (K)", "apply prime_field_polynomial_aligned_add_transport", "exact hu", "exact hv", "exact ht", "specialize prime_field_polynomial_equivalent_symmetric (ab)", "specialize prime_field_polynomial_equivalent_symmetric (ac)", "specialize prime_field_polynomial_equivalent_symmetric (L)", "specialize prime_field_polynomial_equivalent_symmetric (ub)", "specialize prime_field_polynomial_equivalent_symmetric (uc)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "apply prime_field_polynomial_equivalent_symmetric", "exact hc_left", "specialize prime_field_polynomial_equivalent_symmetric (bb)", "specialize prime_field_polynomial_equivalent_symmetric (bc)", "specialize prime_field_polynomial_equivalent_symmetric (M)", "specialize prime_field_polynomial_equivalent_symmetric (vb)", "specialize prime_field_polynomial_equivalent_symmetric (vc)", "specialize prime_field_polynomial_equivalent_symmetric (K)", "apply prime_field_polynomial_equivalent_symmetric", "exact hc_right", "exact hr", "exact h", "have he : forall pfrep_power_realize_output_equivalent pfrep_left_realize_output_equivalent pfrep_right_realize_output_equivalent. ((exists pfrep_position_realize_output_equivalentfirst. ((pfrep_position_realize_output_equivalentfirst+S (pfrep_power_realize_output_equivalent)=(K)) /\\ ((((exists ff_h_pfp_realize_output_equivalentfirstentry. ff_h_pfp_realize_output_equivalentfirstentry + S (pfrep_left_realize_output_equivalent) = S ((S (pfrep_position_realize_output_equivalentfirst)) * x1)) /\\ exists ff_q_pfp_realize_output_equivalentfirstentry. x = ff_q_pfp_realize_output_equivalentfirstentry * S ((S (pfrep_position_realize_output_equivalentfirst)) * x1) + (pfrep_left_realize_output_equivalent)))))) \\/ (((exists pfrep_gap_realize_output_equivalentfirstoutside. pfrep_gap_realize_output_equivalentfirstoutside+(K)=(pfrep_power_realize_output_equivalent)) /\\ (((pfrep_left_realize_output_equivalent)=0))))) -> ((exists pfrep_position_realize_output_equivalentsecond. ((pfrep_position_realize_output_equivalentsecond+S (pfrep_power_realize_output_equivalent)=(K)) /\\ ((((exists ff_h_pfp_realize_output_equivalentsecondentry. ff_h_pfp_realize_output_equivalentsecondentry + S (pfrep_right_realize_output_equivalent) = S ((S (pfrep_position_realize_output_equivalentsecond)) * tc)) /\\ exists ff_q_pfp_realize_output_equivalentsecondentry. tb = ff_q_pfp_realize_output_equivalentsecondentry * S ((S (pfrep_position_realize_output_equivalentsecond)) * tc) + (pfrep_right_realize_output_equivalent)))))) \\/ (((exists pfrep_gap_realize_output_equivalentsecondoutside. pfrep_gap_realize_output_equivalentsecondoutside+(K)=(pfrep_power_realize_output_equivalent)) /\\ (((pfrep_right_realize_output_equivalent)=0))))) -> pfrep_left_realize_output_equivalent=pfrep_right_realize_output_equivalent", "specialize prime_field_polynomial_aligned_add_functional (p)", "specialize prime_field_polynomial_aligned_add_functional (ub)", "specialize prime_field_polynomial_aligned_add_functional (uc)", "specialize prime_field_polynomial_aligned_add_functional (K)", "specialize prime_field_polynomial_aligned_add_functional (vb)", "specialize prime_field_polynomial_aligned_add_functional (vc)", "specialize prime_field_polynomial_aligned_add_functional (K)", "specialize prime_field_polynomial_aligned_add_functional (x)", "specialize prime_field_polynomial_aligned_add_functional (x1)", "specialize prime_field_polynomial_aligned_add_functional (K)", "specialize prime_field_polynomial_aligned_add_functional (tb)", "specialize prime_field_polynomial_aligned_add_functional (tc)", "specialize prime_field_polynomial_aligned_add_functional (K)", "apply prime_field_polynomial_aligned_add_functional", "exact hp", "exact hzgraph", "exact htgraph", "specialize prime_field_polynomial_add_transport (p)", "specialize prime_field_polynomial_add_transport (ub)", "specialize prime_field_polynomial_add_transport (uc)", "specialize prime_field_polynomial_add_transport (vb)", "specialize prime_field_polynomial_add_transport (vc)", "specialize prime_field_polynomial_add_transport (x)", "specialize prime_field_polynomial_add_transport (x1)", "specialize prime_field_polynomial_add_transport (ub)", "specialize prime_field_polynomial_add_transport (uc)", "specialize prime_field_polynomial_add_transport (vb)", "specialize prime_field_polynomial_add_transport (vc)", "specialize prime_field_polynomial_add_transport (tb)", "specialize prime_field_polynomial_add_transport (tc)", "specialize prime_field_polynomial_add_transport (K)", "apply prime_field_polynomial_add_transport", "intro i", "intro a", "intro hi", "intro ha", "exact ha", "intro i", "intro a", "intro hi", "intro ha", "exact ha", "specialize prime_field_polynomial_equivalent_implies_equal_same_length (x)", "specialize prime_field_polynomial_equivalent_implies_equal_same_length (x1)", "specialize prime_field_polynomial_equivalent_implies_equal_same_length (tb)", "specialize prime_field_polynomial_equivalent_implies_equal_same_length (tc)", "specialize prime_field_polynomial_equivalent_implies_equal_same_length (K)", "apply prime_field_polynomial_equivalent_implies_equal_same_length", "exact he", "exact hz_witness_witness" ], "script_sha256": "80141751625ac1394b29411c6440880f331cb44360eae15548a5092726065f5a", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_aligned_add_candidate.py", "sha256": "a05bb4f5c4230ca05f51690d3ab82e33ff4596af65176874e25fbe38cf87a0db" }, "statement": "forall p ab ac L bb bc M rb rc N ub uc vb vc tb tc K. (~((p) = 1) /\\ forall pfa_factor_left_realize_prime pfa_factor_right_realize_prime. (p) = pfa_factor_left_realize_prime * pfa_factor_right_realize_prime -> pfa_factor_left_realize_prime = 1 \\/ pfa_factor_right_realize_prime = 1) -> (((forall fom_index_pfp_realize_original_left_bounded. (exists fom_gap_pfp_realize_original_left_bounded_index_bound. fom_gap_pfp_realize_original_left_bounded_index_bound + S (fom_index_pfp_realize_original_left_bounded) = L) -> exists fom_value_pfp_realize_original_left_bounded. ((((exists fom_beta_height_pfp_realize_original_left_bounded_entry. fom_beta_height_pfp_realize_original_left_bounded_entry + S (fom_value_pfp_realize_original_left_bounded) = S ((S (fom_index_pfp_realize_original_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_realize_original_left_bounded_entry. ab = fom_beta_quotient_pfp_realize_original_left_bounded_entry * S ((S (fom_index_pfp_realize_original_left_bounded)) * ac) + (fom_value_pfp_realize_original_left_bounded))) /\\ (exists fom_gap_pfp_realize_original_left_bounded_value_bound. fom_gap_pfp_realize_original_left_bounded_value_bound + S (fom_value_pfp_realize_original_left_bounded) = p))) /\\ (((forall fom_index_pfp_realize_original_right_bounded. (exists fom_gap_pfp_realize_original_right_bounded_index_bound. fom_gap_pfp_realize_original_right_bounded_index_bound + S (fom_index_pfp_realize_original_right_bounded) = M) -> exists fom_value_pfp_realize_original_right_bounded. ((((exists fom_beta_height_pfp_realize_original_right_bounded_entry. fom_beta_height_pfp_realize_original_right_bounded_entry + S (fom_value_pfp_realize_original_right_bounded) = S ((S (fom_index_pfp_realize_original_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_realize_original_right_bounded_entry. bb = fom_beta_quotient_pfp_realize_original_right_bounded_entry * S ((S (fom_index_pfp_realize_original_right_bounded)) * bc) + (fom_value_pfp_realize_original_right_bounded))) /\\ (exists fom_gap_pfp_realize_original_right_bounded_value_bound. fom_gap_pfp_realize_original_right_bounded_value_bound + S (fom_value_pfp_realize_original_right_bounded) = p))) /\\ (((forall fom_index_pfp_realize_original_result_bounded. (exists fom_gap_pfp_realize_original_result_bounded_index_bound. fom_gap_pfp_realize_original_result_bounded_index_bound + S (fom_index_pfp_realize_original_result_bounded) = N) -> exists fom_value_pfp_realize_original_result_bounded. ((((exists fom_beta_height_pfp_realize_original_result_bounded_entry. fom_beta_height_pfp_realize_original_result_bounded_entry + S (fom_value_pfp_realize_original_result_bounded) = S ((S (fom_index_pfp_realize_original_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_realize_original_result_bounded_entry. rb = fom_beta_quotient_pfp_realize_original_result_bounded_entry * S ((S (fom_index_pfp_realize_original_result_bounded)) * rc) + (fom_value_pfp_realize_original_result_bounded))) /\\ (exists fom_gap_pfp_realize_original_result_bounded_value_bound. fom_gap_pfp_realize_original_result_bounded_value_bound + S (fom_value_pfp_realize_original_result_bounded) = p))) /\\ ((exists pfaa_left_b_realize_original pfaa_left_c_realize_original pfaa_right_b_realize_original pfaa_right_c_realize_original pfaa_sum_b_realize_original pfaa_sum_c_realize_original pfaa_length_realize_original. ((((forall pfrep_power_realize_original_witness_common_left pfrep_left_realize_original_witness_common_left pfrep_right_realize_original_witness_common_left. ((exists pfrep_position_realize_original_witness_common_leftfirst. ((pfrep_position_realize_original_witness_common_leftfirst+S (pfrep_power_realize_original_witness_common_left)=(L)) /\\ ((((exists ff_h_pfp_realize_original_witness_common_leftfirstentry. ff_h_pfp_realize_original_witness_common_leftfirstentry + S (pfrep_left_realize_original_witness_common_left) = S ((S (pfrep_position_realize_original_witness_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_realize_original_witness_common_leftfirstentry. ab = ff_q_pfp_realize_original_witness_common_leftfirstentry * S ((S (pfrep_position_realize_original_witness_common_leftfirst)) * ac) + (pfrep_left_realize_original_witness_common_left)))))) \\/ (((exists pfrep_gap_realize_original_witness_common_leftfirstoutside. pfrep_gap_realize_original_witness_common_leftfirstoutside+(L)=(pfrep_power_realize_original_witness_common_left)) /\\ (((pfrep_left_realize_original_witness_common_left)=0))))) -> ((exists pfrep_position_realize_original_witness_common_leftsecond. ((pfrep_position_realize_original_witness_common_leftsecond+S (pfrep_power_realize_original_witness_common_left)=(pfaa_length_realize_original)) /\\ ((((exists ff_h_pfp_realize_original_witness_common_leftsecondentry. ff_h_pfp_realize_original_witness_common_leftsecondentry + S (pfrep_right_realize_original_witness_common_left) = S ((S (pfrep_position_realize_original_witness_common_leftsecond)) * pfaa_left_c_realize_original)) /\\ exists ff_q_pfp_realize_original_witness_common_leftsecondentry. pfaa_left_b_realize_original = ff_q_pfp_realize_original_witness_common_leftsecondentry * S ((S (pfrep_position_realize_original_witness_common_leftsecond)) * pfaa_left_c_realize_original) + (pfrep_right_realize_original_witness_common_left)))))) \\/ (((exists pfrep_gap_realize_original_witness_common_leftsecondoutside. pfrep_gap_realize_original_witness_common_leftsecondoutside+(pfaa_length_realize_original)=(pfrep_power_realize_original_witness_common_left)) /\\ (((pfrep_right_realize_original_witness_common_left)=0))))) -> pfrep_left_realize_original_witness_common_left=pfrep_right_realize_original_witness_common_left) /\\ ((forall pfrep_power_realize_original_witness_common_right pfrep_left_realize_original_witness_common_right pfrep_right_realize_original_witness_common_right. ((exists pfrep_position_realize_original_witness_common_rightfirst. ((pfrep_position_realize_original_witness_common_rightfirst+S (pfrep_power_realize_original_witness_common_right)=(M)) /\\ ((((exists ff_h_pfp_realize_original_witness_common_rightfirstentry. ff_h_pfp_realize_original_witness_common_rightfirstentry + S (pfrep_left_realize_original_witness_common_right) = S ((S (pfrep_position_realize_original_witness_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_realize_original_witness_common_rightfirstentry. bb = ff_q_pfp_realize_original_witness_common_rightfirstentry * S ((S (pfrep_position_realize_original_witness_common_rightfirst)) * bc) + (pfrep_left_realize_original_witness_common_right)))))) \\/ (((exists pfrep_gap_realize_original_witness_common_rightfirstoutside. pfrep_gap_realize_original_witness_common_rightfirstoutside+(M)=(pfrep_power_realize_original_witness_common_right)) /\\ (((pfrep_left_realize_original_witness_common_right)=0))))) -> ((exists pfrep_position_realize_original_witness_common_rightsecond. ((pfrep_position_realize_original_witness_common_rightsecond+S (pfrep_power_realize_original_witness_common_right)=(pfaa_length_realize_original)) /\\ ((((exists ff_h_pfp_realize_original_witness_common_rightsecondentry. ff_h_pfp_realize_original_witness_common_rightsecondentry + S (pfrep_right_realize_original_witness_common_right) = S ((S (pfrep_position_realize_original_witness_common_rightsecond)) * pfaa_right_c_realize_original)) /\\ exists ff_q_pfp_realize_original_witness_common_rightsecondentry. pfaa_right_b_realize_original = ff_q_pfp_realize_original_witness_common_rightsecondentry * S ((S (pfrep_position_realize_original_witness_common_rightsecond)) * pfaa_right_c_realize_original) + (pfrep_right_realize_original_witness_common_right)))))) \\/ (((exists pfrep_gap_realize_original_witness_common_rightsecondoutside. pfrep_gap_realize_original_witness_common_rightsecondoutside+(pfaa_length_realize_original)=(pfrep_power_realize_original_witness_common_right)) /\\ (((pfrep_right_realize_original_witness_common_right)=0))))) -> pfrep_left_realize_original_witness_common_right=pfrep_right_realize_original_witness_common_right)))) /\\ (((forall pfp_index_realize_original_witness_operation. (exists pfa_gap_realize_original_witness_operationindex. pfa_gap_realize_original_witness_operationindex + S (pfp_index_realize_original_witness_operation) = (pfaa_length_realize_original)) -> exists pfp_left_realize_original_witness_operation pfp_right_realize_original_witness_operation pfp_value_realize_original_witness_operation. ((((exists ff_h_pfp_realize_original_witness_operationleft. ff_h_pfp_realize_original_witness_operationleft + S (pfp_left_realize_original_witness_operation) = S ((S (pfp_index_realize_original_witness_operation)) * pfaa_left_c_realize_original)) /\\ exists ff_q_pfp_realize_original_witness_operationleft. pfaa_left_b_realize_original = ff_q_pfp_realize_original_witness_operationleft * S ((S (pfp_index_realize_original_witness_operation)) * pfaa_left_c_realize_original) + (pfp_left_realize_original_witness_operation))) /\\ (((((exists ff_h_pfp_realize_original_witness_operationright. ff_h_pfp_realize_original_witness_operationright + S (pfp_right_realize_original_witness_operation) = S ((S (pfp_index_realize_original_witness_operation)) * pfaa_right_c_realize_original)) /\\ exists ff_q_pfp_realize_original_witness_operationright. pfaa_right_b_realize_original = ff_q_pfp_realize_original_witness_operationright * S ((S (pfp_index_realize_original_witness_operation)) * pfaa_right_c_realize_original) + (pfp_right_realize_original_witness_operation))) /\\ (((((exists ff_h_pfp_realize_original_witness_operationtarget. ff_h_pfp_realize_original_witness_operationtarget + S (pfp_value_realize_original_witness_operation) = S ((S (pfp_index_realize_original_witness_operation)) * pfaa_sum_c_realize_original)) /\\ exists ff_q_pfp_realize_original_witness_operationtarget. pfaa_sum_b_realize_original = ff_q_pfp_realize_original_witness_operationtarget * S ((S (pfp_index_realize_original_witness_operation)) * pfaa_sum_c_realize_original) + (pfp_value_realize_original_witness_operation))) /\\ ((((exists pfa_gap_realize_original_witness_operationoperationleft. pfa_gap_realize_original_witness_operationoperationleft + S (pfp_left_realize_original_witness_operation) = (p)) /\\ (((exists pfa_gap_realize_original_witness_operationoperationright. pfa_gap_realize_original_witness_operationoperationright + S (pfp_right_realize_original_witness_operation) = (p)) /\\ ((((exists pfa_gap_realize_original_witness_operationoperationresultbound. pfa_gap_realize_original_witness_operationoperationresultbound + S (pfp_value_realize_original_witness_operation) = (p)) /\\ ((exists pfa_offset_left_realize_original_witness_operationoperationresultcongruence pfa_offset_right_realize_original_witness_operationoperationresultcongruence. ((pfp_left_realize_original_witness_operation) + (pfp_right_realize_original_witness_operation)) + (p) * pfa_offset_left_realize_original_witness_operationoperationresultcongruence = (pfp_value_realize_original_witness_operation) + (p) * pfa_offset_right_realize_original_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_realize_original_witness_output pfrep_left_realize_original_witness_output pfrep_right_realize_original_witness_output. ((exists pfrep_position_realize_original_witness_outputfirst. ((pfrep_position_realize_original_witness_outputfirst+S (pfrep_power_realize_original_witness_output)=(pfaa_length_realize_original)) /\\ ((((exists ff_h_pfp_realize_original_witness_outputfirstentry. ff_h_pfp_realize_original_witness_outputfirstentry + S (pfrep_left_realize_original_witness_output) = S ((S (pfrep_position_realize_original_witness_outputfirst)) * pfaa_sum_c_realize_original)) /\\ exists ff_q_pfp_realize_original_witness_outputfirstentry. pfaa_sum_b_realize_original = ff_q_pfp_realize_original_witness_outputfirstentry * S ((S (pfrep_position_realize_original_witness_outputfirst)) * pfaa_sum_c_realize_original) + (pfrep_left_realize_original_witness_output)))))) \\/ (((exists pfrep_gap_realize_original_witness_outputfirstoutside. pfrep_gap_realize_original_witness_outputfirstoutside+(pfaa_length_realize_original)=(pfrep_power_realize_original_witness_output)) /\\ (((pfrep_left_realize_original_witness_output)=0))))) -> ((exists pfrep_position_realize_original_witness_outputsecond. ((pfrep_position_realize_original_witness_outputsecond+S (pfrep_power_realize_original_witness_output)=(N)) /\\ ((((exists ff_h_pfp_realize_original_witness_outputsecondentry. ff_h_pfp_realize_original_witness_outputsecondentry + S (pfrep_right_realize_original_witness_output) = S ((S (pfrep_position_realize_original_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_realize_original_witness_outputsecondentry. rb = ff_q_pfp_realize_original_witness_outputsecondentry * S ((S (pfrep_position_realize_original_witness_outputsecond)) * rc) + (pfrep_right_realize_original_witness_output)))))) \\/ (((exists pfrep_gap_realize_original_witness_outputsecondoutside. pfrep_gap_realize_original_witness_outputsecondoutside+(N)=(pfrep_power_realize_original_witness_output)) /\\ (((pfrep_right_realize_original_witness_output)=0))))) -> pfrep_left_realize_original_witness_output=pfrep_right_realize_original_witness_output))))))))))))) -> (forall fom_index_pfp_realize_bound_0. (exists fom_gap_pfp_realize_bound_0_index_bound. fom_gap_pfp_realize_bound_0_index_bound + S (fom_index_pfp_realize_bound_0) = K) -> exists fom_value_pfp_realize_bound_0. ((((exists fom_beta_height_pfp_realize_bound_0_entry. fom_beta_height_pfp_realize_bound_0_entry + S (fom_value_pfp_realize_bound_0) = S ((S (fom_index_pfp_realize_bound_0)) * uc)) /\\ exists fom_beta_quotient_pfp_realize_bound_0_entry. ub = fom_beta_quotient_pfp_realize_bound_0_entry * S ((S (fom_index_pfp_realize_bound_0)) * uc) + (fom_value_pfp_realize_bound_0))) /\\ (exists fom_gap_pfp_realize_bound_0_value_bound. fom_gap_pfp_realize_bound_0_value_bound + S (fom_value_pfp_realize_bound_0) = p))) -> (forall fom_index_pfp_realize_bound_1. (exists fom_gap_pfp_realize_bound_1_index_bound. fom_gap_pfp_realize_bound_1_index_bound + S (fom_index_pfp_realize_bound_1) = K) -> exists fom_value_pfp_realize_bound_1. ((((exists fom_beta_height_pfp_realize_bound_1_entry. fom_beta_height_pfp_realize_bound_1_entry + S (fom_value_pfp_realize_bound_1) = S ((S (fom_index_pfp_realize_bound_1)) * vc)) /\\ exists fom_beta_quotient_pfp_realize_bound_1_entry. vb = fom_beta_quotient_pfp_realize_bound_1_entry * S ((S (fom_index_pfp_realize_bound_1)) * vc) + (fom_value_pfp_realize_bound_1))) /\\ (exists fom_gap_pfp_realize_bound_1_value_bound. fom_gap_pfp_realize_bound_1_value_bound + S (fom_value_pfp_realize_bound_1) = p))) -> (forall fom_index_pfp_realize_bound_2. (exists fom_gap_pfp_realize_bound_2_index_bound. fom_gap_pfp_realize_bound_2_index_bound + S (fom_index_pfp_realize_bound_2) = K) -> exists fom_value_pfp_realize_bound_2. ((((exists fom_beta_height_pfp_realize_bound_2_entry. fom_beta_height_pfp_realize_bound_2_entry + S (fom_value_pfp_realize_bound_2) = S ((S (fom_index_pfp_realize_bound_2)) * tc)) /\\ exists fom_beta_quotient_pfp_realize_bound_2_entry. tb = fom_beta_quotient_pfp_realize_bound_2_entry * S ((S (fom_index_pfp_realize_bound_2)) * tc) + (fom_value_pfp_realize_bound_2))) /\\ (exists fom_gap_pfp_realize_bound_2_value_bound. fom_gap_pfp_realize_bound_2_value_bound + S (fom_value_pfp_realize_bound_2) = p))) -> (((forall pfrep_power_realize_common_left pfrep_left_realize_common_left pfrep_right_realize_common_left. ((exists pfrep_position_realize_common_leftfirst. ((pfrep_position_realize_common_leftfirst+S (pfrep_power_realize_common_left)=(L)) /\\ ((((exists ff_h_pfp_realize_common_leftfirstentry. ff_h_pfp_realize_common_leftfirstentry + S (pfrep_left_realize_common_left) = S ((S (pfrep_position_realize_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_realize_common_leftfirstentry. ab = ff_q_pfp_realize_common_leftfirstentry * S ((S (pfrep_position_realize_common_leftfirst)) * ac) + (pfrep_left_realize_common_left)))))) \\/ (((exists pfrep_gap_realize_common_leftfirstoutside. pfrep_gap_realize_common_leftfirstoutside+(L)=(pfrep_power_realize_common_left)) /\\ (((pfrep_left_realize_common_left)=0))))) -> ((exists pfrep_position_realize_common_leftsecond. ((pfrep_position_realize_common_leftsecond+S (pfrep_power_realize_common_left)=(K)) /\\ ((((exists ff_h_pfp_realize_common_leftsecondentry. ff_h_pfp_realize_common_leftsecondentry + S (pfrep_right_realize_common_left) = S ((S (pfrep_position_realize_common_leftsecond)) * uc)) /\\ exists ff_q_pfp_realize_common_leftsecondentry. ub = ff_q_pfp_realize_common_leftsecondentry * S ((S (pfrep_position_realize_common_leftsecond)) * uc) + (pfrep_right_realize_common_left)))))) \\/ (((exists pfrep_gap_realize_common_leftsecondoutside. pfrep_gap_realize_common_leftsecondoutside+(K)=(pfrep_power_realize_common_left)) /\\ (((pfrep_right_realize_common_left)=0))))) -> pfrep_left_realize_common_left=pfrep_right_realize_common_left) /\\ ((forall pfrep_power_realize_common_right pfrep_left_realize_common_right pfrep_right_realize_common_right. ((exists pfrep_position_realize_common_rightfirst. ((pfrep_position_realize_common_rightfirst+S (pfrep_power_realize_common_right)=(M)) /\\ ((((exists ff_h_pfp_realize_common_rightfirstentry. ff_h_pfp_realize_common_rightfirstentry + S (pfrep_left_realize_common_right) = S ((S (pfrep_position_realize_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_realize_common_rightfirstentry. bb = ff_q_pfp_realize_common_rightfirstentry * S ((S (pfrep_position_realize_common_rightfirst)) * bc) + (pfrep_left_realize_common_right)))))) \\/ (((exists pfrep_gap_realize_common_rightfirstoutside. pfrep_gap_realize_common_rightfirstoutside+(M)=(pfrep_power_realize_common_right)) /\\ (((pfrep_left_realize_common_right)=0))))) -> ((exists pfrep_position_realize_common_rightsecond. ((pfrep_position_realize_common_rightsecond+S (pfrep_power_realize_common_right)=(K)) /\\ ((((exists ff_h_pfp_realize_common_rightsecondentry. ff_h_pfp_realize_common_rightsecondentry + S (pfrep_right_realize_common_right) = S ((S (pfrep_position_realize_common_rightsecond)) * vc)) /\\ exists ff_q_pfp_realize_common_rightsecondentry. vb = ff_q_pfp_realize_common_rightsecondentry * S ((S (pfrep_position_realize_common_rightsecond)) * vc) + (pfrep_right_realize_common_right)))))) \\/ (((exists pfrep_gap_realize_common_rightsecondoutside. pfrep_gap_realize_common_rightsecondoutside+(K)=(pfrep_power_realize_common_right)) /\\ (((pfrep_right_realize_common_right)=0))))) -> pfrep_left_realize_common_right=pfrep_right_realize_common_right)))) -> (forall pfrep_power_realize_output pfrep_left_realize_output pfrep_right_realize_output. ((exists pfrep_position_realize_outputfirst. ((pfrep_position_realize_outputfirst+S (pfrep_power_realize_output)=(N)) /\\ ((((exists ff_h_pfp_realize_outputfirstentry. ff_h_pfp_realize_outputfirstentry + S (pfrep_left_realize_output) = S ((S (pfrep_position_realize_outputfirst)) * rc)) /\\ exists ff_q_pfp_realize_outputfirstentry. rb = ff_q_pfp_realize_outputfirstentry * S ((S (pfrep_position_realize_outputfirst)) * rc) + (pfrep_left_realize_output)))))) \\/ (((exists pfrep_gap_realize_outputfirstoutside. pfrep_gap_realize_outputfirstoutside+(N)=(pfrep_power_realize_output)) /\\ (((pfrep_left_realize_output)=0))))) -> ((exists pfrep_position_realize_outputsecond. ((pfrep_position_realize_outputsecond+S (pfrep_power_realize_output)=(K)) /\\ ((((exists ff_h_pfp_realize_outputsecondentry. ff_h_pfp_realize_outputsecondentry + S (pfrep_right_realize_output) = S ((S (pfrep_position_realize_outputsecond)) * tc)) /\\ exists ff_q_pfp_realize_outputsecondentry. tb = ff_q_pfp_realize_outputsecondentry * S ((S (pfrep_position_realize_outputsecond)) * tc) + (pfrep_right_realize_output)))))) \\/ (((exists pfrep_gap_realize_outputsecondoutside. pfrep_gap_realize_outputsecondoutside+(K)=(pfrep_power_realize_output)) /\\ (((pfrep_right_realize_output)=0))))) -> pfrep_left_realize_output=pfrep_right_realize_output) -> (forall pfp_index_realize_actual_operation. (exists pfa_gap_realize_actual_operationindex. pfa_gap_realize_actual_operationindex + S (pfp_index_realize_actual_operation) = (K)) -> exists pfp_left_realize_actual_operation pfp_right_realize_actual_operation pfp_value_realize_actual_operation. ((((exists ff_h_pfp_realize_actual_operationleft. ff_h_pfp_realize_actual_operationleft + S (pfp_left_realize_actual_operation) = S ((S (pfp_index_realize_actual_operation)) * uc)) /\\ exists ff_q_pfp_realize_actual_operationleft. ub = ff_q_pfp_realize_actual_operationleft * S ((S (pfp_index_realize_actual_operation)) * uc) + (pfp_left_realize_actual_operation))) /\\ (((((exists ff_h_pfp_realize_actual_operationright. ff_h_pfp_realize_actual_operationright + S (pfp_right_realize_actual_operation) = S ((S (pfp_index_realize_actual_operation)) * vc)) /\\ exists ff_q_pfp_realize_actual_operationright. vb = ff_q_pfp_realize_actual_operationright * S ((S (pfp_index_realize_actual_operation)) * vc) + (pfp_right_realize_actual_operation))) /\\ (((((exists ff_h_pfp_realize_actual_operationtarget. ff_h_pfp_realize_actual_operationtarget + S (pfp_value_realize_actual_operation) = S ((S (pfp_index_realize_actual_operation)) * tc)) /\\ exists ff_q_pfp_realize_actual_operationtarget. tb = ff_q_pfp_realize_actual_operationtarget * S ((S (pfp_index_realize_actual_operation)) * tc) + (pfp_value_realize_actual_operation))) /\\ ((((exists pfa_gap_realize_actual_operationoperationleft. pfa_gap_realize_actual_operationoperationleft + S (pfp_left_realize_actual_operation) = (p)) /\\ (((exists pfa_gap_realize_actual_operationoperationright. pfa_gap_realize_actual_operationoperationright + S (pfp_right_realize_actual_operation) = (p)) /\\ ((((exists pfa_gap_realize_actual_operationoperationresultbound. pfa_gap_realize_actual_operationoperationresultbound + S (pfp_value_realize_actual_operation) = (p)) /\\ ((exists pfa_offset_left_realize_actual_operationoperationresultcongruence pfa_offset_right_realize_actual_operationoperationresultcongruence. 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(forall pfs_index_sub_fixed_input. (exists pfa_gap_sub_fixed_inputindex. pfa_gap_sub_fixed_inputindex + S (pfs_index_sub_fixed_input) = (K)) -> exists pfs_left_sub_fixed_input pfs_right_sub_fixed_input pfs_result_sub_fixed_input. ((((exists ff_h_pfp_sub_fixed_inputleft. ff_h_pfp_sub_fixed_inputleft + S (pfs_left_sub_fixed_input) = S ((S (pfs_index_sub_fixed_input)) * ac)) /\\ exists ff_q_pfp_sub_fixed_inputleft. ab = ff_q_pfp_sub_fixed_inputleft * S ((S (pfs_index_sub_fixed_input)) * ac) + (pfs_left_sub_fixed_input))) /\\ (((((exists ff_h_pfp_sub_fixed_inputright. ff_h_pfp_sub_fixed_inputright + S (pfs_right_sub_fixed_input) = S ((S (pfs_index_sub_fixed_input)) * bc)) /\\ exists ff_q_pfp_sub_fixed_inputright. bb = ff_q_pfp_sub_fixed_inputright * S ((S (pfs_index_sub_fixed_input)) * bc) + (pfs_right_sub_fixed_input))) /\\ (((((exists ff_h_pfp_sub_fixed_inputresult. ff_h_pfp_sub_fixed_inputresult + S (pfs_result_sub_fixed_input) = S ((S (pfs_index_sub_fixed_input)) * rc)) /\\ exists ff_q_pfp_sub_fixed_inputresult. rb = ff_q_pfp_sub_fixed_inputresult * S ((S (pfs_index_sub_fixed_input)) * rc) + (pfs_result_sub_fixed_input))) /\\ ((((exists pfa_gap_sub_fixed_inputoperationleft. pfa_gap_sub_fixed_inputoperationleft + S (pfs_right_sub_fixed_input) = (p)) /\\ (((exists pfa_gap_sub_fixed_inputoperationright. pfa_gap_sub_fixed_inputoperationright + S (pfs_result_sub_fixed_input) = (p)) /\\ ((((exists pfa_gap_sub_fixed_inputoperationresultbound. pfa_gap_sub_fixed_inputoperationresultbound + S (pfs_left_sub_fixed_input) = (p)) /\\ ((exists pfa_offset_left_sub_fixed_inputoperationresultcongruence pfa_offset_right_sub_fixed_inputoperationresultcongruence. ((pfs_right_sub_fixed_input) + (pfs_result_sub_fixed_input)) + (p) * pfa_offset_left_sub_fixed_inputoperationresultcongruence = (pfs_left_sub_fixed_input) + (p) * pfa_offset_right_sub_fixed_inputoperationresultcongruence)))))))))))))))) -> (((forall fom_index_pfp_sub_fixed_result_left_bounded. (exists fom_gap_pfp_sub_fixed_result_left_bounded_index_bound. fom_gap_pfp_sub_fixed_result_left_bounded_index_bound + S (fom_index_pfp_sub_fixed_result_left_bounded) = K) -> exists fom_value_pfp_sub_fixed_result_left_bounded. ((((exists fom_beta_height_pfp_sub_fixed_result_left_bounded_entry. fom_beta_height_pfp_sub_fixed_result_left_bounded_entry + S (fom_value_pfp_sub_fixed_result_left_bounded) = S ((S (fom_index_pfp_sub_fixed_result_left_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_sub_fixed_result_left_bounded_entry. bb = fom_beta_quotient_pfp_sub_fixed_result_left_bounded_entry * S ((S (fom_index_pfp_sub_fixed_result_left_bounded)) * bc) + (fom_value_pfp_sub_fixed_result_left_bounded))) /\\ (exists fom_gap_pfp_sub_fixed_result_left_bounded_value_bound. fom_gap_pfp_sub_fixed_result_left_bounded_value_bound + S (fom_value_pfp_sub_fixed_result_left_bounded) = p))) /\\ (((forall fom_index_pfp_sub_fixed_result_right_bounded. (exists fom_gap_pfp_sub_fixed_result_right_bounded_index_bound. fom_gap_pfp_sub_fixed_result_right_bounded_index_bound + S (fom_index_pfp_sub_fixed_result_right_bounded) = K) -> exists fom_value_pfp_sub_fixed_result_right_bounded. ((((exists fom_beta_height_pfp_sub_fixed_result_right_bounded_entry. fom_beta_height_pfp_sub_fixed_result_right_bounded_entry + S (fom_value_pfp_sub_fixed_result_right_bounded) = S ((S (fom_index_pfp_sub_fixed_result_right_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_sub_fixed_result_right_bounded_entry. rb = fom_beta_quotient_pfp_sub_fixed_result_right_bounded_entry * S ((S (fom_index_pfp_sub_fixed_result_right_bounded)) * rc) + (fom_value_pfp_sub_fixed_result_right_bounded))) /\\ (exists fom_gap_pfp_sub_fixed_result_right_bounded_value_bound. fom_gap_pfp_sub_fixed_result_right_bounded_value_bound + S (fom_value_pfp_sub_fixed_result_right_bounded) = p))) /\\ (((forall fom_index_pfp_sub_fixed_result_result_bounded. (exists fom_gap_pfp_sub_fixed_result_result_bounded_index_bound. fom_gap_pfp_sub_fixed_result_result_bounded_index_bound + S (fom_index_pfp_sub_fixed_result_result_bounded) = K) -> exists fom_value_pfp_sub_fixed_result_result_bounded. ((((exists fom_beta_height_pfp_sub_fixed_result_result_bounded_entry. fom_beta_height_pfp_sub_fixed_result_result_bounded_entry + S (fom_value_pfp_sub_fixed_result_result_bounded) = S ((S (fom_index_pfp_sub_fixed_result_result_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_sub_fixed_result_result_bounded_entry. ab = fom_beta_quotient_pfp_sub_fixed_result_result_bounded_entry * S ((S (fom_index_pfp_sub_fixed_result_result_bounded)) * ac) + (fom_value_pfp_sub_fixed_result_result_bounded))) /\\ (exists fom_gap_pfp_sub_fixed_result_result_bounded_value_bound. fom_gap_pfp_sub_fixed_result_result_bounded_value_bound + S (fom_value_pfp_sub_fixed_result_result_bounded) = p))) /\\ ((exists pfaa_left_b_sub_fixed_result pfaa_left_c_sub_fixed_result pfaa_right_b_sub_fixed_result pfaa_right_c_sub_fixed_result pfaa_sum_b_sub_fixed_result pfaa_sum_c_sub_fixed_result pfaa_length_sub_fixed_result. ((((forall pfrep_power_sub_fixed_result_witness_common_left pfrep_left_sub_fixed_result_witness_common_left pfrep_right_sub_fixed_result_witness_common_left. ((exists pfrep_position_sub_fixed_result_witness_common_leftfirst. ((pfrep_position_sub_fixed_result_witness_common_leftfirst+S (pfrep_power_sub_fixed_result_witness_common_left)=(K)) /\\ ((((exists ff_h_pfp_sub_fixed_result_witness_common_leftfirstentry. ff_h_pfp_sub_fixed_result_witness_common_leftfirstentry + S (pfrep_left_sub_fixed_result_witness_common_left) = S ((S (pfrep_position_sub_fixed_result_witness_common_leftfirst)) * bc)) /\\ exists ff_q_pfp_sub_fixed_result_witness_common_leftfirstentry. bb = ff_q_pfp_sub_fixed_result_witness_common_leftfirstentry * S ((S (pfrep_position_sub_fixed_result_witness_common_leftfirst)) * bc) + (pfrep_left_sub_fixed_result_witness_common_left)))))) \\/ (((exists pfrep_gap_sub_fixed_result_witness_common_leftfirstoutside. pfrep_gap_sub_fixed_result_witness_common_leftfirstoutside+(K)=(pfrep_power_sub_fixed_result_witness_common_left)) /\\ (((pfrep_left_sub_fixed_result_witness_common_left)=0))))) -> ((exists pfrep_position_sub_fixed_result_witness_common_leftsecond. ((pfrep_position_sub_fixed_result_witness_common_leftsecond+S (pfrep_power_sub_fixed_result_witness_common_left)=(pfaa_length_sub_fixed_result)) /\\ ((((exists ff_h_pfp_sub_fixed_result_witness_common_leftsecondentry. ff_h_pfp_sub_fixed_result_witness_common_leftsecondentry + S (pfrep_right_sub_fixed_result_witness_common_left) = S ((S (pfrep_position_sub_fixed_result_witness_common_leftsecond)) * pfaa_left_c_sub_fixed_result)) /\\ exists ff_q_pfp_sub_fixed_result_witness_common_leftsecondentry. pfaa_left_b_sub_fixed_result = ff_q_pfp_sub_fixed_result_witness_common_leftsecondentry * S ((S (pfrep_position_sub_fixed_result_witness_common_leftsecond)) * pfaa_left_c_sub_fixed_result) + (pfrep_right_sub_fixed_result_witness_common_left)))))) \\/ (((exists pfrep_gap_sub_fixed_result_witness_common_leftsecondoutside. pfrep_gap_sub_fixed_result_witness_common_leftsecondoutside+(pfaa_length_sub_fixed_result)=(pfrep_power_sub_fixed_result_witness_common_left)) /\\ (((pfrep_right_sub_fixed_result_witness_common_left)=0))))) -> pfrep_left_sub_fixed_result_witness_common_left=pfrep_right_sub_fixed_result_witness_common_left) /\\ ((forall pfrep_power_sub_fixed_result_witness_common_right pfrep_left_sub_fixed_result_witness_common_right pfrep_right_sub_fixed_result_witness_common_right. ((exists pfrep_position_sub_fixed_result_witness_common_rightfirst. ((pfrep_position_sub_fixed_result_witness_common_rightfirst+S (pfrep_power_sub_fixed_result_witness_common_right)=(K)) /\\ ((((exists ff_h_pfp_sub_fixed_result_witness_common_rightfirstentry. ff_h_pfp_sub_fixed_result_witness_common_rightfirstentry + S (pfrep_left_sub_fixed_result_witness_common_right) = S ((S (pfrep_position_sub_fixed_result_witness_common_rightfirst)) * rc)) /\\ exists ff_q_pfp_sub_fixed_result_witness_common_rightfirstentry. rb = ff_q_pfp_sub_fixed_result_witness_common_rightfirstentry * S ((S (pfrep_position_sub_fixed_result_witness_common_rightfirst)) * rc) + (pfrep_left_sub_fixed_result_witness_common_right)))))) \\/ (((exists pfrep_gap_sub_fixed_result_witness_common_rightfirstoutside. pfrep_gap_sub_fixed_result_witness_common_rightfirstoutside+(K)=(pfrep_power_sub_fixed_result_witness_common_right)) /\\ (((pfrep_left_sub_fixed_result_witness_common_right)=0))))) -> ((exists pfrep_position_sub_fixed_result_witness_common_rightsecond. ((pfrep_position_sub_fixed_result_witness_common_rightsecond+S (pfrep_power_sub_fixed_result_witness_common_right)=(pfaa_length_sub_fixed_result)) /\\ ((((exists ff_h_pfp_sub_fixed_result_witness_common_rightsecondentry. ff_h_pfp_sub_fixed_result_witness_common_rightsecondentry + S (pfrep_right_sub_fixed_result_witness_common_right) = S ((S (pfrep_position_sub_fixed_result_witness_common_rightsecond)) * pfaa_right_c_sub_fixed_result)) /\\ exists ff_q_pfp_sub_fixed_result_witness_common_rightsecondentry. pfaa_right_b_sub_fixed_result = ff_q_pfp_sub_fixed_result_witness_common_rightsecondentry * S ((S (pfrep_position_sub_fixed_result_witness_common_rightsecond)) * pfaa_right_c_sub_fixed_result) + (pfrep_right_sub_fixed_result_witness_common_right)))))) \\/ (((exists pfrep_gap_sub_fixed_result_witness_common_rightsecondoutside. pfrep_gap_sub_fixed_result_witness_common_rightsecondoutside+(pfaa_length_sub_fixed_result)=(pfrep_power_sub_fixed_result_witness_common_right)) /\\ (((pfrep_right_sub_fixed_result_witness_common_right)=0))))) -> pfrep_left_sub_fixed_result_witness_common_right=pfrep_right_sub_fixed_result_witness_common_right)))) /\\ (((forall pfp_index_sub_fixed_result_witness_operation. (exists pfa_gap_sub_fixed_result_witness_operationindex. pfa_gap_sub_fixed_result_witness_operationindex + S (pfp_index_sub_fixed_result_witness_operation) = (pfaa_length_sub_fixed_result)) -> exists pfp_left_sub_fixed_result_witness_operation pfp_right_sub_fixed_result_witness_operation pfp_value_sub_fixed_result_witness_operation. ((((exists ff_h_pfp_sub_fixed_result_witness_operationleft. ff_h_pfp_sub_fixed_result_witness_operationleft + S (pfp_left_sub_fixed_result_witness_operation) = S ((S (pfp_index_sub_fixed_result_witness_operation)) * pfaa_left_c_sub_fixed_result)) /\\ exists ff_q_pfp_sub_fixed_result_witness_operationleft. pfaa_left_b_sub_fixed_result = ff_q_pfp_sub_fixed_result_witness_operationleft * S ((S (pfp_index_sub_fixed_result_witness_operation)) * pfaa_left_c_sub_fixed_result) + (pfp_left_sub_fixed_result_witness_operation))) /\\ (((((exists ff_h_pfp_sub_fixed_result_witness_operationright. ff_h_pfp_sub_fixed_result_witness_operationright + S (pfp_right_sub_fixed_result_witness_operation) = S ((S (pfp_index_sub_fixed_result_witness_operation)) * pfaa_right_c_sub_fixed_result)) /\\ exists ff_q_pfp_sub_fixed_result_witness_operationright. pfaa_right_b_sub_fixed_result = ff_q_pfp_sub_fixed_result_witness_operationright * S ((S (pfp_index_sub_fixed_result_witness_operation)) * pfaa_right_c_sub_fixed_result) + (pfp_right_sub_fixed_result_witness_operation))) /\\ (((((exists ff_h_pfp_sub_fixed_result_witness_operationtarget. ff_h_pfp_sub_fixed_result_witness_operationtarget + S (pfp_value_sub_fixed_result_witness_operation) = S ((S (pfp_index_sub_fixed_result_witness_operation)) * pfaa_sum_c_sub_fixed_result)) /\\ exists ff_q_pfp_sub_fixed_result_witness_operationtarget. pfaa_sum_b_sub_fixed_result = ff_q_pfp_sub_fixed_result_witness_operationtarget * S ((S (pfp_index_sub_fixed_result_witness_operation)) * pfaa_sum_c_sub_fixed_result) + (pfp_value_sub_fixed_result_witness_operation))) /\\ ((((exists pfa_gap_sub_fixed_result_witness_operationoperationleft. pfa_gap_sub_fixed_result_witness_operationoperationleft + S (pfp_left_sub_fixed_result_witness_operation) = (p)) /\\ (((exists pfa_gap_sub_fixed_result_witness_operationoperationright. pfa_gap_sub_fixed_result_witness_operationoperationright + S (pfp_right_sub_fixed_result_witness_operation) = (p)) /\\ ((((exists pfa_gap_sub_fixed_result_witness_operationoperationresultbound. pfa_gap_sub_fixed_result_witness_operationoperationresultbound + S (pfp_value_sub_fixed_result_witness_operation) = (p)) /\\ ((exists pfa_offset_left_sub_fixed_result_witness_operationoperationresultcongruence pfa_offset_right_sub_fixed_result_witness_operationoperationresultcongruence. ((pfp_left_sub_fixed_result_witness_operation) + (pfp_right_sub_fixed_result_witness_operation)) + (p) * pfa_offset_left_sub_fixed_result_witness_operationoperationresultcongruence = (pfp_value_sub_fixed_result_witness_operation) + (p) * pfa_offset_right_sub_fixed_result_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_sub_fixed_result_witness_output pfrep_left_sub_fixed_result_witness_output pfrep_right_sub_fixed_result_witness_output. ((exists pfrep_position_sub_fixed_result_witness_outputfirst. ((pfrep_position_sub_fixed_result_witness_outputfirst+S (pfrep_power_sub_fixed_result_witness_output)=(pfaa_length_sub_fixed_result)) /\\ ((((exists ff_h_pfp_sub_fixed_result_witness_outputfirstentry. ff_h_pfp_sub_fixed_result_witness_outputfirstentry + S (pfrep_left_sub_fixed_result_witness_output) = S ((S (pfrep_position_sub_fixed_result_witness_outputfirst)) * pfaa_sum_c_sub_fixed_result)) /\\ exists ff_q_pfp_sub_fixed_result_witness_outputfirstentry. pfaa_sum_b_sub_fixed_result = ff_q_pfp_sub_fixed_result_witness_outputfirstentry * S ((S (pfrep_position_sub_fixed_result_witness_outputfirst)) * pfaa_sum_c_sub_fixed_result) + (pfrep_left_sub_fixed_result_witness_output)))))) \\/ (((exists pfrep_gap_sub_fixed_result_witness_outputfirstoutside. pfrep_gap_sub_fixed_result_witness_outputfirstoutside+(pfaa_length_sub_fixed_result)=(pfrep_power_sub_fixed_result_witness_output)) /\\ (((pfrep_left_sub_fixed_result_witness_output)=0))))) -> ((exists pfrep_position_sub_fixed_result_witness_outputsecond. ((pfrep_position_sub_fixed_result_witness_outputsecond+S (pfrep_power_sub_fixed_result_witness_output)=(K)) /\\ ((((exists ff_h_pfp_sub_fixed_result_witness_outputsecondentry. ff_h_pfp_sub_fixed_result_witness_outputsecondentry + S (pfrep_right_sub_fixed_result_witness_output) = S ((S (pfrep_position_sub_fixed_result_witness_outputsecond)) * ac)) /\\ exists ff_q_pfp_sub_fixed_result_witness_outputsecondentry. ab = ff_q_pfp_sub_fixed_result_witness_outputsecondentry * S ((S (pfrep_position_sub_fixed_result_witness_outputsecond)) * ac) + (pfrep_right_sub_fixed_result_witness_output)))))) \\/ (((exists pfrep_gap_sub_fixed_result_witness_outputsecondoutside. pfrep_gap_sub_fixed_result_witness_outputsecondoutside+(K)=(pfrep_power_sub_fixed_result_witness_output)) /\\ (((pfrep_right_sub_fixed_result_witness_output)=0))))) -> pfrep_left_sub_fixed_result_witness_output=pfrep_right_sub_fixed_result_witness_output)))))))))))))", "statement_sha256": "3122386d4be93f7e4bca06128ec30ae0e3334dd046f69bb995b602499ae49804", "summary": "A genuine fixed-length subtraction is an aligned subtraction because its actual coefficient graph supplies B+R=A.", "summary_sha256": "440a3cd31ba3f7d3dabc18f0b41d9213a6a5ae4287231e3711debed50a7d74b3" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "b475bc6cc17cab06ab83573498ce4105f375a5138aac324edb764888343caee0", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 352, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "a68de84439afb5f6dd87f1d47449c0bce8dd53a66346c00cc1b7645fb80b2390", "test_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85" }, "body_checked": true, 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((forall fom_index_pfp_sub_exists_left_bound. (exists fom_gap_pfp_sub_exists_left_bound_index_bound. fom_gap_pfp_sub_exists_left_bound_index_bound + S (fom_index_pfp_sub_exists_left_bound) = L+M) -> exists fom_value_pfp_sub_exists_left_bound. ((((exists fom_beta_height_pfp_sub_exists_left_bound_entry. fom_beta_height_pfp_sub_exists_left_bound_entry + S (fom_value_pfp_sub_exists_left_bound) = S ((S (fom_index_pfp_sub_exists_left_bound)) * uc)) /\\ exists fom_beta_quotient_pfp_sub_exists_left_bound_entry. ub = fom_beta_quotient_pfp_sub_exists_left_bound_entry * S ((S (fom_index_pfp_sub_exists_left_bound)) * uc) + (fom_value_pfp_sub_exists_left_bound))) /\\ (exists fom_gap_pfp_sub_exists_left_bound_value_bound. fom_gap_pfp_sub_exists_left_bound_value_bound + S (fom_value_pfp_sub_exists_left_bound) = p))) /\\ (((forall fom_index_pfp_sub_exists_right_bound. (exists fom_gap_pfp_sub_exists_right_bound_index_bound. fom_gap_pfp_sub_exists_right_bound_index_bound + S (fom_index_pfp_sub_exists_right_bound) = L+M) -> exists fom_value_pfp_sub_exists_right_bound. ((((exists fom_beta_height_pfp_sub_exists_right_bound_entry. fom_beta_height_pfp_sub_exists_right_bound_entry + S (fom_value_pfp_sub_exists_right_bound) = S ((S (fom_index_pfp_sub_exists_right_bound)) * vc)) /\\ exists fom_beta_quotient_pfp_sub_exists_right_bound_entry. vb = fom_beta_quotient_pfp_sub_exists_right_bound_entry * S ((S (fom_index_pfp_sub_exists_right_bound)) * vc) + (fom_value_pfp_sub_exists_right_bound))) /\\ (exists fom_gap_pfp_sub_exists_right_bound_value_bound. fom_gap_pfp_sub_exists_right_bound_value_bound + S (fom_value_pfp_sub_exists_right_bound) = p))) /\\ ((((forall pfrep_power_sub_exists_common_left pfrep_left_sub_exists_common_left pfrep_right_sub_exists_common_left. ((exists pfrep_position_sub_exists_common_leftfirst. ((pfrep_position_sub_exists_common_leftfirst+S (pfrep_power_sub_exists_common_left)=(L)) /\\ ((((exists ff_h_pfp_sub_exists_common_leftfirstentry. ff_h_pfp_sub_exists_common_leftfirstentry + S (pfrep_left_sub_exists_common_left) = S ((S (pfrep_position_sub_exists_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_sub_exists_common_leftfirstentry. ab = ff_q_pfp_sub_exists_common_leftfirstentry * S ((S (pfrep_position_sub_exists_common_leftfirst)) * ac) + (pfrep_left_sub_exists_common_left)))))) \\/ (((exists pfrep_gap_sub_exists_common_leftfirstoutside. pfrep_gap_sub_exists_common_leftfirstoutside+(L)=(pfrep_power_sub_exists_common_left)) /\\ (((pfrep_left_sub_exists_common_left)=0))))) -> ((exists pfrep_position_sub_exists_common_leftsecond. ((pfrep_position_sub_exists_common_leftsecond+S (pfrep_power_sub_exists_common_left)=(L+M)) /\\ ((((exists ff_h_pfp_sub_exists_common_leftsecondentry. ff_h_pfp_sub_exists_common_leftsecondentry + S (pfrep_right_sub_exists_common_left) = S ((S (pfrep_position_sub_exists_common_leftsecond)) * uc)) /\\ exists ff_q_pfp_sub_exists_common_leftsecondentry. ub = ff_q_pfp_sub_exists_common_leftsecondentry * S ((S (pfrep_position_sub_exists_common_leftsecond)) * uc) + (pfrep_right_sub_exists_common_left)))))) \\/ (((exists pfrep_gap_sub_exists_common_leftsecondoutside. pfrep_gap_sub_exists_common_leftsecondoutside+(L+M)=(pfrep_power_sub_exists_common_left)) /\\ (((pfrep_right_sub_exists_common_left)=0))))) -> pfrep_left_sub_exists_common_left=pfrep_right_sub_exists_common_left) /\\ ((forall pfrep_power_sub_exists_common_right pfrep_left_sub_exists_common_right pfrep_right_sub_exists_common_right. ((exists pfrep_position_sub_exists_common_rightfirst. ((pfrep_position_sub_exists_common_rightfirst+S (pfrep_power_sub_exists_common_right)=(M)) /\\ ((((exists ff_h_pfp_sub_exists_common_rightfirstentry. ff_h_pfp_sub_exists_common_rightfirstentry + S (pfrep_left_sub_exists_common_right) = S ((S (pfrep_position_sub_exists_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_sub_exists_common_rightfirstentry. bb = ff_q_pfp_sub_exists_common_rightfirstentry * S ((S (pfrep_position_sub_exists_common_rightfirst)) * bc) + (pfrep_left_sub_exists_common_right)))))) \\/ (((exists pfrep_gap_sub_exists_common_rightfirstoutside. pfrep_gap_sub_exists_common_rightfirstoutside+(M)=(pfrep_power_sub_exists_common_right)) /\\ (((pfrep_left_sub_exists_common_right)=0))))) -> ((exists pfrep_position_sub_exists_common_rightsecond. ((pfrep_position_sub_exists_common_rightsecond+S (pfrep_power_sub_exists_common_right)=(L+M)) /\\ ((((exists ff_h_pfp_sub_exists_common_rightsecondentry. ff_h_pfp_sub_exists_common_rightsecondentry + S (pfrep_right_sub_exists_common_right) = S ((S (pfrep_position_sub_exists_common_rightsecond)) * vc)) /\\ exists ff_q_pfp_sub_exists_common_rightsecondentry. vb = ff_q_pfp_sub_exists_common_rightsecondentry * S ((S (pfrep_position_sub_exists_common_rightsecond)) * vc) + (pfrep_right_sub_exists_common_right)))))) \\/ (((exists pfrep_gap_sub_exists_common_rightsecondoutside. pfrep_gap_sub_exists_common_rightsecondoutside+(L+M)=(pfrep_power_sub_exists_common_right)) /\\ (((pfrep_right_sub_exists_common_right)=0))))) -> pfrep_left_sub_exists_common_right=pfrep_right_sub_exists_common_right))))))))", "specialize prime_field_polynomial_common_representatives_exists (p)", "specialize prime_field_polynomial_common_representatives_exists (ab)", "specialize prime_field_polynomial_common_representatives_exists (ac)", "specialize prime_field_polynomial_common_representatives_exists (L)", "specialize prime_field_polynomial_common_representatives_exists (bb)", "specialize prime_field_polynomial_common_representatives_exists (bc)", "specialize prime_field_polynomial_common_representatives_exists (M)", "apply prime_field_polynomial_common_representatives_exists", "exact hp", "exact ha", "exact hb", "cases hc", "cases hc_witness", "cases hc_witness_witness", "cases hc_witness_witness_witness", "cases hc_witness_witness_witness_witness", "cases hc_witness_witness_witness_witness_right", "cases hc_witness_witness_witness_witness_right_right", "have hd : exists rb rc. forall pfs_index_sub_exists_difference. (exists pfa_gap_sub_exists_differenceindex. pfa_gap_sub_exists_differenceindex + S (pfs_index_sub_exists_difference) = (L+M)) -> exists pfs_left_sub_exists_difference pfs_right_sub_exists_difference pfs_result_sub_exists_difference. ((((exists ff_h_pfp_sub_exists_differenceleft. ff_h_pfp_sub_exists_differenceleft + S (pfs_left_sub_exists_difference) = S ((S (pfs_index_sub_exists_difference)) * x1)) /\\ exists ff_q_pfp_sub_exists_differenceleft. x = ff_q_pfp_sub_exists_differenceleft * S ((S (pfs_index_sub_exists_difference)) * x1) + (pfs_left_sub_exists_difference))) /\\ (((((exists ff_h_pfp_sub_exists_differenceright. ff_h_pfp_sub_exists_differenceright + S (pfs_right_sub_exists_difference) = S ((S (pfs_index_sub_exists_difference)) * x3)) /\\ exists ff_q_pfp_sub_exists_differenceright. x2 = ff_q_pfp_sub_exists_differenceright * S ((S (pfs_index_sub_exists_difference)) * x3) + (pfs_right_sub_exists_difference))) /\\ (((((exists ff_h_pfp_sub_exists_differenceresult. ff_h_pfp_sub_exists_differenceresult + S (pfs_result_sub_exists_difference) = S ((S (pfs_index_sub_exists_difference)) * rc)) /\\ exists ff_q_pfp_sub_exists_differenceresult. rb = ff_q_pfp_sub_exists_differenceresult * S ((S (pfs_index_sub_exists_difference)) * rc) + (pfs_result_sub_exists_difference))) /\\ ((((exists pfa_gap_sub_exists_differenceoperationleft. pfa_gap_sub_exists_differenceoperationleft + S (pfs_right_sub_exists_difference) = (p)) /\\ (((exists pfa_gap_sub_exists_differenceoperationright. pfa_gap_sub_exists_differenceoperationright + S (pfs_result_sub_exists_difference) = (p)) /\\ ((((exists pfa_gap_sub_exists_differenceoperationresultbound. pfa_gap_sub_exists_differenceoperationresultbound + S (pfs_left_sub_exists_difference) = (p)) /\\ ((exists pfa_offset_left_sub_exists_differenceoperationresultcongruence pfa_offset_right_sub_exists_differenceoperationresultcongruence. ((pfs_right_sub_exists_difference) + (pfs_result_sub_exists_difference)) + (p) * pfa_offset_left_sub_exists_differenceoperationresultcongruence = (pfs_left_sub_exists_difference) + (p) * pfa_offset_right_sub_exists_differenceoperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_subtract_exists (p)", "specialize prime_field_polynomial_subtract_exists (x)", "specialize prime_field_polynomial_subtract_exists (x1)", "specialize prime_field_polynomial_subtract_exists (x2)", "specialize prime_field_polynomial_subtract_exists (x3)", "specialize prime_field_polynomial_subtract_exists (L+M)", "apply prime_field_polynomial_subtract_exists", "exact hp", "exact hc_witness_witness_witness_witness_left", "exact hc_witness_witness_witness_witness_right_left", "cases hd", "cases hd_witness", "have hs : forall pfp_index_sub_exists_recover. (exists pfa_gap_sub_exists_recoverindex. pfa_gap_sub_exists_recoverindex + S (pfp_index_sub_exists_recover) = (L+M)) -> exists pfp_left_sub_exists_recover pfp_right_sub_exists_recover pfp_value_sub_exists_recover. ((((exists ff_h_pfp_sub_exists_recoverleft. ff_h_pfp_sub_exists_recoverleft + S (pfp_left_sub_exists_recover) = S ((S (pfp_index_sub_exists_recover)) * x3)) /\\ exists ff_q_pfp_sub_exists_recoverleft. x2 = ff_q_pfp_sub_exists_recoverleft * S ((S (pfp_index_sub_exists_recover)) * x3) + (pfp_left_sub_exists_recover))) /\\ (((((exists ff_h_pfp_sub_exists_recoverright. ff_h_pfp_sub_exists_recoverright + S (pfp_right_sub_exists_recover) = S ((S (pfp_index_sub_exists_recover)) * x5)) /\\ exists ff_q_pfp_sub_exists_recoverright. x4 = ff_q_pfp_sub_exists_recoverright * S ((S (pfp_index_sub_exists_recover)) * x5) + (pfp_right_sub_exists_recover))) /\\ (((((exists ff_h_pfp_sub_exists_recovertarget. ff_h_pfp_sub_exists_recovertarget + S (pfp_value_sub_exists_recover) = S ((S (pfp_index_sub_exists_recover)) * x1)) /\\ exists ff_q_pfp_sub_exists_recovertarget. x = ff_q_pfp_sub_exists_recovertarget * S ((S (pfp_index_sub_exists_recover)) * x1) + (pfp_value_sub_exists_recover))) /\\ ((((exists pfa_gap_sub_exists_recoveroperationleft. pfa_gap_sub_exists_recoveroperationleft + S (pfp_left_sub_exists_recover) = (p)) /\\ (((exists pfa_gap_sub_exists_recoveroperationright. pfa_gap_sub_exists_recoveroperationright + S (pfp_right_sub_exists_recover) = (p)) /\\ ((((exists pfa_gap_sub_exists_recoveroperationresultbound. pfa_gap_sub_exists_recoveroperationresultbound + S (pfp_value_sub_exists_recover) = (p)) /\\ ((exists pfa_offset_left_sub_exists_recoveroperationresultcongruence pfa_offset_right_sub_exists_recoveroperationresultcongruence. ((pfp_left_sub_exists_recover) + (pfp_right_sub_exists_recover)) + (p) * pfa_offset_left_sub_exists_recoveroperationresultcongruence = (pfp_value_sub_exists_recover) + (p) * pfa_offset_right_sub_exists_recoveroperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_subtract_recover_add (p)", "specialize prime_field_polynomial_subtract_recover_add (x)", "specialize prime_field_polynomial_subtract_recover_add (x1)", "specialize prime_field_polynomial_subtract_recover_add (x2)", "specialize prime_field_polynomial_subtract_recover_add (x3)", "specialize prime_field_polynomial_subtract_recover_add (x4)", "specialize prime_field_polynomial_subtract_recover_add (x5)", "specialize prime_field_polynomial_subtract_recover_add (L+M)", "apply prime_field_polynomial_subtract_recover_add", "exact hd_witness_witness", "have hbound : ((forall fom_index_pfp_sub_exists_sum_left. (exists fom_gap_pfp_sub_exists_sum_left_index_bound. fom_gap_pfp_sub_exists_sum_left_index_bound + S (fom_index_pfp_sub_exists_sum_left) = L+M) -> exists fom_value_pfp_sub_exists_sum_left. ((((exists fom_beta_height_pfp_sub_exists_sum_left_entry. fom_beta_height_pfp_sub_exists_sum_left_entry + S (fom_value_pfp_sub_exists_sum_left) = S ((S (fom_index_pfp_sub_exists_sum_left)) * x3)) /\\ exists fom_beta_quotient_pfp_sub_exists_sum_left_entry. x2 = fom_beta_quotient_pfp_sub_exists_sum_left_entry * S ((S (fom_index_pfp_sub_exists_sum_left)) * x3) + (fom_value_pfp_sub_exists_sum_left))) /\\ (exists fom_gap_pfp_sub_exists_sum_left_value_bound. fom_gap_pfp_sub_exists_sum_left_value_bound + S (fom_value_pfp_sub_exists_sum_left) = p))) /\\ (((forall fom_index_pfp_sub_exists_sum_right. (exists fom_gap_pfp_sub_exists_sum_right_index_bound. fom_gap_pfp_sub_exists_sum_right_index_bound + S (fom_index_pfp_sub_exists_sum_right) = L+M) -> exists fom_value_pfp_sub_exists_sum_right. ((((exists fom_beta_height_pfp_sub_exists_sum_right_entry. fom_beta_height_pfp_sub_exists_sum_right_entry + S (fom_value_pfp_sub_exists_sum_right) = S ((S (fom_index_pfp_sub_exists_sum_right)) * x5)) /\\ exists fom_beta_quotient_pfp_sub_exists_sum_right_entry. x4 = fom_beta_quotient_pfp_sub_exists_sum_right_entry * S ((S (fom_index_pfp_sub_exists_sum_right)) * x5) + (fom_value_pfp_sub_exists_sum_right))) /\\ (exists fom_gap_pfp_sub_exists_sum_right_value_bound. fom_gap_pfp_sub_exists_sum_right_value_bound + S (fom_value_pfp_sub_exists_sum_right) = p))) /\\ ((forall fom_index_pfp_sub_exists_sum_result. (exists fom_gap_pfp_sub_exists_sum_result_index_bound. fom_gap_pfp_sub_exists_sum_result_index_bound + S (fom_index_pfp_sub_exists_sum_result) = L+M) -> exists fom_value_pfp_sub_exists_sum_result. ((((exists fom_beta_height_pfp_sub_exists_sum_result_entry. fom_beta_height_pfp_sub_exists_sum_result_entry + S (fom_value_pfp_sub_exists_sum_result) = S ((S (fom_index_pfp_sub_exists_sum_result)) * x1)) /\\ exists fom_beta_quotient_pfp_sub_exists_sum_result_entry. x = fom_beta_quotient_pfp_sub_exists_sum_result_entry * S ((S (fom_index_pfp_sub_exists_sum_result)) * x1) + (fom_value_pfp_sub_exists_sum_result))) /\\ (exists fom_gap_pfp_sub_exists_sum_result_value_bound. fom_gap_pfp_sub_exists_sum_result_value_bound + S (fom_value_pfp_sub_exists_sum_result) = p)))))))", "specialize prime_field_polynomial_add_bounded (p)", "specialize prime_field_polynomial_add_bounded (x2)", "specialize prime_field_polynomial_add_bounded (x3)", "specialize prime_field_polynomial_add_bounded (x4)", "specialize prime_field_polynomial_add_bounded (x5)", "specialize prime_field_polynomial_add_bounded (x)", "specialize prime_field_polynomial_add_bounded (x1)", "specialize prime_field_polynomial_add_bounded (L+M)", "apply prime_field_polynomial_add_bounded", "exact hs", "cases hbound", "cases hbound_right", "exists x4", "exists x5", "specialize prime_field_polynomial_aligned_add_from_common (p)", "specialize prime_field_polynomial_aligned_add_from_common (bb)", "specialize prime_field_polynomial_aligned_add_from_common (bc)", "specialize prime_field_polynomial_aligned_add_from_common (M)", "specialize prime_field_polynomial_aligned_add_from_common (x4)", "specialize prime_field_polynomial_aligned_add_from_common (x5)", "specialize prime_field_polynomial_aligned_add_from_common (L+M)", "specialize prime_field_polynomial_aligned_add_from_common (ab)", "specialize prime_field_polynomial_aligned_add_from_common (ac)", "specialize prime_field_polynomial_aligned_add_from_common (L)", "specialize prime_field_polynomial_aligned_add_from_common (x2)", "specialize prime_field_polynomial_aligned_add_from_common (x3)", "specialize prime_field_polynomial_aligned_add_from_common (x4)", "specialize prime_field_polynomial_aligned_add_from_common (x5)", "specialize prime_field_polynomial_aligned_add_from_common (x)", "specialize prime_field_polynomial_aligned_add_from_common (x1)", "specialize prime_field_polynomial_aligned_add_from_common (L+M)", "apply prime_field_polynomial_aligned_add_from_common", "exact hb", "exact hbound_right_left", "exact ha", "split", "exact hc_witness_witness_witness_witness_right_right_right", "specialize prime_field_polynomial_power_coefficient_functional (x4)", "specialize prime_field_polynomial_power_coefficient_functional (x5)", "specialize prime_field_polynomial_power_coefficient_functional (L+M)", "apply prime_field_polynomial_power_coefficient_functional", "exact hs", "specialize prime_field_polynomial_equivalent_symmetric (ab)", "specialize prime_field_polynomial_equivalent_symmetric (ac)", "specialize prime_field_polynomial_equivalent_symmetric (L)", "specialize prime_field_polynomial_equivalent_symmetric (x)", "specialize prime_field_polynomial_equivalent_symmetric (x1)", "specialize prime_field_polynomial_equivalent_symmetric (L+M)", "apply prime_field_polynomial_equivalent_symmetric", "exact hc_witness_witness_witness_witness_right_right_left" ], "script_sha256": "4e27d5602500874782ed3d60ee94e783bf34dc1babe22964bee1259c6e5faef8", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_aligned_algebra_candidate.py", "sha256": "a68de84439afb5f6dd87f1d47449c0bce8dd53a66346c00cc1b7645fb80b2390" }, "statement": "forall p ab ac L bb bc M. (~((p) = 1) /\\ forall pfa_factor_left_sub_exists_prime pfa_factor_right_sub_exists_prime. (p) = pfa_factor_left_sub_exists_prime * pfa_factor_right_sub_exists_prime -> pfa_factor_left_sub_exists_prime = 1 \\/ pfa_factor_right_sub_exists_prime = 1) -> (forall fom_index_pfp_sub_exists_A. (exists fom_gap_pfp_sub_exists_A_index_bound. fom_gap_pfp_sub_exists_A_index_bound + S (fom_index_pfp_sub_exists_A) = L) -> exists fom_value_pfp_sub_exists_A. ((((exists fom_beta_height_pfp_sub_exists_A_entry. fom_beta_height_pfp_sub_exists_A_entry + S (fom_value_pfp_sub_exists_A) = S ((S (fom_index_pfp_sub_exists_A)) * ac)) /\\ exists fom_beta_quotient_pfp_sub_exists_A_entry. ab = fom_beta_quotient_pfp_sub_exists_A_entry * S ((S (fom_index_pfp_sub_exists_A)) * ac) + (fom_value_pfp_sub_exists_A))) /\\ (exists fom_gap_pfp_sub_exists_A_value_bound. fom_gap_pfp_sub_exists_A_value_bound + S (fom_value_pfp_sub_exists_A) = p))) -> (forall fom_index_pfp_sub_exists_B. (exists fom_gap_pfp_sub_exists_B_index_bound. fom_gap_pfp_sub_exists_B_index_bound + S (fom_index_pfp_sub_exists_B) = M) -> exists fom_value_pfp_sub_exists_B. ((((exists fom_beta_height_pfp_sub_exists_B_entry. fom_beta_height_pfp_sub_exists_B_entry + S (fom_value_pfp_sub_exists_B) = S ((S (fom_index_pfp_sub_exists_B)) * bc)) /\\ exists fom_beta_quotient_pfp_sub_exists_B_entry. bb = fom_beta_quotient_pfp_sub_exists_B_entry * S ((S (fom_index_pfp_sub_exists_B)) * bc) + (fom_value_pfp_sub_exists_B))) /\\ (exists fom_gap_pfp_sub_exists_B_value_bound. fom_gap_pfp_sub_exists_B_value_bound + S (fom_value_pfp_sub_exists_B) = p))) -> (exists rb rc. ((forall fom_index_pfp_sub_exists_result_left_bounded. (exists fom_gap_pfp_sub_exists_result_left_bounded_index_bound. fom_gap_pfp_sub_exists_result_left_bounded_index_bound + S (fom_index_pfp_sub_exists_result_left_bounded) = M) -> exists fom_value_pfp_sub_exists_result_left_bounded. ((((exists fom_beta_height_pfp_sub_exists_result_left_bounded_entry. fom_beta_height_pfp_sub_exists_result_left_bounded_entry + S (fom_value_pfp_sub_exists_result_left_bounded) = S ((S (fom_index_pfp_sub_exists_result_left_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_sub_exists_result_left_bounded_entry. bb = fom_beta_quotient_pfp_sub_exists_result_left_bounded_entry * S ((S (fom_index_pfp_sub_exists_result_left_bounded)) * bc) + (fom_value_pfp_sub_exists_result_left_bounded))) /\\ (exists fom_gap_pfp_sub_exists_result_left_bounded_value_bound. fom_gap_pfp_sub_exists_result_left_bounded_value_bound + S (fom_value_pfp_sub_exists_result_left_bounded) = p))) /\\ (((forall fom_index_pfp_sub_exists_result_right_bounded. (exists fom_gap_pfp_sub_exists_result_right_bounded_index_bound. fom_gap_pfp_sub_exists_result_right_bounded_index_bound + S (fom_index_pfp_sub_exists_result_right_bounded) = L+M) -> exists fom_value_pfp_sub_exists_result_right_bounded. ((((exists fom_beta_height_pfp_sub_exists_result_right_bounded_entry. fom_beta_height_pfp_sub_exists_result_right_bounded_entry + S (fom_value_pfp_sub_exists_result_right_bounded) = S ((S (fom_index_pfp_sub_exists_result_right_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_sub_exists_result_right_bounded_entry. rb = fom_beta_quotient_pfp_sub_exists_result_right_bounded_entry * S ((S (fom_index_pfp_sub_exists_result_right_bounded)) * rc) + (fom_value_pfp_sub_exists_result_right_bounded))) /\\ (exists fom_gap_pfp_sub_exists_result_right_bounded_value_bound. fom_gap_pfp_sub_exists_result_right_bounded_value_bound + S (fom_value_pfp_sub_exists_result_right_bounded) = p))) /\\ (((forall fom_index_pfp_sub_exists_result_result_bounded. (exists fom_gap_pfp_sub_exists_result_result_bounded_index_bound. fom_gap_pfp_sub_exists_result_result_bounded_index_bound + S (fom_index_pfp_sub_exists_result_result_bounded) = L) -> exists fom_value_pfp_sub_exists_result_result_bounded. ((((exists fom_beta_height_pfp_sub_exists_result_result_bounded_entry. fom_beta_height_pfp_sub_exists_result_result_bounded_entry + S (fom_value_pfp_sub_exists_result_result_bounded) = S ((S (fom_index_pfp_sub_exists_result_result_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_sub_exists_result_result_bounded_entry. ab = fom_beta_quotient_pfp_sub_exists_result_result_bounded_entry * S ((S (fom_index_pfp_sub_exists_result_result_bounded)) * ac) + (fom_value_pfp_sub_exists_result_result_bounded))) /\\ (exists fom_gap_pfp_sub_exists_result_result_bounded_value_bound. fom_gap_pfp_sub_exists_result_result_bounded_value_bound + S (fom_value_pfp_sub_exists_result_result_bounded) = p))) /\\ ((exists pfaa_left_b_sub_exists_result pfaa_left_c_sub_exists_result pfaa_right_b_sub_exists_result pfaa_right_c_sub_exists_result pfaa_sum_b_sub_exists_result pfaa_sum_c_sub_exists_result pfaa_length_sub_exists_result. ((((forall pfrep_power_sub_exists_result_witness_common_left pfrep_left_sub_exists_result_witness_common_left pfrep_right_sub_exists_result_witness_common_left. ((exists pfrep_position_sub_exists_result_witness_common_leftfirst. ((pfrep_position_sub_exists_result_witness_common_leftfirst+S (pfrep_power_sub_exists_result_witness_common_left)=(M)) /\\ ((((exists ff_h_pfp_sub_exists_result_witness_common_leftfirstentry. ff_h_pfp_sub_exists_result_witness_common_leftfirstentry + S (pfrep_left_sub_exists_result_witness_common_left) = S ((S (pfrep_position_sub_exists_result_witness_common_leftfirst)) * bc)) /\\ exists ff_q_pfp_sub_exists_result_witness_common_leftfirstentry. bb = ff_q_pfp_sub_exists_result_witness_common_leftfirstentry * S ((S (pfrep_position_sub_exists_result_witness_common_leftfirst)) * bc) + (pfrep_left_sub_exists_result_witness_common_left)))))) \\/ (((exists pfrep_gap_sub_exists_result_witness_common_leftfirstoutside. pfrep_gap_sub_exists_result_witness_common_leftfirstoutside+(M)=(pfrep_power_sub_exists_result_witness_common_left)) /\\ (((pfrep_left_sub_exists_result_witness_common_left)=0))))) -> ((exists pfrep_position_sub_exists_result_witness_common_leftsecond. ((pfrep_position_sub_exists_result_witness_common_leftsecond+S (pfrep_power_sub_exists_result_witness_common_left)=(pfaa_length_sub_exists_result)) /\\ ((((exists ff_h_pfp_sub_exists_result_witness_common_leftsecondentry. ff_h_pfp_sub_exists_result_witness_common_leftsecondentry + S (pfrep_right_sub_exists_result_witness_common_left) = S ((S (pfrep_position_sub_exists_result_witness_common_leftsecond)) * pfaa_left_c_sub_exists_result)) /\\ exists ff_q_pfp_sub_exists_result_witness_common_leftsecondentry. pfaa_left_b_sub_exists_result = ff_q_pfp_sub_exists_result_witness_common_leftsecondentry * S ((S (pfrep_position_sub_exists_result_witness_common_leftsecond)) * pfaa_left_c_sub_exists_result) + (pfrep_right_sub_exists_result_witness_common_left)))))) \\/ (((exists pfrep_gap_sub_exists_result_witness_common_leftsecondoutside. pfrep_gap_sub_exists_result_witness_common_leftsecondoutside+(pfaa_length_sub_exists_result)=(pfrep_power_sub_exists_result_witness_common_left)) /\\ (((pfrep_right_sub_exists_result_witness_common_left)=0))))) -> pfrep_left_sub_exists_result_witness_common_left=pfrep_right_sub_exists_result_witness_common_left) /\\ ((forall pfrep_power_sub_exists_result_witness_common_right pfrep_left_sub_exists_result_witness_common_right pfrep_right_sub_exists_result_witness_common_right. ((exists pfrep_position_sub_exists_result_witness_common_rightfirst. ((pfrep_position_sub_exists_result_witness_common_rightfirst+S (pfrep_power_sub_exists_result_witness_common_right)=(L+M)) /\\ ((((exists ff_h_pfp_sub_exists_result_witness_common_rightfirstentry. ff_h_pfp_sub_exists_result_witness_common_rightfirstentry + S (pfrep_left_sub_exists_result_witness_common_right) = S ((S (pfrep_position_sub_exists_result_witness_common_rightfirst)) * rc)) /\\ exists ff_q_pfp_sub_exists_result_witness_common_rightfirstentry. rb = ff_q_pfp_sub_exists_result_witness_common_rightfirstentry * S ((S (pfrep_position_sub_exists_result_witness_common_rightfirst)) * rc) + (pfrep_left_sub_exists_result_witness_common_right)))))) \\/ (((exists pfrep_gap_sub_exists_result_witness_common_rightfirstoutside. pfrep_gap_sub_exists_result_witness_common_rightfirstoutside+(L+M)=(pfrep_power_sub_exists_result_witness_common_right)) /\\ (((pfrep_left_sub_exists_result_witness_common_right)=0))))) -> ((exists pfrep_position_sub_exists_result_witness_common_rightsecond. ((pfrep_position_sub_exists_result_witness_common_rightsecond+S (pfrep_power_sub_exists_result_witness_common_right)=(pfaa_length_sub_exists_result)) /\\ ((((exists ff_h_pfp_sub_exists_result_witness_common_rightsecondentry. ff_h_pfp_sub_exists_result_witness_common_rightsecondentry + S (pfrep_right_sub_exists_result_witness_common_right) = S ((S (pfrep_position_sub_exists_result_witness_common_rightsecond)) * pfaa_right_c_sub_exists_result)) /\\ exists ff_q_pfp_sub_exists_result_witness_common_rightsecondentry. pfaa_right_b_sub_exists_result = ff_q_pfp_sub_exists_result_witness_common_rightsecondentry * S ((S (pfrep_position_sub_exists_result_witness_common_rightsecond)) * pfaa_right_c_sub_exists_result) + (pfrep_right_sub_exists_result_witness_common_right)))))) \\/ (((exists pfrep_gap_sub_exists_result_witness_common_rightsecondoutside. pfrep_gap_sub_exists_result_witness_common_rightsecondoutside+(pfaa_length_sub_exists_result)=(pfrep_power_sub_exists_result_witness_common_right)) /\\ (((pfrep_right_sub_exists_result_witness_common_right)=0))))) -> pfrep_left_sub_exists_result_witness_common_right=pfrep_right_sub_exists_result_witness_common_right)))) /\\ (((forall pfp_index_sub_exists_result_witness_operation. (exists pfa_gap_sub_exists_result_witness_operationindex. pfa_gap_sub_exists_result_witness_operationindex + S (pfp_index_sub_exists_result_witness_operation) = (pfaa_length_sub_exists_result)) -> exists pfp_left_sub_exists_result_witness_operation pfp_right_sub_exists_result_witness_operation pfp_value_sub_exists_result_witness_operation. ((((exists ff_h_pfp_sub_exists_result_witness_operationleft. ff_h_pfp_sub_exists_result_witness_operationleft + S (pfp_left_sub_exists_result_witness_operation) = S ((S (pfp_index_sub_exists_result_witness_operation)) * pfaa_left_c_sub_exists_result)) /\\ exists ff_q_pfp_sub_exists_result_witness_operationleft. pfaa_left_b_sub_exists_result = ff_q_pfp_sub_exists_result_witness_operationleft * S ((S (pfp_index_sub_exists_result_witness_operation)) * pfaa_left_c_sub_exists_result) + (pfp_left_sub_exists_result_witness_operation))) /\\ (((((exists ff_h_pfp_sub_exists_result_witness_operationright. ff_h_pfp_sub_exists_result_witness_operationright + S (pfp_right_sub_exists_result_witness_operation) = S ((S (pfp_index_sub_exists_result_witness_operation)) * pfaa_right_c_sub_exists_result)) /\\ exists ff_q_pfp_sub_exists_result_witness_operationright. pfaa_right_b_sub_exists_result = ff_q_pfp_sub_exists_result_witness_operationright * S ((S (pfp_index_sub_exists_result_witness_operation)) * pfaa_right_c_sub_exists_result) + (pfp_right_sub_exists_result_witness_operation))) /\\ (((((exists ff_h_pfp_sub_exists_result_witness_operationtarget. ff_h_pfp_sub_exists_result_witness_operationtarget + S (pfp_value_sub_exists_result_witness_operation) = S ((S (pfp_index_sub_exists_result_witness_operation)) * pfaa_sum_c_sub_exists_result)) /\\ exists ff_q_pfp_sub_exists_result_witness_operationtarget. pfaa_sum_b_sub_exists_result = ff_q_pfp_sub_exists_result_witness_operationtarget * S ((S (pfp_index_sub_exists_result_witness_operation)) * pfaa_sum_c_sub_exists_result) + (pfp_value_sub_exists_result_witness_operation))) /\\ ((((exists pfa_gap_sub_exists_result_witness_operationoperationleft. pfa_gap_sub_exists_result_witness_operationoperationleft + S (pfp_left_sub_exists_result_witness_operation) = (p)) /\\ (((exists pfa_gap_sub_exists_result_witness_operationoperationright. pfa_gap_sub_exists_result_witness_operationoperationright + S (pfp_right_sub_exists_result_witness_operation) = (p)) /\\ ((((exists pfa_gap_sub_exists_result_witness_operationoperationresultbound. pfa_gap_sub_exists_result_witness_operationoperationresultbound + S (pfp_value_sub_exists_result_witness_operation) = (p)) /\\ ((exists pfa_offset_left_sub_exists_result_witness_operationoperationresultcongruence pfa_offset_right_sub_exists_result_witness_operationoperationresultcongruence. ((pfp_left_sub_exists_result_witness_operation) + (pfp_right_sub_exists_result_witness_operation)) + (p) * pfa_offset_left_sub_exists_result_witness_operationoperationresultcongruence = (pfp_value_sub_exists_result_witness_operation) + (p) * pfa_offset_right_sub_exists_result_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_sub_exists_result_witness_output pfrep_left_sub_exists_result_witness_output pfrep_right_sub_exists_result_witness_output. ((exists pfrep_position_sub_exists_result_witness_outputfirst. ((pfrep_position_sub_exists_result_witness_outputfirst+S (pfrep_power_sub_exists_result_witness_output)=(pfaa_length_sub_exists_result)) /\\ ((((exists ff_h_pfp_sub_exists_result_witness_outputfirstentry. ff_h_pfp_sub_exists_result_witness_outputfirstentry + S (pfrep_left_sub_exists_result_witness_output) = S ((S (pfrep_position_sub_exists_result_witness_outputfirst)) * pfaa_sum_c_sub_exists_result)) /\\ exists ff_q_pfp_sub_exists_result_witness_outputfirstentry. pfaa_sum_b_sub_exists_result = ff_q_pfp_sub_exists_result_witness_outputfirstentry * S ((S (pfrep_position_sub_exists_result_witness_outputfirst)) * pfaa_sum_c_sub_exists_result) + (pfrep_left_sub_exists_result_witness_output)))))) \\/ (((exists pfrep_gap_sub_exists_result_witness_outputfirstoutside. pfrep_gap_sub_exists_result_witness_outputfirstoutside+(pfaa_length_sub_exists_result)=(pfrep_power_sub_exists_result_witness_output)) /\\ (((pfrep_left_sub_exists_result_witness_output)=0))))) -> ((exists pfrep_position_sub_exists_result_witness_outputsecond. ((pfrep_position_sub_exists_result_witness_outputsecond+S (pfrep_power_sub_exists_result_witness_output)=(L)) /\\ ((((exists ff_h_pfp_sub_exists_result_witness_outputsecondentry. ff_h_pfp_sub_exists_result_witness_outputsecondentry + S (pfrep_right_sub_exists_result_witness_output) = S ((S (pfrep_position_sub_exists_result_witness_outputsecond)) * ac)) /\\ exists ff_q_pfp_sub_exists_result_witness_outputsecondentry. ab = ff_q_pfp_sub_exists_result_witness_outputsecondentry * S ((S (pfrep_position_sub_exists_result_witness_outputsecond)) * ac) + (pfrep_right_sub_exists_result_witness_output)))))) \\/ (((exists pfrep_gap_sub_exists_result_witness_outputsecondoutside. pfrep_gap_sub_exists_result_witness_outputsecondoutside+(L)=(pfrep_power_sub_exists_result_witness_output)) /\\ (((pfrep_right_sub_exists_result_witness_output)=0))))) -> pfrep_left_sub_exists_result_witness_output=pfrep_right_sub_exists_result_witness_output)))))))))))))", "statement_sha256": "a77e628e8832ce3840fcff85ab49fa7eea4ff135f5a01be3e4fc1729f97346cf", "summary": "Construct actual aligned field subtraction at length L+M, using real canonical common representatives and the genuine solution B+R=A.", "summary_sha256": "5ed046ca723c02810233ceaa29d5fac5055780c0623fd6b5747d5217bacf2299" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "2b360160446044df320e97a2776957577a2af03c0541952cb25c1246391f88c7", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 357, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "a68de84439afb5f6dd87f1d47449c0bce8dd53a66346c00cc1b7645fb80b2390", "test_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85" }, "body_checked": true, "body_receipt": { "command_count": 279, "dependency_count": 11, "dne_command_count": 0, "name": "prime_field_polynomial_aligned_add_cancel_left", "proof_depth": 67, "proof_edges": 352, "proof_nodes": 353, "proof_objects": 353, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "prime_field_polynomial_aligned_add_bounded", "prime_field_polynomial_bounded_representative_at_length_exists", "le_add_right", "le_refl", "le_trans", "prime_field_polynomial_aligned_add_realize", "prime_field_polynomial_equivalent_transitive", "prime_field_polynomial_equal_implies_equivalent", "prime_field_polynomial_equivalent_symmetric", "prime_field_polynomial_subtract_functional", "prime_field_polynomial_subtract_from_add" ], "dependencies_sha256": "ae80cfd2961f794e78ff08d75138f546b37864ea3814b0b0a2ae430043f26599", "empty_context_closure": { "body_proof_depth": 67, "body_proof_nodes": 353, "bundle_campaign": "polynomial-gcd-bezout", "bundle_dependency_edge_count": 1578, "bundle_node_count": 493, "bundle_node_id": 357, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "bundle_root_id": 492, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "acfdd4a7c10fa06393b7d533f4604f8f5bc53a71d49f43bf2596bf509d8cb777", "status": "checked" }, "enrollment_index": 4161, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "a68de84439afb5f6dd87f1d47449c0bce8dd53a66346c00cc1b7645fb80b2390", "kind": "alpha_v34_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_aligned_algebra_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85", "kind": "alpha_v34_frontier_executable_audit", "path": "peano-lab/py/tests/test_campaign_research_v34_closure.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "kind": "alpha_v34_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-field-polynomial-gcd-bezout-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "kind": "alpha_v34_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=357]" }, { "document_sha256": "81943a545e7b9f667ad758c24de14bb47280e8cf3fbb3ca714efec55fb096408", "kind": "alpha_v34_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "2577565f7db87bf74aff73dd23b430c1d12c5453c9c46c3d306d2fec4d0a70a9", "membership": "alpha_only", "name": "prime_field_polynomial_aligned_add_cancel_left", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro M", "intro cb", "intro cc", "intro J", "intro rb", "intro rc", "intro N", "intro hp", "intro hb", "intro hc", "have hbbound : ((forall fom_index_pfp_hbbound_0. (exists fom_gap_pfp_hbbound_0_index_bound. fom_gap_pfp_hbbound_0_index_bound + S (fom_index_pfp_hbbound_0) = L) -> exists fom_value_pfp_hbbound_0. ((((exists fom_beta_height_pfp_hbbound_0_entry. fom_beta_height_pfp_hbbound_0_entry + S (fom_value_pfp_hbbound_0) = S ((S (fom_index_pfp_hbbound_0)) * ac)) /\\ exists fom_beta_quotient_pfp_hbbound_0_entry. ab = fom_beta_quotient_pfp_hbbound_0_entry * S ((S (fom_index_pfp_hbbound_0)) * ac) + (fom_value_pfp_hbbound_0))) /\\ (exists fom_gap_pfp_hbbound_0_value_bound. fom_gap_pfp_hbbound_0_value_bound + S (fom_value_pfp_hbbound_0) = p))) /\\ (((forall fom_index_pfp_hbbound_1. (exists fom_gap_pfp_hbbound_1_index_bound. fom_gap_pfp_hbbound_1_index_bound + S (fom_index_pfp_hbbound_1) = M) -> exists fom_value_pfp_hbbound_1. ((((exists fom_beta_height_pfp_hbbound_1_entry. fom_beta_height_pfp_hbbound_1_entry + S (fom_value_pfp_hbbound_1) = S ((S (fom_index_pfp_hbbound_1)) * bc)) /\\ exists fom_beta_quotient_pfp_hbbound_1_entry. bb = fom_beta_quotient_pfp_hbbound_1_entry * S ((S (fom_index_pfp_hbbound_1)) * bc) + (fom_value_pfp_hbbound_1))) /\\ (exists fom_gap_pfp_hbbound_1_value_bound. fom_gap_pfp_hbbound_1_value_bound + S (fom_value_pfp_hbbound_1) = p))) /\\ ((forall fom_index_pfp_hbbound_2. (exists fom_gap_pfp_hbbound_2_index_bound. fom_gap_pfp_hbbound_2_index_bound + S (fom_index_pfp_hbbound_2) = N) -> exists fom_value_pfp_hbbound_2. ((((exists fom_beta_height_pfp_hbbound_2_entry. fom_beta_height_pfp_hbbound_2_entry + S (fom_value_pfp_hbbound_2) = S ((S (fom_index_pfp_hbbound_2)) * rc)) /\\ exists fom_beta_quotient_pfp_hbbound_2_entry. rb = fom_beta_quotient_pfp_hbbound_2_entry * S ((S (fom_index_pfp_hbbound_2)) * rc) + (fom_value_pfp_hbbound_2))) /\\ (exists fom_gap_pfp_hbbound_2_value_bound. fom_gap_pfp_hbbound_2_value_bound + S (fom_value_pfp_hbbound_2) = p)))))))", "specialize prime_field_polynomial_aligned_add_bounded (p)", "specialize prime_field_polynomial_aligned_add_bounded (ab)", "specialize prime_field_polynomial_aligned_add_bounded (ac)", "specialize prime_field_polynomial_aligned_add_bounded (L)", "specialize prime_field_polynomial_aligned_add_bounded (bb)", "specialize prime_field_polynomial_aligned_add_bounded (bc)", "specialize prime_field_polynomial_aligned_add_bounded (M)", "specialize prime_field_polynomial_aligned_add_bounded (rb)", "specialize prime_field_polynomial_aligned_add_bounded (rc)", "specialize prime_field_polynomial_aligned_add_bounded (N)", "apply prime_field_polynomial_aligned_add_bounded", "exact hb", "cases hbbound", "cases hbbound_right", "have hcbound : ((forall fom_index_pfp_hcbound_0. (exists fom_gap_pfp_hcbound_0_index_bound. fom_gap_pfp_hcbound_0_index_bound + S (fom_index_pfp_hcbound_0) = L) -> exists fom_value_pfp_hcbound_0. ((((exists fom_beta_height_pfp_hcbound_0_entry. fom_beta_height_pfp_hcbound_0_entry + S (fom_value_pfp_hcbound_0) = S ((S (fom_index_pfp_hcbound_0)) * ac)) /\\ exists fom_beta_quotient_pfp_hcbound_0_entry. ab = fom_beta_quotient_pfp_hcbound_0_entry * S ((S (fom_index_pfp_hcbound_0)) * ac) + (fom_value_pfp_hcbound_0))) /\\ (exists fom_gap_pfp_hcbound_0_value_bound. fom_gap_pfp_hcbound_0_value_bound + S (fom_value_pfp_hcbound_0) = p))) /\\ (((forall fom_index_pfp_hcbound_1. (exists fom_gap_pfp_hcbound_1_index_bound. fom_gap_pfp_hcbound_1_index_bound + S (fom_index_pfp_hcbound_1) = J) -> exists fom_value_pfp_hcbound_1. ((((exists fom_beta_height_pfp_hcbound_1_entry. fom_beta_height_pfp_hcbound_1_entry + S (fom_value_pfp_hcbound_1) = S ((S (fom_index_pfp_hcbound_1)) * cc)) /\\ exists fom_beta_quotient_pfp_hcbound_1_entry. cb = fom_beta_quotient_pfp_hcbound_1_entry * S ((S (fom_index_pfp_hcbound_1)) * cc) + (fom_value_pfp_hcbound_1))) /\\ (exists fom_gap_pfp_hcbound_1_value_bound. fom_gap_pfp_hcbound_1_value_bound + S (fom_value_pfp_hcbound_1) = p))) /\\ ((forall fom_index_pfp_hcbound_2. (exists fom_gap_pfp_hcbound_2_index_bound. fom_gap_pfp_hcbound_2_index_bound + S (fom_index_pfp_hcbound_2) = N) -> exists fom_value_pfp_hcbound_2. ((((exists fom_beta_height_pfp_hcbound_2_entry. fom_beta_height_pfp_hcbound_2_entry + S (fom_value_pfp_hcbound_2) = S ((S (fom_index_pfp_hcbound_2)) * rc)) /\\ exists fom_beta_quotient_pfp_hcbound_2_entry. rb = fom_beta_quotient_pfp_hcbound_2_entry * S ((S (fom_index_pfp_hcbound_2)) * rc) + (fom_value_pfp_hcbound_2))) /\\ (exists fom_gap_pfp_hcbound_2_value_bound. fom_gap_pfp_hcbound_2_value_bound + S (fom_value_pfp_hcbound_2) = p)))))))", "specialize prime_field_polynomial_aligned_add_bounded (p)", "specialize prime_field_polynomial_aligned_add_bounded (ab)", "specialize prime_field_polynomial_aligned_add_bounded (ac)", "specialize prime_field_polynomial_aligned_add_bounded (L)", "specialize prime_field_polynomial_aligned_add_bounded (cb)", "specialize prime_field_polynomial_aligned_add_bounded (cc)", "specialize prime_field_polynomial_aligned_add_bounded (J)", "specialize prime_field_polynomial_aligned_add_bounded (rb)", "specialize prime_field_polynomial_aligned_add_bounded (rc)", "specialize prime_field_polynomial_aligned_add_bounded (N)", "apply prime_field_polynomial_aligned_add_bounded", "exact hc", "cases hcbound", "cases hcbound_right", "have cancel_representative_0 : exists cancel_representative_0_code cancel_representative_0_scale. ((forall fom_index_pfp_cancel_representative_0_bounded. (exists fom_gap_pfp_cancel_representative_0_bounded_index_bound. fom_gap_pfp_cancel_representative_0_bounded_index_bound + S (fom_index_pfp_cancel_representative_0_bounded) = (L)+((M)+((J)+(N)))) -> exists fom_value_pfp_cancel_representative_0_bounded. ((((exists fom_beta_height_pfp_cancel_representative_0_bounded_entry. fom_beta_height_pfp_cancel_representative_0_bounded_entry + S (fom_value_pfp_cancel_representative_0_bounded) = S ((S (fom_index_pfp_cancel_representative_0_bounded)) * cancel_representative_0_scale)) /\\ exists fom_beta_quotient_pfp_cancel_representative_0_bounded_entry. cancel_representative_0_code = fom_beta_quotient_pfp_cancel_representative_0_bounded_entry * S ((S (fom_index_pfp_cancel_representative_0_bounded)) * cancel_representative_0_scale) + (fom_value_pfp_cancel_representative_0_bounded))) /\\ (exists fom_gap_pfp_cancel_representative_0_bounded_value_bound. fom_gap_pfp_cancel_representative_0_bounded_value_bound + S (fom_value_pfp_cancel_representative_0_bounded) = p))) /\\ ((forall pfrep_power_cancel_representative_0_equivalent pfrep_left_cancel_representative_0_equivalent pfrep_right_cancel_representative_0_equivalent. ((exists pfrep_position_cancel_representative_0_equivalentfirst. ((pfrep_position_cancel_representative_0_equivalentfirst+S (pfrep_power_cancel_representative_0_equivalent)=(L)) /\\ ((((exists ff_h_pfp_cancel_representative_0_equivalentfirstentry. ff_h_pfp_cancel_representative_0_equivalentfirstentry + S (pfrep_left_cancel_representative_0_equivalent) = S ((S (pfrep_position_cancel_representative_0_equivalentfirst)) * ac)) /\\ exists ff_q_pfp_cancel_representative_0_equivalentfirstentry. ab = ff_q_pfp_cancel_representative_0_equivalentfirstentry * S ((S (pfrep_position_cancel_representative_0_equivalentfirst)) * ac) + (pfrep_left_cancel_representative_0_equivalent)))))) \\/ (((exists pfrep_gap_cancel_representative_0_equivalentfirstoutside. pfrep_gap_cancel_representative_0_equivalentfirstoutside+(L)=(pfrep_power_cancel_representative_0_equivalent)) /\\ (((pfrep_left_cancel_representative_0_equivalent)=0))))) -> ((exists pfrep_position_cancel_representative_0_equivalentsecond. ((pfrep_position_cancel_representative_0_equivalentsecond+S (pfrep_power_cancel_representative_0_equivalent)=((L)+((M)+((J)+(N))))) /\\ ((((exists ff_h_pfp_cancel_representative_0_equivalentsecondentry. ff_h_pfp_cancel_representative_0_equivalentsecondentry + S (pfrep_right_cancel_representative_0_equivalent) = S ((S (pfrep_position_cancel_representative_0_equivalentsecond)) * cancel_representative_0_scale)) /\\ exists ff_q_pfp_cancel_representative_0_equivalentsecondentry. cancel_representative_0_code = ff_q_pfp_cancel_representative_0_equivalentsecondentry * S ((S (pfrep_position_cancel_representative_0_equivalentsecond)) * cancel_representative_0_scale) + (pfrep_right_cancel_representative_0_equivalent)))))) \\/ (((exists pfrep_gap_cancel_representative_0_equivalentsecondoutside. pfrep_gap_cancel_representative_0_equivalentsecondoutside+((L)+((M)+((J)+(N))))=(pfrep_power_cancel_representative_0_equivalent)) /\\ (((pfrep_right_cancel_representative_0_equivalent)=0))))) -> pfrep_left_cancel_representative_0_equivalent=pfrep_right_cancel_representative_0_equivalent)))", "specialize prime_field_polynomial_bounded_representative_at_length_exists (p)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (ab)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (ac)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (L)", "specialize prime_field_polynomial_bounded_representative_at_length_exists ((L)+((M)+((J)+(N))))", "apply prime_field_polynomial_bounded_representative_at_length_exists", "exact hp", "exact hbbound_left", "specialize le_add_right (L)", "specialize le_add_right ((M)+((J)+(N)))", "apply le_add_right", "cases cancel_representative_0", "cases cancel_representative_0_witness", "cases cancel_representative_0_witness_witness", "have cancel_representative_1 : exists cancel_representative_1_code cancel_representative_1_scale. ((forall fom_index_pfp_cancel_representative_1_bounded. (exists fom_gap_pfp_cancel_representative_1_bounded_index_bound. fom_gap_pfp_cancel_representative_1_bounded_index_bound + S (fom_index_pfp_cancel_representative_1_bounded) = (L)+((M)+((J)+(N)))) -> exists fom_value_pfp_cancel_representative_1_bounded. ((((exists fom_beta_height_pfp_cancel_representative_1_bounded_entry. fom_beta_height_pfp_cancel_representative_1_bounded_entry + S (fom_value_pfp_cancel_representative_1_bounded) = S ((S (fom_index_pfp_cancel_representative_1_bounded)) * cancel_representative_1_scale)) /\\ exists fom_beta_quotient_pfp_cancel_representative_1_bounded_entry. cancel_representative_1_code = fom_beta_quotient_pfp_cancel_representative_1_bounded_entry * S ((S (fom_index_pfp_cancel_representative_1_bounded)) * cancel_representative_1_scale) + (fom_value_pfp_cancel_representative_1_bounded))) /\\ (exists fom_gap_pfp_cancel_representative_1_bounded_value_bound. fom_gap_pfp_cancel_representative_1_bounded_value_bound + S (fom_value_pfp_cancel_representative_1_bounded) = p))) /\\ ((forall pfrep_power_cancel_representative_1_equivalent pfrep_left_cancel_representative_1_equivalent pfrep_right_cancel_representative_1_equivalent. ((exists pfrep_position_cancel_representative_1_equivalentfirst. ((pfrep_position_cancel_representative_1_equivalentfirst+S (pfrep_power_cancel_representative_1_equivalent)=(M)) /\\ ((((exists ff_h_pfp_cancel_representative_1_equivalentfirstentry. ff_h_pfp_cancel_representative_1_equivalentfirstentry + S (pfrep_left_cancel_representative_1_equivalent) = S ((S (pfrep_position_cancel_representative_1_equivalentfirst)) * bc)) /\\ exists ff_q_pfp_cancel_representative_1_equivalentfirstentry. bb = ff_q_pfp_cancel_representative_1_equivalentfirstentry * S ((S (pfrep_position_cancel_representative_1_equivalentfirst)) * bc) + (pfrep_left_cancel_representative_1_equivalent)))))) \\/ (((exists pfrep_gap_cancel_representative_1_equivalentfirstoutside. pfrep_gap_cancel_representative_1_equivalentfirstoutside+(M)=(pfrep_power_cancel_representative_1_equivalent)) /\\ (((pfrep_left_cancel_representative_1_equivalent)=0))))) -> ((exists pfrep_position_cancel_representative_1_equivalentsecond. ((pfrep_position_cancel_representative_1_equivalentsecond+S (pfrep_power_cancel_representative_1_equivalent)=((L)+((M)+((J)+(N))))) /\\ ((((exists ff_h_pfp_cancel_representative_1_equivalentsecondentry. ff_h_pfp_cancel_representative_1_equivalentsecondentry + S (pfrep_right_cancel_representative_1_equivalent) = S ((S (pfrep_position_cancel_representative_1_equivalentsecond)) * cancel_representative_1_scale)) /\\ exists ff_q_pfp_cancel_representative_1_equivalentsecondentry. cancel_representative_1_code = ff_q_pfp_cancel_representative_1_equivalentsecondentry * S ((S (pfrep_position_cancel_representative_1_equivalentsecond)) * cancel_representative_1_scale) + (pfrep_right_cancel_representative_1_equivalent)))))) \\/ (((exists pfrep_gap_cancel_representative_1_equivalentsecondoutside. pfrep_gap_cancel_representative_1_equivalentsecondoutside+((L)+((M)+((J)+(N))))=(pfrep_power_cancel_representative_1_equivalent)) /\\ (((pfrep_right_cancel_representative_1_equivalent)=0))))) -> pfrep_left_cancel_representative_1_equivalent=pfrep_right_cancel_representative_1_equivalent)))", "specialize prime_field_polynomial_bounded_representative_at_length_exists (p)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (bb)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (bc)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (M)", "specialize prime_field_polynomial_bounded_representative_at_length_exists ((L)+((M)+((J)+(N))))", "apply prime_field_polynomial_bounded_representative_at_length_exists", "exact hp", "exact hbbound_right_left", "have length_bound_cancel_representative_1 : exists pfrep_gap_cancel_representative_1_inner. pfrep_gap_cancel_representative_1_inner+(M)=((M)+((J)+(N)))", "specialize le_add_right (M)", "specialize le_add_right ((J)+(N))", "apply le_add_right", "specialize le_trans (M)", "specialize le_trans ((M)+((J)+(N)))", "specialize le_trans ((L)+((M)+((J)+(N))))", "apply le_trans", "exact length_bound_cancel_representative_1", "exists L", "refl", "cases cancel_representative_1", "cases cancel_representative_1_witness", "cases cancel_representative_1_witness_witness", "have cancel_representative_2 : exists cancel_representative_2_code cancel_representative_2_scale. ((forall fom_index_pfp_cancel_representative_2_bounded. (exists fom_gap_pfp_cancel_representative_2_bounded_index_bound. fom_gap_pfp_cancel_representative_2_bounded_index_bound + S (fom_index_pfp_cancel_representative_2_bounded) = (L)+((M)+((J)+(N)))) -> exists fom_value_pfp_cancel_representative_2_bounded. ((((exists fom_beta_height_pfp_cancel_representative_2_bounded_entry. fom_beta_height_pfp_cancel_representative_2_bounded_entry + S (fom_value_pfp_cancel_representative_2_bounded) = S ((S (fom_index_pfp_cancel_representative_2_bounded)) * cancel_representative_2_scale)) /\\ exists fom_beta_quotient_pfp_cancel_representative_2_bounded_entry. cancel_representative_2_code = fom_beta_quotient_pfp_cancel_representative_2_bounded_entry * S ((S (fom_index_pfp_cancel_representative_2_bounded)) * cancel_representative_2_scale) + (fom_value_pfp_cancel_representative_2_bounded))) /\\ (exists fom_gap_pfp_cancel_representative_2_bounded_value_bound. fom_gap_pfp_cancel_representative_2_bounded_value_bound + S (fom_value_pfp_cancel_representative_2_bounded) = p))) /\\ ((forall pfrep_power_cancel_representative_2_equivalent pfrep_left_cancel_representative_2_equivalent pfrep_right_cancel_representative_2_equivalent. ((exists pfrep_position_cancel_representative_2_equivalentfirst. ((pfrep_position_cancel_representative_2_equivalentfirst+S (pfrep_power_cancel_representative_2_equivalent)=(J)) /\\ ((((exists ff_h_pfp_cancel_representative_2_equivalentfirstentry. ff_h_pfp_cancel_representative_2_equivalentfirstentry + S (pfrep_left_cancel_representative_2_equivalent) = S ((S (pfrep_position_cancel_representative_2_equivalentfirst)) * cc)) /\\ exists ff_q_pfp_cancel_representative_2_equivalentfirstentry. cb = ff_q_pfp_cancel_representative_2_equivalentfirstentry * S ((S (pfrep_position_cancel_representative_2_equivalentfirst)) * cc) + (pfrep_left_cancel_representative_2_equivalent)))))) \\/ (((exists pfrep_gap_cancel_representative_2_equivalentfirstoutside. pfrep_gap_cancel_representative_2_equivalentfirstoutside+(J)=(pfrep_power_cancel_representative_2_equivalent)) /\\ (((pfrep_left_cancel_representative_2_equivalent)=0))))) -> ((exists pfrep_position_cancel_representative_2_equivalentsecond. ((pfrep_position_cancel_representative_2_equivalentsecond+S (pfrep_power_cancel_representative_2_equivalent)=((L)+((M)+((J)+(N))))) /\\ ((((exists ff_h_pfp_cancel_representative_2_equivalentsecondentry. ff_h_pfp_cancel_representative_2_equivalentsecondentry + S (pfrep_right_cancel_representative_2_equivalent) = S ((S (pfrep_position_cancel_representative_2_equivalentsecond)) * cancel_representative_2_scale)) /\\ exists ff_q_pfp_cancel_representative_2_equivalentsecondentry. cancel_representative_2_code = ff_q_pfp_cancel_representative_2_equivalentsecondentry * S ((S (pfrep_position_cancel_representative_2_equivalentsecond)) * cancel_representative_2_scale) + (pfrep_right_cancel_representative_2_equivalent)))))) \\/ (((exists pfrep_gap_cancel_representative_2_equivalentsecondoutside. pfrep_gap_cancel_representative_2_equivalentsecondoutside+((L)+((M)+((J)+(N))))=(pfrep_power_cancel_representative_2_equivalent)) /\\ (((pfrep_right_cancel_representative_2_equivalent)=0))))) -> pfrep_left_cancel_representative_2_equivalent=pfrep_right_cancel_representative_2_equivalent)))", "specialize prime_field_polynomial_bounded_representative_at_length_exists (p)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (cb)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (cc)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (J)", "specialize prime_field_polynomial_bounded_representative_at_length_exists ((L)+((M)+((J)+(N))))", "apply prime_field_polynomial_bounded_representative_at_length_exists", "exact hp", "exact hcbound_right_left", "have length_bound_cancel_representative_2 : exists pfrep_gap_cancel_representative_2_inner. pfrep_gap_cancel_representative_2_inner+(J)=((M)+((J)+(N)))", "have length_bound_cancel_representative_2_inner : exists pfrep_gap_cancel_representative_2_inner_inner. pfrep_gap_cancel_representative_2_inner_inner+(J)=((J)+(N))", "specialize le_add_right (J)", "specialize le_add_right (N)", "apply le_add_right", "specialize le_trans (J)", "specialize le_trans ((J)+(N))", "specialize le_trans ((M)+((J)+(N)))", "apply le_trans", "exact length_bound_cancel_representative_2_inner", "exists M", "refl", "specialize le_trans (J)", "specialize le_trans ((M)+((J)+(N)))", "specialize le_trans ((L)+((M)+((J)+(N))))", "apply le_trans", "exact length_bound_cancel_representative_2", "exists L", "refl", "cases cancel_representative_2", "cases cancel_representative_2_witness", "cases cancel_representative_2_witness_witness", "have cancel_representative_3 : exists cancel_representative_3_code cancel_representative_3_scale. ((forall fom_index_pfp_cancel_representative_3_bounded. (exists fom_gap_pfp_cancel_representative_3_bounded_index_bound. fom_gap_pfp_cancel_representative_3_bounded_index_bound + S (fom_index_pfp_cancel_representative_3_bounded) = (L)+((M)+((J)+(N)))) -> exists fom_value_pfp_cancel_representative_3_bounded. ((((exists fom_beta_height_pfp_cancel_representative_3_bounded_entry. fom_beta_height_pfp_cancel_representative_3_bounded_entry + S (fom_value_pfp_cancel_representative_3_bounded) = S ((S (fom_index_pfp_cancel_representative_3_bounded)) * cancel_representative_3_scale)) /\\ exists fom_beta_quotient_pfp_cancel_representative_3_bounded_entry. cancel_representative_3_code = fom_beta_quotient_pfp_cancel_representative_3_bounded_entry * S ((S (fom_index_pfp_cancel_representative_3_bounded)) * cancel_representative_3_scale) + (fom_value_pfp_cancel_representative_3_bounded))) /\\ (exists fom_gap_pfp_cancel_representative_3_bounded_value_bound. fom_gap_pfp_cancel_representative_3_bounded_value_bound + S (fom_value_pfp_cancel_representative_3_bounded) = p))) /\\ ((forall pfrep_power_cancel_representative_3_equivalent pfrep_left_cancel_representative_3_equivalent pfrep_right_cancel_representative_3_equivalent. ((exists pfrep_position_cancel_representative_3_equivalentfirst. ((pfrep_position_cancel_representative_3_equivalentfirst+S (pfrep_power_cancel_representative_3_equivalent)=(N)) /\\ ((((exists ff_h_pfp_cancel_representative_3_equivalentfirstentry. ff_h_pfp_cancel_representative_3_equivalentfirstentry + S (pfrep_left_cancel_representative_3_equivalent) = S ((S (pfrep_position_cancel_representative_3_equivalentfirst)) * rc)) /\\ exists ff_q_pfp_cancel_representative_3_equivalentfirstentry. rb = ff_q_pfp_cancel_representative_3_equivalentfirstentry * S ((S (pfrep_position_cancel_representative_3_equivalentfirst)) * rc) + (pfrep_left_cancel_representative_3_equivalent)))))) \\/ (((exists pfrep_gap_cancel_representative_3_equivalentfirstoutside. pfrep_gap_cancel_representative_3_equivalentfirstoutside+(N)=(pfrep_power_cancel_representative_3_equivalent)) /\\ (((pfrep_left_cancel_representative_3_equivalent)=0))))) -> ((exists pfrep_position_cancel_representative_3_equivalentsecond. ((pfrep_position_cancel_representative_3_equivalentsecond+S (pfrep_power_cancel_representative_3_equivalent)=((L)+((M)+((J)+(N))))) /\\ ((((exists ff_h_pfp_cancel_representative_3_equivalentsecondentry. ff_h_pfp_cancel_representative_3_equivalentsecondentry + S (pfrep_right_cancel_representative_3_equivalent) = S ((S (pfrep_position_cancel_representative_3_equivalentsecond)) * cancel_representative_3_scale)) /\\ exists ff_q_pfp_cancel_representative_3_equivalentsecondentry. cancel_representative_3_code = ff_q_pfp_cancel_representative_3_equivalentsecondentry * S ((S (pfrep_position_cancel_representative_3_equivalentsecond)) * cancel_representative_3_scale) + (pfrep_right_cancel_representative_3_equivalent)))))) \\/ (((exists pfrep_gap_cancel_representative_3_equivalentsecondoutside. pfrep_gap_cancel_representative_3_equivalentsecondoutside+((L)+((M)+((J)+(N))))=(pfrep_power_cancel_representative_3_equivalent)) /\\ (((pfrep_right_cancel_representative_3_equivalent)=0))))) -> pfrep_left_cancel_representative_3_equivalent=pfrep_right_cancel_representative_3_equivalent)))", "specialize prime_field_polynomial_bounded_representative_at_length_exists (p)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (rb)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (rc)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (N)", "specialize prime_field_polynomial_bounded_representative_at_length_exists ((L)+((M)+((J)+(N))))", "apply prime_field_polynomial_bounded_representative_at_length_exists", "exact hp", "exact hbbound_right_right", "have length_bound_cancel_representative_3 : exists pfrep_gap_cancel_representative_3_inner. pfrep_gap_cancel_representative_3_inner+(N)=((M)+((J)+(N)))", "have length_bound_cancel_representative_3_inner : exists pfrep_gap_cancel_representative_3_inner_inner. pfrep_gap_cancel_representative_3_inner_inner+(N)=((J)+(N))", "have length_bound_cancel_representative_3_inner_inner : exists pfrep_gap_cancel_representative_3_inner_inner_inner. pfrep_gap_cancel_representative_3_inner_inner_inner+(N)=(N)", "specialize le_refl (N)", "apply le_refl", "specialize le_trans (N)", "specialize le_trans (N)", "specialize le_trans ((J)+(N))", "apply le_trans", "exact length_bound_cancel_representative_3_inner_inner", "exists J", "refl", "specialize le_trans (N)", "specialize le_trans ((J)+(N))", "specialize le_trans ((M)+((J)+(N)))", "apply le_trans", "exact length_bound_cancel_representative_3_inner", "exists M", "refl", "specialize le_trans (N)", "specialize le_trans ((M)+((J)+(N)))", "specialize le_trans ((L)+((M)+((J)+(N))))", "apply le_trans", "exact length_bound_cancel_representative_3", "exists L", "refl", "cases cancel_representative_3", "cases cancel_representative_3_witness", "cases cancel_representative_3_witness_witness", "have hfirst : forall pfp_index_hfirst_operation. (exists pfa_gap_hfirst_operationindex. pfa_gap_hfirst_operationindex + S (pfp_index_hfirst_operation) = ((L)+((M)+((J)+(N))))) -> exists pfp_left_hfirst_operation pfp_right_hfirst_operation pfp_value_hfirst_operation. ((((exists ff_h_pfp_hfirst_operationleft. ff_h_pfp_hfirst_operationleft + S (pfp_left_hfirst_operation) = S ((S (pfp_index_hfirst_operation)) * x1)) /\\ exists ff_q_pfp_hfirst_operationleft. x = ff_q_pfp_hfirst_operationleft * S ((S (pfp_index_hfirst_operation)) * x1) + (pfp_left_hfirst_operation))) /\\ (((((exists ff_h_pfp_hfirst_operationright. ff_h_pfp_hfirst_operationright + S (pfp_right_hfirst_operation) = S ((S (pfp_index_hfirst_operation)) * x3)) /\\ exists ff_q_pfp_hfirst_operationright. x2 = ff_q_pfp_hfirst_operationright * S ((S (pfp_index_hfirst_operation)) * x3) + (pfp_right_hfirst_operation))) /\\ (((((exists ff_h_pfp_hfirst_operationtarget. ff_h_pfp_hfirst_operationtarget + S (pfp_value_hfirst_operation) = S ((S (pfp_index_hfirst_operation)) * x7)) /\\ exists ff_q_pfp_hfirst_operationtarget. x6 = ff_q_pfp_hfirst_operationtarget * S ((S (pfp_index_hfirst_operation)) * x7) + (pfp_value_hfirst_operation))) /\\ ((((exists pfa_gap_hfirst_operationoperationleft. pfa_gap_hfirst_operationoperationleft + S (pfp_left_hfirst_operation) = (p)) /\\ (((exists pfa_gap_hfirst_operationoperationright. pfa_gap_hfirst_operationoperationright + S (pfp_right_hfirst_operation) = (p)) /\\ ((((exists pfa_gap_hfirst_operationoperationresultbound. pfa_gap_hfirst_operationoperationresultbound + S (pfp_value_hfirst_operation) = (p)) /\\ ((exists pfa_offset_left_hfirst_operationoperationresultcongruence pfa_offset_right_hfirst_operationoperationresultcongruence. ((pfp_left_hfirst_operation) + (pfp_right_hfirst_operation)) + (p) * pfa_offset_left_hfirst_operationoperationresultcongruence = (pfp_value_hfirst_operation) + (p) * pfa_offset_right_hfirst_operationoperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_aligned_add_realize (p)", "specialize prime_field_polynomial_aligned_add_realize (ab)", "specialize prime_field_polynomial_aligned_add_realize (ac)", "specialize prime_field_polynomial_aligned_add_realize (L)", "specialize prime_field_polynomial_aligned_add_realize (bb)", "specialize prime_field_polynomial_aligned_add_realize (bc)", "specialize prime_field_polynomial_aligned_add_realize (M)", "specialize prime_field_polynomial_aligned_add_realize (rb)", "specialize prime_field_polynomial_aligned_add_realize (rc)", "specialize prime_field_polynomial_aligned_add_realize (N)", "specialize prime_field_polynomial_aligned_add_realize (x)", "specialize prime_field_polynomial_aligned_add_realize (x1)", "specialize prime_field_polynomial_aligned_add_realize (x2)", "specialize prime_field_polynomial_aligned_add_realize (x3)", "specialize prime_field_polynomial_aligned_add_realize (x6)", "specialize prime_field_polynomial_aligned_add_realize (x7)", "specialize prime_field_polynomial_aligned_add_realize ((L)+((M)+((J)+(N))))", "apply prime_field_polynomial_aligned_add_realize", "exact hp", "exact hb", "exact cancel_representative_0_witness_witness_left", "exact cancel_representative_1_witness_witness_left", "exact cancel_representative_3_witness_witness_left", "split", "exact cancel_representative_0_witness_witness_right", "exact cancel_representative_1_witness_witness_right", "exact cancel_representative_3_witness_witness_right", "have hsecond : forall pfp_index_hsecond_operation. (exists pfa_gap_hsecond_operationindex. pfa_gap_hsecond_operationindex + S (pfp_index_hsecond_operation) = ((L)+((M)+((J)+(N))))) -> exists pfp_left_hsecond_operation pfp_right_hsecond_operation pfp_value_hsecond_operation. ((((exists ff_h_pfp_hsecond_operationleft. ff_h_pfp_hsecond_operationleft + S (pfp_left_hsecond_operation) = S ((S (pfp_index_hsecond_operation)) * x1)) /\\ exists ff_q_pfp_hsecond_operationleft. x = ff_q_pfp_hsecond_operationleft * S ((S (pfp_index_hsecond_operation)) * x1) + (pfp_left_hsecond_operation))) /\\ (((((exists ff_h_pfp_hsecond_operationright. ff_h_pfp_hsecond_operationright + S (pfp_right_hsecond_operation) = S ((S (pfp_index_hsecond_operation)) * x5)) /\\ exists ff_q_pfp_hsecond_operationright. x4 = ff_q_pfp_hsecond_operationright * S ((S (pfp_index_hsecond_operation)) * x5) + (pfp_right_hsecond_operation))) /\\ (((((exists ff_h_pfp_hsecond_operationtarget. ff_h_pfp_hsecond_operationtarget + S (pfp_value_hsecond_operation) = S ((S (pfp_index_hsecond_operation)) * x7)) /\\ exists ff_q_pfp_hsecond_operationtarget. x6 = ff_q_pfp_hsecond_operationtarget * S ((S (pfp_index_hsecond_operation)) * x7) + (pfp_value_hsecond_operation))) /\\ ((((exists pfa_gap_hsecond_operationoperationleft. pfa_gap_hsecond_operationoperationleft + S (pfp_left_hsecond_operation) = (p)) /\\ (((exists pfa_gap_hsecond_operationoperationright. pfa_gap_hsecond_operationoperationright + S (pfp_right_hsecond_operation) = (p)) /\\ ((((exists pfa_gap_hsecond_operationoperationresultbound. pfa_gap_hsecond_operationoperationresultbound + S (pfp_value_hsecond_operation) = (p)) /\\ ((exists pfa_offset_left_hsecond_operationoperationresultcongruence pfa_offset_right_hsecond_operationoperationresultcongruence. ((pfp_left_hsecond_operation) + (pfp_right_hsecond_operation)) + (p) * pfa_offset_left_hsecond_operationoperationresultcongruence = (pfp_value_hsecond_operation) + (p) * pfa_offset_right_hsecond_operationoperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_aligned_add_realize (p)", "specialize prime_field_polynomial_aligned_add_realize (ab)", "specialize prime_field_polynomial_aligned_add_realize (ac)", "specialize prime_field_polynomial_aligned_add_realize (L)", "specialize prime_field_polynomial_aligned_add_realize (cb)", "specialize prime_field_polynomial_aligned_add_realize (cc)", "specialize prime_field_polynomial_aligned_add_realize (J)", "specialize prime_field_polynomial_aligned_add_realize (rb)", "specialize prime_field_polynomial_aligned_add_realize (rc)", "specialize prime_field_polynomial_aligned_add_realize (N)", "specialize prime_field_polynomial_aligned_add_realize (x)", "specialize prime_field_polynomial_aligned_add_realize (x1)", "specialize prime_field_polynomial_aligned_add_realize (x4)", "specialize prime_field_polynomial_aligned_add_realize (x5)", "specialize prime_field_polynomial_aligned_add_realize (x6)", "specialize prime_field_polynomial_aligned_add_realize (x7)", "specialize prime_field_polynomial_aligned_add_realize ((L)+((M)+((J)+(N))))", "apply prime_field_polynomial_aligned_add_realize", "exact hp", "exact hc", "exact cancel_representative_0_witness_witness_left", "exact cancel_representative_2_witness_witness_left", "exact cancel_representative_3_witness_witness_left", "split", "exact cancel_representative_0_witness_witness_right", "exact cancel_representative_2_witness_witness_right", "exact cancel_representative_3_witness_witness_right", "have heq : forall mdr_i_pfp_cancel_actual_outputs mdr_a_pfp_cancel_actual_outputs. (exists mdr_gap_pfp_cancel_actual_outputsb. mdr_gap_pfp_cancel_actual_outputsb + S (mdr_i_pfp_cancel_actual_outputs) = ((L)+((M)+((J)+(N))))) -> (((exists ff_h_mdr_pfp_cancel_actual_outputso. ff_h_mdr_pfp_cancel_actual_outputso + S (mdr_a_pfp_cancel_actual_outputs) = S ((S (mdr_i_pfp_cancel_actual_outputs)) * x3)) /\\ exists ff_q_mdr_pfp_cancel_actual_outputso. x2 = ff_q_mdr_pfp_cancel_actual_outputso * S ((S (mdr_i_pfp_cancel_actual_outputs)) * x3) + (mdr_a_pfp_cancel_actual_outputs))) -> (((exists ff_h_mdr_pfp_cancel_actual_outputsn. ff_h_mdr_pfp_cancel_actual_outputsn + S (mdr_a_pfp_cancel_actual_outputs) = S ((S (mdr_i_pfp_cancel_actual_outputs)) * x5)) /\\ exists ff_q_mdr_pfp_cancel_actual_outputsn. x4 = ff_q_mdr_pfp_cancel_actual_outputsn * S ((S (mdr_i_pfp_cancel_actual_outputs)) * x5) + (mdr_a_pfp_cancel_actual_outputs)))", "specialize prime_field_polynomial_subtract_functional (p)", "specialize prime_field_polynomial_subtract_functional (x6)", "specialize prime_field_polynomial_subtract_functional (x7)", "specialize prime_field_polynomial_subtract_functional (x)", "specialize prime_field_polynomial_subtract_functional (x1)", "specialize prime_field_polynomial_subtract_functional (x2)", "specialize prime_field_polynomial_subtract_functional (x3)", "specialize prime_field_polynomial_subtract_functional (x4)", "specialize prime_field_polynomial_subtract_functional (x5)", "specialize prime_field_polynomial_subtract_functional ((L)+((M)+((J)+(N))))", "apply prime_field_polynomial_subtract_functional", "specialize prime_field_polynomial_subtract_from_add (p)", "specialize prime_field_polynomial_subtract_from_add (x6)", "specialize prime_field_polynomial_subtract_from_add (x7)", "specialize prime_field_polynomial_subtract_from_add (x)", "specialize prime_field_polynomial_subtract_from_add (x1)", "specialize prime_field_polynomial_subtract_from_add (x2)", "specialize prime_field_polynomial_subtract_from_add (x3)", "specialize prime_field_polynomial_subtract_from_add ((L)+((M)+((J)+(N))))", "apply prime_field_polynomial_subtract_from_add", "exact hfirst", "specialize prime_field_polynomial_subtract_from_add (p)", "specialize prime_field_polynomial_subtract_from_add (x6)", "specialize prime_field_polynomial_subtract_from_add (x7)", "specialize prime_field_polynomial_subtract_from_add (x)", "specialize prime_field_polynomial_subtract_from_add (x1)", "specialize prime_field_polynomial_subtract_from_add (x4)", "specialize prime_field_polynomial_subtract_from_add (x5)", "specialize prime_field_polynomial_subtract_from_add ((L)+((M)+((J)+(N))))", "apply prime_field_polynomial_subtract_from_add", "exact hsecond", "have cancel_middle : forall pfrep_power_cancel_middle_result pfrep_left_cancel_middle_result pfrep_right_cancel_middle_result. ((exists pfrep_position_cancel_middle_resultfirst. ((pfrep_position_cancel_middle_resultfirst+S (pfrep_power_cancel_middle_result)=(M)) /\\ ((((exists ff_h_pfp_cancel_middle_resultfirstentry. ff_h_pfp_cancel_middle_resultfirstentry + S (pfrep_left_cancel_middle_result) = S ((S (pfrep_position_cancel_middle_resultfirst)) * bc)) /\\ exists ff_q_pfp_cancel_middle_resultfirstentry. bb = ff_q_pfp_cancel_middle_resultfirstentry * S ((S (pfrep_position_cancel_middle_resultfirst)) * bc) + (pfrep_left_cancel_middle_result)))))) \\/ (((exists pfrep_gap_cancel_middle_resultfirstoutside. pfrep_gap_cancel_middle_resultfirstoutside+(M)=(pfrep_power_cancel_middle_result)) /\\ (((pfrep_left_cancel_middle_result)=0))))) -> ((exists pfrep_position_cancel_middle_resultsecond. ((pfrep_position_cancel_middle_resultsecond+S (pfrep_power_cancel_middle_result)=((L)+((M)+((J)+(N))))) /\\ ((((exists ff_h_pfp_cancel_middle_resultsecondentry. ff_h_pfp_cancel_middle_resultsecondentry + S (pfrep_right_cancel_middle_result) = S ((S (pfrep_position_cancel_middle_resultsecond)) * x5)) /\\ exists ff_q_pfp_cancel_middle_resultsecondentry. x4 = ff_q_pfp_cancel_middle_resultsecondentry * S ((S (pfrep_position_cancel_middle_resultsecond)) * x5) + (pfrep_right_cancel_middle_result)))))) \\/ (((exists pfrep_gap_cancel_middle_resultsecondoutside. pfrep_gap_cancel_middle_resultsecondoutside+((L)+((M)+((J)+(N))))=(pfrep_power_cancel_middle_result)) /\\ (((pfrep_right_cancel_middle_result)=0))))) -> pfrep_left_cancel_middle_result=pfrep_right_cancel_middle_result", "specialize prime_field_polynomial_equivalent_transitive (bb)", "specialize prime_field_polynomial_equivalent_transitive (bc)", "specialize prime_field_polynomial_equivalent_transitive (M)", "specialize prime_field_polynomial_equivalent_transitive (x2)", "specialize prime_field_polynomial_equivalent_transitive (x3)", "specialize prime_field_polynomial_equivalent_transitive ((L)+((M)+((J)+(N))))", "specialize prime_field_polynomial_equivalent_transitive (x4)", "specialize prime_field_polynomial_equivalent_transitive (x5)", "specialize prime_field_polynomial_equivalent_transitive ((L)+((M)+((J)+(N))))", "apply prime_field_polynomial_equivalent_transitive", "exact cancel_representative_1_witness_witness_right", "specialize prime_field_polynomial_equal_implies_equivalent (x2)", "specialize prime_field_polynomial_equal_implies_equivalent (x3)", "specialize prime_field_polynomial_equal_implies_equivalent (x4)", "specialize prime_field_polynomial_equal_implies_equivalent (x5)", "specialize prime_field_polynomial_equal_implies_equivalent ((L)+((M)+((J)+(N))))", "apply prime_field_polynomial_equal_implies_equivalent", "exact heq", "specialize prime_field_polynomial_equivalent_transitive (bb)", "specialize prime_field_polynomial_equivalent_transitive (bc)", "specialize prime_field_polynomial_equivalent_transitive (M)", "specialize prime_field_polynomial_equivalent_transitive (x4)", "specialize prime_field_polynomial_equivalent_transitive (x5)", "specialize prime_field_polynomial_equivalent_transitive ((L)+((M)+((J)+(N))))", "specialize prime_field_polynomial_equivalent_transitive (cb)", "specialize prime_field_polynomial_equivalent_transitive (cc)", "specialize prime_field_polynomial_equivalent_transitive (J)", "apply prime_field_polynomial_equivalent_transitive", "exact cancel_middle", "specialize prime_field_polynomial_equivalent_symmetric (cb)", "specialize prime_field_polynomial_equivalent_symmetric (cc)", "specialize prime_field_polynomial_equivalent_symmetric (J)", "specialize prime_field_polynomial_equivalent_symmetric (x4)", "specialize prime_field_polynomial_equivalent_symmetric (x5)", "specialize prime_field_polynomial_equivalent_symmetric ((L)+((M)+((J)+(N))))", "apply prime_field_polynomial_equivalent_symmetric", "exact cancel_representative_2_witness_witness_right" ], "script_sha256": "0189fdb405b534f72f871ef7ac74f19b0c81255fafbe6072abfa284808ef242c", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_aligned_algebra_candidate.py", "sha256": "a68de84439afb5f6dd87f1d47449c0bce8dd53a66346c00cc1b7645fb80b2390" }, "statement": "forall p ab ac L bb bc M cb cc J rb rc N. (~((p) = 1) /\\ forall pfa_factor_left_cancel_prime pfa_factor_right_cancel_prime. (p) = pfa_factor_left_cancel_prime * pfa_factor_right_cancel_prime -> pfa_factor_left_cancel_prime = 1 \\/ pfa_factor_right_cancel_prime = 1) -> (((forall fom_index_pfp_cancel_first_left_bounded. (exists fom_gap_pfp_cancel_first_left_bounded_index_bound. fom_gap_pfp_cancel_first_left_bounded_index_bound + S (fom_index_pfp_cancel_first_left_bounded) = L) -> exists fom_value_pfp_cancel_first_left_bounded. ((((exists fom_beta_height_pfp_cancel_first_left_bounded_entry. fom_beta_height_pfp_cancel_first_left_bounded_entry + S (fom_value_pfp_cancel_first_left_bounded) = S ((S (fom_index_pfp_cancel_first_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_cancel_first_left_bounded_entry. ab = fom_beta_quotient_pfp_cancel_first_left_bounded_entry * S ((S (fom_index_pfp_cancel_first_left_bounded)) * ac) + (fom_value_pfp_cancel_first_left_bounded))) /\\ (exists fom_gap_pfp_cancel_first_left_bounded_value_bound. fom_gap_pfp_cancel_first_left_bounded_value_bound + S (fom_value_pfp_cancel_first_left_bounded) = p))) /\\ (((forall fom_index_pfp_cancel_first_right_bounded. (exists fom_gap_pfp_cancel_first_right_bounded_index_bound. fom_gap_pfp_cancel_first_right_bounded_index_bound + S (fom_index_pfp_cancel_first_right_bounded) = M) -> exists fom_value_pfp_cancel_first_right_bounded. ((((exists fom_beta_height_pfp_cancel_first_right_bounded_entry. fom_beta_height_pfp_cancel_first_right_bounded_entry + S (fom_value_pfp_cancel_first_right_bounded) = S ((S (fom_index_pfp_cancel_first_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_cancel_first_right_bounded_entry. bb = fom_beta_quotient_pfp_cancel_first_right_bounded_entry * S ((S (fom_index_pfp_cancel_first_right_bounded)) * bc) + (fom_value_pfp_cancel_first_right_bounded))) /\\ (exists fom_gap_pfp_cancel_first_right_bounded_value_bound. fom_gap_pfp_cancel_first_right_bounded_value_bound + S (fom_value_pfp_cancel_first_right_bounded) = p))) /\\ (((forall fom_index_pfp_cancel_first_result_bounded. (exists fom_gap_pfp_cancel_first_result_bounded_index_bound. fom_gap_pfp_cancel_first_result_bounded_index_bound + S (fom_index_pfp_cancel_first_result_bounded) = N) -> exists fom_value_pfp_cancel_first_result_bounded. ((((exists fom_beta_height_pfp_cancel_first_result_bounded_entry. fom_beta_height_pfp_cancel_first_result_bounded_entry + S (fom_value_pfp_cancel_first_result_bounded) = S ((S (fom_index_pfp_cancel_first_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_cancel_first_result_bounded_entry. rb = fom_beta_quotient_pfp_cancel_first_result_bounded_entry * S ((S (fom_index_pfp_cancel_first_result_bounded)) * rc) + (fom_value_pfp_cancel_first_result_bounded))) /\\ (exists fom_gap_pfp_cancel_first_result_bounded_value_bound. fom_gap_pfp_cancel_first_result_bounded_value_bound + S (fom_value_pfp_cancel_first_result_bounded) = p))) /\\ ((exists pfaa_left_b_cancel_first pfaa_left_c_cancel_first pfaa_right_b_cancel_first pfaa_right_c_cancel_first pfaa_sum_b_cancel_first pfaa_sum_c_cancel_first pfaa_length_cancel_first. ((((forall pfrep_power_cancel_first_witness_common_left pfrep_left_cancel_first_witness_common_left pfrep_right_cancel_first_witness_common_left. ((exists pfrep_position_cancel_first_witness_common_leftfirst. ((pfrep_position_cancel_first_witness_common_leftfirst+S (pfrep_power_cancel_first_witness_common_left)=(L)) /\\ ((((exists ff_h_pfp_cancel_first_witness_common_leftfirstentry. ff_h_pfp_cancel_first_witness_common_leftfirstentry + S (pfrep_left_cancel_first_witness_common_left) = S ((S (pfrep_position_cancel_first_witness_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_cancel_first_witness_common_leftfirstentry. ab = ff_q_pfp_cancel_first_witness_common_leftfirstentry * S ((S (pfrep_position_cancel_first_witness_common_leftfirst)) * ac) + (pfrep_left_cancel_first_witness_common_left)))))) \\/ (((exists pfrep_gap_cancel_first_witness_common_leftfirstoutside. pfrep_gap_cancel_first_witness_common_leftfirstoutside+(L)=(pfrep_power_cancel_first_witness_common_left)) /\\ (((pfrep_left_cancel_first_witness_common_left)=0))))) -> ((exists pfrep_position_cancel_first_witness_common_leftsecond. ((pfrep_position_cancel_first_witness_common_leftsecond+S (pfrep_power_cancel_first_witness_common_left)=(pfaa_length_cancel_first)) /\\ ((((exists ff_h_pfp_cancel_first_witness_common_leftsecondentry. ff_h_pfp_cancel_first_witness_common_leftsecondentry + S (pfrep_right_cancel_first_witness_common_left) = S ((S (pfrep_position_cancel_first_witness_common_leftsecond)) * pfaa_left_c_cancel_first)) /\\ exists ff_q_pfp_cancel_first_witness_common_leftsecondentry. pfaa_left_b_cancel_first = ff_q_pfp_cancel_first_witness_common_leftsecondentry * S ((S (pfrep_position_cancel_first_witness_common_leftsecond)) * pfaa_left_c_cancel_first) + (pfrep_right_cancel_first_witness_common_left)))))) \\/ (((exists pfrep_gap_cancel_first_witness_common_leftsecondoutside. pfrep_gap_cancel_first_witness_common_leftsecondoutside+(pfaa_length_cancel_first)=(pfrep_power_cancel_first_witness_common_left)) /\\ (((pfrep_right_cancel_first_witness_common_left)=0))))) -> pfrep_left_cancel_first_witness_common_left=pfrep_right_cancel_first_witness_common_left) /\\ ((forall pfrep_power_cancel_first_witness_common_right pfrep_left_cancel_first_witness_common_right pfrep_right_cancel_first_witness_common_right. ((exists pfrep_position_cancel_first_witness_common_rightfirst. ((pfrep_position_cancel_first_witness_common_rightfirst+S (pfrep_power_cancel_first_witness_common_right)=(M)) /\\ ((((exists ff_h_pfp_cancel_first_witness_common_rightfirstentry. ff_h_pfp_cancel_first_witness_common_rightfirstentry + S (pfrep_left_cancel_first_witness_common_right) = S ((S (pfrep_position_cancel_first_witness_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_cancel_first_witness_common_rightfirstentry. bb = ff_q_pfp_cancel_first_witness_common_rightfirstentry * S ((S (pfrep_position_cancel_first_witness_common_rightfirst)) * bc) + (pfrep_left_cancel_first_witness_common_right)))))) \\/ (((exists pfrep_gap_cancel_first_witness_common_rightfirstoutside. pfrep_gap_cancel_first_witness_common_rightfirstoutside+(M)=(pfrep_power_cancel_first_witness_common_right)) /\\ (((pfrep_left_cancel_first_witness_common_right)=0))))) -> ((exists pfrep_position_cancel_first_witness_common_rightsecond. ((pfrep_position_cancel_first_witness_common_rightsecond+S (pfrep_power_cancel_first_witness_common_right)=(pfaa_length_cancel_first)) /\\ ((((exists ff_h_pfp_cancel_first_witness_common_rightsecondentry. ff_h_pfp_cancel_first_witness_common_rightsecondentry + S (pfrep_right_cancel_first_witness_common_right) = S ((S (pfrep_position_cancel_first_witness_common_rightsecond)) * pfaa_right_c_cancel_first)) /\\ exists ff_q_pfp_cancel_first_witness_common_rightsecondentry. pfaa_right_b_cancel_first = ff_q_pfp_cancel_first_witness_common_rightsecondentry * S ((S (pfrep_position_cancel_first_witness_common_rightsecond)) * pfaa_right_c_cancel_first) + (pfrep_right_cancel_first_witness_common_right)))))) \\/ (((exists pfrep_gap_cancel_first_witness_common_rightsecondoutside. pfrep_gap_cancel_first_witness_common_rightsecondoutside+(pfaa_length_cancel_first)=(pfrep_power_cancel_first_witness_common_right)) /\\ (((pfrep_right_cancel_first_witness_common_right)=0))))) -> pfrep_left_cancel_first_witness_common_right=pfrep_right_cancel_first_witness_common_right)))) /\\ (((forall pfp_index_cancel_first_witness_operation. (exists pfa_gap_cancel_first_witness_operationindex. pfa_gap_cancel_first_witness_operationindex + S (pfp_index_cancel_first_witness_operation) = (pfaa_length_cancel_first)) -> exists pfp_left_cancel_first_witness_operation pfp_right_cancel_first_witness_operation pfp_value_cancel_first_witness_operation. ((((exists ff_h_pfp_cancel_first_witness_operationleft. ff_h_pfp_cancel_first_witness_operationleft + S (pfp_left_cancel_first_witness_operation) = S ((S (pfp_index_cancel_first_witness_operation)) * pfaa_left_c_cancel_first)) /\\ exists ff_q_pfp_cancel_first_witness_operationleft. pfaa_left_b_cancel_first = ff_q_pfp_cancel_first_witness_operationleft * S ((S (pfp_index_cancel_first_witness_operation)) * pfaa_left_c_cancel_first) + (pfp_left_cancel_first_witness_operation))) /\\ (((((exists ff_h_pfp_cancel_first_witness_operationright. ff_h_pfp_cancel_first_witness_operationright + S (pfp_right_cancel_first_witness_operation) = S ((S (pfp_index_cancel_first_witness_operation)) * pfaa_right_c_cancel_first)) /\\ exists ff_q_pfp_cancel_first_witness_operationright. pfaa_right_b_cancel_first = ff_q_pfp_cancel_first_witness_operationright * S ((S (pfp_index_cancel_first_witness_operation)) * pfaa_right_c_cancel_first) + (pfp_right_cancel_first_witness_operation))) /\\ (((((exists ff_h_pfp_cancel_first_witness_operationtarget. ff_h_pfp_cancel_first_witness_operationtarget + S (pfp_value_cancel_first_witness_operation) = S ((S (pfp_index_cancel_first_witness_operation)) * pfaa_sum_c_cancel_first)) /\\ exists ff_q_pfp_cancel_first_witness_operationtarget. pfaa_sum_b_cancel_first = ff_q_pfp_cancel_first_witness_operationtarget * S ((S (pfp_index_cancel_first_witness_operation)) * pfaa_sum_c_cancel_first) + (pfp_value_cancel_first_witness_operation))) /\\ ((((exists pfa_gap_cancel_first_witness_operationoperationleft. pfa_gap_cancel_first_witness_operationoperationleft + S (pfp_left_cancel_first_witness_operation) = (p)) /\\ (((exists pfa_gap_cancel_first_witness_operationoperationright. pfa_gap_cancel_first_witness_operationoperationright + S (pfp_right_cancel_first_witness_operation) = (p)) /\\ ((((exists pfa_gap_cancel_first_witness_operationoperationresultbound. pfa_gap_cancel_first_witness_operationoperationresultbound + S (pfp_value_cancel_first_witness_operation) = (p)) /\\ ((exists pfa_offset_left_cancel_first_witness_operationoperationresultcongruence pfa_offset_right_cancel_first_witness_operationoperationresultcongruence. ((pfp_left_cancel_first_witness_operation) + (pfp_right_cancel_first_witness_operation)) + (p) * pfa_offset_left_cancel_first_witness_operationoperationresultcongruence = (pfp_value_cancel_first_witness_operation) + (p) * pfa_offset_right_cancel_first_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_cancel_first_witness_output pfrep_left_cancel_first_witness_output pfrep_right_cancel_first_witness_output. ((exists pfrep_position_cancel_first_witness_outputfirst. ((pfrep_position_cancel_first_witness_outputfirst+S (pfrep_power_cancel_first_witness_output)=(pfaa_length_cancel_first)) /\\ ((((exists ff_h_pfp_cancel_first_witness_outputfirstentry. ff_h_pfp_cancel_first_witness_outputfirstentry + S (pfrep_left_cancel_first_witness_output) = S ((S (pfrep_position_cancel_first_witness_outputfirst)) * pfaa_sum_c_cancel_first)) /\\ exists ff_q_pfp_cancel_first_witness_outputfirstentry. pfaa_sum_b_cancel_first = ff_q_pfp_cancel_first_witness_outputfirstentry * S ((S (pfrep_position_cancel_first_witness_outputfirst)) * pfaa_sum_c_cancel_first) + (pfrep_left_cancel_first_witness_output)))))) \\/ (((exists pfrep_gap_cancel_first_witness_outputfirstoutside. pfrep_gap_cancel_first_witness_outputfirstoutside+(pfaa_length_cancel_first)=(pfrep_power_cancel_first_witness_output)) /\\ (((pfrep_left_cancel_first_witness_output)=0))))) -> ((exists pfrep_position_cancel_first_witness_outputsecond. ((pfrep_position_cancel_first_witness_outputsecond+S (pfrep_power_cancel_first_witness_output)=(N)) /\\ ((((exists ff_h_pfp_cancel_first_witness_outputsecondentry. ff_h_pfp_cancel_first_witness_outputsecondentry + S (pfrep_right_cancel_first_witness_output) = S ((S (pfrep_position_cancel_first_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_cancel_first_witness_outputsecondentry. rb = ff_q_pfp_cancel_first_witness_outputsecondentry * S ((S (pfrep_position_cancel_first_witness_outputsecond)) * rc) + (pfrep_right_cancel_first_witness_output)))))) \\/ (((exists pfrep_gap_cancel_first_witness_outputsecondoutside. pfrep_gap_cancel_first_witness_outputsecondoutside+(N)=(pfrep_power_cancel_first_witness_output)) /\\ (((pfrep_right_cancel_first_witness_output)=0))))) -> pfrep_left_cancel_first_witness_output=pfrep_right_cancel_first_witness_output))))))))))))) -> (((forall fom_index_pfp_cancel_second_left_bounded. (exists fom_gap_pfp_cancel_second_left_bounded_index_bound. fom_gap_pfp_cancel_second_left_bounded_index_bound + S (fom_index_pfp_cancel_second_left_bounded) = L) -> exists fom_value_pfp_cancel_second_left_bounded. ((((exists fom_beta_height_pfp_cancel_second_left_bounded_entry. fom_beta_height_pfp_cancel_second_left_bounded_entry + S (fom_value_pfp_cancel_second_left_bounded) = S ((S (fom_index_pfp_cancel_second_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_cancel_second_left_bounded_entry. ab = fom_beta_quotient_pfp_cancel_second_left_bounded_entry * S ((S (fom_index_pfp_cancel_second_left_bounded)) * ac) + (fom_value_pfp_cancel_second_left_bounded))) /\\ (exists fom_gap_pfp_cancel_second_left_bounded_value_bound. fom_gap_pfp_cancel_second_left_bounded_value_bound + S (fom_value_pfp_cancel_second_left_bounded) = p))) /\\ (((forall fom_index_pfp_cancel_second_right_bounded. (exists fom_gap_pfp_cancel_second_right_bounded_index_bound. fom_gap_pfp_cancel_second_right_bounded_index_bound + S (fom_index_pfp_cancel_second_right_bounded) = J) -> exists fom_value_pfp_cancel_second_right_bounded. ((((exists fom_beta_height_pfp_cancel_second_right_bounded_entry. fom_beta_height_pfp_cancel_second_right_bounded_entry + S (fom_value_pfp_cancel_second_right_bounded) = S ((S (fom_index_pfp_cancel_second_right_bounded)) * cc)) /\\ exists fom_beta_quotient_pfp_cancel_second_right_bounded_entry. cb = fom_beta_quotient_pfp_cancel_second_right_bounded_entry * S ((S (fom_index_pfp_cancel_second_right_bounded)) * cc) + (fom_value_pfp_cancel_second_right_bounded))) /\\ (exists fom_gap_pfp_cancel_second_right_bounded_value_bound. fom_gap_pfp_cancel_second_right_bounded_value_bound + S (fom_value_pfp_cancel_second_right_bounded) = p))) /\\ (((forall fom_index_pfp_cancel_second_result_bounded. (exists fom_gap_pfp_cancel_second_result_bounded_index_bound. fom_gap_pfp_cancel_second_result_bounded_index_bound + S (fom_index_pfp_cancel_second_result_bounded) = N) -> exists fom_value_pfp_cancel_second_result_bounded. ((((exists fom_beta_height_pfp_cancel_second_result_bounded_entry. fom_beta_height_pfp_cancel_second_result_bounded_entry + S (fom_value_pfp_cancel_second_result_bounded) = S ((S (fom_index_pfp_cancel_second_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_cancel_second_result_bounded_entry. rb = fom_beta_quotient_pfp_cancel_second_result_bounded_entry * S ((S (fom_index_pfp_cancel_second_result_bounded)) * rc) + (fom_value_pfp_cancel_second_result_bounded))) /\\ (exists fom_gap_pfp_cancel_second_result_bounded_value_bound. fom_gap_pfp_cancel_second_result_bounded_value_bound + S (fom_value_pfp_cancel_second_result_bounded) = p))) /\\ ((exists pfaa_left_b_cancel_second pfaa_left_c_cancel_second pfaa_right_b_cancel_second pfaa_right_c_cancel_second pfaa_sum_b_cancel_second pfaa_sum_c_cancel_second pfaa_length_cancel_second. ((((forall pfrep_power_cancel_second_witness_common_left pfrep_left_cancel_second_witness_common_left pfrep_right_cancel_second_witness_common_left. ((exists pfrep_position_cancel_second_witness_common_leftfirst. ((pfrep_position_cancel_second_witness_common_leftfirst+S (pfrep_power_cancel_second_witness_common_left)=(L)) /\\ ((((exists ff_h_pfp_cancel_second_witness_common_leftfirstentry. ff_h_pfp_cancel_second_witness_common_leftfirstentry + S (pfrep_left_cancel_second_witness_common_left) = S ((S (pfrep_position_cancel_second_witness_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_cancel_second_witness_common_leftfirstentry. ab = ff_q_pfp_cancel_second_witness_common_leftfirstentry * S ((S (pfrep_position_cancel_second_witness_common_leftfirst)) * ac) + (pfrep_left_cancel_second_witness_common_left)))))) \\/ (((exists pfrep_gap_cancel_second_witness_common_leftfirstoutside. pfrep_gap_cancel_second_witness_common_leftfirstoutside+(L)=(pfrep_power_cancel_second_witness_common_left)) /\\ (((pfrep_left_cancel_second_witness_common_left)=0))))) -> ((exists pfrep_position_cancel_second_witness_common_leftsecond. ((pfrep_position_cancel_second_witness_common_leftsecond+S (pfrep_power_cancel_second_witness_common_left)=(pfaa_length_cancel_second)) /\\ ((((exists ff_h_pfp_cancel_second_witness_common_leftsecondentry. ff_h_pfp_cancel_second_witness_common_leftsecondentry + S (pfrep_right_cancel_second_witness_common_left) = S ((S (pfrep_position_cancel_second_witness_common_leftsecond)) * pfaa_left_c_cancel_second)) /\\ exists ff_q_pfp_cancel_second_witness_common_leftsecondentry. pfaa_left_b_cancel_second = ff_q_pfp_cancel_second_witness_common_leftsecondentry * S ((S (pfrep_position_cancel_second_witness_common_leftsecond)) * pfaa_left_c_cancel_second) + (pfrep_right_cancel_second_witness_common_left)))))) \\/ (((exists pfrep_gap_cancel_second_witness_common_leftsecondoutside. pfrep_gap_cancel_second_witness_common_leftsecondoutside+(pfaa_length_cancel_second)=(pfrep_power_cancel_second_witness_common_left)) /\\ (((pfrep_right_cancel_second_witness_common_left)=0))))) -> pfrep_left_cancel_second_witness_common_left=pfrep_right_cancel_second_witness_common_left) /\\ ((forall pfrep_power_cancel_second_witness_common_right pfrep_left_cancel_second_witness_common_right pfrep_right_cancel_second_witness_common_right. ((exists pfrep_position_cancel_second_witness_common_rightfirst. ((pfrep_position_cancel_second_witness_common_rightfirst+S (pfrep_power_cancel_second_witness_common_right)=(J)) /\\ ((((exists ff_h_pfp_cancel_second_witness_common_rightfirstentry. ff_h_pfp_cancel_second_witness_common_rightfirstentry + S (pfrep_left_cancel_second_witness_common_right) = S ((S (pfrep_position_cancel_second_witness_common_rightfirst)) * cc)) /\\ exists ff_q_pfp_cancel_second_witness_common_rightfirstentry. cb = ff_q_pfp_cancel_second_witness_common_rightfirstentry * S ((S (pfrep_position_cancel_second_witness_common_rightfirst)) * cc) + (pfrep_left_cancel_second_witness_common_right)))))) \\/ (((exists pfrep_gap_cancel_second_witness_common_rightfirstoutside. pfrep_gap_cancel_second_witness_common_rightfirstoutside+(J)=(pfrep_power_cancel_second_witness_common_right)) /\\ (((pfrep_left_cancel_second_witness_common_right)=0))))) -> ((exists pfrep_position_cancel_second_witness_common_rightsecond. ((pfrep_position_cancel_second_witness_common_rightsecond+S (pfrep_power_cancel_second_witness_common_right)=(pfaa_length_cancel_second)) /\\ ((((exists ff_h_pfp_cancel_second_witness_common_rightsecondentry. ff_h_pfp_cancel_second_witness_common_rightsecondentry + S (pfrep_right_cancel_second_witness_common_right) = S ((S (pfrep_position_cancel_second_witness_common_rightsecond)) * pfaa_right_c_cancel_second)) /\\ exists ff_q_pfp_cancel_second_witness_common_rightsecondentry. pfaa_right_b_cancel_second = ff_q_pfp_cancel_second_witness_common_rightsecondentry * S ((S (pfrep_position_cancel_second_witness_common_rightsecond)) * pfaa_right_c_cancel_second) + (pfrep_right_cancel_second_witness_common_right)))))) \\/ (((exists pfrep_gap_cancel_second_witness_common_rightsecondoutside. pfrep_gap_cancel_second_witness_common_rightsecondoutside+(pfaa_length_cancel_second)=(pfrep_power_cancel_second_witness_common_right)) /\\ (((pfrep_right_cancel_second_witness_common_right)=0))))) -> pfrep_left_cancel_second_witness_common_right=pfrep_right_cancel_second_witness_common_right)))) /\\ (((forall pfp_index_cancel_second_witness_operation. (exists pfa_gap_cancel_second_witness_operationindex. pfa_gap_cancel_second_witness_operationindex + S (pfp_index_cancel_second_witness_operation) = (pfaa_length_cancel_second)) -> exists pfp_left_cancel_second_witness_operation pfp_right_cancel_second_witness_operation pfp_value_cancel_second_witness_operation. ((((exists ff_h_pfp_cancel_second_witness_operationleft. ff_h_pfp_cancel_second_witness_operationleft + S (pfp_left_cancel_second_witness_operation) = S ((S (pfp_index_cancel_second_witness_operation)) * pfaa_left_c_cancel_second)) /\\ exists ff_q_pfp_cancel_second_witness_operationleft. pfaa_left_b_cancel_second = ff_q_pfp_cancel_second_witness_operationleft * S ((S (pfp_index_cancel_second_witness_operation)) * pfaa_left_c_cancel_second) + (pfp_left_cancel_second_witness_operation))) /\\ (((((exists ff_h_pfp_cancel_second_witness_operationright. ff_h_pfp_cancel_second_witness_operationright + S (pfp_right_cancel_second_witness_operation) = S ((S (pfp_index_cancel_second_witness_operation)) * pfaa_right_c_cancel_second)) /\\ exists ff_q_pfp_cancel_second_witness_operationright. pfaa_right_b_cancel_second = ff_q_pfp_cancel_second_witness_operationright * S ((S (pfp_index_cancel_second_witness_operation)) * pfaa_right_c_cancel_second) + (pfp_right_cancel_second_witness_operation))) /\\ (((((exists ff_h_pfp_cancel_second_witness_operationtarget. ff_h_pfp_cancel_second_witness_operationtarget + S (pfp_value_cancel_second_witness_operation) = S ((S (pfp_index_cancel_second_witness_operation)) * pfaa_sum_c_cancel_second)) /\\ exists ff_q_pfp_cancel_second_witness_operationtarget. pfaa_sum_b_cancel_second = ff_q_pfp_cancel_second_witness_operationtarget * S ((S (pfp_index_cancel_second_witness_operation)) * pfaa_sum_c_cancel_second) + (pfp_value_cancel_second_witness_operation))) /\\ ((((exists pfa_gap_cancel_second_witness_operationoperationleft. pfa_gap_cancel_second_witness_operationoperationleft + S (pfp_left_cancel_second_witness_operation) = (p)) /\\ (((exists pfa_gap_cancel_second_witness_operationoperationright. pfa_gap_cancel_second_witness_operationoperationright + S (pfp_right_cancel_second_witness_operation) = (p)) /\\ ((((exists pfa_gap_cancel_second_witness_operationoperationresultbound. pfa_gap_cancel_second_witness_operationoperationresultbound + S (pfp_value_cancel_second_witness_operation) = (p)) /\\ ((exists pfa_offset_left_cancel_second_witness_operationoperationresultcongruence pfa_offset_right_cancel_second_witness_operationoperationresultcongruence. ((pfp_left_cancel_second_witness_operation) + (pfp_right_cancel_second_witness_operation)) + (p) * pfa_offset_left_cancel_second_witness_operationoperationresultcongruence = (pfp_value_cancel_second_witness_operation) + (p) * pfa_offset_right_cancel_second_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_cancel_second_witness_output pfrep_left_cancel_second_witness_output pfrep_right_cancel_second_witness_output. ((exists pfrep_position_cancel_second_witness_outputfirst. ((pfrep_position_cancel_second_witness_outputfirst+S (pfrep_power_cancel_second_witness_output)=(pfaa_length_cancel_second)) /\\ ((((exists ff_h_pfp_cancel_second_witness_outputfirstentry. ff_h_pfp_cancel_second_witness_outputfirstentry + S (pfrep_left_cancel_second_witness_output) = S ((S (pfrep_position_cancel_second_witness_outputfirst)) * pfaa_sum_c_cancel_second)) /\\ exists ff_q_pfp_cancel_second_witness_outputfirstentry. pfaa_sum_b_cancel_second = ff_q_pfp_cancel_second_witness_outputfirstentry * S ((S (pfrep_position_cancel_second_witness_outputfirst)) * pfaa_sum_c_cancel_second) + (pfrep_left_cancel_second_witness_output)))))) \\/ (((exists pfrep_gap_cancel_second_witness_outputfirstoutside. pfrep_gap_cancel_second_witness_outputfirstoutside+(pfaa_length_cancel_second)=(pfrep_power_cancel_second_witness_output)) /\\ (((pfrep_left_cancel_second_witness_output)=0))))) -> ((exists pfrep_position_cancel_second_witness_outputsecond. ((pfrep_position_cancel_second_witness_outputsecond+S (pfrep_power_cancel_second_witness_output)=(N)) /\\ ((((exists ff_h_pfp_cancel_second_witness_outputsecondentry. ff_h_pfp_cancel_second_witness_outputsecondentry + S (pfrep_right_cancel_second_witness_output) = S ((S (pfrep_position_cancel_second_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_cancel_second_witness_outputsecondentry. rb = ff_q_pfp_cancel_second_witness_outputsecondentry * S ((S (pfrep_position_cancel_second_witness_outputsecond)) * rc) + (pfrep_right_cancel_second_witness_output)))))) \\/ (((exists pfrep_gap_cancel_second_witness_outputsecondoutside. pfrep_gap_cancel_second_witness_outputsecondoutside+(N)=(pfrep_power_cancel_second_witness_output)) /\\ (((pfrep_right_cancel_second_witness_output)=0))))) -> pfrep_left_cancel_second_witness_output=pfrep_right_cancel_second_witness_output))))))))))))) -> (forall pfrep_power_cancel_result pfrep_left_cancel_result pfrep_right_cancel_result. ((exists pfrep_position_cancel_resultfirst. ((pfrep_position_cancel_resultfirst+S (pfrep_power_cancel_result)=(M)) /\\ ((((exists ff_h_pfp_cancel_resultfirstentry. ff_h_pfp_cancel_resultfirstentry + S (pfrep_left_cancel_result) = S ((S (pfrep_position_cancel_resultfirst)) * bc)) /\\ exists ff_q_pfp_cancel_resultfirstentry. bb = ff_q_pfp_cancel_resultfirstentry * S ((S (pfrep_position_cancel_resultfirst)) * bc) + (pfrep_left_cancel_result)))))) \\/ (((exists pfrep_gap_cancel_resultfirstoutside. pfrep_gap_cancel_resultfirstoutside+(M)=(pfrep_power_cancel_result)) /\\ (((pfrep_left_cancel_result)=0))))) -> ((exists pfrep_position_cancel_resultsecond. ((pfrep_position_cancel_resultsecond+S (pfrep_power_cancel_result)=(J)) /\\ ((((exists ff_h_pfp_cancel_resultsecondentry. ff_h_pfp_cancel_resultsecondentry + S (pfrep_right_cancel_result) = S ((S (pfrep_position_cancel_resultsecond)) * cc)) /\\ exists ff_q_pfp_cancel_resultsecondentry. cb = ff_q_pfp_cancel_resultsecondentry * S ((S (pfrep_position_cancel_resultsecond)) * cc) + (pfrep_right_cancel_result)))))) \\/ (((exists pfrep_gap_cancel_resultsecondoutside. pfrep_gap_cancel_resultsecondoutside+(J)=(pfrep_power_cancel_result)) /\\ (((pfrep_right_cancel_result)=0))))) -> pfrep_left_cancel_result=pfrep_right_cancel_result)", "statement_sha256": "acfdd4a7c10fa06393b7d533f4604f8f5bc53a71d49f43bf2596bf509d8cb777", "summary": "Cancel a common addend from actual aligned sums: construct four real common-length prefixes, realize both additions and apply checked coefficient subtraction functionality.", "summary_sha256": "1ad2ab1daa59ed8601007cfdd64eae2f8e62d5a0b129d598a060d605aa931d16" }, { 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(exists fom_gap_pfp_hab_bounded_0_index_bound. fom_gap_pfp_hab_bounded_0_index_bound + S (fom_index_pfp_hab_bounded_0) = La) -> exists fom_value_pfp_hab_bounded_0. ((((exists fom_beta_height_pfp_hab_bounded_0_entry. fom_beta_height_pfp_hab_bounded_0_entry + S (fom_value_pfp_hab_bounded_0) = S ((S (fom_index_pfp_hab_bounded_0)) * ac)) /\\ exists fom_beta_quotient_pfp_hab_bounded_0_entry. ab = fom_beta_quotient_pfp_hab_bounded_0_entry * S ((S (fom_index_pfp_hab_bounded_0)) * ac) + (fom_value_pfp_hab_bounded_0))) /\\ (exists fom_gap_pfp_hab_bounded_0_value_bound. fom_gap_pfp_hab_bounded_0_value_bound + S (fom_value_pfp_hab_bounded_0) = p))) /\\ (((forall fom_index_pfp_hab_bounded_1. (exists fom_gap_pfp_hab_bounded_1_index_bound. fom_gap_pfp_hab_bounded_1_index_bound + S (fom_index_pfp_hab_bounded_1) = Lb) -> exists fom_value_pfp_hab_bounded_1. ((((exists fom_beta_height_pfp_hab_bounded_1_entry. fom_beta_height_pfp_hab_bounded_1_entry + S (fom_value_pfp_hab_bounded_1) = S ((S (fom_index_pfp_hab_bounded_1)) * bc)) /\\ exists fom_beta_quotient_pfp_hab_bounded_1_entry. bb = fom_beta_quotient_pfp_hab_bounded_1_entry * S ((S (fom_index_pfp_hab_bounded_1)) * bc) + (fom_value_pfp_hab_bounded_1))) /\\ (exists fom_gap_pfp_hab_bounded_1_value_bound. fom_gap_pfp_hab_bounded_1_value_bound + S (fom_value_pfp_hab_bounded_1) = p))) /\\ ((forall fom_index_pfp_hab_bounded_2. (exists fom_gap_pfp_hab_bounded_2_index_bound. fom_gap_pfp_hab_bounded_2_index_bound + S (fom_index_pfp_hab_bounded_2) = Lu) -> exists fom_value_pfp_hab_bounded_2. ((((exists fom_beta_height_pfp_hab_bounded_2_entry. fom_beta_height_pfp_hab_bounded_2_entry + S (fom_value_pfp_hab_bounded_2) = S ((S (fom_index_pfp_hab_bounded_2)) * uc)) /\\ exists fom_beta_quotient_pfp_hab_bounded_2_entry. ub = fom_beta_quotient_pfp_hab_bounded_2_entry * S ((S (fom_index_pfp_hab_bounded_2)) * uc) + (fom_value_pfp_hab_bounded_2))) /\\ (exists fom_gap_pfp_hab_bounded_2_value_bound. fom_gap_pfp_hab_bounded_2_value_bound + S (fom_value_pfp_hab_bounded_2) = p)))))))", "specialize prime_field_polynomial_aligned_add_bounded (p)", "specialize prime_field_polynomial_aligned_add_bounded (ab)", "specialize prime_field_polynomial_aligned_add_bounded (ac)", "specialize prime_field_polynomial_aligned_add_bounded (La)", "specialize prime_field_polynomial_aligned_add_bounded (bb)", "specialize prime_field_polynomial_aligned_add_bounded (bc)", "specialize prime_field_polynomial_aligned_add_bounded (Lb)", "specialize prime_field_polynomial_aligned_add_bounded (ub)", "specialize prime_field_polynomial_aligned_add_bounded (uc)", "specialize prime_field_polynomial_aligned_add_bounded (Lu)", "apply prime_field_polynomial_aligned_add_bounded", "exact hab", "cases hab_bounded", "cases hab_bounded_right", "have hleft_bounded : ((forall fom_index_pfp_hleft_bounded_0. (exists fom_gap_pfp_hleft_bounded_0_index_bound. fom_gap_pfp_hleft_bounded_0_index_bound + S (fom_index_pfp_hleft_bounded_0) = Lu) -> exists fom_value_pfp_hleft_bounded_0. ((((exists fom_beta_height_pfp_hleft_bounded_0_entry. fom_beta_height_pfp_hleft_bounded_0_entry + S (fom_value_pfp_hleft_bounded_0) = S ((S (fom_index_pfp_hleft_bounded_0)) * uc)) /\\ exists fom_beta_quotient_pfp_hleft_bounded_0_entry. ub = fom_beta_quotient_pfp_hleft_bounded_0_entry * S ((S (fom_index_pfp_hleft_bounded_0)) * uc) + (fom_value_pfp_hleft_bounded_0))) /\\ (exists fom_gap_pfp_hleft_bounded_0_value_bound. fom_gap_pfp_hleft_bounded_0_value_bound + S (fom_value_pfp_hleft_bounded_0) = p))) /\\ (((forall fom_index_pfp_hleft_bounded_1. (exists fom_gap_pfp_hleft_bounded_1_index_bound. fom_gap_pfp_hleft_bounded_1_index_bound + S (fom_index_pfp_hleft_bounded_1) = Lc) -> exists fom_value_pfp_hleft_bounded_1. ((((exists fom_beta_height_pfp_hleft_bounded_1_entry. fom_beta_height_pfp_hleft_bounded_1_entry + S (fom_value_pfp_hleft_bounded_1) = S ((S (fom_index_pfp_hleft_bounded_1)) * cc)) /\\ exists fom_beta_quotient_pfp_hleft_bounded_1_entry. cb = fom_beta_quotient_pfp_hleft_bounded_1_entry * S ((S (fom_index_pfp_hleft_bounded_1)) * cc) + (fom_value_pfp_hleft_bounded_1))) /\\ (exists fom_gap_pfp_hleft_bounded_1_value_bound. fom_gap_pfp_hleft_bounded_1_value_bound + S (fom_value_pfp_hleft_bounded_1) = p))) /\\ ((forall fom_index_pfp_hleft_bounded_2. (exists fom_gap_pfp_hleft_bounded_2_index_bound. fom_gap_pfp_hleft_bounded_2_index_bound + S (fom_index_pfp_hleft_bounded_2) = Lr) -> exists fom_value_pfp_hleft_bounded_2. ((((exists fom_beta_height_pfp_hleft_bounded_2_entry. fom_beta_height_pfp_hleft_bounded_2_entry + S (fom_value_pfp_hleft_bounded_2) = S ((S (fom_index_pfp_hleft_bounded_2)) * rc)) /\\ exists fom_beta_quotient_pfp_hleft_bounded_2_entry. rb = fom_beta_quotient_pfp_hleft_bounded_2_entry * S ((S (fom_index_pfp_hleft_bounded_2)) * rc) + (fom_value_pfp_hleft_bounded_2))) /\\ (exists fom_gap_pfp_hleft_bounded_2_value_bound. fom_gap_pfp_hleft_bounded_2_value_bound + S (fom_value_pfp_hleft_bounded_2) = p)))))))", "specialize prime_field_polynomial_aligned_add_bounded (p)", "specialize prime_field_polynomial_aligned_add_bounded (ub)", "specialize prime_field_polynomial_aligned_add_bounded (uc)", "specialize prime_field_polynomial_aligned_add_bounded (Lu)", "specialize prime_field_polynomial_aligned_add_bounded (cb)", "specialize prime_field_polynomial_aligned_add_bounded (cc)", "specialize prime_field_polynomial_aligned_add_bounded (Lc)", "specialize prime_field_polynomial_aligned_add_bounded (rb)", "specialize prime_field_polynomial_aligned_add_bounded (rc)", "specialize prime_field_polynomial_aligned_add_bounded (Lr)", "apply prime_field_polynomial_aligned_add_bounded", "exact hleft", "cases hleft_bounded", "cases hleft_bounded_right", "have hbc_bounded : ((forall fom_index_pfp_hbc_bounded_0. (exists fom_gap_pfp_hbc_bounded_0_index_bound. fom_gap_pfp_hbc_bounded_0_index_bound + S (fom_index_pfp_hbc_bounded_0) = Lb) -> exists fom_value_pfp_hbc_bounded_0. ((((exists fom_beta_height_pfp_hbc_bounded_0_entry. fom_beta_height_pfp_hbc_bounded_0_entry + S (fom_value_pfp_hbc_bounded_0) = S ((S (fom_index_pfp_hbc_bounded_0)) * bc)) /\\ exists fom_beta_quotient_pfp_hbc_bounded_0_entry. bb = fom_beta_quotient_pfp_hbc_bounded_0_entry * S ((S (fom_index_pfp_hbc_bounded_0)) * bc) + (fom_value_pfp_hbc_bounded_0))) /\\ (exists fom_gap_pfp_hbc_bounded_0_value_bound. fom_gap_pfp_hbc_bounded_0_value_bound + S (fom_value_pfp_hbc_bounded_0) = p))) /\\ (((forall fom_index_pfp_hbc_bounded_1. (exists fom_gap_pfp_hbc_bounded_1_index_bound. fom_gap_pfp_hbc_bounded_1_index_bound + S (fom_index_pfp_hbc_bounded_1) = Lc) -> exists fom_value_pfp_hbc_bounded_1. ((((exists fom_beta_height_pfp_hbc_bounded_1_entry. fom_beta_height_pfp_hbc_bounded_1_entry + S (fom_value_pfp_hbc_bounded_1) = S ((S (fom_index_pfp_hbc_bounded_1)) * cc)) /\\ exists fom_beta_quotient_pfp_hbc_bounded_1_entry. cb = fom_beta_quotient_pfp_hbc_bounded_1_entry * S ((S (fom_index_pfp_hbc_bounded_1)) * cc) + (fom_value_pfp_hbc_bounded_1))) /\\ (exists fom_gap_pfp_hbc_bounded_1_value_bound. fom_gap_pfp_hbc_bounded_1_value_bound + S (fom_value_pfp_hbc_bounded_1) = p))) /\\ ((forall fom_index_pfp_hbc_bounded_2. (exists fom_gap_pfp_hbc_bounded_2_index_bound. fom_gap_pfp_hbc_bounded_2_index_bound + S (fom_index_pfp_hbc_bounded_2) = Lv) -> exists fom_value_pfp_hbc_bounded_2. ((((exists fom_beta_height_pfp_hbc_bounded_2_entry. fom_beta_height_pfp_hbc_bounded_2_entry + S (fom_value_pfp_hbc_bounded_2) = S ((S (fom_index_pfp_hbc_bounded_2)) * vc)) /\\ exists fom_beta_quotient_pfp_hbc_bounded_2_entry. vb = fom_beta_quotient_pfp_hbc_bounded_2_entry * S ((S (fom_index_pfp_hbc_bounded_2)) * vc) + (fom_value_pfp_hbc_bounded_2))) /\\ (exists fom_gap_pfp_hbc_bounded_2_value_bound. fom_gap_pfp_hbc_bounded_2_value_bound + S (fom_value_pfp_hbc_bounded_2) = p)))))))", "specialize prime_field_polynomial_aligned_add_bounded (p)", "specialize prime_field_polynomial_aligned_add_bounded (bb)", "specialize prime_field_polynomial_aligned_add_bounded (bc)", "specialize prime_field_polynomial_aligned_add_bounded (Lb)", "specialize prime_field_polynomial_aligned_add_bounded (cb)", "specialize prime_field_polynomial_aligned_add_bounded (cc)", "specialize prime_field_polynomial_aligned_add_bounded (Lc)", "specialize prime_field_polynomial_aligned_add_bounded (vb)", "specialize prime_field_polynomial_aligned_add_bounded (vc)", "specialize prime_field_polynomial_aligned_add_bounded (Lv)", "apply prime_field_polynomial_aligned_add_bounded", "exact hbc", "cases hbc_bounded", "cases hbc_bounded_right", "have hright_bounded : ((forall fom_index_pfp_hright_bounded_0. (exists fom_gap_pfp_hright_bounded_0_index_bound. fom_gap_pfp_hright_bounded_0_index_bound + S (fom_index_pfp_hright_bounded_0) = La) -> exists fom_value_pfp_hright_bounded_0. ((((exists fom_beta_height_pfp_hright_bounded_0_entry. fom_beta_height_pfp_hright_bounded_0_entry + S (fom_value_pfp_hright_bounded_0) = S ((S (fom_index_pfp_hright_bounded_0)) * ac)) /\\ exists fom_beta_quotient_pfp_hright_bounded_0_entry. ab = fom_beta_quotient_pfp_hright_bounded_0_entry * S ((S (fom_index_pfp_hright_bounded_0)) * ac) + (fom_value_pfp_hright_bounded_0))) /\\ (exists fom_gap_pfp_hright_bounded_0_value_bound. fom_gap_pfp_hright_bounded_0_value_bound + S (fom_value_pfp_hright_bounded_0) = p))) /\\ (((forall fom_index_pfp_hright_bounded_1. (exists fom_gap_pfp_hright_bounded_1_index_bound. fom_gap_pfp_hright_bounded_1_index_bound + S (fom_index_pfp_hright_bounded_1) = Lv) -> exists fom_value_pfp_hright_bounded_1. ((((exists fom_beta_height_pfp_hright_bounded_1_entry. fom_beta_height_pfp_hright_bounded_1_entry + S (fom_value_pfp_hright_bounded_1) = S ((S (fom_index_pfp_hright_bounded_1)) * vc)) /\\ exists fom_beta_quotient_pfp_hright_bounded_1_entry. vb = fom_beta_quotient_pfp_hright_bounded_1_entry * S ((S (fom_index_pfp_hright_bounded_1)) * vc) + (fom_value_pfp_hright_bounded_1))) /\\ (exists fom_gap_pfp_hright_bounded_1_value_bound. fom_gap_pfp_hright_bounded_1_value_bound + S (fom_value_pfp_hright_bounded_1) = p))) /\\ ((forall fom_index_pfp_hright_bounded_2. (exists fom_gap_pfp_hright_bounded_2_index_bound. fom_gap_pfp_hright_bounded_2_index_bound + S (fom_index_pfp_hright_bounded_2) = Ls) -> exists fom_value_pfp_hright_bounded_2. ((((exists fom_beta_height_pfp_hright_bounded_2_entry. fom_beta_height_pfp_hright_bounded_2_entry + S (fom_value_pfp_hright_bounded_2) = S ((S (fom_index_pfp_hright_bounded_2)) * sc)) /\\ exists fom_beta_quotient_pfp_hright_bounded_2_entry. sb = fom_beta_quotient_pfp_hright_bounded_2_entry * S ((S (fom_index_pfp_hright_bounded_2)) * sc) + (fom_value_pfp_hright_bounded_2))) /\\ (exists fom_gap_pfp_hright_bounded_2_value_bound. fom_gap_pfp_hright_bounded_2_value_bound + S (fom_value_pfp_hright_bounded_2) = p)))))))", "specialize prime_field_polynomial_aligned_add_bounded (p)", "specialize prime_field_polynomial_aligned_add_bounded (ab)", "specialize prime_field_polynomial_aligned_add_bounded (ac)", "specialize prime_field_polynomial_aligned_add_bounded (La)", "specialize prime_field_polynomial_aligned_add_bounded (vb)", "specialize prime_field_polynomial_aligned_add_bounded (vc)", "specialize prime_field_polynomial_aligned_add_bounded (Lv)", "specialize prime_field_polynomial_aligned_add_bounded (sb)", "specialize prime_field_polynomial_aligned_add_bounded (sc)", "specialize prime_field_polynomial_aligned_add_bounded (Ls)", "apply prime_field_polynomial_aligned_add_bounded", "exact hright", "cases hright_bounded", "cases hright_bounded_right", "have associative_representative_0 : exists associative_representative_0_code associative_representative_0_scale. ((forall fom_index_pfp_associative_representative_0_bounded. (exists fom_gap_pfp_associative_representative_0_bounded_index_bound. fom_gap_pfp_associative_representative_0_bounded_index_bound + S (fom_index_pfp_associative_representative_0_bounded) = (La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))) -> exists fom_value_pfp_associative_representative_0_bounded. ((((exists fom_beta_height_pfp_associative_representative_0_bounded_entry. fom_beta_height_pfp_associative_representative_0_bounded_entry + S (fom_value_pfp_associative_representative_0_bounded) = S ((S (fom_index_pfp_associative_representative_0_bounded)) * associative_representative_0_scale)) /\\ exists fom_beta_quotient_pfp_associative_representative_0_bounded_entry. associative_representative_0_code = fom_beta_quotient_pfp_associative_representative_0_bounded_entry * S ((S (fom_index_pfp_associative_representative_0_bounded)) * associative_representative_0_scale) + (fom_value_pfp_associative_representative_0_bounded))) /\\ (exists fom_gap_pfp_associative_representative_0_bounded_value_bound. fom_gap_pfp_associative_representative_0_bounded_value_bound + S (fom_value_pfp_associative_representative_0_bounded) = p))) /\\ ((forall pfrep_power_associative_representative_0_equivalent pfrep_left_associative_representative_0_equivalent pfrep_right_associative_representative_0_equivalent. ((exists pfrep_position_associative_representative_0_equivalentfirst. ((pfrep_position_associative_representative_0_equivalentfirst+S (pfrep_power_associative_representative_0_equivalent)=(La)) /\\ ((((exists ff_h_pfp_associative_representative_0_equivalentfirstentry. ff_h_pfp_associative_representative_0_equivalentfirstentry + S (pfrep_left_associative_representative_0_equivalent) = S ((S (pfrep_position_associative_representative_0_equivalentfirst)) * ac)) /\\ exists ff_q_pfp_associative_representative_0_equivalentfirstentry. ab = ff_q_pfp_associative_representative_0_equivalentfirstentry * S ((S (pfrep_position_associative_representative_0_equivalentfirst)) * ac) + (pfrep_left_associative_representative_0_equivalent)))))) \\/ (((exists pfrep_gap_associative_representative_0_equivalentfirstoutside. pfrep_gap_associative_representative_0_equivalentfirstoutside+(La)=(pfrep_power_associative_representative_0_equivalent)) /\\ (((pfrep_left_associative_representative_0_equivalent)=0))))) -> ((exists pfrep_position_associative_representative_0_equivalentsecond. ((pfrep_position_associative_representative_0_equivalentsecond+S (pfrep_power_associative_representative_0_equivalent)=((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))) /\\ ((((exists ff_h_pfp_associative_representative_0_equivalentsecondentry. ff_h_pfp_associative_representative_0_equivalentsecondentry + S (pfrep_right_associative_representative_0_equivalent) = S ((S (pfrep_position_associative_representative_0_equivalentsecond)) * associative_representative_0_scale)) /\\ exists ff_q_pfp_associative_representative_0_equivalentsecondentry. associative_representative_0_code = ff_q_pfp_associative_representative_0_equivalentsecondentry * S ((S (pfrep_position_associative_representative_0_equivalentsecond)) * associative_representative_0_scale) + (pfrep_right_associative_representative_0_equivalent)))))) \\/ (((exists pfrep_gap_associative_representative_0_equivalentsecondoutside. pfrep_gap_associative_representative_0_equivalentsecondoutside+((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))=(pfrep_power_associative_representative_0_equivalent)) /\\ (((pfrep_right_associative_representative_0_equivalent)=0))))) -> pfrep_left_associative_representative_0_equivalent=pfrep_right_associative_representative_0_equivalent)))", "specialize prime_field_polynomial_bounded_representative_at_length_exists (p)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (ab)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (ac)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (La)", "specialize prime_field_polynomial_bounded_representative_at_length_exists ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply prime_field_polynomial_bounded_representative_at_length_exists", "exact hp", "exact hab_bounded_left", "specialize le_add_right (La)", "specialize le_add_right ((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "apply le_add_right", "cases associative_representative_0", "cases associative_representative_0_witness", "cases associative_representative_0_witness_witness", "have associative_representative_1 : exists associative_representative_1_code associative_representative_1_scale. ((forall fom_index_pfp_associative_representative_1_bounded. (exists fom_gap_pfp_associative_representative_1_bounded_index_bound. fom_gap_pfp_associative_representative_1_bounded_index_bound + S (fom_index_pfp_associative_representative_1_bounded) = (La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))) -> exists fom_value_pfp_associative_representative_1_bounded. ((((exists fom_beta_height_pfp_associative_representative_1_bounded_entry. fom_beta_height_pfp_associative_representative_1_bounded_entry + S (fom_value_pfp_associative_representative_1_bounded) = S ((S (fom_index_pfp_associative_representative_1_bounded)) * associative_representative_1_scale)) /\\ exists fom_beta_quotient_pfp_associative_representative_1_bounded_entry. associative_representative_1_code = fom_beta_quotient_pfp_associative_representative_1_bounded_entry * S ((S (fom_index_pfp_associative_representative_1_bounded)) * associative_representative_1_scale) + (fom_value_pfp_associative_representative_1_bounded))) /\\ (exists fom_gap_pfp_associative_representative_1_bounded_value_bound. fom_gap_pfp_associative_representative_1_bounded_value_bound + S (fom_value_pfp_associative_representative_1_bounded) = p))) /\\ ((forall pfrep_power_associative_representative_1_equivalent pfrep_left_associative_representative_1_equivalent pfrep_right_associative_representative_1_equivalent. ((exists pfrep_position_associative_representative_1_equivalentfirst. ((pfrep_position_associative_representative_1_equivalentfirst+S (pfrep_power_associative_representative_1_equivalent)=(Lb)) /\\ ((((exists ff_h_pfp_associative_representative_1_equivalentfirstentry. ff_h_pfp_associative_representative_1_equivalentfirstentry + S (pfrep_left_associative_representative_1_equivalent) = S ((S (pfrep_position_associative_representative_1_equivalentfirst)) * bc)) /\\ exists ff_q_pfp_associative_representative_1_equivalentfirstentry. bb = ff_q_pfp_associative_representative_1_equivalentfirstentry * S ((S (pfrep_position_associative_representative_1_equivalentfirst)) * bc) + (pfrep_left_associative_representative_1_equivalent)))))) \\/ (((exists pfrep_gap_associative_representative_1_equivalentfirstoutside. pfrep_gap_associative_representative_1_equivalentfirstoutside+(Lb)=(pfrep_power_associative_representative_1_equivalent)) /\\ (((pfrep_left_associative_representative_1_equivalent)=0))))) -> ((exists pfrep_position_associative_representative_1_equivalentsecond. ((pfrep_position_associative_representative_1_equivalentsecond+S (pfrep_power_associative_representative_1_equivalent)=((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))) /\\ ((((exists ff_h_pfp_associative_representative_1_equivalentsecondentry. ff_h_pfp_associative_representative_1_equivalentsecondentry + S (pfrep_right_associative_representative_1_equivalent) = S ((S (pfrep_position_associative_representative_1_equivalentsecond)) * associative_representative_1_scale)) /\\ exists ff_q_pfp_associative_representative_1_equivalentsecondentry. associative_representative_1_code = ff_q_pfp_associative_representative_1_equivalentsecondentry * S ((S (pfrep_position_associative_representative_1_equivalentsecond)) * associative_representative_1_scale) + (pfrep_right_associative_representative_1_equivalent)))))) \\/ (((exists pfrep_gap_associative_representative_1_equivalentsecondoutside. pfrep_gap_associative_representative_1_equivalentsecondoutside+((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))=(pfrep_power_associative_representative_1_equivalent)) /\\ (((pfrep_right_associative_representative_1_equivalent)=0))))) -> pfrep_left_associative_representative_1_equivalent=pfrep_right_associative_representative_1_equivalent)))", "specialize prime_field_polynomial_bounded_representative_at_length_exists (p)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (bb)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (bc)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (Lb)", "specialize prime_field_polynomial_bounded_representative_at_length_exists ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply prime_field_polynomial_bounded_representative_at_length_exists", "exact hp", "exact hab_bounded_right_left", "have length_bound_associative_representative_1 : exists pfrep_gap_associative_representative_1_inner. pfrep_gap_associative_representative_1_inner+(Lb)=((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "specialize le_add_right (Lb)", "specialize le_add_right ((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))", "apply le_add_right", "specialize le_trans (Lb)", "specialize le_trans ((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "specialize le_trans ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply le_trans", "exact length_bound_associative_representative_1", "exists La", "refl", "cases associative_representative_1", "cases associative_representative_1_witness", "cases associative_representative_1_witness_witness", "have associative_representative_2 : exists associative_representative_2_code associative_representative_2_scale. ((forall fom_index_pfp_associative_representative_2_bounded. (exists fom_gap_pfp_associative_representative_2_bounded_index_bound. fom_gap_pfp_associative_representative_2_bounded_index_bound + S (fom_index_pfp_associative_representative_2_bounded) = (La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))) -> exists fom_value_pfp_associative_representative_2_bounded. ((((exists fom_beta_height_pfp_associative_representative_2_bounded_entry. fom_beta_height_pfp_associative_representative_2_bounded_entry + S (fom_value_pfp_associative_representative_2_bounded) = S ((S (fom_index_pfp_associative_representative_2_bounded)) * associative_representative_2_scale)) /\\ exists fom_beta_quotient_pfp_associative_representative_2_bounded_entry. associative_representative_2_code = fom_beta_quotient_pfp_associative_representative_2_bounded_entry * S ((S (fom_index_pfp_associative_representative_2_bounded)) * associative_representative_2_scale) + (fom_value_pfp_associative_representative_2_bounded))) /\\ (exists fom_gap_pfp_associative_representative_2_bounded_value_bound. fom_gap_pfp_associative_representative_2_bounded_value_bound + S (fom_value_pfp_associative_representative_2_bounded) = p))) /\\ ((forall pfrep_power_associative_representative_2_equivalent pfrep_left_associative_representative_2_equivalent pfrep_right_associative_representative_2_equivalent. ((exists pfrep_position_associative_representative_2_equivalentfirst. ((pfrep_position_associative_representative_2_equivalentfirst+S (pfrep_power_associative_representative_2_equivalent)=(Lc)) /\\ ((((exists ff_h_pfp_associative_representative_2_equivalentfirstentry. ff_h_pfp_associative_representative_2_equivalentfirstentry + S (pfrep_left_associative_representative_2_equivalent) = S ((S (pfrep_position_associative_representative_2_equivalentfirst)) * cc)) /\\ exists ff_q_pfp_associative_representative_2_equivalentfirstentry. cb = ff_q_pfp_associative_representative_2_equivalentfirstentry * S ((S (pfrep_position_associative_representative_2_equivalentfirst)) * cc) + (pfrep_left_associative_representative_2_equivalent)))))) \\/ (((exists pfrep_gap_associative_representative_2_equivalentfirstoutside. pfrep_gap_associative_representative_2_equivalentfirstoutside+(Lc)=(pfrep_power_associative_representative_2_equivalent)) /\\ (((pfrep_left_associative_representative_2_equivalent)=0))))) -> ((exists pfrep_position_associative_representative_2_equivalentsecond. ((pfrep_position_associative_representative_2_equivalentsecond+S (pfrep_power_associative_representative_2_equivalent)=((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))) /\\ ((((exists ff_h_pfp_associative_representative_2_equivalentsecondentry. ff_h_pfp_associative_representative_2_equivalentsecondentry + S (pfrep_right_associative_representative_2_equivalent) = S ((S (pfrep_position_associative_representative_2_equivalentsecond)) * associative_representative_2_scale)) /\\ exists ff_q_pfp_associative_representative_2_equivalentsecondentry. associative_representative_2_code = ff_q_pfp_associative_representative_2_equivalentsecondentry * S ((S (pfrep_position_associative_representative_2_equivalentsecond)) * associative_representative_2_scale) + (pfrep_right_associative_representative_2_equivalent)))))) \\/ (((exists pfrep_gap_associative_representative_2_equivalentsecondoutside. pfrep_gap_associative_representative_2_equivalentsecondoutside+((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))=(pfrep_power_associative_representative_2_equivalent)) /\\ (((pfrep_right_associative_representative_2_equivalent)=0))))) -> pfrep_left_associative_representative_2_equivalent=pfrep_right_associative_representative_2_equivalent)))", "specialize prime_field_polynomial_bounded_representative_at_length_exists (p)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (cb)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (cc)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (Lc)", "specialize prime_field_polynomial_bounded_representative_at_length_exists ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply prime_field_polynomial_bounded_representative_at_length_exists", "exact hp", "exact hbc_bounded_right_left", "have length_bound_associative_representative_2 : exists pfrep_gap_associative_representative_2_inner. pfrep_gap_associative_representative_2_inner+(Lc)=((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "have length_bound_associative_representative_2_inner : exists pfrep_gap_associative_representative_2_inner_inner. pfrep_gap_associative_representative_2_inner_inner+(Lc)=((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))", "specialize le_add_right (Lc)", "specialize le_add_right ((Lu)+((Lv)+((Lr)+(Ls))))", "apply le_add_right", "specialize le_trans (Lc)", "specialize le_trans ((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))", "specialize le_trans ((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "apply le_trans", "exact length_bound_associative_representative_2_inner", "exists Lb", "refl", "specialize le_trans (Lc)", "specialize le_trans ((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "specialize le_trans ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply le_trans", "exact length_bound_associative_representative_2", "exists La", "refl", "cases associative_representative_2", "cases associative_representative_2_witness", "cases associative_representative_2_witness_witness", "have associative_representative_3 : exists associative_representative_3_code associative_representative_3_scale. ((forall fom_index_pfp_associative_representative_3_bounded. (exists fom_gap_pfp_associative_representative_3_bounded_index_bound. fom_gap_pfp_associative_representative_3_bounded_index_bound + S (fom_index_pfp_associative_representative_3_bounded) = (La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))) -> exists fom_value_pfp_associative_representative_3_bounded. ((((exists fom_beta_height_pfp_associative_representative_3_bounded_entry. fom_beta_height_pfp_associative_representative_3_bounded_entry + S (fom_value_pfp_associative_representative_3_bounded) = S ((S (fom_index_pfp_associative_representative_3_bounded)) * associative_representative_3_scale)) /\\ exists fom_beta_quotient_pfp_associative_representative_3_bounded_entry. associative_representative_3_code = fom_beta_quotient_pfp_associative_representative_3_bounded_entry * S ((S (fom_index_pfp_associative_representative_3_bounded)) * associative_representative_3_scale) + (fom_value_pfp_associative_representative_3_bounded))) /\\ (exists fom_gap_pfp_associative_representative_3_bounded_value_bound. fom_gap_pfp_associative_representative_3_bounded_value_bound + S (fom_value_pfp_associative_representative_3_bounded) = p))) /\\ ((forall pfrep_power_associative_representative_3_equivalent pfrep_left_associative_representative_3_equivalent pfrep_right_associative_representative_3_equivalent. ((exists pfrep_position_associative_representative_3_equivalentfirst. ((pfrep_position_associative_representative_3_equivalentfirst+S (pfrep_power_associative_representative_3_equivalent)=(Lu)) /\\ ((((exists ff_h_pfp_associative_representative_3_equivalentfirstentry. ff_h_pfp_associative_representative_3_equivalentfirstentry + S (pfrep_left_associative_representative_3_equivalent) = S ((S (pfrep_position_associative_representative_3_equivalentfirst)) * uc)) /\\ exists ff_q_pfp_associative_representative_3_equivalentfirstentry. ub = ff_q_pfp_associative_representative_3_equivalentfirstentry * S ((S (pfrep_position_associative_representative_3_equivalentfirst)) * uc) + (pfrep_left_associative_representative_3_equivalent)))))) \\/ (((exists pfrep_gap_associative_representative_3_equivalentfirstoutside. pfrep_gap_associative_representative_3_equivalentfirstoutside+(Lu)=(pfrep_power_associative_representative_3_equivalent)) /\\ (((pfrep_left_associative_representative_3_equivalent)=0))))) -> ((exists pfrep_position_associative_representative_3_equivalentsecond. ((pfrep_position_associative_representative_3_equivalentsecond+S (pfrep_power_associative_representative_3_equivalent)=((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))) /\\ ((((exists ff_h_pfp_associative_representative_3_equivalentsecondentry. ff_h_pfp_associative_representative_3_equivalentsecondentry + S (pfrep_right_associative_representative_3_equivalent) = S ((S (pfrep_position_associative_representative_3_equivalentsecond)) * associative_representative_3_scale)) /\\ exists ff_q_pfp_associative_representative_3_equivalentsecondentry. associative_representative_3_code = ff_q_pfp_associative_representative_3_equivalentsecondentry * S ((S (pfrep_position_associative_representative_3_equivalentsecond)) * associative_representative_3_scale) + (pfrep_right_associative_representative_3_equivalent)))))) \\/ (((exists pfrep_gap_associative_representative_3_equivalentsecondoutside. pfrep_gap_associative_representative_3_equivalentsecondoutside+((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))=(pfrep_power_associative_representative_3_equivalent)) /\\ (((pfrep_right_associative_representative_3_equivalent)=0))))) -> pfrep_left_associative_representative_3_equivalent=pfrep_right_associative_representative_3_equivalent)))", "specialize prime_field_polynomial_bounded_representative_at_length_exists (p)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (ub)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (uc)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (Lu)", "specialize prime_field_polynomial_bounded_representative_at_length_exists ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply prime_field_polynomial_bounded_representative_at_length_exists", "exact hp", "exact hab_bounded_right_right", "have length_bound_associative_representative_3 : exists pfrep_gap_associative_representative_3_inner. pfrep_gap_associative_representative_3_inner+(Lu)=((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "have length_bound_associative_representative_3_inner : exists pfrep_gap_associative_representative_3_inner_inner. pfrep_gap_associative_representative_3_inner_inner+(Lu)=((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))", "have length_bound_associative_representative_3_inner_inner : exists pfrep_gap_associative_representative_3_inner_inner_inner. pfrep_gap_associative_representative_3_inner_inner_inner+(Lu)=((Lu)+((Lv)+((Lr)+(Ls))))", "specialize le_add_right (Lu)", "specialize le_add_right ((Lv)+((Lr)+(Ls)))", "apply le_add_right", "specialize le_trans (Lu)", "specialize le_trans ((Lu)+((Lv)+((Lr)+(Ls))))", "specialize le_trans ((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))", "apply le_trans", "exact length_bound_associative_representative_3_inner_inner", "exists Lc", "refl", "specialize le_trans (Lu)", "specialize le_trans ((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))", "specialize le_trans ((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "apply le_trans", "exact length_bound_associative_representative_3_inner", "exists Lb", "refl", "specialize le_trans (Lu)", "specialize le_trans ((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "specialize le_trans ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply le_trans", "exact length_bound_associative_representative_3", "exists La", "refl", "cases associative_representative_3", "cases associative_representative_3_witness", "cases associative_representative_3_witness_witness", "have associative_representative_4 : exists associative_representative_4_code associative_representative_4_scale. ((forall fom_index_pfp_associative_representative_4_bounded. (exists fom_gap_pfp_associative_representative_4_bounded_index_bound. fom_gap_pfp_associative_representative_4_bounded_index_bound + S (fom_index_pfp_associative_representative_4_bounded) = (La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))) -> exists fom_value_pfp_associative_representative_4_bounded. ((((exists fom_beta_height_pfp_associative_representative_4_bounded_entry. fom_beta_height_pfp_associative_representative_4_bounded_entry + S (fom_value_pfp_associative_representative_4_bounded) = S ((S (fom_index_pfp_associative_representative_4_bounded)) * associative_representative_4_scale)) /\\ exists fom_beta_quotient_pfp_associative_representative_4_bounded_entry. associative_representative_4_code = fom_beta_quotient_pfp_associative_representative_4_bounded_entry * S ((S (fom_index_pfp_associative_representative_4_bounded)) * associative_representative_4_scale) + (fom_value_pfp_associative_representative_4_bounded))) /\\ (exists fom_gap_pfp_associative_representative_4_bounded_value_bound. fom_gap_pfp_associative_representative_4_bounded_value_bound + S (fom_value_pfp_associative_representative_4_bounded) = p))) /\\ ((forall pfrep_power_associative_representative_4_equivalent pfrep_left_associative_representative_4_equivalent pfrep_right_associative_representative_4_equivalent. ((exists pfrep_position_associative_representative_4_equivalentfirst. ((pfrep_position_associative_representative_4_equivalentfirst+S (pfrep_power_associative_representative_4_equivalent)=(Lv)) /\\ ((((exists ff_h_pfp_associative_representative_4_equivalentfirstentry. ff_h_pfp_associative_representative_4_equivalentfirstentry + S (pfrep_left_associative_representative_4_equivalent) = S ((S (pfrep_position_associative_representative_4_equivalentfirst)) * vc)) /\\ exists ff_q_pfp_associative_representative_4_equivalentfirstentry. vb = ff_q_pfp_associative_representative_4_equivalentfirstentry * S ((S (pfrep_position_associative_representative_4_equivalentfirst)) * vc) + (pfrep_left_associative_representative_4_equivalent)))))) \\/ (((exists pfrep_gap_associative_representative_4_equivalentfirstoutside. pfrep_gap_associative_representative_4_equivalentfirstoutside+(Lv)=(pfrep_power_associative_representative_4_equivalent)) /\\ (((pfrep_left_associative_representative_4_equivalent)=0))))) -> ((exists pfrep_position_associative_representative_4_equivalentsecond. ((pfrep_position_associative_representative_4_equivalentsecond+S (pfrep_power_associative_representative_4_equivalent)=((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))) /\\ ((((exists ff_h_pfp_associative_representative_4_equivalentsecondentry. ff_h_pfp_associative_representative_4_equivalentsecondentry + S (pfrep_right_associative_representative_4_equivalent) = S ((S (pfrep_position_associative_representative_4_equivalentsecond)) * associative_representative_4_scale)) /\\ exists ff_q_pfp_associative_representative_4_equivalentsecondentry. associative_representative_4_code = ff_q_pfp_associative_representative_4_equivalentsecondentry * S ((S (pfrep_position_associative_representative_4_equivalentsecond)) * associative_representative_4_scale) + (pfrep_right_associative_representative_4_equivalent)))))) \\/ (((exists pfrep_gap_associative_representative_4_equivalentsecondoutside. pfrep_gap_associative_representative_4_equivalentsecondoutside+((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))=(pfrep_power_associative_representative_4_equivalent)) /\\ (((pfrep_right_associative_representative_4_equivalent)=0))))) -> pfrep_left_associative_representative_4_equivalent=pfrep_right_associative_representative_4_equivalent)))", "specialize prime_field_polynomial_bounded_representative_at_length_exists (p)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (vb)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (vc)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (Lv)", "specialize prime_field_polynomial_bounded_representative_at_length_exists ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply prime_field_polynomial_bounded_representative_at_length_exists", "exact hp", "exact hbc_bounded_right_right", "have length_bound_associative_representative_4 : exists pfrep_gap_associative_representative_4_inner. pfrep_gap_associative_representative_4_inner+(Lv)=((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "have length_bound_associative_representative_4_inner : exists pfrep_gap_associative_representative_4_inner_inner. pfrep_gap_associative_representative_4_inner_inner+(Lv)=((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))", "have length_bound_associative_representative_4_inner_inner : exists pfrep_gap_associative_representative_4_inner_inner_inner. pfrep_gap_associative_representative_4_inner_inner_inner+(Lv)=((Lu)+((Lv)+((Lr)+(Ls))))", "have length_bound_associative_representative_4_inner_inner_inner : exists pfrep_gap_associative_representative_4_inner_inner_inner_inner. pfrep_gap_associative_representative_4_inner_inner_inner_inner+(Lv)=((Lv)+((Lr)+(Ls)))", "specialize le_add_right (Lv)", "specialize le_add_right ((Lr)+(Ls))", "apply le_add_right", "specialize le_trans (Lv)", "specialize le_trans ((Lv)+((Lr)+(Ls)))", "specialize le_trans ((Lu)+((Lv)+((Lr)+(Ls))))", "apply le_trans", "exact length_bound_associative_representative_4_inner_inner_inner", "exists Lu", "refl", "specialize le_trans (Lv)", "specialize le_trans ((Lu)+((Lv)+((Lr)+(Ls))))", "specialize le_trans ((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))", "apply le_trans", "exact length_bound_associative_representative_4_inner_inner", "exists Lc", "refl", "specialize le_trans (Lv)", "specialize le_trans ((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))", "specialize le_trans ((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "apply le_trans", "exact length_bound_associative_representative_4_inner", "exists Lb", "refl", "specialize le_trans (Lv)", "specialize le_trans ((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "specialize le_trans ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply le_trans", "exact length_bound_associative_representative_4", "exists La", "refl", "cases associative_representative_4", "cases associative_representative_4_witness", "cases associative_representative_4_witness_witness", "have associative_representative_5 : exists associative_representative_5_code associative_representative_5_scale. ((forall fom_index_pfp_associative_representative_5_bounded. (exists fom_gap_pfp_associative_representative_5_bounded_index_bound. fom_gap_pfp_associative_representative_5_bounded_index_bound + S (fom_index_pfp_associative_representative_5_bounded) = (La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))) -> exists fom_value_pfp_associative_representative_5_bounded. ((((exists fom_beta_height_pfp_associative_representative_5_bounded_entry. fom_beta_height_pfp_associative_representative_5_bounded_entry + S (fom_value_pfp_associative_representative_5_bounded) = S ((S (fom_index_pfp_associative_representative_5_bounded)) * associative_representative_5_scale)) /\\ exists fom_beta_quotient_pfp_associative_representative_5_bounded_entry. associative_representative_5_code = fom_beta_quotient_pfp_associative_representative_5_bounded_entry * S ((S (fom_index_pfp_associative_representative_5_bounded)) * associative_representative_5_scale) + (fom_value_pfp_associative_representative_5_bounded))) /\\ (exists fom_gap_pfp_associative_representative_5_bounded_value_bound. fom_gap_pfp_associative_representative_5_bounded_value_bound + S (fom_value_pfp_associative_representative_5_bounded) = p))) /\\ ((forall pfrep_power_associative_representative_5_equivalent pfrep_left_associative_representative_5_equivalent pfrep_right_associative_representative_5_equivalent. ((exists pfrep_position_associative_representative_5_equivalentfirst. ((pfrep_position_associative_representative_5_equivalentfirst+S (pfrep_power_associative_representative_5_equivalent)=(Lr)) /\\ ((((exists ff_h_pfp_associative_representative_5_equivalentfirstentry. ff_h_pfp_associative_representative_5_equivalentfirstentry + S (pfrep_left_associative_representative_5_equivalent) = S ((S (pfrep_position_associative_representative_5_equivalentfirst)) * rc)) /\\ exists ff_q_pfp_associative_representative_5_equivalentfirstentry. rb = ff_q_pfp_associative_representative_5_equivalentfirstentry * S ((S (pfrep_position_associative_representative_5_equivalentfirst)) * rc) + (pfrep_left_associative_representative_5_equivalent)))))) \\/ (((exists pfrep_gap_associative_representative_5_equivalentfirstoutside. pfrep_gap_associative_representative_5_equivalentfirstoutside+(Lr)=(pfrep_power_associative_representative_5_equivalent)) /\\ (((pfrep_left_associative_representative_5_equivalent)=0))))) -> ((exists pfrep_position_associative_representative_5_equivalentsecond. ((pfrep_position_associative_representative_5_equivalentsecond+S (pfrep_power_associative_representative_5_equivalent)=((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))) /\\ ((((exists ff_h_pfp_associative_representative_5_equivalentsecondentry. ff_h_pfp_associative_representative_5_equivalentsecondentry + S (pfrep_right_associative_representative_5_equivalent) = S ((S (pfrep_position_associative_representative_5_equivalentsecond)) * associative_representative_5_scale)) /\\ exists ff_q_pfp_associative_representative_5_equivalentsecondentry. associative_representative_5_code = ff_q_pfp_associative_representative_5_equivalentsecondentry * S ((S (pfrep_position_associative_representative_5_equivalentsecond)) * associative_representative_5_scale) + (pfrep_right_associative_representative_5_equivalent)))))) \\/ (((exists pfrep_gap_associative_representative_5_equivalentsecondoutside. pfrep_gap_associative_representative_5_equivalentsecondoutside+((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))=(pfrep_power_associative_representative_5_equivalent)) /\\ (((pfrep_right_associative_representative_5_equivalent)=0))))) -> pfrep_left_associative_representative_5_equivalent=pfrep_right_associative_representative_5_equivalent)))", "specialize prime_field_polynomial_bounded_representative_at_length_exists (p)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (rb)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (rc)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (Lr)", "specialize prime_field_polynomial_bounded_representative_at_length_exists ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply prime_field_polynomial_bounded_representative_at_length_exists", "exact hp", "exact hleft_bounded_right_right", "have length_bound_associative_representative_5 : exists pfrep_gap_associative_representative_5_inner. pfrep_gap_associative_representative_5_inner+(Lr)=((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "have length_bound_associative_representative_5_inner : exists pfrep_gap_associative_representative_5_inner_inner. pfrep_gap_associative_representative_5_inner_inner+(Lr)=((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))", "have length_bound_associative_representative_5_inner_inner : exists pfrep_gap_associative_representative_5_inner_inner_inner. pfrep_gap_associative_representative_5_inner_inner_inner+(Lr)=((Lu)+((Lv)+((Lr)+(Ls))))", "have length_bound_associative_representative_5_inner_inner_inner : exists pfrep_gap_associative_representative_5_inner_inner_inner_inner. pfrep_gap_associative_representative_5_inner_inner_inner_inner+(Lr)=((Lv)+((Lr)+(Ls)))", "have length_bound_associative_representative_5_inner_inner_inner_inner : exists pfrep_gap_associative_representative_5_inner_inner_inner_inner_inner. pfrep_gap_associative_representative_5_inner_inner_inner_inner_inner+(Lr)=((Lr)+(Ls))", "specialize le_add_right (Lr)", "specialize le_add_right (Ls)", "apply le_add_right", "specialize le_trans (Lr)", "specialize le_trans ((Lr)+(Ls))", "specialize le_trans ((Lv)+((Lr)+(Ls)))", "apply le_trans", "exact length_bound_associative_representative_5_inner_inner_inner_inner", "exists Lv", "refl", "specialize le_trans (Lr)", "specialize le_trans ((Lv)+((Lr)+(Ls)))", "specialize le_trans ((Lu)+((Lv)+((Lr)+(Ls))))", "apply le_trans", "exact length_bound_associative_representative_5_inner_inner_inner", "exists Lu", "refl", "specialize le_trans (Lr)", "specialize le_trans ((Lu)+((Lv)+((Lr)+(Ls))))", "specialize le_trans ((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))", "apply le_trans", "exact length_bound_associative_representative_5_inner_inner", "exists Lc", "refl", "specialize le_trans (Lr)", "specialize le_trans ((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))", "specialize le_trans ((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "apply le_trans", "exact length_bound_associative_representative_5_inner", "exists Lb", "refl", "specialize le_trans (Lr)", "specialize le_trans ((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "specialize le_trans ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply le_trans", "exact length_bound_associative_representative_5", "exists La", "refl", "cases associative_representative_5", "cases associative_representative_5_witness", "cases associative_representative_5_witness_witness", "have associative_representative_6 : exists associative_representative_6_code associative_representative_6_scale. ((forall fom_index_pfp_associative_representative_6_bounded. (exists fom_gap_pfp_associative_representative_6_bounded_index_bound. fom_gap_pfp_associative_representative_6_bounded_index_bound + S (fom_index_pfp_associative_representative_6_bounded) = (La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))) -> exists fom_value_pfp_associative_representative_6_bounded. ((((exists fom_beta_height_pfp_associative_representative_6_bounded_entry. fom_beta_height_pfp_associative_representative_6_bounded_entry + S (fom_value_pfp_associative_representative_6_bounded) = S ((S (fom_index_pfp_associative_representative_6_bounded)) * associative_representative_6_scale)) /\\ exists fom_beta_quotient_pfp_associative_representative_6_bounded_entry. associative_representative_6_code = fom_beta_quotient_pfp_associative_representative_6_bounded_entry * S ((S (fom_index_pfp_associative_representative_6_bounded)) * associative_representative_6_scale) + (fom_value_pfp_associative_representative_6_bounded))) /\\ (exists fom_gap_pfp_associative_representative_6_bounded_value_bound. fom_gap_pfp_associative_representative_6_bounded_value_bound + S (fom_value_pfp_associative_representative_6_bounded) = p))) /\\ ((forall pfrep_power_associative_representative_6_equivalent pfrep_left_associative_representative_6_equivalent pfrep_right_associative_representative_6_equivalent. ((exists pfrep_position_associative_representative_6_equivalentfirst. ((pfrep_position_associative_representative_6_equivalentfirst+S (pfrep_power_associative_representative_6_equivalent)=(Ls)) /\\ ((((exists ff_h_pfp_associative_representative_6_equivalentfirstentry. ff_h_pfp_associative_representative_6_equivalentfirstentry + S (pfrep_left_associative_representative_6_equivalent) = S ((S (pfrep_position_associative_representative_6_equivalentfirst)) * sc)) /\\ exists ff_q_pfp_associative_representative_6_equivalentfirstentry. sb = ff_q_pfp_associative_representative_6_equivalentfirstentry * S ((S (pfrep_position_associative_representative_6_equivalentfirst)) * sc) + (pfrep_left_associative_representative_6_equivalent)))))) \\/ (((exists pfrep_gap_associative_representative_6_equivalentfirstoutside. pfrep_gap_associative_representative_6_equivalentfirstoutside+(Ls)=(pfrep_power_associative_representative_6_equivalent)) /\\ (((pfrep_left_associative_representative_6_equivalent)=0))))) -> ((exists pfrep_position_associative_representative_6_equivalentsecond. ((pfrep_position_associative_representative_6_equivalentsecond+S (pfrep_power_associative_representative_6_equivalent)=((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))) /\\ ((((exists ff_h_pfp_associative_representative_6_equivalentsecondentry. ff_h_pfp_associative_representative_6_equivalentsecondentry + S (pfrep_right_associative_representative_6_equivalent) = S ((S (pfrep_position_associative_representative_6_equivalentsecond)) * associative_representative_6_scale)) /\\ exists ff_q_pfp_associative_representative_6_equivalentsecondentry. associative_representative_6_code = ff_q_pfp_associative_representative_6_equivalentsecondentry * S ((S (pfrep_position_associative_representative_6_equivalentsecond)) * associative_representative_6_scale) + (pfrep_right_associative_representative_6_equivalent)))))) \\/ (((exists pfrep_gap_associative_representative_6_equivalentsecondoutside. pfrep_gap_associative_representative_6_equivalentsecondoutside+((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))=(pfrep_power_associative_representative_6_equivalent)) /\\ (((pfrep_right_associative_representative_6_equivalent)=0))))) -> pfrep_left_associative_representative_6_equivalent=pfrep_right_associative_representative_6_equivalent)))", "specialize prime_field_polynomial_bounded_representative_at_length_exists (p)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (sb)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (sc)", "specialize prime_field_polynomial_bounded_representative_at_length_exists (Ls)", "specialize prime_field_polynomial_bounded_representative_at_length_exists ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply prime_field_polynomial_bounded_representative_at_length_exists", "exact hp", "exact hright_bounded_right_right", "have length_bound_associative_representative_6 : exists pfrep_gap_associative_representative_6_inner. pfrep_gap_associative_representative_6_inner+(Ls)=((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "have length_bound_associative_representative_6_inner : exists pfrep_gap_associative_representative_6_inner_inner. pfrep_gap_associative_representative_6_inner_inner+(Ls)=((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))", "have length_bound_associative_representative_6_inner_inner : exists pfrep_gap_associative_representative_6_inner_inner_inner. pfrep_gap_associative_representative_6_inner_inner_inner+(Ls)=((Lu)+((Lv)+((Lr)+(Ls))))", "have length_bound_associative_representative_6_inner_inner_inner : exists pfrep_gap_associative_representative_6_inner_inner_inner_inner. pfrep_gap_associative_representative_6_inner_inner_inner_inner+(Ls)=((Lv)+((Lr)+(Ls)))", "have length_bound_associative_representative_6_inner_inner_inner_inner : exists pfrep_gap_associative_representative_6_inner_inner_inner_inner_inner. pfrep_gap_associative_representative_6_inner_inner_inner_inner_inner+(Ls)=((Lr)+(Ls))", "have length_bound_associative_representative_6_inner_inner_inner_inner_inner : exists pfrep_gap_associative_representative_6_inner_inner_inner_inner_inner_inner. pfrep_gap_associative_representative_6_inner_inner_inner_inner_inner_inner+(Ls)=(Ls)", "specialize le_refl (Ls)", "apply le_refl", "specialize le_trans (Ls)", "specialize le_trans (Ls)", "specialize le_trans ((Lr)+(Ls))", "apply le_trans", "exact length_bound_associative_representative_6_inner_inner_inner_inner_inner", "exists Lr", "refl", "specialize le_trans (Ls)", "specialize le_trans ((Lr)+(Ls))", "specialize le_trans ((Lv)+((Lr)+(Ls)))", "apply le_trans", "exact length_bound_associative_representative_6_inner_inner_inner_inner", "exists Lv", "refl", "specialize le_trans (Ls)", "specialize le_trans ((Lv)+((Lr)+(Ls)))", "specialize le_trans ((Lu)+((Lv)+((Lr)+(Ls))))", "apply le_trans", "exact length_bound_associative_representative_6_inner_inner_inner", "exists Lu", "refl", "specialize le_trans (Ls)", "specialize le_trans ((Lu)+((Lv)+((Lr)+(Ls))))", "specialize le_trans ((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))", "apply le_trans", "exact length_bound_associative_representative_6_inner_inner", "exists Lc", "refl", "specialize le_trans (Ls)", "specialize le_trans ((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))", "specialize le_trans ((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "apply le_trans", "exact length_bound_associative_representative_6_inner", "exists Lb", "refl", "specialize le_trans (Ls)", "specialize le_trans ((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls))))))", "specialize le_trans ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply le_trans", "exact length_bound_associative_representative_6", "exists La", "refl", "cases associative_representative_6", "cases associative_representative_6_witness", "cases associative_representative_6_witness_witness", "have hab_actual : forall pfp_index_hab_actual_operation. (exists pfa_gap_hab_actual_operationindex. pfa_gap_hab_actual_operationindex + S (pfp_index_hab_actual_operation) = ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))) -> exists pfp_left_hab_actual_operation pfp_right_hab_actual_operation pfp_value_hab_actual_operation. ((((exists ff_h_pfp_hab_actual_operationleft. ff_h_pfp_hab_actual_operationleft + S (pfp_left_hab_actual_operation) = S ((S (pfp_index_hab_actual_operation)) * x1)) /\\ exists ff_q_pfp_hab_actual_operationleft. x = ff_q_pfp_hab_actual_operationleft * S ((S (pfp_index_hab_actual_operation)) * x1) + (pfp_left_hab_actual_operation))) /\\ (((((exists ff_h_pfp_hab_actual_operationright. ff_h_pfp_hab_actual_operationright + S (pfp_right_hab_actual_operation) = S ((S (pfp_index_hab_actual_operation)) * x3)) /\\ exists ff_q_pfp_hab_actual_operationright. x2 = ff_q_pfp_hab_actual_operationright * S ((S (pfp_index_hab_actual_operation)) * x3) + (pfp_right_hab_actual_operation))) /\\ (((((exists ff_h_pfp_hab_actual_operationtarget. ff_h_pfp_hab_actual_operationtarget + S (pfp_value_hab_actual_operation) = S ((S (pfp_index_hab_actual_operation)) * x7)) /\\ exists ff_q_pfp_hab_actual_operationtarget. x6 = ff_q_pfp_hab_actual_operationtarget * S ((S (pfp_index_hab_actual_operation)) * x7) + (pfp_value_hab_actual_operation))) /\\ ((((exists pfa_gap_hab_actual_operationoperationleft. pfa_gap_hab_actual_operationoperationleft + S (pfp_left_hab_actual_operation) = (p)) /\\ (((exists pfa_gap_hab_actual_operationoperationright. pfa_gap_hab_actual_operationoperationright + S (pfp_right_hab_actual_operation) = (p)) /\\ ((((exists pfa_gap_hab_actual_operationoperationresultbound. pfa_gap_hab_actual_operationoperationresultbound + S (pfp_value_hab_actual_operation) = (p)) /\\ ((exists pfa_offset_left_hab_actual_operationoperationresultcongruence pfa_offset_right_hab_actual_operationoperationresultcongruence. ((pfp_left_hab_actual_operation) + (pfp_right_hab_actual_operation)) + (p) * pfa_offset_left_hab_actual_operationoperationresultcongruence = (pfp_value_hab_actual_operation) + (p) * pfa_offset_right_hab_actual_operationoperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_aligned_add_realize (p)", "specialize prime_field_polynomial_aligned_add_realize (ab)", "specialize prime_field_polynomial_aligned_add_realize (ac)", "specialize prime_field_polynomial_aligned_add_realize (La)", "specialize prime_field_polynomial_aligned_add_realize (bb)", "specialize prime_field_polynomial_aligned_add_realize (bc)", "specialize prime_field_polynomial_aligned_add_realize (Lb)", "specialize prime_field_polynomial_aligned_add_realize (ub)", "specialize prime_field_polynomial_aligned_add_realize (uc)", "specialize prime_field_polynomial_aligned_add_realize (Lu)", "specialize prime_field_polynomial_aligned_add_realize (x)", "specialize prime_field_polynomial_aligned_add_realize (x1)", "specialize prime_field_polynomial_aligned_add_realize (x2)", "specialize prime_field_polynomial_aligned_add_realize (x3)", "specialize prime_field_polynomial_aligned_add_realize (x6)", "specialize prime_field_polynomial_aligned_add_realize (x7)", "specialize prime_field_polynomial_aligned_add_realize ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply prime_field_polynomial_aligned_add_realize", "exact hp", "exact hab", "exact associative_representative_0_witness_witness_left", "exact associative_representative_1_witness_witness_left", "exact associative_representative_3_witness_witness_left", "split", "exact associative_representative_0_witness_witness_right", "exact associative_representative_1_witness_witness_right", "exact associative_representative_3_witness_witness_right", "have hleft_actual : forall pfp_index_hleft_actual_operation. (exists pfa_gap_hleft_actual_operationindex. pfa_gap_hleft_actual_operationindex + S (pfp_index_hleft_actual_operation) = ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))) -> exists pfp_left_hleft_actual_operation pfp_right_hleft_actual_operation pfp_value_hleft_actual_operation. ((((exists ff_h_pfp_hleft_actual_operationleft. ff_h_pfp_hleft_actual_operationleft + S (pfp_left_hleft_actual_operation) = S ((S (pfp_index_hleft_actual_operation)) * x7)) /\\ exists ff_q_pfp_hleft_actual_operationleft. x6 = ff_q_pfp_hleft_actual_operationleft * S ((S (pfp_index_hleft_actual_operation)) * x7) + (pfp_left_hleft_actual_operation))) /\\ (((((exists ff_h_pfp_hleft_actual_operationright. ff_h_pfp_hleft_actual_operationright + S (pfp_right_hleft_actual_operation) = S ((S (pfp_index_hleft_actual_operation)) * x5)) /\\ exists ff_q_pfp_hleft_actual_operationright. x4 = ff_q_pfp_hleft_actual_operationright * S ((S (pfp_index_hleft_actual_operation)) * x5) + (pfp_right_hleft_actual_operation))) /\\ (((((exists ff_h_pfp_hleft_actual_operationtarget. ff_h_pfp_hleft_actual_operationtarget + S (pfp_value_hleft_actual_operation) = S ((S (pfp_index_hleft_actual_operation)) * x11)) /\\ exists ff_q_pfp_hleft_actual_operationtarget. x10 = ff_q_pfp_hleft_actual_operationtarget * S ((S (pfp_index_hleft_actual_operation)) * x11) + (pfp_value_hleft_actual_operation))) /\\ ((((exists pfa_gap_hleft_actual_operationoperationleft. pfa_gap_hleft_actual_operationoperationleft + S (pfp_left_hleft_actual_operation) = (p)) /\\ (((exists pfa_gap_hleft_actual_operationoperationright. pfa_gap_hleft_actual_operationoperationright + S (pfp_right_hleft_actual_operation) = (p)) /\\ ((((exists pfa_gap_hleft_actual_operationoperationresultbound. pfa_gap_hleft_actual_operationoperationresultbound + S (pfp_value_hleft_actual_operation) = (p)) /\\ ((exists pfa_offset_left_hleft_actual_operationoperationresultcongruence pfa_offset_right_hleft_actual_operationoperationresultcongruence. ((pfp_left_hleft_actual_operation) + (pfp_right_hleft_actual_operation)) + (p) * pfa_offset_left_hleft_actual_operationoperationresultcongruence = (pfp_value_hleft_actual_operation) + (p) * pfa_offset_right_hleft_actual_operationoperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_aligned_add_realize (p)", "specialize prime_field_polynomial_aligned_add_realize (ub)", "specialize prime_field_polynomial_aligned_add_realize (uc)", "specialize prime_field_polynomial_aligned_add_realize (Lu)", "specialize prime_field_polynomial_aligned_add_realize (cb)", "specialize prime_field_polynomial_aligned_add_realize (cc)", "specialize prime_field_polynomial_aligned_add_realize (Lc)", "specialize prime_field_polynomial_aligned_add_realize (rb)", "specialize prime_field_polynomial_aligned_add_realize (rc)", "specialize prime_field_polynomial_aligned_add_realize (Lr)", "specialize prime_field_polynomial_aligned_add_realize (x6)", "specialize prime_field_polynomial_aligned_add_realize (x7)", "specialize prime_field_polynomial_aligned_add_realize (x4)", "specialize prime_field_polynomial_aligned_add_realize (x5)", "specialize prime_field_polynomial_aligned_add_realize (x10)", "specialize prime_field_polynomial_aligned_add_realize (x11)", "specialize prime_field_polynomial_aligned_add_realize ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply prime_field_polynomial_aligned_add_realize", "exact hp", "exact hleft", "exact associative_representative_3_witness_witness_left", "exact associative_representative_2_witness_witness_left", "exact associative_representative_5_witness_witness_left", "split", "exact associative_representative_3_witness_witness_right", "exact associative_representative_2_witness_witness_right", "exact associative_representative_5_witness_witness_right", "have hbc_actual : forall pfp_index_hbc_actual_operation. (exists pfa_gap_hbc_actual_operationindex. pfa_gap_hbc_actual_operationindex + S (pfp_index_hbc_actual_operation) = ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))) -> exists pfp_left_hbc_actual_operation pfp_right_hbc_actual_operation pfp_value_hbc_actual_operation. ((((exists ff_h_pfp_hbc_actual_operationleft. ff_h_pfp_hbc_actual_operationleft + S (pfp_left_hbc_actual_operation) = S ((S (pfp_index_hbc_actual_operation)) * x3)) /\\ exists ff_q_pfp_hbc_actual_operationleft. x2 = ff_q_pfp_hbc_actual_operationleft * S ((S (pfp_index_hbc_actual_operation)) * x3) + (pfp_left_hbc_actual_operation))) /\\ (((((exists ff_h_pfp_hbc_actual_operationright. ff_h_pfp_hbc_actual_operationright + S (pfp_right_hbc_actual_operation) = S ((S (pfp_index_hbc_actual_operation)) * x5)) /\\ exists ff_q_pfp_hbc_actual_operationright. x4 = ff_q_pfp_hbc_actual_operationright * S ((S (pfp_index_hbc_actual_operation)) * x5) + (pfp_right_hbc_actual_operation))) /\\ (((((exists ff_h_pfp_hbc_actual_operationtarget. ff_h_pfp_hbc_actual_operationtarget + S (pfp_value_hbc_actual_operation) = S ((S (pfp_index_hbc_actual_operation)) * x9)) /\\ exists ff_q_pfp_hbc_actual_operationtarget. x8 = ff_q_pfp_hbc_actual_operationtarget * S ((S (pfp_index_hbc_actual_operation)) * x9) + (pfp_value_hbc_actual_operation))) /\\ ((((exists pfa_gap_hbc_actual_operationoperationleft. pfa_gap_hbc_actual_operationoperationleft + S (pfp_left_hbc_actual_operation) = (p)) /\\ (((exists pfa_gap_hbc_actual_operationoperationright. pfa_gap_hbc_actual_operationoperationright + S (pfp_right_hbc_actual_operation) = (p)) /\\ ((((exists pfa_gap_hbc_actual_operationoperationresultbound. pfa_gap_hbc_actual_operationoperationresultbound + S (pfp_value_hbc_actual_operation) = (p)) /\\ ((exists pfa_offset_left_hbc_actual_operationoperationresultcongruence pfa_offset_right_hbc_actual_operationoperationresultcongruence. ((pfp_left_hbc_actual_operation) + (pfp_right_hbc_actual_operation)) + (p) * pfa_offset_left_hbc_actual_operationoperationresultcongruence = (pfp_value_hbc_actual_operation) + (p) * pfa_offset_right_hbc_actual_operationoperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_aligned_add_realize (p)", "specialize prime_field_polynomial_aligned_add_realize (bb)", "specialize prime_field_polynomial_aligned_add_realize (bc)", "specialize prime_field_polynomial_aligned_add_realize (Lb)", "specialize prime_field_polynomial_aligned_add_realize (cb)", "specialize prime_field_polynomial_aligned_add_realize (cc)", "specialize prime_field_polynomial_aligned_add_realize (Lc)", "specialize prime_field_polynomial_aligned_add_realize (vb)", "specialize prime_field_polynomial_aligned_add_realize (vc)", "specialize prime_field_polynomial_aligned_add_realize (Lv)", "specialize prime_field_polynomial_aligned_add_realize (x2)", "specialize prime_field_polynomial_aligned_add_realize (x3)", "specialize prime_field_polynomial_aligned_add_realize (x4)", "specialize prime_field_polynomial_aligned_add_realize (x5)", "specialize prime_field_polynomial_aligned_add_realize (x8)", "specialize prime_field_polynomial_aligned_add_realize (x9)", "specialize prime_field_polynomial_aligned_add_realize ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply prime_field_polynomial_aligned_add_realize", "exact hp", "exact hbc", "exact associative_representative_1_witness_witness_left", "exact associative_representative_2_witness_witness_left", "exact associative_representative_4_witness_witness_left", "split", "exact associative_representative_1_witness_witness_right", "exact associative_representative_2_witness_witness_right", "exact associative_representative_4_witness_witness_right", "have hright_actual : forall pfp_index_hright_actual_operation. (exists pfa_gap_hright_actual_operationindex. pfa_gap_hright_actual_operationindex + S (pfp_index_hright_actual_operation) = ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))) -> exists pfp_left_hright_actual_operation pfp_right_hright_actual_operation pfp_value_hright_actual_operation. ((((exists ff_h_pfp_hright_actual_operationleft. ff_h_pfp_hright_actual_operationleft + S (pfp_left_hright_actual_operation) = S ((S (pfp_index_hright_actual_operation)) * x1)) /\\ exists ff_q_pfp_hright_actual_operationleft. x = ff_q_pfp_hright_actual_operationleft * S ((S (pfp_index_hright_actual_operation)) * x1) + (pfp_left_hright_actual_operation))) /\\ (((((exists ff_h_pfp_hright_actual_operationright. ff_h_pfp_hright_actual_operationright + S (pfp_right_hright_actual_operation) = S ((S (pfp_index_hright_actual_operation)) * x9)) /\\ exists ff_q_pfp_hright_actual_operationright. x8 = ff_q_pfp_hright_actual_operationright * S ((S (pfp_index_hright_actual_operation)) * x9) + (pfp_right_hright_actual_operation))) /\\ (((((exists ff_h_pfp_hright_actual_operationtarget. ff_h_pfp_hright_actual_operationtarget + S (pfp_value_hright_actual_operation) = S ((S (pfp_index_hright_actual_operation)) * x13)) /\\ exists ff_q_pfp_hright_actual_operationtarget. x12 = ff_q_pfp_hright_actual_operationtarget * S ((S (pfp_index_hright_actual_operation)) * x13) + (pfp_value_hright_actual_operation))) /\\ ((((exists pfa_gap_hright_actual_operationoperationleft. pfa_gap_hright_actual_operationoperationleft + S (pfp_left_hright_actual_operation) = (p)) /\\ (((exists pfa_gap_hright_actual_operationoperationright. pfa_gap_hright_actual_operationoperationright + S (pfp_right_hright_actual_operation) = (p)) /\\ ((((exists pfa_gap_hright_actual_operationoperationresultbound. pfa_gap_hright_actual_operationoperationresultbound + S (pfp_value_hright_actual_operation) = (p)) /\\ ((exists pfa_offset_left_hright_actual_operationoperationresultcongruence pfa_offset_right_hright_actual_operationoperationresultcongruence. ((pfp_left_hright_actual_operation) + (pfp_right_hright_actual_operation)) + (p) * pfa_offset_left_hright_actual_operationoperationresultcongruence = (pfp_value_hright_actual_operation) + (p) * pfa_offset_right_hright_actual_operationoperationresultcongruence)))))))))))))))", "specialize prime_field_polynomial_aligned_add_realize (p)", "specialize prime_field_polynomial_aligned_add_realize (ab)", "specialize prime_field_polynomial_aligned_add_realize (ac)", "specialize prime_field_polynomial_aligned_add_realize (La)", "specialize prime_field_polynomial_aligned_add_realize (vb)", "specialize prime_field_polynomial_aligned_add_realize (vc)", "specialize prime_field_polynomial_aligned_add_realize (Lv)", "specialize prime_field_polynomial_aligned_add_realize (sb)", "specialize prime_field_polynomial_aligned_add_realize (sc)", "specialize prime_field_polynomial_aligned_add_realize (Ls)", "specialize prime_field_polynomial_aligned_add_realize (x)", "specialize prime_field_polynomial_aligned_add_realize (x1)", "specialize prime_field_polynomial_aligned_add_realize (x8)", "specialize prime_field_polynomial_aligned_add_realize (x9)", "specialize prime_field_polynomial_aligned_add_realize (x12)", "specialize prime_field_polynomial_aligned_add_realize (x13)", "specialize prime_field_polynomial_aligned_add_realize ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply prime_field_polynomial_aligned_add_realize", "exact hp", "exact hright", "exact associative_representative_0_witness_witness_left", "exact associative_representative_4_witness_witness_left", "exact associative_representative_6_witness_witness_left", "split", "exact associative_representative_0_witness_witness_right", "exact associative_representative_4_witness_witness_right", "exact associative_representative_6_witness_witness_right", "have heq : forall mdr_i_pfp_associative_actual_outputs mdr_a_pfp_associative_actual_outputs. (exists mdr_gap_pfp_associative_actual_outputsb. mdr_gap_pfp_associative_actual_outputsb + S (mdr_i_pfp_associative_actual_outputs) = ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))) -> (((exists ff_h_mdr_pfp_associative_actual_outputso. ff_h_mdr_pfp_associative_actual_outputso + S (mdr_a_pfp_associative_actual_outputs) = S ((S (mdr_i_pfp_associative_actual_outputs)) * x11)) /\\ exists ff_q_mdr_pfp_associative_actual_outputso. x10 = ff_q_mdr_pfp_associative_actual_outputso * S ((S (mdr_i_pfp_associative_actual_outputs)) * x11) + (mdr_a_pfp_associative_actual_outputs))) -> (((exists ff_h_mdr_pfp_associative_actual_outputsn. ff_h_mdr_pfp_associative_actual_outputsn + S (mdr_a_pfp_associative_actual_outputs) = S ((S (mdr_i_pfp_associative_actual_outputs)) * x13)) /\\ exists ff_q_mdr_pfp_associative_actual_outputsn. x12 = ff_q_mdr_pfp_associative_actual_outputsn * S ((S (mdr_i_pfp_associative_actual_outputs)) * x13) + (mdr_a_pfp_associative_actual_outputs)))", "specialize prime_field_polynomial_add_associative (p)", "specialize prime_field_polynomial_add_associative (x)", "specialize prime_field_polynomial_add_associative (x1)", "specialize prime_field_polynomial_add_associative (x2)", "specialize prime_field_polynomial_add_associative (x3)", "specialize prime_field_polynomial_add_associative (x4)", "specialize prime_field_polynomial_add_associative (x5)", "specialize prime_field_polynomial_add_associative (x6)", "specialize prime_field_polynomial_add_associative (x7)", "specialize prime_field_polynomial_add_associative (x8)", "specialize prime_field_polynomial_add_associative (x9)", "specialize prime_field_polynomial_add_associative (x10)", "specialize prime_field_polynomial_add_associative (x11)", "specialize prime_field_polynomial_add_associative (x12)", "specialize prime_field_polynomial_add_associative (x13)", "specialize prime_field_polynomial_add_associative ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply prime_field_polynomial_add_associative", "exact hab_actual", "exact hleft_actual", "exact hbc_actual", "exact hright_actual", "have associative_middle : forall pfrep_power_associative_middle_result pfrep_left_associative_middle_result pfrep_right_associative_middle_result. ((exists pfrep_position_associative_middle_resultfirst. ((pfrep_position_associative_middle_resultfirst+S (pfrep_power_associative_middle_result)=(Lr)) /\\ ((((exists ff_h_pfp_associative_middle_resultfirstentry. ff_h_pfp_associative_middle_resultfirstentry + S (pfrep_left_associative_middle_result) = S ((S (pfrep_position_associative_middle_resultfirst)) * rc)) /\\ exists ff_q_pfp_associative_middle_resultfirstentry. rb = ff_q_pfp_associative_middle_resultfirstentry * S ((S (pfrep_position_associative_middle_resultfirst)) * rc) + (pfrep_left_associative_middle_result)))))) \\/ (((exists pfrep_gap_associative_middle_resultfirstoutside. pfrep_gap_associative_middle_resultfirstoutside+(Lr)=(pfrep_power_associative_middle_result)) /\\ (((pfrep_left_associative_middle_result)=0))))) -> ((exists pfrep_position_associative_middle_resultsecond. ((pfrep_position_associative_middle_resultsecond+S (pfrep_power_associative_middle_result)=((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))) /\\ ((((exists ff_h_pfp_associative_middle_resultsecondentry. ff_h_pfp_associative_middle_resultsecondentry + S (pfrep_right_associative_middle_result) = S ((S (pfrep_position_associative_middle_resultsecond)) * x13)) /\\ exists ff_q_pfp_associative_middle_resultsecondentry. x12 = ff_q_pfp_associative_middle_resultsecondentry * S ((S (pfrep_position_associative_middle_resultsecond)) * x13) + (pfrep_right_associative_middle_result)))))) \\/ (((exists pfrep_gap_associative_middle_resultsecondoutside. pfrep_gap_associative_middle_resultsecondoutside+((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))=(pfrep_power_associative_middle_result)) /\\ (((pfrep_right_associative_middle_result)=0))))) -> pfrep_left_associative_middle_result=pfrep_right_associative_middle_result", "specialize prime_field_polynomial_equivalent_transitive (rb)", "specialize prime_field_polynomial_equivalent_transitive (rc)", "specialize prime_field_polynomial_equivalent_transitive (Lr)", "specialize prime_field_polynomial_equivalent_transitive (x10)", "specialize prime_field_polynomial_equivalent_transitive (x11)", "specialize prime_field_polynomial_equivalent_transitive ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "specialize prime_field_polynomial_equivalent_transitive (x12)", "specialize prime_field_polynomial_equivalent_transitive (x13)", "specialize prime_field_polynomial_equivalent_transitive ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply prime_field_polynomial_equivalent_transitive", "exact associative_representative_5_witness_witness_right", "specialize prime_field_polynomial_equal_implies_equivalent (x10)", "specialize prime_field_polynomial_equal_implies_equivalent (x11)", "specialize prime_field_polynomial_equal_implies_equivalent (x12)", "specialize prime_field_polynomial_equal_implies_equivalent (x13)", "specialize prime_field_polynomial_equal_implies_equivalent ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply prime_field_polynomial_equal_implies_equivalent", "exact heq", "specialize prime_field_polynomial_equivalent_transitive (rb)", "specialize prime_field_polynomial_equivalent_transitive (rc)", "specialize prime_field_polynomial_equivalent_transitive (Lr)", "specialize prime_field_polynomial_equivalent_transitive (x12)", "specialize prime_field_polynomial_equivalent_transitive (x13)", "specialize prime_field_polynomial_equivalent_transitive ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "specialize prime_field_polynomial_equivalent_transitive (sb)", "specialize prime_field_polynomial_equivalent_transitive (sc)", "specialize prime_field_polynomial_equivalent_transitive (Ls)", "apply prime_field_polynomial_equivalent_transitive", "exact associative_middle", "specialize prime_field_polynomial_equivalent_symmetric (sb)", "specialize prime_field_polynomial_equivalent_symmetric (sc)", "specialize prime_field_polynomial_equivalent_symmetric (Ls)", "specialize prime_field_polynomial_equivalent_symmetric (x12)", "specialize prime_field_polynomial_equivalent_symmetric (x13)", "specialize prime_field_polynomial_equivalent_symmetric ((La)+((Lb)+((Lc)+((Lu)+((Lv)+((Lr)+(Ls)))))))", "apply prime_field_polynomial_equivalent_symmetric", "exact associative_representative_6_witness_witness_right" ], "script_sha256": "8c74ad0e3bf3268660ff5f327f85dd0618140796715236c2d30a3de275782d4b", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_aligned_algebra_candidate.py", "sha256": "a68de84439afb5f6dd87f1d47449c0bce8dd53a66346c00cc1b7645fb80b2390" }, "statement": "forall p ab ac La bb bc Lb cb cc Lc ub uc Lu vb vc Lv rb rc Lr sb sc Ls. (~((p) = 1) /\\ forall pfa_factor_left_associative_prime pfa_factor_right_associative_prime. (p) = pfa_factor_left_associative_prime * pfa_factor_right_associative_prime -> pfa_factor_left_associative_prime = 1 \\/ pfa_factor_right_associative_prime = 1) -> (((forall fom_index_pfp_associative_0_left_bounded. (exists fom_gap_pfp_associative_0_left_bounded_index_bound. fom_gap_pfp_associative_0_left_bounded_index_bound + S (fom_index_pfp_associative_0_left_bounded) = La) -> exists fom_value_pfp_associative_0_left_bounded. ((((exists fom_beta_height_pfp_associative_0_left_bounded_entry. fom_beta_height_pfp_associative_0_left_bounded_entry + S (fom_value_pfp_associative_0_left_bounded) = S ((S (fom_index_pfp_associative_0_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_associative_0_left_bounded_entry. ab = fom_beta_quotient_pfp_associative_0_left_bounded_entry * S ((S (fom_index_pfp_associative_0_left_bounded)) * ac) + (fom_value_pfp_associative_0_left_bounded))) /\\ (exists fom_gap_pfp_associative_0_left_bounded_value_bound. fom_gap_pfp_associative_0_left_bounded_value_bound + S (fom_value_pfp_associative_0_left_bounded) = p))) /\\ (((forall fom_index_pfp_associative_0_right_bounded. (exists fom_gap_pfp_associative_0_right_bounded_index_bound. fom_gap_pfp_associative_0_right_bounded_index_bound + S (fom_index_pfp_associative_0_right_bounded) = Lb) -> exists fom_value_pfp_associative_0_right_bounded. ((((exists fom_beta_height_pfp_associative_0_right_bounded_entry. fom_beta_height_pfp_associative_0_right_bounded_entry + S (fom_value_pfp_associative_0_right_bounded) = S ((S (fom_index_pfp_associative_0_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_associative_0_right_bounded_entry. bb = fom_beta_quotient_pfp_associative_0_right_bounded_entry * S ((S (fom_index_pfp_associative_0_right_bounded)) * bc) + (fom_value_pfp_associative_0_right_bounded))) /\\ (exists fom_gap_pfp_associative_0_right_bounded_value_bound. fom_gap_pfp_associative_0_right_bounded_value_bound + S (fom_value_pfp_associative_0_right_bounded) = p))) /\\ (((forall fom_index_pfp_associative_0_result_bounded. (exists fom_gap_pfp_associative_0_result_bounded_index_bound. fom_gap_pfp_associative_0_result_bounded_index_bound + S (fom_index_pfp_associative_0_result_bounded) = Lu) -> exists fom_value_pfp_associative_0_result_bounded. ((((exists fom_beta_height_pfp_associative_0_result_bounded_entry. fom_beta_height_pfp_associative_0_result_bounded_entry + S (fom_value_pfp_associative_0_result_bounded) = S ((S (fom_index_pfp_associative_0_result_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_associative_0_result_bounded_entry. ub = fom_beta_quotient_pfp_associative_0_result_bounded_entry * S ((S (fom_index_pfp_associative_0_result_bounded)) * uc) + (fom_value_pfp_associative_0_result_bounded))) /\\ (exists fom_gap_pfp_associative_0_result_bounded_value_bound. fom_gap_pfp_associative_0_result_bounded_value_bound + S (fom_value_pfp_associative_0_result_bounded) = p))) /\\ ((exists pfaa_left_b_associative_0 pfaa_left_c_associative_0 pfaa_right_b_associative_0 pfaa_right_c_associative_0 pfaa_sum_b_associative_0 pfaa_sum_c_associative_0 pfaa_length_associative_0. ((((forall pfrep_power_associative_0_witness_common_left pfrep_left_associative_0_witness_common_left pfrep_right_associative_0_witness_common_left. ((exists pfrep_position_associative_0_witness_common_leftfirst. ((pfrep_position_associative_0_witness_common_leftfirst+S (pfrep_power_associative_0_witness_common_left)=(La)) /\\ ((((exists ff_h_pfp_associative_0_witness_common_leftfirstentry. ff_h_pfp_associative_0_witness_common_leftfirstentry + S (pfrep_left_associative_0_witness_common_left) = S ((S (pfrep_position_associative_0_witness_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_associative_0_witness_common_leftfirstentry. ab = ff_q_pfp_associative_0_witness_common_leftfirstentry * S ((S (pfrep_position_associative_0_witness_common_leftfirst)) * ac) + (pfrep_left_associative_0_witness_common_left)))))) \\/ (((exists pfrep_gap_associative_0_witness_common_leftfirstoutside. pfrep_gap_associative_0_witness_common_leftfirstoutside+(La)=(pfrep_power_associative_0_witness_common_left)) /\\ (((pfrep_left_associative_0_witness_common_left)=0))))) -> ((exists pfrep_position_associative_0_witness_common_leftsecond. ((pfrep_position_associative_0_witness_common_leftsecond+S (pfrep_power_associative_0_witness_common_left)=(pfaa_length_associative_0)) /\\ ((((exists ff_h_pfp_associative_0_witness_common_leftsecondentry. ff_h_pfp_associative_0_witness_common_leftsecondentry + S (pfrep_right_associative_0_witness_common_left) = S ((S (pfrep_position_associative_0_witness_common_leftsecond)) * pfaa_left_c_associative_0)) /\\ exists ff_q_pfp_associative_0_witness_common_leftsecondentry. pfaa_left_b_associative_0 = ff_q_pfp_associative_0_witness_common_leftsecondentry * S ((S (pfrep_position_associative_0_witness_common_leftsecond)) * pfaa_left_c_associative_0) + (pfrep_right_associative_0_witness_common_left)))))) \\/ (((exists pfrep_gap_associative_0_witness_common_leftsecondoutside. pfrep_gap_associative_0_witness_common_leftsecondoutside+(pfaa_length_associative_0)=(pfrep_power_associative_0_witness_common_left)) /\\ (((pfrep_right_associative_0_witness_common_left)=0))))) -> pfrep_left_associative_0_witness_common_left=pfrep_right_associative_0_witness_common_left) /\\ ((forall pfrep_power_associative_0_witness_common_right pfrep_left_associative_0_witness_common_right pfrep_right_associative_0_witness_common_right. ((exists pfrep_position_associative_0_witness_common_rightfirst. ((pfrep_position_associative_0_witness_common_rightfirst+S (pfrep_power_associative_0_witness_common_right)=(Lb)) /\\ ((((exists ff_h_pfp_associative_0_witness_common_rightfirstentry. ff_h_pfp_associative_0_witness_common_rightfirstentry + S (pfrep_left_associative_0_witness_common_right) = S ((S (pfrep_position_associative_0_witness_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_associative_0_witness_common_rightfirstentry. bb = ff_q_pfp_associative_0_witness_common_rightfirstentry * S ((S (pfrep_position_associative_0_witness_common_rightfirst)) * bc) + (pfrep_left_associative_0_witness_common_right)))))) \\/ (((exists pfrep_gap_associative_0_witness_common_rightfirstoutside. pfrep_gap_associative_0_witness_common_rightfirstoutside+(Lb)=(pfrep_power_associative_0_witness_common_right)) /\\ (((pfrep_left_associative_0_witness_common_right)=0))))) -> ((exists pfrep_position_associative_0_witness_common_rightsecond. ((pfrep_position_associative_0_witness_common_rightsecond+S (pfrep_power_associative_0_witness_common_right)=(pfaa_length_associative_0)) /\\ ((((exists ff_h_pfp_associative_0_witness_common_rightsecondentry. ff_h_pfp_associative_0_witness_common_rightsecondentry + S (pfrep_right_associative_0_witness_common_right) = S ((S (pfrep_position_associative_0_witness_common_rightsecond)) * pfaa_right_c_associative_0)) /\\ exists ff_q_pfp_associative_0_witness_common_rightsecondentry. pfaa_right_b_associative_0 = ff_q_pfp_associative_0_witness_common_rightsecondentry * S ((S (pfrep_position_associative_0_witness_common_rightsecond)) * pfaa_right_c_associative_0) + (pfrep_right_associative_0_witness_common_right)))))) \\/ (((exists pfrep_gap_associative_0_witness_common_rightsecondoutside. pfrep_gap_associative_0_witness_common_rightsecondoutside+(pfaa_length_associative_0)=(pfrep_power_associative_0_witness_common_right)) /\\ (((pfrep_right_associative_0_witness_common_right)=0))))) -> pfrep_left_associative_0_witness_common_right=pfrep_right_associative_0_witness_common_right)))) /\\ (((forall pfp_index_associative_0_witness_operation. (exists pfa_gap_associative_0_witness_operationindex. pfa_gap_associative_0_witness_operationindex + S (pfp_index_associative_0_witness_operation) = (pfaa_length_associative_0)) -> exists pfp_left_associative_0_witness_operation pfp_right_associative_0_witness_operation pfp_value_associative_0_witness_operation. ((((exists ff_h_pfp_associative_0_witness_operationleft. ff_h_pfp_associative_0_witness_operationleft + S (pfp_left_associative_0_witness_operation) = S ((S (pfp_index_associative_0_witness_operation)) * pfaa_left_c_associative_0)) /\\ exists ff_q_pfp_associative_0_witness_operationleft. pfaa_left_b_associative_0 = ff_q_pfp_associative_0_witness_operationleft * S ((S (pfp_index_associative_0_witness_operation)) * pfaa_left_c_associative_0) + (pfp_left_associative_0_witness_operation))) /\\ (((((exists ff_h_pfp_associative_0_witness_operationright. ff_h_pfp_associative_0_witness_operationright + S (pfp_right_associative_0_witness_operation) = S ((S (pfp_index_associative_0_witness_operation)) * pfaa_right_c_associative_0)) /\\ exists ff_q_pfp_associative_0_witness_operationright. pfaa_right_b_associative_0 = ff_q_pfp_associative_0_witness_operationright * S ((S (pfp_index_associative_0_witness_operation)) * pfaa_right_c_associative_0) + (pfp_right_associative_0_witness_operation))) /\\ (((((exists ff_h_pfp_associative_0_witness_operationtarget. ff_h_pfp_associative_0_witness_operationtarget + S (pfp_value_associative_0_witness_operation) = S ((S (pfp_index_associative_0_witness_operation)) * pfaa_sum_c_associative_0)) /\\ exists ff_q_pfp_associative_0_witness_operationtarget. pfaa_sum_b_associative_0 = ff_q_pfp_associative_0_witness_operationtarget * S ((S (pfp_index_associative_0_witness_operation)) * pfaa_sum_c_associative_0) + (pfp_value_associative_0_witness_operation))) /\\ ((((exists pfa_gap_associative_0_witness_operationoperationleft. pfa_gap_associative_0_witness_operationoperationleft + S (pfp_left_associative_0_witness_operation) = (p)) /\\ (((exists pfa_gap_associative_0_witness_operationoperationright. pfa_gap_associative_0_witness_operationoperationright + S (pfp_right_associative_0_witness_operation) = (p)) /\\ ((((exists pfa_gap_associative_0_witness_operationoperationresultbound. pfa_gap_associative_0_witness_operationoperationresultbound + S (pfp_value_associative_0_witness_operation) = (p)) /\\ ((exists pfa_offset_left_associative_0_witness_operationoperationresultcongruence pfa_offset_right_associative_0_witness_operationoperationresultcongruence. ((pfp_left_associative_0_witness_operation) + (pfp_right_associative_0_witness_operation)) + (p) * pfa_offset_left_associative_0_witness_operationoperationresultcongruence = (pfp_value_associative_0_witness_operation) + (p) * pfa_offset_right_associative_0_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_associative_0_witness_output pfrep_left_associative_0_witness_output pfrep_right_associative_0_witness_output. ((exists pfrep_position_associative_0_witness_outputfirst. ((pfrep_position_associative_0_witness_outputfirst+S (pfrep_power_associative_0_witness_output)=(pfaa_length_associative_0)) /\\ ((((exists ff_h_pfp_associative_0_witness_outputfirstentry. ff_h_pfp_associative_0_witness_outputfirstentry + S (pfrep_left_associative_0_witness_output) = S ((S (pfrep_position_associative_0_witness_outputfirst)) * pfaa_sum_c_associative_0)) /\\ exists ff_q_pfp_associative_0_witness_outputfirstentry. pfaa_sum_b_associative_0 = ff_q_pfp_associative_0_witness_outputfirstentry * S ((S (pfrep_position_associative_0_witness_outputfirst)) * pfaa_sum_c_associative_0) + (pfrep_left_associative_0_witness_output)))))) \\/ (((exists pfrep_gap_associative_0_witness_outputfirstoutside. pfrep_gap_associative_0_witness_outputfirstoutside+(pfaa_length_associative_0)=(pfrep_power_associative_0_witness_output)) /\\ (((pfrep_left_associative_0_witness_output)=0))))) -> ((exists pfrep_position_associative_0_witness_outputsecond. ((pfrep_position_associative_0_witness_outputsecond+S (pfrep_power_associative_0_witness_output)=(Lu)) /\\ ((((exists ff_h_pfp_associative_0_witness_outputsecondentry. ff_h_pfp_associative_0_witness_outputsecondentry + S (pfrep_right_associative_0_witness_output) = S ((S (pfrep_position_associative_0_witness_outputsecond)) * uc)) /\\ exists ff_q_pfp_associative_0_witness_outputsecondentry. ub = ff_q_pfp_associative_0_witness_outputsecondentry * S ((S (pfrep_position_associative_0_witness_outputsecond)) * uc) + (pfrep_right_associative_0_witness_output)))))) \\/ (((exists pfrep_gap_associative_0_witness_outputsecondoutside. pfrep_gap_associative_0_witness_outputsecondoutside+(Lu)=(pfrep_power_associative_0_witness_output)) /\\ (((pfrep_right_associative_0_witness_output)=0))))) -> pfrep_left_associative_0_witness_output=pfrep_right_associative_0_witness_output))))))))))))) -> (((forall fom_index_pfp_associative_1_left_bounded. (exists fom_gap_pfp_associative_1_left_bounded_index_bound. fom_gap_pfp_associative_1_left_bounded_index_bound + S (fom_index_pfp_associative_1_left_bounded) = Lu) -> exists fom_value_pfp_associative_1_left_bounded. ((((exists fom_beta_height_pfp_associative_1_left_bounded_entry. fom_beta_height_pfp_associative_1_left_bounded_entry + S (fom_value_pfp_associative_1_left_bounded) = S ((S (fom_index_pfp_associative_1_left_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_associative_1_left_bounded_entry. ub = fom_beta_quotient_pfp_associative_1_left_bounded_entry * S ((S (fom_index_pfp_associative_1_left_bounded)) * uc) + (fom_value_pfp_associative_1_left_bounded))) /\\ (exists fom_gap_pfp_associative_1_left_bounded_value_bound. fom_gap_pfp_associative_1_left_bounded_value_bound + S (fom_value_pfp_associative_1_left_bounded) = p))) /\\ (((forall fom_index_pfp_associative_1_right_bounded. (exists fom_gap_pfp_associative_1_right_bounded_index_bound. fom_gap_pfp_associative_1_right_bounded_index_bound + S (fom_index_pfp_associative_1_right_bounded) = Lc) -> exists fom_value_pfp_associative_1_right_bounded. ((((exists fom_beta_height_pfp_associative_1_right_bounded_entry. fom_beta_height_pfp_associative_1_right_bounded_entry + S (fom_value_pfp_associative_1_right_bounded) = S ((S (fom_index_pfp_associative_1_right_bounded)) * cc)) /\\ exists fom_beta_quotient_pfp_associative_1_right_bounded_entry. cb = fom_beta_quotient_pfp_associative_1_right_bounded_entry * S ((S (fom_index_pfp_associative_1_right_bounded)) * cc) + (fom_value_pfp_associative_1_right_bounded))) /\\ (exists fom_gap_pfp_associative_1_right_bounded_value_bound. fom_gap_pfp_associative_1_right_bounded_value_bound + S (fom_value_pfp_associative_1_right_bounded) = p))) /\\ (((forall fom_index_pfp_associative_1_result_bounded. (exists fom_gap_pfp_associative_1_result_bounded_index_bound. fom_gap_pfp_associative_1_result_bounded_index_bound + S (fom_index_pfp_associative_1_result_bounded) = Lr) -> exists fom_value_pfp_associative_1_result_bounded. ((((exists fom_beta_height_pfp_associative_1_result_bounded_entry. fom_beta_height_pfp_associative_1_result_bounded_entry + S (fom_value_pfp_associative_1_result_bounded) = S ((S (fom_index_pfp_associative_1_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_associative_1_result_bounded_entry. rb = fom_beta_quotient_pfp_associative_1_result_bounded_entry * S ((S (fom_index_pfp_associative_1_result_bounded)) * rc) + (fom_value_pfp_associative_1_result_bounded))) /\\ (exists fom_gap_pfp_associative_1_result_bounded_value_bound. fom_gap_pfp_associative_1_result_bounded_value_bound + S (fom_value_pfp_associative_1_result_bounded) = p))) /\\ ((exists pfaa_left_b_associative_1 pfaa_left_c_associative_1 pfaa_right_b_associative_1 pfaa_right_c_associative_1 pfaa_sum_b_associative_1 pfaa_sum_c_associative_1 pfaa_length_associative_1. ((((forall pfrep_power_associative_1_witness_common_left pfrep_left_associative_1_witness_common_left pfrep_right_associative_1_witness_common_left. ((exists pfrep_position_associative_1_witness_common_leftfirst. ((pfrep_position_associative_1_witness_common_leftfirst+S (pfrep_power_associative_1_witness_common_left)=(Lu)) /\\ ((((exists ff_h_pfp_associative_1_witness_common_leftfirstentry. ff_h_pfp_associative_1_witness_common_leftfirstentry + S (pfrep_left_associative_1_witness_common_left) = S ((S (pfrep_position_associative_1_witness_common_leftfirst)) * uc)) /\\ exists ff_q_pfp_associative_1_witness_common_leftfirstentry. ub = ff_q_pfp_associative_1_witness_common_leftfirstentry * S ((S (pfrep_position_associative_1_witness_common_leftfirst)) * uc) + (pfrep_left_associative_1_witness_common_left)))))) \\/ (((exists pfrep_gap_associative_1_witness_common_leftfirstoutside. pfrep_gap_associative_1_witness_common_leftfirstoutside+(Lu)=(pfrep_power_associative_1_witness_common_left)) /\\ (((pfrep_left_associative_1_witness_common_left)=0))))) -> ((exists pfrep_position_associative_1_witness_common_leftsecond. ((pfrep_position_associative_1_witness_common_leftsecond+S (pfrep_power_associative_1_witness_common_left)=(pfaa_length_associative_1)) /\\ ((((exists ff_h_pfp_associative_1_witness_common_leftsecondentry. ff_h_pfp_associative_1_witness_common_leftsecondentry + S (pfrep_right_associative_1_witness_common_left) = S ((S (pfrep_position_associative_1_witness_common_leftsecond)) * pfaa_left_c_associative_1)) /\\ exists ff_q_pfp_associative_1_witness_common_leftsecondentry. pfaa_left_b_associative_1 = ff_q_pfp_associative_1_witness_common_leftsecondentry * S ((S (pfrep_position_associative_1_witness_common_leftsecond)) * pfaa_left_c_associative_1) + (pfrep_right_associative_1_witness_common_left)))))) \\/ (((exists pfrep_gap_associative_1_witness_common_leftsecondoutside. pfrep_gap_associative_1_witness_common_leftsecondoutside+(pfaa_length_associative_1)=(pfrep_power_associative_1_witness_common_left)) /\\ (((pfrep_right_associative_1_witness_common_left)=0))))) -> pfrep_left_associative_1_witness_common_left=pfrep_right_associative_1_witness_common_left) /\\ ((forall pfrep_power_associative_1_witness_common_right pfrep_left_associative_1_witness_common_right pfrep_right_associative_1_witness_common_right. ((exists pfrep_position_associative_1_witness_common_rightfirst. ((pfrep_position_associative_1_witness_common_rightfirst+S (pfrep_power_associative_1_witness_common_right)=(Lc)) /\\ ((((exists ff_h_pfp_associative_1_witness_common_rightfirstentry. ff_h_pfp_associative_1_witness_common_rightfirstentry + S (pfrep_left_associative_1_witness_common_right) = S ((S (pfrep_position_associative_1_witness_common_rightfirst)) * cc)) /\\ exists ff_q_pfp_associative_1_witness_common_rightfirstentry. cb = ff_q_pfp_associative_1_witness_common_rightfirstentry * S ((S (pfrep_position_associative_1_witness_common_rightfirst)) * cc) + (pfrep_left_associative_1_witness_common_right)))))) \\/ (((exists pfrep_gap_associative_1_witness_common_rightfirstoutside. pfrep_gap_associative_1_witness_common_rightfirstoutside+(Lc)=(pfrep_power_associative_1_witness_common_right)) /\\ (((pfrep_left_associative_1_witness_common_right)=0))))) -> ((exists pfrep_position_associative_1_witness_common_rightsecond. ((pfrep_position_associative_1_witness_common_rightsecond+S (pfrep_power_associative_1_witness_common_right)=(pfaa_length_associative_1)) /\\ ((((exists ff_h_pfp_associative_1_witness_common_rightsecondentry. ff_h_pfp_associative_1_witness_common_rightsecondentry + S (pfrep_right_associative_1_witness_common_right) = S ((S (pfrep_position_associative_1_witness_common_rightsecond)) * pfaa_right_c_associative_1)) /\\ exists ff_q_pfp_associative_1_witness_common_rightsecondentry. pfaa_right_b_associative_1 = ff_q_pfp_associative_1_witness_common_rightsecondentry * S ((S (pfrep_position_associative_1_witness_common_rightsecond)) * pfaa_right_c_associative_1) + (pfrep_right_associative_1_witness_common_right)))))) \\/ (((exists pfrep_gap_associative_1_witness_common_rightsecondoutside. pfrep_gap_associative_1_witness_common_rightsecondoutside+(pfaa_length_associative_1)=(pfrep_power_associative_1_witness_common_right)) /\\ (((pfrep_right_associative_1_witness_common_right)=0))))) -> pfrep_left_associative_1_witness_common_right=pfrep_right_associative_1_witness_common_right)))) /\\ (((forall pfp_index_associative_1_witness_operation. (exists pfa_gap_associative_1_witness_operationindex. pfa_gap_associative_1_witness_operationindex + S (pfp_index_associative_1_witness_operation) = (pfaa_length_associative_1)) -> exists pfp_left_associative_1_witness_operation pfp_right_associative_1_witness_operation pfp_value_associative_1_witness_operation. ((((exists ff_h_pfp_associative_1_witness_operationleft. ff_h_pfp_associative_1_witness_operationleft + S (pfp_left_associative_1_witness_operation) = S ((S (pfp_index_associative_1_witness_operation)) * pfaa_left_c_associative_1)) /\\ exists ff_q_pfp_associative_1_witness_operationleft. pfaa_left_b_associative_1 = ff_q_pfp_associative_1_witness_operationleft * S ((S (pfp_index_associative_1_witness_operation)) * pfaa_left_c_associative_1) + (pfp_left_associative_1_witness_operation))) /\\ (((((exists ff_h_pfp_associative_1_witness_operationright. ff_h_pfp_associative_1_witness_operationright + S (pfp_right_associative_1_witness_operation) = S ((S (pfp_index_associative_1_witness_operation)) * pfaa_right_c_associative_1)) /\\ exists ff_q_pfp_associative_1_witness_operationright. pfaa_right_b_associative_1 = ff_q_pfp_associative_1_witness_operationright * S ((S (pfp_index_associative_1_witness_operation)) * pfaa_right_c_associative_1) + (pfp_right_associative_1_witness_operation))) /\\ (((((exists ff_h_pfp_associative_1_witness_operationtarget. ff_h_pfp_associative_1_witness_operationtarget + S (pfp_value_associative_1_witness_operation) = S ((S (pfp_index_associative_1_witness_operation)) * pfaa_sum_c_associative_1)) /\\ exists ff_q_pfp_associative_1_witness_operationtarget. pfaa_sum_b_associative_1 = ff_q_pfp_associative_1_witness_operationtarget * S ((S (pfp_index_associative_1_witness_operation)) * pfaa_sum_c_associative_1) + (pfp_value_associative_1_witness_operation))) /\\ ((((exists pfa_gap_associative_1_witness_operationoperationleft. pfa_gap_associative_1_witness_operationoperationleft + S (pfp_left_associative_1_witness_operation) = (p)) /\\ (((exists pfa_gap_associative_1_witness_operationoperationright. pfa_gap_associative_1_witness_operationoperationright + S (pfp_right_associative_1_witness_operation) = (p)) /\\ ((((exists pfa_gap_associative_1_witness_operationoperationresultbound. pfa_gap_associative_1_witness_operationoperationresultbound + S (pfp_value_associative_1_witness_operation) = (p)) /\\ ((exists pfa_offset_left_associative_1_witness_operationoperationresultcongruence pfa_offset_right_associative_1_witness_operationoperationresultcongruence. ((pfp_left_associative_1_witness_operation) + (pfp_right_associative_1_witness_operation)) + (p) * pfa_offset_left_associative_1_witness_operationoperationresultcongruence = (pfp_value_associative_1_witness_operation) + (p) * pfa_offset_right_associative_1_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_associative_1_witness_output pfrep_left_associative_1_witness_output pfrep_right_associative_1_witness_output. ((exists pfrep_position_associative_1_witness_outputfirst. ((pfrep_position_associative_1_witness_outputfirst+S (pfrep_power_associative_1_witness_output)=(pfaa_length_associative_1)) /\\ ((((exists ff_h_pfp_associative_1_witness_outputfirstentry. ff_h_pfp_associative_1_witness_outputfirstentry + S (pfrep_left_associative_1_witness_output) = S ((S (pfrep_position_associative_1_witness_outputfirst)) * pfaa_sum_c_associative_1)) /\\ exists ff_q_pfp_associative_1_witness_outputfirstentry. pfaa_sum_b_associative_1 = ff_q_pfp_associative_1_witness_outputfirstentry * S ((S (pfrep_position_associative_1_witness_outputfirst)) * pfaa_sum_c_associative_1) + (pfrep_left_associative_1_witness_output)))))) \\/ (((exists pfrep_gap_associative_1_witness_outputfirstoutside. pfrep_gap_associative_1_witness_outputfirstoutside+(pfaa_length_associative_1)=(pfrep_power_associative_1_witness_output)) /\\ (((pfrep_left_associative_1_witness_output)=0))))) -> ((exists pfrep_position_associative_1_witness_outputsecond. ((pfrep_position_associative_1_witness_outputsecond+S (pfrep_power_associative_1_witness_output)=(Lr)) /\\ ((((exists ff_h_pfp_associative_1_witness_outputsecondentry. ff_h_pfp_associative_1_witness_outputsecondentry + S (pfrep_right_associative_1_witness_output) = S ((S (pfrep_position_associative_1_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_associative_1_witness_outputsecondentry. rb = ff_q_pfp_associative_1_witness_outputsecondentry * S ((S (pfrep_position_associative_1_witness_outputsecond)) * rc) + (pfrep_right_associative_1_witness_output)))))) \\/ (((exists pfrep_gap_associative_1_witness_outputsecondoutside. pfrep_gap_associative_1_witness_outputsecondoutside+(Lr)=(pfrep_power_associative_1_witness_output)) /\\ (((pfrep_right_associative_1_witness_output)=0))))) -> pfrep_left_associative_1_witness_output=pfrep_right_associative_1_witness_output))))))))))))) -> (((forall fom_index_pfp_associative_2_left_bounded. (exists fom_gap_pfp_associative_2_left_bounded_index_bound. fom_gap_pfp_associative_2_left_bounded_index_bound + S (fom_index_pfp_associative_2_left_bounded) = Lb) -> exists fom_value_pfp_associative_2_left_bounded. ((((exists fom_beta_height_pfp_associative_2_left_bounded_entry. fom_beta_height_pfp_associative_2_left_bounded_entry + S (fom_value_pfp_associative_2_left_bounded) = S ((S (fom_index_pfp_associative_2_left_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_associative_2_left_bounded_entry. bb = fom_beta_quotient_pfp_associative_2_left_bounded_entry * S ((S (fom_index_pfp_associative_2_left_bounded)) * bc) + (fom_value_pfp_associative_2_left_bounded))) /\\ (exists fom_gap_pfp_associative_2_left_bounded_value_bound. fom_gap_pfp_associative_2_left_bounded_value_bound + S (fom_value_pfp_associative_2_left_bounded) = p))) /\\ (((forall fom_index_pfp_associative_2_right_bounded. (exists fom_gap_pfp_associative_2_right_bounded_index_bound. fom_gap_pfp_associative_2_right_bounded_index_bound + S (fom_index_pfp_associative_2_right_bounded) = Lc) -> exists fom_value_pfp_associative_2_right_bounded. ((((exists fom_beta_height_pfp_associative_2_right_bounded_entry. fom_beta_height_pfp_associative_2_right_bounded_entry + S (fom_value_pfp_associative_2_right_bounded) = S ((S (fom_index_pfp_associative_2_right_bounded)) * cc)) /\\ exists fom_beta_quotient_pfp_associative_2_right_bounded_entry. cb = fom_beta_quotient_pfp_associative_2_right_bounded_entry * S ((S (fom_index_pfp_associative_2_right_bounded)) * cc) + (fom_value_pfp_associative_2_right_bounded))) /\\ (exists fom_gap_pfp_associative_2_right_bounded_value_bound. fom_gap_pfp_associative_2_right_bounded_value_bound + S (fom_value_pfp_associative_2_right_bounded) = p))) /\\ (((forall fom_index_pfp_associative_2_result_bounded. (exists fom_gap_pfp_associative_2_result_bounded_index_bound. fom_gap_pfp_associative_2_result_bounded_index_bound + S (fom_index_pfp_associative_2_result_bounded) = Lv) -> exists fom_value_pfp_associative_2_result_bounded. ((((exists fom_beta_height_pfp_associative_2_result_bounded_entry. fom_beta_height_pfp_associative_2_result_bounded_entry + S (fom_value_pfp_associative_2_result_bounded) = S ((S (fom_index_pfp_associative_2_result_bounded)) * vc)) /\\ exists fom_beta_quotient_pfp_associative_2_result_bounded_entry. vb = fom_beta_quotient_pfp_associative_2_result_bounded_entry * S ((S (fom_index_pfp_associative_2_result_bounded)) * vc) + (fom_value_pfp_associative_2_result_bounded))) /\\ (exists fom_gap_pfp_associative_2_result_bounded_value_bound. fom_gap_pfp_associative_2_result_bounded_value_bound + S (fom_value_pfp_associative_2_result_bounded) = p))) /\\ ((exists pfaa_left_b_associative_2 pfaa_left_c_associative_2 pfaa_right_b_associative_2 pfaa_right_c_associative_2 pfaa_sum_b_associative_2 pfaa_sum_c_associative_2 pfaa_length_associative_2. ((((forall pfrep_power_associative_2_witness_common_left pfrep_left_associative_2_witness_common_left pfrep_right_associative_2_witness_common_left. ((exists pfrep_position_associative_2_witness_common_leftfirst. ((pfrep_position_associative_2_witness_common_leftfirst+S (pfrep_power_associative_2_witness_common_left)=(Lb)) /\\ ((((exists ff_h_pfp_associative_2_witness_common_leftfirstentry. ff_h_pfp_associative_2_witness_common_leftfirstentry + S (pfrep_left_associative_2_witness_common_left) = S ((S (pfrep_position_associative_2_witness_common_leftfirst)) * bc)) /\\ exists ff_q_pfp_associative_2_witness_common_leftfirstentry. bb = ff_q_pfp_associative_2_witness_common_leftfirstentry * S ((S (pfrep_position_associative_2_witness_common_leftfirst)) * bc) + (pfrep_left_associative_2_witness_common_left)))))) \\/ (((exists pfrep_gap_associative_2_witness_common_leftfirstoutside. pfrep_gap_associative_2_witness_common_leftfirstoutside+(Lb)=(pfrep_power_associative_2_witness_common_left)) /\\ (((pfrep_left_associative_2_witness_common_left)=0))))) -> ((exists pfrep_position_associative_2_witness_common_leftsecond. ((pfrep_position_associative_2_witness_common_leftsecond+S (pfrep_power_associative_2_witness_common_left)=(pfaa_length_associative_2)) /\\ ((((exists ff_h_pfp_associative_2_witness_common_leftsecondentry. ff_h_pfp_associative_2_witness_common_leftsecondentry + S (pfrep_right_associative_2_witness_common_left) = S ((S (pfrep_position_associative_2_witness_common_leftsecond)) * pfaa_left_c_associative_2)) /\\ exists ff_q_pfp_associative_2_witness_common_leftsecondentry. pfaa_left_b_associative_2 = ff_q_pfp_associative_2_witness_common_leftsecondentry * S ((S (pfrep_position_associative_2_witness_common_leftsecond)) * pfaa_left_c_associative_2) + (pfrep_right_associative_2_witness_common_left)))))) \\/ (((exists pfrep_gap_associative_2_witness_common_leftsecondoutside. pfrep_gap_associative_2_witness_common_leftsecondoutside+(pfaa_length_associative_2)=(pfrep_power_associative_2_witness_common_left)) /\\ (((pfrep_right_associative_2_witness_common_left)=0))))) -> pfrep_left_associative_2_witness_common_left=pfrep_right_associative_2_witness_common_left) /\\ ((forall pfrep_power_associative_2_witness_common_right pfrep_left_associative_2_witness_common_right pfrep_right_associative_2_witness_common_right. ((exists pfrep_position_associative_2_witness_common_rightfirst. ((pfrep_position_associative_2_witness_common_rightfirst+S (pfrep_power_associative_2_witness_common_right)=(Lc)) /\\ ((((exists ff_h_pfp_associative_2_witness_common_rightfirstentry. ff_h_pfp_associative_2_witness_common_rightfirstentry + S (pfrep_left_associative_2_witness_common_right) = S ((S (pfrep_position_associative_2_witness_common_rightfirst)) * cc)) /\\ exists ff_q_pfp_associative_2_witness_common_rightfirstentry. cb = ff_q_pfp_associative_2_witness_common_rightfirstentry * S ((S (pfrep_position_associative_2_witness_common_rightfirst)) * cc) + (pfrep_left_associative_2_witness_common_right)))))) \\/ (((exists pfrep_gap_associative_2_witness_common_rightfirstoutside. pfrep_gap_associative_2_witness_common_rightfirstoutside+(Lc)=(pfrep_power_associative_2_witness_common_right)) /\\ (((pfrep_left_associative_2_witness_common_right)=0))))) -> ((exists pfrep_position_associative_2_witness_common_rightsecond. ((pfrep_position_associative_2_witness_common_rightsecond+S (pfrep_power_associative_2_witness_common_right)=(pfaa_length_associative_2)) /\\ ((((exists ff_h_pfp_associative_2_witness_common_rightsecondentry. ff_h_pfp_associative_2_witness_common_rightsecondentry + S (pfrep_right_associative_2_witness_common_right) = S ((S (pfrep_position_associative_2_witness_common_rightsecond)) * pfaa_right_c_associative_2)) /\\ exists ff_q_pfp_associative_2_witness_common_rightsecondentry. pfaa_right_b_associative_2 = ff_q_pfp_associative_2_witness_common_rightsecondentry * S ((S (pfrep_position_associative_2_witness_common_rightsecond)) * pfaa_right_c_associative_2) + (pfrep_right_associative_2_witness_common_right)))))) \\/ (((exists pfrep_gap_associative_2_witness_common_rightsecondoutside. pfrep_gap_associative_2_witness_common_rightsecondoutside+(pfaa_length_associative_2)=(pfrep_power_associative_2_witness_common_right)) /\\ (((pfrep_right_associative_2_witness_common_right)=0))))) -> pfrep_left_associative_2_witness_common_right=pfrep_right_associative_2_witness_common_right)))) /\\ (((forall pfp_index_associative_2_witness_operation. (exists pfa_gap_associative_2_witness_operationindex. pfa_gap_associative_2_witness_operationindex + S (pfp_index_associative_2_witness_operation) = (pfaa_length_associative_2)) -> exists pfp_left_associative_2_witness_operation pfp_right_associative_2_witness_operation pfp_value_associative_2_witness_operation. ((((exists ff_h_pfp_associative_2_witness_operationleft. ff_h_pfp_associative_2_witness_operationleft + S (pfp_left_associative_2_witness_operation) = S ((S (pfp_index_associative_2_witness_operation)) * pfaa_left_c_associative_2)) /\\ exists ff_q_pfp_associative_2_witness_operationleft. pfaa_left_b_associative_2 = ff_q_pfp_associative_2_witness_operationleft * S ((S (pfp_index_associative_2_witness_operation)) * pfaa_left_c_associative_2) + (pfp_left_associative_2_witness_operation))) /\\ (((((exists ff_h_pfp_associative_2_witness_operationright. ff_h_pfp_associative_2_witness_operationright + S (pfp_right_associative_2_witness_operation) = S ((S (pfp_index_associative_2_witness_operation)) * pfaa_right_c_associative_2)) /\\ exists ff_q_pfp_associative_2_witness_operationright. pfaa_right_b_associative_2 = ff_q_pfp_associative_2_witness_operationright * S ((S (pfp_index_associative_2_witness_operation)) * pfaa_right_c_associative_2) + (pfp_right_associative_2_witness_operation))) /\\ (((((exists ff_h_pfp_associative_2_witness_operationtarget. ff_h_pfp_associative_2_witness_operationtarget + S (pfp_value_associative_2_witness_operation) = S ((S (pfp_index_associative_2_witness_operation)) * pfaa_sum_c_associative_2)) /\\ exists ff_q_pfp_associative_2_witness_operationtarget. pfaa_sum_b_associative_2 = ff_q_pfp_associative_2_witness_operationtarget * S ((S (pfp_index_associative_2_witness_operation)) * pfaa_sum_c_associative_2) + (pfp_value_associative_2_witness_operation))) /\\ ((((exists pfa_gap_associative_2_witness_operationoperationleft. pfa_gap_associative_2_witness_operationoperationleft + S (pfp_left_associative_2_witness_operation) = (p)) /\\ (((exists pfa_gap_associative_2_witness_operationoperationright. pfa_gap_associative_2_witness_operationoperationright + S (pfp_right_associative_2_witness_operation) = (p)) /\\ ((((exists pfa_gap_associative_2_witness_operationoperationresultbound. pfa_gap_associative_2_witness_operationoperationresultbound + S (pfp_value_associative_2_witness_operation) = (p)) /\\ ((exists pfa_offset_left_associative_2_witness_operationoperationresultcongruence pfa_offset_right_associative_2_witness_operationoperationresultcongruence. ((pfp_left_associative_2_witness_operation) + (pfp_right_associative_2_witness_operation)) + (p) * pfa_offset_left_associative_2_witness_operationoperationresultcongruence = (pfp_value_associative_2_witness_operation) + (p) * pfa_offset_right_associative_2_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_associative_2_witness_output pfrep_left_associative_2_witness_output pfrep_right_associative_2_witness_output. ((exists pfrep_position_associative_2_witness_outputfirst. ((pfrep_position_associative_2_witness_outputfirst+S (pfrep_power_associative_2_witness_output)=(pfaa_length_associative_2)) /\\ ((((exists ff_h_pfp_associative_2_witness_outputfirstentry. ff_h_pfp_associative_2_witness_outputfirstentry + S (pfrep_left_associative_2_witness_output) = S ((S (pfrep_position_associative_2_witness_outputfirst)) * pfaa_sum_c_associative_2)) /\\ exists ff_q_pfp_associative_2_witness_outputfirstentry. pfaa_sum_b_associative_2 = ff_q_pfp_associative_2_witness_outputfirstentry * S ((S (pfrep_position_associative_2_witness_outputfirst)) * pfaa_sum_c_associative_2) + (pfrep_left_associative_2_witness_output)))))) \\/ (((exists pfrep_gap_associative_2_witness_outputfirstoutside. pfrep_gap_associative_2_witness_outputfirstoutside+(pfaa_length_associative_2)=(pfrep_power_associative_2_witness_output)) /\\ (((pfrep_left_associative_2_witness_output)=0))))) -> ((exists pfrep_position_associative_2_witness_outputsecond. ((pfrep_position_associative_2_witness_outputsecond+S (pfrep_power_associative_2_witness_output)=(Lv)) /\\ ((((exists ff_h_pfp_associative_2_witness_outputsecondentry. ff_h_pfp_associative_2_witness_outputsecondentry + S (pfrep_right_associative_2_witness_output) = S ((S (pfrep_position_associative_2_witness_outputsecond)) * vc)) /\\ exists ff_q_pfp_associative_2_witness_outputsecondentry. vb = ff_q_pfp_associative_2_witness_outputsecondentry * S ((S (pfrep_position_associative_2_witness_outputsecond)) * vc) + (pfrep_right_associative_2_witness_output)))))) \\/ (((exists pfrep_gap_associative_2_witness_outputsecondoutside. pfrep_gap_associative_2_witness_outputsecondoutside+(Lv)=(pfrep_power_associative_2_witness_output)) /\\ (((pfrep_right_associative_2_witness_output)=0))))) -> pfrep_left_associative_2_witness_output=pfrep_right_associative_2_witness_output))))))))))))) -> (((forall fom_index_pfp_associative_3_left_bounded. (exists fom_gap_pfp_associative_3_left_bounded_index_bound. fom_gap_pfp_associative_3_left_bounded_index_bound + S (fom_index_pfp_associative_3_left_bounded) = La) -> exists fom_value_pfp_associative_3_left_bounded. ((((exists fom_beta_height_pfp_associative_3_left_bounded_entry. fom_beta_height_pfp_associative_3_left_bounded_entry + S (fom_value_pfp_associative_3_left_bounded) = S ((S (fom_index_pfp_associative_3_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_associative_3_left_bounded_entry. ab = fom_beta_quotient_pfp_associative_3_left_bounded_entry * S ((S (fom_index_pfp_associative_3_left_bounded)) * ac) + (fom_value_pfp_associative_3_left_bounded))) /\\ (exists fom_gap_pfp_associative_3_left_bounded_value_bound. fom_gap_pfp_associative_3_left_bounded_value_bound + S (fom_value_pfp_associative_3_left_bounded) = p))) /\\ (((forall fom_index_pfp_associative_3_right_bounded. (exists fom_gap_pfp_associative_3_right_bounded_index_bound. fom_gap_pfp_associative_3_right_bounded_index_bound + S (fom_index_pfp_associative_3_right_bounded) = Lv) -> exists fom_value_pfp_associative_3_right_bounded. ((((exists fom_beta_height_pfp_associative_3_right_bounded_entry. fom_beta_height_pfp_associative_3_right_bounded_entry + S (fom_value_pfp_associative_3_right_bounded) = S ((S (fom_index_pfp_associative_3_right_bounded)) * vc)) /\\ exists fom_beta_quotient_pfp_associative_3_right_bounded_entry. vb = fom_beta_quotient_pfp_associative_3_right_bounded_entry * S ((S (fom_index_pfp_associative_3_right_bounded)) * vc) + (fom_value_pfp_associative_3_right_bounded))) /\\ (exists fom_gap_pfp_associative_3_right_bounded_value_bound. fom_gap_pfp_associative_3_right_bounded_value_bound + S (fom_value_pfp_associative_3_right_bounded) = p))) /\\ (((forall fom_index_pfp_associative_3_result_bounded. (exists fom_gap_pfp_associative_3_result_bounded_index_bound. fom_gap_pfp_associative_3_result_bounded_index_bound + S (fom_index_pfp_associative_3_result_bounded) = Ls) -> exists fom_value_pfp_associative_3_result_bounded. ((((exists fom_beta_height_pfp_associative_3_result_bounded_entry. fom_beta_height_pfp_associative_3_result_bounded_entry + S (fom_value_pfp_associative_3_result_bounded) = S ((S (fom_index_pfp_associative_3_result_bounded)) * sc)) /\\ exists fom_beta_quotient_pfp_associative_3_result_bounded_entry. sb = fom_beta_quotient_pfp_associative_3_result_bounded_entry * S ((S (fom_index_pfp_associative_3_result_bounded)) * sc) + (fom_value_pfp_associative_3_result_bounded))) /\\ (exists fom_gap_pfp_associative_3_result_bounded_value_bound. fom_gap_pfp_associative_3_result_bounded_value_bound + S (fom_value_pfp_associative_3_result_bounded) = p))) /\\ ((exists pfaa_left_b_associative_3 pfaa_left_c_associative_3 pfaa_right_b_associative_3 pfaa_right_c_associative_3 pfaa_sum_b_associative_3 pfaa_sum_c_associative_3 pfaa_length_associative_3. ((((forall pfrep_power_associative_3_witness_common_left pfrep_left_associative_3_witness_common_left pfrep_right_associative_3_witness_common_left. ((exists pfrep_position_associative_3_witness_common_leftfirst. ((pfrep_position_associative_3_witness_common_leftfirst+S (pfrep_power_associative_3_witness_common_left)=(La)) /\\ ((((exists ff_h_pfp_associative_3_witness_common_leftfirstentry. ff_h_pfp_associative_3_witness_common_leftfirstentry + S (pfrep_left_associative_3_witness_common_left) = S ((S (pfrep_position_associative_3_witness_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_associative_3_witness_common_leftfirstentry. ab = ff_q_pfp_associative_3_witness_common_leftfirstentry * S ((S (pfrep_position_associative_3_witness_common_leftfirst)) * ac) + (pfrep_left_associative_3_witness_common_left)))))) \\/ (((exists pfrep_gap_associative_3_witness_common_leftfirstoutside. pfrep_gap_associative_3_witness_common_leftfirstoutside+(La)=(pfrep_power_associative_3_witness_common_left)) /\\ (((pfrep_left_associative_3_witness_common_left)=0))))) -> ((exists pfrep_position_associative_3_witness_common_leftsecond. ((pfrep_position_associative_3_witness_common_leftsecond+S (pfrep_power_associative_3_witness_common_left)=(pfaa_length_associative_3)) /\\ ((((exists ff_h_pfp_associative_3_witness_common_leftsecondentry. ff_h_pfp_associative_3_witness_common_leftsecondentry + S (pfrep_right_associative_3_witness_common_left) = S ((S (pfrep_position_associative_3_witness_common_leftsecond)) * pfaa_left_c_associative_3)) /\\ exists ff_q_pfp_associative_3_witness_common_leftsecondentry. pfaa_left_b_associative_3 = ff_q_pfp_associative_3_witness_common_leftsecondentry * S ((S (pfrep_position_associative_3_witness_common_leftsecond)) * pfaa_left_c_associative_3) + (pfrep_right_associative_3_witness_common_left)))))) \\/ (((exists pfrep_gap_associative_3_witness_common_leftsecondoutside. pfrep_gap_associative_3_witness_common_leftsecondoutside+(pfaa_length_associative_3)=(pfrep_power_associative_3_witness_common_left)) /\\ (((pfrep_right_associative_3_witness_common_left)=0))))) -> pfrep_left_associative_3_witness_common_left=pfrep_right_associative_3_witness_common_left) /\\ ((forall pfrep_power_associative_3_witness_common_right pfrep_left_associative_3_witness_common_right pfrep_right_associative_3_witness_common_right. ((exists pfrep_position_associative_3_witness_common_rightfirst. ((pfrep_position_associative_3_witness_common_rightfirst+S (pfrep_power_associative_3_witness_common_right)=(Lv)) /\\ ((((exists ff_h_pfp_associative_3_witness_common_rightfirstentry. ff_h_pfp_associative_3_witness_common_rightfirstentry + S (pfrep_left_associative_3_witness_common_right) = S ((S (pfrep_position_associative_3_witness_common_rightfirst)) * vc)) /\\ exists ff_q_pfp_associative_3_witness_common_rightfirstentry. vb = ff_q_pfp_associative_3_witness_common_rightfirstentry * S ((S (pfrep_position_associative_3_witness_common_rightfirst)) * vc) + (pfrep_left_associative_3_witness_common_right)))))) \\/ (((exists pfrep_gap_associative_3_witness_common_rightfirstoutside. pfrep_gap_associative_3_witness_common_rightfirstoutside+(Lv)=(pfrep_power_associative_3_witness_common_right)) /\\ (((pfrep_left_associative_3_witness_common_right)=0))))) -> ((exists pfrep_position_associative_3_witness_common_rightsecond. ((pfrep_position_associative_3_witness_common_rightsecond+S (pfrep_power_associative_3_witness_common_right)=(pfaa_length_associative_3)) /\\ ((((exists ff_h_pfp_associative_3_witness_common_rightsecondentry. ff_h_pfp_associative_3_witness_common_rightsecondentry + S (pfrep_right_associative_3_witness_common_right) = S ((S (pfrep_position_associative_3_witness_common_rightsecond)) * pfaa_right_c_associative_3)) /\\ exists ff_q_pfp_associative_3_witness_common_rightsecondentry. pfaa_right_b_associative_3 = ff_q_pfp_associative_3_witness_common_rightsecondentry * S ((S (pfrep_position_associative_3_witness_common_rightsecond)) * pfaa_right_c_associative_3) + (pfrep_right_associative_3_witness_common_right)))))) \\/ (((exists pfrep_gap_associative_3_witness_common_rightsecondoutside. pfrep_gap_associative_3_witness_common_rightsecondoutside+(pfaa_length_associative_3)=(pfrep_power_associative_3_witness_common_right)) /\\ (((pfrep_right_associative_3_witness_common_right)=0))))) -> pfrep_left_associative_3_witness_common_right=pfrep_right_associative_3_witness_common_right)))) /\\ (((forall pfp_index_associative_3_witness_operation. (exists pfa_gap_associative_3_witness_operationindex. pfa_gap_associative_3_witness_operationindex + S (pfp_index_associative_3_witness_operation) = (pfaa_length_associative_3)) -> exists pfp_left_associative_3_witness_operation pfp_right_associative_3_witness_operation pfp_value_associative_3_witness_operation. ((((exists ff_h_pfp_associative_3_witness_operationleft. ff_h_pfp_associative_3_witness_operationleft + S (pfp_left_associative_3_witness_operation) = S ((S (pfp_index_associative_3_witness_operation)) * pfaa_left_c_associative_3)) /\\ exists ff_q_pfp_associative_3_witness_operationleft. pfaa_left_b_associative_3 = ff_q_pfp_associative_3_witness_operationleft * S ((S (pfp_index_associative_3_witness_operation)) * pfaa_left_c_associative_3) + (pfp_left_associative_3_witness_operation))) /\\ (((((exists ff_h_pfp_associative_3_witness_operationright. ff_h_pfp_associative_3_witness_operationright + S (pfp_right_associative_3_witness_operation) = S ((S (pfp_index_associative_3_witness_operation)) * pfaa_right_c_associative_3)) /\\ exists ff_q_pfp_associative_3_witness_operationright. pfaa_right_b_associative_3 = ff_q_pfp_associative_3_witness_operationright * S ((S (pfp_index_associative_3_witness_operation)) * pfaa_right_c_associative_3) + (pfp_right_associative_3_witness_operation))) /\\ (((((exists ff_h_pfp_associative_3_witness_operationtarget. ff_h_pfp_associative_3_witness_operationtarget + S (pfp_value_associative_3_witness_operation) = S ((S (pfp_index_associative_3_witness_operation)) * pfaa_sum_c_associative_3)) /\\ exists ff_q_pfp_associative_3_witness_operationtarget. pfaa_sum_b_associative_3 = ff_q_pfp_associative_3_witness_operationtarget * S ((S (pfp_index_associative_3_witness_operation)) * pfaa_sum_c_associative_3) + (pfp_value_associative_3_witness_operation))) /\\ ((((exists pfa_gap_associative_3_witness_operationoperationleft. pfa_gap_associative_3_witness_operationoperationleft + S (pfp_left_associative_3_witness_operation) = (p)) /\\ (((exists pfa_gap_associative_3_witness_operationoperationright. pfa_gap_associative_3_witness_operationoperationright + S (pfp_right_associative_3_witness_operation) = (p)) /\\ ((((exists pfa_gap_associative_3_witness_operationoperationresultbound. pfa_gap_associative_3_witness_operationoperationresultbound + S (pfp_value_associative_3_witness_operation) = (p)) /\\ ((exists pfa_offset_left_associative_3_witness_operationoperationresultcongruence pfa_offset_right_associative_3_witness_operationoperationresultcongruence. ((pfp_left_associative_3_witness_operation) + (pfp_right_associative_3_witness_operation)) + (p) * pfa_offset_left_associative_3_witness_operationoperationresultcongruence = (pfp_value_associative_3_witness_operation) + (p) * pfa_offset_right_associative_3_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_associative_3_witness_output pfrep_left_associative_3_witness_output pfrep_right_associative_3_witness_output. ((exists pfrep_position_associative_3_witness_outputfirst. ((pfrep_position_associative_3_witness_outputfirst+S (pfrep_power_associative_3_witness_output)=(pfaa_length_associative_3)) /\\ ((((exists ff_h_pfp_associative_3_witness_outputfirstentry. ff_h_pfp_associative_3_witness_outputfirstentry + S (pfrep_left_associative_3_witness_output) = S ((S (pfrep_position_associative_3_witness_outputfirst)) * pfaa_sum_c_associative_3)) /\\ exists ff_q_pfp_associative_3_witness_outputfirstentry. pfaa_sum_b_associative_3 = ff_q_pfp_associative_3_witness_outputfirstentry * S ((S (pfrep_position_associative_3_witness_outputfirst)) * pfaa_sum_c_associative_3) + (pfrep_left_associative_3_witness_output)))))) \\/ (((exists pfrep_gap_associative_3_witness_outputfirstoutside. pfrep_gap_associative_3_witness_outputfirstoutside+(pfaa_length_associative_3)=(pfrep_power_associative_3_witness_output)) /\\ (((pfrep_left_associative_3_witness_output)=0))))) -> ((exists pfrep_position_associative_3_witness_outputsecond. ((pfrep_position_associative_3_witness_outputsecond+S (pfrep_power_associative_3_witness_output)=(Ls)) /\\ ((((exists ff_h_pfp_associative_3_witness_outputsecondentry. ff_h_pfp_associative_3_witness_outputsecondentry + S (pfrep_right_associative_3_witness_output) = S ((S (pfrep_position_associative_3_witness_outputsecond)) * sc)) /\\ exists ff_q_pfp_associative_3_witness_outputsecondentry. sb = ff_q_pfp_associative_3_witness_outputsecondentry * S ((S (pfrep_position_associative_3_witness_outputsecond)) * sc) + (pfrep_right_associative_3_witness_output)))))) \\/ (((exists pfrep_gap_associative_3_witness_outputsecondoutside. pfrep_gap_associative_3_witness_outputsecondoutside+(Ls)=(pfrep_power_associative_3_witness_output)) /\\ (((pfrep_right_associative_3_witness_output)=0))))) -> pfrep_left_associative_3_witness_output=pfrep_right_associative_3_witness_output))))))))))))) -> (forall pfrep_power_associative_result pfrep_left_associative_result pfrep_right_associative_result. ((exists pfrep_position_associative_resultfirst. ((pfrep_position_associative_resultfirst+S (pfrep_power_associative_result)=(Lr)) /\\ ((((exists ff_h_pfp_associative_resultfirstentry. ff_h_pfp_associative_resultfirstentry + S (pfrep_left_associative_result) = S ((S (pfrep_position_associative_resultfirst)) * rc)) /\\ exists ff_q_pfp_associative_resultfirstentry. rb = ff_q_pfp_associative_resultfirstentry * S ((S (pfrep_position_associative_resultfirst)) * rc) + (pfrep_left_associative_result)))))) \\/ (((exists pfrep_gap_associative_resultfirstoutside. pfrep_gap_associative_resultfirstoutside+(Lr)=(pfrep_power_associative_result)) /\\ (((pfrep_left_associative_result)=0))))) -> ((exists pfrep_position_associative_resultsecond. ((pfrep_position_associative_resultsecond+S (pfrep_power_associative_result)=(Ls)) /\\ ((((exists ff_h_pfp_associative_resultsecondentry. ff_h_pfp_associative_resultsecondentry + S (pfrep_right_associative_result) = S ((S (pfrep_position_associative_resultsecond)) * sc)) /\\ exists ff_q_pfp_associative_resultsecondentry. sb = ff_q_pfp_associative_resultsecondentry * S ((S (pfrep_position_associative_resultsecond)) * sc) + (pfrep_right_associative_result)))))) \\/ (((exists pfrep_gap_associative_resultsecondoutside. pfrep_gap_associative_resultsecondoutside+(Ls)=(pfrep_power_associative_result)) /\\ (((pfrep_right_associative_result)=0))))) -> pfrep_left_associative_result=pfrep_right_associative_result)", "statement_sha256": "afe1e2da4cc26be6c5b25def47b9e27bd8ca47a707ea17b1ea99313d9ab72160", "summary": "Both actual bracketings of three independently sized polynomials give formally equivalent outputs; all seven comparison prefixes and all four coefficient operations are genuinely 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(~((p) = 1) /\\ forall pfa_factor_left_sub_function_prime pfa_factor_right_sub_function_prime. (p) = pfa_factor_left_sub_function_prime * pfa_factor_right_sub_function_prime -> pfa_factor_left_sub_function_prime = 1 \\/ pfa_factor_right_sub_function_prime = 1) -> (((forall fom_index_pfp_sub_function_first_left_bounded. (exists fom_gap_pfp_sub_function_first_left_bounded_index_bound. fom_gap_pfp_sub_function_first_left_bounded_index_bound + S (fom_index_pfp_sub_function_first_left_bounded) = M) -> exists fom_value_pfp_sub_function_first_left_bounded. ((((exists fom_beta_height_pfp_sub_function_first_left_bounded_entry. fom_beta_height_pfp_sub_function_first_left_bounded_entry + S (fom_value_pfp_sub_function_first_left_bounded) = S ((S (fom_index_pfp_sub_function_first_left_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_sub_function_first_left_bounded_entry. bb = fom_beta_quotient_pfp_sub_function_first_left_bounded_entry * S ((S (fom_index_pfp_sub_function_first_left_bounded)) * bc) + (fom_value_pfp_sub_function_first_left_bounded))) /\\ (exists fom_gap_pfp_sub_function_first_left_bounded_value_bound. fom_gap_pfp_sub_function_first_left_bounded_value_bound + S (fom_value_pfp_sub_function_first_left_bounded) = p))) /\\ (((forall fom_index_pfp_sub_function_first_right_bounded. (exists fom_gap_pfp_sub_function_first_right_bounded_index_bound. fom_gap_pfp_sub_function_first_right_bounded_index_bound + S (fom_index_pfp_sub_function_first_right_bounded) = N) -> exists fom_value_pfp_sub_function_first_right_bounded. ((((exists fom_beta_height_pfp_sub_function_first_right_bounded_entry. fom_beta_height_pfp_sub_function_first_right_bounded_entry + S (fom_value_pfp_sub_function_first_right_bounded) = S ((S (fom_index_pfp_sub_function_first_right_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_sub_function_first_right_bounded_entry. rb = fom_beta_quotient_pfp_sub_function_first_right_bounded_entry * S ((S (fom_index_pfp_sub_function_first_right_bounded)) * rc) + (fom_value_pfp_sub_function_first_right_bounded))) /\\ (exists fom_gap_pfp_sub_function_first_right_bounded_value_bound. fom_gap_pfp_sub_function_first_right_bounded_value_bound + S (fom_value_pfp_sub_function_first_right_bounded) = p))) /\\ (((forall fom_index_pfp_sub_function_first_result_bounded. (exists fom_gap_pfp_sub_function_first_result_bounded_index_bound. fom_gap_pfp_sub_function_first_result_bounded_index_bound + S (fom_index_pfp_sub_function_first_result_bounded) = L) -> exists fom_value_pfp_sub_function_first_result_bounded. ((((exists fom_beta_height_pfp_sub_function_first_result_bounded_entry. fom_beta_height_pfp_sub_function_first_result_bounded_entry + S (fom_value_pfp_sub_function_first_result_bounded) = S ((S (fom_index_pfp_sub_function_first_result_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_sub_function_first_result_bounded_entry. ab = fom_beta_quotient_pfp_sub_function_first_result_bounded_entry * S ((S (fom_index_pfp_sub_function_first_result_bounded)) * ac) + (fom_value_pfp_sub_function_first_result_bounded))) /\\ (exists fom_gap_pfp_sub_function_first_result_bounded_value_bound. fom_gap_pfp_sub_function_first_result_bounded_value_bound + S (fom_value_pfp_sub_function_first_result_bounded) = p))) /\\ ((exists pfaa_left_b_sub_function_first pfaa_left_c_sub_function_first pfaa_right_b_sub_function_first pfaa_right_c_sub_function_first pfaa_sum_b_sub_function_first pfaa_sum_c_sub_function_first pfaa_length_sub_function_first. ((((forall pfrep_power_sub_function_first_witness_common_left pfrep_left_sub_function_first_witness_common_left pfrep_right_sub_function_first_witness_common_left. ((exists pfrep_position_sub_function_first_witness_common_leftfirst. ((pfrep_position_sub_function_first_witness_common_leftfirst+S (pfrep_power_sub_function_first_witness_common_left)=(M)) /\\ ((((exists ff_h_pfp_sub_function_first_witness_common_leftfirstentry. ff_h_pfp_sub_function_first_witness_common_leftfirstentry + S (pfrep_left_sub_function_first_witness_common_left) = S ((S (pfrep_position_sub_function_first_witness_common_leftfirst)) * bc)) /\\ exists ff_q_pfp_sub_function_first_witness_common_leftfirstentry. bb = ff_q_pfp_sub_function_first_witness_common_leftfirstentry * S ((S (pfrep_position_sub_function_first_witness_common_leftfirst)) * bc) + (pfrep_left_sub_function_first_witness_common_left)))))) \\/ (((exists pfrep_gap_sub_function_first_witness_common_leftfirstoutside. pfrep_gap_sub_function_first_witness_common_leftfirstoutside+(M)=(pfrep_power_sub_function_first_witness_common_left)) /\\ (((pfrep_left_sub_function_first_witness_common_left)=0))))) -> ((exists pfrep_position_sub_function_first_witness_common_leftsecond. ((pfrep_position_sub_function_first_witness_common_leftsecond+S (pfrep_power_sub_function_first_witness_common_left)=(pfaa_length_sub_function_first)) /\\ ((((exists ff_h_pfp_sub_function_first_witness_common_leftsecondentry. ff_h_pfp_sub_function_first_witness_common_leftsecondentry + S (pfrep_right_sub_function_first_witness_common_left) = S ((S (pfrep_position_sub_function_first_witness_common_leftsecond)) * pfaa_left_c_sub_function_first)) /\\ exists ff_q_pfp_sub_function_first_witness_common_leftsecondentry. pfaa_left_b_sub_function_first = ff_q_pfp_sub_function_first_witness_common_leftsecondentry * S ((S (pfrep_position_sub_function_first_witness_common_leftsecond)) * pfaa_left_c_sub_function_first) + (pfrep_right_sub_function_first_witness_common_left)))))) \\/ (((exists pfrep_gap_sub_function_first_witness_common_leftsecondoutside. pfrep_gap_sub_function_first_witness_common_leftsecondoutside+(pfaa_length_sub_function_first)=(pfrep_power_sub_function_first_witness_common_left)) /\\ (((pfrep_right_sub_function_first_witness_common_left)=0))))) -> pfrep_left_sub_function_first_witness_common_left=pfrep_right_sub_function_first_witness_common_left) /\\ ((forall pfrep_power_sub_function_first_witness_common_right pfrep_left_sub_function_first_witness_common_right pfrep_right_sub_function_first_witness_common_right. ((exists pfrep_position_sub_function_first_witness_common_rightfirst. ((pfrep_position_sub_function_first_witness_common_rightfirst+S (pfrep_power_sub_function_first_witness_common_right)=(N)) /\\ ((((exists ff_h_pfp_sub_function_first_witness_common_rightfirstentry. ff_h_pfp_sub_function_first_witness_common_rightfirstentry + S (pfrep_left_sub_function_first_witness_common_right) = S ((S (pfrep_position_sub_function_first_witness_common_rightfirst)) * rc)) /\\ exists ff_q_pfp_sub_function_first_witness_common_rightfirstentry. rb = ff_q_pfp_sub_function_first_witness_common_rightfirstentry * S ((S (pfrep_position_sub_function_first_witness_common_rightfirst)) * rc) + (pfrep_left_sub_function_first_witness_common_right)))))) \\/ (((exists pfrep_gap_sub_function_first_witness_common_rightfirstoutside. pfrep_gap_sub_function_first_witness_common_rightfirstoutside+(N)=(pfrep_power_sub_function_first_witness_common_right)) /\\ (((pfrep_left_sub_function_first_witness_common_right)=0))))) -> ((exists pfrep_position_sub_function_first_witness_common_rightsecond. ((pfrep_position_sub_function_first_witness_common_rightsecond+S (pfrep_power_sub_function_first_witness_common_right)=(pfaa_length_sub_function_first)) /\\ ((((exists ff_h_pfp_sub_function_first_witness_common_rightsecondentry. ff_h_pfp_sub_function_first_witness_common_rightsecondentry + S (pfrep_right_sub_function_first_witness_common_right) = S ((S (pfrep_position_sub_function_first_witness_common_rightsecond)) * pfaa_right_c_sub_function_first)) /\\ exists ff_q_pfp_sub_function_first_witness_common_rightsecondentry. pfaa_right_b_sub_function_first = ff_q_pfp_sub_function_first_witness_common_rightsecondentry * S ((S (pfrep_position_sub_function_first_witness_common_rightsecond)) * pfaa_right_c_sub_function_first) + (pfrep_right_sub_function_first_witness_common_right)))))) \\/ (((exists pfrep_gap_sub_function_first_witness_common_rightsecondoutside. pfrep_gap_sub_function_first_witness_common_rightsecondoutside+(pfaa_length_sub_function_first)=(pfrep_power_sub_function_first_witness_common_right)) /\\ (((pfrep_right_sub_function_first_witness_common_right)=0))))) -> pfrep_left_sub_function_first_witness_common_right=pfrep_right_sub_function_first_witness_common_right)))) /\\ (((forall pfp_index_sub_function_first_witness_operation. (exists pfa_gap_sub_function_first_witness_operationindex. pfa_gap_sub_function_first_witness_operationindex + S (pfp_index_sub_function_first_witness_operation) = (pfaa_length_sub_function_first)) -> exists pfp_left_sub_function_first_witness_operation pfp_right_sub_function_first_witness_operation pfp_value_sub_function_first_witness_operation. ((((exists ff_h_pfp_sub_function_first_witness_operationleft. ff_h_pfp_sub_function_first_witness_operationleft + S (pfp_left_sub_function_first_witness_operation) = S ((S (pfp_index_sub_function_first_witness_operation)) * pfaa_left_c_sub_function_first)) /\\ exists ff_q_pfp_sub_function_first_witness_operationleft. pfaa_left_b_sub_function_first = ff_q_pfp_sub_function_first_witness_operationleft * S ((S (pfp_index_sub_function_first_witness_operation)) * pfaa_left_c_sub_function_first) + (pfp_left_sub_function_first_witness_operation))) /\\ (((((exists ff_h_pfp_sub_function_first_witness_operationright. ff_h_pfp_sub_function_first_witness_operationright + S (pfp_right_sub_function_first_witness_operation) = S ((S (pfp_index_sub_function_first_witness_operation)) * pfaa_right_c_sub_function_first)) /\\ exists ff_q_pfp_sub_function_first_witness_operationright. pfaa_right_b_sub_function_first = ff_q_pfp_sub_function_first_witness_operationright * S ((S (pfp_index_sub_function_first_witness_operation)) * pfaa_right_c_sub_function_first) + (pfp_right_sub_function_first_witness_operation))) /\\ (((((exists ff_h_pfp_sub_function_first_witness_operationtarget. ff_h_pfp_sub_function_first_witness_operationtarget + S (pfp_value_sub_function_first_witness_operation) = S ((S (pfp_index_sub_function_first_witness_operation)) * pfaa_sum_c_sub_function_first)) /\\ exists ff_q_pfp_sub_function_first_witness_operationtarget. pfaa_sum_b_sub_function_first = ff_q_pfp_sub_function_first_witness_operationtarget * S ((S (pfp_index_sub_function_first_witness_operation)) * pfaa_sum_c_sub_function_first) + (pfp_value_sub_function_first_witness_operation))) /\\ ((((exists pfa_gap_sub_function_first_witness_operationoperationleft. pfa_gap_sub_function_first_witness_operationoperationleft + S (pfp_left_sub_function_first_witness_operation) = (p)) /\\ (((exists pfa_gap_sub_function_first_witness_operationoperationright. pfa_gap_sub_function_first_witness_operationoperationright + S (pfp_right_sub_function_first_witness_operation) = (p)) /\\ ((((exists pfa_gap_sub_function_first_witness_operationoperationresultbound. pfa_gap_sub_function_first_witness_operationoperationresultbound + S (pfp_value_sub_function_first_witness_operation) = (p)) /\\ ((exists pfa_offset_left_sub_function_first_witness_operationoperationresultcongruence pfa_offset_right_sub_function_first_witness_operationoperationresultcongruence. ((pfp_left_sub_function_first_witness_operation) + (pfp_right_sub_function_first_witness_operation)) + (p) * pfa_offset_left_sub_function_first_witness_operationoperationresultcongruence = (pfp_value_sub_function_first_witness_operation) + (p) * pfa_offset_right_sub_function_first_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_sub_function_first_witness_output pfrep_left_sub_function_first_witness_output pfrep_right_sub_function_first_witness_output. ((exists pfrep_position_sub_function_first_witness_outputfirst. ((pfrep_position_sub_function_first_witness_outputfirst+S (pfrep_power_sub_function_first_witness_output)=(pfaa_length_sub_function_first)) /\\ ((((exists ff_h_pfp_sub_function_first_witness_outputfirstentry. ff_h_pfp_sub_function_first_witness_outputfirstentry + S (pfrep_left_sub_function_first_witness_output) = S ((S (pfrep_position_sub_function_first_witness_outputfirst)) * pfaa_sum_c_sub_function_first)) /\\ exists ff_q_pfp_sub_function_first_witness_outputfirstentry. pfaa_sum_b_sub_function_first = ff_q_pfp_sub_function_first_witness_outputfirstentry * S ((S (pfrep_position_sub_function_first_witness_outputfirst)) * pfaa_sum_c_sub_function_first) + (pfrep_left_sub_function_first_witness_output)))))) \\/ (((exists pfrep_gap_sub_function_first_witness_outputfirstoutside. pfrep_gap_sub_function_first_witness_outputfirstoutside+(pfaa_length_sub_function_first)=(pfrep_power_sub_function_first_witness_output)) /\\ (((pfrep_left_sub_function_first_witness_output)=0))))) -> ((exists pfrep_position_sub_function_first_witness_outputsecond. ((pfrep_position_sub_function_first_witness_outputsecond+S (pfrep_power_sub_function_first_witness_output)=(L)) /\\ ((((exists ff_h_pfp_sub_function_first_witness_outputsecondentry. ff_h_pfp_sub_function_first_witness_outputsecondentry + S (pfrep_right_sub_function_first_witness_output) = S ((S (pfrep_position_sub_function_first_witness_outputsecond)) * ac)) /\\ exists ff_q_pfp_sub_function_first_witness_outputsecondentry. ab = ff_q_pfp_sub_function_first_witness_outputsecondentry * S ((S (pfrep_position_sub_function_first_witness_outputsecond)) * ac) + (pfrep_right_sub_function_first_witness_output)))))) \\/ (((exists pfrep_gap_sub_function_first_witness_outputsecondoutside. pfrep_gap_sub_function_first_witness_outputsecondoutside+(L)=(pfrep_power_sub_function_first_witness_output)) /\\ (((pfrep_right_sub_function_first_witness_output)=0))))) -> pfrep_left_sub_function_first_witness_output=pfrep_right_sub_function_first_witness_output))))))))))))) -> (((forall fom_index_pfp_sub_function_second_left_bounded. (exists fom_gap_pfp_sub_function_second_left_bounded_index_bound. fom_gap_pfp_sub_function_second_left_bounded_index_bound + S (fom_index_pfp_sub_function_second_left_bounded) = M) -> exists fom_value_pfp_sub_function_second_left_bounded. ((((exists fom_beta_height_pfp_sub_function_second_left_bounded_entry. fom_beta_height_pfp_sub_function_second_left_bounded_entry + S (fom_value_pfp_sub_function_second_left_bounded) = S ((S (fom_index_pfp_sub_function_second_left_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_sub_function_second_left_bounded_entry. bb = fom_beta_quotient_pfp_sub_function_second_left_bounded_entry * S ((S (fom_index_pfp_sub_function_second_left_bounded)) * bc) + (fom_value_pfp_sub_function_second_left_bounded))) /\\ (exists fom_gap_pfp_sub_function_second_left_bounded_value_bound. fom_gap_pfp_sub_function_second_left_bounded_value_bound + S (fom_value_pfp_sub_function_second_left_bounded) = p))) /\\ (((forall fom_index_pfp_sub_function_second_right_bounded. (exists fom_gap_pfp_sub_function_second_right_bounded_index_bound. fom_gap_pfp_sub_function_second_right_bounded_index_bound + S (fom_index_pfp_sub_function_second_right_bounded) = J) -> exists fom_value_pfp_sub_function_second_right_bounded. ((((exists fom_beta_height_pfp_sub_function_second_right_bounded_entry. fom_beta_height_pfp_sub_function_second_right_bounded_entry + S (fom_value_pfp_sub_function_second_right_bounded) = S ((S (fom_index_pfp_sub_function_second_right_bounded)) * sc)) /\\ exists fom_beta_quotient_pfp_sub_function_second_right_bounded_entry. sb = fom_beta_quotient_pfp_sub_function_second_right_bounded_entry * S ((S (fom_index_pfp_sub_function_second_right_bounded)) * sc) + (fom_value_pfp_sub_function_second_right_bounded))) /\\ (exists fom_gap_pfp_sub_function_second_right_bounded_value_bound. fom_gap_pfp_sub_function_second_right_bounded_value_bound + S (fom_value_pfp_sub_function_second_right_bounded) = p))) /\\ (((forall fom_index_pfp_sub_function_second_result_bounded. (exists fom_gap_pfp_sub_function_second_result_bounded_index_bound. fom_gap_pfp_sub_function_second_result_bounded_index_bound + S (fom_index_pfp_sub_function_second_result_bounded) = L) -> exists fom_value_pfp_sub_function_second_result_bounded. ((((exists fom_beta_height_pfp_sub_function_second_result_bounded_entry. fom_beta_height_pfp_sub_function_second_result_bounded_entry + S (fom_value_pfp_sub_function_second_result_bounded) = S ((S (fom_index_pfp_sub_function_second_result_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_sub_function_second_result_bounded_entry. ab = fom_beta_quotient_pfp_sub_function_second_result_bounded_entry * S ((S (fom_index_pfp_sub_function_second_result_bounded)) * ac) + (fom_value_pfp_sub_function_second_result_bounded))) /\\ (exists fom_gap_pfp_sub_function_second_result_bounded_value_bound. fom_gap_pfp_sub_function_second_result_bounded_value_bound + S (fom_value_pfp_sub_function_second_result_bounded) = p))) /\\ ((exists pfaa_left_b_sub_function_second pfaa_left_c_sub_function_second pfaa_right_b_sub_function_second pfaa_right_c_sub_function_second pfaa_sum_b_sub_function_second pfaa_sum_c_sub_function_second pfaa_length_sub_function_second. ((((forall pfrep_power_sub_function_second_witness_common_left pfrep_left_sub_function_second_witness_common_left pfrep_right_sub_function_second_witness_common_left. ((exists pfrep_position_sub_function_second_witness_common_leftfirst. ((pfrep_position_sub_function_second_witness_common_leftfirst+S (pfrep_power_sub_function_second_witness_common_left)=(M)) /\\ ((((exists ff_h_pfp_sub_function_second_witness_common_leftfirstentry. ff_h_pfp_sub_function_second_witness_common_leftfirstentry + S (pfrep_left_sub_function_second_witness_common_left) = S ((S (pfrep_position_sub_function_second_witness_common_leftfirst)) * bc)) /\\ exists ff_q_pfp_sub_function_second_witness_common_leftfirstentry. bb = ff_q_pfp_sub_function_second_witness_common_leftfirstentry * S ((S (pfrep_position_sub_function_second_witness_common_leftfirst)) * bc) + (pfrep_left_sub_function_second_witness_common_left)))))) \\/ (((exists pfrep_gap_sub_function_second_witness_common_leftfirstoutside. pfrep_gap_sub_function_second_witness_common_leftfirstoutside+(M)=(pfrep_power_sub_function_second_witness_common_left)) /\\ (((pfrep_left_sub_function_second_witness_common_left)=0))))) -> ((exists pfrep_position_sub_function_second_witness_common_leftsecond. ((pfrep_position_sub_function_second_witness_common_leftsecond+S (pfrep_power_sub_function_second_witness_common_left)=(pfaa_length_sub_function_second)) /\\ ((((exists ff_h_pfp_sub_function_second_witness_common_leftsecondentry. ff_h_pfp_sub_function_second_witness_common_leftsecondentry + S (pfrep_right_sub_function_second_witness_common_left) = S ((S (pfrep_position_sub_function_second_witness_common_leftsecond)) * pfaa_left_c_sub_function_second)) /\\ exists ff_q_pfp_sub_function_second_witness_common_leftsecondentry. pfaa_left_b_sub_function_second = ff_q_pfp_sub_function_second_witness_common_leftsecondentry * S ((S (pfrep_position_sub_function_second_witness_common_leftsecond)) * pfaa_left_c_sub_function_second) + (pfrep_right_sub_function_second_witness_common_left)))))) \\/ (((exists pfrep_gap_sub_function_second_witness_common_leftsecondoutside. pfrep_gap_sub_function_second_witness_common_leftsecondoutside+(pfaa_length_sub_function_second)=(pfrep_power_sub_function_second_witness_common_left)) /\\ (((pfrep_right_sub_function_second_witness_common_left)=0))))) -> pfrep_left_sub_function_second_witness_common_left=pfrep_right_sub_function_second_witness_common_left) /\\ ((forall pfrep_power_sub_function_second_witness_common_right pfrep_left_sub_function_second_witness_common_right pfrep_right_sub_function_second_witness_common_right. ((exists pfrep_position_sub_function_second_witness_common_rightfirst. ((pfrep_position_sub_function_second_witness_common_rightfirst+S (pfrep_power_sub_function_second_witness_common_right)=(J)) /\\ ((((exists ff_h_pfp_sub_function_second_witness_common_rightfirstentry. ff_h_pfp_sub_function_second_witness_common_rightfirstentry + S (pfrep_left_sub_function_second_witness_common_right) = S ((S (pfrep_position_sub_function_second_witness_common_rightfirst)) * sc)) /\\ exists ff_q_pfp_sub_function_second_witness_common_rightfirstentry. sb = ff_q_pfp_sub_function_second_witness_common_rightfirstentry * S ((S (pfrep_position_sub_function_second_witness_common_rightfirst)) * sc) + (pfrep_left_sub_function_second_witness_common_right)))))) \\/ (((exists pfrep_gap_sub_function_second_witness_common_rightfirstoutside. pfrep_gap_sub_function_second_witness_common_rightfirstoutside+(J)=(pfrep_power_sub_function_second_witness_common_right)) /\\ (((pfrep_left_sub_function_second_witness_common_right)=0))))) -> ((exists pfrep_position_sub_function_second_witness_common_rightsecond. ((pfrep_position_sub_function_second_witness_common_rightsecond+S (pfrep_power_sub_function_second_witness_common_right)=(pfaa_length_sub_function_second)) /\\ ((((exists ff_h_pfp_sub_function_second_witness_common_rightsecondentry. ff_h_pfp_sub_function_second_witness_common_rightsecondentry + S (pfrep_right_sub_function_second_witness_common_right) = S ((S (pfrep_position_sub_function_second_witness_common_rightsecond)) * pfaa_right_c_sub_function_second)) /\\ exists ff_q_pfp_sub_function_second_witness_common_rightsecondentry. pfaa_right_b_sub_function_second = ff_q_pfp_sub_function_second_witness_common_rightsecondentry * S ((S (pfrep_position_sub_function_second_witness_common_rightsecond)) * pfaa_right_c_sub_function_second) + (pfrep_right_sub_function_second_witness_common_right)))))) \\/ (((exists pfrep_gap_sub_function_second_witness_common_rightsecondoutside. pfrep_gap_sub_function_second_witness_common_rightsecondoutside+(pfaa_length_sub_function_second)=(pfrep_power_sub_function_second_witness_common_right)) /\\ (((pfrep_right_sub_function_second_witness_common_right)=0))))) -> pfrep_left_sub_function_second_witness_common_right=pfrep_right_sub_function_second_witness_common_right)))) /\\ (((forall pfp_index_sub_function_second_witness_operation. (exists pfa_gap_sub_function_second_witness_operationindex. pfa_gap_sub_function_second_witness_operationindex + S (pfp_index_sub_function_second_witness_operation) = (pfaa_length_sub_function_second)) -> exists pfp_left_sub_function_second_witness_operation pfp_right_sub_function_second_witness_operation pfp_value_sub_function_second_witness_operation. ((((exists ff_h_pfp_sub_function_second_witness_operationleft. ff_h_pfp_sub_function_second_witness_operationleft + S (pfp_left_sub_function_second_witness_operation) = S ((S (pfp_index_sub_function_second_witness_operation)) * pfaa_left_c_sub_function_second)) /\\ exists ff_q_pfp_sub_function_second_witness_operationleft. pfaa_left_b_sub_function_second = ff_q_pfp_sub_function_second_witness_operationleft * S ((S (pfp_index_sub_function_second_witness_operation)) * pfaa_left_c_sub_function_second) + (pfp_left_sub_function_second_witness_operation))) /\\ (((((exists ff_h_pfp_sub_function_second_witness_operationright. ff_h_pfp_sub_function_second_witness_operationright + S (pfp_right_sub_function_second_witness_operation) = S ((S (pfp_index_sub_function_second_witness_operation)) * pfaa_right_c_sub_function_second)) /\\ exists ff_q_pfp_sub_function_second_witness_operationright. pfaa_right_b_sub_function_second = ff_q_pfp_sub_function_second_witness_operationright * S ((S (pfp_index_sub_function_second_witness_operation)) * pfaa_right_c_sub_function_second) + (pfp_right_sub_function_second_witness_operation))) /\\ (((((exists ff_h_pfp_sub_function_second_witness_operationtarget. ff_h_pfp_sub_function_second_witness_operationtarget + S (pfp_value_sub_function_second_witness_operation) = S ((S (pfp_index_sub_function_second_witness_operation)) * pfaa_sum_c_sub_function_second)) /\\ exists ff_q_pfp_sub_function_second_witness_operationtarget. pfaa_sum_b_sub_function_second = ff_q_pfp_sub_function_second_witness_operationtarget * S ((S (pfp_index_sub_function_second_witness_operation)) * pfaa_sum_c_sub_function_second) + (pfp_value_sub_function_second_witness_operation))) /\\ ((((exists pfa_gap_sub_function_second_witness_operationoperationleft. pfa_gap_sub_function_second_witness_operationoperationleft + S (pfp_left_sub_function_second_witness_operation) = (p)) /\\ (((exists pfa_gap_sub_function_second_witness_operationoperationright. pfa_gap_sub_function_second_witness_operationoperationright + S (pfp_right_sub_function_second_witness_operation) = (p)) /\\ ((((exists pfa_gap_sub_function_second_witness_operationoperationresultbound. pfa_gap_sub_function_second_witness_operationoperationresultbound + S (pfp_value_sub_function_second_witness_operation) = (p)) /\\ ((exists pfa_offset_left_sub_function_second_witness_operationoperationresultcongruence pfa_offset_right_sub_function_second_witness_operationoperationresultcongruence. ((pfp_left_sub_function_second_witness_operation) + (pfp_right_sub_function_second_witness_operation)) + (p) * pfa_offset_left_sub_function_second_witness_operationoperationresultcongruence = (pfp_value_sub_function_second_witness_operation) + (p) * pfa_offset_right_sub_function_second_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_sub_function_second_witness_output pfrep_left_sub_function_second_witness_output pfrep_right_sub_function_second_witness_output. ((exists pfrep_position_sub_function_second_witness_outputfirst. ((pfrep_position_sub_function_second_witness_outputfirst+S (pfrep_power_sub_function_second_witness_output)=(pfaa_length_sub_function_second)) /\\ ((((exists ff_h_pfp_sub_function_second_witness_outputfirstentry. ff_h_pfp_sub_function_second_witness_outputfirstentry + S (pfrep_left_sub_function_second_witness_output) = S ((S (pfrep_position_sub_function_second_witness_outputfirst)) * pfaa_sum_c_sub_function_second)) /\\ exists ff_q_pfp_sub_function_second_witness_outputfirstentry. pfaa_sum_b_sub_function_second = ff_q_pfp_sub_function_second_witness_outputfirstentry * S ((S (pfrep_position_sub_function_second_witness_outputfirst)) * pfaa_sum_c_sub_function_second) + (pfrep_left_sub_function_second_witness_output)))))) \\/ (((exists pfrep_gap_sub_function_second_witness_outputfirstoutside. pfrep_gap_sub_function_second_witness_outputfirstoutside+(pfaa_length_sub_function_second)=(pfrep_power_sub_function_second_witness_output)) /\\ (((pfrep_left_sub_function_second_witness_output)=0))))) -> ((exists pfrep_position_sub_function_second_witness_outputsecond. ((pfrep_position_sub_function_second_witness_outputsecond+S (pfrep_power_sub_function_second_witness_output)=(L)) /\\ ((((exists ff_h_pfp_sub_function_second_witness_outputsecondentry. ff_h_pfp_sub_function_second_witness_outputsecondentry + S (pfrep_right_sub_function_second_witness_output) = S ((S (pfrep_position_sub_function_second_witness_outputsecond)) * ac)) /\\ exists ff_q_pfp_sub_function_second_witness_outputsecondentry. ab = ff_q_pfp_sub_function_second_witness_outputsecondentry * S ((S (pfrep_position_sub_function_second_witness_outputsecond)) * ac) + (pfrep_right_sub_function_second_witness_output)))))) \\/ (((exists pfrep_gap_sub_function_second_witness_outputsecondoutside. pfrep_gap_sub_function_second_witness_outputsecondoutside+(L)=(pfrep_power_sub_function_second_witness_output)) /\\ (((pfrep_right_sub_function_second_witness_output)=0))))) -> pfrep_left_sub_function_second_witness_output=pfrep_right_sub_function_second_witness_output))))))))))))) -> (forall pfrep_power_sub_function_result pfrep_left_sub_function_result pfrep_right_sub_function_result. ((exists pfrep_position_sub_function_resultfirst. ((pfrep_position_sub_function_resultfirst+S (pfrep_power_sub_function_result)=(N)) /\\ ((((exists ff_h_pfp_sub_function_resultfirstentry. ff_h_pfp_sub_function_resultfirstentry + S (pfrep_left_sub_function_result) = S ((S (pfrep_position_sub_function_resultfirst)) * rc)) /\\ exists ff_q_pfp_sub_function_resultfirstentry. rb = ff_q_pfp_sub_function_resultfirstentry * S ((S (pfrep_position_sub_function_resultfirst)) * rc) + (pfrep_left_sub_function_result)))))) \\/ (((exists pfrep_gap_sub_function_resultfirstoutside. pfrep_gap_sub_function_resultfirstoutside+(N)=(pfrep_power_sub_function_result)) /\\ (((pfrep_left_sub_function_result)=0))))) -> ((exists pfrep_position_sub_function_resultsecond. ((pfrep_position_sub_function_resultsecond+S (pfrep_power_sub_function_result)=(J)) /\\ ((((exists ff_h_pfp_sub_function_resultsecondentry. ff_h_pfp_sub_function_resultsecondentry + S (pfrep_right_sub_function_result) = S ((S (pfrep_position_sub_function_resultsecond)) * sc)) /\\ exists ff_q_pfp_sub_function_resultsecondentry. sb = ff_q_pfp_sub_function_resultsecondentry * S ((S (pfrep_position_sub_function_resultsecond)) * sc) + (pfrep_right_sub_function_result)))))) \\/ (((exists pfrep_gap_sub_function_resultsecondoutside. pfrep_gap_sub_function_resultsecondoutside+(J)=(pfrep_power_sub_function_result)) /\\ (((pfrep_right_sub_function_result)=0))))) -> pfrep_left_sub_function_result=pfrep_right_sub_function_result)", "statement_sha256": "1025f30027f56856f3370a9d951e7ed68e7b83c785a30164ee5a868824667813", "summary": "Actual aligned subtraction has a formally unique result, including unequal lengths and unrelated beta encodings.", "summary_sha256": 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87, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "prime_field_polynomial_add_bounded", "prime_field_polynomial_trim_output_coefficients", "prime_field_polynomial_equivalent_symmetric", "prime_field_polynomial_trim_equivalent", "prime_field_polynomial_power_coefficient_functional" ], "dependencies_sha256": "f2c8cea210b5cf706320c4fb6c14064ab3f1f80daf83b1ee08a4277db64b9252", "empty_context_closure": { "body_proof_depth": 47, "body_proof_nodes": 87, "bundle_campaign": "polynomial-gcd-bezout", "bundle_dependency_edge_count": 1578, "bundle_node_count": 493, "bundle_node_id": 366, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "bundle_root_id": 492, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "closure_kind": "dependency_closed_bundle_node", "digest_kind": 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"artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "b16df2a62306a042fd1e0a1a8b7e9dfbe016218692f080a886aba8edf1242971", "membership": "alpha_only", "name": "prime_field_polynomial_add_trim_aligned", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro pb", "intro pc", "intro N", "intro xb", "intro xc", "intro ub", "intro uc", "intro ab", "intro ac", "intro L", "intro t", "intro rb", "intro rc", "intro R", "intro hproduct", "intro hequivalent", "intro hadd", "intro htrim", "have hbounds : ((forall fom_index_pfp_trim_aligned_left_bound. (exists fom_gap_pfp_trim_aligned_left_bound_index_bound. fom_gap_pfp_trim_aligned_left_bound_index_bound + S (fom_index_pfp_trim_aligned_left_bound) = L) -> exists fom_value_pfp_trim_aligned_left_bound. ((((exists fom_beta_height_pfp_trim_aligned_left_bound_entry. fom_beta_height_pfp_trim_aligned_left_bound_entry + S (fom_value_pfp_trim_aligned_left_bound) = S ((S (fom_index_pfp_trim_aligned_left_bound)) * xc)) /\\ exists fom_beta_quotient_pfp_trim_aligned_left_bound_entry. xb = fom_beta_quotient_pfp_trim_aligned_left_bound_entry * S ((S (fom_index_pfp_trim_aligned_left_bound)) * xc) + (fom_value_pfp_trim_aligned_left_bound))) /\\ (exists fom_gap_pfp_trim_aligned_left_bound_value_bound. fom_gap_pfp_trim_aligned_left_bound_value_bound + S (fom_value_pfp_trim_aligned_left_bound) = p))) /\\ (((forall fom_index_pfp_trim_aligned_right_bound. (exists fom_gap_pfp_trim_aligned_right_bound_index_bound. fom_gap_pfp_trim_aligned_right_bound_index_bound + S (fom_index_pfp_trim_aligned_right_bound) = L) -> exists fom_value_pfp_trim_aligned_right_bound. ((((exists fom_beta_height_pfp_trim_aligned_right_bound_entry. fom_beta_height_pfp_trim_aligned_right_bound_entry + S (fom_value_pfp_trim_aligned_right_bound) = S ((S (fom_index_pfp_trim_aligned_right_bound)) * uc)) /\\ exists fom_beta_quotient_pfp_trim_aligned_right_bound_entry. ub = fom_beta_quotient_pfp_trim_aligned_right_bound_entry * S ((S (fom_index_pfp_trim_aligned_right_bound)) * uc) + (fom_value_pfp_trim_aligned_right_bound))) /\\ (exists fom_gap_pfp_trim_aligned_right_bound_value_bound. fom_gap_pfp_trim_aligned_right_bound_value_bound + S (fom_value_pfp_trim_aligned_right_bound) = p))) /\\ ((forall fom_index_pfp_trim_aligned_sum_bound. (exists fom_gap_pfp_trim_aligned_sum_bound_index_bound. fom_gap_pfp_trim_aligned_sum_bound_index_bound + S (fom_index_pfp_trim_aligned_sum_bound) = L) -> exists fom_value_pfp_trim_aligned_sum_bound. ((((exists fom_beta_height_pfp_trim_aligned_sum_bound_entry. fom_beta_height_pfp_trim_aligned_sum_bound_entry + S (fom_value_pfp_trim_aligned_sum_bound) = S ((S (fom_index_pfp_trim_aligned_sum_bound)) * ac)) /\\ exists fom_beta_quotient_pfp_trim_aligned_sum_bound_entry. ab = fom_beta_quotient_pfp_trim_aligned_sum_bound_entry * S ((S (fom_index_pfp_trim_aligned_sum_bound)) * ac) + (fom_value_pfp_trim_aligned_sum_bound))) /\\ (exists fom_gap_pfp_trim_aligned_sum_bound_value_bound. fom_gap_pfp_trim_aligned_sum_bound_value_bound + S (fom_value_pfp_trim_aligned_sum_bound) = p)))))))", "specialize prime_field_polynomial_add_bounded (p)", "specialize prime_field_polynomial_add_bounded (xb)", "specialize prime_field_polynomial_add_bounded (xc)", "specialize prime_field_polynomial_add_bounded (ub)", "specialize prime_field_polynomial_add_bounded (uc)", "specialize prime_field_polynomial_add_bounded (ab)", "specialize prime_field_polynomial_add_bounded (ac)", "specialize prime_field_polynomial_add_bounded (L)", "apply prime_field_polynomial_add_bounded", "exact hadd", "cases hbounds", "cases hbounds_right", "have hremainder : forall fom_index_pfp_trim_aligned_remainder_bound. (exists fom_gap_pfp_trim_aligned_remainder_bound_index_bound. fom_gap_pfp_trim_aligned_remainder_bound_index_bound + S (fom_index_pfp_trim_aligned_remainder_bound) = R) -> exists fom_value_pfp_trim_aligned_remainder_bound. ((((exists fom_beta_height_pfp_trim_aligned_remainder_bound_entry. fom_beta_height_pfp_trim_aligned_remainder_bound_entry + S (fom_value_pfp_trim_aligned_remainder_bound) = S ((S (fom_index_pfp_trim_aligned_remainder_bound)) * rc)) /\\ exists fom_beta_quotient_pfp_trim_aligned_remainder_bound_entry. rb = fom_beta_quotient_pfp_trim_aligned_remainder_bound_entry * S ((S (fom_index_pfp_trim_aligned_remainder_bound)) * rc) + (fom_value_pfp_trim_aligned_remainder_bound))) /\\ (exists fom_gap_pfp_trim_aligned_remainder_bound_value_bound. fom_gap_pfp_trim_aligned_remainder_bound_value_bound + S (fom_value_pfp_trim_aligned_remainder_bound) = p))", "specialize prime_field_polynomial_trim_output_coefficients (p)", "specialize prime_field_polynomial_trim_output_coefficients (ub)", "specialize prime_field_polynomial_trim_output_coefficients (uc)", "specialize prime_field_polynomial_trim_output_coefficients (L)", "specialize prime_field_polynomial_trim_output_coefficients (t)", "specialize prime_field_polynomial_trim_output_coefficients (rb)", "specialize prime_field_polynomial_trim_output_coefficients (rc)", "specialize prime_field_polynomial_trim_output_coefficients (R)", "apply prime_field_polynomial_trim_output_coefficients", "exact htrim", "have hreverse : forall pfrep_power_trim_aligned_reverse pfrep_left_trim_aligned_reverse pfrep_right_trim_aligned_reverse. ((exists pfrep_position_trim_aligned_reversefirst. ((pfrep_position_trim_aligned_reversefirst+S (pfrep_power_trim_aligned_reverse)=(R)) /\\ ((((exists ff_h_pfp_trim_aligned_reversefirstentry. ff_h_pfp_trim_aligned_reversefirstentry + S (pfrep_left_trim_aligned_reverse) = S ((S (pfrep_position_trim_aligned_reversefirst)) * rc)) /\\ exists ff_q_pfp_trim_aligned_reversefirstentry. rb = ff_q_pfp_trim_aligned_reversefirstentry * S ((S (pfrep_position_trim_aligned_reversefirst)) * rc) + (pfrep_left_trim_aligned_reverse)))))) \\/ (((exists pfrep_gap_trim_aligned_reversefirstoutside. pfrep_gap_trim_aligned_reversefirstoutside+(R)=(pfrep_power_trim_aligned_reverse)) /\\ (((pfrep_left_trim_aligned_reverse)=0))))) -> ((exists pfrep_position_trim_aligned_reversesecond. ((pfrep_position_trim_aligned_reversesecond+S (pfrep_power_trim_aligned_reverse)=(L)) /\\ ((((exists ff_h_pfp_trim_aligned_reversesecondentry. ff_h_pfp_trim_aligned_reversesecondentry + S (pfrep_right_trim_aligned_reverse) = S ((S (pfrep_position_trim_aligned_reversesecond)) * uc)) /\\ exists ff_q_pfp_trim_aligned_reversesecondentry. ub = ff_q_pfp_trim_aligned_reversesecondentry * S ((S (pfrep_position_trim_aligned_reversesecond)) * uc) + (pfrep_right_trim_aligned_reverse)))))) \\/ (((exists pfrep_gap_trim_aligned_reversesecondoutside. pfrep_gap_trim_aligned_reversesecondoutside+(L)=(pfrep_power_trim_aligned_reverse)) /\\ (((pfrep_right_trim_aligned_reverse)=0))))) -> pfrep_left_trim_aligned_reverse=pfrep_right_trim_aligned_reverse", "specialize prime_field_polynomial_equivalent_symmetric (ub)", "specialize prime_field_polynomial_equivalent_symmetric (uc)", "specialize prime_field_polynomial_equivalent_symmetric (L)", "specialize prime_field_polynomial_equivalent_symmetric (rb)", "specialize prime_field_polynomial_equivalent_symmetric (rc)", "specialize prime_field_polynomial_equivalent_symmetric (R)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_trim_equivalent (p)", "specialize prime_field_polynomial_trim_equivalent (ub)", "specialize prime_field_polynomial_trim_equivalent (uc)", "specialize prime_field_polynomial_trim_equivalent (L)", "specialize prime_field_polynomial_trim_equivalent (t)", "specialize prime_field_polynomial_trim_equivalent (rb)", "specialize prime_field_polynomial_trim_equivalent (rc)", "specialize prime_field_polynomial_trim_equivalent (R)", "apply prime_field_polynomial_trim_equivalent", "exact htrim", "split", "exact hproduct", "split", "exact hremainder", "split", "exact hbounds_right_right", "exists xb", "exists xc", "exists ub", "exists uc", "exists ab", "exists ac", "exists L", "split", "split", "exact hequivalent", "exact hreverse", "split", "exact hadd", "specialize prime_field_polynomial_power_coefficient_functional (ab)", "specialize prime_field_polynomial_power_coefficient_functional (ac)", "specialize prime_field_polynomial_power_coefficient_functional (L)", "apply prime_field_polynomial_power_coefficient_functional" ], "script_sha256": "c1fb4cfa7ad2a2aa190a354c7f9d4388d74ac801ed4ff1636b60d87517588782", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_euclidean_identity_candidate.py", "sha256": "8efdcd2abf2143891b79edcb3fc90d7126ae69507c1c631ed33b497172ffdb77" }, "statement": "forall p pb pc N xb xc ub uc ab ac L t rb rc R. (forall fom_index_pfp_trim_aligned_product_bounded. (exists fom_gap_pfp_trim_aligned_product_bounded_index_bound. fom_gap_pfp_trim_aligned_product_bounded_index_bound + S (fom_index_pfp_trim_aligned_product_bounded) = N) -> exists fom_value_pfp_trim_aligned_product_bounded. ((((exists fom_beta_height_pfp_trim_aligned_product_bounded_entry. fom_beta_height_pfp_trim_aligned_product_bounded_entry + S (fom_value_pfp_trim_aligned_product_bounded) = S ((S (fom_index_pfp_trim_aligned_product_bounded)) * pc)) /\\ exists fom_beta_quotient_pfp_trim_aligned_product_bounded_entry. pb = fom_beta_quotient_pfp_trim_aligned_product_bounded_entry * S ((S (fom_index_pfp_trim_aligned_product_bounded)) * pc) + (fom_value_pfp_trim_aligned_product_bounded))) /\\ (exists fom_gap_pfp_trim_aligned_product_bounded_value_bound. fom_gap_pfp_trim_aligned_product_bounded_value_bound + S (fom_value_pfp_trim_aligned_product_bounded) = p))) -> (forall pfrep_power_trim_aligned_product pfrep_left_trim_aligned_product pfrep_right_trim_aligned_product. ((exists pfrep_position_trim_aligned_productfirst. ((pfrep_position_trim_aligned_productfirst+S (pfrep_power_trim_aligned_product)=(N)) /\\ ((((exists ff_h_pfp_trim_aligned_productfirstentry. ff_h_pfp_trim_aligned_productfirstentry + S (pfrep_left_trim_aligned_product) = S ((S (pfrep_position_trim_aligned_productfirst)) * pc)) /\\ exists ff_q_pfp_trim_aligned_productfirstentry. pb = ff_q_pfp_trim_aligned_productfirstentry * S ((S (pfrep_position_trim_aligned_productfirst)) * pc) + (pfrep_left_trim_aligned_product)))))) \\/ (((exists pfrep_gap_trim_aligned_productfirstoutside. pfrep_gap_trim_aligned_productfirstoutside+(N)=(pfrep_power_trim_aligned_product)) /\\ (((pfrep_left_trim_aligned_product)=0))))) -> ((exists pfrep_position_trim_aligned_productsecond. ((pfrep_position_trim_aligned_productsecond+S (pfrep_power_trim_aligned_product)=(L)) /\\ ((((exists ff_h_pfp_trim_aligned_productsecondentry. ff_h_pfp_trim_aligned_productsecondentry + S (pfrep_right_trim_aligned_product) = S ((S (pfrep_position_trim_aligned_productsecond)) * xc)) /\\ exists ff_q_pfp_trim_aligned_productsecondentry. xb = ff_q_pfp_trim_aligned_productsecondentry * S ((S (pfrep_position_trim_aligned_productsecond)) * xc) + (pfrep_right_trim_aligned_product)))))) \\/ (((exists pfrep_gap_trim_aligned_productsecondoutside. pfrep_gap_trim_aligned_productsecondoutside+(L)=(pfrep_power_trim_aligned_product)) /\\ (((pfrep_right_trim_aligned_product)=0))))) -> pfrep_left_trim_aligned_product=pfrep_right_trim_aligned_product) -> (forall pfp_index_trim_aligned_sum. (exists pfa_gap_trim_aligned_sumindex. pfa_gap_trim_aligned_sumindex + S (pfp_index_trim_aligned_sum) = (L)) -> exists pfp_left_trim_aligned_sum pfp_right_trim_aligned_sum pfp_value_trim_aligned_sum. ((((exists ff_h_pfp_trim_aligned_sumleft. ff_h_pfp_trim_aligned_sumleft + S (pfp_left_trim_aligned_sum) = S ((S (pfp_index_trim_aligned_sum)) * xc)) /\\ exists ff_q_pfp_trim_aligned_sumleft. xb = ff_q_pfp_trim_aligned_sumleft * S ((S (pfp_index_trim_aligned_sum)) * xc) + (pfp_left_trim_aligned_sum))) /\\ (((((exists ff_h_pfp_trim_aligned_sumright. ff_h_pfp_trim_aligned_sumright + S (pfp_right_trim_aligned_sum) = S ((S (pfp_index_trim_aligned_sum)) * uc)) /\\ exists ff_q_pfp_trim_aligned_sumright. ub = ff_q_pfp_trim_aligned_sumright * S ((S (pfp_index_trim_aligned_sum)) * uc) + (pfp_right_trim_aligned_sum))) /\\ (((((exists ff_h_pfp_trim_aligned_sumtarget. ff_h_pfp_trim_aligned_sumtarget + S (pfp_value_trim_aligned_sum) = S ((S (pfp_index_trim_aligned_sum)) * ac)) /\\ exists ff_q_pfp_trim_aligned_sumtarget. ab = ff_q_pfp_trim_aligned_sumtarget * S ((S (pfp_index_trim_aligned_sum)) * ac) + (pfp_value_trim_aligned_sum))) /\\ ((((exists pfa_gap_trim_aligned_sumoperationleft. pfa_gap_trim_aligned_sumoperationleft + S (pfp_left_trim_aligned_sum) = (p)) /\\ (((exists pfa_gap_trim_aligned_sumoperationright. pfa_gap_trim_aligned_sumoperationright + S (pfp_right_trim_aligned_sum) = (p)) /\\ ((((exists pfa_gap_trim_aligned_sumoperationresultbound. pfa_gap_trim_aligned_sumoperationresultbound + S (pfp_value_trim_aligned_sum) = (p)) /\\ ((exists pfa_offset_left_trim_aligned_sumoperationresultcongruence pfa_offset_right_trim_aligned_sumoperationresultcongruence. ((pfp_left_trim_aligned_sum) + (pfp_right_trim_aligned_sum)) + (p) * pfa_offset_left_trim_aligned_sumoperationresultcongruence = (pfp_value_trim_aligned_sum) + (p) * pfa_offset_right_trim_aligned_sumoperationresultcongruence)))))))))))))))) -> ((((L)=(t)+(R)) /\\ (((forall fom_index_pfp_trim_aligned_triminput. (exists fom_gap_pfp_trim_aligned_triminput_index_bound. fom_gap_pfp_trim_aligned_triminput_index_bound + S (fom_index_pfp_trim_aligned_triminput) = L) -> exists fom_value_pfp_trim_aligned_triminput. ((((exists fom_beta_height_pfp_trim_aligned_triminput_entry. fom_beta_height_pfp_trim_aligned_triminput_entry + S (fom_value_pfp_trim_aligned_triminput) = S ((S (fom_index_pfp_trim_aligned_triminput)) * uc)) /\\ exists fom_beta_quotient_pfp_trim_aligned_triminput_entry. ub = fom_beta_quotient_pfp_trim_aligned_triminput_entry * S ((S (fom_index_pfp_trim_aligned_triminput)) * uc) + (fom_value_pfp_trim_aligned_triminput))) /\\ (exists fom_gap_pfp_trim_aligned_triminput_value_bound. fom_gap_pfp_trim_aligned_triminput_value_bound + S (fom_value_pfp_trim_aligned_triminput) = p))) /\\ (((forall pfp_repeat_index_trim_aligned_trimremoved. (exists pfa_gap_trim_aligned_trimremovedindex. pfa_gap_trim_aligned_trimremovedindex + S (pfp_repeat_index_trim_aligned_trimremoved) = (t)) -> (((exists ff_h_pfp_trim_aligned_trimremovedentry. ff_h_pfp_trim_aligned_trimremovedentry + S (0) = S ((S (pfp_repeat_index_trim_aligned_trimremoved)) * uc)) /\\ exists ff_q_pfp_trim_aligned_trimremovedentry. ub = ff_q_pfp_trim_aligned_trimremovedentry * S ((S (pfp_repeat_index_trim_aligned_trimremoved)) * uc) + (0)))) /\\ (((forall pftrim_index_trim_aligned_trimsuffix pftrim_value_trim_aligned_trimsuffix. (exists pfa_gap_trim_aligned_trimsuffixbound. pfa_gap_trim_aligned_trimsuffixbound + S (pftrim_index_trim_aligned_trimsuffix) = (R)) -> (((exists ff_h_pfp_trim_aligned_trimsuffixsource. ff_h_pfp_trim_aligned_trimsuffixsource + S (pftrim_value_trim_aligned_trimsuffix) = S ((S ((t)+pftrim_index_trim_aligned_trimsuffix)) * uc)) /\\ exists ff_q_pfp_trim_aligned_trimsuffixsource. ub = ff_q_pfp_trim_aligned_trimsuffixsource * S ((S ((t)+pftrim_index_trim_aligned_trimsuffix)) * uc) + (pftrim_value_trim_aligned_trimsuffix))) -> (((exists ff_h_pfp_trim_aligned_trimsuffixoutput. ff_h_pfp_trim_aligned_trimsuffixoutput + S (pftrim_value_trim_aligned_trimsuffix) = S ((S (pftrim_index_trim_aligned_trimsuffix)) * rc)) /\\ exists ff_q_pfp_trim_aligned_trimsuffixoutput. rb = ff_q_pfp_trim_aligned_trimsuffixoutput * S ((S (pftrim_index_trim_aligned_trimsuffix)) * rc) + (pftrim_value_trim_aligned_trimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_trim_aligned_trimnormal. ((((exists ff_h_pfp_trim_aligned_trimnormalentry. ff_h_pfp_trim_aligned_trimnormalentry + S (pftrim_leading_trim_aligned_trimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_trim_aligned_trimnormalentry. rb = ff_q_pfp_trim_aligned_trimnormalentry * S ((S (0)) * rc) + (pftrim_leading_trim_aligned_trimnormal))) /\\ ((~(pftrim_leading_trim_aligned_trimnormal=0))))))))))))))) -> (((forall fom_index_pfp_trim_aligned_result_left_bounded. (exists fom_gap_pfp_trim_aligned_result_left_bounded_index_bound. fom_gap_pfp_trim_aligned_result_left_bounded_index_bound + S (fom_index_pfp_trim_aligned_result_left_bounded) = N) -> exists fom_value_pfp_trim_aligned_result_left_bounded. ((((exists fom_beta_height_pfp_trim_aligned_result_left_bounded_entry. fom_beta_height_pfp_trim_aligned_result_left_bounded_entry + S (fom_value_pfp_trim_aligned_result_left_bounded) = S ((S (fom_index_pfp_trim_aligned_result_left_bounded)) * pc)) /\\ exists fom_beta_quotient_pfp_trim_aligned_result_left_bounded_entry. pb = fom_beta_quotient_pfp_trim_aligned_result_left_bounded_entry * S ((S (fom_index_pfp_trim_aligned_result_left_bounded)) * pc) + (fom_value_pfp_trim_aligned_result_left_bounded))) /\\ (exists fom_gap_pfp_trim_aligned_result_left_bounded_value_bound. fom_gap_pfp_trim_aligned_result_left_bounded_value_bound + S (fom_value_pfp_trim_aligned_result_left_bounded) = p))) /\\ (((forall fom_index_pfp_trim_aligned_result_right_bounded. (exists fom_gap_pfp_trim_aligned_result_right_bounded_index_bound. fom_gap_pfp_trim_aligned_result_right_bounded_index_bound + S (fom_index_pfp_trim_aligned_result_right_bounded) = R) -> exists fom_value_pfp_trim_aligned_result_right_bounded. ((((exists fom_beta_height_pfp_trim_aligned_result_right_bounded_entry. fom_beta_height_pfp_trim_aligned_result_right_bounded_entry + S (fom_value_pfp_trim_aligned_result_right_bounded) = S ((S (fom_index_pfp_trim_aligned_result_right_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_trim_aligned_result_right_bounded_entry. rb = fom_beta_quotient_pfp_trim_aligned_result_right_bounded_entry * S ((S (fom_index_pfp_trim_aligned_result_right_bounded)) * rc) + (fom_value_pfp_trim_aligned_result_right_bounded))) /\\ (exists fom_gap_pfp_trim_aligned_result_right_bounded_value_bound. fom_gap_pfp_trim_aligned_result_right_bounded_value_bound + S (fom_value_pfp_trim_aligned_result_right_bounded) = p))) /\\ (((forall fom_index_pfp_trim_aligned_result_result_bounded. (exists fom_gap_pfp_trim_aligned_result_result_bounded_index_bound. fom_gap_pfp_trim_aligned_result_result_bounded_index_bound + S (fom_index_pfp_trim_aligned_result_result_bounded) = L) -> exists fom_value_pfp_trim_aligned_result_result_bounded. ((((exists fom_beta_height_pfp_trim_aligned_result_result_bounded_entry. fom_beta_height_pfp_trim_aligned_result_result_bounded_entry + S (fom_value_pfp_trim_aligned_result_result_bounded) = S ((S (fom_index_pfp_trim_aligned_result_result_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_trim_aligned_result_result_bounded_entry. ab = fom_beta_quotient_pfp_trim_aligned_result_result_bounded_entry * S ((S (fom_index_pfp_trim_aligned_result_result_bounded)) * ac) + (fom_value_pfp_trim_aligned_result_result_bounded))) /\\ (exists fom_gap_pfp_trim_aligned_result_result_bounded_value_bound. fom_gap_pfp_trim_aligned_result_result_bounded_value_bound + S (fom_value_pfp_trim_aligned_result_result_bounded) = p))) /\\ ((exists pfaa_left_b_trim_aligned_result pfaa_left_c_trim_aligned_result pfaa_right_b_trim_aligned_result pfaa_right_c_trim_aligned_result pfaa_sum_b_trim_aligned_result pfaa_sum_c_trim_aligned_result pfaa_length_trim_aligned_result. ((((forall pfrep_power_trim_aligned_result_witness_common_left pfrep_left_trim_aligned_result_witness_common_left pfrep_right_trim_aligned_result_witness_common_left. ((exists pfrep_position_trim_aligned_result_witness_common_leftfirst. ((pfrep_position_trim_aligned_result_witness_common_leftfirst+S (pfrep_power_trim_aligned_result_witness_common_left)=(N)) /\\ ((((exists ff_h_pfp_trim_aligned_result_witness_common_leftfirstentry. ff_h_pfp_trim_aligned_result_witness_common_leftfirstentry + S (pfrep_left_trim_aligned_result_witness_common_left) = S ((S (pfrep_position_trim_aligned_result_witness_common_leftfirst)) * pc)) /\\ exists ff_q_pfp_trim_aligned_result_witness_common_leftfirstentry. pb = ff_q_pfp_trim_aligned_result_witness_common_leftfirstentry * S ((S (pfrep_position_trim_aligned_result_witness_common_leftfirst)) * pc) + (pfrep_left_trim_aligned_result_witness_common_left)))))) \\/ (((exists pfrep_gap_trim_aligned_result_witness_common_leftfirstoutside. pfrep_gap_trim_aligned_result_witness_common_leftfirstoutside+(N)=(pfrep_power_trim_aligned_result_witness_common_left)) /\\ (((pfrep_left_trim_aligned_result_witness_common_left)=0))))) -> ((exists pfrep_position_trim_aligned_result_witness_common_leftsecond. ((pfrep_position_trim_aligned_result_witness_common_leftsecond+S (pfrep_power_trim_aligned_result_witness_common_left)=(pfaa_length_trim_aligned_result)) /\\ ((((exists ff_h_pfp_trim_aligned_result_witness_common_leftsecondentry. ff_h_pfp_trim_aligned_result_witness_common_leftsecondentry + S (pfrep_right_trim_aligned_result_witness_common_left) = S ((S (pfrep_position_trim_aligned_result_witness_common_leftsecond)) * pfaa_left_c_trim_aligned_result)) /\\ exists ff_q_pfp_trim_aligned_result_witness_common_leftsecondentry. pfaa_left_b_trim_aligned_result = ff_q_pfp_trim_aligned_result_witness_common_leftsecondentry * S ((S (pfrep_position_trim_aligned_result_witness_common_leftsecond)) * pfaa_left_c_trim_aligned_result) + (pfrep_right_trim_aligned_result_witness_common_left)))))) \\/ (((exists pfrep_gap_trim_aligned_result_witness_common_leftsecondoutside. pfrep_gap_trim_aligned_result_witness_common_leftsecondoutside+(pfaa_length_trim_aligned_result)=(pfrep_power_trim_aligned_result_witness_common_left)) /\\ (((pfrep_right_trim_aligned_result_witness_common_left)=0))))) -> pfrep_left_trim_aligned_result_witness_common_left=pfrep_right_trim_aligned_result_witness_common_left) /\\ ((forall pfrep_power_trim_aligned_result_witness_common_right pfrep_left_trim_aligned_result_witness_common_right pfrep_right_trim_aligned_result_witness_common_right. ((exists pfrep_position_trim_aligned_result_witness_common_rightfirst. ((pfrep_position_trim_aligned_result_witness_common_rightfirst+S (pfrep_power_trim_aligned_result_witness_common_right)=(R)) /\\ ((((exists ff_h_pfp_trim_aligned_result_witness_common_rightfirstentry. ff_h_pfp_trim_aligned_result_witness_common_rightfirstentry + S (pfrep_left_trim_aligned_result_witness_common_right) = S ((S (pfrep_position_trim_aligned_result_witness_common_rightfirst)) * rc)) /\\ exists ff_q_pfp_trim_aligned_result_witness_common_rightfirstentry. rb = ff_q_pfp_trim_aligned_result_witness_common_rightfirstentry * S ((S (pfrep_position_trim_aligned_result_witness_common_rightfirst)) * rc) + (pfrep_left_trim_aligned_result_witness_common_right)))))) \\/ (((exists pfrep_gap_trim_aligned_result_witness_common_rightfirstoutside. pfrep_gap_trim_aligned_result_witness_common_rightfirstoutside+(R)=(pfrep_power_trim_aligned_result_witness_common_right)) /\\ (((pfrep_left_trim_aligned_result_witness_common_right)=0))))) -> ((exists pfrep_position_trim_aligned_result_witness_common_rightsecond. ((pfrep_position_trim_aligned_result_witness_common_rightsecond+S (pfrep_power_trim_aligned_result_witness_common_right)=(pfaa_length_trim_aligned_result)) /\\ ((((exists ff_h_pfp_trim_aligned_result_witness_common_rightsecondentry. ff_h_pfp_trim_aligned_result_witness_common_rightsecondentry + S (pfrep_right_trim_aligned_result_witness_common_right) = S ((S (pfrep_position_trim_aligned_result_witness_common_rightsecond)) * pfaa_right_c_trim_aligned_result)) /\\ exists ff_q_pfp_trim_aligned_result_witness_common_rightsecondentry. pfaa_right_b_trim_aligned_result = ff_q_pfp_trim_aligned_result_witness_common_rightsecondentry * S ((S (pfrep_position_trim_aligned_result_witness_common_rightsecond)) * pfaa_right_c_trim_aligned_result) + (pfrep_right_trim_aligned_result_witness_common_right)))))) \\/ (((exists pfrep_gap_trim_aligned_result_witness_common_rightsecondoutside. pfrep_gap_trim_aligned_result_witness_common_rightsecondoutside+(pfaa_length_trim_aligned_result)=(pfrep_power_trim_aligned_result_witness_common_right)) /\\ (((pfrep_right_trim_aligned_result_witness_common_right)=0))))) -> pfrep_left_trim_aligned_result_witness_common_right=pfrep_right_trim_aligned_result_witness_common_right)))) /\\ (((forall pfp_index_trim_aligned_result_witness_operation. (exists pfa_gap_trim_aligned_result_witness_operationindex. pfa_gap_trim_aligned_result_witness_operationindex + S (pfp_index_trim_aligned_result_witness_operation) = (pfaa_length_trim_aligned_result)) -> exists pfp_left_trim_aligned_result_witness_operation pfp_right_trim_aligned_result_witness_operation pfp_value_trim_aligned_result_witness_operation. ((((exists ff_h_pfp_trim_aligned_result_witness_operationleft. ff_h_pfp_trim_aligned_result_witness_operationleft + S (pfp_left_trim_aligned_result_witness_operation) = S ((S (pfp_index_trim_aligned_result_witness_operation)) * pfaa_left_c_trim_aligned_result)) /\\ exists ff_q_pfp_trim_aligned_result_witness_operationleft. pfaa_left_b_trim_aligned_result = ff_q_pfp_trim_aligned_result_witness_operationleft * S ((S (pfp_index_trim_aligned_result_witness_operation)) * pfaa_left_c_trim_aligned_result) + (pfp_left_trim_aligned_result_witness_operation))) /\\ (((((exists ff_h_pfp_trim_aligned_result_witness_operationright. ff_h_pfp_trim_aligned_result_witness_operationright + S (pfp_right_trim_aligned_result_witness_operation) = S ((S (pfp_index_trim_aligned_result_witness_operation)) * pfaa_right_c_trim_aligned_result)) /\\ exists ff_q_pfp_trim_aligned_result_witness_operationright. pfaa_right_b_trim_aligned_result = ff_q_pfp_trim_aligned_result_witness_operationright * S ((S (pfp_index_trim_aligned_result_witness_operation)) * pfaa_right_c_trim_aligned_result) + (pfp_right_trim_aligned_result_witness_operation))) /\\ (((((exists ff_h_pfp_trim_aligned_result_witness_operationtarget. ff_h_pfp_trim_aligned_result_witness_operationtarget + S (pfp_value_trim_aligned_result_witness_operation) = S ((S (pfp_index_trim_aligned_result_witness_operation)) * pfaa_sum_c_trim_aligned_result)) /\\ exists ff_q_pfp_trim_aligned_result_witness_operationtarget. pfaa_sum_b_trim_aligned_result = ff_q_pfp_trim_aligned_result_witness_operationtarget * S ((S (pfp_index_trim_aligned_result_witness_operation)) * pfaa_sum_c_trim_aligned_result) + (pfp_value_trim_aligned_result_witness_operation))) /\\ ((((exists pfa_gap_trim_aligned_result_witness_operationoperationleft. pfa_gap_trim_aligned_result_witness_operationoperationleft + S (pfp_left_trim_aligned_result_witness_operation) = (p)) /\\ (((exists pfa_gap_trim_aligned_result_witness_operationoperationright. pfa_gap_trim_aligned_result_witness_operationoperationright + S (pfp_right_trim_aligned_result_witness_operation) = (p)) /\\ ((((exists pfa_gap_trim_aligned_result_witness_operationoperationresultbound. pfa_gap_trim_aligned_result_witness_operationoperationresultbound + S (pfp_value_trim_aligned_result_witness_operation) = (p)) /\\ ((exists pfa_offset_left_trim_aligned_result_witness_operationoperationresultcongruence pfa_offset_right_trim_aligned_result_witness_operationoperationresultcongruence. ((pfp_left_trim_aligned_result_witness_operation) + (pfp_right_trim_aligned_result_witness_operation)) + (p) * pfa_offset_left_trim_aligned_result_witness_operationoperationresultcongruence = (pfp_value_trim_aligned_result_witness_operation) + (p) * pfa_offset_right_trim_aligned_result_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_trim_aligned_result_witness_output pfrep_left_trim_aligned_result_witness_output pfrep_right_trim_aligned_result_witness_output. ((exists pfrep_position_trim_aligned_result_witness_outputfirst. ((pfrep_position_trim_aligned_result_witness_outputfirst+S (pfrep_power_trim_aligned_result_witness_output)=(pfaa_length_trim_aligned_result)) /\\ ((((exists ff_h_pfp_trim_aligned_result_witness_outputfirstentry. ff_h_pfp_trim_aligned_result_witness_outputfirstentry + S (pfrep_left_trim_aligned_result_witness_output) = S ((S (pfrep_position_trim_aligned_result_witness_outputfirst)) * pfaa_sum_c_trim_aligned_result)) /\\ exists ff_q_pfp_trim_aligned_result_witness_outputfirstentry. pfaa_sum_b_trim_aligned_result = ff_q_pfp_trim_aligned_result_witness_outputfirstentry * S ((S (pfrep_position_trim_aligned_result_witness_outputfirst)) * pfaa_sum_c_trim_aligned_result) + (pfrep_left_trim_aligned_result_witness_output)))))) \\/ (((exists pfrep_gap_trim_aligned_result_witness_outputfirstoutside. pfrep_gap_trim_aligned_result_witness_outputfirstoutside+(pfaa_length_trim_aligned_result)=(pfrep_power_trim_aligned_result_witness_output)) /\\ (((pfrep_left_trim_aligned_result_witness_output)=0))))) -> ((exists pfrep_position_trim_aligned_result_witness_outputsecond. ((pfrep_position_trim_aligned_result_witness_outputsecond+S (pfrep_power_trim_aligned_result_witness_output)=(L)) /\\ ((((exists ff_h_pfp_trim_aligned_result_witness_outputsecondentry. ff_h_pfp_trim_aligned_result_witness_outputsecondentry + S (pfrep_right_trim_aligned_result_witness_output) = S ((S (pfrep_position_trim_aligned_result_witness_outputsecond)) * ac)) /\\ exists ff_q_pfp_trim_aligned_result_witness_outputsecondentry. ab = ff_q_pfp_trim_aligned_result_witness_outputsecondentry * S ((S (pfrep_position_trim_aligned_result_witness_outputsecond)) * ac) + (pfrep_right_trim_aligned_result_witness_output)))))) \\/ (((exists pfrep_gap_trim_aligned_result_witness_outputsecondoutside. pfrep_gap_trim_aligned_result_witness_outputsecondoutside+(L)=(pfrep_power_trim_aligned_result_witness_output)) /\\ (((pfrep_right_trim_aligned_result_witness_output)=0))))) -> pfrep_left_trim_aligned_result_witness_output=pfrep_right_trim_aligned_result_witness_output)))))))))))))", "statement_sha256": 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"intro q", "intro rb", "intro rc", "intro R", "intro hp", "intro he", "have hidentity : exists pfd_identity_pb_execution_aligned_identity pfd_identity_pc_execution_aligned_identity pfd_identity_ub_execution_aligned_identity pfd_identity_uc_execution_aligned_identity pfd_identity_t_execution_aligned_identity. ((((((q)=0) /\\ ((forall pfp_repeat_index_execution_aligned_identityempty. (exists pfa_gap_execution_aligned_identityemptyindex. pfa_gap_execution_aligned_identityemptyindex + S (pfp_repeat_index_execution_aligned_identityempty) = (L)) -> (((exists ff_h_pfp_execution_aligned_identityemptyentry. ff_h_pfp_execution_aligned_identityemptyentry + S (0) = S ((S (pfp_repeat_index_execution_aligned_identityempty)) * pfd_identity_pc_execution_aligned_identity)) /\\ exists ff_q_pfp_execution_aligned_identityemptyentry. pfd_identity_pb_execution_aligned_identity = ff_q_pfp_execution_aligned_identityemptyentry * S ((S (pfp_repeat_index_execution_aligned_identityempty)) * pfd_identity_pc_execution_aligned_identity) + (0))))))) \\/ (((~((q)=0)) /\\ ((((forall fom_index_pfp_execution_aligned_identityproductleft. (exists fom_gap_pfp_execution_aligned_identityproductleft_index_bound. fom_gap_pfp_execution_aligned_identityproductleft_index_bound + S (fom_index_pfp_execution_aligned_identityproductleft) = q) -> exists fom_value_pfp_execution_aligned_identityproductleft. ((((exists fom_beta_height_pfp_execution_aligned_identityproductleft_entry. fom_beta_height_pfp_execution_aligned_identityproductleft_entry + S (fom_value_pfp_execution_aligned_identityproductleft) = S ((S (fom_index_pfp_execution_aligned_identityproductleft)) * qc)) /\\ exists fom_beta_quotient_pfp_execution_aligned_identityproductleft_entry. qb = fom_beta_quotient_pfp_execution_aligned_identityproductleft_entry * S ((S (fom_index_pfp_execution_aligned_identityproductleft)) * qc) + (fom_value_pfp_execution_aligned_identityproductleft))) /\\ (exists fom_gap_pfp_execution_aligned_identityproductleft_value_bound. fom_gap_pfp_execution_aligned_identityproductleft_value_bound + S (fom_value_pfp_execution_aligned_identityproductleft) = p))) /\\ (((forall fom_index_pfp_execution_aligned_identityproductright. (exists fom_gap_pfp_execution_aligned_identityproductright_index_bound. fom_gap_pfp_execution_aligned_identityproductright_index_bound + S (fom_index_pfp_execution_aligned_identityproductright) = S (d)) -> exists fom_value_pfp_execution_aligned_identityproductright. ((((exists fom_beta_height_pfp_execution_aligned_identityproductright_entry. fom_beta_height_pfp_execution_aligned_identityproductright_entry + S (fom_value_pfp_execution_aligned_identityproductright) = S ((S (fom_index_pfp_execution_aligned_identityproductright)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_aligned_identityproductright_entry. bb = fom_beta_quotient_pfp_execution_aligned_identityproductright_entry * S ((S (fom_index_pfp_execution_aligned_identityproductright)) * bc) + (fom_value_pfp_execution_aligned_identityproductright))) /\\ (exists fom_gap_pfp_execution_aligned_identityproductright_value_bound. fom_gap_pfp_execution_aligned_identityproductright_value_bound + S (fom_value_pfp_execution_aligned_identityproductright) = p))) /\\ (((((((q)=0 \\/ (S (d))=0) /\\ (((L)=0)))) \\/ (((~((q)=0)) /\\ (((~((S (d))=0)) /\\ (((q)+(S (d))=S (L)))))))) /\\ ((forall pfc_index_execution_aligned_identityproductcoefficients. (exists pfa_gap_execution_aligned_identityproductcoefficientsbound. pfa_gap_execution_aligned_identityproductcoefficientsbound + S (pfc_index_execution_aligned_identityproductcoefficients) = (L)) -> exists pfc_value_execution_aligned_identityproductcoefficients. ((((exists ff_h_pfp_execution_aligned_identityproductcoefficientsentry. ff_h_pfp_execution_aligned_identityproductcoefficientsentry + S (pfc_value_execution_aligned_identityproductcoefficients) = S ((S (pfc_index_execution_aligned_identityproductcoefficients)) * pfd_identity_pc_execution_aligned_identity)) /\\ exists ff_q_pfp_execution_aligned_identityproductcoefficientsentry. pfd_identity_pb_execution_aligned_identity = ff_q_pfp_execution_aligned_identityproductcoefficientsentry * S ((S (pfc_index_execution_aligned_identityproductcoefficients)) * pfd_identity_pc_execution_aligned_identity) + (pfc_value_execution_aligned_identityproductcoefficients))) /\\ ((exists pfc_terms_code_execution_aligned_identityproductcoefficientscoefficient pfc_terms_scale_execution_aligned_identityproductcoefficientscoefficient pfc_natural_sum_execution_aligned_identityproductcoefficientscoefficient. ((forall pfc_index_execution_aligned_identityproductcoefficientscoefficientdiagonal. (exists pfa_gap_execution_aligned_identityproductcoefficientscoefficientdiagonalbound. pfa_gap_execution_aligned_identityproductcoefficientscoefficientdiagonalbound + S (pfc_index_execution_aligned_identityproductcoefficientscoefficientdiagonal) = (S (pfc_index_execution_aligned_identityproductcoefficients))) -> exists pfc_value_execution_aligned_identityproductcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_aligned_identityproductcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_aligned_identityproductcoefficientscoefficientdiagonalentry + S (pfc_value_execution_aligned_identityproductcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_aligned_identityproductcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_aligned_identityproductcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_aligned_identityproductcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_aligned_identityproductcoefficientscoefficient = ff_q_pfp_execution_aligned_identityproductcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_aligned_identityproductcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_aligned_identityproductcoefficientscoefficient) + (pfc_value_execution_aligned_identityproductcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_aligned_identityproductcoefficientscoefficientdiagonalterm pfc_left_execution_aligned_identityproductcoefficientscoefficientdiagonalterm pfc_right_execution_aligned_identityproductcoefficientscoefficientdiagonalterm. (((pfc_index_execution_aligned_identityproductcoefficientscoefficientdiagonal)+pfc_complement_execution_aligned_identityproductcoefficientscoefficientdiagonalterm=(pfc_index_execution_aligned_identityproductcoefficients)) /\\ ((((((exists pfa_gap_execution_aligned_identityproductcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_aligned_identityproductcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_aligned_identityproductcoefficientscoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_execution_aligned_identityproductcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_aligned_identityproductcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_aligned_identityproductcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_aligned_identityproductcoefficientscoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_aligned_identityproductcoefficientscoefficientdiagonaltermleftentry. qb = ff_q_pfp_execution_aligned_identityproductcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_aligned_identityproductcoefficientscoefficientdiagonal)) * qc) + (pfc_left_execution_aligned_identityproductcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_aligned_identityproductcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_aligned_identityproductcoefficientscoefficientdiagonaltermleftoutside+(q)=(pfc_index_execution_aligned_identityproductcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_aligned_identityproductcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_aligned_identityproductcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_aligned_identityproductcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_aligned_identityproductcoefficientscoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_aligned_identityproductcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_aligned_identityproductcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_aligned_identityproductcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_aligned_identityproductcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_aligned_identityproductcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_aligned_identityproductcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_aligned_identityproductcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_execution_aligned_identityproductcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_aligned_identityproductcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_aligned_identityproductcoefficientscoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_aligned_identityproductcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_aligned_identityproductcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_aligned_identityproductcoefficientscoefficientdiagonal)=pfc_left_execution_aligned_identityproductcoefficientscoefficientdiagonalterm*pfc_right_execution_aligned_identityproductcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_aligned_identityproductcoefficientscoefficientsum fs_v_pfc_execution_aligned_identityproductcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_start. fs_h_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_aligned_identityproductcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_start. fs_u_pfc_execution_aligned_identityproductcoefficientscoefficientsum = fs_q_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_aligned_identityproductcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_aligned_identityproductcoefficientscoefficient) = S ((S (S (pfc_index_execution_aligned_identityproductcoefficients))) * fs_v_pfc_execution_aligned_identityproductcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_aligned_identityproductcoefficientscoefficientsum = fs_q_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_aligned_identityproductcoefficients))) * fs_v_pfc_execution_aligned_identityproductcoefficientscoefficientsum) + (pfc_natural_sum_execution_aligned_identityproductcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps = S (pfc_index_execution_aligned_identityproductcoefficients)) -> exists fs_a_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps fs_r_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps fs_s_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_aligned_identityproductcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_aligned_identityproductcoefficientscoefficient = fs_q_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_aligned_identityproductcoefficientscoefficient) + (fs_a_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_aligned_identityproductcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_aligned_identityproductcoefficientscoefficientsum = fs_q_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_aligned_identityproductcoefficientscoefficientsum) + (fs_r_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_aligned_identityproductcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_aligned_identityproductcoefficientscoefficientsum = fs_q_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_aligned_identityproductcoefficientscoefficientsum) + (fs_s_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_aligned_identityproductcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_aligned_identityproductcoefficientscoefficientresiduebound. pfa_gap_execution_aligned_identityproductcoefficientscoefficientresiduebound + S (pfc_value_execution_aligned_identityproductcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_aligned_identityproductcoefficientscoefficientresiduecongruence pfa_offset_right_execution_aligned_identityproductcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_aligned_identityproductcoefficientscoefficient) + (p) * pfa_offset_left_execution_aligned_identityproductcoefficientscoefficientresiduecongruence = (pfc_value_execution_aligned_identityproductcoefficients) + (p) * pfa_offset_right_execution_aligned_identityproductcoefficientscoefficientresiduecongruence))))))))))))))))))))))) /\\ (((forall pfp_index_execution_aligned_identityaddition. (exists pfa_gap_execution_aligned_identityadditionindex. pfa_gap_execution_aligned_identityadditionindex + S (pfp_index_execution_aligned_identityaddition) = (L)) -> exists pfp_left_execution_aligned_identityaddition pfp_right_execution_aligned_identityaddition pfp_value_execution_aligned_identityaddition. ((((exists ff_h_pfp_execution_aligned_identityadditionleft. ff_h_pfp_execution_aligned_identityadditionleft + S (pfp_left_execution_aligned_identityaddition) = S ((S (pfp_index_execution_aligned_identityaddition)) * pfd_identity_pc_execution_aligned_identity)) /\\ exists ff_q_pfp_execution_aligned_identityadditionleft. pfd_identity_pb_execution_aligned_identity = ff_q_pfp_execution_aligned_identityadditionleft * S ((S (pfp_index_execution_aligned_identityaddition)) * pfd_identity_pc_execution_aligned_identity) + (pfp_left_execution_aligned_identityaddition))) /\\ (((((exists ff_h_pfp_execution_aligned_identityadditionright. ff_h_pfp_execution_aligned_identityadditionright + S (pfp_right_execution_aligned_identityaddition) = S ((S (pfp_index_execution_aligned_identityaddition)) * pfd_identity_uc_execution_aligned_identity)) /\\ exists ff_q_pfp_execution_aligned_identityadditionright. pfd_identity_ub_execution_aligned_identity = ff_q_pfp_execution_aligned_identityadditionright * S ((S (pfp_index_execution_aligned_identityaddition)) * pfd_identity_uc_execution_aligned_identity) + (pfp_right_execution_aligned_identityaddition))) /\\ (((((exists ff_h_pfp_execution_aligned_identityadditiontarget. ff_h_pfp_execution_aligned_identityadditiontarget + S (pfp_value_execution_aligned_identityaddition) = S ((S (pfp_index_execution_aligned_identityaddition)) * ac)) /\\ exists ff_q_pfp_execution_aligned_identityadditiontarget. ab = ff_q_pfp_execution_aligned_identityadditiontarget * S ((S (pfp_index_execution_aligned_identityaddition)) * ac) + (pfp_value_execution_aligned_identityaddition))) /\\ ((((exists pfa_gap_execution_aligned_identityadditionoperationleft. pfa_gap_execution_aligned_identityadditionoperationleft + S (pfp_left_execution_aligned_identityaddition) = (p)) /\\ (((exists pfa_gap_execution_aligned_identityadditionoperationright. pfa_gap_execution_aligned_identityadditionoperationright + S (pfp_right_execution_aligned_identityaddition) = (p)) /\\ ((((exists pfa_gap_execution_aligned_identityadditionoperationresultbound. pfa_gap_execution_aligned_identityadditionoperationresultbound + S (pfp_value_execution_aligned_identityaddition) = (p)) /\\ ((exists pfa_offset_left_execution_aligned_identityadditionoperationresultcongruence pfa_offset_right_execution_aligned_identityadditionoperationresultcongruence. ((pfp_left_execution_aligned_identityaddition) + (pfp_right_execution_aligned_identityaddition)) + (p) * pfa_offset_left_execution_aligned_identityadditionoperationresultcongruence = (pfp_value_execution_aligned_identityaddition) + (p) * pfa_offset_right_execution_aligned_identityadditionoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_identity_t_execution_aligned_identity)+(R)) /\\ (((forall fom_index_pfp_execution_aligned_identityremainderinput. (exists fom_gap_pfp_execution_aligned_identityremainderinput_index_bound. fom_gap_pfp_execution_aligned_identityremainderinput_index_bound + S (fom_index_pfp_execution_aligned_identityremainderinput) = L) -> exists fom_value_pfp_execution_aligned_identityremainderinput. ((((exists fom_beta_height_pfp_execution_aligned_identityremainderinput_entry. fom_beta_height_pfp_execution_aligned_identityremainderinput_entry + S (fom_value_pfp_execution_aligned_identityremainderinput) = S ((S (fom_index_pfp_execution_aligned_identityremainderinput)) * pfd_identity_uc_execution_aligned_identity)) /\\ exists fom_beta_quotient_pfp_execution_aligned_identityremainderinput_entry. pfd_identity_ub_execution_aligned_identity = fom_beta_quotient_pfp_execution_aligned_identityremainderinput_entry * S ((S (fom_index_pfp_execution_aligned_identityremainderinput)) * pfd_identity_uc_execution_aligned_identity) + (fom_value_pfp_execution_aligned_identityremainderinput))) /\\ (exists fom_gap_pfp_execution_aligned_identityremainderinput_value_bound. fom_gap_pfp_execution_aligned_identityremainderinput_value_bound + S (fom_value_pfp_execution_aligned_identityremainderinput) = p))) /\\ (((forall pfp_repeat_index_execution_aligned_identityremainderremoved. (exists pfa_gap_execution_aligned_identityremainderremovedindex. pfa_gap_execution_aligned_identityremainderremovedindex + S (pfp_repeat_index_execution_aligned_identityremainderremoved) = (pfd_identity_t_execution_aligned_identity)) -> (((exists ff_h_pfp_execution_aligned_identityremainderremovedentry. ff_h_pfp_execution_aligned_identityremainderremovedentry + S (0) = S ((S (pfp_repeat_index_execution_aligned_identityremainderremoved)) * pfd_identity_uc_execution_aligned_identity)) /\\ exists ff_q_pfp_execution_aligned_identityremainderremovedentry. pfd_identity_ub_execution_aligned_identity = ff_q_pfp_execution_aligned_identityremainderremovedentry * S ((S (pfp_repeat_index_execution_aligned_identityremainderremoved)) * pfd_identity_uc_execution_aligned_identity) + (0)))) /\\ (((forall pftrim_index_execution_aligned_identityremaindersuffix pftrim_value_execution_aligned_identityremaindersuffix. (exists pfa_gap_execution_aligned_identityremaindersuffixbound. pfa_gap_execution_aligned_identityremaindersuffixbound + S (pftrim_index_execution_aligned_identityremaindersuffix) = (R)) -> (((exists ff_h_pfp_execution_aligned_identityremaindersuffixsource. ff_h_pfp_execution_aligned_identityremaindersuffixsource + S (pftrim_value_execution_aligned_identityremaindersuffix) = S ((S ((pfd_identity_t_execution_aligned_identity)+pftrim_index_execution_aligned_identityremaindersuffix)) * pfd_identity_uc_execution_aligned_identity)) /\\ exists ff_q_pfp_execution_aligned_identityremaindersuffixsource. pfd_identity_ub_execution_aligned_identity = ff_q_pfp_execution_aligned_identityremaindersuffixsource * S ((S ((pfd_identity_t_execution_aligned_identity)+pftrim_index_execution_aligned_identityremaindersuffix)) * pfd_identity_uc_execution_aligned_identity) + (pftrim_value_execution_aligned_identityremaindersuffix))) -> (((exists ff_h_pfp_execution_aligned_identityremaindersuffixoutput. ff_h_pfp_execution_aligned_identityremaindersuffixoutput + S (pftrim_value_execution_aligned_identityremaindersuffix) = S ((S (pftrim_index_execution_aligned_identityremaindersuffix)) * rc)) /\\ exists ff_q_pfp_execution_aligned_identityremaindersuffixoutput. rb = ff_q_pfp_execution_aligned_identityremaindersuffixoutput * S ((S (pftrim_index_execution_aligned_identityremaindersuffix)) * rc) + (pftrim_value_execution_aligned_identityremaindersuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_execution_aligned_identityremaindernormal. ((((exists ff_h_pfp_execution_aligned_identityremaindernormalentry. ff_h_pfp_execution_aligned_identityremaindernormalentry + S (pftrim_leading_execution_aligned_identityremaindernormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_execution_aligned_identityremaindernormalentry. rb = ff_q_pfp_execution_aligned_identityremaindernormalentry * S ((S (0)) * rc) + (pftrim_leading_execution_aligned_identityremaindernormal))) /\\ ((~(pftrim_leading_execution_aligned_identityremaindernormal=0)))))))))))))))))))", "specialize prime_field_polynomial_division_coefficient_identity (p)", "specialize prime_field_polynomial_division_coefficient_identity (ab)", "specialize prime_field_polynomial_division_coefficient_identity (ac)", "specialize prime_field_polynomial_division_coefficient_identity (L)", "specialize prime_field_polynomial_division_coefficient_identity (bb)", "specialize prime_field_polynomial_division_coefficient_identity (bc)", "specialize prime_field_polynomial_division_coefficient_identity (d)", "specialize prime_field_polynomial_division_coefficient_identity (qb)", "specialize prime_field_polynomial_division_coefficient_identity (qc)", "specialize prime_field_polynomial_division_coefficient_identity (q)", "specialize prime_field_polynomial_division_coefficient_identity (rb)", "specialize prime_field_polynomial_division_coefficient_identity (rc)", "specialize prime_field_polynomial_division_coefficient_identity (R)", "apply prime_field_polynomial_division_coefficient_identity", "exact hp", "exact he", "cases hidentity", "cases hidentity_witness", "cases hidentity_witness_witness", "cases hidentity_witness_witness_witness", "cases hidentity_witness_witness_witness_witness", "cases hidentity_witness_witness_witness_witness_witness", "cases hidentity_witness_witness_witness_witness_witness_right", "cases he", "cases he_right", "cases he_right_right", "have hbounds : ((forall fom_index_pfp_execution_aligned_ambient_bound. (exists fom_gap_pfp_execution_aligned_ambient_bound_index_bound. fom_gap_pfp_execution_aligned_ambient_bound_index_bound + S (fom_index_pfp_execution_aligned_ambient_bound) = L) -> exists fom_value_pfp_execution_aligned_ambient_bound. ((((exists fom_beta_height_pfp_execution_aligned_ambient_bound_entry. fom_beta_height_pfp_execution_aligned_ambient_bound_entry + S (fom_value_pfp_execution_aligned_ambient_bound) = S ((S (fom_index_pfp_execution_aligned_ambient_bound)) * x1)) /\\ exists fom_beta_quotient_pfp_execution_aligned_ambient_bound_entry. x = fom_beta_quotient_pfp_execution_aligned_ambient_bound_entry * S ((S (fom_index_pfp_execution_aligned_ambient_bound)) * x1) + (fom_value_pfp_execution_aligned_ambient_bound))) /\\ (exists fom_gap_pfp_execution_aligned_ambient_bound_value_bound. fom_gap_pfp_execution_aligned_ambient_bound_value_bound + S (fom_value_pfp_execution_aligned_ambient_bound) = p))) /\\ (((forall fom_index_pfp_execution_aligned_residual_bound. (exists fom_gap_pfp_execution_aligned_residual_bound_index_bound. fom_gap_pfp_execution_aligned_residual_bound_index_bound + S (fom_index_pfp_execution_aligned_residual_bound) = L) -> exists fom_value_pfp_execution_aligned_residual_bound. ((((exists fom_beta_height_pfp_execution_aligned_residual_bound_entry. fom_beta_height_pfp_execution_aligned_residual_bound_entry + S (fom_value_pfp_execution_aligned_residual_bound) = S ((S (fom_index_pfp_execution_aligned_residual_bound)) * x3)) /\\ exists fom_beta_quotient_pfp_execution_aligned_residual_bound_entry. x2 = fom_beta_quotient_pfp_execution_aligned_residual_bound_entry * S ((S (fom_index_pfp_execution_aligned_residual_bound)) * x3) + (fom_value_pfp_execution_aligned_residual_bound))) /\\ (exists fom_gap_pfp_execution_aligned_residual_bound_value_bound. fom_gap_pfp_execution_aligned_residual_bound_value_bound + S (fom_value_pfp_execution_aligned_residual_bound) = p))) /\\ ((forall fom_index_pfp_execution_aligned_input_bound. (exists fom_gap_pfp_execution_aligned_input_bound_index_bound. fom_gap_pfp_execution_aligned_input_bound_index_bound + S (fom_index_pfp_execution_aligned_input_bound) = L) -> exists fom_value_pfp_execution_aligned_input_bound. ((((exists fom_beta_height_pfp_execution_aligned_input_bound_entry. fom_beta_height_pfp_execution_aligned_input_bound_entry + S (fom_value_pfp_execution_aligned_input_bound) = S ((S (fom_index_pfp_execution_aligned_input_bound)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_aligned_input_bound_entry. ab = fom_beta_quotient_pfp_execution_aligned_input_bound_entry * S ((S (fom_index_pfp_execution_aligned_input_bound)) * ac) + (fom_value_pfp_execution_aligned_input_bound))) /\\ (exists fom_gap_pfp_execution_aligned_input_bound_value_bound. fom_gap_pfp_execution_aligned_input_bound_value_bound + S (fom_value_pfp_execution_aligned_input_bound) = p)))))))", "specialize prime_field_polynomial_add_bounded (p)", "specialize prime_field_polynomial_add_bounded (x)", "specialize prime_field_polynomial_add_bounded (x1)", "specialize prime_field_polynomial_add_bounded (x2)", "specialize prime_field_polynomial_add_bounded (x3)", "specialize prime_field_polynomial_add_bounded (ab)", "specialize prime_field_polynomial_add_bounded (ac)", "specialize prime_field_polynomial_add_bounded (L)", "apply prime_field_polynomial_add_bounded", "exact hidentity_witness_witness_witness_witness_witness_right_left", "cases hbounds", "cases hbounds_right", "cases hidentity_witness_witness_witness_witness_witness_left", "cases hidentity_witness_witness_witness_witness_witness_left_left", "exists 0", "exists 0", "exists 0", "split", "specialize prime_field_polynomial_convolution_empty (p)", "specialize prime_field_polynomial_convolution_empty (qb)", "specialize prime_field_polynomial_convolution_empty (qc)", "specialize prime_field_polynomial_convolution_empty (q)", "specialize prime_field_polynomial_convolution_empty (bb)", "specialize prime_field_polynomial_convolution_empty (bc)", "specialize prime_field_polynomial_convolution_empty (S d)", "specialize prime_field_polynomial_convolution_empty (0)", "specialize prime_field_polynomial_convolution_empty (0)", "apply prime_field_polynomial_convolution_empty", "rewrite hidentity_witness_witness_witness_witness_witness_left_left_left", "intro empty_i", "intro empty_hi", "exfalso", "specialize lt_not_le (empty_i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact empty_hi", "specialize zero_le (empty_i)", "apply zero_le", "exact he_right_left", "left", "exact hidentity_witness_witness_witness_witness_witness_left_left_left", "specialize prime_field_polynomial_add_trim_aligned (p)", "specialize prime_field_polynomial_add_trim_aligned (0)", "specialize prime_field_polynomial_add_trim_aligned (0)", "specialize prime_field_polynomial_add_trim_aligned (0)", "specialize prime_field_polynomial_add_trim_aligned (x)", "specialize prime_field_polynomial_add_trim_aligned (x1)", "specialize prime_field_polynomial_add_trim_aligned (x2)", "specialize prime_field_polynomial_add_trim_aligned (x3)", "specialize prime_field_polynomial_add_trim_aligned (ab)", "specialize prime_field_polynomial_add_trim_aligned (ac)", "specialize prime_field_polynomial_add_trim_aligned (L)", "specialize prime_field_polynomial_add_trim_aligned (x4)", "specialize prime_field_polynomial_add_trim_aligned (rb)", "specialize prime_field_polynomial_add_trim_aligned (rc)", "specialize prime_field_polynomial_add_trim_aligned (R)", "apply prime_field_polynomial_add_trim_aligned", "intro zero_i", "intro zero_hi", "exfalso", "specialize lt_not_le (zero_i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact zero_hi", "specialize zero_le (zero_i)", "apply zero_le", "specialize prime_field_polynomial_equivalent_symmetric (x)", "specialize prime_field_polynomial_equivalent_symmetric (x1)", "specialize prime_field_polynomial_equivalent_symmetric (L)", "specialize prime_field_polynomial_equivalent_symmetric (0)", "specialize prime_field_polynomial_equivalent_symmetric (0)", "specialize prime_field_polynomial_equivalent_symmetric (0)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (x)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (x1)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (L)", "apply prime_field_polynomial_zero_prefix_equivalent_empty", "exact hidentity_witness_witness_witness_witness_witness_left_left_right", "exact hidentity_witness_witness_witness_witness_witness_right_left", "exact hidentity_witness_witness_witness_witness_witness_right_right", "cases hidentity_witness_witness_witness_witness_witness_left_right", "exists x", "exists x1", "exists L", "split", "exact hidentity_witness_witness_witness_witness_witness_left_right_right", "specialize prime_field_polynomial_add_trim_aligned (p)", "specialize prime_field_polynomial_add_trim_aligned (x)", "specialize prime_field_polynomial_add_trim_aligned (x1)", "specialize prime_field_polynomial_add_trim_aligned (L)", "specialize prime_field_polynomial_add_trim_aligned (x)", "specialize prime_field_polynomial_add_trim_aligned (x1)", "specialize prime_field_polynomial_add_trim_aligned (x2)", "specialize prime_field_polynomial_add_trim_aligned (x3)", "specialize prime_field_polynomial_add_trim_aligned (ab)", "specialize prime_field_polynomial_add_trim_aligned (ac)", "specialize prime_field_polynomial_add_trim_aligned (L)", "specialize prime_field_polynomial_add_trim_aligned (x4)", "specialize prime_field_polynomial_add_trim_aligned (rb)", "specialize prime_field_polynomial_add_trim_aligned (rc)", "specialize prime_field_polynomial_add_trim_aligned (R)", "apply prime_field_polynomial_add_trim_aligned", "exact hbounds_left", "specialize prime_field_polynomial_power_coefficient_functional (x)", "specialize prime_field_polynomial_power_coefficient_functional (x1)", "specialize prime_field_polynomial_power_coefficient_functional (L)", "apply prime_field_polynomial_power_coefficient_functional", "exact hidentity_witness_witness_witness_witness_witness_right_left", "exact hidentity_witness_witness_witness_witness_witness_right_right" ], "script_sha256": "19c027a3950e190682cd192022b5911072b9bc0ca1aa75fc1ef5866a2f075616", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_euclidean_identity_candidate.py", "sha256": "8efdcd2abf2143891b79edcb3fc90d7126ae69507c1c631ed33b497172ffdb77" }, "statement": "forall p ab ac L bb bc d qb qc q rb rc R. (~((p) = 1) /\\ forall pfa_factor_left_execution_aligned_prime pfa_factor_right_execution_aligned_prime. (p) = pfa_factor_left_execution_aligned_prime * pfa_factor_right_execution_aligned_prime -> pfa_factor_left_execution_aligned_prime = 1 \\/ pfa_factor_right_execution_aligned_prime = 1) -> (((forall fom_index_pfp_execution_aligned_sourceinput. (exists fom_gap_pfp_execution_aligned_sourceinput_index_bound. fom_gap_pfp_execution_aligned_sourceinput_index_bound + S (fom_index_pfp_execution_aligned_sourceinput) = L) -> exists fom_value_pfp_execution_aligned_sourceinput. ((((exists fom_beta_height_pfp_execution_aligned_sourceinput_entry. fom_beta_height_pfp_execution_aligned_sourceinput_entry + S (fom_value_pfp_execution_aligned_sourceinput) = S ((S (fom_index_pfp_execution_aligned_sourceinput)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_aligned_sourceinput_entry. ab = fom_beta_quotient_pfp_execution_aligned_sourceinput_entry * S ((S (fom_index_pfp_execution_aligned_sourceinput)) * ac) + (fom_value_pfp_execution_aligned_sourceinput))) /\\ (exists fom_gap_pfp_execution_aligned_sourceinput_value_bound. fom_gap_pfp_execution_aligned_sourceinput_value_bound + S (fom_value_pfp_execution_aligned_sourceinput) = p))) /\\ (((forall fom_index_pfp_execution_aligned_sourcedivisor. (exists fom_gap_pfp_execution_aligned_sourcedivisor_index_bound. fom_gap_pfp_execution_aligned_sourcedivisor_index_bound + S (fom_index_pfp_execution_aligned_sourcedivisor) = S (d)) -> exists fom_value_pfp_execution_aligned_sourcedivisor. ((((exists fom_beta_height_pfp_execution_aligned_sourcedivisor_entry. fom_beta_height_pfp_execution_aligned_sourcedivisor_entry + S (fom_value_pfp_execution_aligned_sourcedivisor) = S ((S (fom_index_pfp_execution_aligned_sourcedivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_aligned_sourcedivisor_entry. bb = fom_beta_quotient_pfp_execution_aligned_sourcedivisor_entry * S ((S (fom_index_pfp_execution_aligned_sourcedivisor)) * bc) + (fom_value_pfp_execution_aligned_sourcedivisor))) /\\ (exists fom_gap_pfp_execution_aligned_sourcedivisor_value_bound. fom_gap_pfp_execution_aligned_sourcedivisor_value_bound + S (fom_value_pfp_execution_aligned_sourcedivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_execution_aligned_sourcelengthshort. pfc_gap_execution_aligned_sourcelengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((exists pfd_head_execution_aligned_source pfd_inverse_execution_aligned_source pfd_product_code_execution_aligned_source pfd_product_scale_execution_aligned_source pfd_residual_code_execution_aligned_source pfd_residual_scale_execution_aligned_source pfd_cut_execution_aligned_source. ((((exists ff_h_pfp_execution_aligned_sourcehead. ff_h_pfp_execution_aligned_sourcehead + S (pfd_head_execution_aligned_source) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_execution_aligned_sourcehead. bb = ff_q_pfp_execution_aligned_sourcehead * S ((S (0)) * bc) + (pfd_head_execution_aligned_source))) /\\ (((((~((pfd_head_execution_aligned_source) = 0)) /\\ ((((exists pfa_gap_execution_aligned_sourceinversemultiplicationleft. pfa_gap_execution_aligned_sourceinversemultiplicationleft + S (pfd_head_execution_aligned_source) = (p)) /\\ (((exists pfa_gap_execution_aligned_sourceinversemultiplicationright. pfa_gap_execution_aligned_sourceinversemultiplicationright + S (pfd_inverse_execution_aligned_source) = (p)) /\\ ((((exists pfa_gap_execution_aligned_sourceinversemultiplicationresultbound. pfa_gap_execution_aligned_sourceinversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_execution_aligned_sourceinversemultiplicationresultcongruence pfa_offset_right_execution_aligned_sourceinversemultiplicationresultcongruence. ((pfd_head_execution_aligned_source) * (pfd_inverse_execution_aligned_source)) + (p) * pfa_offset_left_execution_aligned_sourceinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_execution_aligned_sourceinversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_execution_aligned_sourcequotient. (exists pfa_gap_execution_aligned_sourcequotientbound. pfa_gap_execution_aligned_sourcequotientbound + S (pfd_index_execution_aligned_sourcequotient) = (q)) -> exists pfd_value_execution_aligned_sourcequotient. ((((exists ff_h_pfp_execution_aligned_sourcequotiententry. ff_h_pfp_execution_aligned_sourcequotiententry + S (pfd_value_execution_aligned_sourcequotient) = S ((S (pfd_index_execution_aligned_sourcequotient)) * qc)) /\\ exists ff_q_pfp_execution_aligned_sourcequotiententry. qb = ff_q_pfp_execution_aligned_sourcequotiententry * S ((S (pfd_index_execution_aligned_sourcequotient)) * qc) + (pfd_value_execution_aligned_sourcequotient))) /\\ ((exists pfd_input_execution_aligned_sourcequotientstep pfd_previous_execution_aligned_sourcequotientstep pfd_difference_execution_aligned_sourcequotientstep. ((((exists ff_h_pfp_execution_aligned_sourcequotientstepinput. ff_h_pfp_execution_aligned_sourcequotientstepinput + S (pfd_input_execution_aligned_sourcequotientstep) = S ((S (pfd_index_execution_aligned_sourcequotient)) * ac)) /\\ exists ff_q_pfp_execution_aligned_sourcequotientstepinput. ab = ff_q_pfp_execution_aligned_sourcequotientstepinput * S ((S (pfd_index_execution_aligned_sourcequotient)) * ac) + (pfd_input_execution_aligned_sourcequotientstep))) /\\ (((exists pfc_terms_code_execution_aligned_sourcequotientstepprevious pfc_terms_scale_execution_aligned_sourcequotientstepprevious pfc_natural_sum_execution_aligned_sourcequotientstepprevious. ((forall pfc_index_execution_aligned_sourcequotientsteppreviousdiagonal. (exists pfa_gap_execution_aligned_sourcequotientsteppreviousdiagonalbound. pfa_gap_execution_aligned_sourcequotientsteppreviousdiagonalbound + S (pfc_index_execution_aligned_sourcequotientsteppreviousdiagonal) = (S (pfd_index_execution_aligned_sourcequotient))) -> exists pfc_value_execution_aligned_sourcequotientsteppreviousdiagonal. ((((exists ff_h_pfp_execution_aligned_sourcequotientsteppreviousdiagonalentry. ff_h_pfp_execution_aligned_sourcequotientsteppreviousdiagonalentry + S (pfc_value_execution_aligned_sourcequotientsteppreviousdiagonal) = S ((S (pfc_index_execution_aligned_sourcequotientsteppreviousdiagonal)) * pfc_terms_scale_execution_aligned_sourcequotientstepprevious)) /\\ exists ff_q_pfp_execution_aligned_sourcequotientsteppreviousdiagonalentry. pfc_terms_code_execution_aligned_sourcequotientstepprevious = ff_q_pfp_execution_aligned_sourcequotientsteppreviousdiagonalentry * S ((S (pfc_index_execution_aligned_sourcequotientsteppreviousdiagonal)) * pfc_terms_scale_execution_aligned_sourcequotientstepprevious) + (pfc_value_execution_aligned_sourcequotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_execution_aligned_sourcequotientsteppreviousdiagonalterm pfc_left_execution_aligned_sourcequotientsteppreviousdiagonalterm pfc_right_execution_aligned_sourcequotientsteppreviousdiagonalterm. (((pfc_index_execution_aligned_sourcequotientsteppreviousdiagonal)+pfc_complement_execution_aligned_sourcequotientsteppreviousdiagonalterm=(pfd_index_execution_aligned_sourcequotient)) /\\ ((((((exists pfa_gap_execution_aligned_sourcequotientsteppreviousdiagonaltermleftinside. pfa_gap_execution_aligned_sourcequotientsteppreviousdiagonaltermleftinside + S (pfc_index_execution_aligned_sourcequotientsteppreviousdiagonal) = (pfd_index_execution_aligned_sourcequotient)) /\\ ((((exists ff_h_pfp_execution_aligned_sourcequotientsteppreviousdiagonaltermleftentry. ff_h_pfp_execution_aligned_sourcequotientsteppreviousdiagonaltermleftentry + S (pfc_left_execution_aligned_sourcequotientsteppreviousdiagonalterm) = S ((S (pfc_index_execution_aligned_sourcequotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_aligned_sourcequotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_execution_aligned_sourcequotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_execution_aligned_sourcequotientsteppreviousdiagonal)) * qc) + (pfc_left_execution_aligned_sourcequotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_aligned_sourcequotientsteppreviousdiagonaltermleftoutside. pfc_gap_execution_aligned_sourcequotientsteppreviousdiagonaltermleftoutside+(pfd_index_execution_aligned_sourcequotient)=(pfc_index_execution_aligned_sourcequotientsteppreviousdiagonal)) /\\ (((pfc_left_execution_aligned_sourcequotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_aligned_sourcequotientsteppreviousdiagonaltermrightinside. pfa_gap_execution_aligned_sourcequotientsteppreviousdiagonaltermrightinside + S (pfc_complement_execution_aligned_sourcequotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_aligned_sourcequotientsteppreviousdiagonaltermrightentry. ff_h_pfp_execution_aligned_sourcequotientsteppreviousdiagonaltermrightentry + S (pfc_right_execution_aligned_sourcequotientsteppreviousdiagonalterm) = S ((S (pfc_complement_execution_aligned_sourcequotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_aligned_sourcequotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_execution_aligned_sourcequotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_execution_aligned_sourcequotientsteppreviousdiagonalterm)) * bc) + (pfc_right_execution_aligned_sourcequotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_aligned_sourcequotientsteppreviousdiagonaltermrightoutside. pfc_gap_execution_aligned_sourcequotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_aligned_sourcequotientsteppreviousdiagonalterm)) /\\ (((pfc_right_execution_aligned_sourcequotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_execution_aligned_sourcequotientsteppreviousdiagonal)=pfc_left_execution_aligned_sourcequotientsteppreviousdiagonalterm*pfc_right_execution_aligned_sourcequotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_aligned_sourcequotientstepprevioussum fs_v_pfc_execution_aligned_sourcequotientstepprevioussum. ((((exists fs_h_pfc_execution_aligned_sourcequotientstepprevioussum_body_start. fs_h_pfc_execution_aligned_sourcequotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_aligned_sourcequotientstepprevioussum)) /\\ exists fs_q_pfc_execution_aligned_sourcequotientstepprevioussum_body_start. fs_u_pfc_execution_aligned_sourcequotientstepprevioussum = fs_q_pfc_execution_aligned_sourcequotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_execution_aligned_sourcequotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_execution_aligned_sourcequotientstepprevioussum_body_terminal. fs_h_pfc_execution_aligned_sourcequotientstepprevioussum_body_terminal + S (pfc_natural_sum_execution_aligned_sourcequotientstepprevious) = S ((S (S (pfd_index_execution_aligned_sourcequotient))) * fs_v_pfc_execution_aligned_sourcequotientstepprevioussum)) /\\ exists fs_q_pfc_execution_aligned_sourcequotientstepprevioussum_body_terminal. fs_u_pfc_execution_aligned_sourcequotientstepprevioussum = fs_q_pfc_execution_aligned_sourcequotientstepprevioussum_body_terminal * S ((S (S (pfd_index_execution_aligned_sourcequotient))) * fs_v_pfc_execution_aligned_sourcequotientstepprevioussum) + (pfc_natural_sum_execution_aligned_sourcequotientstepprevious))) /\\ forall fs_i_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps. (exists fs_lt_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps_bound. fs_lt_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps_bound + S fs_i_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps = S (pfd_index_execution_aligned_sourcequotient)) -> exists fs_a_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps fs_r_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps fs_s_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps. ((((exists fs_h_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps_summand. fs_h_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps_summand + S (fs_a_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_aligned_sourcequotientstepprevious)) /\\ exists fs_q_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps_summand. pfc_terms_code_execution_aligned_sourcequotientstepprevious = fs_q_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_aligned_sourcequotientstepprevious) + (fs_a_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps_partial. fs_h_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps_partial + S (fs_r_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps)) * fs_v_pfc_execution_aligned_sourcequotientstepprevioussum)) /\\ exists fs_q_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps_partial. fs_u_pfc_execution_aligned_sourcequotientstepprevioussum = fs_q_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps)) * fs_v_pfc_execution_aligned_sourcequotientstepprevioussum) + (fs_r_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps_successor. fs_h_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps_successor + S (fs_s_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps)) * fs_v_pfc_execution_aligned_sourcequotientstepprevioussum)) /\\ exists fs_q_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps_successor. fs_u_pfc_execution_aligned_sourcequotientstepprevioussum = fs_q_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps)) * fs_v_pfc_execution_aligned_sourcequotientstepprevioussum) + (fs_s_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps))) /\\ fs_s_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps = fs_r_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps + fs_a_pfc_execution_aligned_sourcequotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_execution_aligned_sourcequotientsteppreviousresiduebound. pfa_gap_execution_aligned_sourcequotientsteppreviousresiduebound + S (pfd_previous_execution_aligned_sourcequotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_aligned_sourcequotientsteppreviousresiduecongruence pfa_offset_right_execution_aligned_sourcequotientsteppreviousresiduecongruence. (pfc_natural_sum_execution_aligned_sourcequotientstepprevious) + (p) * pfa_offset_left_execution_aligned_sourcequotientsteppreviousresiduecongruence = (pfd_previous_execution_aligned_sourcequotientstep) + (p) * pfa_offset_right_execution_aligned_sourcequotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_execution_aligned_sourcequotientstepsubtractleft. pfa_gap_execution_aligned_sourcequotientstepsubtractleft + S (pfd_previous_execution_aligned_sourcequotientstep) = (p)) /\\ (((exists pfa_gap_execution_aligned_sourcequotientstepsubtractright. pfa_gap_execution_aligned_sourcequotientstepsubtractright + S (pfd_difference_execution_aligned_sourcequotientstep) = (p)) /\\ ((((exists pfa_gap_execution_aligned_sourcequotientstepsubtractresultbound. pfa_gap_execution_aligned_sourcequotientstepsubtractresultbound + S (pfd_input_execution_aligned_sourcequotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_aligned_sourcequotientstepsubtractresultcongruence pfa_offset_right_execution_aligned_sourcequotientstepsubtractresultcongruence. ((pfd_previous_execution_aligned_sourcequotientstep) + (pfd_difference_execution_aligned_sourcequotientstep)) + (p) * pfa_offset_left_execution_aligned_sourcequotientstepsubtractresultcongruence = (pfd_input_execution_aligned_sourcequotientstep) + (p) * pfa_offset_right_execution_aligned_sourcequotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_execution_aligned_sourcequotientstepmultiplyleft. pfa_gap_execution_aligned_sourcequotientstepmultiplyleft + S (pfd_inverse_execution_aligned_source) = (p)) /\\ (((exists pfa_gap_execution_aligned_sourcequotientstepmultiplyright. pfa_gap_execution_aligned_sourcequotientstepmultiplyright + S (pfd_difference_execution_aligned_sourcequotientstep) = (p)) /\\ ((((exists pfa_gap_execution_aligned_sourcequotientstepmultiplyresultbound. pfa_gap_execution_aligned_sourcequotientstepmultiplyresultbound + S (pfd_value_execution_aligned_sourcequotient) = (p)) /\\ ((exists pfa_offset_left_execution_aligned_sourcequotientstepmultiplyresultcongruence pfa_offset_right_execution_aligned_sourcequotientstepmultiplyresultcongruence. ((pfd_inverse_execution_aligned_source) * (pfd_difference_execution_aligned_sourcequotientstep)) + (p) * pfa_offset_left_execution_aligned_sourcequotientstepmultiplyresultcongruence = (pfd_value_execution_aligned_sourcequotient) + (p) * pfa_offset_right_execution_aligned_sourcequotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_execution_aligned_sourceproduct. (exists pfa_gap_execution_aligned_sourceproductbound. pfa_gap_execution_aligned_sourceproductbound + S (pfc_index_execution_aligned_sourceproduct) = (L)) -> exists pfc_value_execution_aligned_sourceproduct. ((((exists ff_h_pfp_execution_aligned_sourceproductentry. ff_h_pfp_execution_aligned_sourceproductentry + S (pfc_value_execution_aligned_sourceproduct) = S ((S (pfc_index_execution_aligned_sourceproduct)) * pfd_product_scale_execution_aligned_source)) /\\ exists ff_q_pfp_execution_aligned_sourceproductentry. pfd_product_code_execution_aligned_source = ff_q_pfp_execution_aligned_sourceproductentry * S ((S (pfc_index_execution_aligned_sourceproduct)) * pfd_product_scale_execution_aligned_source) + (pfc_value_execution_aligned_sourceproduct))) /\\ ((exists pfc_terms_code_execution_aligned_sourceproductcoefficient pfc_terms_scale_execution_aligned_sourceproductcoefficient pfc_natural_sum_execution_aligned_sourceproductcoefficient. ((forall pfc_index_execution_aligned_sourceproductcoefficientdiagonal. (exists pfa_gap_execution_aligned_sourceproductcoefficientdiagonalbound. pfa_gap_execution_aligned_sourceproductcoefficientdiagonalbound + S (pfc_index_execution_aligned_sourceproductcoefficientdiagonal) = (S (pfc_index_execution_aligned_sourceproduct))) -> exists pfc_value_execution_aligned_sourceproductcoefficientdiagonal. ((((exists ff_h_pfp_execution_aligned_sourceproductcoefficientdiagonalentry. ff_h_pfp_execution_aligned_sourceproductcoefficientdiagonalentry + S (pfc_value_execution_aligned_sourceproductcoefficientdiagonal) = S ((S (pfc_index_execution_aligned_sourceproductcoefficientdiagonal)) * pfc_terms_scale_execution_aligned_sourceproductcoefficient)) /\\ exists ff_q_pfp_execution_aligned_sourceproductcoefficientdiagonalentry. pfc_terms_code_execution_aligned_sourceproductcoefficient = ff_q_pfp_execution_aligned_sourceproductcoefficientdiagonalentry * S ((S (pfc_index_execution_aligned_sourceproductcoefficientdiagonal)) * pfc_terms_scale_execution_aligned_sourceproductcoefficient) + (pfc_value_execution_aligned_sourceproductcoefficientdiagonal))) /\\ ((exists pfc_complement_execution_aligned_sourceproductcoefficientdiagonalterm pfc_left_execution_aligned_sourceproductcoefficientdiagonalterm pfc_right_execution_aligned_sourceproductcoefficientdiagonalterm. (((pfc_index_execution_aligned_sourceproductcoefficientdiagonal)+pfc_complement_execution_aligned_sourceproductcoefficientdiagonalterm=(pfc_index_execution_aligned_sourceproduct)) /\\ ((((((exists pfa_gap_execution_aligned_sourceproductcoefficientdiagonaltermleftinside. pfa_gap_execution_aligned_sourceproductcoefficientdiagonaltermleftinside + S (pfc_index_execution_aligned_sourceproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_execution_aligned_sourceproductcoefficientdiagonaltermleftentry. ff_h_pfp_execution_aligned_sourceproductcoefficientdiagonaltermleftentry + S (pfc_left_execution_aligned_sourceproductcoefficientdiagonalterm) = S ((S (pfc_index_execution_aligned_sourceproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_aligned_sourceproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_execution_aligned_sourceproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_aligned_sourceproductcoefficientdiagonal)) * qc) + (pfc_left_execution_aligned_sourceproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_aligned_sourceproductcoefficientdiagonaltermleftoutside. pfc_gap_execution_aligned_sourceproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_execution_aligned_sourceproductcoefficientdiagonal)) /\\ (((pfc_left_execution_aligned_sourceproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_aligned_sourceproductcoefficientdiagonaltermrightinside. pfa_gap_execution_aligned_sourceproductcoefficientdiagonaltermrightinside + S (pfc_complement_execution_aligned_sourceproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_aligned_sourceproductcoefficientdiagonaltermrightentry. ff_h_pfp_execution_aligned_sourceproductcoefficientdiagonaltermrightentry + S (pfc_right_execution_aligned_sourceproductcoefficientdiagonalterm) = S ((S (pfc_complement_execution_aligned_sourceproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_aligned_sourceproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_aligned_sourceproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_aligned_sourceproductcoefficientdiagonalterm)) * bc) + (pfc_right_execution_aligned_sourceproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_aligned_sourceproductcoefficientdiagonaltermrightoutside. pfc_gap_execution_aligned_sourceproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_aligned_sourceproductcoefficientdiagonalterm)) /\\ (((pfc_right_execution_aligned_sourceproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_aligned_sourceproductcoefficientdiagonal)=pfc_left_execution_aligned_sourceproductcoefficientdiagonalterm*pfc_right_execution_aligned_sourceproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_aligned_sourceproductcoefficientsum fs_v_pfc_execution_aligned_sourceproductcoefficientsum. ((((exists fs_h_pfc_execution_aligned_sourceproductcoefficientsum_body_start. fs_h_pfc_execution_aligned_sourceproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_aligned_sourceproductcoefficientsum)) /\\ exists fs_q_pfc_execution_aligned_sourceproductcoefficientsum_body_start. fs_u_pfc_execution_aligned_sourceproductcoefficientsum = fs_q_pfc_execution_aligned_sourceproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_aligned_sourceproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_aligned_sourceproductcoefficientsum_body_terminal. fs_h_pfc_execution_aligned_sourceproductcoefficientsum_body_terminal + S (pfc_natural_sum_execution_aligned_sourceproductcoefficient) = S ((S (S (pfc_index_execution_aligned_sourceproduct))) * fs_v_pfc_execution_aligned_sourceproductcoefficientsum)) /\\ exists fs_q_pfc_execution_aligned_sourceproductcoefficientsum_body_terminal. fs_u_pfc_execution_aligned_sourceproductcoefficientsum = fs_q_pfc_execution_aligned_sourceproductcoefficientsum_body_terminal * S ((S (S (pfc_index_execution_aligned_sourceproduct))) * fs_v_pfc_execution_aligned_sourceproductcoefficientsum) + (pfc_natural_sum_execution_aligned_sourceproductcoefficient))) /\\ forall fs_i_pfc_execution_aligned_sourceproductcoefficientsum_body_steps. (exists fs_lt_pfc_execution_aligned_sourceproductcoefficientsum_body_steps_bound. fs_lt_pfc_execution_aligned_sourceproductcoefficientsum_body_steps_bound + S fs_i_pfc_execution_aligned_sourceproductcoefficientsum_body_steps = S (pfc_index_execution_aligned_sourceproduct)) -> exists fs_a_pfc_execution_aligned_sourceproductcoefficientsum_body_steps fs_r_pfc_execution_aligned_sourceproductcoefficientsum_body_steps fs_s_pfc_execution_aligned_sourceproductcoefficientsum_body_steps. ((((exists fs_h_pfc_execution_aligned_sourceproductcoefficientsum_body_steps_summand. fs_h_pfc_execution_aligned_sourceproductcoefficientsum_body_steps_summand + S (fs_a_pfc_execution_aligned_sourceproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_aligned_sourceproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_aligned_sourceproductcoefficient)) /\\ exists fs_q_pfc_execution_aligned_sourceproductcoefficientsum_body_steps_summand. pfc_terms_code_execution_aligned_sourceproductcoefficient = fs_q_pfc_execution_aligned_sourceproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_aligned_sourceproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_aligned_sourceproductcoefficient) + (fs_a_pfc_execution_aligned_sourceproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_aligned_sourceproductcoefficientsum_body_steps_partial. fs_h_pfc_execution_aligned_sourceproductcoefficientsum_body_steps_partial + S (fs_r_pfc_execution_aligned_sourceproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_aligned_sourceproductcoefficientsum_body_steps)) * fs_v_pfc_execution_aligned_sourceproductcoefficientsum)) /\\ exists fs_q_pfc_execution_aligned_sourceproductcoefficientsum_body_steps_partial. fs_u_pfc_execution_aligned_sourceproductcoefficientsum = fs_q_pfc_execution_aligned_sourceproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_aligned_sourceproductcoefficientsum_body_steps)) * fs_v_pfc_execution_aligned_sourceproductcoefficientsum) + (fs_r_pfc_execution_aligned_sourceproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_aligned_sourceproductcoefficientsum_body_steps_successor. fs_h_pfc_execution_aligned_sourceproductcoefficientsum_body_steps_successor + S (fs_s_pfc_execution_aligned_sourceproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_aligned_sourceproductcoefficientsum_body_steps)) * fs_v_pfc_execution_aligned_sourceproductcoefficientsum)) /\\ exists fs_q_pfc_execution_aligned_sourceproductcoefficientsum_body_steps_successor. fs_u_pfc_execution_aligned_sourceproductcoefficientsum = fs_q_pfc_execution_aligned_sourceproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_aligned_sourceproductcoefficientsum_body_steps)) * fs_v_pfc_execution_aligned_sourceproductcoefficientsum) + (fs_s_pfc_execution_aligned_sourceproductcoefficientsum_body_steps))) /\\ fs_s_pfc_execution_aligned_sourceproductcoefficientsum_body_steps = fs_r_pfc_execution_aligned_sourceproductcoefficientsum_body_steps + fs_a_pfc_execution_aligned_sourceproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_aligned_sourceproductcoefficientresiduebound. pfa_gap_execution_aligned_sourceproductcoefficientresiduebound + S (pfc_value_execution_aligned_sourceproduct) = (p)) /\\ ((exists pfa_offset_left_execution_aligned_sourceproductcoefficientresiduecongruence pfa_offset_right_execution_aligned_sourceproductcoefficientresiduecongruence. (pfc_natural_sum_execution_aligned_sourceproductcoefficient) + (p) * pfa_offset_left_execution_aligned_sourceproductcoefficientresiduecongruence = (pfc_value_execution_aligned_sourceproduct) + (p) * pfa_offset_right_execution_aligned_sourceproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_execution_aligned_sourcedifference. (exists pfa_gap_execution_aligned_sourcedifferenceindex. pfa_gap_execution_aligned_sourcedifferenceindex + S (pfs_index_execution_aligned_sourcedifference) = (L)) -> exists pfs_left_execution_aligned_sourcedifference pfs_right_execution_aligned_sourcedifference pfs_result_execution_aligned_sourcedifference. ((((exists ff_h_pfp_execution_aligned_sourcedifferenceleft. ff_h_pfp_execution_aligned_sourcedifferenceleft + S (pfs_left_execution_aligned_sourcedifference) = S ((S (pfs_index_execution_aligned_sourcedifference)) * ac)) /\\ exists ff_q_pfp_execution_aligned_sourcedifferenceleft. ab = ff_q_pfp_execution_aligned_sourcedifferenceleft * S ((S (pfs_index_execution_aligned_sourcedifference)) * ac) + (pfs_left_execution_aligned_sourcedifference))) /\\ (((((exists ff_h_pfp_execution_aligned_sourcedifferenceright. ff_h_pfp_execution_aligned_sourcedifferenceright + S (pfs_right_execution_aligned_sourcedifference) = S ((S (pfs_index_execution_aligned_sourcedifference)) * pfd_product_scale_execution_aligned_source)) /\\ exists ff_q_pfp_execution_aligned_sourcedifferenceright. pfd_product_code_execution_aligned_source = ff_q_pfp_execution_aligned_sourcedifferenceright * S ((S (pfs_index_execution_aligned_sourcedifference)) * pfd_product_scale_execution_aligned_source) + (pfs_right_execution_aligned_sourcedifference))) /\\ (((((exists ff_h_pfp_execution_aligned_sourcedifferenceresult. ff_h_pfp_execution_aligned_sourcedifferenceresult + S (pfs_result_execution_aligned_sourcedifference) = S ((S (pfs_index_execution_aligned_sourcedifference)) * pfd_residual_scale_execution_aligned_source)) /\\ exists ff_q_pfp_execution_aligned_sourcedifferenceresult. pfd_residual_code_execution_aligned_source = ff_q_pfp_execution_aligned_sourcedifferenceresult * S ((S (pfs_index_execution_aligned_sourcedifference)) * pfd_residual_scale_execution_aligned_source) + (pfs_result_execution_aligned_sourcedifference))) /\\ ((((exists pfa_gap_execution_aligned_sourcedifferenceoperationleft. pfa_gap_execution_aligned_sourcedifferenceoperationleft + S (pfs_right_execution_aligned_sourcedifference) = (p)) /\\ (((exists pfa_gap_execution_aligned_sourcedifferenceoperationright. pfa_gap_execution_aligned_sourcedifferenceoperationright + S (pfs_result_execution_aligned_sourcedifference) = (p)) /\\ ((((exists pfa_gap_execution_aligned_sourcedifferenceoperationresultbound. pfa_gap_execution_aligned_sourcedifferenceoperationresultbound + S (pfs_left_execution_aligned_sourcedifference) = (p)) /\\ ((exists pfa_offset_left_execution_aligned_sourcedifferenceoperationresultcongruence pfa_offset_right_execution_aligned_sourcedifferenceoperationresultcongruence. ((pfs_right_execution_aligned_sourcedifference) + (pfs_result_execution_aligned_sourcedifference)) + (p) * pfa_offset_left_execution_aligned_sourcedifferenceoperationresultcongruence = (pfs_left_execution_aligned_sourcedifference) + (p) * pfa_offset_right_execution_aligned_sourcedifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_execution_aligned_source)+(R)) /\\ (((forall fom_index_pfp_execution_aligned_sourcetriminput. (exists fom_gap_pfp_execution_aligned_sourcetriminput_index_bound. fom_gap_pfp_execution_aligned_sourcetriminput_index_bound + S (fom_index_pfp_execution_aligned_sourcetriminput) = L) -> exists fom_value_pfp_execution_aligned_sourcetriminput. ((((exists fom_beta_height_pfp_execution_aligned_sourcetriminput_entry. fom_beta_height_pfp_execution_aligned_sourcetriminput_entry + S (fom_value_pfp_execution_aligned_sourcetriminput) = S ((S (fom_index_pfp_execution_aligned_sourcetriminput)) * pfd_residual_scale_execution_aligned_source)) /\\ exists fom_beta_quotient_pfp_execution_aligned_sourcetriminput_entry. pfd_residual_code_execution_aligned_source = fom_beta_quotient_pfp_execution_aligned_sourcetriminput_entry * S ((S (fom_index_pfp_execution_aligned_sourcetriminput)) * pfd_residual_scale_execution_aligned_source) + (fom_value_pfp_execution_aligned_sourcetriminput))) /\\ (exists fom_gap_pfp_execution_aligned_sourcetriminput_value_bound. fom_gap_pfp_execution_aligned_sourcetriminput_value_bound + S (fom_value_pfp_execution_aligned_sourcetriminput) = p))) /\\ (((forall pfp_repeat_index_execution_aligned_sourcetrimremoved. (exists pfa_gap_execution_aligned_sourcetrimremovedindex. pfa_gap_execution_aligned_sourcetrimremovedindex + S (pfp_repeat_index_execution_aligned_sourcetrimremoved) = (pfd_cut_execution_aligned_source)) -> (((exists ff_h_pfp_execution_aligned_sourcetrimremovedentry. ff_h_pfp_execution_aligned_sourcetrimremovedentry + S (0) = S ((S (pfp_repeat_index_execution_aligned_sourcetrimremoved)) * pfd_residual_scale_execution_aligned_source)) /\\ exists ff_q_pfp_execution_aligned_sourcetrimremovedentry. pfd_residual_code_execution_aligned_source = ff_q_pfp_execution_aligned_sourcetrimremovedentry * S ((S (pfp_repeat_index_execution_aligned_sourcetrimremoved)) * pfd_residual_scale_execution_aligned_source) + (0)))) /\\ (((forall pftrim_index_execution_aligned_sourcetrimsuffix pftrim_value_execution_aligned_sourcetrimsuffix. (exists pfa_gap_execution_aligned_sourcetrimsuffixbound. pfa_gap_execution_aligned_sourcetrimsuffixbound + S (pftrim_index_execution_aligned_sourcetrimsuffix) = (R)) -> (((exists ff_h_pfp_execution_aligned_sourcetrimsuffixsource. ff_h_pfp_execution_aligned_sourcetrimsuffixsource + S (pftrim_value_execution_aligned_sourcetrimsuffix) = S ((S ((pfd_cut_execution_aligned_source)+pftrim_index_execution_aligned_sourcetrimsuffix)) * pfd_residual_scale_execution_aligned_source)) /\\ exists ff_q_pfp_execution_aligned_sourcetrimsuffixsource. pfd_residual_code_execution_aligned_source = ff_q_pfp_execution_aligned_sourcetrimsuffixsource * S ((S ((pfd_cut_execution_aligned_source)+pftrim_index_execution_aligned_sourcetrimsuffix)) * pfd_residual_scale_execution_aligned_source) + (pftrim_value_execution_aligned_sourcetrimsuffix))) -> (((exists ff_h_pfp_execution_aligned_sourcetrimsuffixoutput. ff_h_pfp_execution_aligned_sourcetrimsuffixoutput + S (pftrim_value_execution_aligned_sourcetrimsuffix) = S ((S (pftrim_index_execution_aligned_sourcetrimsuffix)) * rc)) /\\ exists ff_q_pfp_execution_aligned_sourcetrimsuffixoutput. rb = ff_q_pfp_execution_aligned_sourcetrimsuffixoutput * S ((S (pftrim_index_execution_aligned_sourcetrimsuffix)) * rc) + (pftrim_value_execution_aligned_sourcetrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_execution_aligned_sourcetrimnormal. ((((exists ff_h_pfp_execution_aligned_sourcetrimnormalentry. ff_h_pfp_execution_aligned_sourcetrimnormalentry + S (pftrim_leading_execution_aligned_sourcetrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_execution_aligned_sourcetrimnormalentry. rb = ff_q_pfp_execution_aligned_sourcetrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_execution_aligned_sourcetrimnormal))) /\\ ((~(pftrim_leading_execution_aligned_sourcetrimnormal=0))))))))))))))))))))))))))))))))) -> (exists pb pc N. ((((forall fom_index_pfp_execution_aligned_productleft. (exists fom_gap_pfp_execution_aligned_productleft_index_bound. fom_gap_pfp_execution_aligned_productleft_index_bound + S (fom_index_pfp_execution_aligned_productleft) = q) -> exists fom_value_pfp_execution_aligned_productleft. ((((exists fom_beta_height_pfp_execution_aligned_productleft_entry. fom_beta_height_pfp_execution_aligned_productleft_entry + S (fom_value_pfp_execution_aligned_productleft) = S ((S (fom_index_pfp_execution_aligned_productleft)) * qc)) /\\ exists fom_beta_quotient_pfp_execution_aligned_productleft_entry. qb = fom_beta_quotient_pfp_execution_aligned_productleft_entry * S ((S (fom_index_pfp_execution_aligned_productleft)) * qc) + (fom_value_pfp_execution_aligned_productleft))) /\\ (exists fom_gap_pfp_execution_aligned_productleft_value_bound. fom_gap_pfp_execution_aligned_productleft_value_bound + S (fom_value_pfp_execution_aligned_productleft) = p))) /\\ (((forall fom_index_pfp_execution_aligned_productright. (exists fom_gap_pfp_execution_aligned_productright_index_bound. fom_gap_pfp_execution_aligned_productright_index_bound + S (fom_index_pfp_execution_aligned_productright) = S d) -> exists fom_value_pfp_execution_aligned_productright. ((((exists fom_beta_height_pfp_execution_aligned_productright_entry. fom_beta_height_pfp_execution_aligned_productright_entry + S (fom_value_pfp_execution_aligned_productright) = S ((S (fom_index_pfp_execution_aligned_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_aligned_productright_entry. bb = fom_beta_quotient_pfp_execution_aligned_productright_entry * S ((S (fom_index_pfp_execution_aligned_productright)) * bc) + (fom_value_pfp_execution_aligned_productright))) /\\ (exists fom_gap_pfp_execution_aligned_productright_value_bound. fom_gap_pfp_execution_aligned_productright_value_bound + S (fom_value_pfp_execution_aligned_productright) = p))) /\\ (((((((q)=0 \\/ (S d)=0) /\\ (((N)=0)))) \\/ (((~((q)=0)) /\\ (((~((S d)=0)) /\\ (((q)+(S d)=S (N)))))))) /\\ ((forall pfc_index_execution_aligned_productcoefficients. (exists pfa_gap_execution_aligned_productcoefficientsbound. pfa_gap_execution_aligned_productcoefficientsbound + S (pfc_index_execution_aligned_productcoefficients) = (N)) -> exists pfc_value_execution_aligned_productcoefficients. ((((exists ff_h_pfp_execution_aligned_productcoefficientsentry. ff_h_pfp_execution_aligned_productcoefficientsentry + S (pfc_value_execution_aligned_productcoefficients) = S ((S (pfc_index_execution_aligned_productcoefficients)) * pc)) /\\ exists ff_q_pfp_execution_aligned_productcoefficientsentry. pb = ff_q_pfp_execution_aligned_productcoefficientsentry * S ((S (pfc_index_execution_aligned_productcoefficients)) * pc) + (pfc_value_execution_aligned_productcoefficients))) /\\ ((exists pfc_terms_code_execution_aligned_productcoefficientscoefficient pfc_terms_scale_execution_aligned_productcoefficientscoefficient pfc_natural_sum_execution_aligned_productcoefficientscoefficient. ((forall pfc_index_execution_aligned_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_aligned_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_aligned_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_aligned_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_aligned_productcoefficients))) -> exists pfc_value_execution_aligned_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_aligned_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_aligned_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_aligned_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_aligned_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_aligned_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_aligned_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_aligned_productcoefficientscoefficient = ff_q_pfp_execution_aligned_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_aligned_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_aligned_productcoefficientscoefficient) + (pfc_value_execution_aligned_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_aligned_productcoefficientscoefficientdiagonalterm pfc_left_execution_aligned_productcoefficientscoefficientdiagonalterm pfc_right_execution_aligned_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_aligned_productcoefficientscoefficientdiagonal)+pfc_complement_execution_aligned_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_aligned_productcoefficients)) /\\ ((((((exists pfa_gap_execution_aligned_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_aligned_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_aligned_productcoefficientscoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_execution_aligned_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_aligned_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_aligned_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_aligned_productcoefficientscoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_aligned_productcoefficientscoefficientdiagonaltermleftentry. qb = ff_q_pfp_execution_aligned_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_aligned_productcoefficientscoefficientdiagonal)) * qc) + (pfc_left_execution_aligned_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_aligned_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_aligned_productcoefficientscoefficientdiagonaltermleftoutside+(q)=(pfc_index_execution_aligned_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_aligned_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_aligned_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_aligned_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_aligned_productcoefficientscoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_execution_aligned_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_aligned_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_aligned_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_aligned_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_aligned_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_aligned_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_aligned_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_execution_aligned_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_aligned_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_aligned_productcoefficientscoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_execution_aligned_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_aligned_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_aligned_productcoefficientscoefficientdiagonal)=pfc_left_execution_aligned_productcoefficientscoefficientdiagonalterm*pfc_right_execution_aligned_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_aligned_productcoefficientscoefficientsum fs_v_pfc_execution_aligned_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_aligned_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_aligned_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_aligned_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_aligned_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_aligned_productcoefficientscoefficientsum = fs_q_pfc_execution_aligned_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_aligned_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_aligned_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_aligned_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_aligned_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_aligned_productcoefficients))) * fs_v_pfc_execution_aligned_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_aligned_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_aligned_productcoefficientscoefficientsum = fs_q_pfc_execution_aligned_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_aligned_productcoefficients))) * fs_v_pfc_execution_aligned_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_aligned_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_aligned_productcoefficients)) -> exists fs_a_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_aligned_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_aligned_productcoefficientscoefficient = fs_q_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_aligned_productcoefficientscoefficient) + (fs_a_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_aligned_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_aligned_productcoefficientscoefficientsum = fs_q_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_aligned_productcoefficientscoefficientsum) + (fs_r_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_aligned_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_aligned_productcoefficientscoefficientsum = fs_q_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_aligned_productcoefficientscoefficientsum) + (fs_s_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_aligned_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_aligned_productcoefficientscoefficientresiduebound. pfa_gap_execution_aligned_productcoefficientscoefficientresiduebound + S (pfc_value_execution_aligned_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_aligned_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_aligned_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_aligned_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_aligned_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_aligned_productcoefficients) + (p) * pfa_offset_right_execution_aligned_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_execution_aligned_sum_left_bounded. (exists fom_gap_pfp_execution_aligned_sum_left_bounded_index_bound. fom_gap_pfp_execution_aligned_sum_left_bounded_index_bound + S (fom_index_pfp_execution_aligned_sum_left_bounded) = N) -> exists fom_value_pfp_execution_aligned_sum_left_bounded. ((((exists fom_beta_height_pfp_execution_aligned_sum_left_bounded_entry. fom_beta_height_pfp_execution_aligned_sum_left_bounded_entry + S (fom_value_pfp_execution_aligned_sum_left_bounded) = S ((S (fom_index_pfp_execution_aligned_sum_left_bounded)) * pc)) /\\ exists fom_beta_quotient_pfp_execution_aligned_sum_left_bounded_entry. pb = fom_beta_quotient_pfp_execution_aligned_sum_left_bounded_entry * S ((S (fom_index_pfp_execution_aligned_sum_left_bounded)) * pc) + (fom_value_pfp_execution_aligned_sum_left_bounded))) /\\ (exists fom_gap_pfp_execution_aligned_sum_left_bounded_value_bound. fom_gap_pfp_execution_aligned_sum_left_bounded_value_bound + S (fom_value_pfp_execution_aligned_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_execution_aligned_sum_right_bounded. (exists fom_gap_pfp_execution_aligned_sum_right_bounded_index_bound. fom_gap_pfp_execution_aligned_sum_right_bounded_index_bound + S (fom_index_pfp_execution_aligned_sum_right_bounded) = R) -> exists fom_value_pfp_execution_aligned_sum_right_bounded. ((((exists fom_beta_height_pfp_execution_aligned_sum_right_bounded_entry. fom_beta_height_pfp_execution_aligned_sum_right_bounded_entry + S (fom_value_pfp_execution_aligned_sum_right_bounded) = S ((S (fom_index_pfp_execution_aligned_sum_right_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_execution_aligned_sum_right_bounded_entry. rb = fom_beta_quotient_pfp_execution_aligned_sum_right_bounded_entry * S ((S (fom_index_pfp_execution_aligned_sum_right_bounded)) * rc) + (fom_value_pfp_execution_aligned_sum_right_bounded))) /\\ (exists fom_gap_pfp_execution_aligned_sum_right_bounded_value_bound. fom_gap_pfp_execution_aligned_sum_right_bounded_value_bound + S (fom_value_pfp_execution_aligned_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_execution_aligned_sum_result_bounded. (exists fom_gap_pfp_execution_aligned_sum_result_bounded_index_bound. fom_gap_pfp_execution_aligned_sum_result_bounded_index_bound + S (fom_index_pfp_execution_aligned_sum_result_bounded) = L) -> exists fom_value_pfp_execution_aligned_sum_result_bounded. ((((exists fom_beta_height_pfp_execution_aligned_sum_result_bounded_entry. fom_beta_height_pfp_execution_aligned_sum_result_bounded_entry + S (fom_value_pfp_execution_aligned_sum_result_bounded) = S ((S (fom_index_pfp_execution_aligned_sum_result_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_aligned_sum_result_bounded_entry. ab = fom_beta_quotient_pfp_execution_aligned_sum_result_bounded_entry * S ((S (fom_index_pfp_execution_aligned_sum_result_bounded)) * ac) + (fom_value_pfp_execution_aligned_sum_result_bounded))) /\\ (exists fom_gap_pfp_execution_aligned_sum_result_bounded_value_bound. fom_gap_pfp_execution_aligned_sum_result_bounded_value_bound + S (fom_value_pfp_execution_aligned_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_execution_aligned_sum pfaa_left_c_execution_aligned_sum pfaa_right_b_execution_aligned_sum pfaa_right_c_execution_aligned_sum pfaa_sum_b_execution_aligned_sum pfaa_sum_c_execution_aligned_sum pfaa_length_execution_aligned_sum. ((((forall pfrep_power_execution_aligned_sum_witness_common_left pfrep_left_execution_aligned_sum_witness_common_left pfrep_right_execution_aligned_sum_witness_common_left. ((exists pfrep_position_execution_aligned_sum_witness_common_leftfirst. ((pfrep_position_execution_aligned_sum_witness_common_leftfirst+S (pfrep_power_execution_aligned_sum_witness_common_left)=(N)) /\\ ((((exists ff_h_pfp_execution_aligned_sum_witness_common_leftfirstentry. ff_h_pfp_execution_aligned_sum_witness_common_leftfirstentry + S (pfrep_left_execution_aligned_sum_witness_common_left) = S ((S (pfrep_position_execution_aligned_sum_witness_common_leftfirst)) * pc)) /\\ exists ff_q_pfp_execution_aligned_sum_witness_common_leftfirstentry. pb = ff_q_pfp_execution_aligned_sum_witness_common_leftfirstentry * S ((S (pfrep_position_execution_aligned_sum_witness_common_leftfirst)) * pc) + (pfrep_left_execution_aligned_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_execution_aligned_sum_witness_common_leftfirstoutside. pfrep_gap_execution_aligned_sum_witness_common_leftfirstoutside+(N)=(pfrep_power_execution_aligned_sum_witness_common_left)) /\\ (((pfrep_left_execution_aligned_sum_witness_common_left)=0))))) -> ((exists pfrep_position_execution_aligned_sum_witness_common_leftsecond. ((pfrep_position_execution_aligned_sum_witness_common_leftsecond+S (pfrep_power_execution_aligned_sum_witness_common_left)=(pfaa_length_execution_aligned_sum)) /\\ ((((exists ff_h_pfp_execution_aligned_sum_witness_common_leftsecondentry. ff_h_pfp_execution_aligned_sum_witness_common_leftsecondentry + S (pfrep_right_execution_aligned_sum_witness_common_left) = S ((S (pfrep_position_execution_aligned_sum_witness_common_leftsecond)) * pfaa_left_c_execution_aligned_sum)) /\\ exists ff_q_pfp_execution_aligned_sum_witness_common_leftsecondentry. pfaa_left_b_execution_aligned_sum = ff_q_pfp_execution_aligned_sum_witness_common_leftsecondentry * S ((S (pfrep_position_execution_aligned_sum_witness_common_leftsecond)) * pfaa_left_c_execution_aligned_sum) + (pfrep_right_execution_aligned_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_execution_aligned_sum_witness_common_leftsecondoutside. pfrep_gap_execution_aligned_sum_witness_common_leftsecondoutside+(pfaa_length_execution_aligned_sum)=(pfrep_power_execution_aligned_sum_witness_common_left)) /\\ (((pfrep_right_execution_aligned_sum_witness_common_left)=0))))) -> pfrep_left_execution_aligned_sum_witness_common_left=pfrep_right_execution_aligned_sum_witness_common_left) /\\ ((forall pfrep_power_execution_aligned_sum_witness_common_right pfrep_left_execution_aligned_sum_witness_common_right pfrep_right_execution_aligned_sum_witness_common_right. ((exists pfrep_position_execution_aligned_sum_witness_common_rightfirst. ((pfrep_position_execution_aligned_sum_witness_common_rightfirst+S (pfrep_power_execution_aligned_sum_witness_common_right)=(R)) /\\ ((((exists ff_h_pfp_execution_aligned_sum_witness_common_rightfirstentry. ff_h_pfp_execution_aligned_sum_witness_common_rightfirstentry + S (pfrep_left_execution_aligned_sum_witness_common_right) = S ((S (pfrep_position_execution_aligned_sum_witness_common_rightfirst)) * rc)) /\\ exists ff_q_pfp_execution_aligned_sum_witness_common_rightfirstentry. rb = ff_q_pfp_execution_aligned_sum_witness_common_rightfirstentry * S ((S (pfrep_position_execution_aligned_sum_witness_common_rightfirst)) * rc) + (pfrep_left_execution_aligned_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_execution_aligned_sum_witness_common_rightfirstoutside. pfrep_gap_execution_aligned_sum_witness_common_rightfirstoutside+(R)=(pfrep_power_execution_aligned_sum_witness_common_right)) /\\ (((pfrep_left_execution_aligned_sum_witness_common_right)=0))))) -> ((exists pfrep_position_execution_aligned_sum_witness_common_rightsecond. ((pfrep_position_execution_aligned_sum_witness_common_rightsecond+S (pfrep_power_execution_aligned_sum_witness_common_right)=(pfaa_length_execution_aligned_sum)) /\\ ((((exists ff_h_pfp_execution_aligned_sum_witness_common_rightsecondentry. ff_h_pfp_execution_aligned_sum_witness_common_rightsecondentry + S (pfrep_right_execution_aligned_sum_witness_common_right) = S ((S (pfrep_position_execution_aligned_sum_witness_common_rightsecond)) * pfaa_right_c_execution_aligned_sum)) /\\ exists ff_q_pfp_execution_aligned_sum_witness_common_rightsecondentry. pfaa_right_b_execution_aligned_sum = ff_q_pfp_execution_aligned_sum_witness_common_rightsecondentry * S ((S (pfrep_position_execution_aligned_sum_witness_common_rightsecond)) * pfaa_right_c_execution_aligned_sum) + (pfrep_right_execution_aligned_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_execution_aligned_sum_witness_common_rightsecondoutside. pfrep_gap_execution_aligned_sum_witness_common_rightsecondoutside+(pfaa_length_execution_aligned_sum)=(pfrep_power_execution_aligned_sum_witness_common_right)) /\\ (((pfrep_right_execution_aligned_sum_witness_common_right)=0))))) -> pfrep_left_execution_aligned_sum_witness_common_right=pfrep_right_execution_aligned_sum_witness_common_right)))) /\\ (((forall pfp_index_execution_aligned_sum_witness_operation. (exists pfa_gap_execution_aligned_sum_witness_operationindex. pfa_gap_execution_aligned_sum_witness_operationindex + S (pfp_index_execution_aligned_sum_witness_operation) = (pfaa_length_execution_aligned_sum)) -> exists pfp_left_execution_aligned_sum_witness_operation pfp_right_execution_aligned_sum_witness_operation pfp_value_execution_aligned_sum_witness_operation. ((((exists ff_h_pfp_execution_aligned_sum_witness_operationleft. ff_h_pfp_execution_aligned_sum_witness_operationleft + S (pfp_left_execution_aligned_sum_witness_operation) = S ((S (pfp_index_execution_aligned_sum_witness_operation)) * pfaa_left_c_execution_aligned_sum)) /\\ exists ff_q_pfp_execution_aligned_sum_witness_operationleft. pfaa_left_b_execution_aligned_sum = ff_q_pfp_execution_aligned_sum_witness_operationleft * S ((S (pfp_index_execution_aligned_sum_witness_operation)) * pfaa_left_c_execution_aligned_sum) + (pfp_left_execution_aligned_sum_witness_operation))) /\\ (((((exists ff_h_pfp_execution_aligned_sum_witness_operationright. ff_h_pfp_execution_aligned_sum_witness_operationright + S (pfp_right_execution_aligned_sum_witness_operation) = S ((S (pfp_index_execution_aligned_sum_witness_operation)) * pfaa_right_c_execution_aligned_sum)) /\\ exists ff_q_pfp_execution_aligned_sum_witness_operationright. pfaa_right_b_execution_aligned_sum = ff_q_pfp_execution_aligned_sum_witness_operationright * S ((S (pfp_index_execution_aligned_sum_witness_operation)) * pfaa_right_c_execution_aligned_sum) + (pfp_right_execution_aligned_sum_witness_operation))) /\\ (((((exists ff_h_pfp_execution_aligned_sum_witness_operationtarget. ff_h_pfp_execution_aligned_sum_witness_operationtarget + S (pfp_value_execution_aligned_sum_witness_operation) = S ((S (pfp_index_execution_aligned_sum_witness_operation)) * pfaa_sum_c_execution_aligned_sum)) /\\ exists ff_q_pfp_execution_aligned_sum_witness_operationtarget. pfaa_sum_b_execution_aligned_sum = ff_q_pfp_execution_aligned_sum_witness_operationtarget * S ((S (pfp_index_execution_aligned_sum_witness_operation)) * pfaa_sum_c_execution_aligned_sum) + (pfp_value_execution_aligned_sum_witness_operation))) /\\ ((((exists pfa_gap_execution_aligned_sum_witness_operationoperationleft. pfa_gap_execution_aligned_sum_witness_operationoperationleft + S (pfp_left_execution_aligned_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_execution_aligned_sum_witness_operationoperationright. pfa_gap_execution_aligned_sum_witness_operationoperationright + S (pfp_right_execution_aligned_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_execution_aligned_sum_witness_operationoperationresultbound. pfa_gap_execution_aligned_sum_witness_operationoperationresultbound + S (pfp_value_execution_aligned_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_execution_aligned_sum_witness_operationoperationresultcongruence pfa_offset_right_execution_aligned_sum_witness_operationoperationresultcongruence. ((pfp_left_execution_aligned_sum_witness_operation) + (pfp_right_execution_aligned_sum_witness_operation)) + (p) * pfa_offset_left_execution_aligned_sum_witness_operationoperationresultcongruence = (pfp_value_execution_aligned_sum_witness_operation) + (p) * pfa_offset_right_execution_aligned_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_execution_aligned_sum_witness_output pfrep_left_execution_aligned_sum_witness_output pfrep_right_execution_aligned_sum_witness_output. ((exists pfrep_position_execution_aligned_sum_witness_outputfirst. ((pfrep_position_execution_aligned_sum_witness_outputfirst+S (pfrep_power_execution_aligned_sum_witness_output)=(pfaa_length_execution_aligned_sum)) /\\ ((((exists ff_h_pfp_execution_aligned_sum_witness_outputfirstentry. ff_h_pfp_execution_aligned_sum_witness_outputfirstentry + S (pfrep_left_execution_aligned_sum_witness_output) = S ((S (pfrep_position_execution_aligned_sum_witness_outputfirst)) * pfaa_sum_c_execution_aligned_sum)) /\\ exists ff_q_pfp_execution_aligned_sum_witness_outputfirstentry. pfaa_sum_b_execution_aligned_sum = ff_q_pfp_execution_aligned_sum_witness_outputfirstentry * S ((S (pfrep_position_execution_aligned_sum_witness_outputfirst)) * pfaa_sum_c_execution_aligned_sum) + (pfrep_left_execution_aligned_sum_witness_output)))))) \\/ (((exists pfrep_gap_execution_aligned_sum_witness_outputfirstoutside. pfrep_gap_execution_aligned_sum_witness_outputfirstoutside+(pfaa_length_execution_aligned_sum)=(pfrep_power_execution_aligned_sum_witness_output)) /\\ (((pfrep_left_execution_aligned_sum_witness_output)=0))))) -> ((exists pfrep_position_execution_aligned_sum_witness_outputsecond. ((pfrep_position_execution_aligned_sum_witness_outputsecond+S (pfrep_power_execution_aligned_sum_witness_output)=(L)) /\\ ((((exists ff_h_pfp_execution_aligned_sum_witness_outputsecondentry. ff_h_pfp_execution_aligned_sum_witness_outputsecondentry + S (pfrep_right_execution_aligned_sum_witness_output) = S ((S (pfrep_position_execution_aligned_sum_witness_outputsecond)) * ac)) /\\ exists ff_q_pfp_execution_aligned_sum_witness_outputsecondentry. ab = ff_q_pfp_execution_aligned_sum_witness_outputsecondentry * S ((S (pfrep_position_execution_aligned_sum_witness_outputsecond)) * ac) + (pfrep_right_execution_aligned_sum_witness_output)))))) \\/ (((exists pfrep_gap_execution_aligned_sum_witness_outputsecondoutside. pfrep_gap_execution_aligned_sum_witness_outputsecondoutside+(L)=(pfrep_power_execution_aligned_sum_witness_output)) /\\ (((pfrep_right_execution_aligned_sum_witness_output)=0))))) -> 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"intro wc", "intro N", "intro db", "intro dc", "intro J", "intro pb", "intro pc", "intro H", "intro qb", "intro qc", "intro I", "intro rb", "intro rc", "intro K", "intro hp", "intro hs", "intro hP", "intro hQ", "intro hR", "have hp0 : ~(p=0)", "intro hpzero", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hpzero", "have hfactor : ((forall fom_index_pfp_aligned_left_pbleft. (exists fom_gap_pfp_aligned_left_pbleft_index_bound. fom_gap_pfp_aligned_left_pbleft_index_bound + S (fom_index_pfp_aligned_left_pbleft) = J) -> exists fom_value_pfp_aligned_left_pbleft. ((((exists fom_beta_height_pfp_aligned_left_pbleft_entry. fom_beta_height_pfp_aligned_left_pbleft_entry + S (fom_value_pfp_aligned_left_pbleft) = S ((S (fom_index_pfp_aligned_left_pbleft)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_left_pbleft_entry. db = fom_beta_quotient_pfp_aligned_left_pbleft_entry * S ((S (fom_index_pfp_aligned_left_pbleft)) * dc) + (fom_value_pfp_aligned_left_pbleft))) /\\ (exists fom_gap_pfp_aligned_left_pbleft_value_bound. fom_gap_pfp_aligned_left_pbleft_value_bound + S (fom_value_pfp_aligned_left_pbleft) = p))) /\\ (((forall fom_index_pfp_aligned_left_pbright. (exists fom_gap_pfp_aligned_left_pbright_index_bound. fom_gap_pfp_aligned_left_pbright_index_bound + S (fom_index_pfp_aligned_left_pbright) = L) -> exists fom_value_pfp_aligned_left_pbright. ((((exists fom_beta_height_pfp_aligned_left_pbright_entry. fom_beta_height_pfp_aligned_left_pbright_entry + S (fom_value_pfp_aligned_left_pbright) = S ((S (fom_index_pfp_aligned_left_pbright)) * uc)) /\\ exists fom_beta_quotient_pfp_aligned_left_pbright_entry. ub = fom_beta_quotient_pfp_aligned_left_pbright_entry * S ((S (fom_index_pfp_aligned_left_pbright)) * uc) + (fom_value_pfp_aligned_left_pbright))) /\\ (exists fom_gap_pfp_aligned_left_pbright_value_bound. fom_gap_pfp_aligned_left_pbright_value_bound + S (fom_value_pfp_aligned_left_pbright) = p))) /\\ (((((((J)=0 \\/ (L)=0) /\\ (((H)=0)))) \\/ (((~((J)=0)) /\\ (((~((L)=0)) /\\ (((J)+(L)=S (H)))))))) /\\ ((forall pfc_index_aligned_left_pbcoefficients. (exists pfa_gap_aligned_left_pbcoefficientsbound. pfa_gap_aligned_left_pbcoefficientsbound + S (pfc_index_aligned_left_pbcoefficients) = (H)) -> exists pfc_value_aligned_left_pbcoefficients. ((((exists ff_h_pfp_aligned_left_pbcoefficientsentry. ff_h_pfp_aligned_left_pbcoefficientsentry + S (pfc_value_aligned_left_pbcoefficients) = S ((S (pfc_index_aligned_left_pbcoefficients)) * pc)) /\\ exists ff_q_pfp_aligned_left_pbcoefficientsentry. pb = ff_q_pfp_aligned_left_pbcoefficientsentry * S ((S (pfc_index_aligned_left_pbcoefficients)) * pc) + (pfc_value_aligned_left_pbcoefficients))) /\\ ((exists pfc_terms_code_aligned_left_pbcoefficientscoefficient pfc_terms_scale_aligned_left_pbcoefficientscoefficient pfc_natural_sum_aligned_left_pbcoefficientscoefficient. ((forall pfc_index_aligned_left_pbcoefficientscoefficientdiagonal. (exists pfa_gap_aligned_left_pbcoefficientscoefficientdiagonalbound. pfa_gap_aligned_left_pbcoefficientscoefficientdiagonalbound + S (pfc_index_aligned_left_pbcoefficientscoefficientdiagonal) = (S (pfc_index_aligned_left_pbcoefficients))) -> exists pfc_value_aligned_left_pbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_aligned_left_pbcoefficientscoefficientdiagonalentry. ff_h_pfp_aligned_left_pbcoefficientscoefficientdiagonalentry + S (pfc_value_aligned_left_pbcoefficientscoefficientdiagonal) = S ((S (pfc_index_aligned_left_pbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_left_pbcoefficientscoefficient)) /\\ exists ff_q_pfp_aligned_left_pbcoefficientscoefficientdiagonalentry. pfc_terms_code_aligned_left_pbcoefficientscoefficient = ff_q_pfp_aligned_left_pbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_aligned_left_pbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_left_pbcoefficientscoefficient) + (pfc_value_aligned_left_pbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_aligned_left_pbcoefficientscoefficientdiagonalterm pfc_left_aligned_left_pbcoefficientscoefficientdiagonalterm pfc_right_aligned_left_pbcoefficientscoefficientdiagonalterm. (((pfc_index_aligned_left_pbcoefficientscoefficientdiagonal)+pfc_complement_aligned_left_pbcoefficientscoefficientdiagonalterm=(pfc_index_aligned_left_pbcoefficients)) /\\ ((((((exists pfa_gap_aligned_left_pbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_aligned_left_pbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_aligned_left_pbcoefficientscoefficientdiagonal) = (J)) /\\ ((((exists ff_h_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_aligned_left_pbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_aligned_left_pbcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_aligned_left_pbcoefficientscoefficientdiagonal)) * dc) + (pfc_left_aligned_left_pbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_left_pbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_aligned_left_pbcoefficientscoefficientdiagonaltermleftoutside+(J)=(pfc_index_aligned_left_pbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_aligned_left_pbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_aligned_left_pbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_aligned_left_pbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_aligned_left_pbcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_aligned_left_pbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_aligned_left_pbcoefficientscoefficientdiagonalterm)) * uc)) /\\ exists ff_q_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermrightentry. ub = ff_q_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_aligned_left_pbcoefficientscoefficientdiagonalterm)) * uc) + (pfc_right_aligned_left_pbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_left_pbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_aligned_left_pbcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_aligned_left_pbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_aligned_left_pbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_aligned_left_pbcoefficientscoefficientdiagonal)=pfc_left_aligned_left_pbcoefficientscoefficientdiagonalterm*pfc_right_aligned_left_pbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_aligned_left_pbcoefficientscoefficientsum fs_v_pfc_aligned_left_pbcoefficientscoefficientsum. ((((exists fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_start. fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_start. fs_u_pfc_aligned_left_pbcoefficientscoefficientsum = fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_terminal. fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_aligned_left_pbcoefficientscoefficient) = S ((S (S (pfc_index_aligned_left_pbcoefficients))) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_terminal. fs_u_pfc_aligned_left_pbcoefficientscoefficientsum = fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_aligned_left_pbcoefficients))) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum) + (pfc_natural_sum_aligned_left_pbcoefficientscoefficient))) /\\ forall fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps = S (pfc_index_aligned_left_pbcoefficients)) -> exists fs_a_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps fs_r_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps fs_s_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_left_pbcoefficientscoefficient)) /\\ exists fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_aligned_left_pbcoefficientscoefficient = fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_left_pbcoefficientscoefficient) + (fs_a_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_aligned_left_pbcoefficientscoefficientsum = fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum) + (fs_r_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_aligned_left_pbcoefficientscoefficientsum = fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum) + (fs_s_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps = fs_r_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps + fs_a_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_aligned_left_pbcoefficientscoefficientresiduebound. pfa_gap_aligned_left_pbcoefficientscoefficientresiduebound + S (pfc_value_aligned_left_pbcoefficients) = (p)) /\\ ((exists pfa_offset_left_aligned_left_pbcoefficientscoefficientresiduecongruence pfa_offset_right_aligned_left_pbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_aligned_left_pbcoefficientscoefficient) + (p) * pfa_offset_left_aligned_left_pbcoefficientscoefficientresiduecongruence = (pfc_value_aligned_left_pbcoefficients) + (p) * pfa_offset_right_aligned_left_pbcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hP", "cases hfactor", "cases hs", "cases hs_right", "cases hs_right_right", "cases hs_right_right_right", "cases hs_right_right_right_witness", "cases hs_right_right_right_witness_witness", "cases hs_right_right_right_witness_witness_witness", "cases hs_right_right_right_witness_witness_witness_witness", "cases hs_right_right_right_witness_witness_witness_witness_witness", "cases hs_right_right_right_witness_witness_witness_witness_witness_witness", "cases hs_right_right_right_witness_witness_witness_witness_witness_witness_witness", "cases hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_right", "cases hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_left", "have hproducts : exists O PB PC QB QC RB RC. ((((forall fom_index_pfp_aligned_left_constructed_PBleft. (exists fom_gap_pfp_aligned_left_constructed_PBleft_index_bound. fom_gap_pfp_aligned_left_constructed_PBleft_index_bound + S (fom_index_pfp_aligned_left_constructed_PBleft) = J) -> exists fom_value_pfp_aligned_left_constructed_PBleft. ((((exists fom_beta_height_pfp_aligned_left_constructed_PBleft_entry. fom_beta_height_pfp_aligned_left_constructed_PBleft_entry + S (fom_value_pfp_aligned_left_constructed_PBleft) = S ((S (fom_index_pfp_aligned_left_constructed_PBleft)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_left_constructed_PBleft_entry. db = fom_beta_quotient_pfp_aligned_left_constructed_PBleft_entry * S ((S (fom_index_pfp_aligned_left_constructed_PBleft)) * dc) + (fom_value_pfp_aligned_left_constructed_PBleft))) /\\ (exists fom_gap_pfp_aligned_left_constructed_PBleft_value_bound. fom_gap_pfp_aligned_left_constructed_PBleft_value_bound + S (fom_value_pfp_aligned_left_constructed_PBleft) = p))) /\\ (((forall fom_index_pfp_aligned_left_constructed_PBright. (exists fom_gap_pfp_aligned_left_constructed_PBright_index_bound. fom_gap_pfp_aligned_left_constructed_PBright_index_bound + S (fom_index_pfp_aligned_left_constructed_PBright) = x6) -> exists fom_value_pfp_aligned_left_constructed_PBright. ((((exists fom_beta_height_pfp_aligned_left_constructed_PBright_entry. fom_beta_height_pfp_aligned_left_constructed_PBright_entry + S (fom_value_pfp_aligned_left_constructed_PBright) = S ((S (fom_index_pfp_aligned_left_constructed_PBright)) * x1)) /\\ exists fom_beta_quotient_pfp_aligned_left_constructed_PBright_entry. x = fom_beta_quotient_pfp_aligned_left_constructed_PBright_entry * S ((S (fom_index_pfp_aligned_left_constructed_PBright)) * x1) + (fom_value_pfp_aligned_left_constructed_PBright))) /\\ (exists fom_gap_pfp_aligned_left_constructed_PBright_value_bound. fom_gap_pfp_aligned_left_constructed_PBright_value_bound + S (fom_value_pfp_aligned_left_constructed_PBright) = p))) /\\ (((((((J)=0 \\/ (x6)=0) /\\ (((O)=0)))) \\/ (((~((J)=0)) /\\ (((~((x6)=0)) /\\ (((J)+(x6)=S (O)))))))) /\\ ((forall pfc_index_aligned_left_constructed_PBcoefficients. (exists pfa_gap_aligned_left_constructed_PBcoefficientsbound. pfa_gap_aligned_left_constructed_PBcoefficientsbound + S (pfc_index_aligned_left_constructed_PBcoefficients) = (O)) -> exists pfc_value_aligned_left_constructed_PBcoefficients. ((((exists ff_h_pfp_aligned_left_constructed_PBcoefficientsentry. ff_h_pfp_aligned_left_constructed_PBcoefficientsentry + S (pfc_value_aligned_left_constructed_PBcoefficients) = S ((S (pfc_index_aligned_left_constructed_PBcoefficients)) * PC)) /\\ exists ff_q_pfp_aligned_left_constructed_PBcoefficientsentry. PB = ff_q_pfp_aligned_left_constructed_PBcoefficientsentry * S ((S (pfc_index_aligned_left_constructed_PBcoefficients)) * PC) + (pfc_value_aligned_left_constructed_PBcoefficients))) /\\ ((exists pfc_terms_code_aligned_left_constructed_PBcoefficientscoefficient pfc_terms_scale_aligned_left_constructed_PBcoefficientscoefficient pfc_natural_sum_aligned_left_constructed_PBcoefficientscoefficient. ((forall pfc_index_aligned_left_constructed_PBcoefficientscoefficientdiagonal. (exists pfa_gap_aligned_left_constructed_PBcoefficientscoefficientdiagonalbound. pfa_gap_aligned_left_constructed_PBcoefficientscoefficientdiagonalbound + S (pfc_index_aligned_left_constructed_PBcoefficientscoefficientdiagonal) = (S (pfc_index_aligned_left_constructed_PBcoefficients))) -> exists pfc_value_aligned_left_constructed_PBcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_aligned_left_constructed_PBcoefficientscoefficientdiagonalentry. ff_h_pfp_aligned_left_constructed_PBcoefficientscoefficientdiagonalentry + S (pfc_value_aligned_left_constructed_PBcoefficientscoefficientdiagonal) = S ((S (pfc_index_aligned_left_constructed_PBcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_left_constructed_PBcoefficientscoefficient)) /\\ exists ff_q_pfp_aligned_left_constructed_PBcoefficientscoefficientdiagonalentry. pfc_terms_code_aligned_left_constructed_PBcoefficientscoefficient = ff_q_pfp_aligned_left_constructed_PBcoefficientscoefficientdiagonalentry * S ((S (pfc_index_aligned_left_constructed_PBcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_left_constructed_PBcoefficientscoefficient) + (pfc_value_aligned_left_constructed_PBcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm pfc_left_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm pfc_right_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm. (((pfc_index_aligned_left_constructed_PBcoefficientscoefficientdiagonal)+pfc_complement_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm=(pfc_index_aligned_left_constructed_PBcoefficients)) /\\ ((((((exists pfa_gap_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermleftinside. pfa_gap_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_aligned_left_constructed_PBcoefficientscoefficientdiagonal) = (J)) /\\ ((((exists ff_h_pfp_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_aligned_left_constructed_PBcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_aligned_left_constructed_PBcoefficientscoefficientdiagonal)) * dc) + (pfc_left_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermleftoutside+(J)=(pfc_index_aligned_left_constructed_PBcoefficientscoefficientdiagonal)) /\\ (((pfc_left_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermrightinside. pfa_gap_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm) = (x6)) /\\ ((((exists ff_h_pfp_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermrightentry. x = ff_q_pfp_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm)) * x1) + (pfc_right_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_aligned_left_constructed_PBcoefficientscoefficientdiagonaltermrightoutside+(x6)=(pfc_complement_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_aligned_left_constructed_PBcoefficientscoefficientdiagonal)=pfc_left_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm*pfc_right_aligned_left_constructed_PBcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_aligned_left_constructed_PBcoefficientscoefficientsum fs_v_pfc_aligned_left_constructed_PBcoefficientscoefficientsum. ((((exists fs_h_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_start. fs_h_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_aligned_left_constructed_PBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_start. fs_u_pfc_aligned_left_constructed_PBcoefficientscoefficientsum = fs_q_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_aligned_left_constructed_PBcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_terminal. fs_h_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_aligned_left_constructed_PBcoefficientscoefficient) = S ((S (S (pfc_index_aligned_left_constructed_PBcoefficients))) * fs_v_pfc_aligned_left_constructed_PBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_terminal. fs_u_pfc_aligned_left_constructed_PBcoefficientscoefficientsum = fs_q_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_aligned_left_constructed_PBcoefficients))) * fs_v_pfc_aligned_left_constructed_PBcoefficientscoefficientsum) + (pfc_natural_sum_aligned_left_constructed_PBcoefficientscoefficient))) /\\ forall fs_i_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps = S (pfc_index_aligned_left_constructed_PBcoefficients)) -> exists fs_a_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps fs_r_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps fs_s_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_left_constructed_PBcoefficientscoefficient)) /\\ exists fs_q_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_aligned_left_constructed_PBcoefficientscoefficient = fs_q_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_left_constructed_PBcoefficientscoefficient) + (fs_a_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_constructed_PBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_aligned_left_constructed_PBcoefficientscoefficientsum = fs_q_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_constructed_PBcoefficientscoefficientsum) + (fs_r_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_constructed_PBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_aligned_left_constructed_PBcoefficientscoefficientsum = fs_q_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_constructed_PBcoefficientscoefficientsum) + (fs_s_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps = fs_r_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps + fs_a_pfc_aligned_left_constructed_PBcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_aligned_left_constructed_PBcoefficientscoefficientresiduebound. pfa_gap_aligned_left_constructed_PBcoefficientscoefficientresiduebound + S (pfc_value_aligned_left_constructed_PBcoefficients) = (p)) /\\ ((exists pfa_offset_left_aligned_left_constructed_PBcoefficientscoefficientresiduecongruence pfa_offset_right_aligned_left_constructed_PBcoefficientscoefficientresiduecongruence. (pfc_natural_sum_aligned_left_constructed_PBcoefficientscoefficient) + (p) * pfa_offset_left_aligned_left_constructed_PBcoefficientscoefficientresiduecongruence = (pfc_value_aligned_left_constructed_PBcoefficients) + (p) * pfa_offset_right_aligned_left_constructed_PBcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_aligned_left_constructed_QBleft. (exists fom_gap_pfp_aligned_left_constructed_QBleft_index_bound. fom_gap_pfp_aligned_left_constructed_QBleft_index_bound + S (fom_index_pfp_aligned_left_constructed_QBleft) = J) -> exists fom_value_pfp_aligned_left_constructed_QBleft. ((((exists fom_beta_height_pfp_aligned_left_constructed_QBleft_entry. fom_beta_height_pfp_aligned_left_constructed_QBleft_entry + S (fom_value_pfp_aligned_left_constructed_QBleft) = S ((S (fom_index_pfp_aligned_left_constructed_QBleft)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_left_constructed_QBleft_entry. db = fom_beta_quotient_pfp_aligned_left_constructed_QBleft_entry * S ((S (fom_index_pfp_aligned_left_constructed_QBleft)) * dc) + (fom_value_pfp_aligned_left_constructed_QBleft))) /\\ (exists fom_gap_pfp_aligned_left_constructed_QBleft_value_bound. fom_gap_pfp_aligned_left_constructed_QBleft_value_bound + S (fom_value_pfp_aligned_left_constructed_QBleft) = p))) /\\ (((forall fom_index_pfp_aligned_left_constructed_QBright. (exists fom_gap_pfp_aligned_left_constructed_QBright_index_bound. fom_gap_pfp_aligned_left_constructed_QBright_index_bound + S (fom_index_pfp_aligned_left_constructed_QBright) = x6) -> exists fom_value_pfp_aligned_left_constructed_QBright. ((((exists fom_beta_height_pfp_aligned_left_constructed_QBright_entry. fom_beta_height_pfp_aligned_left_constructed_QBright_entry + S (fom_value_pfp_aligned_left_constructed_QBright) = S ((S (fom_index_pfp_aligned_left_constructed_QBright)) * x3)) /\\ exists fom_beta_quotient_pfp_aligned_left_constructed_QBright_entry. x2 = fom_beta_quotient_pfp_aligned_left_constructed_QBright_entry * S ((S (fom_index_pfp_aligned_left_constructed_QBright)) * x3) + (fom_value_pfp_aligned_left_constructed_QBright))) /\\ (exists fom_gap_pfp_aligned_left_constructed_QBright_value_bound. fom_gap_pfp_aligned_left_constructed_QBright_value_bound + S (fom_value_pfp_aligned_left_constructed_QBright) = p))) /\\ (((((((J)=0 \\/ (x6)=0) /\\ (((O)=0)))) \\/ (((~((J)=0)) /\\ (((~((x6)=0)) /\\ (((J)+(x6)=S (O)))))))) /\\ ((forall pfc_index_aligned_left_constructed_QBcoefficients. (exists pfa_gap_aligned_left_constructed_QBcoefficientsbound. pfa_gap_aligned_left_constructed_QBcoefficientsbound + S (pfc_index_aligned_left_constructed_QBcoefficients) = (O)) -> exists pfc_value_aligned_left_constructed_QBcoefficients. ((((exists ff_h_pfp_aligned_left_constructed_QBcoefficientsentry. ff_h_pfp_aligned_left_constructed_QBcoefficientsentry + S (pfc_value_aligned_left_constructed_QBcoefficients) = S ((S (pfc_index_aligned_left_constructed_QBcoefficients)) * QC)) /\\ exists ff_q_pfp_aligned_left_constructed_QBcoefficientsentry. QB = ff_q_pfp_aligned_left_constructed_QBcoefficientsentry * S ((S (pfc_index_aligned_left_constructed_QBcoefficients)) * QC) + (pfc_value_aligned_left_constructed_QBcoefficients))) /\\ ((exists pfc_terms_code_aligned_left_constructed_QBcoefficientscoefficient pfc_terms_scale_aligned_left_constructed_QBcoefficientscoefficient pfc_natural_sum_aligned_left_constructed_QBcoefficientscoefficient. ((forall pfc_index_aligned_left_constructed_QBcoefficientscoefficientdiagonal. (exists pfa_gap_aligned_left_constructed_QBcoefficientscoefficientdiagonalbound. pfa_gap_aligned_left_constructed_QBcoefficientscoefficientdiagonalbound + S (pfc_index_aligned_left_constructed_QBcoefficientscoefficientdiagonal) = (S (pfc_index_aligned_left_constructed_QBcoefficients))) -> exists pfc_value_aligned_left_constructed_QBcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_aligned_left_constructed_QBcoefficientscoefficientdiagonalentry. ff_h_pfp_aligned_left_constructed_QBcoefficientscoefficientdiagonalentry + S (pfc_value_aligned_left_constructed_QBcoefficientscoefficientdiagonal) = S ((S (pfc_index_aligned_left_constructed_QBcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_left_constructed_QBcoefficientscoefficient)) /\\ exists ff_q_pfp_aligned_left_constructed_QBcoefficientscoefficientdiagonalentry. pfc_terms_code_aligned_left_constructed_QBcoefficientscoefficient = ff_q_pfp_aligned_left_constructed_QBcoefficientscoefficientdiagonalentry * S ((S (pfc_index_aligned_left_constructed_QBcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_left_constructed_QBcoefficientscoefficient) + (pfc_value_aligned_left_constructed_QBcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm pfc_left_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm pfc_right_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm. (((pfc_index_aligned_left_constructed_QBcoefficientscoefficientdiagonal)+pfc_complement_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm=(pfc_index_aligned_left_constructed_QBcoefficients)) /\\ ((((((exists pfa_gap_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermleftinside. pfa_gap_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_aligned_left_constructed_QBcoefficientscoefficientdiagonal) = (J)) /\\ ((((exists ff_h_pfp_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_aligned_left_constructed_QBcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_aligned_left_constructed_QBcoefficientscoefficientdiagonal)) * dc) + (pfc_left_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermleftoutside+(J)=(pfc_index_aligned_left_constructed_QBcoefficientscoefficientdiagonal)) /\\ (((pfc_left_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermrightinside. pfa_gap_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm) = (x6)) /\\ ((((exists ff_h_pfp_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm)) * x3)) /\\ exists ff_q_pfp_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermrightentry. x2 = ff_q_pfp_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm)) * x3) + (pfc_right_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_aligned_left_constructed_QBcoefficientscoefficientdiagonaltermrightoutside+(x6)=(pfc_complement_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_aligned_left_constructed_QBcoefficientscoefficientdiagonal)=pfc_left_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm*pfc_right_aligned_left_constructed_QBcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_aligned_left_constructed_QBcoefficientscoefficientsum fs_v_pfc_aligned_left_constructed_QBcoefficientscoefficientsum. ((((exists fs_h_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_start. fs_h_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_aligned_left_constructed_QBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_start. fs_u_pfc_aligned_left_constructed_QBcoefficientscoefficientsum = fs_q_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_aligned_left_constructed_QBcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_terminal. fs_h_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_aligned_left_constructed_QBcoefficientscoefficient) = S ((S (S (pfc_index_aligned_left_constructed_QBcoefficients))) * fs_v_pfc_aligned_left_constructed_QBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_terminal. fs_u_pfc_aligned_left_constructed_QBcoefficientscoefficientsum = fs_q_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_aligned_left_constructed_QBcoefficients))) * fs_v_pfc_aligned_left_constructed_QBcoefficientscoefficientsum) + (pfc_natural_sum_aligned_left_constructed_QBcoefficientscoefficient))) /\\ forall fs_i_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps = S (pfc_index_aligned_left_constructed_QBcoefficients)) -> exists fs_a_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps fs_r_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps fs_s_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_left_constructed_QBcoefficientscoefficient)) /\\ exists fs_q_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_aligned_left_constructed_QBcoefficientscoefficient = fs_q_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_left_constructed_QBcoefficientscoefficient) + (fs_a_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_constructed_QBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_aligned_left_constructed_QBcoefficientscoefficientsum = fs_q_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_constructed_QBcoefficientscoefficientsum) + (fs_r_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_constructed_QBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_aligned_left_constructed_QBcoefficientscoefficientsum = fs_q_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_constructed_QBcoefficientscoefficientsum) + (fs_s_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps = fs_r_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps + fs_a_pfc_aligned_left_constructed_QBcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_aligned_left_constructed_QBcoefficientscoefficientresiduebound. pfa_gap_aligned_left_constructed_QBcoefficientscoefficientresiduebound + S (pfc_value_aligned_left_constructed_QBcoefficients) = (p)) /\\ ((exists pfa_offset_left_aligned_left_constructed_QBcoefficientscoefficientresiduecongruence pfa_offset_right_aligned_left_constructed_QBcoefficientscoefficientresiduecongruence. (pfc_natural_sum_aligned_left_constructed_QBcoefficientscoefficient) + (p) * pfa_offset_left_aligned_left_constructed_QBcoefficientscoefficientresiduecongruence = (pfc_value_aligned_left_constructed_QBcoefficients) + (p) * pfa_offset_right_aligned_left_constructed_QBcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_aligned_left_constructed_RBleft. (exists fom_gap_pfp_aligned_left_constructed_RBleft_index_bound. fom_gap_pfp_aligned_left_constructed_RBleft_index_bound + S (fom_index_pfp_aligned_left_constructed_RBleft) = J) -> exists fom_value_pfp_aligned_left_constructed_RBleft. ((((exists fom_beta_height_pfp_aligned_left_constructed_RBleft_entry. fom_beta_height_pfp_aligned_left_constructed_RBleft_entry + S (fom_value_pfp_aligned_left_constructed_RBleft) = S ((S (fom_index_pfp_aligned_left_constructed_RBleft)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_left_constructed_RBleft_entry. db = fom_beta_quotient_pfp_aligned_left_constructed_RBleft_entry * S ((S (fom_index_pfp_aligned_left_constructed_RBleft)) * dc) + (fom_value_pfp_aligned_left_constructed_RBleft))) /\\ (exists fom_gap_pfp_aligned_left_constructed_RBleft_value_bound. fom_gap_pfp_aligned_left_constructed_RBleft_value_bound + S (fom_value_pfp_aligned_left_constructed_RBleft) = p))) /\\ (((forall fom_index_pfp_aligned_left_constructed_RBright. (exists fom_gap_pfp_aligned_left_constructed_RBright_index_bound. fom_gap_pfp_aligned_left_constructed_RBright_index_bound + S (fom_index_pfp_aligned_left_constructed_RBright) = x6) -> exists fom_value_pfp_aligned_left_constructed_RBright. ((((exists fom_beta_height_pfp_aligned_left_constructed_RBright_entry. fom_beta_height_pfp_aligned_left_constructed_RBright_entry + S (fom_value_pfp_aligned_left_constructed_RBright) = S ((S (fom_index_pfp_aligned_left_constructed_RBright)) * x5)) /\\ exists fom_beta_quotient_pfp_aligned_left_constructed_RBright_entry. x4 = fom_beta_quotient_pfp_aligned_left_constructed_RBright_entry * S ((S (fom_index_pfp_aligned_left_constructed_RBright)) * x5) + (fom_value_pfp_aligned_left_constructed_RBright))) /\\ (exists fom_gap_pfp_aligned_left_constructed_RBright_value_bound. fom_gap_pfp_aligned_left_constructed_RBright_value_bound + S (fom_value_pfp_aligned_left_constructed_RBright) = p))) /\\ (((((((J)=0 \\/ (x6)=0) /\\ (((O)=0)))) \\/ (((~((J)=0)) /\\ (((~((x6)=0)) /\\ (((J)+(x6)=S (O)))))))) /\\ ((forall pfc_index_aligned_left_constructed_RBcoefficients. (exists pfa_gap_aligned_left_constructed_RBcoefficientsbound. pfa_gap_aligned_left_constructed_RBcoefficientsbound + S (pfc_index_aligned_left_constructed_RBcoefficients) = (O)) -> exists pfc_value_aligned_left_constructed_RBcoefficients. ((((exists ff_h_pfp_aligned_left_constructed_RBcoefficientsentry. ff_h_pfp_aligned_left_constructed_RBcoefficientsentry + S (pfc_value_aligned_left_constructed_RBcoefficients) = S ((S (pfc_index_aligned_left_constructed_RBcoefficients)) * RC)) /\\ exists ff_q_pfp_aligned_left_constructed_RBcoefficientsentry. RB = ff_q_pfp_aligned_left_constructed_RBcoefficientsentry * S ((S (pfc_index_aligned_left_constructed_RBcoefficients)) * RC) + (pfc_value_aligned_left_constructed_RBcoefficients))) /\\ ((exists pfc_terms_code_aligned_left_constructed_RBcoefficientscoefficient pfc_terms_scale_aligned_left_constructed_RBcoefficientscoefficient pfc_natural_sum_aligned_left_constructed_RBcoefficientscoefficient. ((forall pfc_index_aligned_left_constructed_RBcoefficientscoefficientdiagonal. (exists pfa_gap_aligned_left_constructed_RBcoefficientscoefficientdiagonalbound. pfa_gap_aligned_left_constructed_RBcoefficientscoefficientdiagonalbound + S (pfc_index_aligned_left_constructed_RBcoefficientscoefficientdiagonal) = (S (pfc_index_aligned_left_constructed_RBcoefficients))) -> exists pfc_value_aligned_left_constructed_RBcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_aligned_left_constructed_RBcoefficientscoefficientdiagonalentry. ff_h_pfp_aligned_left_constructed_RBcoefficientscoefficientdiagonalentry + S (pfc_value_aligned_left_constructed_RBcoefficientscoefficientdiagonal) = S ((S (pfc_index_aligned_left_constructed_RBcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_left_constructed_RBcoefficientscoefficient)) /\\ exists ff_q_pfp_aligned_left_constructed_RBcoefficientscoefficientdiagonalentry. pfc_terms_code_aligned_left_constructed_RBcoefficientscoefficient = ff_q_pfp_aligned_left_constructed_RBcoefficientscoefficientdiagonalentry * S ((S (pfc_index_aligned_left_constructed_RBcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_left_constructed_RBcoefficientscoefficient) + (pfc_value_aligned_left_constructed_RBcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm pfc_left_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm pfc_right_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm. (((pfc_index_aligned_left_constructed_RBcoefficientscoefficientdiagonal)+pfc_complement_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm=(pfc_index_aligned_left_constructed_RBcoefficients)) /\\ ((((((exists pfa_gap_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermleftinside. pfa_gap_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_aligned_left_constructed_RBcoefficientscoefficientdiagonal) = (J)) /\\ ((((exists ff_h_pfp_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_aligned_left_constructed_RBcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_aligned_left_constructed_RBcoefficientscoefficientdiagonal)) * dc) + (pfc_left_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermleftoutside+(J)=(pfc_index_aligned_left_constructed_RBcoefficientscoefficientdiagonal)) /\\ (((pfc_left_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermrightinside. pfa_gap_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm) = (x6)) /\\ ((((exists ff_h_pfp_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm)) * x5)) /\\ exists ff_q_pfp_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermrightentry. x4 = ff_q_pfp_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm)) * x5) + (pfc_right_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_aligned_left_constructed_RBcoefficientscoefficientdiagonaltermrightoutside+(x6)=(pfc_complement_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_aligned_left_constructed_RBcoefficientscoefficientdiagonal)=pfc_left_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm*pfc_right_aligned_left_constructed_RBcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_aligned_left_constructed_RBcoefficientscoefficientsum fs_v_pfc_aligned_left_constructed_RBcoefficientscoefficientsum. ((((exists fs_h_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_start. fs_h_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_aligned_left_constructed_RBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_start. fs_u_pfc_aligned_left_constructed_RBcoefficientscoefficientsum = fs_q_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_aligned_left_constructed_RBcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_terminal. fs_h_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_aligned_left_constructed_RBcoefficientscoefficient) = S ((S (S (pfc_index_aligned_left_constructed_RBcoefficients))) * fs_v_pfc_aligned_left_constructed_RBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_terminal. fs_u_pfc_aligned_left_constructed_RBcoefficientscoefficientsum = fs_q_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_aligned_left_constructed_RBcoefficients))) * fs_v_pfc_aligned_left_constructed_RBcoefficientscoefficientsum) + (pfc_natural_sum_aligned_left_constructed_RBcoefficientscoefficient))) /\\ forall fs_i_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps = S (pfc_index_aligned_left_constructed_RBcoefficients)) -> exists fs_a_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps fs_r_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps fs_s_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_left_constructed_RBcoefficientscoefficient)) /\\ exists fs_q_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_aligned_left_constructed_RBcoefficientscoefficient = fs_q_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_left_constructed_RBcoefficientscoefficient) + (fs_a_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_constructed_RBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_aligned_left_constructed_RBcoefficientscoefficientsum = fs_q_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_constructed_RBcoefficientscoefficientsum) + (fs_r_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_constructed_RBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_aligned_left_constructed_RBcoefficientscoefficientsum = fs_q_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_constructed_RBcoefficientscoefficientsum) + (fs_s_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps = fs_r_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps + fs_a_pfc_aligned_left_constructed_RBcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_aligned_left_constructed_RBcoefficientscoefficientresiduebound. pfa_gap_aligned_left_constructed_RBcoefficientscoefficientresiduebound + S (pfc_value_aligned_left_constructed_RBcoefficients) = (p)) /\\ ((exists pfa_offset_left_aligned_left_constructed_RBcoefficientscoefficientresiduecongruence pfa_offset_right_aligned_left_constructed_RBcoefficientscoefficientresiduecongruence. (pfc_natural_sum_aligned_left_constructed_RBcoefficientscoefficient) + (p) * pfa_offset_left_aligned_left_constructed_RBcoefficientscoefficientresiduecongruence = (pfc_value_aligned_left_constructed_RBcoefficients) + (p) * pfa_offset_right_aligned_left_constructed_RBcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfp_index_aligned_left_constructed_sum. (exists pfa_gap_aligned_left_constructed_sumindex. pfa_gap_aligned_left_constructed_sumindex + S (pfp_index_aligned_left_constructed_sum) = (O)) -> exists pfp_left_aligned_left_constructed_sum pfp_right_aligned_left_constructed_sum pfp_value_aligned_left_constructed_sum. ((((exists ff_h_pfp_aligned_left_constructed_sumleft. ff_h_pfp_aligned_left_constructed_sumleft + S (pfp_left_aligned_left_constructed_sum) = S ((S (pfp_index_aligned_left_constructed_sum)) * PC)) /\\ exists ff_q_pfp_aligned_left_constructed_sumleft. PB = ff_q_pfp_aligned_left_constructed_sumleft * S ((S (pfp_index_aligned_left_constructed_sum)) * PC) + (pfp_left_aligned_left_constructed_sum))) /\\ (((((exists ff_h_pfp_aligned_left_constructed_sumright. ff_h_pfp_aligned_left_constructed_sumright + S (pfp_right_aligned_left_constructed_sum) = S ((S (pfp_index_aligned_left_constructed_sum)) * QC)) /\\ exists ff_q_pfp_aligned_left_constructed_sumright. QB = ff_q_pfp_aligned_left_constructed_sumright * S ((S (pfp_index_aligned_left_constructed_sum)) * QC) + (pfp_right_aligned_left_constructed_sum))) /\\ (((((exists ff_h_pfp_aligned_left_constructed_sumtarget. ff_h_pfp_aligned_left_constructed_sumtarget + S (pfp_value_aligned_left_constructed_sum) = S ((S (pfp_index_aligned_left_constructed_sum)) * RC)) /\\ exists ff_q_pfp_aligned_left_constructed_sumtarget. RB = ff_q_pfp_aligned_left_constructed_sumtarget * S ((S (pfp_index_aligned_left_constructed_sum)) * RC) + (pfp_value_aligned_left_constructed_sum))) /\\ ((((exists pfa_gap_aligned_left_constructed_sumoperationleft. pfa_gap_aligned_left_constructed_sumoperationleft + S (pfp_left_aligned_left_constructed_sum) = (p)) /\\ (((exists pfa_gap_aligned_left_constructed_sumoperationright. pfa_gap_aligned_left_constructed_sumoperationright + S (pfp_right_aligned_left_constructed_sum) = (p)) /\\ ((((exists pfa_gap_aligned_left_constructed_sumoperationresultbound. pfa_gap_aligned_left_constructed_sumoperationresultbound + S (pfp_value_aligned_left_constructed_sum) = (p)) /\\ ((exists pfa_offset_left_aligned_left_constructed_sumoperationresultcongruence pfa_offset_right_aligned_left_constructed_sumoperationresultcongruence. ((pfp_left_aligned_left_constructed_sum) + (pfp_right_aligned_left_constructed_sum)) + (p) * pfa_offset_left_aligned_left_constructed_sumoperationresultcongruence = (pfp_value_aligned_left_constructed_sum) + (p) * pfa_offset_right_aligned_left_constructed_sumoperationresultcongruence))))))))))))))))))))))", "specialize prime_field_polynomial_left_distributive_products_exists (p)", "specialize prime_field_polynomial_left_distributive_products_exists (x)", "specialize prime_field_polynomial_left_distributive_products_exists (x1)", "specialize prime_field_polynomial_left_distributive_products_exists (x2)", "specialize prime_field_polynomial_left_distributive_products_exists (x3)", "specialize prime_field_polynomial_left_distributive_products_exists (x4)", "specialize prime_field_polynomial_left_distributive_products_exists (x5)", "specialize prime_field_polynomial_left_distributive_products_exists (x6)", "specialize prime_field_polynomial_left_distributive_products_exists (db)", "specialize prime_field_polynomial_left_distributive_products_exists (dc)", "specialize prime_field_polynomial_left_distributive_products_exists (J)", "apply prime_field_polynomial_left_distributive_products_exists", "exact hp0", "exact hfactor_left", "exact hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_left", "cases hproducts", "cases hproducts_witness", "cases hproducts_witness_witness", "cases hproducts_witness_witness_witness", "cases hproducts_witness_witness_witness_witness", "cases hproducts_witness_witness_witness_witness_witness", "cases hproducts_witness_witness_witness_witness_witness_witness", "cases hproducts_witness_witness_witness_witness_witness_witness_witness", "cases hproducts_witness_witness_witness_witness_witness_witness_witness_right", "cases hproducts_witness_witness_witness_witness_witness_witness_witness_right_right", "specialize prime_field_polynomial_aligned_add_from_common (p)", "specialize prime_field_polynomial_aligned_add_from_common (pb)", "specialize prime_field_polynomial_aligned_add_from_common (pc)", "specialize prime_field_polynomial_aligned_add_from_common (H)", "specialize prime_field_polynomial_aligned_add_from_common (qb)", "specialize prime_field_polynomial_aligned_add_from_common (qc)", "specialize prime_field_polynomial_aligned_add_from_common (I)", "specialize prime_field_polynomial_aligned_add_from_common (rb)", "specialize prime_field_polynomial_aligned_add_from_common (rc)", "specialize prime_field_polynomial_aligned_add_from_common (K)", "specialize prime_field_polynomial_aligned_add_from_common (x8)", "specialize prime_field_polynomial_aligned_add_from_common (x9)", "specialize prime_field_polynomial_aligned_add_from_common (x10)", "specialize prime_field_polynomial_aligned_add_from_common (x11)", "specialize prime_field_polynomial_aligned_add_from_common (x12)", "specialize prime_field_polynomial_aligned_add_from_common (x13)", "specialize prime_field_polynomial_aligned_add_from_common (x7)", "apply prime_field_polynomial_aligned_add_from_common", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (db)", "specialize prime_field_polynomial_convolution_bounded (dc)", "specialize prime_field_polynomial_convolution_bounded (J)", "specialize prime_field_polynomial_convolution_bounded (ub)", "specialize prime_field_polynomial_convolution_bounded (uc)", "specialize prime_field_polynomial_convolution_bounded (L)", "specialize prime_field_polynomial_convolution_bounded (pb)", "specialize prime_field_polynomial_convolution_bounded (pc)", "specialize prime_field_polynomial_convolution_bounded (H)", "apply prime_field_polynomial_convolution_bounded", "exact hP", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (db)", "specialize prime_field_polynomial_convolution_bounded (dc)", "specialize prime_field_polynomial_convolution_bounded (J)", "specialize prime_field_polynomial_convolution_bounded (vb)", "specialize prime_field_polynomial_convolution_bounded (vc)", "specialize prime_field_polynomial_convolution_bounded (M)", "specialize prime_field_polynomial_convolution_bounded (qb)", "specialize prime_field_polynomial_convolution_bounded (qc)", "specialize prime_field_polynomial_convolution_bounded (I)", "apply prime_field_polynomial_convolution_bounded", "exact hQ", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (db)", "specialize prime_field_polynomial_convolution_bounded (dc)", "specialize prime_field_polynomial_convolution_bounded (J)", "specialize prime_field_polynomial_convolution_bounded (wb)", "specialize prime_field_polynomial_convolution_bounded (wc)", "specialize prime_field_polynomial_convolution_bounded (N)", "specialize prime_field_polynomial_convolution_bounded (rb)", "specialize prime_field_polynomial_convolution_bounded (rc)", "specialize prime_field_polynomial_convolution_bounded (K)", "apply prime_field_polynomial_convolution_bounded", "exact hR", "split", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (p)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (db)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (dc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (J)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (ub)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (uc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (L)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (pb)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (pc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (H)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x1)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x6)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x8)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x9)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x7)", "apply prime_field_polynomial_convolution_equivalent_congruent_right", "exact hp0", "exact hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_left_left", "exact hP", "exact hproducts_witness_witness_witness_witness_witness_witness_witness_left", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (p)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (db)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (dc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (J)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (vb)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (vc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (M)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (qb)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (qc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (I)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x2)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x3)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x6)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x10)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x11)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x7)", "apply prime_field_polynomial_convolution_equivalent_congruent_right", "exact hp0", "exact hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_left_right", "exact hQ", "exact hproducts_witness_witness_witness_witness_witness_witness_witness_right_left", "exact hproducts_witness_witness_witness_witness_witness_witness_witness_right_right_right", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (p)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (db)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (dc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (J)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x4)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x5)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x6)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x12)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x13)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x7)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (wb)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (wc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (N)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (rb)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (rc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (K)", "apply prime_field_polynomial_convolution_equivalent_congruent_right", "exact hp0", "exact hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right", "exact hproducts_witness_witness_witness_witness_witness_witness_witness_right_right_left", "exact hR" ], "script_sha256": "34687df65cc738cc2404cbca27ddf82ce0cbddf803dc098c593e40b99121dfcd", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_aligned_distributivity_candidate.py", "sha256": "7d535939e24fe6d82158c485533b2ff6934f4d897b6141fde6c50b4fec9788ba" }, "statement": "forall p ub uc L vb vc M wb wc N db dc J pb pc H qb qc I rb rc K. (~((p) = 1) /\\ forall pfa_factor_left_aligned_left_prime pfa_factor_right_aligned_left_prime. (p) = pfa_factor_left_aligned_left_prime * pfa_factor_right_aligned_left_prime -> pfa_factor_left_aligned_left_prime = 1 \\/ pfa_factor_right_aligned_left_prime = 1) -> (((forall fom_index_pfp_aligned_left_input_left_bounded. (exists fom_gap_pfp_aligned_left_input_left_bounded_index_bound. fom_gap_pfp_aligned_left_input_left_bounded_index_bound + S (fom_index_pfp_aligned_left_input_left_bounded) = L) -> exists fom_value_pfp_aligned_left_input_left_bounded. ((((exists fom_beta_height_pfp_aligned_left_input_left_bounded_entry. fom_beta_height_pfp_aligned_left_input_left_bounded_entry + S (fom_value_pfp_aligned_left_input_left_bounded) = S ((S (fom_index_pfp_aligned_left_input_left_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_aligned_left_input_left_bounded_entry. ub = fom_beta_quotient_pfp_aligned_left_input_left_bounded_entry * S ((S (fom_index_pfp_aligned_left_input_left_bounded)) * uc) + (fom_value_pfp_aligned_left_input_left_bounded))) /\\ (exists fom_gap_pfp_aligned_left_input_left_bounded_value_bound. fom_gap_pfp_aligned_left_input_left_bounded_value_bound + S (fom_value_pfp_aligned_left_input_left_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_left_input_right_bounded. (exists fom_gap_pfp_aligned_left_input_right_bounded_index_bound. fom_gap_pfp_aligned_left_input_right_bounded_index_bound + S (fom_index_pfp_aligned_left_input_right_bounded) = M) -> exists fom_value_pfp_aligned_left_input_right_bounded. ((((exists fom_beta_height_pfp_aligned_left_input_right_bounded_entry. fom_beta_height_pfp_aligned_left_input_right_bounded_entry + S (fom_value_pfp_aligned_left_input_right_bounded) = S ((S (fom_index_pfp_aligned_left_input_right_bounded)) * vc)) /\\ exists fom_beta_quotient_pfp_aligned_left_input_right_bounded_entry. vb = fom_beta_quotient_pfp_aligned_left_input_right_bounded_entry * S ((S (fom_index_pfp_aligned_left_input_right_bounded)) * vc) + (fom_value_pfp_aligned_left_input_right_bounded))) /\\ (exists fom_gap_pfp_aligned_left_input_right_bounded_value_bound. fom_gap_pfp_aligned_left_input_right_bounded_value_bound + S (fom_value_pfp_aligned_left_input_right_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_left_input_result_bounded. (exists fom_gap_pfp_aligned_left_input_result_bounded_index_bound. fom_gap_pfp_aligned_left_input_result_bounded_index_bound + S (fom_index_pfp_aligned_left_input_result_bounded) = N) -> exists fom_value_pfp_aligned_left_input_result_bounded. ((((exists fom_beta_height_pfp_aligned_left_input_result_bounded_entry. fom_beta_height_pfp_aligned_left_input_result_bounded_entry + S (fom_value_pfp_aligned_left_input_result_bounded) = S ((S (fom_index_pfp_aligned_left_input_result_bounded)) * wc)) /\\ exists fom_beta_quotient_pfp_aligned_left_input_result_bounded_entry. wb = fom_beta_quotient_pfp_aligned_left_input_result_bounded_entry * S ((S (fom_index_pfp_aligned_left_input_result_bounded)) * wc) + (fom_value_pfp_aligned_left_input_result_bounded))) /\\ (exists fom_gap_pfp_aligned_left_input_result_bounded_value_bound. fom_gap_pfp_aligned_left_input_result_bounded_value_bound + S (fom_value_pfp_aligned_left_input_result_bounded) = p))) /\\ ((exists pfaa_left_b_aligned_left_input pfaa_left_c_aligned_left_input pfaa_right_b_aligned_left_input pfaa_right_c_aligned_left_input pfaa_sum_b_aligned_left_input pfaa_sum_c_aligned_left_input pfaa_length_aligned_left_input. ((((forall pfrep_power_aligned_left_input_witness_common_left pfrep_left_aligned_left_input_witness_common_left pfrep_right_aligned_left_input_witness_common_left. ((exists pfrep_position_aligned_left_input_witness_common_leftfirst. ((pfrep_position_aligned_left_input_witness_common_leftfirst+S (pfrep_power_aligned_left_input_witness_common_left)=(L)) /\\ ((((exists ff_h_pfp_aligned_left_input_witness_common_leftfirstentry. ff_h_pfp_aligned_left_input_witness_common_leftfirstentry + S (pfrep_left_aligned_left_input_witness_common_left) = S ((S (pfrep_position_aligned_left_input_witness_common_leftfirst)) * uc)) /\\ exists ff_q_pfp_aligned_left_input_witness_common_leftfirstentry. ub = ff_q_pfp_aligned_left_input_witness_common_leftfirstentry * S ((S (pfrep_position_aligned_left_input_witness_common_leftfirst)) * uc) + (pfrep_left_aligned_left_input_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_left_input_witness_common_leftfirstoutside. pfrep_gap_aligned_left_input_witness_common_leftfirstoutside+(L)=(pfrep_power_aligned_left_input_witness_common_left)) /\\ (((pfrep_left_aligned_left_input_witness_common_left)=0))))) -> ((exists pfrep_position_aligned_left_input_witness_common_leftsecond. ((pfrep_position_aligned_left_input_witness_common_leftsecond+S (pfrep_power_aligned_left_input_witness_common_left)=(pfaa_length_aligned_left_input)) /\\ ((((exists ff_h_pfp_aligned_left_input_witness_common_leftsecondentry. ff_h_pfp_aligned_left_input_witness_common_leftsecondentry + S (pfrep_right_aligned_left_input_witness_common_left) = S ((S (pfrep_position_aligned_left_input_witness_common_leftsecond)) * pfaa_left_c_aligned_left_input)) /\\ exists ff_q_pfp_aligned_left_input_witness_common_leftsecondentry. pfaa_left_b_aligned_left_input = ff_q_pfp_aligned_left_input_witness_common_leftsecondentry * S ((S (pfrep_position_aligned_left_input_witness_common_leftsecond)) * pfaa_left_c_aligned_left_input) + (pfrep_right_aligned_left_input_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_left_input_witness_common_leftsecondoutside. pfrep_gap_aligned_left_input_witness_common_leftsecondoutside+(pfaa_length_aligned_left_input)=(pfrep_power_aligned_left_input_witness_common_left)) /\\ (((pfrep_right_aligned_left_input_witness_common_left)=0))))) -> pfrep_left_aligned_left_input_witness_common_left=pfrep_right_aligned_left_input_witness_common_left) /\\ ((forall pfrep_power_aligned_left_input_witness_common_right pfrep_left_aligned_left_input_witness_common_right pfrep_right_aligned_left_input_witness_common_right. ((exists pfrep_position_aligned_left_input_witness_common_rightfirst. ((pfrep_position_aligned_left_input_witness_common_rightfirst+S (pfrep_power_aligned_left_input_witness_common_right)=(M)) /\\ ((((exists ff_h_pfp_aligned_left_input_witness_common_rightfirstentry. ff_h_pfp_aligned_left_input_witness_common_rightfirstentry + S (pfrep_left_aligned_left_input_witness_common_right) = S ((S (pfrep_position_aligned_left_input_witness_common_rightfirst)) * vc)) /\\ exists ff_q_pfp_aligned_left_input_witness_common_rightfirstentry. vb = ff_q_pfp_aligned_left_input_witness_common_rightfirstentry * S ((S (pfrep_position_aligned_left_input_witness_common_rightfirst)) * vc) + (pfrep_left_aligned_left_input_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_left_input_witness_common_rightfirstoutside. pfrep_gap_aligned_left_input_witness_common_rightfirstoutside+(M)=(pfrep_power_aligned_left_input_witness_common_right)) /\\ (((pfrep_left_aligned_left_input_witness_common_right)=0))))) -> ((exists pfrep_position_aligned_left_input_witness_common_rightsecond. ((pfrep_position_aligned_left_input_witness_common_rightsecond+S (pfrep_power_aligned_left_input_witness_common_right)=(pfaa_length_aligned_left_input)) /\\ ((((exists ff_h_pfp_aligned_left_input_witness_common_rightsecondentry. ff_h_pfp_aligned_left_input_witness_common_rightsecondentry + S (pfrep_right_aligned_left_input_witness_common_right) = S ((S (pfrep_position_aligned_left_input_witness_common_rightsecond)) * pfaa_right_c_aligned_left_input)) /\\ exists ff_q_pfp_aligned_left_input_witness_common_rightsecondentry. pfaa_right_b_aligned_left_input = ff_q_pfp_aligned_left_input_witness_common_rightsecondentry * S ((S (pfrep_position_aligned_left_input_witness_common_rightsecond)) * pfaa_right_c_aligned_left_input) + (pfrep_right_aligned_left_input_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_left_input_witness_common_rightsecondoutside. pfrep_gap_aligned_left_input_witness_common_rightsecondoutside+(pfaa_length_aligned_left_input)=(pfrep_power_aligned_left_input_witness_common_right)) /\\ (((pfrep_right_aligned_left_input_witness_common_right)=0))))) -> pfrep_left_aligned_left_input_witness_common_right=pfrep_right_aligned_left_input_witness_common_right)))) /\\ (((forall pfp_index_aligned_left_input_witness_operation. (exists pfa_gap_aligned_left_input_witness_operationindex. pfa_gap_aligned_left_input_witness_operationindex + S (pfp_index_aligned_left_input_witness_operation) = (pfaa_length_aligned_left_input)) -> exists pfp_left_aligned_left_input_witness_operation pfp_right_aligned_left_input_witness_operation pfp_value_aligned_left_input_witness_operation. ((((exists ff_h_pfp_aligned_left_input_witness_operationleft. ff_h_pfp_aligned_left_input_witness_operationleft + S (pfp_left_aligned_left_input_witness_operation) = S ((S (pfp_index_aligned_left_input_witness_operation)) * pfaa_left_c_aligned_left_input)) /\\ exists ff_q_pfp_aligned_left_input_witness_operationleft. pfaa_left_b_aligned_left_input = ff_q_pfp_aligned_left_input_witness_operationleft * S ((S (pfp_index_aligned_left_input_witness_operation)) * pfaa_left_c_aligned_left_input) + (pfp_left_aligned_left_input_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_left_input_witness_operationright. ff_h_pfp_aligned_left_input_witness_operationright + S (pfp_right_aligned_left_input_witness_operation) = S ((S (pfp_index_aligned_left_input_witness_operation)) * pfaa_right_c_aligned_left_input)) /\\ exists ff_q_pfp_aligned_left_input_witness_operationright. pfaa_right_b_aligned_left_input = ff_q_pfp_aligned_left_input_witness_operationright * S ((S (pfp_index_aligned_left_input_witness_operation)) * pfaa_right_c_aligned_left_input) + (pfp_right_aligned_left_input_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_left_input_witness_operationtarget. ff_h_pfp_aligned_left_input_witness_operationtarget + S (pfp_value_aligned_left_input_witness_operation) = S ((S (pfp_index_aligned_left_input_witness_operation)) * pfaa_sum_c_aligned_left_input)) /\\ exists ff_q_pfp_aligned_left_input_witness_operationtarget. pfaa_sum_b_aligned_left_input = ff_q_pfp_aligned_left_input_witness_operationtarget * S ((S (pfp_index_aligned_left_input_witness_operation)) * pfaa_sum_c_aligned_left_input) + (pfp_value_aligned_left_input_witness_operation))) /\\ ((((exists pfa_gap_aligned_left_input_witness_operationoperationleft. pfa_gap_aligned_left_input_witness_operationoperationleft + S (pfp_left_aligned_left_input_witness_operation) = (p)) /\\ (((exists pfa_gap_aligned_left_input_witness_operationoperationright. pfa_gap_aligned_left_input_witness_operationoperationright + S (pfp_right_aligned_left_input_witness_operation) = (p)) /\\ ((((exists pfa_gap_aligned_left_input_witness_operationoperationresultbound. pfa_gap_aligned_left_input_witness_operationoperationresultbound + S (pfp_value_aligned_left_input_witness_operation) = (p)) /\\ ((exists pfa_offset_left_aligned_left_input_witness_operationoperationresultcongruence pfa_offset_right_aligned_left_input_witness_operationoperationresultcongruence. ((pfp_left_aligned_left_input_witness_operation) + (pfp_right_aligned_left_input_witness_operation)) + (p) * pfa_offset_left_aligned_left_input_witness_operationoperationresultcongruence = (pfp_value_aligned_left_input_witness_operation) + (p) * pfa_offset_right_aligned_left_input_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_aligned_left_input_witness_output pfrep_left_aligned_left_input_witness_output pfrep_right_aligned_left_input_witness_output. ((exists pfrep_position_aligned_left_input_witness_outputfirst. ((pfrep_position_aligned_left_input_witness_outputfirst+S (pfrep_power_aligned_left_input_witness_output)=(pfaa_length_aligned_left_input)) /\\ ((((exists ff_h_pfp_aligned_left_input_witness_outputfirstentry. ff_h_pfp_aligned_left_input_witness_outputfirstentry + S (pfrep_left_aligned_left_input_witness_output) = S ((S (pfrep_position_aligned_left_input_witness_outputfirst)) * pfaa_sum_c_aligned_left_input)) /\\ exists ff_q_pfp_aligned_left_input_witness_outputfirstentry. pfaa_sum_b_aligned_left_input = ff_q_pfp_aligned_left_input_witness_outputfirstentry * S ((S (pfrep_position_aligned_left_input_witness_outputfirst)) * pfaa_sum_c_aligned_left_input) + (pfrep_left_aligned_left_input_witness_output)))))) \\/ (((exists pfrep_gap_aligned_left_input_witness_outputfirstoutside. pfrep_gap_aligned_left_input_witness_outputfirstoutside+(pfaa_length_aligned_left_input)=(pfrep_power_aligned_left_input_witness_output)) /\\ (((pfrep_left_aligned_left_input_witness_output)=0))))) -> ((exists pfrep_position_aligned_left_input_witness_outputsecond. ((pfrep_position_aligned_left_input_witness_outputsecond+S (pfrep_power_aligned_left_input_witness_output)=(N)) /\\ ((((exists ff_h_pfp_aligned_left_input_witness_outputsecondentry. ff_h_pfp_aligned_left_input_witness_outputsecondentry + S (pfrep_right_aligned_left_input_witness_output) = S ((S (pfrep_position_aligned_left_input_witness_outputsecond)) * wc)) /\\ exists ff_q_pfp_aligned_left_input_witness_outputsecondentry. wb = ff_q_pfp_aligned_left_input_witness_outputsecondentry * S ((S (pfrep_position_aligned_left_input_witness_outputsecond)) * wc) + (pfrep_right_aligned_left_input_witness_output)))))) \\/ (((exists pfrep_gap_aligned_left_input_witness_outputsecondoutside. pfrep_gap_aligned_left_input_witness_outputsecondoutside+(N)=(pfrep_power_aligned_left_input_witness_output)) /\\ (((pfrep_right_aligned_left_input_witness_output)=0))))) -> pfrep_left_aligned_left_input_witness_output=pfrep_right_aligned_left_input_witness_output))))))))))))) -> (((forall fom_index_pfp_aligned_left_pbleft. (exists fom_gap_pfp_aligned_left_pbleft_index_bound. fom_gap_pfp_aligned_left_pbleft_index_bound + S (fom_index_pfp_aligned_left_pbleft) = J) -> exists fom_value_pfp_aligned_left_pbleft. ((((exists fom_beta_height_pfp_aligned_left_pbleft_entry. fom_beta_height_pfp_aligned_left_pbleft_entry + S (fom_value_pfp_aligned_left_pbleft) = S ((S (fom_index_pfp_aligned_left_pbleft)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_left_pbleft_entry. db = fom_beta_quotient_pfp_aligned_left_pbleft_entry * S ((S (fom_index_pfp_aligned_left_pbleft)) * dc) + (fom_value_pfp_aligned_left_pbleft))) /\\ (exists fom_gap_pfp_aligned_left_pbleft_value_bound. fom_gap_pfp_aligned_left_pbleft_value_bound + S (fom_value_pfp_aligned_left_pbleft) = p))) /\\ (((forall fom_index_pfp_aligned_left_pbright. (exists fom_gap_pfp_aligned_left_pbright_index_bound. fom_gap_pfp_aligned_left_pbright_index_bound + S (fom_index_pfp_aligned_left_pbright) = L) -> exists fom_value_pfp_aligned_left_pbright. ((((exists fom_beta_height_pfp_aligned_left_pbright_entry. fom_beta_height_pfp_aligned_left_pbright_entry + S (fom_value_pfp_aligned_left_pbright) = S ((S (fom_index_pfp_aligned_left_pbright)) * uc)) /\\ exists fom_beta_quotient_pfp_aligned_left_pbright_entry. ub = fom_beta_quotient_pfp_aligned_left_pbright_entry * S ((S (fom_index_pfp_aligned_left_pbright)) * uc) + (fom_value_pfp_aligned_left_pbright))) /\\ (exists fom_gap_pfp_aligned_left_pbright_value_bound. fom_gap_pfp_aligned_left_pbright_value_bound + S (fom_value_pfp_aligned_left_pbright) = p))) /\\ (((((((J)=0 \\/ (L)=0) /\\ (((H)=0)))) \\/ (((~((J)=0)) /\\ (((~((L)=0)) /\\ (((J)+(L)=S (H)))))))) /\\ ((forall pfc_index_aligned_left_pbcoefficients. (exists pfa_gap_aligned_left_pbcoefficientsbound. pfa_gap_aligned_left_pbcoefficientsbound + S (pfc_index_aligned_left_pbcoefficients) = (H)) -> exists pfc_value_aligned_left_pbcoefficients. ((((exists ff_h_pfp_aligned_left_pbcoefficientsentry. ff_h_pfp_aligned_left_pbcoefficientsentry + S (pfc_value_aligned_left_pbcoefficients) = S ((S (pfc_index_aligned_left_pbcoefficients)) * pc)) /\\ exists ff_q_pfp_aligned_left_pbcoefficientsentry. pb = ff_q_pfp_aligned_left_pbcoefficientsentry * S ((S (pfc_index_aligned_left_pbcoefficients)) * pc) + (pfc_value_aligned_left_pbcoefficients))) /\\ ((exists pfc_terms_code_aligned_left_pbcoefficientscoefficient pfc_terms_scale_aligned_left_pbcoefficientscoefficient pfc_natural_sum_aligned_left_pbcoefficientscoefficient. ((forall pfc_index_aligned_left_pbcoefficientscoefficientdiagonal. (exists pfa_gap_aligned_left_pbcoefficientscoefficientdiagonalbound. pfa_gap_aligned_left_pbcoefficientscoefficientdiagonalbound + S (pfc_index_aligned_left_pbcoefficientscoefficientdiagonal) = (S (pfc_index_aligned_left_pbcoefficients))) -> exists pfc_value_aligned_left_pbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_aligned_left_pbcoefficientscoefficientdiagonalentry. ff_h_pfp_aligned_left_pbcoefficientscoefficientdiagonalentry + S (pfc_value_aligned_left_pbcoefficientscoefficientdiagonal) = S ((S (pfc_index_aligned_left_pbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_left_pbcoefficientscoefficient)) /\\ exists ff_q_pfp_aligned_left_pbcoefficientscoefficientdiagonalentry. pfc_terms_code_aligned_left_pbcoefficientscoefficient = ff_q_pfp_aligned_left_pbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_aligned_left_pbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_left_pbcoefficientscoefficient) + (pfc_value_aligned_left_pbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_aligned_left_pbcoefficientscoefficientdiagonalterm pfc_left_aligned_left_pbcoefficientscoefficientdiagonalterm pfc_right_aligned_left_pbcoefficientscoefficientdiagonalterm. (((pfc_index_aligned_left_pbcoefficientscoefficientdiagonal)+pfc_complement_aligned_left_pbcoefficientscoefficientdiagonalterm=(pfc_index_aligned_left_pbcoefficients)) /\\ ((((((exists pfa_gap_aligned_left_pbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_aligned_left_pbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_aligned_left_pbcoefficientscoefficientdiagonal) = (J)) /\\ ((((exists ff_h_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_aligned_left_pbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_aligned_left_pbcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_aligned_left_pbcoefficientscoefficientdiagonal)) * dc) + (pfc_left_aligned_left_pbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_left_pbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_aligned_left_pbcoefficientscoefficientdiagonaltermleftoutside+(J)=(pfc_index_aligned_left_pbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_aligned_left_pbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_aligned_left_pbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_aligned_left_pbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_aligned_left_pbcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_aligned_left_pbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_aligned_left_pbcoefficientscoefficientdiagonalterm)) * uc)) /\\ exists ff_q_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermrightentry. ub = ff_q_pfp_aligned_left_pbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_aligned_left_pbcoefficientscoefficientdiagonalterm)) * uc) + (pfc_right_aligned_left_pbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_left_pbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_aligned_left_pbcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_aligned_left_pbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_aligned_left_pbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_aligned_left_pbcoefficientscoefficientdiagonal)=pfc_left_aligned_left_pbcoefficientscoefficientdiagonalterm*pfc_right_aligned_left_pbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_aligned_left_pbcoefficientscoefficientsum fs_v_pfc_aligned_left_pbcoefficientscoefficientsum. ((((exists fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_start. fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_start. fs_u_pfc_aligned_left_pbcoefficientscoefficientsum = fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_terminal. fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_aligned_left_pbcoefficientscoefficient) = S ((S (S (pfc_index_aligned_left_pbcoefficients))) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_terminal. fs_u_pfc_aligned_left_pbcoefficientscoefficientsum = fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_aligned_left_pbcoefficients))) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum) + (pfc_natural_sum_aligned_left_pbcoefficientscoefficient))) /\\ forall fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps = S (pfc_index_aligned_left_pbcoefficients)) -> exists fs_a_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps fs_r_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps fs_s_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_left_pbcoefficientscoefficient)) /\\ exists fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_aligned_left_pbcoefficientscoefficient = fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_left_pbcoefficientscoefficient) + (fs_a_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_aligned_left_pbcoefficientscoefficientsum = fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum) + (fs_r_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_aligned_left_pbcoefficientscoefficientsum = fs_q_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_pbcoefficientscoefficientsum) + (fs_s_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps = fs_r_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps + fs_a_pfc_aligned_left_pbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_aligned_left_pbcoefficientscoefficientresiduebound. pfa_gap_aligned_left_pbcoefficientscoefficientresiduebound + S (pfc_value_aligned_left_pbcoefficients) = (p)) /\\ ((exists pfa_offset_left_aligned_left_pbcoefficientscoefficientresiduecongruence pfa_offset_right_aligned_left_pbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_aligned_left_pbcoefficientscoefficient) + (p) * pfa_offset_left_aligned_left_pbcoefficientscoefficientresiduecongruence = (pfc_value_aligned_left_pbcoefficients) + (p) * pfa_offset_right_aligned_left_pbcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_aligned_left_qbleft. (exists fom_gap_pfp_aligned_left_qbleft_index_bound. fom_gap_pfp_aligned_left_qbleft_index_bound + S (fom_index_pfp_aligned_left_qbleft) = J) -> exists fom_value_pfp_aligned_left_qbleft. ((((exists fom_beta_height_pfp_aligned_left_qbleft_entry. fom_beta_height_pfp_aligned_left_qbleft_entry + S (fom_value_pfp_aligned_left_qbleft) = S ((S (fom_index_pfp_aligned_left_qbleft)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_left_qbleft_entry. db = fom_beta_quotient_pfp_aligned_left_qbleft_entry * S ((S (fom_index_pfp_aligned_left_qbleft)) * dc) + (fom_value_pfp_aligned_left_qbleft))) /\\ (exists fom_gap_pfp_aligned_left_qbleft_value_bound. fom_gap_pfp_aligned_left_qbleft_value_bound + S (fom_value_pfp_aligned_left_qbleft) = p))) /\\ (((forall fom_index_pfp_aligned_left_qbright. (exists fom_gap_pfp_aligned_left_qbright_index_bound. fom_gap_pfp_aligned_left_qbright_index_bound + S (fom_index_pfp_aligned_left_qbright) = M) -> exists fom_value_pfp_aligned_left_qbright. ((((exists fom_beta_height_pfp_aligned_left_qbright_entry. fom_beta_height_pfp_aligned_left_qbright_entry + S (fom_value_pfp_aligned_left_qbright) = S ((S (fom_index_pfp_aligned_left_qbright)) * vc)) /\\ exists fom_beta_quotient_pfp_aligned_left_qbright_entry. vb = fom_beta_quotient_pfp_aligned_left_qbright_entry * S ((S (fom_index_pfp_aligned_left_qbright)) * vc) + (fom_value_pfp_aligned_left_qbright))) /\\ (exists fom_gap_pfp_aligned_left_qbright_value_bound. fom_gap_pfp_aligned_left_qbright_value_bound + S (fom_value_pfp_aligned_left_qbright) = p))) /\\ (((((((J)=0 \\/ (M)=0) /\\ (((I)=0)))) \\/ (((~((J)=0)) /\\ (((~((M)=0)) /\\ (((J)+(M)=S (I)))))))) /\\ ((forall pfc_index_aligned_left_qbcoefficients. (exists pfa_gap_aligned_left_qbcoefficientsbound. pfa_gap_aligned_left_qbcoefficientsbound + S (pfc_index_aligned_left_qbcoefficients) = (I)) -> exists pfc_value_aligned_left_qbcoefficients. ((((exists ff_h_pfp_aligned_left_qbcoefficientsentry. ff_h_pfp_aligned_left_qbcoefficientsentry + S (pfc_value_aligned_left_qbcoefficients) = S ((S (pfc_index_aligned_left_qbcoefficients)) * qc)) /\\ exists ff_q_pfp_aligned_left_qbcoefficientsentry. qb = ff_q_pfp_aligned_left_qbcoefficientsentry * S ((S (pfc_index_aligned_left_qbcoefficients)) * qc) + (pfc_value_aligned_left_qbcoefficients))) /\\ ((exists pfc_terms_code_aligned_left_qbcoefficientscoefficient pfc_terms_scale_aligned_left_qbcoefficientscoefficient pfc_natural_sum_aligned_left_qbcoefficientscoefficient. ((forall pfc_index_aligned_left_qbcoefficientscoefficientdiagonal. (exists pfa_gap_aligned_left_qbcoefficientscoefficientdiagonalbound. pfa_gap_aligned_left_qbcoefficientscoefficientdiagonalbound + S (pfc_index_aligned_left_qbcoefficientscoefficientdiagonal) = (S (pfc_index_aligned_left_qbcoefficients))) -> exists pfc_value_aligned_left_qbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_aligned_left_qbcoefficientscoefficientdiagonalentry. ff_h_pfp_aligned_left_qbcoefficientscoefficientdiagonalentry + S (pfc_value_aligned_left_qbcoefficientscoefficientdiagonal) = S ((S (pfc_index_aligned_left_qbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_left_qbcoefficientscoefficient)) /\\ exists ff_q_pfp_aligned_left_qbcoefficientscoefficientdiagonalentry. pfc_terms_code_aligned_left_qbcoefficientscoefficient = ff_q_pfp_aligned_left_qbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_aligned_left_qbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_left_qbcoefficientscoefficient) + (pfc_value_aligned_left_qbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_aligned_left_qbcoefficientscoefficientdiagonalterm pfc_left_aligned_left_qbcoefficientscoefficientdiagonalterm pfc_right_aligned_left_qbcoefficientscoefficientdiagonalterm. (((pfc_index_aligned_left_qbcoefficientscoefficientdiagonal)+pfc_complement_aligned_left_qbcoefficientscoefficientdiagonalterm=(pfc_index_aligned_left_qbcoefficients)) /\\ ((((((exists pfa_gap_aligned_left_qbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_aligned_left_qbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_aligned_left_qbcoefficientscoefficientdiagonal) = (J)) /\\ ((((exists ff_h_pfp_aligned_left_qbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_aligned_left_qbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_aligned_left_qbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_aligned_left_qbcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_aligned_left_qbcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_aligned_left_qbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_aligned_left_qbcoefficientscoefficientdiagonal)) * dc) + (pfc_left_aligned_left_qbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_left_qbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_aligned_left_qbcoefficientscoefficientdiagonaltermleftoutside+(J)=(pfc_index_aligned_left_qbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_aligned_left_qbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_aligned_left_qbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_aligned_left_qbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_aligned_left_qbcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_aligned_left_qbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_aligned_left_qbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_aligned_left_qbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_aligned_left_qbcoefficientscoefficientdiagonalterm)) * vc)) /\\ exists ff_q_pfp_aligned_left_qbcoefficientscoefficientdiagonaltermrightentry. vb = ff_q_pfp_aligned_left_qbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_aligned_left_qbcoefficientscoefficientdiagonalterm)) * vc) + (pfc_right_aligned_left_qbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_left_qbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_aligned_left_qbcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_aligned_left_qbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_aligned_left_qbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_aligned_left_qbcoefficientscoefficientdiagonal)=pfc_left_aligned_left_qbcoefficientscoefficientdiagonalterm*pfc_right_aligned_left_qbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_aligned_left_qbcoefficientscoefficientsum fs_v_pfc_aligned_left_qbcoefficientscoefficientsum. ((((exists fs_h_pfc_aligned_left_qbcoefficientscoefficientsum_body_start. fs_h_pfc_aligned_left_qbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_aligned_left_qbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_qbcoefficientscoefficientsum_body_start. fs_u_pfc_aligned_left_qbcoefficientscoefficientsum = fs_q_pfc_aligned_left_qbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_aligned_left_qbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_aligned_left_qbcoefficientscoefficientsum_body_terminal. fs_h_pfc_aligned_left_qbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_aligned_left_qbcoefficientscoefficient) = S ((S (S (pfc_index_aligned_left_qbcoefficients))) * fs_v_pfc_aligned_left_qbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_qbcoefficientscoefficientsum_body_terminal. fs_u_pfc_aligned_left_qbcoefficientscoefficientsum = fs_q_pfc_aligned_left_qbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_aligned_left_qbcoefficients))) * fs_v_pfc_aligned_left_qbcoefficientscoefficientsum) + (pfc_natural_sum_aligned_left_qbcoefficientscoefficient))) /\\ forall fs_i_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps = S (pfc_index_aligned_left_qbcoefficients)) -> exists fs_a_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps fs_r_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps fs_s_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_left_qbcoefficientscoefficient)) /\\ exists fs_q_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_aligned_left_qbcoefficientscoefficient = fs_q_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_left_qbcoefficientscoefficient) + (fs_a_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_qbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_aligned_left_qbcoefficientscoefficientsum = fs_q_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_qbcoefficientscoefficientsum) + (fs_r_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_qbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_aligned_left_qbcoefficientscoefficientsum = fs_q_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_qbcoefficientscoefficientsum) + (fs_s_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps = fs_r_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps + fs_a_pfc_aligned_left_qbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_aligned_left_qbcoefficientscoefficientresiduebound. pfa_gap_aligned_left_qbcoefficientscoefficientresiduebound + S (pfc_value_aligned_left_qbcoefficients) = (p)) /\\ ((exists pfa_offset_left_aligned_left_qbcoefficientscoefficientresiduecongruence pfa_offset_right_aligned_left_qbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_aligned_left_qbcoefficientscoefficient) + (p) * pfa_offset_left_aligned_left_qbcoefficientscoefficientresiduecongruence = (pfc_value_aligned_left_qbcoefficients) + (p) * pfa_offset_right_aligned_left_qbcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_aligned_left_rbleft. (exists fom_gap_pfp_aligned_left_rbleft_index_bound. fom_gap_pfp_aligned_left_rbleft_index_bound + S (fom_index_pfp_aligned_left_rbleft) = J) -> exists fom_value_pfp_aligned_left_rbleft. ((((exists fom_beta_height_pfp_aligned_left_rbleft_entry. fom_beta_height_pfp_aligned_left_rbleft_entry + S (fom_value_pfp_aligned_left_rbleft) = S ((S (fom_index_pfp_aligned_left_rbleft)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_left_rbleft_entry. db = fom_beta_quotient_pfp_aligned_left_rbleft_entry * S ((S (fom_index_pfp_aligned_left_rbleft)) * dc) + (fom_value_pfp_aligned_left_rbleft))) /\\ (exists fom_gap_pfp_aligned_left_rbleft_value_bound. fom_gap_pfp_aligned_left_rbleft_value_bound + S (fom_value_pfp_aligned_left_rbleft) = p))) /\\ (((forall fom_index_pfp_aligned_left_rbright. (exists fom_gap_pfp_aligned_left_rbright_index_bound. fom_gap_pfp_aligned_left_rbright_index_bound + S (fom_index_pfp_aligned_left_rbright) = N) -> exists fom_value_pfp_aligned_left_rbright. ((((exists fom_beta_height_pfp_aligned_left_rbright_entry. fom_beta_height_pfp_aligned_left_rbright_entry + S (fom_value_pfp_aligned_left_rbright) = S ((S (fom_index_pfp_aligned_left_rbright)) * wc)) /\\ exists fom_beta_quotient_pfp_aligned_left_rbright_entry. wb = fom_beta_quotient_pfp_aligned_left_rbright_entry * S ((S (fom_index_pfp_aligned_left_rbright)) * wc) + (fom_value_pfp_aligned_left_rbright))) /\\ (exists fom_gap_pfp_aligned_left_rbright_value_bound. fom_gap_pfp_aligned_left_rbright_value_bound + S (fom_value_pfp_aligned_left_rbright) = p))) /\\ (((((((J)=0 \\/ (N)=0) /\\ (((K)=0)))) \\/ (((~((J)=0)) /\\ (((~((N)=0)) /\\ (((J)+(N)=S (K)))))))) /\\ ((forall pfc_index_aligned_left_rbcoefficients. (exists pfa_gap_aligned_left_rbcoefficientsbound. pfa_gap_aligned_left_rbcoefficientsbound + S (pfc_index_aligned_left_rbcoefficients) = (K)) -> exists pfc_value_aligned_left_rbcoefficients. ((((exists ff_h_pfp_aligned_left_rbcoefficientsentry. ff_h_pfp_aligned_left_rbcoefficientsentry + S (pfc_value_aligned_left_rbcoefficients) = S ((S (pfc_index_aligned_left_rbcoefficients)) * rc)) /\\ exists ff_q_pfp_aligned_left_rbcoefficientsentry. rb = ff_q_pfp_aligned_left_rbcoefficientsentry * S ((S (pfc_index_aligned_left_rbcoefficients)) * rc) + (pfc_value_aligned_left_rbcoefficients))) /\\ ((exists pfc_terms_code_aligned_left_rbcoefficientscoefficient pfc_terms_scale_aligned_left_rbcoefficientscoefficient pfc_natural_sum_aligned_left_rbcoefficientscoefficient. ((forall pfc_index_aligned_left_rbcoefficientscoefficientdiagonal. (exists pfa_gap_aligned_left_rbcoefficientscoefficientdiagonalbound. pfa_gap_aligned_left_rbcoefficientscoefficientdiagonalbound + S (pfc_index_aligned_left_rbcoefficientscoefficientdiagonal) = (S (pfc_index_aligned_left_rbcoefficients))) -> exists pfc_value_aligned_left_rbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_aligned_left_rbcoefficientscoefficientdiagonalentry. ff_h_pfp_aligned_left_rbcoefficientscoefficientdiagonalentry + S (pfc_value_aligned_left_rbcoefficientscoefficientdiagonal) = S ((S (pfc_index_aligned_left_rbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_left_rbcoefficientscoefficient)) /\\ exists ff_q_pfp_aligned_left_rbcoefficientscoefficientdiagonalentry. pfc_terms_code_aligned_left_rbcoefficientscoefficient = ff_q_pfp_aligned_left_rbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_aligned_left_rbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_left_rbcoefficientscoefficient) + (pfc_value_aligned_left_rbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_aligned_left_rbcoefficientscoefficientdiagonalterm pfc_left_aligned_left_rbcoefficientscoefficientdiagonalterm pfc_right_aligned_left_rbcoefficientscoefficientdiagonalterm. (((pfc_index_aligned_left_rbcoefficientscoefficientdiagonal)+pfc_complement_aligned_left_rbcoefficientscoefficientdiagonalterm=(pfc_index_aligned_left_rbcoefficients)) /\\ ((((((exists pfa_gap_aligned_left_rbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_aligned_left_rbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_aligned_left_rbcoefficientscoefficientdiagonal) = (J)) /\\ ((((exists ff_h_pfp_aligned_left_rbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_aligned_left_rbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_aligned_left_rbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_aligned_left_rbcoefficientscoefficientdiagonal)) * dc)) /\\ exists ff_q_pfp_aligned_left_rbcoefficientscoefficientdiagonaltermleftentry. db = ff_q_pfp_aligned_left_rbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_aligned_left_rbcoefficientscoefficientdiagonal)) * dc) + (pfc_left_aligned_left_rbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_left_rbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_aligned_left_rbcoefficientscoefficientdiagonaltermleftoutside+(J)=(pfc_index_aligned_left_rbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_aligned_left_rbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_aligned_left_rbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_aligned_left_rbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_aligned_left_rbcoefficientscoefficientdiagonalterm) = (N)) /\\ ((((exists ff_h_pfp_aligned_left_rbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_aligned_left_rbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_aligned_left_rbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_aligned_left_rbcoefficientscoefficientdiagonalterm)) * wc)) /\\ exists ff_q_pfp_aligned_left_rbcoefficientscoefficientdiagonaltermrightentry. wb = ff_q_pfp_aligned_left_rbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_aligned_left_rbcoefficientscoefficientdiagonalterm)) * wc) + (pfc_right_aligned_left_rbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_left_rbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_aligned_left_rbcoefficientscoefficientdiagonaltermrightoutside+(N)=(pfc_complement_aligned_left_rbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_aligned_left_rbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_aligned_left_rbcoefficientscoefficientdiagonal)=pfc_left_aligned_left_rbcoefficientscoefficientdiagonalterm*pfc_right_aligned_left_rbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_aligned_left_rbcoefficientscoefficientsum fs_v_pfc_aligned_left_rbcoefficientscoefficientsum. ((((exists fs_h_pfc_aligned_left_rbcoefficientscoefficientsum_body_start. fs_h_pfc_aligned_left_rbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_aligned_left_rbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_rbcoefficientscoefficientsum_body_start. fs_u_pfc_aligned_left_rbcoefficientscoefficientsum = fs_q_pfc_aligned_left_rbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_aligned_left_rbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_aligned_left_rbcoefficientscoefficientsum_body_terminal. fs_h_pfc_aligned_left_rbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_aligned_left_rbcoefficientscoefficient) = S ((S (S (pfc_index_aligned_left_rbcoefficients))) * fs_v_pfc_aligned_left_rbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_rbcoefficientscoefficientsum_body_terminal. fs_u_pfc_aligned_left_rbcoefficientscoefficientsum = fs_q_pfc_aligned_left_rbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_aligned_left_rbcoefficients))) * fs_v_pfc_aligned_left_rbcoefficientscoefficientsum) + (pfc_natural_sum_aligned_left_rbcoefficientscoefficient))) /\\ forall fs_i_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps = S (pfc_index_aligned_left_rbcoefficients)) -> exists fs_a_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps fs_r_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps fs_s_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_left_rbcoefficientscoefficient)) /\\ exists fs_q_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_aligned_left_rbcoefficientscoefficient = fs_q_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_left_rbcoefficientscoefficient) + (fs_a_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_rbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_aligned_left_rbcoefficientscoefficientsum = fs_q_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_rbcoefficientscoefficientsum) + (fs_r_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_rbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_aligned_left_rbcoefficientscoefficientsum = fs_q_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_left_rbcoefficientscoefficientsum) + (fs_s_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps = fs_r_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps + fs_a_pfc_aligned_left_rbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_aligned_left_rbcoefficientscoefficientresiduebound. pfa_gap_aligned_left_rbcoefficientscoefficientresiduebound + S (pfc_value_aligned_left_rbcoefficients) = (p)) /\\ ((exists pfa_offset_left_aligned_left_rbcoefficientscoefficientresiduecongruence pfa_offset_right_aligned_left_rbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_aligned_left_rbcoefficientscoefficient) + (p) * pfa_offset_left_aligned_left_rbcoefficientscoefficientresiduecongruence = (pfc_value_aligned_left_rbcoefficients) + (p) * pfa_offset_right_aligned_left_rbcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_aligned_left_result_left_bounded. (exists fom_gap_pfp_aligned_left_result_left_bounded_index_bound. fom_gap_pfp_aligned_left_result_left_bounded_index_bound + S (fom_index_pfp_aligned_left_result_left_bounded) = H) -> exists fom_value_pfp_aligned_left_result_left_bounded. ((((exists fom_beta_height_pfp_aligned_left_result_left_bounded_entry. fom_beta_height_pfp_aligned_left_result_left_bounded_entry + S (fom_value_pfp_aligned_left_result_left_bounded) = S ((S (fom_index_pfp_aligned_left_result_left_bounded)) * pc)) /\\ exists fom_beta_quotient_pfp_aligned_left_result_left_bounded_entry. pb = fom_beta_quotient_pfp_aligned_left_result_left_bounded_entry * S ((S (fom_index_pfp_aligned_left_result_left_bounded)) * pc) + (fom_value_pfp_aligned_left_result_left_bounded))) /\\ (exists fom_gap_pfp_aligned_left_result_left_bounded_value_bound. fom_gap_pfp_aligned_left_result_left_bounded_value_bound + S (fom_value_pfp_aligned_left_result_left_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_left_result_right_bounded. (exists fom_gap_pfp_aligned_left_result_right_bounded_index_bound. fom_gap_pfp_aligned_left_result_right_bounded_index_bound + S (fom_index_pfp_aligned_left_result_right_bounded) = I) -> exists fom_value_pfp_aligned_left_result_right_bounded. ((((exists fom_beta_height_pfp_aligned_left_result_right_bounded_entry. fom_beta_height_pfp_aligned_left_result_right_bounded_entry + S (fom_value_pfp_aligned_left_result_right_bounded) = S ((S (fom_index_pfp_aligned_left_result_right_bounded)) * qc)) /\\ exists fom_beta_quotient_pfp_aligned_left_result_right_bounded_entry. qb = fom_beta_quotient_pfp_aligned_left_result_right_bounded_entry * S ((S (fom_index_pfp_aligned_left_result_right_bounded)) * qc) + (fom_value_pfp_aligned_left_result_right_bounded))) /\\ (exists fom_gap_pfp_aligned_left_result_right_bounded_value_bound. fom_gap_pfp_aligned_left_result_right_bounded_value_bound + S (fom_value_pfp_aligned_left_result_right_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_left_result_result_bounded. (exists fom_gap_pfp_aligned_left_result_result_bounded_index_bound. fom_gap_pfp_aligned_left_result_result_bounded_index_bound + S (fom_index_pfp_aligned_left_result_result_bounded) = K) -> exists fom_value_pfp_aligned_left_result_result_bounded. ((((exists fom_beta_height_pfp_aligned_left_result_result_bounded_entry. fom_beta_height_pfp_aligned_left_result_result_bounded_entry + S (fom_value_pfp_aligned_left_result_result_bounded) = S ((S (fom_index_pfp_aligned_left_result_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_aligned_left_result_result_bounded_entry. rb = fom_beta_quotient_pfp_aligned_left_result_result_bounded_entry * S ((S (fom_index_pfp_aligned_left_result_result_bounded)) * rc) + (fom_value_pfp_aligned_left_result_result_bounded))) /\\ (exists fom_gap_pfp_aligned_left_result_result_bounded_value_bound. fom_gap_pfp_aligned_left_result_result_bounded_value_bound + S (fom_value_pfp_aligned_left_result_result_bounded) = p))) /\\ ((exists pfaa_left_b_aligned_left_result pfaa_left_c_aligned_left_result pfaa_right_b_aligned_left_result pfaa_right_c_aligned_left_result pfaa_sum_b_aligned_left_result pfaa_sum_c_aligned_left_result pfaa_length_aligned_left_result. ((((forall pfrep_power_aligned_left_result_witness_common_left pfrep_left_aligned_left_result_witness_common_left pfrep_right_aligned_left_result_witness_common_left. ((exists pfrep_position_aligned_left_result_witness_common_leftfirst. ((pfrep_position_aligned_left_result_witness_common_leftfirst+S (pfrep_power_aligned_left_result_witness_common_left)=(H)) /\\ ((((exists ff_h_pfp_aligned_left_result_witness_common_leftfirstentry. ff_h_pfp_aligned_left_result_witness_common_leftfirstentry + S (pfrep_left_aligned_left_result_witness_common_left) = S ((S (pfrep_position_aligned_left_result_witness_common_leftfirst)) * pc)) /\\ exists ff_q_pfp_aligned_left_result_witness_common_leftfirstentry. pb = ff_q_pfp_aligned_left_result_witness_common_leftfirstentry * S ((S (pfrep_position_aligned_left_result_witness_common_leftfirst)) * pc) + (pfrep_left_aligned_left_result_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_left_result_witness_common_leftfirstoutside. pfrep_gap_aligned_left_result_witness_common_leftfirstoutside+(H)=(pfrep_power_aligned_left_result_witness_common_left)) /\\ (((pfrep_left_aligned_left_result_witness_common_left)=0))))) -> ((exists pfrep_position_aligned_left_result_witness_common_leftsecond. ((pfrep_position_aligned_left_result_witness_common_leftsecond+S (pfrep_power_aligned_left_result_witness_common_left)=(pfaa_length_aligned_left_result)) /\\ ((((exists ff_h_pfp_aligned_left_result_witness_common_leftsecondentry. ff_h_pfp_aligned_left_result_witness_common_leftsecondentry + S (pfrep_right_aligned_left_result_witness_common_left) = S ((S (pfrep_position_aligned_left_result_witness_common_leftsecond)) * pfaa_left_c_aligned_left_result)) /\\ exists ff_q_pfp_aligned_left_result_witness_common_leftsecondentry. pfaa_left_b_aligned_left_result = ff_q_pfp_aligned_left_result_witness_common_leftsecondentry * S ((S (pfrep_position_aligned_left_result_witness_common_leftsecond)) * pfaa_left_c_aligned_left_result) + (pfrep_right_aligned_left_result_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_left_result_witness_common_leftsecondoutside. pfrep_gap_aligned_left_result_witness_common_leftsecondoutside+(pfaa_length_aligned_left_result)=(pfrep_power_aligned_left_result_witness_common_left)) /\\ (((pfrep_right_aligned_left_result_witness_common_left)=0))))) -> pfrep_left_aligned_left_result_witness_common_left=pfrep_right_aligned_left_result_witness_common_left) /\\ ((forall pfrep_power_aligned_left_result_witness_common_right pfrep_left_aligned_left_result_witness_common_right pfrep_right_aligned_left_result_witness_common_right. ((exists pfrep_position_aligned_left_result_witness_common_rightfirst. ((pfrep_position_aligned_left_result_witness_common_rightfirst+S (pfrep_power_aligned_left_result_witness_common_right)=(I)) /\\ ((((exists ff_h_pfp_aligned_left_result_witness_common_rightfirstentry. ff_h_pfp_aligned_left_result_witness_common_rightfirstentry + S (pfrep_left_aligned_left_result_witness_common_right) = S ((S (pfrep_position_aligned_left_result_witness_common_rightfirst)) * qc)) /\\ exists ff_q_pfp_aligned_left_result_witness_common_rightfirstentry. qb = ff_q_pfp_aligned_left_result_witness_common_rightfirstentry * S ((S (pfrep_position_aligned_left_result_witness_common_rightfirst)) * qc) + (pfrep_left_aligned_left_result_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_left_result_witness_common_rightfirstoutside. pfrep_gap_aligned_left_result_witness_common_rightfirstoutside+(I)=(pfrep_power_aligned_left_result_witness_common_right)) /\\ (((pfrep_left_aligned_left_result_witness_common_right)=0))))) -> ((exists pfrep_position_aligned_left_result_witness_common_rightsecond. ((pfrep_position_aligned_left_result_witness_common_rightsecond+S (pfrep_power_aligned_left_result_witness_common_right)=(pfaa_length_aligned_left_result)) /\\ ((((exists ff_h_pfp_aligned_left_result_witness_common_rightsecondentry. ff_h_pfp_aligned_left_result_witness_common_rightsecondentry + S (pfrep_right_aligned_left_result_witness_common_right) = S ((S (pfrep_position_aligned_left_result_witness_common_rightsecond)) * pfaa_right_c_aligned_left_result)) /\\ exists ff_q_pfp_aligned_left_result_witness_common_rightsecondentry. pfaa_right_b_aligned_left_result = ff_q_pfp_aligned_left_result_witness_common_rightsecondentry * S ((S (pfrep_position_aligned_left_result_witness_common_rightsecond)) * pfaa_right_c_aligned_left_result) + (pfrep_right_aligned_left_result_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_left_result_witness_common_rightsecondoutside. pfrep_gap_aligned_left_result_witness_common_rightsecondoutside+(pfaa_length_aligned_left_result)=(pfrep_power_aligned_left_result_witness_common_right)) /\\ (((pfrep_right_aligned_left_result_witness_common_right)=0))))) -> pfrep_left_aligned_left_result_witness_common_right=pfrep_right_aligned_left_result_witness_common_right)))) /\\ (((forall pfp_index_aligned_left_result_witness_operation. (exists pfa_gap_aligned_left_result_witness_operationindex. pfa_gap_aligned_left_result_witness_operationindex + S (pfp_index_aligned_left_result_witness_operation) = (pfaa_length_aligned_left_result)) -> exists pfp_left_aligned_left_result_witness_operation pfp_right_aligned_left_result_witness_operation pfp_value_aligned_left_result_witness_operation. ((((exists ff_h_pfp_aligned_left_result_witness_operationleft. ff_h_pfp_aligned_left_result_witness_operationleft + S (pfp_left_aligned_left_result_witness_operation) = S ((S (pfp_index_aligned_left_result_witness_operation)) * pfaa_left_c_aligned_left_result)) /\\ exists ff_q_pfp_aligned_left_result_witness_operationleft. pfaa_left_b_aligned_left_result = ff_q_pfp_aligned_left_result_witness_operationleft * S ((S (pfp_index_aligned_left_result_witness_operation)) * pfaa_left_c_aligned_left_result) + (pfp_left_aligned_left_result_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_left_result_witness_operationright. ff_h_pfp_aligned_left_result_witness_operationright + S (pfp_right_aligned_left_result_witness_operation) = S ((S (pfp_index_aligned_left_result_witness_operation)) * pfaa_right_c_aligned_left_result)) /\\ exists ff_q_pfp_aligned_left_result_witness_operationright. pfaa_right_b_aligned_left_result = ff_q_pfp_aligned_left_result_witness_operationright * S ((S (pfp_index_aligned_left_result_witness_operation)) * pfaa_right_c_aligned_left_result) + (pfp_right_aligned_left_result_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_left_result_witness_operationtarget. ff_h_pfp_aligned_left_result_witness_operationtarget + S (pfp_value_aligned_left_result_witness_operation) = S ((S (pfp_index_aligned_left_result_witness_operation)) * pfaa_sum_c_aligned_left_result)) /\\ exists ff_q_pfp_aligned_left_result_witness_operationtarget. pfaa_sum_b_aligned_left_result = ff_q_pfp_aligned_left_result_witness_operationtarget * S ((S (pfp_index_aligned_left_result_witness_operation)) * pfaa_sum_c_aligned_left_result) + (pfp_value_aligned_left_result_witness_operation))) /\\ ((((exists pfa_gap_aligned_left_result_witness_operationoperationleft. pfa_gap_aligned_left_result_witness_operationoperationleft + S (pfp_left_aligned_left_result_witness_operation) = (p)) /\\ (((exists pfa_gap_aligned_left_result_witness_operationoperationright. pfa_gap_aligned_left_result_witness_operationoperationright + S (pfp_right_aligned_left_result_witness_operation) = (p)) /\\ ((((exists pfa_gap_aligned_left_result_witness_operationoperationresultbound. pfa_gap_aligned_left_result_witness_operationoperationresultbound + S (pfp_value_aligned_left_result_witness_operation) = (p)) /\\ ((exists pfa_offset_left_aligned_left_result_witness_operationoperationresultcongruence pfa_offset_right_aligned_left_result_witness_operationoperationresultcongruence. ((pfp_left_aligned_left_result_witness_operation) + (pfp_right_aligned_left_result_witness_operation)) + (p) * pfa_offset_left_aligned_left_result_witness_operationoperationresultcongruence = (pfp_value_aligned_left_result_witness_operation) + (p) * pfa_offset_right_aligned_left_result_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_aligned_left_result_witness_output pfrep_left_aligned_left_result_witness_output pfrep_right_aligned_left_result_witness_output. ((exists pfrep_position_aligned_left_result_witness_outputfirst. ((pfrep_position_aligned_left_result_witness_outputfirst+S (pfrep_power_aligned_left_result_witness_output)=(pfaa_length_aligned_left_result)) /\\ ((((exists ff_h_pfp_aligned_left_result_witness_outputfirstentry. ff_h_pfp_aligned_left_result_witness_outputfirstentry + S (pfrep_left_aligned_left_result_witness_output) = S ((S (pfrep_position_aligned_left_result_witness_outputfirst)) * pfaa_sum_c_aligned_left_result)) /\\ exists ff_q_pfp_aligned_left_result_witness_outputfirstentry. pfaa_sum_b_aligned_left_result = ff_q_pfp_aligned_left_result_witness_outputfirstentry * S ((S (pfrep_position_aligned_left_result_witness_outputfirst)) * pfaa_sum_c_aligned_left_result) + (pfrep_left_aligned_left_result_witness_output)))))) \\/ (((exists pfrep_gap_aligned_left_result_witness_outputfirstoutside. pfrep_gap_aligned_left_result_witness_outputfirstoutside+(pfaa_length_aligned_left_result)=(pfrep_power_aligned_left_result_witness_output)) /\\ (((pfrep_left_aligned_left_result_witness_output)=0))))) -> ((exists pfrep_position_aligned_left_result_witness_outputsecond. ((pfrep_position_aligned_left_result_witness_outputsecond+S (pfrep_power_aligned_left_result_witness_output)=(K)) /\\ ((((exists ff_h_pfp_aligned_left_result_witness_outputsecondentry. ff_h_pfp_aligned_left_result_witness_outputsecondentry + S (pfrep_right_aligned_left_result_witness_output) = S ((S (pfrep_position_aligned_left_result_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_aligned_left_result_witness_outputsecondentry. rb = ff_q_pfp_aligned_left_result_witness_outputsecondentry * S ((S (pfrep_position_aligned_left_result_witness_outputsecond)) * rc) + (pfrep_right_aligned_left_result_witness_output)))))) \\/ (((exists pfrep_gap_aligned_left_result_witness_outputsecondoutside. pfrep_gap_aligned_left_result_witness_outputsecondoutside+(K)=(pfrep_power_aligned_left_result_witness_output)) /\\ (((pfrep_right_aligned_left_result_witness_output)=0))))) -> pfrep_left_aligned_left_result_witness_output=pfrep_right_aligned_left_result_witness_output)))))))))))))", "statement_sha256": 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"intro db", "intro dc", "intro J", "intro pb", "intro pc", "intro H", "intro qb", "intro qc", "intro I", "intro rb", "intro rc", "intro K", "intro hp", "intro hs", "intro hP", "intro hQ", "intro hR", "have hp0 : ~(p=0)", "intro hpzero", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hpzero", "have hfactor : ((forall fom_index_pfp_aligned_right_pbleft. (exists fom_gap_pfp_aligned_right_pbleft_index_bound. fom_gap_pfp_aligned_right_pbleft_index_bound + S (fom_index_pfp_aligned_right_pbleft) = L) -> exists fom_value_pfp_aligned_right_pbleft. ((((exists fom_beta_height_pfp_aligned_right_pbleft_entry. fom_beta_height_pfp_aligned_right_pbleft_entry + S (fom_value_pfp_aligned_right_pbleft) = S ((S (fom_index_pfp_aligned_right_pbleft)) * uc)) /\\ exists fom_beta_quotient_pfp_aligned_right_pbleft_entry. ub = fom_beta_quotient_pfp_aligned_right_pbleft_entry * S ((S (fom_index_pfp_aligned_right_pbleft)) * uc) + (fom_value_pfp_aligned_right_pbleft))) /\\ (exists fom_gap_pfp_aligned_right_pbleft_value_bound. fom_gap_pfp_aligned_right_pbleft_value_bound + S (fom_value_pfp_aligned_right_pbleft) = p))) /\\ (((forall fom_index_pfp_aligned_right_pbright. (exists fom_gap_pfp_aligned_right_pbright_index_bound. fom_gap_pfp_aligned_right_pbright_index_bound + S (fom_index_pfp_aligned_right_pbright) = J) -> exists fom_value_pfp_aligned_right_pbright. ((((exists fom_beta_height_pfp_aligned_right_pbright_entry. fom_beta_height_pfp_aligned_right_pbright_entry + S (fom_value_pfp_aligned_right_pbright) = S ((S (fom_index_pfp_aligned_right_pbright)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_right_pbright_entry. db = fom_beta_quotient_pfp_aligned_right_pbright_entry * S ((S (fom_index_pfp_aligned_right_pbright)) * dc) + (fom_value_pfp_aligned_right_pbright))) /\\ (exists fom_gap_pfp_aligned_right_pbright_value_bound. fom_gap_pfp_aligned_right_pbright_value_bound + S (fom_value_pfp_aligned_right_pbright) = p))) /\\ (((((((L)=0 \\/ (J)=0) /\\ (((H)=0)))) \\/ (((~((L)=0)) /\\ (((~((J)=0)) /\\ (((L)+(J)=S (H)))))))) /\\ ((forall pfc_index_aligned_right_pbcoefficients. (exists pfa_gap_aligned_right_pbcoefficientsbound. pfa_gap_aligned_right_pbcoefficientsbound + S (pfc_index_aligned_right_pbcoefficients) = (H)) -> exists pfc_value_aligned_right_pbcoefficients. ((((exists ff_h_pfp_aligned_right_pbcoefficientsentry. ff_h_pfp_aligned_right_pbcoefficientsentry + S (pfc_value_aligned_right_pbcoefficients) = S ((S (pfc_index_aligned_right_pbcoefficients)) * pc)) /\\ exists ff_q_pfp_aligned_right_pbcoefficientsentry. pb = ff_q_pfp_aligned_right_pbcoefficientsentry * S ((S (pfc_index_aligned_right_pbcoefficients)) * pc) + (pfc_value_aligned_right_pbcoefficients))) /\\ ((exists pfc_terms_code_aligned_right_pbcoefficientscoefficient pfc_terms_scale_aligned_right_pbcoefficientscoefficient pfc_natural_sum_aligned_right_pbcoefficientscoefficient. ((forall pfc_index_aligned_right_pbcoefficientscoefficientdiagonal. (exists pfa_gap_aligned_right_pbcoefficientscoefficientdiagonalbound. pfa_gap_aligned_right_pbcoefficientscoefficientdiagonalbound + S (pfc_index_aligned_right_pbcoefficientscoefficientdiagonal) = (S (pfc_index_aligned_right_pbcoefficients))) -> exists pfc_value_aligned_right_pbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_aligned_right_pbcoefficientscoefficientdiagonalentry. ff_h_pfp_aligned_right_pbcoefficientscoefficientdiagonalentry + S (pfc_value_aligned_right_pbcoefficientscoefficientdiagonal) = S ((S (pfc_index_aligned_right_pbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_right_pbcoefficientscoefficient)) /\\ exists ff_q_pfp_aligned_right_pbcoefficientscoefficientdiagonalentry. pfc_terms_code_aligned_right_pbcoefficientscoefficient = ff_q_pfp_aligned_right_pbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_aligned_right_pbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_right_pbcoefficientscoefficient) + (pfc_value_aligned_right_pbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_aligned_right_pbcoefficientscoefficientdiagonalterm pfc_left_aligned_right_pbcoefficientscoefficientdiagonalterm pfc_right_aligned_right_pbcoefficientscoefficientdiagonalterm. (((pfc_index_aligned_right_pbcoefficientscoefficientdiagonal)+pfc_complement_aligned_right_pbcoefficientscoefficientdiagonalterm=(pfc_index_aligned_right_pbcoefficients)) /\\ ((((((exists pfa_gap_aligned_right_pbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_aligned_right_pbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_aligned_right_pbcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_aligned_right_pbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_aligned_right_pbcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_aligned_right_pbcoefficientscoefficientdiagonal)) * uc) + (pfc_left_aligned_right_pbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_right_pbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_aligned_right_pbcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_aligned_right_pbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_aligned_right_pbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_aligned_right_pbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_aligned_right_pbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_aligned_right_pbcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_aligned_right_pbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_aligned_right_pbcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_aligned_right_pbcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_aligned_right_pbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_right_pbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_aligned_right_pbcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_aligned_right_pbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_aligned_right_pbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_aligned_right_pbcoefficientscoefficientdiagonal)=pfc_left_aligned_right_pbcoefficientscoefficientdiagonalterm*pfc_right_aligned_right_pbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_aligned_right_pbcoefficientscoefficientsum fs_v_pfc_aligned_right_pbcoefficientscoefficientsum. ((((exists fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_start. fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_start. fs_u_pfc_aligned_right_pbcoefficientscoefficientsum = fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_terminal. fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_aligned_right_pbcoefficientscoefficient) = S ((S (S (pfc_index_aligned_right_pbcoefficients))) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_terminal. fs_u_pfc_aligned_right_pbcoefficientscoefficientsum = fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_aligned_right_pbcoefficients))) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum) + (pfc_natural_sum_aligned_right_pbcoefficientscoefficient))) /\\ forall fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps = S (pfc_index_aligned_right_pbcoefficients)) -> exists fs_a_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps fs_r_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps fs_s_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_right_pbcoefficientscoefficient)) /\\ exists fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_aligned_right_pbcoefficientscoefficient = fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_right_pbcoefficientscoefficient) + (fs_a_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_aligned_right_pbcoefficientscoefficientsum = fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum) + (fs_r_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_aligned_right_pbcoefficientscoefficientsum = fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum) + (fs_s_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps = fs_r_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps + fs_a_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_aligned_right_pbcoefficientscoefficientresiduebound. pfa_gap_aligned_right_pbcoefficientscoefficientresiduebound + S (pfc_value_aligned_right_pbcoefficients) = (p)) /\\ ((exists pfa_offset_left_aligned_right_pbcoefficientscoefficientresiduecongruence pfa_offset_right_aligned_right_pbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_aligned_right_pbcoefficientscoefficient) + (p) * pfa_offset_left_aligned_right_pbcoefficientscoefficientresiduecongruence = (pfc_value_aligned_right_pbcoefficients) + (p) * pfa_offset_right_aligned_right_pbcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hP", "cases hfactor", "cases hfactor_right", "cases hs", "cases hs_right", "cases hs_right_right", "cases hs_right_right_right", "cases hs_right_right_right_witness", "cases hs_right_right_right_witness_witness", "cases hs_right_right_right_witness_witness_witness", "cases hs_right_right_right_witness_witness_witness_witness", "cases hs_right_right_right_witness_witness_witness_witness_witness", "cases hs_right_right_right_witness_witness_witness_witness_witness_witness", "cases hs_right_right_right_witness_witness_witness_witness_witness_witness_witness", "cases hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_right", "cases hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_left", "have hproducts : exists O PB PC QB QC RB RC. ((((forall fom_index_pfp_aligned_right_constructed_PBleft. (exists fom_gap_pfp_aligned_right_constructed_PBleft_index_bound. fom_gap_pfp_aligned_right_constructed_PBleft_index_bound + S (fom_index_pfp_aligned_right_constructed_PBleft) = x6) -> exists fom_value_pfp_aligned_right_constructed_PBleft. ((((exists fom_beta_height_pfp_aligned_right_constructed_PBleft_entry. fom_beta_height_pfp_aligned_right_constructed_PBleft_entry + S (fom_value_pfp_aligned_right_constructed_PBleft) = S ((S (fom_index_pfp_aligned_right_constructed_PBleft)) * x1)) /\\ exists fom_beta_quotient_pfp_aligned_right_constructed_PBleft_entry. x = fom_beta_quotient_pfp_aligned_right_constructed_PBleft_entry * S ((S (fom_index_pfp_aligned_right_constructed_PBleft)) * x1) + (fom_value_pfp_aligned_right_constructed_PBleft))) /\\ (exists fom_gap_pfp_aligned_right_constructed_PBleft_value_bound. fom_gap_pfp_aligned_right_constructed_PBleft_value_bound + S (fom_value_pfp_aligned_right_constructed_PBleft) = p))) /\\ (((forall fom_index_pfp_aligned_right_constructed_PBright. (exists fom_gap_pfp_aligned_right_constructed_PBright_index_bound. fom_gap_pfp_aligned_right_constructed_PBright_index_bound + S (fom_index_pfp_aligned_right_constructed_PBright) = J) -> exists fom_value_pfp_aligned_right_constructed_PBright. ((((exists fom_beta_height_pfp_aligned_right_constructed_PBright_entry. fom_beta_height_pfp_aligned_right_constructed_PBright_entry + S (fom_value_pfp_aligned_right_constructed_PBright) = S ((S (fom_index_pfp_aligned_right_constructed_PBright)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_right_constructed_PBright_entry. db = fom_beta_quotient_pfp_aligned_right_constructed_PBright_entry * S ((S (fom_index_pfp_aligned_right_constructed_PBright)) * dc) + (fom_value_pfp_aligned_right_constructed_PBright))) /\\ (exists fom_gap_pfp_aligned_right_constructed_PBright_value_bound. fom_gap_pfp_aligned_right_constructed_PBright_value_bound + S (fom_value_pfp_aligned_right_constructed_PBright) = p))) /\\ (((((((x6)=0 \\/ (J)=0) /\\ (((O)=0)))) \\/ (((~((x6)=0)) /\\ (((~((J)=0)) /\\ (((x6)+(J)=S (O)))))))) /\\ ((forall pfc_index_aligned_right_constructed_PBcoefficients. (exists pfa_gap_aligned_right_constructed_PBcoefficientsbound. pfa_gap_aligned_right_constructed_PBcoefficientsbound + S (pfc_index_aligned_right_constructed_PBcoefficients) = (O)) -> exists pfc_value_aligned_right_constructed_PBcoefficients. ((((exists ff_h_pfp_aligned_right_constructed_PBcoefficientsentry. ff_h_pfp_aligned_right_constructed_PBcoefficientsentry + S (pfc_value_aligned_right_constructed_PBcoefficients) = S ((S (pfc_index_aligned_right_constructed_PBcoefficients)) * PC)) /\\ exists ff_q_pfp_aligned_right_constructed_PBcoefficientsentry. PB = ff_q_pfp_aligned_right_constructed_PBcoefficientsentry * S ((S (pfc_index_aligned_right_constructed_PBcoefficients)) * PC) + (pfc_value_aligned_right_constructed_PBcoefficients))) /\\ ((exists pfc_terms_code_aligned_right_constructed_PBcoefficientscoefficient pfc_terms_scale_aligned_right_constructed_PBcoefficientscoefficient pfc_natural_sum_aligned_right_constructed_PBcoefficientscoefficient. ((forall pfc_index_aligned_right_constructed_PBcoefficientscoefficientdiagonal. (exists pfa_gap_aligned_right_constructed_PBcoefficientscoefficientdiagonalbound. pfa_gap_aligned_right_constructed_PBcoefficientscoefficientdiagonalbound + S (pfc_index_aligned_right_constructed_PBcoefficientscoefficientdiagonal) = (S (pfc_index_aligned_right_constructed_PBcoefficients))) -> exists pfc_value_aligned_right_constructed_PBcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_aligned_right_constructed_PBcoefficientscoefficientdiagonalentry. ff_h_pfp_aligned_right_constructed_PBcoefficientscoefficientdiagonalentry + S (pfc_value_aligned_right_constructed_PBcoefficientscoefficientdiagonal) = S ((S (pfc_index_aligned_right_constructed_PBcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_right_constructed_PBcoefficientscoefficient)) /\\ exists ff_q_pfp_aligned_right_constructed_PBcoefficientscoefficientdiagonalentry. pfc_terms_code_aligned_right_constructed_PBcoefficientscoefficient = ff_q_pfp_aligned_right_constructed_PBcoefficientscoefficientdiagonalentry * S ((S (pfc_index_aligned_right_constructed_PBcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_right_constructed_PBcoefficientscoefficient) + (pfc_value_aligned_right_constructed_PBcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm pfc_left_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm pfc_right_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm. (((pfc_index_aligned_right_constructed_PBcoefficientscoefficientdiagonal)+pfc_complement_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm=(pfc_index_aligned_right_constructed_PBcoefficients)) /\\ ((((((exists pfa_gap_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermleftinside. pfa_gap_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_aligned_right_constructed_PBcoefficientscoefficientdiagonal) = (x6)) /\\ ((((exists ff_h_pfp_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_aligned_right_constructed_PBcoefficientscoefficientdiagonal)) * x1)) /\\ exists ff_q_pfp_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermleftentry. x = ff_q_pfp_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_aligned_right_constructed_PBcoefficientscoefficientdiagonal)) * x1) + (pfc_left_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermleftoutside+(x6)=(pfc_index_aligned_right_constructed_PBcoefficientscoefficientdiagonal)) /\\ (((pfc_left_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermrightinside. pfa_gap_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_aligned_right_constructed_PBcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_aligned_right_constructed_PBcoefficientscoefficientdiagonal)=pfc_left_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm*pfc_right_aligned_right_constructed_PBcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_aligned_right_constructed_PBcoefficientscoefficientsum fs_v_pfc_aligned_right_constructed_PBcoefficientscoefficientsum. ((((exists fs_h_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_start. fs_h_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_aligned_right_constructed_PBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_start. fs_u_pfc_aligned_right_constructed_PBcoefficientscoefficientsum = fs_q_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_aligned_right_constructed_PBcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_terminal. fs_h_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_aligned_right_constructed_PBcoefficientscoefficient) = S ((S (S (pfc_index_aligned_right_constructed_PBcoefficients))) * fs_v_pfc_aligned_right_constructed_PBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_terminal. fs_u_pfc_aligned_right_constructed_PBcoefficientscoefficientsum = fs_q_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_aligned_right_constructed_PBcoefficients))) * fs_v_pfc_aligned_right_constructed_PBcoefficientscoefficientsum) + (pfc_natural_sum_aligned_right_constructed_PBcoefficientscoefficient))) /\\ forall fs_i_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps = S (pfc_index_aligned_right_constructed_PBcoefficients)) -> exists fs_a_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps fs_r_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps fs_s_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_right_constructed_PBcoefficientscoefficient)) /\\ exists fs_q_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_aligned_right_constructed_PBcoefficientscoefficient = fs_q_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_right_constructed_PBcoefficientscoefficient) + (fs_a_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_constructed_PBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_aligned_right_constructed_PBcoefficientscoefficientsum = fs_q_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_constructed_PBcoefficientscoefficientsum) + (fs_r_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_constructed_PBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_aligned_right_constructed_PBcoefficientscoefficientsum = fs_q_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_constructed_PBcoefficientscoefficientsum) + (fs_s_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps = fs_r_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps + fs_a_pfc_aligned_right_constructed_PBcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_aligned_right_constructed_PBcoefficientscoefficientresiduebound. pfa_gap_aligned_right_constructed_PBcoefficientscoefficientresiduebound + S (pfc_value_aligned_right_constructed_PBcoefficients) = (p)) /\\ ((exists pfa_offset_left_aligned_right_constructed_PBcoefficientscoefficientresiduecongruence pfa_offset_right_aligned_right_constructed_PBcoefficientscoefficientresiduecongruence. (pfc_natural_sum_aligned_right_constructed_PBcoefficientscoefficient) + (p) * pfa_offset_left_aligned_right_constructed_PBcoefficientscoefficientresiduecongruence = (pfc_value_aligned_right_constructed_PBcoefficients) + (p) * pfa_offset_right_aligned_right_constructed_PBcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_aligned_right_constructed_QBleft. (exists fom_gap_pfp_aligned_right_constructed_QBleft_index_bound. fom_gap_pfp_aligned_right_constructed_QBleft_index_bound + S (fom_index_pfp_aligned_right_constructed_QBleft) = x6) -> exists fom_value_pfp_aligned_right_constructed_QBleft. ((((exists fom_beta_height_pfp_aligned_right_constructed_QBleft_entry. fom_beta_height_pfp_aligned_right_constructed_QBleft_entry + S (fom_value_pfp_aligned_right_constructed_QBleft) = S ((S (fom_index_pfp_aligned_right_constructed_QBleft)) * x3)) /\\ exists fom_beta_quotient_pfp_aligned_right_constructed_QBleft_entry. x2 = fom_beta_quotient_pfp_aligned_right_constructed_QBleft_entry * S ((S (fom_index_pfp_aligned_right_constructed_QBleft)) * x3) + (fom_value_pfp_aligned_right_constructed_QBleft))) /\\ (exists fom_gap_pfp_aligned_right_constructed_QBleft_value_bound. fom_gap_pfp_aligned_right_constructed_QBleft_value_bound + S (fom_value_pfp_aligned_right_constructed_QBleft) = p))) /\\ (((forall fom_index_pfp_aligned_right_constructed_QBright. (exists fom_gap_pfp_aligned_right_constructed_QBright_index_bound. fom_gap_pfp_aligned_right_constructed_QBright_index_bound + S (fom_index_pfp_aligned_right_constructed_QBright) = J) -> exists fom_value_pfp_aligned_right_constructed_QBright. ((((exists fom_beta_height_pfp_aligned_right_constructed_QBright_entry. fom_beta_height_pfp_aligned_right_constructed_QBright_entry + S (fom_value_pfp_aligned_right_constructed_QBright) = S ((S (fom_index_pfp_aligned_right_constructed_QBright)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_right_constructed_QBright_entry. db = fom_beta_quotient_pfp_aligned_right_constructed_QBright_entry * S ((S (fom_index_pfp_aligned_right_constructed_QBright)) * dc) + (fom_value_pfp_aligned_right_constructed_QBright))) /\\ (exists fom_gap_pfp_aligned_right_constructed_QBright_value_bound. fom_gap_pfp_aligned_right_constructed_QBright_value_bound + S (fom_value_pfp_aligned_right_constructed_QBright) = p))) /\\ (((((((x6)=0 \\/ (J)=0) /\\ (((O)=0)))) \\/ (((~((x6)=0)) /\\ (((~((J)=0)) /\\ (((x6)+(J)=S (O)))))))) /\\ ((forall pfc_index_aligned_right_constructed_QBcoefficients. (exists pfa_gap_aligned_right_constructed_QBcoefficientsbound. pfa_gap_aligned_right_constructed_QBcoefficientsbound + S (pfc_index_aligned_right_constructed_QBcoefficients) = (O)) -> exists pfc_value_aligned_right_constructed_QBcoefficients. ((((exists ff_h_pfp_aligned_right_constructed_QBcoefficientsentry. ff_h_pfp_aligned_right_constructed_QBcoefficientsentry + S (pfc_value_aligned_right_constructed_QBcoefficients) = S ((S (pfc_index_aligned_right_constructed_QBcoefficients)) * QC)) /\\ exists ff_q_pfp_aligned_right_constructed_QBcoefficientsentry. QB = ff_q_pfp_aligned_right_constructed_QBcoefficientsentry * S ((S (pfc_index_aligned_right_constructed_QBcoefficients)) * QC) + (pfc_value_aligned_right_constructed_QBcoefficients))) /\\ ((exists pfc_terms_code_aligned_right_constructed_QBcoefficientscoefficient pfc_terms_scale_aligned_right_constructed_QBcoefficientscoefficient pfc_natural_sum_aligned_right_constructed_QBcoefficientscoefficient. ((forall pfc_index_aligned_right_constructed_QBcoefficientscoefficientdiagonal. (exists pfa_gap_aligned_right_constructed_QBcoefficientscoefficientdiagonalbound. pfa_gap_aligned_right_constructed_QBcoefficientscoefficientdiagonalbound + S (pfc_index_aligned_right_constructed_QBcoefficientscoefficientdiagonal) = (S (pfc_index_aligned_right_constructed_QBcoefficients))) -> exists pfc_value_aligned_right_constructed_QBcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_aligned_right_constructed_QBcoefficientscoefficientdiagonalentry. ff_h_pfp_aligned_right_constructed_QBcoefficientscoefficientdiagonalentry + S (pfc_value_aligned_right_constructed_QBcoefficientscoefficientdiagonal) = S ((S (pfc_index_aligned_right_constructed_QBcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_right_constructed_QBcoefficientscoefficient)) /\\ exists ff_q_pfp_aligned_right_constructed_QBcoefficientscoefficientdiagonalentry. pfc_terms_code_aligned_right_constructed_QBcoefficientscoefficient = ff_q_pfp_aligned_right_constructed_QBcoefficientscoefficientdiagonalentry * S ((S (pfc_index_aligned_right_constructed_QBcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_right_constructed_QBcoefficientscoefficient) + (pfc_value_aligned_right_constructed_QBcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm pfc_left_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm pfc_right_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm. (((pfc_index_aligned_right_constructed_QBcoefficientscoefficientdiagonal)+pfc_complement_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm=(pfc_index_aligned_right_constructed_QBcoefficients)) /\\ ((((((exists pfa_gap_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermleftinside. pfa_gap_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_aligned_right_constructed_QBcoefficientscoefficientdiagonal) = (x6)) /\\ ((((exists ff_h_pfp_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_aligned_right_constructed_QBcoefficientscoefficientdiagonal)) * x3)) /\\ exists ff_q_pfp_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermleftentry. x2 = ff_q_pfp_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_aligned_right_constructed_QBcoefficientscoefficientdiagonal)) * x3) + (pfc_left_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermleftoutside+(x6)=(pfc_index_aligned_right_constructed_QBcoefficientscoefficientdiagonal)) /\\ (((pfc_left_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermrightinside. pfa_gap_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_aligned_right_constructed_QBcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_aligned_right_constructed_QBcoefficientscoefficientdiagonal)=pfc_left_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm*pfc_right_aligned_right_constructed_QBcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_aligned_right_constructed_QBcoefficientscoefficientsum fs_v_pfc_aligned_right_constructed_QBcoefficientscoefficientsum. ((((exists fs_h_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_start. fs_h_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_aligned_right_constructed_QBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_start. fs_u_pfc_aligned_right_constructed_QBcoefficientscoefficientsum = fs_q_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_aligned_right_constructed_QBcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_terminal. fs_h_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_aligned_right_constructed_QBcoefficientscoefficient) = S ((S (S (pfc_index_aligned_right_constructed_QBcoefficients))) * fs_v_pfc_aligned_right_constructed_QBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_terminal. fs_u_pfc_aligned_right_constructed_QBcoefficientscoefficientsum = fs_q_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_aligned_right_constructed_QBcoefficients))) * fs_v_pfc_aligned_right_constructed_QBcoefficientscoefficientsum) + (pfc_natural_sum_aligned_right_constructed_QBcoefficientscoefficient))) /\\ forall fs_i_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps = S (pfc_index_aligned_right_constructed_QBcoefficients)) -> exists fs_a_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps fs_r_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps fs_s_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_right_constructed_QBcoefficientscoefficient)) /\\ exists fs_q_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_aligned_right_constructed_QBcoefficientscoefficient = fs_q_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_right_constructed_QBcoefficientscoefficient) + (fs_a_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_constructed_QBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_aligned_right_constructed_QBcoefficientscoefficientsum = fs_q_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_constructed_QBcoefficientscoefficientsum) + (fs_r_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_constructed_QBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_aligned_right_constructed_QBcoefficientscoefficientsum = fs_q_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_constructed_QBcoefficientscoefficientsum) + (fs_s_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps = fs_r_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps + fs_a_pfc_aligned_right_constructed_QBcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_aligned_right_constructed_QBcoefficientscoefficientresiduebound. pfa_gap_aligned_right_constructed_QBcoefficientscoefficientresiduebound + S (pfc_value_aligned_right_constructed_QBcoefficients) = (p)) /\\ ((exists pfa_offset_left_aligned_right_constructed_QBcoefficientscoefficientresiduecongruence pfa_offset_right_aligned_right_constructed_QBcoefficientscoefficientresiduecongruence. (pfc_natural_sum_aligned_right_constructed_QBcoefficientscoefficient) + (p) * pfa_offset_left_aligned_right_constructed_QBcoefficientscoefficientresiduecongruence = (pfc_value_aligned_right_constructed_QBcoefficients) + (p) * pfa_offset_right_aligned_right_constructed_QBcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_aligned_right_constructed_RBleft. (exists fom_gap_pfp_aligned_right_constructed_RBleft_index_bound. fom_gap_pfp_aligned_right_constructed_RBleft_index_bound + S (fom_index_pfp_aligned_right_constructed_RBleft) = x6) -> exists fom_value_pfp_aligned_right_constructed_RBleft. ((((exists fom_beta_height_pfp_aligned_right_constructed_RBleft_entry. fom_beta_height_pfp_aligned_right_constructed_RBleft_entry + S (fom_value_pfp_aligned_right_constructed_RBleft) = S ((S (fom_index_pfp_aligned_right_constructed_RBleft)) * x5)) /\\ exists fom_beta_quotient_pfp_aligned_right_constructed_RBleft_entry. x4 = fom_beta_quotient_pfp_aligned_right_constructed_RBleft_entry * S ((S (fom_index_pfp_aligned_right_constructed_RBleft)) * x5) + (fom_value_pfp_aligned_right_constructed_RBleft))) /\\ (exists fom_gap_pfp_aligned_right_constructed_RBleft_value_bound. fom_gap_pfp_aligned_right_constructed_RBleft_value_bound + S (fom_value_pfp_aligned_right_constructed_RBleft) = p))) /\\ (((forall fom_index_pfp_aligned_right_constructed_RBright. (exists fom_gap_pfp_aligned_right_constructed_RBright_index_bound. fom_gap_pfp_aligned_right_constructed_RBright_index_bound + S (fom_index_pfp_aligned_right_constructed_RBright) = J) -> exists fom_value_pfp_aligned_right_constructed_RBright. ((((exists fom_beta_height_pfp_aligned_right_constructed_RBright_entry. fom_beta_height_pfp_aligned_right_constructed_RBright_entry + S (fom_value_pfp_aligned_right_constructed_RBright) = S ((S (fom_index_pfp_aligned_right_constructed_RBright)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_right_constructed_RBright_entry. db = fom_beta_quotient_pfp_aligned_right_constructed_RBright_entry * S ((S (fom_index_pfp_aligned_right_constructed_RBright)) * dc) + (fom_value_pfp_aligned_right_constructed_RBright))) /\\ (exists fom_gap_pfp_aligned_right_constructed_RBright_value_bound. fom_gap_pfp_aligned_right_constructed_RBright_value_bound + S (fom_value_pfp_aligned_right_constructed_RBright) = p))) /\\ (((((((x6)=0 \\/ (J)=0) /\\ (((O)=0)))) \\/ (((~((x6)=0)) /\\ (((~((J)=0)) /\\ (((x6)+(J)=S (O)))))))) /\\ ((forall pfc_index_aligned_right_constructed_RBcoefficients. (exists pfa_gap_aligned_right_constructed_RBcoefficientsbound. pfa_gap_aligned_right_constructed_RBcoefficientsbound + S (pfc_index_aligned_right_constructed_RBcoefficients) = (O)) -> exists pfc_value_aligned_right_constructed_RBcoefficients. ((((exists ff_h_pfp_aligned_right_constructed_RBcoefficientsentry. ff_h_pfp_aligned_right_constructed_RBcoefficientsentry + S (pfc_value_aligned_right_constructed_RBcoefficients) = S ((S (pfc_index_aligned_right_constructed_RBcoefficients)) * RC)) /\\ exists ff_q_pfp_aligned_right_constructed_RBcoefficientsentry. RB = ff_q_pfp_aligned_right_constructed_RBcoefficientsentry * S ((S (pfc_index_aligned_right_constructed_RBcoefficients)) * RC) + (pfc_value_aligned_right_constructed_RBcoefficients))) /\\ ((exists pfc_terms_code_aligned_right_constructed_RBcoefficientscoefficient pfc_terms_scale_aligned_right_constructed_RBcoefficientscoefficient pfc_natural_sum_aligned_right_constructed_RBcoefficientscoefficient. ((forall pfc_index_aligned_right_constructed_RBcoefficientscoefficientdiagonal. (exists pfa_gap_aligned_right_constructed_RBcoefficientscoefficientdiagonalbound. pfa_gap_aligned_right_constructed_RBcoefficientscoefficientdiagonalbound + S (pfc_index_aligned_right_constructed_RBcoefficientscoefficientdiagonal) = (S (pfc_index_aligned_right_constructed_RBcoefficients))) -> exists pfc_value_aligned_right_constructed_RBcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_aligned_right_constructed_RBcoefficientscoefficientdiagonalentry. ff_h_pfp_aligned_right_constructed_RBcoefficientscoefficientdiagonalentry + S (pfc_value_aligned_right_constructed_RBcoefficientscoefficientdiagonal) = S ((S (pfc_index_aligned_right_constructed_RBcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_right_constructed_RBcoefficientscoefficient)) /\\ exists ff_q_pfp_aligned_right_constructed_RBcoefficientscoefficientdiagonalentry. pfc_terms_code_aligned_right_constructed_RBcoefficientscoefficient = ff_q_pfp_aligned_right_constructed_RBcoefficientscoefficientdiagonalentry * S ((S (pfc_index_aligned_right_constructed_RBcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_right_constructed_RBcoefficientscoefficient) + (pfc_value_aligned_right_constructed_RBcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm pfc_left_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm pfc_right_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm. (((pfc_index_aligned_right_constructed_RBcoefficientscoefficientdiagonal)+pfc_complement_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm=(pfc_index_aligned_right_constructed_RBcoefficients)) /\\ ((((((exists pfa_gap_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermleftinside. pfa_gap_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_aligned_right_constructed_RBcoefficientscoefficientdiagonal) = (x6)) /\\ ((((exists ff_h_pfp_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_aligned_right_constructed_RBcoefficientscoefficientdiagonal)) * x5)) /\\ exists ff_q_pfp_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermleftentry. x4 = ff_q_pfp_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_aligned_right_constructed_RBcoefficientscoefficientdiagonal)) * x5) + (pfc_left_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermleftoutside+(x6)=(pfc_index_aligned_right_constructed_RBcoefficientscoefficientdiagonal)) /\\ (((pfc_left_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermrightinside. pfa_gap_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_aligned_right_constructed_RBcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_aligned_right_constructed_RBcoefficientscoefficientdiagonal)=pfc_left_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm*pfc_right_aligned_right_constructed_RBcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_aligned_right_constructed_RBcoefficientscoefficientsum fs_v_pfc_aligned_right_constructed_RBcoefficientscoefficientsum. ((((exists fs_h_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_start. fs_h_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_aligned_right_constructed_RBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_start. fs_u_pfc_aligned_right_constructed_RBcoefficientscoefficientsum = fs_q_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_aligned_right_constructed_RBcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_terminal. fs_h_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_aligned_right_constructed_RBcoefficientscoefficient) = S ((S (S (pfc_index_aligned_right_constructed_RBcoefficients))) * fs_v_pfc_aligned_right_constructed_RBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_terminal. fs_u_pfc_aligned_right_constructed_RBcoefficientscoefficientsum = fs_q_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_aligned_right_constructed_RBcoefficients))) * fs_v_pfc_aligned_right_constructed_RBcoefficientscoefficientsum) + (pfc_natural_sum_aligned_right_constructed_RBcoefficientscoefficient))) /\\ forall fs_i_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps = S (pfc_index_aligned_right_constructed_RBcoefficients)) -> exists fs_a_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps fs_r_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps fs_s_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_right_constructed_RBcoefficientscoefficient)) /\\ exists fs_q_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_aligned_right_constructed_RBcoefficientscoefficient = fs_q_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_right_constructed_RBcoefficientscoefficient) + (fs_a_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_constructed_RBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_aligned_right_constructed_RBcoefficientscoefficientsum = fs_q_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_constructed_RBcoefficientscoefficientsum) + (fs_r_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_constructed_RBcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_aligned_right_constructed_RBcoefficientscoefficientsum = fs_q_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_constructed_RBcoefficientscoefficientsum) + (fs_s_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps = fs_r_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps + fs_a_pfc_aligned_right_constructed_RBcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_aligned_right_constructed_RBcoefficientscoefficientresiduebound. pfa_gap_aligned_right_constructed_RBcoefficientscoefficientresiduebound + S (pfc_value_aligned_right_constructed_RBcoefficients) = (p)) /\\ ((exists pfa_offset_left_aligned_right_constructed_RBcoefficientscoefficientresiduecongruence pfa_offset_right_aligned_right_constructed_RBcoefficientscoefficientresiduecongruence. (pfc_natural_sum_aligned_right_constructed_RBcoefficientscoefficient) + (p) * pfa_offset_left_aligned_right_constructed_RBcoefficientscoefficientresiduecongruence = (pfc_value_aligned_right_constructed_RBcoefficients) + (p) * pfa_offset_right_aligned_right_constructed_RBcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfp_index_aligned_right_constructed_sum. (exists pfa_gap_aligned_right_constructed_sumindex. pfa_gap_aligned_right_constructed_sumindex + S (pfp_index_aligned_right_constructed_sum) = (O)) -> exists pfp_left_aligned_right_constructed_sum pfp_right_aligned_right_constructed_sum pfp_value_aligned_right_constructed_sum. ((((exists ff_h_pfp_aligned_right_constructed_sumleft. ff_h_pfp_aligned_right_constructed_sumleft + S (pfp_left_aligned_right_constructed_sum) = S ((S (pfp_index_aligned_right_constructed_sum)) * PC)) /\\ exists ff_q_pfp_aligned_right_constructed_sumleft. PB = ff_q_pfp_aligned_right_constructed_sumleft * S ((S (pfp_index_aligned_right_constructed_sum)) * PC) + (pfp_left_aligned_right_constructed_sum))) /\\ (((((exists ff_h_pfp_aligned_right_constructed_sumright. ff_h_pfp_aligned_right_constructed_sumright + S (pfp_right_aligned_right_constructed_sum) = S ((S (pfp_index_aligned_right_constructed_sum)) * QC)) /\\ exists ff_q_pfp_aligned_right_constructed_sumright. QB = ff_q_pfp_aligned_right_constructed_sumright * S ((S (pfp_index_aligned_right_constructed_sum)) * QC) + (pfp_right_aligned_right_constructed_sum))) /\\ (((((exists ff_h_pfp_aligned_right_constructed_sumtarget. ff_h_pfp_aligned_right_constructed_sumtarget + S (pfp_value_aligned_right_constructed_sum) = S ((S (pfp_index_aligned_right_constructed_sum)) * RC)) /\\ exists ff_q_pfp_aligned_right_constructed_sumtarget. RB = ff_q_pfp_aligned_right_constructed_sumtarget * S ((S (pfp_index_aligned_right_constructed_sum)) * RC) + (pfp_value_aligned_right_constructed_sum))) /\\ ((((exists pfa_gap_aligned_right_constructed_sumoperationleft. pfa_gap_aligned_right_constructed_sumoperationleft + S (pfp_left_aligned_right_constructed_sum) = (p)) /\\ (((exists pfa_gap_aligned_right_constructed_sumoperationright. pfa_gap_aligned_right_constructed_sumoperationright + S (pfp_right_aligned_right_constructed_sum) = (p)) /\\ ((((exists pfa_gap_aligned_right_constructed_sumoperationresultbound. pfa_gap_aligned_right_constructed_sumoperationresultbound + S (pfp_value_aligned_right_constructed_sum) = (p)) /\\ ((exists pfa_offset_left_aligned_right_constructed_sumoperationresultcongruence pfa_offset_right_aligned_right_constructed_sumoperationresultcongruence. ((pfp_left_aligned_right_constructed_sum) + (pfp_right_aligned_right_constructed_sum)) + (p) * pfa_offset_left_aligned_right_constructed_sumoperationresultcongruence = (pfp_value_aligned_right_constructed_sum) + (p) * pfa_offset_right_aligned_right_constructed_sumoperationresultcongruence))))))))))))))))))))))", "specialize prime_field_polynomial_right_distributive_products_exists (p)", "specialize prime_field_polynomial_right_distributive_products_exists (x)", "specialize prime_field_polynomial_right_distributive_products_exists (x1)", "specialize prime_field_polynomial_right_distributive_products_exists (x2)", "specialize prime_field_polynomial_right_distributive_products_exists (x3)", "specialize prime_field_polynomial_right_distributive_products_exists (x4)", "specialize prime_field_polynomial_right_distributive_products_exists (x5)", "specialize prime_field_polynomial_right_distributive_products_exists (x6)", "specialize prime_field_polynomial_right_distributive_products_exists (db)", "specialize prime_field_polynomial_right_distributive_products_exists (dc)", "specialize prime_field_polynomial_right_distributive_products_exists (J)", "apply prime_field_polynomial_right_distributive_products_exists", "exact hp0", "exact hfactor_right_left", "exact hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_left", "cases hproducts", "cases hproducts_witness", "cases hproducts_witness_witness", "cases hproducts_witness_witness_witness", "cases hproducts_witness_witness_witness_witness", "cases hproducts_witness_witness_witness_witness_witness", "cases hproducts_witness_witness_witness_witness_witness_witness", "cases hproducts_witness_witness_witness_witness_witness_witness_witness", "cases hproducts_witness_witness_witness_witness_witness_witness_witness_right", "cases hproducts_witness_witness_witness_witness_witness_witness_witness_right_right", "specialize prime_field_polynomial_aligned_add_from_common (p)", "specialize prime_field_polynomial_aligned_add_from_common (pb)", "specialize prime_field_polynomial_aligned_add_from_common (pc)", "specialize prime_field_polynomial_aligned_add_from_common (H)", "specialize prime_field_polynomial_aligned_add_from_common (qb)", "specialize prime_field_polynomial_aligned_add_from_common (qc)", "specialize prime_field_polynomial_aligned_add_from_common (I)", "specialize prime_field_polynomial_aligned_add_from_common (rb)", "specialize prime_field_polynomial_aligned_add_from_common (rc)", "specialize prime_field_polynomial_aligned_add_from_common (K)", "specialize prime_field_polynomial_aligned_add_from_common (x8)", "specialize prime_field_polynomial_aligned_add_from_common (x9)", "specialize prime_field_polynomial_aligned_add_from_common (x10)", "specialize prime_field_polynomial_aligned_add_from_common (x11)", "specialize prime_field_polynomial_aligned_add_from_common (x12)", "specialize prime_field_polynomial_aligned_add_from_common (x13)", "specialize prime_field_polynomial_aligned_add_from_common (x7)", "apply prime_field_polynomial_aligned_add_from_common", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (ub)", "specialize prime_field_polynomial_convolution_bounded (uc)", "specialize prime_field_polynomial_convolution_bounded (L)", "specialize prime_field_polynomial_convolution_bounded (db)", "specialize prime_field_polynomial_convolution_bounded (dc)", "specialize prime_field_polynomial_convolution_bounded (J)", "specialize prime_field_polynomial_convolution_bounded (pb)", "specialize prime_field_polynomial_convolution_bounded (pc)", "specialize prime_field_polynomial_convolution_bounded (H)", "apply prime_field_polynomial_convolution_bounded", "exact hP", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (vb)", "specialize prime_field_polynomial_convolution_bounded (vc)", "specialize prime_field_polynomial_convolution_bounded (M)", "specialize prime_field_polynomial_convolution_bounded (db)", "specialize prime_field_polynomial_convolution_bounded (dc)", "specialize prime_field_polynomial_convolution_bounded (J)", "specialize prime_field_polynomial_convolution_bounded (qb)", "specialize prime_field_polynomial_convolution_bounded (qc)", "specialize prime_field_polynomial_convolution_bounded (I)", "apply prime_field_polynomial_convolution_bounded", "exact hQ", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (wb)", "specialize prime_field_polynomial_convolution_bounded (wc)", "specialize prime_field_polynomial_convolution_bounded (N)", "specialize prime_field_polynomial_convolution_bounded (db)", "specialize prime_field_polynomial_convolution_bounded (dc)", "specialize prime_field_polynomial_convolution_bounded (J)", "specialize prime_field_polynomial_convolution_bounded (rb)", "specialize prime_field_polynomial_convolution_bounded (rc)", "specialize prime_field_polynomial_convolution_bounded (K)", "apply prime_field_polynomial_convolution_bounded", "exact hR", "split", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (p)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (ub)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (uc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (L)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (db)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (dc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (J)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (pb)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (pc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (H)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x1)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x6)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x8)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x9)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x7)", "apply prime_field_polynomial_convolution_equivalent_congruent_left", "exact hp0", "exact hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_left_left", "exact hP", "exact hproducts_witness_witness_witness_witness_witness_witness_witness_left", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (p)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (vb)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (vc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (M)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (db)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (dc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (J)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (qb)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (qc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (I)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x2)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x3)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x6)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x10)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x11)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x7)", "apply prime_field_polynomial_convolution_equivalent_congruent_left", "exact hp0", "exact hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_left_right", "exact hQ", "exact hproducts_witness_witness_witness_witness_witness_witness_witness_right_left", "exact hproducts_witness_witness_witness_witness_witness_witness_witness_right_right_right", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (p)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x4)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x5)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x6)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (db)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (dc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (J)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x12)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x13)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x7)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (wb)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (wc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (N)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (rb)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (rc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (K)", "apply prime_field_polynomial_convolution_equivalent_congruent_left", "exact hp0", "exact hs_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right", "exact hproducts_witness_witness_witness_witness_witness_witness_witness_right_right_left", "exact hR" ], "script_sha256": "6677195ac0553a0b428cc3a2bcab0fcc1fa7b83866a3a23ad4ec6b2e128f7bfd", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_aligned_distributivity_candidate.py", "sha256": "7d535939e24fe6d82158c485533b2ff6934f4d897b6141fde6c50b4fec9788ba" }, "statement": "forall p ub uc L vb vc M wb wc N db dc J pb pc H qb qc I rb rc K. (~((p) = 1) /\\ forall pfa_factor_left_aligned_right_prime pfa_factor_right_aligned_right_prime. (p) = pfa_factor_left_aligned_right_prime * pfa_factor_right_aligned_right_prime -> pfa_factor_left_aligned_right_prime = 1 \\/ pfa_factor_right_aligned_right_prime = 1) -> (((forall fom_index_pfp_aligned_right_input_left_bounded. (exists fom_gap_pfp_aligned_right_input_left_bounded_index_bound. fom_gap_pfp_aligned_right_input_left_bounded_index_bound + S (fom_index_pfp_aligned_right_input_left_bounded) = L) -> exists fom_value_pfp_aligned_right_input_left_bounded. ((((exists fom_beta_height_pfp_aligned_right_input_left_bounded_entry. fom_beta_height_pfp_aligned_right_input_left_bounded_entry + S (fom_value_pfp_aligned_right_input_left_bounded) = S ((S (fom_index_pfp_aligned_right_input_left_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_aligned_right_input_left_bounded_entry. ub = fom_beta_quotient_pfp_aligned_right_input_left_bounded_entry * S ((S (fom_index_pfp_aligned_right_input_left_bounded)) * uc) + (fom_value_pfp_aligned_right_input_left_bounded))) /\\ (exists fom_gap_pfp_aligned_right_input_left_bounded_value_bound. fom_gap_pfp_aligned_right_input_left_bounded_value_bound + S (fom_value_pfp_aligned_right_input_left_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_right_input_right_bounded. (exists fom_gap_pfp_aligned_right_input_right_bounded_index_bound. fom_gap_pfp_aligned_right_input_right_bounded_index_bound + S (fom_index_pfp_aligned_right_input_right_bounded) = M) -> exists fom_value_pfp_aligned_right_input_right_bounded. ((((exists fom_beta_height_pfp_aligned_right_input_right_bounded_entry. fom_beta_height_pfp_aligned_right_input_right_bounded_entry + S (fom_value_pfp_aligned_right_input_right_bounded) = S ((S (fom_index_pfp_aligned_right_input_right_bounded)) * vc)) /\\ exists fom_beta_quotient_pfp_aligned_right_input_right_bounded_entry. vb = fom_beta_quotient_pfp_aligned_right_input_right_bounded_entry * S ((S (fom_index_pfp_aligned_right_input_right_bounded)) * vc) + (fom_value_pfp_aligned_right_input_right_bounded))) /\\ (exists fom_gap_pfp_aligned_right_input_right_bounded_value_bound. fom_gap_pfp_aligned_right_input_right_bounded_value_bound + S (fom_value_pfp_aligned_right_input_right_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_right_input_result_bounded. (exists fom_gap_pfp_aligned_right_input_result_bounded_index_bound. fom_gap_pfp_aligned_right_input_result_bounded_index_bound + S (fom_index_pfp_aligned_right_input_result_bounded) = N) -> exists fom_value_pfp_aligned_right_input_result_bounded. ((((exists fom_beta_height_pfp_aligned_right_input_result_bounded_entry. fom_beta_height_pfp_aligned_right_input_result_bounded_entry + S (fom_value_pfp_aligned_right_input_result_bounded) = S ((S (fom_index_pfp_aligned_right_input_result_bounded)) * wc)) /\\ exists fom_beta_quotient_pfp_aligned_right_input_result_bounded_entry. wb = fom_beta_quotient_pfp_aligned_right_input_result_bounded_entry * S ((S (fom_index_pfp_aligned_right_input_result_bounded)) * wc) + (fom_value_pfp_aligned_right_input_result_bounded))) /\\ (exists fom_gap_pfp_aligned_right_input_result_bounded_value_bound. fom_gap_pfp_aligned_right_input_result_bounded_value_bound + S (fom_value_pfp_aligned_right_input_result_bounded) = p))) /\\ ((exists pfaa_left_b_aligned_right_input pfaa_left_c_aligned_right_input pfaa_right_b_aligned_right_input pfaa_right_c_aligned_right_input pfaa_sum_b_aligned_right_input pfaa_sum_c_aligned_right_input pfaa_length_aligned_right_input. ((((forall pfrep_power_aligned_right_input_witness_common_left pfrep_left_aligned_right_input_witness_common_left pfrep_right_aligned_right_input_witness_common_left. ((exists pfrep_position_aligned_right_input_witness_common_leftfirst. ((pfrep_position_aligned_right_input_witness_common_leftfirst+S (pfrep_power_aligned_right_input_witness_common_left)=(L)) /\\ ((((exists ff_h_pfp_aligned_right_input_witness_common_leftfirstentry. ff_h_pfp_aligned_right_input_witness_common_leftfirstentry + S (pfrep_left_aligned_right_input_witness_common_left) = S ((S (pfrep_position_aligned_right_input_witness_common_leftfirst)) * uc)) /\\ exists ff_q_pfp_aligned_right_input_witness_common_leftfirstentry. ub = ff_q_pfp_aligned_right_input_witness_common_leftfirstentry * S ((S (pfrep_position_aligned_right_input_witness_common_leftfirst)) * uc) + (pfrep_left_aligned_right_input_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_right_input_witness_common_leftfirstoutside. pfrep_gap_aligned_right_input_witness_common_leftfirstoutside+(L)=(pfrep_power_aligned_right_input_witness_common_left)) /\\ (((pfrep_left_aligned_right_input_witness_common_left)=0))))) -> ((exists pfrep_position_aligned_right_input_witness_common_leftsecond. ((pfrep_position_aligned_right_input_witness_common_leftsecond+S (pfrep_power_aligned_right_input_witness_common_left)=(pfaa_length_aligned_right_input)) /\\ ((((exists ff_h_pfp_aligned_right_input_witness_common_leftsecondentry. ff_h_pfp_aligned_right_input_witness_common_leftsecondentry + S (pfrep_right_aligned_right_input_witness_common_left) = S ((S (pfrep_position_aligned_right_input_witness_common_leftsecond)) * pfaa_left_c_aligned_right_input)) /\\ exists ff_q_pfp_aligned_right_input_witness_common_leftsecondentry. pfaa_left_b_aligned_right_input = ff_q_pfp_aligned_right_input_witness_common_leftsecondentry * S ((S (pfrep_position_aligned_right_input_witness_common_leftsecond)) * pfaa_left_c_aligned_right_input) + (pfrep_right_aligned_right_input_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_right_input_witness_common_leftsecondoutside. pfrep_gap_aligned_right_input_witness_common_leftsecondoutside+(pfaa_length_aligned_right_input)=(pfrep_power_aligned_right_input_witness_common_left)) /\\ (((pfrep_right_aligned_right_input_witness_common_left)=0))))) -> pfrep_left_aligned_right_input_witness_common_left=pfrep_right_aligned_right_input_witness_common_left) /\\ ((forall pfrep_power_aligned_right_input_witness_common_right pfrep_left_aligned_right_input_witness_common_right pfrep_right_aligned_right_input_witness_common_right. ((exists pfrep_position_aligned_right_input_witness_common_rightfirst. ((pfrep_position_aligned_right_input_witness_common_rightfirst+S (pfrep_power_aligned_right_input_witness_common_right)=(M)) /\\ ((((exists ff_h_pfp_aligned_right_input_witness_common_rightfirstentry. ff_h_pfp_aligned_right_input_witness_common_rightfirstentry + S (pfrep_left_aligned_right_input_witness_common_right) = S ((S (pfrep_position_aligned_right_input_witness_common_rightfirst)) * vc)) /\\ exists ff_q_pfp_aligned_right_input_witness_common_rightfirstentry. vb = ff_q_pfp_aligned_right_input_witness_common_rightfirstentry * S ((S (pfrep_position_aligned_right_input_witness_common_rightfirst)) * vc) + (pfrep_left_aligned_right_input_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_right_input_witness_common_rightfirstoutside. pfrep_gap_aligned_right_input_witness_common_rightfirstoutside+(M)=(pfrep_power_aligned_right_input_witness_common_right)) /\\ (((pfrep_left_aligned_right_input_witness_common_right)=0))))) -> ((exists pfrep_position_aligned_right_input_witness_common_rightsecond. ((pfrep_position_aligned_right_input_witness_common_rightsecond+S (pfrep_power_aligned_right_input_witness_common_right)=(pfaa_length_aligned_right_input)) /\\ ((((exists ff_h_pfp_aligned_right_input_witness_common_rightsecondentry. ff_h_pfp_aligned_right_input_witness_common_rightsecondentry + S (pfrep_right_aligned_right_input_witness_common_right) = S ((S (pfrep_position_aligned_right_input_witness_common_rightsecond)) * pfaa_right_c_aligned_right_input)) /\\ exists ff_q_pfp_aligned_right_input_witness_common_rightsecondentry. pfaa_right_b_aligned_right_input = ff_q_pfp_aligned_right_input_witness_common_rightsecondentry * S ((S (pfrep_position_aligned_right_input_witness_common_rightsecond)) * pfaa_right_c_aligned_right_input) + (pfrep_right_aligned_right_input_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_right_input_witness_common_rightsecondoutside. pfrep_gap_aligned_right_input_witness_common_rightsecondoutside+(pfaa_length_aligned_right_input)=(pfrep_power_aligned_right_input_witness_common_right)) /\\ (((pfrep_right_aligned_right_input_witness_common_right)=0))))) -> pfrep_left_aligned_right_input_witness_common_right=pfrep_right_aligned_right_input_witness_common_right)))) /\\ (((forall pfp_index_aligned_right_input_witness_operation. (exists pfa_gap_aligned_right_input_witness_operationindex. pfa_gap_aligned_right_input_witness_operationindex + S (pfp_index_aligned_right_input_witness_operation) = (pfaa_length_aligned_right_input)) -> exists pfp_left_aligned_right_input_witness_operation pfp_right_aligned_right_input_witness_operation pfp_value_aligned_right_input_witness_operation. ((((exists ff_h_pfp_aligned_right_input_witness_operationleft. ff_h_pfp_aligned_right_input_witness_operationleft + S (pfp_left_aligned_right_input_witness_operation) = S ((S (pfp_index_aligned_right_input_witness_operation)) * pfaa_left_c_aligned_right_input)) /\\ exists ff_q_pfp_aligned_right_input_witness_operationleft. pfaa_left_b_aligned_right_input = ff_q_pfp_aligned_right_input_witness_operationleft * S ((S (pfp_index_aligned_right_input_witness_operation)) * pfaa_left_c_aligned_right_input) + (pfp_left_aligned_right_input_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_right_input_witness_operationright. ff_h_pfp_aligned_right_input_witness_operationright + S (pfp_right_aligned_right_input_witness_operation) = S ((S (pfp_index_aligned_right_input_witness_operation)) * pfaa_right_c_aligned_right_input)) /\\ exists ff_q_pfp_aligned_right_input_witness_operationright. pfaa_right_b_aligned_right_input = ff_q_pfp_aligned_right_input_witness_operationright * S ((S (pfp_index_aligned_right_input_witness_operation)) * pfaa_right_c_aligned_right_input) + (pfp_right_aligned_right_input_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_right_input_witness_operationtarget. ff_h_pfp_aligned_right_input_witness_operationtarget + S (pfp_value_aligned_right_input_witness_operation) = S ((S (pfp_index_aligned_right_input_witness_operation)) * pfaa_sum_c_aligned_right_input)) /\\ exists ff_q_pfp_aligned_right_input_witness_operationtarget. pfaa_sum_b_aligned_right_input = ff_q_pfp_aligned_right_input_witness_operationtarget * S ((S (pfp_index_aligned_right_input_witness_operation)) * pfaa_sum_c_aligned_right_input) + (pfp_value_aligned_right_input_witness_operation))) /\\ ((((exists pfa_gap_aligned_right_input_witness_operationoperationleft. pfa_gap_aligned_right_input_witness_operationoperationleft + S (pfp_left_aligned_right_input_witness_operation) = (p)) /\\ (((exists pfa_gap_aligned_right_input_witness_operationoperationright. pfa_gap_aligned_right_input_witness_operationoperationright + S (pfp_right_aligned_right_input_witness_operation) = (p)) /\\ ((((exists pfa_gap_aligned_right_input_witness_operationoperationresultbound. pfa_gap_aligned_right_input_witness_operationoperationresultbound + S (pfp_value_aligned_right_input_witness_operation) = (p)) /\\ ((exists pfa_offset_left_aligned_right_input_witness_operationoperationresultcongruence pfa_offset_right_aligned_right_input_witness_operationoperationresultcongruence. ((pfp_left_aligned_right_input_witness_operation) + (pfp_right_aligned_right_input_witness_operation)) + (p) * pfa_offset_left_aligned_right_input_witness_operationoperationresultcongruence = (pfp_value_aligned_right_input_witness_operation) + (p) * pfa_offset_right_aligned_right_input_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_aligned_right_input_witness_output pfrep_left_aligned_right_input_witness_output pfrep_right_aligned_right_input_witness_output. ((exists pfrep_position_aligned_right_input_witness_outputfirst. ((pfrep_position_aligned_right_input_witness_outputfirst+S (pfrep_power_aligned_right_input_witness_output)=(pfaa_length_aligned_right_input)) /\\ ((((exists ff_h_pfp_aligned_right_input_witness_outputfirstentry. ff_h_pfp_aligned_right_input_witness_outputfirstentry + S (pfrep_left_aligned_right_input_witness_output) = S ((S (pfrep_position_aligned_right_input_witness_outputfirst)) * pfaa_sum_c_aligned_right_input)) /\\ exists ff_q_pfp_aligned_right_input_witness_outputfirstentry. pfaa_sum_b_aligned_right_input = ff_q_pfp_aligned_right_input_witness_outputfirstentry * S ((S (pfrep_position_aligned_right_input_witness_outputfirst)) * pfaa_sum_c_aligned_right_input) + (pfrep_left_aligned_right_input_witness_output)))))) \\/ (((exists pfrep_gap_aligned_right_input_witness_outputfirstoutside. pfrep_gap_aligned_right_input_witness_outputfirstoutside+(pfaa_length_aligned_right_input)=(pfrep_power_aligned_right_input_witness_output)) /\\ (((pfrep_left_aligned_right_input_witness_output)=0))))) -> ((exists pfrep_position_aligned_right_input_witness_outputsecond. ((pfrep_position_aligned_right_input_witness_outputsecond+S (pfrep_power_aligned_right_input_witness_output)=(N)) /\\ ((((exists ff_h_pfp_aligned_right_input_witness_outputsecondentry. ff_h_pfp_aligned_right_input_witness_outputsecondentry + S (pfrep_right_aligned_right_input_witness_output) = S ((S (pfrep_position_aligned_right_input_witness_outputsecond)) * wc)) /\\ exists ff_q_pfp_aligned_right_input_witness_outputsecondentry. wb = ff_q_pfp_aligned_right_input_witness_outputsecondentry * S ((S (pfrep_position_aligned_right_input_witness_outputsecond)) * wc) + (pfrep_right_aligned_right_input_witness_output)))))) \\/ (((exists pfrep_gap_aligned_right_input_witness_outputsecondoutside. pfrep_gap_aligned_right_input_witness_outputsecondoutside+(N)=(pfrep_power_aligned_right_input_witness_output)) /\\ (((pfrep_right_aligned_right_input_witness_output)=0))))) -> pfrep_left_aligned_right_input_witness_output=pfrep_right_aligned_right_input_witness_output))))))))))))) -> (((forall fom_index_pfp_aligned_right_pbleft. (exists fom_gap_pfp_aligned_right_pbleft_index_bound. fom_gap_pfp_aligned_right_pbleft_index_bound + S (fom_index_pfp_aligned_right_pbleft) = L) -> exists fom_value_pfp_aligned_right_pbleft. ((((exists fom_beta_height_pfp_aligned_right_pbleft_entry. fom_beta_height_pfp_aligned_right_pbleft_entry + S (fom_value_pfp_aligned_right_pbleft) = S ((S (fom_index_pfp_aligned_right_pbleft)) * uc)) /\\ exists fom_beta_quotient_pfp_aligned_right_pbleft_entry. ub = fom_beta_quotient_pfp_aligned_right_pbleft_entry * S ((S (fom_index_pfp_aligned_right_pbleft)) * uc) + (fom_value_pfp_aligned_right_pbleft))) /\\ (exists fom_gap_pfp_aligned_right_pbleft_value_bound. fom_gap_pfp_aligned_right_pbleft_value_bound + S (fom_value_pfp_aligned_right_pbleft) = p))) /\\ (((forall fom_index_pfp_aligned_right_pbright. (exists fom_gap_pfp_aligned_right_pbright_index_bound. fom_gap_pfp_aligned_right_pbright_index_bound + S (fom_index_pfp_aligned_right_pbright) = J) -> exists fom_value_pfp_aligned_right_pbright. ((((exists fom_beta_height_pfp_aligned_right_pbright_entry. fom_beta_height_pfp_aligned_right_pbright_entry + S (fom_value_pfp_aligned_right_pbright) = S ((S (fom_index_pfp_aligned_right_pbright)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_right_pbright_entry. db = fom_beta_quotient_pfp_aligned_right_pbright_entry * S ((S (fom_index_pfp_aligned_right_pbright)) * dc) + (fom_value_pfp_aligned_right_pbright))) /\\ (exists fom_gap_pfp_aligned_right_pbright_value_bound. fom_gap_pfp_aligned_right_pbright_value_bound + S (fom_value_pfp_aligned_right_pbright) = p))) /\\ (((((((L)=0 \\/ (J)=0) /\\ (((H)=0)))) \\/ (((~((L)=0)) /\\ (((~((J)=0)) /\\ (((L)+(J)=S (H)))))))) /\\ ((forall pfc_index_aligned_right_pbcoefficients. (exists pfa_gap_aligned_right_pbcoefficientsbound. pfa_gap_aligned_right_pbcoefficientsbound + S (pfc_index_aligned_right_pbcoefficients) = (H)) -> exists pfc_value_aligned_right_pbcoefficients. ((((exists ff_h_pfp_aligned_right_pbcoefficientsentry. ff_h_pfp_aligned_right_pbcoefficientsentry + S (pfc_value_aligned_right_pbcoefficients) = S ((S (pfc_index_aligned_right_pbcoefficients)) * pc)) /\\ exists ff_q_pfp_aligned_right_pbcoefficientsentry. pb = ff_q_pfp_aligned_right_pbcoefficientsentry * S ((S (pfc_index_aligned_right_pbcoefficients)) * pc) + (pfc_value_aligned_right_pbcoefficients))) /\\ ((exists pfc_terms_code_aligned_right_pbcoefficientscoefficient pfc_terms_scale_aligned_right_pbcoefficientscoefficient pfc_natural_sum_aligned_right_pbcoefficientscoefficient. ((forall pfc_index_aligned_right_pbcoefficientscoefficientdiagonal. (exists pfa_gap_aligned_right_pbcoefficientscoefficientdiagonalbound. pfa_gap_aligned_right_pbcoefficientscoefficientdiagonalbound + S (pfc_index_aligned_right_pbcoefficientscoefficientdiagonal) = (S (pfc_index_aligned_right_pbcoefficients))) -> exists pfc_value_aligned_right_pbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_aligned_right_pbcoefficientscoefficientdiagonalentry. ff_h_pfp_aligned_right_pbcoefficientscoefficientdiagonalentry + S (pfc_value_aligned_right_pbcoefficientscoefficientdiagonal) = S ((S (pfc_index_aligned_right_pbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_right_pbcoefficientscoefficient)) /\\ exists ff_q_pfp_aligned_right_pbcoefficientscoefficientdiagonalentry. pfc_terms_code_aligned_right_pbcoefficientscoefficient = ff_q_pfp_aligned_right_pbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_aligned_right_pbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_right_pbcoefficientscoefficient) + (pfc_value_aligned_right_pbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_aligned_right_pbcoefficientscoefficientdiagonalterm pfc_left_aligned_right_pbcoefficientscoefficientdiagonalterm pfc_right_aligned_right_pbcoefficientscoefficientdiagonalterm. (((pfc_index_aligned_right_pbcoefficientscoefficientdiagonal)+pfc_complement_aligned_right_pbcoefficientscoefficientdiagonalterm=(pfc_index_aligned_right_pbcoefficients)) /\\ ((((((exists pfa_gap_aligned_right_pbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_aligned_right_pbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_aligned_right_pbcoefficientscoefficientdiagonal) = (L)) /\\ ((((exists ff_h_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_aligned_right_pbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_aligned_right_pbcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_aligned_right_pbcoefficientscoefficientdiagonal)) * uc) + (pfc_left_aligned_right_pbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_right_pbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_aligned_right_pbcoefficientscoefficientdiagonaltermleftoutside+(L)=(pfc_index_aligned_right_pbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_aligned_right_pbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_aligned_right_pbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_aligned_right_pbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_aligned_right_pbcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_aligned_right_pbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_aligned_right_pbcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_aligned_right_pbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_aligned_right_pbcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_aligned_right_pbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_right_pbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_aligned_right_pbcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_aligned_right_pbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_aligned_right_pbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_aligned_right_pbcoefficientscoefficientdiagonal)=pfc_left_aligned_right_pbcoefficientscoefficientdiagonalterm*pfc_right_aligned_right_pbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_aligned_right_pbcoefficientscoefficientsum fs_v_pfc_aligned_right_pbcoefficientscoefficientsum. ((((exists fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_start. fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_start. fs_u_pfc_aligned_right_pbcoefficientscoefficientsum = fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_terminal. fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_aligned_right_pbcoefficientscoefficient) = S ((S (S (pfc_index_aligned_right_pbcoefficients))) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_terminal. fs_u_pfc_aligned_right_pbcoefficientscoefficientsum = fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_aligned_right_pbcoefficients))) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum) + (pfc_natural_sum_aligned_right_pbcoefficientscoefficient))) /\\ forall fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps = S (pfc_index_aligned_right_pbcoefficients)) -> exists fs_a_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps fs_r_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps fs_s_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_right_pbcoefficientscoefficient)) /\\ exists fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_aligned_right_pbcoefficientscoefficient = fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_right_pbcoefficientscoefficient) + (fs_a_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_aligned_right_pbcoefficientscoefficientsum = fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum) + (fs_r_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_aligned_right_pbcoefficientscoefficientsum = fs_q_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_pbcoefficientscoefficientsum) + (fs_s_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps = fs_r_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps + fs_a_pfc_aligned_right_pbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_aligned_right_pbcoefficientscoefficientresiduebound. pfa_gap_aligned_right_pbcoefficientscoefficientresiduebound + S (pfc_value_aligned_right_pbcoefficients) = (p)) /\\ ((exists pfa_offset_left_aligned_right_pbcoefficientscoefficientresiduecongruence pfa_offset_right_aligned_right_pbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_aligned_right_pbcoefficientscoefficient) + (p) * pfa_offset_left_aligned_right_pbcoefficientscoefficientresiduecongruence = (pfc_value_aligned_right_pbcoefficients) + (p) * pfa_offset_right_aligned_right_pbcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_aligned_right_qbleft. (exists fom_gap_pfp_aligned_right_qbleft_index_bound. fom_gap_pfp_aligned_right_qbleft_index_bound + S (fom_index_pfp_aligned_right_qbleft) = M) -> exists fom_value_pfp_aligned_right_qbleft. ((((exists fom_beta_height_pfp_aligned_right_qbleft_entry. fom_beta_height_pfp_aligned_right_qbleft_entry + S (fom_value_pfp_aligned_right_qbleft) = S ((S (fom_index_pfp_aligned_right_qbleft)) * vc)) /\\ exists fom_beta_quotient_pfp_aligned_right_qbleft_entry. vb = fom_beta_quotient_pfp_aligned_right_qbleft_entry * S ((S (fom_index_pfp_aligned_right_qbleft)) * vc) + (fom_value_pfp_aligned_right_qbleft))) /\\ (exists fom_gap_pfp_aligned_right_qbleft_value_bound. fom_gap_pfp_aligned_right_qbleft_value_bound + S (fom_value_pfp_aligned_right_qbleft) = p))) /\\ (((forall fom_index_pfp_aligned_right_qbright. (exists fom_gap_pfp_aligned_right_qbright_index_bound. fom_gap_pfp_aligned_right_qbright_index_bound + S (fom_index_pfp_aligned_right_qbright) = J) -> exists fom_value_pfp_aligned_right_qbright. ((((exists fom_beta_height_pfp_aligned_right_qbright_entry. fom_beta_height_pfp_aligned_right_qbright_entry + S (fom_value_pfp_aligned_right_qbright) = S ((S (fom_index_pfp_aligned_right_qbright)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_right_qbright_entry. db = fom_beta_quotient_pfp_aligned_right_qbright_entry * S ((S (fom_index_pfp_aligned_right_qbright)) * dc) + (fom_value_pfp_aligned_right_qbright))) /\\ (exists fom_gap_pfp_aligned_right_qbright_value_bound. fom_gap_pfp_aligned_right_qbright_value_bound + S (fom_value_pfp_aligned_right_qbright) = p))) /\\ (((((((M)=0 \\/ (J)=0) /\\ (((I)=0)))) \\/ (((~((M)=0)) /\\ (((~((J)=0)) /\\ (((M)+(J)=S (I)))))))) /\\ ((forall pfc_index_aligned_right_qbcoefficients. (exists pfa_gap_aligned_right_qbcoefficientsbound. pfa_gap_aligned_right_qbcoefficientsbound + S (pfc_index_aligned_right_qbcoefficients) = (I)) -> exists pfc_value_aligned_right_qbcoefficients. ((((exists ff_h_pfp_aligned_right_qbcoefficientsentry. ff_h_pfp_aligned_right_qbcoefficientsentry + S (pfc_value_aligned_right_qbcoefficients) = S ((S (pfc_index_aligned_right_qbcoefficients)) * qc)) /\\ exists ff_q_pfp_aligned_right_qbcoefficientsentry. qb = ff_q_pfp_aligned_right_qbcoefficientsentry * S ((S (pfc_index_aligned_right_qbcoefficients)) * qc) + (pfc_value_aligned_right_qbcoefficients))) /\\ ((exists pfc_terms_code_aligned_right_qbcoefficientscoefficient pfc_terms_scale_aligned_right_qbcoefficientscoefficient pfc_natural_sum_aligned_right_qbcoefficientscoefficient. ((forall pfc_index_aligned_right_qbcoefficientscoefficientdiagonal. (exists pfa_gap_aligned_right_qbcoefficientscoefficientdiagonalbound. pfa_gap_aligned_right_qbcoefficientscoefficientdiagonalbound + S (pfc_index_aligned_right_qbcoefficientscoefficientdiagonal) = (S (pfc_index_aligned_right_qbcoefficients))) -> exists pfc_value_aligned_right_qbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_aligned_right_qbcoefficientscoefficientdiagonalentry. ff_h_pfp_aligned_right_qbcoefficientscoefficientdiagonalentry + S (pfc_value_aligned_right_qbcoefficientscoefficientdiagonal) = S ((S (pfc_index_aligned_right_qbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_right_qbcoefficientscoefficient)) /\\ exists ff_q_pfp_aligned_right_qbcoefficientscoefficientdiagonalentry. pfc_terms_code_aligned_right_qbcoefficientscoefficient = ff_q_pfp_aligned_right_qbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_aligned_right_qbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_right_qbcoefficientscoefficient) + (pfc_value_aligned_right_qbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_aligned_right_qbcoefficientscoefficientdiagonalterm pfc_left_aligned_right_qbcoefficientscoefficientdiagonalterm pfc_right_aligned_right_qbcoefficientscoefficientdiagonalterm. (((pfc_index_aligned_right_qbcoefficientscoefficientdiagonal)+pfc_complement_aligned_right_qbcoefficientscoefficientdiagonalterm=(pfc_index_aligned_right_qbcoefficients)) /\\ ((((((exists pfa_gap_aligned_right_qbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_aligned_right_qbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_aligned_right_qbcoefficientscoefficientdiagonal) = (M)) /\\ ((((exists ff_h_pfp_aligned_right_qbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_aligned_right_qbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_aligned_right_qbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_aligned_right_qbcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_aligned_right_qbcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_aligned_right_qbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_aligned_right_qbcoefficientscoefficientdiagonal)) * vc) + (pfc_left_aligned_right_qbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_right_qbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_aligned_right_qbcoefficientscoefficientdiagonaltermleftoutside+(M)=(pfc_index_aligned_right_qbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_aligned_right_qbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_aligned_right_qbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_aligned_right_qbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_aligned_right_qbcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_aligned_right_qbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_aligned_right_qbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_aligned_right_qbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_aligned_right_qbcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_aligned_right_qbcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_aligned_right_qbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_aligned_right_qbcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_aligned_right_qbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_right_qbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_aligned_right_qbcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_aligned_right_qbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_aligned_right_qbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_aligned_right_qbcoefficientscoefficientdiagonal)=pfc_left_aligned_right_qbcoefficientscoefficientdiagonalterm*pfc_right_aligned_right_qbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_aligned_right_qbcoefficientscoefficientsum fs_v_pfc_aligned_right_qbcoefficientscoefficientsum. ((((exists fs_h_pfc_aligned_right_qbcoefficientscoefficientsum_body_start. fs_h_pfc_aligned_right_qbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_aligned_right_qbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_qbcoefficientscoefficientsum_body_start. fs_u_pfc_aligned_right_qbcoefficientscoefficientsum = fs_q_pfc_aligned_right_qbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_aligned_right_qbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_aligned_right_qbcoefficientscoefficientsum_body_terminal. fs_h_pfc_aligned_right_qbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_aligned_right_qbcoefficientscoefficient) = S ((S (S (pfc_index_aligned_right_qbcoefficients))) * fs_v_pfc_aligned_right_qbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_qbcoefficientscoefficientsum_body_terminal. fs_u_pfc_aligned_right_qbcoefficientscoefficientsum = fs_q_pfc_aligned_right_qbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_aligned_right_qbcoefficients))) * fs_v_pfc_aligned_right_qbcoefficientscoefficientsum) + (pfc_natural_sum_aligned_right_qbcoefficientscoefficient))) /\\ forall fs_i_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps = S (pfc_index_aligned_right_qbcoefficients)) -> exists fs_a_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps fs_r_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps fs_s_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_right_qbcoefficientscoefficient)) /\\ exists fs_q_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_aligned_right_qbcoefficientscoefficient = fs_q_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_right_qbcoefficientscoefficient) + (fs_a_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_qbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_aligned_right_qbcoefficientscoefficientsum = fs_q_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_qbcoefficientscoefficientsum) + (fs_r_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_qbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_aligned_right_qbcoefficientscoefficientsum = fs_q_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_qbcoefficientscoefficientsum) + (fs_s_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps = fs_r_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps + fs_a_pfc_aligned_right_qbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_aligned_right_qbcoefficientscoefficientresiduebound. pfa_gap_aligned_right_qbcoefficientscoefficientresiduebound + S (pfc_value_aligned_right_qbcoefficients) = (p)) /\\ ((exists pfa_offset_left_aligned_right_qbcoefficientscoefficientresiduecongruence pfa_offset_right_aligned_right_qbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_aligned_right_qbcoefficientscoefficient) + (p) * pfa_offset_left_aligned_right_qbcoefficientscoefficientresiduecongruence = (pfc_value_aligned_right_qbcoefficients) + (p) * pfa_offset_right_aligned_right_qbcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_aligned_right_rbleft. (exists fom_gap_pfp_aligned_right_rbleft_index_bound. fom_gap_pfp_aligned_right_rbleft_index_bound + S (fom_index_pfp_aligned_right_rbleft) = N) -> exists fom_value_pfp_aligned_right_rbleft. ((((exists fom_beta_height_pfp_aligned_right_rbleft_entry. fom_beta_height_pfp_aligned_right_rbleft_entry + S (fom_value_pfp_aligned_right_rbleft) = S ((S (fom_index_pfp_aligned_right_rbleft)) * wc)) /\\ exists fom_beta_quotient_pfp_aligned_right_rbleft_entry. wb = fom_beta_quotient_pfp_aligned_right_rbleft_entry * S ((S (fom_index_pfp_aligned_right_rbleft)) * wc) + (fom_value_pfp_aligned_right_rbleft))) /\\ (exists fom_gap_pfp_aligned_right_rbleft_value_bound. fom_gap_pfp_aligned_right_rbleft_value_bound + S (fom_value_pfp_aligned_right_rbleft) = p))) /\\ (((forall fom_index_pfp_aligned_right_rbright. (exists fom_gap_pfp_aligned_right_rbright_index_bound. fom_gap_pfp_aligned_right_rbright_index_bound + S (fom_index_pfp_aligned_right_rbright) = J) -> exists fom_value_pfp_aligned_right_rbright. ((((exists fom_beta_height_pfp_aligned_right_rbright_entry. fom_beta_height_pfp_aligned_right_rbright_entry + S (fom_value_pfp_aligned_right_rbright) = S ((S (fom_index_pfp_aligned_right_rbright)) * dc)) /\\ exists fom_beta_quotient_pfp_aligned_right_rbright_entry. db = fom_beta_quotient_pfp_aligned_right_rbright_entry * S ((S (fom_index_pfp_aligned_right_rbright)) * dc) + (fom_value_pfp_aligned_right_rbright))) /\\ (exists fom_gap_pfp_aligned_right_rbright_value_bound. fom_gap_pfp_aligned_right_rbright_value_bound + S (fom_value_pfp_aligned_right_rbright) = p))) /\\ (((((((N)=0 \\/ (J)=0) /\\ (((K)=0)))) \\/ (((~((N)=0)) /\\ (((~((J)=0)) /\\ (((N)+(J)=S (K)))))))) /\\ ((forall pfc_index_aligned_right_rbcoefficients. (exists pfa_gap_aligned_right_rbcoefficientsbound. pfa_gap_aligned_right_rbcoefficientsbound + S (pfc_index_aligned_right_rbcoefficients) = (K)) -> exists pfc_value_aligned_right_rbcoefficients. ((((exists ff_h_pfp_aligned_right_rbcoefficientsentry. ff_h_pfp_aligned_right_rbcoefficientsentry + S (pfc_value_aligned_right_rbcoefficients) = S ((S (pfc_index_aligned_right_rbcoefficients)) * rc)) /\\ exists ff_q_pfp_aligned_right_rbcoefficientsentry. rb = ff_q_pfp_aligned_right_rbcoefficientsentry * S ((S (pfc_index_aligned_right_rbcoefficients)) * rc) + (pfc_value_aligned_right_rbcoefficients))) /\\ ((exists pfc_terms_code_aligned_right_rbcoefficientscoefficient pfc_terms_scale_aligned_right_rbcoefficientscoefficient pfc_natural_sum_aligned_right_rbcoefficientscoefficient. ((forall pfc_index_aligned_right_rbcoefficientscoefficientdiagonal. (exists pfa_gap_aligned_right_rbcoefficientscoefficientdiagonalbound. pfa_gap_aligned_right_rbcoefficientscoefficientdiagonalbound + S (pfc_index_aligned_right_rbcoefficientscoefficientdiagonal) = (S (pfc_index_aligned_right_rbcoefficients))) -> exists pfc_value_aligned_right_rbcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_aligned_right_rbcoefficientscoefficientdiagonalentry. ff_h_pfp_aligned_right_rbcoefficientscoefficientdiagonalentry + S (pfc_value_aligned_right_rbcoefficientscoefficientdiagonal) = S ((S (pfc_index_aligned_right_rbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_right_rbcoefficientscoefficient)) /\\ exists ff_q_pfp_aligned_right_rbcoefficientscoefficientdiagonalentry. pfc_terms_code_aligned_right_rbcoefficientscoefficient = ff_q_pfp_aligned_right_rbcoefficientscoefficientdiagonalentry * S ((S (pfc_index_aligned_right_rbcoefficientscoefficientdiagonal)) * pfc_terms_scale_aligned_right_rbcoefficientscoefficient) + (pfc_value_aligned_right_rbcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_aligned_right_rbcoefficientscoefficientdiagonalterm pfc_left_aligned_right_rbcoefficientscoefficientdiagonalterm pfc_right_aligned_right_rbcoefficientscoefficientdiagonalterm. (((pfc_index_aligned_right_rbcoefficientscoefficientdiagonal)+pfc_complement_aligned_right_rbcoefficientscoefficientdiagonalterm=(pfc_index_aligned_right_rbcoefficients)) /\\ ((((((exists pfa_gap_aligned_right_rbcoefficientscoefficientdiagonaltermleftinside. pfa_gap_aligned_right_rbcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_aligned_right_rbcoefficientscoefficientdiagonal) = (N)) /\\ ((((exists ff_h_pfp_aligned_right_rbcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_aligned_right_rbcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_aligned_right_rbcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_aligned_right_rbcoefficientscoefficientdiagonal)) * wc)) /\\ exists ff_q_pfp_aligned_right_rbcoefficientscoefficientdiagonaltermleftentry. wb = ff_q_pfp_aligned_right_rbcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_aligned_right_rbcoefficientscoefficientdiagonal)) * wc) + (pfc_left_aligned_right_rbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_right_rbcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_aligned_right_rbcoefficientscoefficientdiagonaltermleftoutside+(N)=(pfc_index_aligned_right_rbcoefficientscoefficientdiagonal)) /\\ (((pfc_left_aligned_right_rbcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_aligned_right_rbcoefficientscoefficientdiagonaltermrightinside. pfa_gap_aligned_right_rbcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_aligned_right_rbcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_aligned_right_rbcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_aligned_right_rbcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_aligned_right_rbcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_aligned_right_rbcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_aligned_right_rbcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_aligned_right_rbcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_aligned_right_rbcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_aligned_right_rbcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_aligned_right_rbcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_aligned_right_rbcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_aligned_right_rbcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_aligned_right_rbcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_aligned_right_rbcoefficientscoefficientdiagonal)=pfc_left_aligned_right_rbcoefficientscoefficientdiagonalterm*pfc_right_aligned_right_rbcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_aligned_right_rbcoefficientscoefficientsum fs_v_pfc_aligned_right_rbcoefficientscoefficientsum. ((((exists fs_h_pfc_aligned_right_rbcoefficientscoefficientsum_body_start. fs_h_pfc_aligned_right_rbcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_aligned_right_rbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_rbcoefficientscoefficientsum_body_start. fs_u_pfc_aligned_right_rbcoefficientscoefficientsum = fs_q_pfc_aligned_right_rbcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_aligned_right_rbcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_aligned_right_rbcoefficientscoefficientsum_body_terminal. fs_h_pfc_aligned_right_rbcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_aligned_right_rbcoefficientscoefficient) = S ((S (S (pfc_index_aligned_right_rbcoefficients))) * fs_v_pfc_aligned_right_rbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_rbcoefficientscoefficientsum_body_terminal. fs_u_pfc_aligned_right_rbcoefficientscoefficientsum = fs_q_pfc_aligned_right_rbcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_aligned_right_rbcoefficients))) * fs_v_pfc_aligned_right_rbcoefficientscoefficientsum) + (pfc_natural_sum_aligned_right_rbcoefficientscoefficient))) /\\ forall fs_i_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps = S (pfc_index_aligned_right_rbcoefficients)) -> exists fs_a_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps fs_r_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps fs_s_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_right_rbcoefficientscoefficient)) /\\ exists fs_q_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_aligned_right_rbcoefficientscoefficient = fs_q_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_aligned_right_rbcoefficientscoefficient) + (fs_a_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_rbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_aligned_right_rbcoefficientscoefficientsum = fs_q_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_rbcoefficientscoefficientsum) + (fs_r_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_rbcoefficientscoefficientsum)) /\\ exists fs_q_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_aligned_right_rbcoefficientscoefficientsum = fs_q_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps)) * fs_v_pfc_aligned_right_rbcoefficientscoefficientsum) + (fs_s_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps = fs_r_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps + fs_a_pfc_aligned_right_rbcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_aligned_right_rbcoefficientscoefficientresiduebound. pfa_gap_aligned_right_rbcoefficientscoefficientresiduebound + S (pfc_value_aligned_right_rbcoefficients) = (p)) /\\ ((exists pfa_offset_left_aligned_right_rbcoefficientscoefficientresiduecongruence pfa_offset_right_aligned_right_rbcoefficientscoefficientresiduecongruence. (pfc_natural_sum_aligned_right_rbcoefficientscoefficient) + (p) * pfa_offset_left_aligned_right_rbcoefficientscoefficientresiduecongruence = (pfc_value_aligned_right_rbcoefficients) + (p) * pfa_offset_right_aligned_right_rbcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_aligned_right_result_left_bounded. (exists fom_gap_pfp_aligned_right_result_left_bounded_index_bound. fom_gap_pfp_aligned_right_result_left_bounded_index_bound + S (fom_index_pfp_aligned_right_result_left_bounded) = H) -> exists fom_value_pfp_aligned_right_result_left_bounded. ((((exists fom_beta_height_pfp_aligned_right_result_left_bounded_entry. fom_beta_height_pfp_aligned_right_result_left_bounded_entry + S (fom_value_pfp_aligned_right_result_left_bounded) = S ((S (fom_index_pfp_aligned_right_result_left_bounded)) * pc)) /\\ exists fom_beta_quotient_pfp_aligned_right_result_left_bounded_entry. pb = fom_beta_quotient_pfp_aligned_right_result_left_bounded_entry * S ((S (fom_index_pfp_aligned_right_result_left_bounded)) * pc) + (fom_value_pfp_aligned_right_result_left_bounded))) /\\ (exists fom_gap_pfp_aligned_right_result_left_bounded_value_bound. fom_gap_pfp_aligned_right_result_left_bounded_value_bound + S (fom_value_pfp_aligned_right_result_left_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_right_result_right_bounded. (exists fom_gap_pfp_aligned_right_result_right_bounded_index_bound. fom_gap_pfp_aligned_right_result_right_bounded_index_bound + S (fom_index_pfp_aligned_right_result_right_bounded) = I) -> exists fom_value_pfp_aligned_right_result_right_bounded. ((((exists fom_beta_height_pfp_aligned_right_result_right_bounded_entry. fom_beta_height_pfp_aligned_right_result_right_bounded_entry + S (fom_value_pfp_aligned_right_result_right_bounded) = S ((S (fom_index_pfp_aligned_right_result_right_bounded)) * qc)) /\\ exists fom_beta_quotient_pfp_aligned_right_result_right_bounded_entry. qb = fom_beta_quotient_pfp_aligned_right_result_right_bounded_entry * S ((S (fom_index_pfp_aligned_right_result_right_bounded)) * qc) + (fom_value_pfp_aligned_right_result_right_bounded))) /\\ (exists fom_gap_pfp_aligned_right_result_right_bounded_value_bound. fom_gap_pfp_aligned_right_result_right_bounded_value_bound + S (fom_value_pfp_aligned_right_result_right_bounded) = p))) /\\ (((forall fom_index_pfp_aligned_right_result_result_bounded. (exists fom_gap_pfp_aligned_right_result_result_bounded_index_bound. fom_gap_pfp_aligned_right_result_result_bounded_index_bound + S (fom_index_pfp_aligned_right_result_result_bounded) = K) -> exists fom_value_pfp_aligned_right_result_result_bounded. ((((exists fom_beta_height_pfp_aligned_right_result_result_bounded_entry. fom_beta_height_pfp_aligned_right_result_result_bounded_entry + S (fom_value_pfp_aligned_right_result_result_bounded) = S ((S (fom_index_pfp_aligned_right_result_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_aligned_right_result_result_bounded_entry. rb = fom_beta_quotient_pfp_aligned_right_result_result_bounded_entry * S ((S (fom_index_pfp_aligned_right_result_result_bounded)) * rc) + (fom_value_pfp_aligned_right_result_result_bounded))) /\\ (exists fom_gap_pfp_aligned_right_result_result_bounded_value_bound. fom_gap_pfp_aligned_right_result_result_bounded_value_bound + S (fom_value_pfp_aligned_right_result_result_bounded) = p))) /\\ ((exists pfaa_left_b_aligned_right_result pfaa_left_c_aligned_right_result pfaa_right_b_aligned_right_result pfaa_right_c_aligned_right_result pfaa_sum_b_aligned_right_result pfaa_sum_c_aligned_right_result pfaa_length_aligned_right_result. ((((forall pfrep_power_aligned_right_result_witness_common_left pfrep_left_aligned_right_result_witness_common_left pfrep_right_aligned_right_result_witness_common_left. ((exists pfrep_position_aligned_right_result_witness_common_leftfirst. ((pfrep_position_aligned_right_result_witness_common_leftfirst+S (pfrep_power_aligned_right_result_witness_common_left)=(H)) /\\ ((((exists ff_h_pfp_aligned_right_result_witness_common_leftfirstentry. ff_h_pfp_aligned_right_result_witness_common_leftfirstentry + S (pfrep_left_aligned_right_result_witness_common_left) = S ((S (pfrep_position_aligned_right_result_witness_common_leftfirst)) * pc)) /\\ exists ff_q_pfp_aligned_right_result_witness_common_leftfirstentry. pb = ff_q_pfp_aligned_right_result_witness_common_leftfirstentry * S ((S (pfrep_position_aligned_right_result_witness_common_leftfirst)) * pc) + (pfrep_left_aligned_right_result_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_right_result_witness_common_leftfirstoutside. pfrep_gap_aligned_right_result_witness_common_leftfirstoutside+(H)=(pfrep_power_aligned_right_result_witness_common_left)) /\\ (((pfrep_left_aligned_right_result_witness_common_left)=0))))) -> ((exists pfrep_position_aligned_right_result_witness_common_leftsecond. ((pfrep_position_aligned_right_result_witness_common_leftsecond+S (pfrep_power_aligned_right_result_witness_common_left)=(pfaa_length_aligned_right_result)) /\\ ((((exists ff_h_pfp_aligned_right_result_witness_common_leftsecondentry. ff_h_pfp_aligned_right_result_witness_common_leftsecondentry + S (pfrep_right_aligned_right_result_witness_common_left) = S ((S (pfrep_position_aligned_right_result_witness_common_leftsecond)) * pfaa_left_c_aligned_right_result)) /\\ exists ff_q_pfp_aligned_right_result_witness_common_leftsecondentry. pfaa_left_b_aligned_right_result = ff_q_pfp_aligned_right_result_witness_common_leftsecondentry * S ((S (pfrep_position_aligned_right_result_witness_common_leftsecond)) * pfaa_left_c_aligned_right_result) + (pfrep_right_aligned_right_result_witness_common_left)))))) \\/ (((exists pfrep_gap_aligned_right_result_witness_common_leftsecondoutside. pfrep_gap_aligned_right_result_witness_common_leftsecondoutside+(pfaa_length_aligned_right_result)=(pfrep_power_aligned_right_result_witness_common_left)) /\\ (((pfrep_right_aligned_right_result_witness_common_left)=0))))) -> pfrep_left_aligned_right_result_witness_common_left=pfrep_right_aligned_right_result_witness_common_left) /\\ ((forall pfrep_power_aligned_right_result_witness_common_right pfrep_left_aligned_right_result_witness_common_right pfrep_right_aligned_right_result_witness_common_right. ((exists pfrep_position_aligned_right_result_witness_common_rightfirst. ((pfrep_position_aligned_right_result_witness_common_rightfirst+S (pfrep_power_aligned_right_result_witness_common_right)=(I)) /\\ ((((exists ff_h_pfp_aligned_right_result_witness_common_rightfirstentry. ff_h_pfp_aligned_right_result_witness_common_rightfirstentry + S (pfrep_left_aligned_right_result_witness_common_right) = S ((S (pfrep_position_aligned_right_result_witness_common_rightfirst)) * qc)) /\\ exists ff_q_pfp_aligned_right_result_witness_common_rightfirstentry. qb = ff_q_pfp_aligned_right_result_witness_common_rightfirstentry * S ((S (pfrep_position_aligned_right_result_witness_common_rightfirst)) * qc) + (pfrep_left_aligned_right_result_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_right_result_witness_common_rightfirstoutside. pfrep_gap_aligned_right_result_witness_common_rightfirstoutside+(I)=(pfrep_power_aligned_right_result_witness_common_right)) /\\ (((pfrep_left_aligned_right_result_witness_common_right)=0))))) -> ((exists pfrep_position_aligned_right_result_witness_common_rightsecond. ((pfrep_position_aligned_right_result_witness_common_rightsecond+S (pfrep_power_aligned_right_result_witness_common_right)=(pfaa_length_aligned_right_result)) /\\ ((((exists ff_h_pfp_aligned_right_result_witness_common_rightsecondentry. ff_h_pfp_aligned_right_result_witness_common_rightsecondentry + S (pfrep_right_aligned_right_result_witness_common_right) = S ((S (pfrep_position_aligned_right_result_witness_common_rightsecond)) * pfaa_right_c_aligned_right_result)) /\\ exists ff_q_pfp_aligned_right_result_witness_common_rightsecondentry. pfaa_right_b_aligned_right_result = ff_q_pfp_aligned_right_result_witness_common_rightsecondentry * S ((S (pfrep_position_aligned_right_result_witness_common_rightsecond)) * pfaa_right_c_aligned_right_result) + (pfrep_right_aligned_right_result_witness_common_right)))))) \\/ (((exists pfrep_gap_aligned_right_result_witness_common_rightsecondoutside. pfrep_gap_aligned_right_result_witness_common_rightsecondoutside+(pfaa_length_aligned_right_result)=(pfrep_power_aligned_right_result_witness_common_right)) /\\ (((pfrep_right_aligned_right_result_witness_common_right)=0))))) -> pfrep_left_aligned_right_result_witness_common_right=pfrep_right_aligned_right_result_witness_common_right)))) /\\ (((forall pfp_index_aligned_right_result_witness_operation. (exists pfa_gap_aligned_right_result_witness_operationindex. pfa_gap_aligned_right_result_witness_operationindex + S (pfp_index_aligned_right_result_witness_operation) = (pfaa_length_aligned_right_result)) -> exists pfp_left_aligned_right_result_witness_operation pfp_right_aligned_right_result_witness_operation pfp_value_aligned_right_result_witness_operation. ((((exists ff_h_pfp_aligned_right_result_witness_operationleft. ff_h_pfp_aligned_right_result_witness_operationleft + S (pfp_left_aligned_right_result_witness_operation) = S ((S (pfp_index_aligned_right_result_witness_operation)) * pfaa_left_c_aligned_right_result)) /\\ exists ff_q_pfp_aligned_right_result_witness_operationleft. pfaa_left_b_aligned_right_result = ff_q_pfp_aligned_right_result_witness_operationleft * S ((S (pfp_index_aligned_right_result_witness_operation)) * pfaa_left_c_aligned_right_result) + (pfp_left_aligned_right_result_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_right_result_witness_operationright. ff_h_pfp_aligned_right_result_witness_operationright + S (pfp_right_aligned_right_result_witness_operation) = S ((S (pfp_index_aligned_right_result_witness_operation)) * pfaa_right_c_aligned_right_result)) /\\ exists ff_q_pfp_aligned_right_result_witness_operationright. pfaa_right_b_aligned_right_result = ff_q_pfp_aligned_right_result_witness_operationright * S ((S (pfp_index_aligned_right_result_witness_operation)) * pfaa_right_c_aligned_right_result) + (pfp_right_aligned_right_result_witness_operation))) /\\ (((((exists ff_h_pfp_aligned_right_result_witness_operationtarget. ff_h_pfp_aligned_right_result_witness_operationtarget + S (pfp_value_aligned_right_result_witness_operation) = S ((S (pfp_index_aligned_right_result_witness_operation)) * pfaa_sum_c_aligned_right_result)) /\\ exists ff_q_pfp_aligned_right_result_witness_operationtarget. pfaa_sum_b_aligned_right_result = ff_q_pfp_aligned_right_result_witness_operationtarget * S ((S (pfp_index_aligned_right_result_witness_operation)) * pfaa_sum_c_aligned_right_result) + (pfp_value_aligned_right_result_witness_operation))) /\\ ((((exists pfa_gap_aligned_right_result_witness_operationoperationleft. pfa_gap_aligned_right_result_witness_operationoperationleft + S (pfp_left_aligned_right_result_witness_operation) = (p)) /\\ (((exists pfa_gap_aligned_right_result_witness_operationoperationright. pfa_gap_aligned_right_result_witness_operationoperationright + S (pfp_right_aligned_right_result_witness_operation) = (p)) /\\ ((((exists pfa_gap_aligned_right_result_witness_operationoperationresultbound. pfa_gap_aligned_right_result_witness_operationoperationresultbound + S (pfp_value_aligned_right_result_witness_operation) = (p)) /\\ ((exists pfa_offset_left_aligned_right_result_witness_operationoperationresultcongruence pfa_offset_right_aligned_right_result_witness_operationoperationresultcongruence. ((pfp_left_aligned_right_result_witness_operation) + (pfp_right_aligned_right_result_witness_operation)) + (p) * pfa_offset_left_aligned_right_result_witness_operationoperationresultcongruence = (pfp_value_aligned_right_result_witness_operation) + (p) * pfa_offset_right_aligned_right_result_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_aligned_right_result_witness_output pfrep_left_aligned_right_result_witness_output pfrep_right_aligned_right_result_witness_output. ((exists pfrep_position_aligned_right_result_witness_outputfirst. ((pfrep_position_aligned_right_result_witness_outputfirst+S (pfrep_power_aligned_right_result_witness_output)=(pfaa_length_aligned_right_result)) /\\ ((((exists ff_h_pfp_aligned_right_result_witness_outputfirstentry. ff_h_pfp_aligned_right_result_witness_outputfirstentry + S (pfrep_left_aligned_right_result_witness_output) = S ((S (pfrep_position_aligned_right_result_witness_outputfirst)) * pfaa_sum_c_aligned_right_result)) /\\ exists ff_q_pfp_aligned_right_result_witness_outputfirstentry. pfaa_sum_b_aligned_right_result = ff_q_pfp_aligned_right_result_witness_outputfirstentry * S ((S (pfrep_position_aligned_right_result_witness_outputfirst)) * pfaa_sum_c_aligned_right_result) + (pfrep_left_aligned_right_result_witness_output)))))) \\/ (((exists pfrep_gap_aligned_right_result_witness_outputfirstoutside. pfrep_gap_aligned_right_result_witness_outputfirstoutside+(pfaa_length_aligned_right_result)=(pfrep_power_aligned_right_result_witness_output)) /\\ (((pfrep_left_aligned_right_result_witness_output)=0))))) -> ((exists pfrep_position_aligned_right_result_witness_outputsecond. ((pfrep_position_aligned_right_result_witness_outputsecond+S (pfrep_power_aligned_right_result_witness_output)=(K)) /\\ ((((exists ff_h_pfp_aligned_right_result_witness_outputsecondentry. ff_h_pfp_aligned_right_result_witness_outputsecondentry + S (pfrep_right_aligned_right_result_witness_output) = S ((S (pfrep_position_aligned_right_result_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_aligned_right_result_witness_outputsecondentry. rb = ff_q_pfp_aligned_right_result_witness_outputsecondentry * S ((S (pfrep_position_aligned_right_result_witness_outputsecond)) * rc) + (pfrep_right_aligned_right_result_witness_output)))))) \\/ (((exists pfrep_gap_aligned_right_result_witness_outputsecondoutside. pfrep_gap_aligned_right_result_witness_outputsecondoutside+(K)=(pfrep_power_aligned_right_result_witness_output)) /\\ (((pfrep_right_aligned_right_result_witness_output)=0))))) -> pfrep_left_aligned_right_result_witness_output=pfrep_right_aligned_right_result_witness_output)))))))))))))", 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(((exists ff_h_pfp_left_constant_first_K. ff_h_pfp_left_constant_first_K + S (k) = S ((S (0)) * kc)) /\\ exists ff_q_pfp_left_constant_first_K. kb = ff_q_pfp_left_constant_first_K * S ((S (0)) * kc) + (k))) -> (exists pfa_gap_left_constant_first_index. pfa_gap_left_constant_first_index + S (i) = (L)) -> (((exists ff_h_pfp_left_constant_first_A. ff_h_pfp_left_constant_first_A + S (a) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_left_constant_first_A. ab = ff_q_pfp_left_constant_first_A * S ((S (i)) * ac) + (a))) -> (exists pfc_complement_left_constant_first_actual pfc_left_left_constant_first_actual pfc_right_left_constant_first_actual. (((0)+pfc_complement_left_constant_first_actual=(i)) /\\ ((((((exists pfa_gap_left_constant_first_actualleftinside. pfa_gap_left_constant_first_actualleftinside + S (0) = (1)) /\\ ((((exists ff_h_pfp_left_constant_first_actualleftentry. ff_h_pfp_left_constant_first_actualleftentry + S (pfc_left_left_constant_first_actual) = S ((S (0)) * kc)) /\\ exists ff_q_pfp_left_constant_first_actualleftentry. kb = ff_q_pfp_left_constant_first_actualleftentry * S ((S (0)) * kc) + (pfc_left_left_constant_first_actual)))))) \\/ (((exists pfc_gap_left_constant_first_actualleftoutside. pfc_gap_left_constant_first_actualleftoutside+(1)=(0)) /\\ (((pfc_left_left_constant_first_actual)=0))))) /\\ ((((((exists pfa_gap_left_constant_first_actualrightinside. pfa_gap_left_constant_first_actualrightinside + S (pfc_complement_left_constant_first_actual) = (L)) /\\ ((((exists ff_h_pfp_left_constant_first_actualrightentry. ff_h_pfp_left_constant_first_actualrightentry + S (pfc_right_left_constant_first_actual) = S ((S (pfc_complement_left_constant_first_actual)) * ac)) /\\ exists ff_q_pfp_left_constant_first_actualrightentry. ab = ff_q_pfp_left_constant_first_actualrightentry * S ((S (pfc_complement_left_constant_first_actual)) * ac) + (pfc_right_left_constant_first_actual)))))) \\/ (((exists pfc_gap_left_constant_first_actualrightoutside. pfc_gap_left_constant_first_actualrightoutside+(L)=(pfc_complement_left_constant_first_actual)) /\\ (((pfc_right_left_constant_first_actual)=0))))) /\\ (((t)=pfc_left_left_constant_first_actual*pfc_right_left_constant_first_actual)))))))) -> (t=k*a)", "statement_sha256": "5c098f6a69ea5c3514dfdde26c586f1af6116063efc7e755c854e6728cf3bdb3", "summary": "The first actual antidiagonal term of a genuine left singleton is the ordered natural product k*a. 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((((exists ff_h_pfp_left_constant_sum_first_entry. ff_h_pfp_left_constant_sum_first_entry + S (t) = S ((S (0)) * dc)) /\\ exists ff_q_pfp_left_constant_sum_first_entry. db = ff_q_pfp_left_constant_sum_first_entry * S ((S (0)) * dc) + (t))) /\\ ((exists pfc_complement_left_constant_sum_first_term pfc_left_left_constant_sum_first_term pfc_right_left_constant_sum_first_term. (((0)+pfc_complement_left_constant_sum_first_term=(i)) /\\ ((((((exists pfa_gap_left_constant_sum_first_termleftinside. pfa_gap_left_constant_sum_first_termleftinside + S (0) = (1)) /\\ ((((exists ff_h_pfp_left_constant_sum_first_termleftentry. ff_h_pfp_left_constant_sum_first_termleftentry + S (pfc_left_left_constant_sum_first_term) = S ((S (0)) * kc)) /\\ exists ff_q_pfp_left_constant_sum_first_termleftentry. kb = ff_q_pfp_left_constant_sum_first_termleftentry * S ((S (0)) * kc) + (pfc_left_left_constant_sum_first_term)))))) \\/ (((exists pfc_gap_left_constant_sum_first_termleftoutside. pfc_gap_left_constant_sum_first_termleftoutside+(1)=(0)) /\\ (((pfc_left_left_constant_sum_first_term)=0))))) /\\ ((((((exists pfa_gap_left_constant_sum_first_termrightinside. pfa_gap_left_constant_sum_first_termrightinside + S (pfc_complement_left_constant_sum_first_term) = (L)) /\\ ((((exists ff_h_pfp_left_constant_sum_first_termrightentry. ff_h_pfp_left_constant_sum_first_termrightentry + S (pfc_right_left_constant_sum_first_term) = S ((S (pfc_complement_left_constant_sum_first_term)) * ac)) /\\ exists ff_q_pfp_left_constant_sum_first_termrightentry. ab = ff_q_pfp_left_constant_sum_first_termrightentry * S ((S (pfc_complement_left_constant_sum_first_term)) * ac) + (pfc_right_left_constant_sum_first_term)))))) \\/ (((exists pfc_gap_left_constant_sum_first_termrightoutside. pfc_gap_left_constant_sum_first_termrightoutside+(L)=(pfc_complement_left_constant_sum_first_term)) /\\ (((pfc_right_left_constant_sum_first_term)=0))))) /\\ (((t)=pfc_left_left_constant_sum_first_term*pfc_right_left_constant_sum_first_term))))))))))", "specialize hd (0)", "apply hd", "exists i", "simp", "cases hv", "cases hv_witness", "have heq : x=k*a", "specialize polynomial_diagonal_left_constant_first_term (k)", "specialize polynomial_diagonal_left_constant_first_term (kb)", "specialize polynomial_diagonal_left_constant_first_term (kc)", "specialize polynomial_diagonal_left_constant_first_term (ab)", "specialize polynomial_diagonal_left_constant_first_term (ac)", "specialize polynomial_diagonal_left_constant_first_term (L)", "specialize polynomial_diagonal_left_constant_first_term (i)", "specialize polynomial_diagonal_left_constant_first_term (a)", "specialize polynomial_diagonal_left_constant_first_term (x)", "apply polynomial_diagonal_left_constant_first_term", "exact hk", "exact hi", "exact ha", "exact hv_witness_right", "rewrite heq at hv_witness_left", "rewrite heq at hv_witness_left", "exact hv_witness_left", "have htail : forall pfpad_tail_index_left_constant_sum_tail. (exists pfa_gap_left_constant_sum_tailbound. pfa_gap_left_constant_sum_tailbound + S (pfpad_tail_index_left_constant_sum_tail) = (i)) -> (((exists ff_h_pfp_left_constant_sum_tailzero. ff_h_pfp_left_constant_sum_tailzero + S (0) = S ((S ((1)+pfpad_tail_index_left_constant_sum_tail)) * dc)) /\\ exists ff_q_pfp_left_constant_sum_tailzero. db = ff_q_pfp_left_constant_sum_tailzero * S ((S ((1)+pfpad_tail_index_left_constant_sum_tail)) * dc) + (0)))", "intro j", "intro hj", "have hv : exists t. ((((exists ff_h_pfp_left_constant_sum_tail_entry. ff_h_pfp_left_constant_sum_tail_entry + S (t) = S ((S (1+j)) * dc)) /\\ exists ff_q_pfp_left_constant_sum_tail_entry. db = ff_q_pfp_left_constant_sum_tail_entry * S ((S (1+j)) * dc) + (t))) /\\ ((exists pfc_complement_left_constant_sum_tail_term pfc_left_left_constant_sum_tail_term pfc_right_left_constant_sum_tail_term. (((1+j)+pfc_complement_left_constant_sum_tail_term=(i)) /\\ ((((((exists pfa_gap_left_constant_sum_tail_termleftinside. pfa_gap_left_constant_sum_tail_termleftinside + S (1+j) = (1)) /\\ ((((exists ff_h_pfp_left_constant_sum_tail_termleftentry. ff_h_pfp_left_constant_sum_tail_termleftentry + S (pfc_left_left_constant_sum_tail_term) = S ((S (1+j)) * kc)) /\\ exists ff_q_pfp_left_constant_sum_tail_termleftentry. kb = ff_q_pfp_left_constant_sum_tail_termleftentry * S ((S (1+j)) * kc) + (pfc_left_left_constant_sum_tail_term)))))) \\/ (((exists pfc_gap_left_constant_sum_tail_termleftoutside. pfc_gap_left_constant_sum_tail_termleftoutside+(1)=(1+j)) /\\ (((pfc_left_left_constant_sum_tail_term)=0))))) /\\ ((((((exists pfa_gap_left_constant_sum_tail_termrightinside. pfa_gap_left_constant_sum_tail_termrightinside + S (pfc_complement_left_constant_sum_tail_term) = (L)) /\\ ((((exists ff_h_pfp_left_constant_sum_tail_termrightentry. ff_h_pfp_left_constant_sum_tail_termrightentry + S (pfc_right_left_constant_sum_tail_term) = S ((S (pfc_complement_left_constant_sum_tail_term)) * ac)) /\\ exists ff_q_pfp_left_constant_sum_tail_termrightentry. ab = ff_q_pfp_left_constant_sum_tail_termrightentry * S ((S (pfc_complement_left_constant_sum_tail_term)) * ac) + (pfc_right_left_constant_sum_tail_term)))))) \\/ (((exists pfc_gap_left_constant_sum_tail_termrightoutside. pfc_gap_left_constant_sum_tail_termrightoutside+(L)=(pfc_complement_left_constant_sum_tail_term)) /\\ (((pfc_right_left_constant_sum_tail_term)=0))))) /\\ (((t)=pfc_left_left_constant_sum_tail_term*pfc_right_left_constant_sum_tail_term))))))))))", "specialize hd (1+j)", "apply hd", "have hindex : 1+j=S j", "simp [add_succ_left,zero_add]", "rewrite hindex", "specialize succ_le_succ (S j)", "specialize succ_le_succ (i)", "apply succ_le_succ", "exact hj", "cases hv", "cases hv_witness", "have heq : x=0", "specialize polynomial_diagonal_left_unit_tail_term (kb)", "specialize polynomial_diagonal_left_unit_tail_term (kc)", "specialize polynomial_diagonal_left_unit_tail_term (ab)", "specialize polynomial_diagonal_left_unit_tail_term (ac)", "specialize polynomial_diagonal_left_unit_tail_term (L)", "specialize polynomial_diagonal_left_unit_tail_term (i)", "specialize polynomial_diagonal_left_unit_tail_term (1+j)", "specialize polynomial_diagonal_left_unit_tail_term (x)", "apply polynomial_diagonal_left_unit_tail_term", "specialize le_add_right (1)", "specialize le_add_right (j)", "apply le_add_right", "exact hv_witness_right", "rewrite heq at hv_witness_left", "rewrite heq at hv_witness_left", "exact hv_witness_left", "have hsingle : exists m. (exists fs_u_pfc_left_constant_single_sum fs_v_pfc_left_constant_single_sum. ((((exists fs_h_pfc_left_constant_single_sum_body_start. fs_h_pfc_left_constant_single_sum_body_start + S (0) = S ((S (0)) * fs_v_pfc_left_constant_single_sum)) /\\ exists fs_q_pfc_left_constant_single_sum_body_start. fs_u_pfc_left_constant_single_sum = fs_q_pfc_left_constant_single_sum_body_start * S ((S (0)) * fs_v_pfc_left_constant_single_sum) + (0))) /\\ ((((exists fs_h_pfc_left_constant_single_sum_body_terminal. fs_h_pfc_left_constant_single_sum_body_terminal + S (m) = S ((S (1)) * fs_v_pfc_left_constant_single_sum)) /\\ exists fs_q_pfc_left_constant_single_sum_body_terminal. fs_u_pfc_left_constant_single_sum = fs_q_pfc_left_constant_single_sum_body_terminal * S ((S (1)) * fs_v_pfc_left_constant_single_sum) + (m))) /\\ forall fs_i_pfc_left_constant_single_sum_body_steps. (exists fs_lt_pfc_left_constant_single_sum_body_steps_bound. fs_lt_pfc_left_constant_single_sum_body_steps_bound + S fs_i_pfc_left_constant_single_sum_body_steps = 1) -> exists fs_a_pfc_left_constant_single_sum_body_steps fs_r_pfc_left_constant_single_sum_body_steps fs_s_pfc_left_constant_single_sum_body_steps. ((((exists fs_h_pfc_left_constant_single_sum_body_steps_summand. fs_h_pfc_left_constant_single_sum_body_steps_summand + S (fs_a_pfc_left_constant_single_sum_body_steps) = S ((S (fs_i_pfc_left_constant_single_sum_body_steps)) * dc)) /\\ exists fs_q_pfc_left_constant_single_sum_body_steps_summand. db = fs_q_pfc_left_constant_single_sum_body_steps_summand * S ((S (fs_i_pfc_left_constant_single_sum_body_steps)) * dc) + (fs_a_pfc_left_constant_single_sum_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_single_sum_body_steps_partial. fs_h_pfc_left_constant_single_sum_body_steps_partial + S (fs_r_pfc_left_constant_single_sum_body_steps) = S ((S (fs_i_pfc_left_constant_single_sum_body_steps)) * fs_v_pfc_left_constant_single_sum)) /\\ exists fs_q_pfc_left_constant_single_sum_body_steps_partial. fs_u_pfc_left_constant_single_sum = fs_q_pfc_left_constant_single_sum_body_steps_partial * S ((S (fs_i_pfc_left_constant_single_sum_body_steps)) * fs_v_pfc_left_constant_single_sum) + (fs_r_pfc_left_constant_single_sum_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_single_sum_body_steps_successor. fs_h_pfc_left_constant_single_sum_body_steps_successor + S (fs_s_pfc_left_constant_single_sum_body_steps) = S ((S (S fs_i_pfc_left_constant_single_sum_body_steps)) * fs_v_pfc_left_constant_single_sum)) /\\ exists fs_q_pfc_left_constant_single_sum_body_steps_successor. fs_u_pfc_left_constant_single_sum = fs_q_pfc_left_constant_single_sum_body_steps_successor * S ((S (S fs_i_pfc_left_constant_single_sum_body_steps)) * fs_v_pfc_left_constant_single_sum) + (fs_s_pfc_left_constant_single_sum_body_steps))) /\\ fs_s_pfc_left_constant_single_sum_body_steps = fs_r_pfc_left_constant_single_sum_body_steps + fs_a_pfc_left_constant_single_sum_body_steps))))))", "specialize beta_sum_exists (db)", "specialize beta_sum_exists (dc)", "specialize beta_sum_exists (1)", "apply beta_sum_exists", "cases hsingle", "have hdecomp : exists t s. ((((exists ff_h_pfp_left_constant_single_entry. ff_h_pfp_left_constant_single_entry + S (t) = S ((S (0)) * dc)) /\\ exists ff_q_pfp_left_constant_single_entry. db = ff_q_pfp_left_constant_single_entry * S ((S (0)) * dc) + (t))) /\\ (((exists fs_u_pfc_left_constant_single_empty fs_v_pfc_left_constant_single_empty. ((((exists fs_h_pfc_left_constant_single_empty_body_start. fs_h_pfc_left_constant_single_empty_body_start + S (0) = S ((S (0)) * fs_v_pfc_left_constant_single_empty)) /\\ exists fs_q_pfc_left_constant_single_empty_body_start. fs_u_pfc_left_constant_single_empty = fs_q_pfc_left_constant_single_empty_body_start * S ((S (0)) * fs_v_pfc_left_constant_single_empty) + (0))) /\\ ((((exists fs_h_pfc_left_constant_single_empty_body_terminal. fs_h_pfc_left_constant_single_empty_body_terminal + S (s) = S ((S (0)) * fs_v_pfc_left_constant_single_empty)) /\\ exists fs_q_pfc_left_constant_single_empty_body_terminal. fs_u_pfc_left_constant_single_empty = fs_q_pfc_left_constant_single_empty_body_terminal * S ((S (0)) * fs_v_pfc_left_constant_single_empty) + (s))) /\\ forall fs_i_pfc_left_constant_single_empty_body_steps. (exists fs_lt_pfc_left_constant_single_empty_body_steps_bound. fs_lt_pfc_left_constant_single_empty_body_steps_bound + S fs_i_pfc_left_constant_single_empty_body_steps = 0) -> exists fs_a_pfc_left_constant_single_empty_body_steps fs_r_pfc_left_constant_single_empty_body_steps fs_s_pfc_left_constant_single_empty_body_steps. ((((exists fs_h_pfc_left_constant_single_empty_body_steps_summand. fs_h_pfc_left_constant_single_empty_body_steps_summand + S (fs_a_pfc_left_constant_single_empty_body_steps) = S ((S (fs_i_pfc_left_constant_single_empty_body_steps)) * dc)) /\\ exists fs_q_pfc_left_constant_single_empty_body_steps_summand. db = fs_q_pfc_left_constant_single_empty_body_steps_summand * S ((S (fs_i_pfc_left_constant_single_empty_body_steps)) * dc) + (fs_a_pfc_left_constant_single_empty_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_single_empty_body_steps_partial. fs_h_pfc_left_constant_single_empty_body_steps_partial + S (fs_r_pfc_left_constant_single_empty_body_steps) = S ((S (fs_i_pfc_left_constant_single_empty_body_steps)) * fs_v_pfc_left_constant_single_empty)) /\\ exists fs_q_pfc_left_constant_single_empty_body_steps_partial. fs_u_pfc_left_constant_single_empty = fs_q_pfc_left_constant_single_empty_body_steps_partial * S ((S (fs_i_pfc_left_constant_single_empty_body_steps)) * fs_v_pfc_left_constant_single_empty) + (fs_r_pfc_left_constant_single_empty_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_single_empty_body_steps_successor. fs_h_pfc_left_constant_single_empty_body_steps_successor + S (fs_s_pfc_left_constant_single_empty_body_steps) = S ((S (S fs_i_pfc_left_constant_single_empty_body_steps)) * fs_v_pfc_left_constant_single_empty)) /\\ exists fs_q_pfc_left_constant_single_empty_body_steps_successor. fs_u_pfc_left_constant_single_empty = fs_q_pfc_left_constant_single_empty_body_steps_successor * S ((S (S fs_i_pfc_left_constant_single_empty_body_steps)) * fs_v_pfc_left_constant_single_empty) + (fs_s_pfc_left_constant_single_empty_body_steps))) /\\ fs_s_pfc_left_constant_single_empty_body_steps = fs_r_pfc_left_constant_single_empty_body_steps + fs_a_pfc_left_constant_single_empty_body_steps)))))) /\\ ((x=s+t)))))", "specialize beta_sum_succ_decompose (db)", "specialize beta_sum_succ_decompose (dc)", "specialize beta_sum_succ_decompose (0)", "specialize beta_sum_succ_decompose (x)", "apply beta_sum_succ_decompose", "exact hsingle_witness", "cases hdecomp", "cases hdecomp_witness", "cases hdecomp_witness_witness", "cases hdecomp_witness_witness_right", "have hzero : x2=0", "specialize beta_sum_zero (db)", "specialize beta_sum_zero (dc)", "specialize beta_sum_zero (x2)", "apply beta_sum_zero", "exact hdecomp_witness_witness_right_left", "have hentry : x1=k*a", "specialize beta_at_unique (db)", "specialize beta_at_unique (dc)", "specialize beta_at_unique (0)", "specialize beta_at_unique (x1)", "specialize beta_at_unique (k*a)", "apply beta_at_unique", "exact hdecomp_witness_witness_left", "exact hhead", "have hvalue : x=k*a", "trans x2+x1", "exact hdecomp_witness_witness_right_right", "rewrite hzero", "trans x1", "apply zero_add", "exact hentry", "trans x", "specialize polynomial_zero_tail_natural_sum_invariant (db)", "specialize polynomial_zero_tail_natural_sum_invariant (dc)", "specialize polynomial_zero_tail_natural_sum_invariant (db)", "specialize polynomial_zero_tail_natural_sum_invariant (dc)", "specialize polynomial_zero_tail_natural_sum_invariant (1)", "specialize polynomial_zero_tail_natural_sum_invariant (i)", "specialize polynomial_zero_tail_natural_sum_invariant (x)", "specialize polynomial_zero_tail_natural_sum_invariant (n)", "apply polynomial_zero_tail_natural_sum_invariant", "intro j0", "intro v0", "intro hj0", "intro hv0", "exact hv0", "exact htail", "exact hsingle_witness", "have hlength : 1+i=S i", "simp [add_succ_left,zero_add]", "rewrite hlength", "rewrite hlength", "rewrite hlength", "exact hs", "exact hvalue" ], "script_sha256": "5b2dff70091b8fdd500c3c5943068ee4ff0c33f99080231e673e8fbc36240d75", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_left_constant_candidate.py", "sha256": "9a7a4de30f5f389bcabc2e6267a0d2cc5dc5f061059dcea303a0a03dab58509a" }, "statement": "forall k kb kc ab ac L i a db dc n. (((exists ff_h_pfp_left_constant_sum_K. ff_h_pfp_left_constant_sum_K + S (k) = S ((S (0)) * kc)) /\\ exists ff_q_pfp_left_constant_sum_K. kb = ff_q_pfp_left_constant_sum_K * S ((S (0)) * kc) + (k))) -> (exists pfa_gap_left_constant_sum_index. pfa_gap_left_constant_sum_index + S (i) = (L)) -> (((exists ff_h_pfp_left_constant_sum_A. ff_h_pfp_left_constant_sum_A + S (a) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_left_constant_sum_A. ab = ff_q_pfp_left_constant_sum_A * S ((S (i)) * ac) + (a))) -> (forall pfc_index_left_constant_sum_diagonal. (exists pfa_gap_left_constant_sum_diagonalbound. pfa_gap_left_constant_sum_diagonalbound + S (pfc_index_left_constant_sum_diagonal) = (S i)) -> exists pfc_value_left_constant_sum_diagonal. ((((exists ff_h_pfp_left_constant_sum_diagonalentry. ff_h_pfp_left_constant_sum_diagonalentry + S (pfc_value_left_constant_sum_diagonal) = S ((S (pfc_index_left_constant_sum_diagonal)) * dc)) /\\ exists ff_q_pfp_left_constant_sum_diagonalentry. db = ff_q_pfp_left_constant_sum_diagonalentry * S ((S (pfc_index_left_constant_sum_diagonal)) * dc) + (pfc_value_left_constant_sum_diagonal))) /\\ ((exists pfc_complement_left_constant_sum_diagonalterm pfc_left_left_constant_sum_diagonalterm pfc_right_left_constant_sum_diagonalterm. (((pfc_index_left_constant_sum_diagonal)+pfc_complement_left_constant_sum_diagonalterm=(i)) /\\ ((((((exists pfa_gap_left_constant_sum_diagonaltermleftinside. pfa_gap_left_constant_sum_diagonaltermleftinside + S (pfc_index_left_constant_sum_diagonal) = (1)) /\\ ((((exists ff_h_pfp_left_constant_sum_diagonaltermleftentry. ff_h_pfp_left_constant_sum_diagonaltermleftentry + S (pfc_left_left_constant_sum_diagonalterm) = S ((S (pfc_index_left_constant_sum_diagonal)) * kc)) /\\ exists ff_q_pfp_left_constant_sum_diagonaltermleftentry. kb = ff_q_pfp_left_constant_sum_diagonaltermleftentry * S ((S (pfc_index_left_constant_sum_diagonal)) * kc) + (pfc_left_left_constant_sum_diagonalterm)))))) \\/ (((exists pfc_gap_left_constant_sum_diagonaltermleftoutside. pfc_gap_left_constant_sum_diagonaltermleftoutside+(1)=(pfc_index_left_constant_sum_diagonal)) /\\ (((pfc_left_left_constant_sum_diagonalterm)=0))))) /\\ ((((((exists pfa_gap_left_constant_sum_diagonaltermrightinside. pfa_gap_left_constant_sum_diagonaltermrightinside + S (pfc_complement_left_constant_sum_diagonalterm) = (L)) /\\ ((((exists ff_h_pfp_left_constant_sum_diagonaltermrightentry. ff_h_pfp_left_constant_sum_diagonaltermrightentry + S (pfc_right_left_constant_sum_diagonalterm) = S ((S (pfc_complement_left_constant_sum_diagonalterm)) * ac)) /\\ exists ff_q_pfp_left_constant_sum_diagonaltermrightentry. ab = ff_q_pfp_left_constant_sum_diagonaltermrightentry * S ((S (pfc_complement_left_constant_sum_diagonalterm)) * ac) + (pfc_right_left_constant_sum_diagonalterm)))))) \\/ (((exists pfc_gap_left_constant_sum_diagonaltermrightoutside. pfc_gap_left_constant_sum_diagonaltermrightoutside+(L)=(pfc_complement_left_constant_sum_diagonalterm)) /\\ (((pfc_right_left_constant_sum_diagonalterm)=0))))) /\\ (((pfc_value_left_constant_sum_diagonal)=pfc_left_left_constant_sum_diagonalterm*pfc_right_left_constant_sum_diagonalterm))))))))))) -> (exists fs_u_pfc_left_constant_sum_actual fs_v_pfc_left_constant_sum_actual. ((((exists fs_h_pfc_left_constant_sum_actual_body_start. fs_h_pfc_left_constant_sum_actual_body_start + S (0) = S ((S (0)) * fs_v_pfc_left_constant_sum_actual)) /\\ exists fs_q_pfc_left_constant_sum_actual_body_start. fs_u_pfc_left_constant_sum_actual = fs_q_pfc_left_constant_sum_actual_body_start * S ((S (0)) * fs_v_pfc_left_constant_sum_actual) + (0))) /\\ ((((exists fs_h_pfc_left_constant_sum_actual_body_terminal. fs_h_pfc_left_constant_sum_actual_body_terminal + S (n) = S ((S (S i)) * fs_v_pfc_left_constant_sum_actual)) /\\ exists fs_q_pfc_left_constant_sum_actual_body_terminal. fs_u_pfc_left_constant_sum_actual = fs_q_pfc_left_constant_sum_actual_body_terminal * S ((S (S i)) * fs_v_pfc_left_constant_sum_actual) + (n))) /\\ forall fs_i_pfc_left_constant_sum_actual_body_steps. (exists fs_lt_pfc_left_constant_sum_actual_body_steps_bound. fs_lt_pfc_left_constant_sum_actual_body_steps_bound + S fs_i_pfc_left_constant_sum_actual_body_steps = S i) -> exists fs_a_pfc_left_constant_sum_actual_body_steps fs_r_pfc_left_constant_sum_actual_body_steps fs_s_pfc_left_constant_sum_actual_body_steps. ((((exists fs_h_pfc_left_constant_sum_actual_body_steps_summand. fs_h_pfc_left_constant_sum_actual_body_steps_summand + S (fs_a_pfc_left_constant_sum_actual_body_steps) = S ((S (fs_i_pfc_left_constant_sum_actual_body_steps)) * dc)) /\\ exists fs_q_pfc_left_constant_sum_actual_body_steps_summand. db = fs_q_pfc_left_constant_sum_actual_body_steps_summand * S ((S (fs_i_pfc_left_constant_sum_actual_body_steps)) * dc) + (fs_a_pfc_left_constant_sum_actual_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_sum_actual_body_steps_partial. fs_h_pfc_left_constant_sum_actual_body_steps_partial + S (fs_r_pfc_left_constant_sum_actual_body_steps) = S ((S (fs_i_pfc_left_constant_sum_actual_body_steps)) * fs_v_pfc_left_constant_sum_actual)) /\\ exists fs_q_pfc_left_constant_sum_actual_body_steps_partial. fs_u_pfc_left_constant_sum_actual = fs_q_pfc_left_constant_sum_actual_body_steps_partial * S ((S (fs_i_pfc_left_constant_sum_actual_body_steps)) * fs_v_pfc_left_constant_sum_actual) + (fs_r_pfc_left_constant_sum_actual_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_sum_actual_body_steps_successor. fs_h_pfc_left_constant_sum_actual_body_steps_successor + S (fs_s_pfc_left_constant_sum_actual_body_steps) = S ((S (S fs_i_pfc_left_constant_sum_actual_body_steps)) * fs_v_pfc_left_constant_sum_actual)) /\\ exists fs_q_pfc_left_constant_sum_actual_body_steps_successor. fs_u_pfc_left_constant_sum_actual = fs_q_pfc_left_constant_sum_actual_body_steps_successor * S ((S (S fs_i_pfc_left_constant_sum_actual_body_steps)) * fs_v_pfc_left_constant_sum_actual) + (fs_s_pfc_left_constant_sum_actual_body_steps))) /\\ fs_s_pfc_left_constant_sum_actual_body_steps = fs_r_pfc_left_constant_sum_actual_body_steps + fs_a_pfc_left_constant_sum_actual_body_steps)))))) -> (n=k*a)", "statement_sha256": "14755e4231073141e181a5d64da95e61423637a369567dacc69d72c986bc8bed", "summary": "The actual finite natural sum equals k*a: construct its one-term sum and use the existing zero-tail invariant for every subsequent summand. 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(((exists ff_h_pfp_left_constant_coefficient_K. ff_h_pfp_left_constant_coefficient_K + S (k) = S ((S (0)) * kc)) /\\ exists ff_q_pfp_left_constant_coefficient_K. kb = ff_q_pfp_left_constant_coefficient_K * S ((S (0)) * kc) + (k))) -> (exists pfa_gap_left_constant_coefficient_index. pfa_gap_left_constant_coefficient_index + S (i) = (L)) -> (((exists ff_h_pfp_left_constant_coefficient_A. ff_h_pfp_left_constant_coefficient_A + S (a) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_left_constant_coefficient_A. ab = ff_q_pfp_left_constant_coefficient_A * S ((S (i)) * ac) + (a))) -> (exists pfa_gap_left_constant_coefficient_k_bound. pfa_gap_left_constant_coefficient_k_bound + S (k) = (p)) -> (exists pfa_gap_left_constant_coefficient_a_bound. pfa_gap_left_constant_coefficient_a_bound + S (a) = (p)) -> (exists pfc_terms_code_left_constant_coefficient_actual pfc_terms_scale_left_constant_coefficient_actual pfc_natural_sum_left_constant_coefficient_actual. ((forall pfc_index_left_constant_coefficient_actualdiagonal. (exists pfa_gap_left_constant_coefficient_actualdiagonalbound. pfa_gap_left_constant_coefficient_actualdiagonalbound + S (pfc_index_left_constant_coefficient_actualdiagonal) = (S (i))) -> exists pfc_value_left_constant_coefficient_actualdiagonal. ((((exists ff_h_pfp_left_constant_coefficient_actualdiagonalentry. ff_h_pfp_left_constant_coefficient_actualdiagonalentry + S (pfc_value_left_constant_coefficient_actualdiagonal) = S ((S (pfc_index_left_constant_coefficient_actualdiagonal)) * pfc_terms_scale_left_constant_coefficient_actual)) /\\ exists ff_q_pfp_left_constant_coefficient_actualdiagonalentry. pfc_terms_code_left_constant_coefficient_actual = ff_q_pfp_left_constant_coefficient_actualdiagonalentry * S ((S (pfc_index_left_constant_coefficient_actualdiagonal)) * pfc_terms_scale_left_constant_coefficient_actual) + (pfc_value_left_constant_coefficient_actualdiagonal))) /\\ ((exists pfc_complement_left_constant_coefficient_actualdiagonalterm pfc_left_left_constant_coefficient_actualdiagonalterm pfc_right_left_constant_coefficient_actualdiagonalterm. (((pfc_index_left_constant_coefficient_actualdiagonal)+pfc_complement_left_constant_coefficient_actualdiagonalterm=(i)) /\\ ((((((exists pfa_gap_left_constant_coefficient_actualdiagonaltermleftinside. pfa_gap_left_constant_coefficient_actualdiagonaltermleftinside + S (pfc_index_left_constant_coefficient_actualdiagonal) = (1)) /\\ ((((exists ff_h_pfp_left_constant_coefficient_actualdiagonaltermleftentry. ff_h_pfp_left_constant_coefficient_actualdiagonaltermleftentry + S (pfc_left_left_constant_coefficient_actualdiagonalterm) = S ((S (pfc_index_left_constant_coefficient_actualdiagonal)) * kc)) /\\ exists ff_q_pfp_left_constant_coefficient_actualdiagonaltermleftentry. kb = ff_q_pfp_left_constant_coefficient_actualdiagonaltermleftentry * S ((S (pfc_index_left_constant_coefficient_actualdiagonal)) * kc) + (pfc_left_left_constant_coefficient_actualdiagonalterm)))))) \\/ (((exists pfc_gap_left_constant_coefficient_actualdiagonaltermleftoutside. pfc_gap_left_constant_coefficient_actualdiagonaltermleftoutside+(1)=(pfc_index_left_constant_coefficient_actualdiagonal)) /\\ (((pfc_left_left_constant_coefficient_actualdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_left_constant_coefficient_actualdiagonaltermrightinside. pfa_gap_left_constant_coefficient_actualdiagonaltermrightinside + S (pfc_complement_left_constant_coefficient_actualdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_left_constant_coefficient_actualdiagonaltermrightentry. ff_h_pfp_left_constant_coefficient_actualdiagonaltermrightentry + S (pfc_right_left_constant_coefficient_actualdiagonalterm) = S ((S (pfc_complement_left_constant_coefficient_actualdiagonalterm)) * ac)) /\\ exists ff_q_pfp_left_constant_coefficient_actualdiagonaltermrightentry. ab = ff_q_pfp_left_constant_coefficient_actualdiagonaltermrightentry * S ((S (pfc_complement_left_constant_coefficient_actualdiagonalterm)) * ac) + (pfc_right_left_constant_coefficient_actualdiagonalterm)))))) \\/ (((exists pfc_gap_left_constant_coefficient_actualdiagonaltermrightoutside. pfc_gap_left_constant_coefficient_actualdiagonaltermrightoutside+(L)=(pfc_complement_left_constant_coefficient_actualdiagonalterm)) /\\ (((pfc_right_left_constant_coefficient_actualdiagonalterm)=0))))) /\\ (((pfc_value_left_constant_coefficient_actualdiagonal)=pfc_left_left_constant_coefficient_actualdiagonalterm*pfc_right_left_constant_coefficient_actualdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_left_constant_coefficient_actualsum fs_v_pfc_left_constant_coefficient_actualsum. ((((exists fs_h_pfc_left_constant_coefficient_actualsum_body_start. fs_h_pfc_left_constant_coefficient_actualsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_left_constant_coefficient_actualsum)) /\\ exists fs_q_pfc_left_constant_coefficient_actualsum_body_start. fs_u_pfc_left_constant_coefficient_actualsum = fs_q_pfc_left_constant_coefficient_actualsum_body_start * S ((S (0)) * fs_v_pfc_left_constant_coefficient_actualsum) + (0))) /\\ ((((exists fs_h_pfc_left_constant_coefficient_actualsum_body_terminal. fs_h_pfc_left_constant_coefficient_actualsum_body_terminal + S (pfc_natural_sum_left_constant_coefficient_actual) = S ((S (S (i))) * fs_v_pfc_left_constant_coefficient_actualsum)) /\\ exists fs_q_pfc_left_constant_coefficient_actualsum_body_terminal. fs_u_pfc_left_constant_coefficient_actualsum = fs_q_pfc_left_constant_coefficient_actualsum_body_terminal * S ((S (S (i))) * fs_v_pfc_left_constant_coefficient_actualsum) + (pfc_natural_sum_left_constant_coefficient_actual))) /\\ forall fs_i_pfc_left_constant_coefficient_actualsum_body_steps. (exists fs_lt_pfc_left_constant_coefficient_actualsum_body_steps_bound. fs_lt_pfc_left_constant_coefficient_actualsum_body_steps_bound + S fs_i_pfc_left_constant_coefficient_actualsum_body_steps = S (i)) -> exists fs_a_pfc_left_constant_coefficient_actualsum_body_steps fs_r_pfc_left_constant_coefficient_actualsum_body_steps fs_s_pfc_left_constant_coefficient_actualsum_body_steps. ((((exists fs_h_pfc_left_constant_coefficient_actualsum_body_steps_summand. fs_h_pfc_left_constant_coefficient_actualsum_body_steps_summand + S (fs_a_pfc_left_constant_coefficient_actualsum_body_steps) = S ((S (fs_i_pfc_left_constant_coefficient_actualsum_body_steps)) * pfc_terms_scale_left_constant_coefficient_actual)) /\\ exists fs_q_pfc_left_constant_coefficient_actualsum_body_steps_summand. pfc_terms_code_left_constant_coefficient_actual = fs_q_pfc_left_constant_coefficient_actualsum_body_steps_summand * S ((S (fs_i_pfc_left_constant_coefficient_actualsum_body_steps)) * pfc_terms_scale_left_constant_coefficient_actual) + (fs_a_pfc_left_constant_coefficient_actualsum_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_coefficient_actualsum_body_steps_partial. fs_h_pfc_left_constant_coefficient_actualsum_body_steps_partial + S (fs_r_pfc_left_constant_coefficient_actualsum_body_steps) = S ((S (fs_i_pfc_left_constant_coefficient_actualsum_body_steps)) * fs_v_pfc_left_constant_coefficient_actualsum)) /\\ exists fs_q_pfc_left_constant_coefficient_actualsum_body_steps_partial. fs_u_pfc_left_constant_coefficient_actualsum = fs_q_pfc_left_constant_coefficient_actualsum_body_steps_partial * S ((S (fs_i_pfc_left_constant_coefficient_actualsum_body_steps)) * fs_v_pfc_left_constant_coefficient_actualsum) + (fs_r_pfc_left_constant_coefficient_actualsum_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_coefficient_actualsum_body_steps_successor. fs_h_pfc_left_constant_coefficient_actualsum_body_steps_successor + S (fs_s_pfc_left_constant_coefficient_actualsum_body_steps) = S ((S (S fs_i_pfc_left_constant_coefficient_actualsum_body_steps)) * fs_v_pfc_left_constant_coefficient_actualsum)) /\\ exists fs_q_pfc_left_constant_coefficient_actualsum_body_steps_successor. fs_u_pfc_left_constant_coefficient_actualsum = fs_q_pfc_left_constant_coefficient_actualsum_body_steps_successor * S ((S (S fs_i_pfc_left_constant_coefficient_actualsum_body_steps)) * fs_v_pfc_left_constant_coefficient_actualsum) + (fs_s_pfc_left_constant_coefficient_actualsum_body_steps))) /\\ fs_s_pfc_left_constant_coefficient_actualsum_body_steps = fs_r_pfc_left_constant_coefficient_actualsum_body_steps + fs_a_pfc_left_constant_coefficient_actualsum_body_steps)))))) /\\ ((((exists pfa_gap_left_constant_coefficient_actualresiduebound. pfa_gap_left_constant_coefficient_actualresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_left_constant_coefficient_actualresiduecongruence pfa_offset_right_left_constant_coefficient_actualresiduecongruence. (pfc_natural_sum_left_constant_coefficient_actual) + (p) * pfa_offset_left_left_constant_coefficient_actualresiduecongruence = (r) + (p) * pfa_offset_right_left_constant_coefficient_actualresiduecongruence))))))))) -> (((exists pfa_gap_left_constant_coefficient_resultleft. pfa_gap_left_constant_coefficient_resultleft + S (k) = (p)) /\\ (((exists pfa_gap_left_constant_coefficient_resultright. pfa_gap_left_constant_coefficient_resultright + S (a) = (p)) /\\ ((((exists pfa_gap_left_constant_coefficient_resultresultbound. pfa_gap_left_constant_coefficient_resultresultbound + S (r) = (p)) /\\ ((exists pfa_offset_left_left_constant_coefficient_resultresultcongruence pfa_offset_right_left_constant_coefficient_resultresultcongruence. 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((exists ff_h_pfp_left_constant_product_A. ff_h_pfp_left_constant_product_A + S (a) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_left_constant_product_A. ab = ff_q_pfp_left_constant_product_A * S ((S (i)) * ac) + (a))", "specialize beta_at_exists (ab)", "specialize beta_at_exists (ac)", "specialize beta_at_exists (i)", "apply beta_at_exists", "cases ha", "have hr : exists r. ((((exists ff_h_pfp_left_constant_product_H. ff_h_pfp_left_constant_product_H + S (r) = S ((S (i)) * hc)) /\\ exists ff_q_pfp_left_constant_product_H. hb = ff_q_pfp_left_constant_product_H * S ((S (i)) * hc) + (r))) /\\ ((exists pfc_terms_code_left_constant_product_coefficient pfc_terms_scale_left_constant_product_coefficient pfc_natural_sum_left_constant_product_coefficient. ((forall pfc_index_left_constant_product_coefficientdiagonal. (exists pfa_gap_left_constant_product_coefficientdiagonalbound. pfa_gap_left_constant_product_coefficientdiagonalbound + S (pfc_index_left_constant_product_coefficientdiagonal) = (S (i))) -> exists pfc_value_left_constant_product_coefficientdiagonal. ((((exists ff_h_pfp_left_constant_product_coefficientdiagonalentry. ff_h_pfp_left_constant_product_coefficientdiagonalentry + S (pfc_value_left_constant_product_coefficientdiagonal) = S ((S (pfc_index_left_constant_product_coefficientdiagonal)) * pfc_terms_scale_left_constant_product_coefficient)) /\\ exists ff_q_pfp_left_constant_product_coefficientdiagonalentry. pfc_terms_code_left_constant_product_coefficient = ff_q_pfp_left_constant_product_coefficientdiagonalentry * S ((S (pfc_index_left_constant_product_coefficientdiagonal)) * pfc_terms_scale_left_constant_product_coefficient) + (pfc_value_left_constant_product_coefficientdiagonal))) /\\ ((exists pfc_complement_left_constant_product_coefficientdiagonalterm pfc_left_left_constant_product_coefficientdiagonalterm pfc_right_left_constant_product_coefficientdiagonalterm. (((pfc_index_left_constant_product_coefficientdiagonal)+pfc_complement_left_constant_product_coefficientdiagonalterm=(i)) /\\ ((((((exists pfa_gap_left_constant_product_coefficientdiagonaltermleftinside. pfa_gap_left_constant_product_coefficientdiagonaltermleftinside + S (pfc_index_left_constant_product_coefficientdiagonal) = (1)) /\\ ((((exists ff_h_pfp_left_constant_product_coefficientdiagonaltermleftentry. ff_h_pfp_left_constant_product_coefficientdiagonaltermleftentry + S (pfc_left_left_constant_product_coefficientdiagonalterm) = S ((S (pfc_index_left_constant_product_coefficientdiagonal)) * kc)) /\\ exists ff_q_pfp_left_constant_product_coefficientdiagonaltermleftentry. kb = ff_q_pfp_left_constant_product_coefficientdiagonaltermleftentry * S ((S (pfc_index_left_constant_product_coefficientdiagonal)) * kc) + (pfc_left_left_constant_product_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_left_constant_product_coefficientdiagonaltermleftoutside. pfc_gap_left_constant_product_coefficientdiagonaltermleftoutside+(1)=(pfc_index_left_constant_product_coefficientdiagonal)) /\\ (((pfc_left_left_constant_product_coefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_left_constant_product_coefficientdiagonaltermrightinside. pfa_gap_left_constant_product_coefficientdiagonaltermrightinside + S (pfc_complement_left_constant_product_coefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_left_constant_product_coefficientdiagonaltermrightentry. ff_h_pfp_left_constant_product_coefficientdiagonaltermrightentry + S (pfc_right_left_constant_product_coefficientdiagonalterm) = S ((S (pfc_complement_left_constant_product_coefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_left_constant_product_coefficientdiagonaltermrightentry. ab = ff_q_pfp_left_constant_product_coefficientdiagonaltermrightentry * S ((S (pfc_complement_left_constant_product_coefficientdiagonalterm)) * ac) + (pfc_right_left_constant_product_coefficientdiagonalterm)))))) \\/ (((exists pfc_gap_left_constant_product_coefficientdiagonaltermrightoutside. pfc_gap_left_constant_product_coefficientdiagonaltermrightoutside+(L)=(pfc_complement_left_constant_product_coefficientdiagonalterm)) /\\ (((pfc_right_left_constant_product_coefficientdiagonalterm)=0))))) /\\ (((pfc_value_left_constant_product_coefficientdiagonal)=pfc_left_left_constant_product_coefficientdiagonalterm*pfc_right_left_constant_product_coefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_left_constant_product_coefficientsum fs_v_pfc_left_constant_product_coefficientsum. ((((exists fs_h_pfc_left_constant_product_coefficientsum_body_start. fs_h_pfc_left_constant_product_coefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_left_constant_product_coefficientsum)) /\\ exists fs_q_pfc_left_constant_product_coefficientsum_body_start. fs_u_pfc_left_constant_product_coefficientsum = fs_q_pfc_left_constant_product_coefficientsum_body_start * S ((S (0)) * fs_v_pfc_left_constant_product_coefficientsum) + (0))) /\\ ((((exists fs_h_pfc_left_constant_product_coefficientsum_body_terminal. fs_h_pfc_left_constant_product_coefficientsum_body_terminal + S (pfc_natural_sum_left_constant_product_coefficient) = S ((S (S (i))) * fs_v_pfc_left_constant_product_coefficientsum)) /\\ exists fs_q_pfc_left_constant_product_coefficientsum_body_terminal. fs_u_pfc_left_constant_product_coefficientsum = fs_q_pfc_left_constant_product_coefficientsum_body_terminal * S ((S (S (i))) * fs_v_pfc_left_constant_product_coefficientsum) + (pfc_natural_sum_left_constant_product_coefficient))) /\\ forall fs_i_pfc_left_constant_product_coefficientsum_body_steps. (exists fs_lt_pfc_left_constant_product_coefficientsum_body_steps_bound. fs_lt_pfc_left_constant_product_coefficientsum_body_steps_bound + S fs_i_pfc_left_constant_product_coefficientsum_body_steps = S (i)) -> exists fs_a_pfc_left_constant_product_coefficientsum_body_steps fs_r_pfc_left_constant_product_coefficientsum_body_steps fs_s_pfc_left_constant_product_coefficientsum_body_steps. ((((exists fs_h_pfc_left_constant_product_coefficientsum_body_steps_summand. fs_h_pfc_left_constant_product_coefficientsum_body_steps_summand + S (fs_a_pfc_left_constant_product_coefficientsum_body_steps) = S ((S (fs_i_pfc_left_constant_product_coefficientsum_body_steps)) * pfc_terms_scale_left_constant_product_coefficient)) /\\ exists fs_q_pfc_left_constant_product_coefficientsum_body_steps_summand. pfc_terms_code_left_constant_product_coefficient = fs_q_pfc_left_constant_product_coefficientsum_body_steps_summand * S ((S (fs_i_pfc_left_constant_product_coefficientsum_body_steps)) * pfc_terms_scale_left_constant_product_coefficient) + (fs_a_pfc_left_constant_product_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_product_coefficientsum_body_steps_partial. fs_h_pfc_left_constant_product_coefficientsum_body_steps_partial + S (fs_r_pfc_left_constant_product_coefficientsum_body_steps) = S ((S (fs_i_pfc_left_constant_product_coefficientsum_body_steps)) * fs_v_pfc_left_constant_product_coefficientsum)) /\\ exists fs_q_pfc_left_constant_product_coefficientsum_body_steps_partial. fs_u_pfc_left_constant_product_coefficientsum = fs_q_pfc_left_constant_product_coefficientsum_body_steps_partial * S ((S (fs_i_pfc_left_constant_product_coefficientsum_body_steps)) * fs_v_pfc_left_constant_product_coefficientsum) + (fs_r_pfc_left_constant_product_coefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_product_coefficientsum_body_steps_successor. fs_h_pfc_left_constant_product_coefficientsum_body_steps_successor + S (fs_s_pfc_left_constant_product_coefficientsum_body_steps) = S ((S (S fs_i_pfc_left_constant_product_coefficientsum_body_steps)) * fs_v_pfc_left_constant_product_coefficientsum)) /\\ exists fs_q_pfc_left_constant_product_coefficientsum_body_steps_successor. fs_u_pfc_left_constant_product_coefficientsum = fs_q_pfc_left_constant_product_coefficientsum_body_steps_successor * S ((S (S fs_i_pfc_left_constant_product_coefficientsum_body_steps)) * fs_v_pfc_left_constant_product_coefficientsum) + (fs_s_pfc_left_constant_product_coefficientsum_body_steps))) /\\ fs_s_pfc_left_constant_product_coefficientsum_body_steps = fs_r_pfc_left_constant_product_coefficientsum_body_steps + fs_a_pfc_left_constant_product_coefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_left_constant_product_coefficientresiduebound. pfa_gap_left_constant_product_coefficientresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_left_constant_product_coefficientresiduecongruence pfa_offset_right_left_constant_product_coefficientresiduecongruence. (pfc_natural_sum_left_constant_product_coefficient) + (p) * pfa_offset_left_left_constant_product_coefficientresiduecongruence = (r) + (p) * pfa_offset_right_left_constant_product_coefficientresiduecongruence)))))))))))", "specialize hproduct_right_right_right (i)", "apply hproduct_right_right_right", "exact hi", "cases hr", "cases hr_witness", "exists x", "exists x1", "split", "exact ha_witness", "split", "exact hr_witness_left", "specialize prime_field_convolution_coefficient_left_constant (p)", "specialize prime_field_convolution_coefficient_left_constant (k)", "specialize prime_field_convolution_coefficient_left_constant (kb)", "specialize prime_field_convolution_coefficient_left_constant (kc)", "specialize prime_field_convolution_coefficient_left_constant (ab)", "specialize prime_field_convolution_coefficient_left_constant (ac)", "specialize prime_field_convolution_coefficient_left_constant (L)", "specialize prime_field_convolution_coefficient_left_constant (i)", "specialize prime_field_convolution_coefficient_left_constant (x)", "specialize prime_field_convolution_coefficient_left_constant (x1)", "apply prime_field_convolution_coefficient_left_constant", "exact hk", "exact hi", "exact ha_witness", "exact hkb", "specialize matrix_rank_bounded_prefix_value (ab)", "specialize matrix_rank_bounded_prefix_value (ac)", "specialize matrix_rank_bounded_prefix_value (L)", "specialize matrix_rank_bounded_prefix_value (p)", "specialize matrix_rank_bounded_prefix_value (i)", "specialize matrix_rank_bounded_prefix_value (x)", "apply matrix_rank_bounded_prefix_value", "exact hproduct_right_left", "exact hi", "exact ha_witness", "exact hr_witness_right" ], "script_sha256": "e1fa5c49bd49f879b11b43b899dac572f18ed2134d48ca3a388d0e3aed98d135", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_left_constant_candidate.py", "sha256": "9a7a4de30f5f389bcabc2e6267a0d2cc5dc5f061059dcea303a0a03dab58509a" }, "statement": "forall p k kb kc ab ac hb hc L. (((exists ff_h_pfp_left_constant_product_K. ff_h_pfp_left_constant_product_K + S (k) = S ((S (0)) * kc)) /\\ exists ff_q_pfp_left_constant_product_K. kb = ff_q_pfp_left_constant_product_K * S ((S (0)) * kc) + (k))) -> (((forall fom_index_pfp_left_constant_product_sourceleft. (exists fom_gap_pfp_left_constant_product_sourceleft_index_bound. fom_gap_pfp_left_constant_product_sourceleft_index_bound + S (fom_index_pfp_left_constant_product_sourceleft) = 1) -> exists fom_value_pfp_left_constant_product_sourceleft. ((((exists fom_beta_height_pfp_left_constant_product_sourceleft_entry. fom_beta_height_pfp_left_constant_product_sourceleft_entry + S (fom_value_pfp_left_constant_product_sourceleft) = S ((S (fom_index_pfp_left_constant_product_sourceleft)) * kc)) /\\ exists fom_beta_quotient_pfp_left_constant_product_sourceleft_entry. kb = fom_beta_quotient_pfp_left_constant_product_sourceleft_entry * S ((S (fom_index_pfp_left_constant_product_sourceleft)) * kc) + (fom_value_pfp_left_constant_product_sourceleft))) /\\ (exists fom_gap_pfp_left_constant_product_sourceleft_value_bound. fom_gap_pfp_left_constant_product_sourceleft_value_bound + S (fom_value_pfp_left_constant_product_sourceleft) = p))) /\\ (((forall fom_index_pfp_left_constant_product_sourceright. (exists fom_gap_pfp_left_constant_product_sourceright_index_bound. fom_gap_pfp_left_constant_product_sourceright_index_bound + S (fom_index_pfp_left_constant_product_sourceright) = L) -> exists fom_value_pfp_left_constant_product_sourceright. ((((exists fom_beta_height_pfp_left_constant_product_sourceright_entry. fom_beta_height_pfp_left_constant_product_sourceright_entry + S (fom_value_pfp_left_constant_product_sourceright) = S ((S (fom_index_pfp_left_constant_product_sourceright)) * ac)) /\\ exists fom_beta_quotient_pfp_left_constant_product_sourceright_entry. ab = fom_beta_quotient_pfp_left_constant_product_sourceright_entry * S ((S (fom_index_pfp_left_constant_product_sourceright)) * ac) + (fom_value_pfp_left_constant_product_sourceright))) /\\ (exists fom_gap_pfp_left_constant_product_sourceright_value_bound. fom_gap_pfp_left_constant_product_sourceright_value_bound + S (fom_value_pfp_left_constant_product_sourceright) = p))) /\\ (((((((1)=0 \\/ (L)=0) /\\ (((L)=0)))) \\/ (((~((1)=0)) /\\ (((~((L)=0)) /\\ (((1)+(L)=S (L)))))))) /\\ ((forall pfc_index_left_constant_product_sourcecoefficients. (exists pfa_gap_left_constant_product_sourcecoefficientsbound. pfa_gap_left_constant_product_sourcecoefficientsbound + S (pfc_index_left_constant_product_sourcecoefficients) = (L)) -> exists pfc_value_left_constant_product_sourcecoefficients. ((((exists ff_h_pfp_left_constant_product_sourcecoefficientsentry. ff_h_pfp_left_constant_product_sourcecoefficientsentry + S (pfc_value_left_constant_product_sourcecoefficients) = S ((S (pfc_index_left_constant_product_sourcecoefficients)) * hc)) /\\ exists ff_q_pfp_left_constant_product_sourcecoefficientsentry. hb = ff_q_pfp_left_constant_product_sourcecoefficientsentry * S ((S (pfc_index_left_constant_product_sourcecoefficients)) * hc) + (pfc_value_left_constant_product_sourcecoefficients))) /\\ ((exists pfc_terms_code_left_constant_product_sourcecoefficientscoefficient pfc_terms_scale_left_constant_product_sourcecoefficientscoefficient pfc_natural_sum_left_constant_product_sourcecoefficientscoefficient. ((forall pfc_index_left_constant_product_sourcecoefficientscoefficientdiagonal. (exists pfa_gap_left_constant_product_sourcecoefficientscoefficientdiagonalbound. pfa_gap_left_constant_product_sourcecoefficientscoefficientdiagonalbound + S (pfc_index_left_constant_product_sourcecoefficientscoefficientdiagonal) = (S (pfc_index_left_constant_product_sourcecoefficients))) -> exists pfc_value_left_constant_product_sourcecoefficientscoefficientdiagonal. ((((exists ff_h_pfp_left_constant_product_sourcecoefficientscoefficientdiagonalentry. ff_h_pfp_left_constant_product_sourcecoefficientscoefficientdiagonalentry + S (pfc_value_left_constant_product_sourcecoefficientscoefficientdiagonal) = S ((S (pfc_index_left_constant_product_sourcecoefficientscoefficientdiagonal)) * pfc_terms_scale_left_constant_product_sourcecoefficientscoefficient)) /\\ exists ff_q_pfp_left_constant_product_sourcecoefficientscoefficientdiagonalentry. pfc_terms_code_left_constant_product_sourcecoefficientscoefficient = ff_q_pfp_left_constant_product_sourcecoefficientscoefficientdiagonalentry * S ((S (pfc_index_left_constant_product_sourcecoefficientscoefficientdiagonal)) * pfc_terms_scale_left_constant_product_sourcecoefficientscoefficient) + (pfc_value_left_constant_product_sourcecoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_left_constant_product_sourcecoefficientscoefficientdiagonalterm pfc_left_left_constant_product_sourcecoefficientscoefficientdiagonalterm pfc_right_left_constant_product_sourcecoefficientscoefficientdiagonalterm. (((pfc_index_left_constant_product_sourcecoefficientscoefficientdiagonal)+pfc_complement_left_constant_product_sourcecoefficientscoefficientdiagonalterm=(pfc_index_left_constant_product_sourcecoefficients)) /\\ ((((((exists pfa_gap_left_constant_product_sourcecoefficientscoefficientdiagonaltermleftinside. pfa_gap_left_constant_product_sourcecoefficientscoefficientdiagonaltermleftinside + S (pfc_index_left_constant_product_sourcecoefficientscoefficientdiagonal) = (1)) /\\ ((((exists ff_h_pfp_left_constant_product_sourcecoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_left_constant_product_sourcecoefficientscoefficientdiagonaltermleftentry + S (pfc_left_left_constant_product_sourcecoefficientscoefficientdiagonalterm) = S ((S (pfc_index_left_constant_product_sourcecoefficientscoefficientdiagonal)) * kc)) /\\ exists ff_q_pfp_left_constant_product_sourcecoefficientscoefficientdiagonaltermleftentry. kb = ff_q_pfp_left_constant_product_sourcecoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_left_constant_product_sourcecoefficientscoefficientdiagonal)) * kc) + (pfc_left_left_constant_product_sourcecoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_left_constant_product_sourcecoefficientscoefficientdiagonaltermleftoutside. pfc_gap_left_constant_product_sourcecoefficientscoefficientdiagonaltermleftoutside+(1)=(pfc_index_left_constant_product_sourcecoefficientscoefficientdiagonal)) /\\ (((pfc_left_left_constant_product_sourcecoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_left_constant_product_sourcecoefficientscoefficientdiagonaltermrightinside. pfa_gap_left_constant_product_sourcecoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_left_constant_product_sourcecoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_left_constant_product_sourcecoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_left_constant_product_sourcecoefficientscoefficientdiagonaltermrightentry + S (pfc_right_left_constant_product_sourcecoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_left_constant_product_sourcecoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_left_constant_product_sourcecoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_left_constant_product_sourcecoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_left_constant_product_sourcecoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_left_constant_product_sourcecoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_left_constant_product_sourcecoefficientscoefficientdiagonaltermrightoutside. pfc_gap_left_constant_product_sourcecoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_left_constant_product_sourcecoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_left_constant_product_sourcecoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_left_constant_product_sourcecoefficientscoefficientdiagonal)=pfc_left_left_constant_product_sourcecoefficientscoefficientdiagonalterm*pfc_right_left_constant_product_sourcecoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_left_constant_product_sourcecoefficientscoefficientsum fs_v_pfc_left_constant_product_sourcecoefficientscoefficientsum. ((((exists fs_h_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_start. fs_h_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_left_constant_product_sourcecoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_start. fs_u_pfc_left_constant_product_sourcecoefficientscoefficientsum = fs_q_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_left_constant_product_sourcecoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_terminal. fs_h_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_left_constant_product_sourcecoefficientscoefficient) = S ((S (S (pfc_index_left_constant_product_sourcecoefficients))) * fs_v_pfc_left_constant_product_sourcecoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_terminal. fs_u_pfc_left_constant_product_sourcecoefficientscoefficientsum = fs_q_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_left_constant_product_sourcecoefficients))) * fs_v_pfc_left_constant_product_sourcecoefficientscoefficientsum) + (pfc_natural_sum_left_constant_product_sourcecoefficientscoefficient))) /\\ forall fs_i_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps = S (pfc_index_left_constant_product_sourcecoefficients)) -> exists fs_a_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps fs_r_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps fs_s_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps_summand. fs_h_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps)) * pfc_terms_scale_left_constant_product_sourcecoefficientscoefficient)) /\\ exists fs_q_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps_summand. pfc_terms_code_left_constant_product_sourcecoefficientscoefficient = fs_q_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps)) * pfc_terms_scale_left_constant_product_sourcecoefficientscoefficient) + (fs_a_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps_partial. fs_h_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_constant_product_sourcecoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps_partial. fs_u_pfc_left_constant_product_sourcecoefficientscoefficientsum = fs_q_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_constant_product_sourcecoefficientscoefficientsum) + (fs_r_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps_successor. fs_h_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_constant_product_sourcecoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps_successor. fs_u_pfc_left_constant_product_sourcecoefficientscoefficientsum = fs_q_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_constant_product_sourcecoefficientscoefficientsum) + (fs_s_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps = fs_r_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps + fs_a_pfc_left_constant_product_sourcecoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_left_constant_product_sourcecoefficientscoefficientresiduebound. pfa_gap_left_constant_product_sourcecoefficientscoefficientresiduebound + S (pfc_value_left_constant_product_sourcecoefficients) = (p)) /\\ ((exists pfa_offset_left_left_constant_product_sourcecoefficientscoefficientresiduecongruence pfa_offset_right_left_constant_product_sourcecoefficientscoefficientresiduecongruence. (pfc_natural_sum_left_constant_product_sourcecoefficientscoefficient) + (p) * pfa_offset_left_left_constant_product_sourcecoefficientscoefficientresiduecongruence = (pfc_value_left_constant_product_sourcecoefficients) + (p) * pfa_offset_right_left_constant_product_sourcecoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((exists pfa_gap_left_constant_product_scalescalar. pfa_gap_left_constant_product_scalescalar + S (k) = (p)) /\\ ((forall pfp_index_left_constant_product_scale. (exists pfa_gap_left_constant_product_scaleindex. pfa_gap_left_constant_product_scaleindex + S (pfp_index_left_constant_product_scale) = (L)) -> exists pfp_source_left_constant_product_scale pfp_value_left_constant_product_scale. ((((exists ff_h_pfp_left_constant_product_scalesource. ff_h_pfp_left_constant_product_scalesource + S (pfp_source_left_constant_product_scale) = S ((S (pfp_index_left_constant_product_scale)) * ac)) /\\ exists ff_q_pfp_left_constant_product_scalesource. ab = ff_q_pfp_left_constant_product_scalesource * S ((S (pfp_index_left_constant_product_scale)) * ac) + (pfp_source_left_constant_product_scale))) /\\ (((((exists ff_h_pfp_left_constant_product_scaletarget. ff_h_pfp_left_constant_product_scaletarget + S (pfp_value_left_constant_product_scale) = S ((S (pfp_index_left_constant_product_scale)) * hc)) /\\ exists ff_q_pfp_left_constant_product_scaletarget. hb = ff_q_pfp_left_constant_product_scaletarget * S ((S (pfp_index_left_constant_product_scale)) * hc) + (pfp_value_left_constant_product_scale))) /\\ ((((exists pfa_gap_left_constant_product_scaleoperationleft. pfa_gap_left_constant_product_scaleoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_left_constant_product_scaleoperationright. pfa_gap_left_constant_product_scaleoperationright + S (pfp_source_left_constant_product_scale) = (p)) /\\ ((((exists pfa_gap_left_constant_product_scaleoperationresultbound. pfa_gap_left_constant_product_scaleoperationresultbound + S (pfp_value_left_constant_product_scale) = (p)) /\\ ((exists pfa_offset_left_left_constant_product_scaleoperationresultcongruence pfa_offset_right_left_constant_product_scaleoperationresultcongruence. ((k) * (pfp_source_left_constant_product_scale)) + (p) * pfa_offset_left_left_constant_product_scaleoperationresultcongruence = (pfp_value_left_constant_product_scale) + (p) * pfa_offset_right_left_constant_product_scaleoperationresultcongruence)))))))))))))))))", "statement_sha256": "c93e29c84d993f933394eb2fc82600d8f3d88f50a06a25ee9d6dc69e6b2141fe", "summary": "An actual length-L product of a canonical left singleton and a length-L prefix yields the existing scalar graph, even when L=0. 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(exists fom_gap_pfp_left_constant_recover_A_index_bound. fom_gap_pfp_left_constant_recover_A_index_bound + S (fom_index_pfp_left_constant_recover_A) = L) -> exists fom_value_pfp_left_constant_recover_A. ((((exists fom_beta_height_pfp_left_constant_recover_A_entry. fom_beta_height_pfp_left_constant_recover_A_entry + S (fom_value_pfp_left_constant_recover_A) = S ((S (fom_index_pfp_left_constant_recover_A)) * ac)) /\\ exists fom_beta_quotient_pfp_left_constant_recover_A_entry. ab = fom_beta_quotient_pfp_left_constant_recover_A_entry * S ((S (fom_index_pfp_left_constant_recover_A)) * ac) + (fom_value_pfp_left_constant_recover_A))) /\\ (exists fom_gap_pfp_left_constant_recover_A_value_bound. fom_gap_pfp_left_constant_recover_A_value_bound + S (fom_value_pfp_left_constant_recover_A) = p))) /\\ ((forall fom_index_pfp_left_constant_recover_H. (exists fom_gap_pfp_left_constant_recover_H_index_bound. fom_gap_pfp_left_constant_recover_H_index_bound + S (fom_index_pfp_left_constant_recover_H) = L) -> exists fom_value_pfp_left_constant_recover_H. ((((exists fom_beta_height_pfp_left_constant_recover_H_entry. fom_beta_height_pfp_left_constant_recover_H_entry + S (fom_value_pfp_left_constant_recover_H) = S ((S (fom_index_pfp_left_constant_recover_H)) * hc)) /\\ exists fom_beta_quotient_pfp_left_constant_recover_H_entry. hb = fom_beta_quotient_pfp_left_constant_recover_H_entry * S ((S (fom_index_pfp_left_constant_recover_H)) * hc) + (fom_value_pfp_left_constant_recover_H))) /\\ (exists fom_gap_pfp_left_constant_recover_H_value_bound. fom_gap_pfp_left_constant_recover_H_value_bound + S (fom_value_pfp_left_constant_recover_H) = p)))))", "specialize prime_field_polynomial_scale_bounded (p)", "specialize prime_field_polynomial_scale_bounded (k)", "specialize prime_field_polynomial_scale_bounded (ab)", "specialize prime_field_polynomial_scale_bounded (ac)", "specialize prime_field_polynomial_scale_bounded (hb)", "specialize prime_field_polynomial_scale_bounded (hc)", "specialize prime_field_polynomial_scale_bounded (L)", "apply prime_field_polynomial_scale_bounded", "exact hs", "cases hbounds", "split", "exact hK", "split", "exact hbounds_left", "split", "have hz : L=0 \\/ ~(L=0)", "specialize eq_decidable (L)", "specialize eq_decidable (0)", "apply eq_decidable", "cases hz", "left", "split", "right", "exact hz_left", "exact hz_left", "right", "split", "intro hbad", "specialize succ_ne_zero (0)", "apply succ_ne_zero", "exact hbad", "split", "exact hz_right", "simp [add_succ_left,zero_add]", "intro i", "intro hi", "have hv : exists a r. ((((exists ff_h_pfp_left_constant_recover_source. ff_h_pfp_left_constant_recover_source + S (a) = S ((S (i)) * ac)) /\\ exists ff_q_pfp_left_constant_recover_source. ab = ff_q_pfp_left_constant_recover_source * S ((S (i)) * ac) + (a))) /\\ (((((exists ff_h_pfp_left_constant_recover_target. ff_h_pfp_left_constant_recover_target + S (r) = S ((S (i)) * hc)) /\\ exists ff_q_pfp_left_constant_recover_target. hb = ff_q_pfp_left_constant_recover_target * S ((S (i)) * hc) + (r))) /\\ ((((exists pfa_gap_left_constant_recover_multiplyleft. pfa_gap_left_constant_recover_multiplyleft + S (k) = (p)) /\\ (((exists pfa_gap_left_constant_recover_multiplyright. pfa_gap_left_constant_recover_multiplyright + S (a) = (p)) /\\ ((((exists pfa_gap_left_constant_recover_multiplyresultbound. pfa_gap_left_constant_recover_multiplyresultbound + S (r) = (p)) /\\ ((exists pfa_offset_left_left_constant_recover_multiplyresultcongruence pfa_offset_right_left_constant_recover_multiplyresultcongruence. ((k) * (a)) + (p) * pfa_offset_left_left_constant_recover_multiplyresultcongruence = (r) + (p) * pfa_offset_right_left_constant_recover_multiplyresultcongruence)))))))))))))", "cases hs", "specialize hs_right (i)", "apply hs_right", "exact hi", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_right", "have hm : ((exists pfa_gap_left_constant_recover_boundsleft. pfa_gap_left_constant_recover_boundsleft + S (k) = (p)) /\\ (((exists pfa_gap_left_constant_recover_boundsright. pfa_gap_left_constant_recover_boundsright + S (x) = (p)) /\\ ((((exists pfa_gap_left_constant_recover_boundsresultbound. pfa_gap_left_constant_recover_boundsresultbound + S (x1) = (p)) /\\ ((exists pfa_offset_left_left_constant_recover_boundsresultcongruence pfa_offset_right_left_constant_recover_boundsresultcongruence. ((k) * (x)) + (p) * pfa_offset_left_left_constant_recover_boundsresultcongruence = (x1) + (p) * pfa_offset_right_left_constant_recover_boundsresultcongruence))))))))", "exact hv_witness_witness_right_right", "cases hm", "cases hm_right", "have hcoefficient : exists r. exists pfc_terms_code_left_constant_recover_actual pfc_terms_scale_left_constant_recover_actual pfc_natural_sum_left_constant_recover_actual. ((forall pfc_index_left_constant_recover_actualdiagonal. (exists pfa_gap_left_constant_recover_actualdiagonalbound. pfa_gap_left_constant_recover_actualdiagonalbound + S (pfc_index_left_constant_recover_actualdiagonal) = (S (i))) -> exists pfc_value_left_constant_recover_actualdiagonal. ((((exists ff_h_pfp_left_constant_recover_actualdiagonalentry. ff_h_pfp_left_constant_recover_actualdiagonalentry + S (pfc_value_left_constant_recover_actualdiagonal) = S ((S (pfc_index_left_constant_recover_actualdiagonal)) * pfc_terms_scale_left_constant_recover_actual)) /\\ exists ff_q_pfp_left_constant_recover_actualdiagonalentry. pfc_terms_code_left_constant_recover_actual = ff_q_pfp_left_constant_recover_actualdiagonalentry * S ((S (pfc_index_left_constant_recover_actualdiagonal)) * pfc_terms_scale_left_constant_recover_actual) + (pfc_value_left_constant_recover_actualdiagonal))) /\\ ((exists pfc_complement_left_constant_recover_actualdiagonalterm pfc_left_left_constant_recover_actualdiagonalterm pfc_right_left_constant_recover_actualdiagonalterm. (((pfc_index_left_constant_recover_actualdiagonal)+pfc_complement_left_constant_recover_actualdiagonalterm=(i)) /\\ ((((((exists pfa_gap_left_constant_recover_actualdiagonaltermleftinside. pfa_gap_left_constant_recover_actualdiagonaltermleftinside + S (pfc_index_left_constant_recover_actualdiagonal) = (1)) /\\ ((((exists ff_h_pfp_left_constant_recover_actualdiagonaltermleftentry. ff_h_pfp_left_constant_recover_actualdiagonaltermleftentry + S (pfc_left_left_constant_recover_actualdiagonalterm) = S ((S (pfc_index_left_constant_recover_actualdiagonal)) * kc)) /\\ exists ff_q_pfp_left_constant_recover_actualdiagonaltermleftentry. kb = ff_q_pfp_left_constant_recover_actualdiagonaltermleftentry * S ((S (pfc_index_left_constant_recover_actualdiagonal)) * kc) + (pfc_left_left_constant_recover_actualdiagonalterm)))))) \\/ (((exists pfc_gap_left_constant_recover_actualdiagonaltermleftoutside. pfc_gap_left_constant_recover_actualdiagonaltermleftoutside+(1)=(pfc_index_left_constant_recover_actualdiagonal)) /\\ (((pfc_left_left_constant_recover_actualdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_left_constant_recover_actualdiagonaltermrightinside. pfa_gap_left_constant_recover_actualdiagonaltermrightinside + S (pfc_complement_left_constant_recover_actualdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_left_constant_recover_actualdiagonaltermrightentry. ff_h_pfp_left_constant_recover_actualdiagonaltermrightentry + S (pfc_right_left_constant_recover_actualdiagonalterm) = S ((S (pfc_complement_left_constant_recover_actualdiagonalterm)) * ac)) /\\ exists ff_q_pfp_left_constant_recover_actualdiagonaltermrightentry. ab = ff_q_pfp_left_constant_recover_actualdiagonaltermrightentry * S ((S (pfc_complement_left_constant_recover_actualdiagonalterm)) * ac) + (pfc_right_left_constant_recover_actualdiagonalterm)))))) \\/ (((exists pfc_gap_left_constant_recover_actualdiagonaltermrightoutside. pfc_gap_left_constant_recover_actualdiagonaltermrightoutside+(L)=(pfc_complement_left_constant_recover_actualdiagonalterm)) /\\ (((pfc_right_left_constant_recover_actualdiagonalterm)=0))))) /\\ (((pfc_value_left_constant_recover_actualdiagonal)=pfc_left_left_constant_recover_actualdiagonalterm*pfc_right_left_constant_recover_actualdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_left_constant_recover_actualsum fs_v_pfc_left_constant_recover_actualsum. ((((exists fs_h_pfc_left_constant_recover_actualsum_body_start. fs_h_pfc_left_constant_recover_actualsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_left_constant_recover_actualsum)) /\\ exists fs_q_pfc_left_constant_recover_actualsum_body_start. fs_u_pfc_left_constant_recover_actualsum = fs_q_pfc_left_constant_recover_actualsum_body_start * S ((S (0)) * fs_v_pfc_left_constant_recover_actualsum) + (0))) /\\ ((((exists fs_h_pfc_left_constant_recover_actualsum_body_terminal. fs_h_pfc_left_constant_recover_actualsum_body_terminal + S (pfc_natural_sum_left_constant_recover_actual) = S ((S (S (i))) * fs_v_pfc_left_constant_recover_actualsum)) /\\ exists fs_q_pfc_left_constant_recover_actualsum_body_terminal. fs_u_pfc_left_constant_recover_actualsum = fs_q_pfc_left_constant_recover_actualsum_body_terminal * S ((S (S (i))) * fs_v_pfc_left_constant_recover_actualsum) + (pfc_natural_sum_left_constant_recover_actual))) /\\ forall fs_i_pfc_left_constant_recover_actualsum_body_steps. (exists fs_lt_pfc_left_constant_recover_actualsum_body_steps_bound. fs_lt_pfc_left_constant_recover_actualsum_body_steps_bound + S fs_i_pfc_left_constant_recover_actualsum_body_steps = S (i)) -> exists fs_a_pfc_left_constant_recover_actualsum_body_steps fs_r_pfc_left_constant_recover_actualsum_body_steps fs_s_pfc_left_constant_recover_actualsum_body_steps. ((((exists fs_h_pfc_left_constant_recover_actualsum_body_steps_summand. fs_h_pfc_left_constant_recover_actualsum_body_steps_summand + S (fs_a_pfc_left_constant_recover_actualsum_body_steps) = S ((S (fs_i_pfc_left_constant_recover_actualsum_body_steps)) * pfc_terms_scale_left_constant_recover_actual)) /\\ exists fs_q_pfc_left_constant_recover_actualsum_body_steps_summand. pfc_terms_code_left_constant_recover_actual = fs_q_pfc_left_constant_recover_actualsum_body_steps_summand * S ((S (fs_i_pfc_left_constant_recover_actualsum_body_steps)) * pfc_terms_scale_left_constant_recover_actual) + (fs_a_pfc_left_constant_recover_actualsum_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_recover_actualsum_body_steps_partial. fs_h_pfc_left_constant_recover_actualsum_body_steps_partial + S (fs_r_pfc_left_constant_recover_actualsum_body_steps) = S ((S (fs_i_pfc_left_constant_recover_actualsum_body_steps)) * fs_v_pfc_left_constant_recover_actualsum)) /\\ exists fs_q_pfc_left_constant_recover_actualsum_body_steps_partial. fs_u_pfc_left_constant_recover_actualsum = fs_q_pfc_left_constant_recover_actualsum_body_steps_partial * S ((S (fs_i_pfc_left_constant_recover_actualsum_body_steps)) * fs_v_pfc_left_constant_recover_actualsum) + (fs_r_pfc_left_constant_recover_actualsum_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_recover_actualsum_body_steps_successor. fs_h_pfc_left_constant_recover_actualsum_body_steps_successor + S (fs_s_pfc_left_constant_recover_actualsum_body_steps) = S ((S (S fs_i_pfc_left_constant_recover_actualsum_body_steps)) * fs_v_pfc_left_constant_recover_actualsum)) /\\ exists fs_q_pfc_left_constant_recover_actualsum_body_steps_successor. fs_u_pfc_left_constant_recover_actualsum = fs_q_pfc_left_constant_recover_actualsum_body_steps_successor * S ((S (S fs_i_pfc_left_constant_recover_actualsum_body_steps)) * fs_v_pfc_left_constant_recover_actualsum) + (fs_s_pfc_left_constant_recover_actualsum_body_steps))) /\\ fs_s_pfc_left_constant_recover_actualsum_body_steps = fs_r_pfc_left_constant_recover_actualsum_body_steps + fs_a_pfc_left_constant_recover_actualsum_body_steps)))))) /\\ ((((exists pfa_gap_left_constant_recover_actualresiduebound. pfa_gap_left_constant_recover_actualresiduebound + S (r) = (p)) /\\ ((exists pfa_offset_left_left_constant_recover_actualresiduecongruence pfa_offset_right_left_constant_recover_actualresiduecongruence. (pfc_natural_sum_left_constant_recover_actual) + (p) * pfa_offset_left_left_constant_recover_actualresiduecongruence = (r) + (p) * pfa_offset_right_left_constant_recover_actualresiduecongruence))))))))", "specialize prime_field_convolution_coefficient_exists (p)", "specialize prime_field_convolution_coefficient_exists (kb)", "specialize prime_field_convolution_coefficient_exists (kc)", "specialize prime_field_convolution_coefficient_exists (1)", "specialize prime_field_convolution_coefficient_exists (ab)", "specialize prime_field_convolution_coefficient_exists (ac)", "specialize prime_field_convolution_coefficient_exists (L)", "specialize prime_field_convolution_coefficient_exists (i)", "apply prime_field_convolution_coefficient_exists", "intro hpzero", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hpzero", "cases hcoefficient", "have heq : x2=x1", "specialize prime_field_multiply_functional (p)", "specialize prime_field_multiply_functional (k)", "specialize prime_field_multiply_functional (x)", "specialize prime_field_multiply_functional (x2)", "specialize prime_field_multiply_functional (x1)", "apply prime_field_multiply_functional", "specialize prime_field_convolution_coefficient_left_constant (p)", "specialize prime_field_convolution_coefficient_left_constant (k)", "specialize prime_field_convolution_coefficient_left_constant (kb)", "specialize prime_field_convolution_coefficient_left_constant (kc)", "specialize prime_field_convolution_coefficient_left_constant (ab)", "specialize prime_field_convolution_coefficient_left_constant (ac)", "specialize prime_field_convolution_coefficient_left_constant (L)", "specialize prime_field_convolution_coefficient_left_constant (i)", "specialize prime_field_convolution_coefficient_left_constant (x)", "specialize prime_field_convolution_coefficient_left_constant (x2)", "apply prime_field_convolution_coefficient_left_constant", "exact hk", "exact hi", "exact hv_witness_witness_left", "exact hm_left", "exact hm_right_left", "exact hcoefficient_witness", "exact hv_witness_witness_right_right", "exists x1", "split", "exact hv_witness_witness_right_left", "rewrite heq at hcoefficient_witness", "rewrite heq at hcoefficient_witness", "exact hcoefficient_witness" ], "script_sha256": "cc2a0f1f3544add41b1a7d1fa6050971e1fff51cec63c8cb709daaaf3873f98e", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_left_constant_candidate.py", "sha256": "9a7a4de30f5f389bcabc2e6267a0d2cc5dc5f061059dcea303a0a03dab58509a" }, "statement": "forall p k kb kc ab ac hb hc L. (~((p) = 1) /\\ forall pfa_factor_left_left_constant_recover_prime pfa_factor_right_left_constant_recover_prime. (p) = pfa_factor_left_left_constant_recover_prime * pfa_factor_right_left_constant_recover_prime -> pfa_factor_left_left_constant_recover_prime = 1 \\/ pfa_factor_right_left_constant_recover_prime = 1) -> (forall fom_index_pfp_left_constant_recover_K. (exists fom_gap_pfp_left_constant_recover_K_index_bound. fom_gap_pfp_left_constant_recover_K_index_bound + S (fom_index_pfp_left_constant_recover_K) = 1) -> exists fom_value_pfp_left_constant_recover_K. ((((exists fom_beta_height_pfp_left_constant_recover_K_entry. fom_beta_height_pfp_left_constant_recover_K_entry + S (fom_value_pfp_left_constant_recover_K) = S ((S (fom_index_pfp_left_constant_recover_K)) * kc)) /\\ exists fom_beta_quotient_pfp_left_constant_recover_K_entry. kb = fom_beta_quotient_pfp_left_constant_recover_K_entry * S ((S (fom_index_pfp_left_constant_recover_K)) * kc) + (fom_value_pfp_left_constant_recover_K))) /\\ (exists fom_gap_pfp_left_constant_recover_K_value_bound. fom_gap_pfp_left_constant_recover_K_value_bound + S (fom_value_pfp_left_constant_recover_K) = p))) -> (((exists ff_h_pfp_left_constant_recover_value. ff_h_pfp_left_constant_recover_value + S (k) = S ((S (0)) * kc)) /\\ exists ff_q_pfp_left_constant_recover_value. kb = ff_q_pfp_left_constant_recover_value * S ((S (0)) * kc) + (k))) -> (((exists pfa_gap_left_constant_recover_scalescalar. pfa_gap_left_constant_recover_scalescalar + S (k) = (p)) /\\ ((forall pfp_index_left_constant_recover_scale. (exists pfa_gap_left_constant_recover_scaleindex. pfa_gap_left_constant_recover_scaleindex + S (pfp_index_left_constant_recover_scale) = (L)) -> exists pfp_source_left_constant_recover_scale pfp_value_left_constant_recover_scale. ((((exists ff_h_pfp_left_constant_recover_scalesource. ff_h_pfp_left_constant_recover_scalesource + S (pfp_source_left_constant_recover_scale) = S ((S (pfp_index_left_constant_recover_scale)) * ac)) /\\ exists ff_q_pfp_left_constant_recover_scalesource. ab = ff_q_pfp_left_constant_recover_scalesource * S ((S (pfp_index_left_constant_recover_scale)) * ac) + (pfp_source_left_constant_recover_scale))) /\\ (((((exists ff_h_pfp_left_constant_recover_scaletarget. ff_h_pfp_left_constant_recover_scaletarget + S (pfp_value_left_constant_recover_scale) = S ((S (pfp_index_left_constant_recover_scale)) * hc)) /\\ exists ff_q_pfp_left_constant_recover_scaletarget. hb = ff_q_pfp_left_constant_recover_scaletarget * S ((S (pfp_index_left_constant_recover_scale)) * hc) + (pfp_value_left_constant_recover_scale))) /\\ ((((exists pfa_gap_left_constant_recover_scaleoperationleft. pfa_gap_left_constant_recover_scaleoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_left_constant_recover_scaleoperationright. pfa_gap_left_constant_recover_scaleoperationright + S (pfp_source_left_constant_recover_scale) = (p)) /\\ ((((exists pfa_gap_left_constant_recover_scaleoperationresultbound. pfa_gap_left_constant_recover_scaleoperationresultbound + S (pfp_value_left_constant_recover_scale) = (p)) /\\ ((exists pfa_offset_left_left_constant_recover_scaleoperationresultcongruence pfa_offset_right_left_constant_recover_scaleoperationresultcongruence. ((k) * (pfp_source_left_constant_recover_scale)) + (p) * pfa_offset_left_left_constant_recover_scaleoperationresultcongruence = (pfp_value_left_constant_recover_scale) + (p) * pfa_offset_right_left_constant_recover_scaleoperationresultcongruence))))))))))))))))) -> (((forall fom_index_pfp_left_constant_recover_productleft. (exists fom_gap_pfp_left_constant_recover_productleft_index_bound. fom_gap_pfp_left_constant_recover_productleft_index_bound + S (fom_index_pfp_left_constant_recover_productleft) = 1) -> exists fom_value_pfp_left_constant_recover_productleft. ((((exists fom_beta_height_pfp_left_constant_recover_productleft_entry. fom_beta_height_pfp_left_constant_recover_productleft_entry + S (fom_value_pfp_left_constant_recover_productleft) = S ((S (fom_index_pfp_left_constant_recover_productleft)) * kc)) /\\ exists fom_beta_quotient_pfp_left_constant_recover_productleft_entry. kb = fom_beta_quotient_pfp_left_constant_recover_productleft_entry * S ((S (fom_index_pfp_left_constant_recover_productleft)) * kc) + (fom_value_pfp_left_constant_recover_productleft))) /\\ (exists fom_gap_pfp_left_constant_recover_productleft_value_bound. fom_gap_pfp_left_constant_recover_productleft_value_bound + S (fom_value_pfp_left_constant_recover_productleft) = p))) /\\ (((forall fom_index_pfp_left_constant_recover_productright. (exists fom_gap_pfp_left_constant_recover_productright_index_bound. fom_gap_pfp_left_constant_recover_productright_index_bound + S (fom_index_pfp_left_constant_recover_productright) = L) -> exists fom_value_pfp_left_constant_recover_productright. ((((exists fom_beta_height_pfp_left_constant_recover_productright_entry. fom_beta_height_pfp_left_constant_recover_productright_entry + S (fom_value_pfp_left_constant_recover_productright) = S ((S (fom_index_pfp_left_constant_recover_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_left_constant_recover_productright_entry. ab = fom_beta_quotient_pfp_left_constant_recover_productright_entry * S ((S (fom_index_pfp_left_constant_recover_productright)) * ac) + (fom_value_pfp_left_constant_recover_productright))) /\\ (exists fom_gap_pfp_left_constant_recover_productright_value_bound. fom_gap_pfp_left_constant_recover_productright_value_bound + S (fom_value_pfp_left_constant_recover_productright) = p))) /\\ (((((((1)=0 \\/ (L)=0) /\\ (((L)=0)))) \\/ (((~((1)=0)) /\\ (((~((L)=0)) /\\ (((1)+(L)=S (L)))))))) /\\ ((forall pfc_index_left_constant_recover_productcoefficients. (exists pfa_gap_left_constant_recover_productcoefficientsbound. pfa_gap_left_constant_recover_productcoefficientsbound + S (pfc_index_left_constant_recover_productcoefficients) = (L)) -> exists pfc_value_left_constant_recover_productcoefficients. ((((exists ff_h_pfp_left_constant_recover_productcoefficientsentry. ff_h_pfp_left_constant_recover_productcoefficientsentry + S (pfc_value_left_constant_recover_productcoefficients) = S ((S (pfc_index_left_constant_recover_productcoefficients)) * hc)) /\\ exists ff_q_pfp_left_constant_recover_productcoefficientsentry. hb = ff_q_pfp_left_constant_recover_productcoefficientsentry * S ((S (pfc_index_left_constant_recover_productcoefficients)) * hc) + (pfc_value_left_constant_recover_productcoefficients))) /\\ ((exists pfc_terms_code_left_constant_recover_productcoefficientscoefficient pfc_terms_scale_left_constant_recover_productcoefficientscoefficient pfc_natural_sum_left_constant_recover_productcoefficientscoefficient. ((forall pfc_index_left_constant_recover_productcoefficientscoefficientdiagonal. (exists pfa_gap_left_constant_recover_productcoefficientscoefficientdiagonalbound. pfa_gap_left_constant_recover_productcoefficientscoefficientdiagonalbound + S (pfc_index_left_constant_recover_productcoefficientscoefficientdiagonal) = (S (pfc_index_left_constant_recover_productcoefficients))) -> exists pfc_value_left_constant_recover_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_left_constant_recover_productcoefficientscoefficientdiagonalentry. ff_h_pfp_left_constant_recover_productcoefficientscoefficientdiagonalentry + S (pfc_value_left_constant_recover_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_left_constant_recover_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_left_constant_recover_productcoefficientscoefficient)) /\\ exists ff_q_pfp_left_constant_recover_productcoefficientscoefficientdiagonalentry. pfc_terms_code_left_constant_recover_productcoefficientscoefficient = ff_q_pfp_left_constant_recover_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_left_constant_recover_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_left_constant_recover_productcoefficientscoefficient) + (pfc_value_left_constant_recover_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_left_constant_recover_productcoefficientscoefficientdiagonalterm pfc_left_left_constant_recover_productcoefficientscoefficientdiagonalterm pfc_right_left_constant_recover_productcoefficientscoefficientdiagonalterm. (((pfc_index_left_constant_recover_productcoefficientscoefficientdiagonal)+pfc_complement_left_constant_recover_productcoefficientscoefficientdiagonalterm=(pfc_index_left_constant_recover_productcoefficients)) /\\ ((((((exists pfa_gap_left_constant_recover_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_left_constant_recover_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_left_constant_recover_productcoefficientscoefficientdiagonal) = (1)) /\\ ((((exists ff_h_pfp_left_constant_recover_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_left_constant_recover_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_left_constant_recover_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_left_constant_recover_productcoefficientscoefficientdiagonal)) * kc)) /\\ exists ff_q_pfp_left_constant_recover_productcoefficientscoefficientdiagonaltermleftentry. kb = ff_q_pfp_left_constant_recover_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_left_constant_recover_productcoefficientscoefficientdiagonal)) * kc) + (pfc_left_left_constant_recover_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_left_constant_recover_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_left_constant_recover_productcoefficientscoefficientdiagonaltermleftoutside+(1)=(pfc_index_left_constant_recover_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_left_constant_recover_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_left_constant_recover_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_left_constant_recover_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_left_constant_recover_productcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_left_constant_recover_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_left_constant_recover_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_left_constant_recover_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_left_constant_recover_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_left_constant_recover_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_left_constant_recover_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_left_constant_recover_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_left_constant_recover_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_left_constant_recover_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_left_constant_recover_productcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_left_constant_recover_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_left_constant_recover_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_left_constant_recover_productcoefficientscoefficientdiagonal)=pfc_left_left_constant_recover_productcoefficientscoefficientdiagonalterm*pfc_right_left_constant_recover_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_left_constant_recover_productcoefficientscoefficientsum fs_v_pfc_left_constant_recover_productcoefficientscoefficientsum. ((((exists fs_h_pfc_left_constant_recover_productcoefficientscoefficientsum_body_start. fs_h_pfc_left_constant_recover_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_left_constant_recover_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_constant_recover_productcoefficientscoefficientsum_body_start. fs_u_pfc_left_constant_recover_productcoefficientscoefficientsum = fs_q_pfc_left_constant_recover_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_left_constant_recover_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_left_constant_recover_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_left_constant_recover_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_left_constant_recover_productcoefficientscoefficient) = S ((S (S (pfc_index_left_constant_recover_productcoefficients))) * fs_v_pfc_left_constant_recover_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_constant_recover_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_left_constant_recover_productcoefficientscoefficientsum = fs_q_pfc_left_constant_recover_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_left_constant_recover_productcoefficients))) * fs_v_pfc_left_constant_recover_productcoefficientscoefficientsum) + (pfc_natural_sum_left_constant_recover_productcoefficientscoefficient))) /\\ forall fs_i_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps = S (pfc_index_left_constant_recover_productcoefficients)) -> exists fs_a_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps fs_r_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps fs_s_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_left_constant_recover_productcoefficientscoefficient)) /\\ exists fs_q_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_left_constant_recover_productcoefficientscoefficient = fs_q_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_left_constant_recover_productcoefficientscoefficient) + (fs_a_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_constant_recover_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_left_constant_recover_productcoefficientscoefficientsum = fs_q_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_constant_recover_productcoefficientscoefficientsum) + (fs_r_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_constant_recover_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_left_constant_recover_productcoefficientscoefficientsum = fs_q_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_constant_recover_productcoefficientscoefficientsum) + (fs_s_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps = fs_r_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps + fs_a_pfc_left_constant_recover_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_left_constant_recover_productcoefficientscoefficientresiduebound. pfa_gap_left_constant_recover_productcoefficientscoefficientresiduebound + S (pfc_value_left_constant_recover_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_left_constant_recover_productcoefficientscoefficientresiduecongruence pfa_offset_right_left_constant_recover_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_left_constant_recover_productcoefficientscoefficient) + (p) * pfa_offset_left_left_constant_recover_productcoefficientscoefficientresiduecongruence = (pfc_value_left_constant_recover_productcoefficients) + (p) * pfa_offset_right_left_constant_recover_productcoefficientscoefficientresiduecongruence)))))))))))))))))))", "statement_sha256": "5d47a2001056d10c3c4e9374ef24ffe4dca62caad61e30010ee5703989b23aea", "summary": "Recover the genuine LEFT-constant convolution on the supplied scalar-output codes. 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((forall fom_index_pfp_left_constant_exists_K. (exists fom_gap_pfp_left_constant_exists_K_index_bound. fom_gap_pfp_left_constant_exists_K_index_bound + S (fom_index_pfp_left_constant_exists_K) = 1) -> exists fom_value_pfp_left_constant_exists_K. ((((exists fom_beta_height_pfp_left_constant_exists_K_entry. fom_beta_height_pfp_left_constant_exists_K_entry + S (fom_value_pfp_left_constant_exists_K) = S ((S (fom_index_pfp_left_constant_exists_K)) * kc)) /\\ exists fom_beta_quotient_pfp_left_constant_exists_K_entry. kb = fom_beta_quotient_pfp_left_constant_exists_K_entry * S ((S (fom_index_pfp_left_constant_exists_K)) * kc) + (fom_value_pfp_left_constant_exists_K))) /\\ (exists fom_gap_pfp_left_constant_exists_K_value_bound. fom_gap_pfp_left_constant_exists_K_value_bound + S (fom_value_pfp_left_constant_exists_K) = p))) /\\ ((forall pfp_repeat_index_left_constant_exists_repeat. (exists pfa_gap_left_constant_exists_repeatindex. pfa_gap_left_constant_exists_repeatindex + S (pfp_repeat_index_left_constant_exists_repeat) = (1)) -> (((exists ff_h_pfp_left_constant_exists_repeatentry. ff_h_pfp_left_constant_exists_repeatentry + S (k) = S ((S (pfp_repeat_index_left_constant_exists_repeat)) * kc)) /\\ exists ff_q_pfp_left_constant_exists_repeatentry. kb = ff_q_pfp_left_constant_exists_repeatentry * S ((S (pfp_repeat_index_left_constant_exists_repeat)) * kc) + (k))))))", "specialize prime_field_polynomial_repeat_exists (p)", "specialize prime_field_polynomial_repeat_exists (k)", "specialize prime_field_polynomial_repeat_exists (1)", "apply prime_field_polynomial_repeat_exists", "exact hk", "cases hK", "cases hK_witness", "cases hK_witness_witness", "have hs : exists hb hc. ((exists pfa_gap_left_constant_exists_scalescalar. pfa_gap_left_constant_exists_scalescalar + S (k) = (p)) /\\ ((forall pfp_index_left_constant_exists_scale. (exists pfa_gap_left_constant_exists_scaleindex. pfa_gap_left_constant_exists_scaleindex + S (pfp_index_left_constant_exists_scale) = (L)) -> exists pfp_source_left_constant_exists_scale pfp_value_left_constant_exists_scale. ((((exists ff_h_pfp_left_constant_exists_scalesource. ff_h_pfp_left_constant_exists_scalesource + S (pfp_source_left_constant_exists_scale) = S ((S (pfp_index_left_constant_exists_scale)) * ac)) /\\ exists ff_q_pfp_left_constant_exists_scalesource. ab = ff_q_pfp_left_constant_exists_scalesource * S ((S (pfp_index_left_constant_exists_scale)) * ac) + (pfp_source_left_constant_exists_scale))) /\\ (((((exists ff_h_pfp_left_constant_exists_scaletarget. ff_h_pfp_left_constant_exists_scaletarget + S (pfp_value_left_constant_exists_scale) = S ((S (pfp_index_left_constant_exists_scale)) * hc)) /\\ exists ff_q_pfp_left_constant_exists_scaletarget. hb = ff_q_pfp_left_constant_exists_scaletarget * S ((S (pfp_index_left_constant_exists_scale)) * hc) + (pfp_value_left_constant_exists_scale))) /\\ ((((exists pfa_gap_left_constant_exists_scaleoperationleft. pfa_gap_left_constant_exists_scaleoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_left_constant_exists_scaleoperationright. pfa_gap_left_constant_exists_scaleoperationright + S (pfp_source_left_constant_exists_scale) = (p)) /\\ ((((exists pfa_gap_left_constant_exists_scaleoperationresultbound. pfa_gap_left_constant_exists_scaleoperationresultbound + S (pfp_value_left_constant_exists_scale) = (p)) /\\ ((exists pfa_offset_left_left_constant_exists_scaleoperationresultcongruence pfa_offset_right_left_constant_exists_scaleoperationresultcongruence. ((k) * (pfp_source_left_constant_exists_scale)) + (p) * pfa_offset_left_left_constant_exists_scaleoperationresultcongruence = (pfp_value_left_constant_exists_scale) + (p) * pfa_offset_right_left_constant_exists_scaleoperationresultcongruence))))))))))))))))", "specialize prime_field_polynomial_scale_exists (p)", "specialize prime_field_polynomial_scale_exists (k)", "specialize prime_field_polynomial_scale_exists (ab)", "specialize prime_field_polynomial_scale_exists (ac)", "specialize prime_field_polynomial_scale_exists (L)", "apply prime_field_polynomial_scale_exists", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hz", "exact hk", "exact ha", "cases hs", "cases hs_witness", "have hentry : ((exists ff_h_pfp_left_constant_exists_head. ff_h_pfp_left_constant_exists_head + S (k) = S ((S (0)) * x1)) /\\ exists ff_q_pfp_left_constant_exists_head. x = ff_q_pfp_left_constant_exists_head * S ((S (0)) * x1) + (k))", "specialize hK_witness_witness_right (0)", "apply hK_witness_witness_right", "exists 0", "apply zero_add", "exists x", "exists x1", "exists x2", "exists x3", "split", "exact hK_witness_witness_left", "split", "exact hentry", "split", "exact hs_witness_witness", "specialize prime_field_polynomial_scale_to_left_constant_product (p)", "specialize prime_field_polynomial_scale_to_left_constant_product (k)", "specialize prime_field_polynomial_scale_to_left_constant_product (x)", "specialize prime_field_polynomial_scale_to_left_constant_product (x1)", "specialize prime_field_polynomial_scale_to_left_constant_product (ab)", "specialize prime_field_polynomial_scale_to_left_constant_product (ac)", "specialize prime_field_polynomial_scale_to_left_constant_product (x2)", "specialize prime_field_polynomial_scale_to_left_constant_product (x3)", "specialize prime_field_polynomial_scale_to_left_constant_product (L)", "apply prime_field_polynomial_scale_to_left_constant_product", "exact hp", "exact hK_witness_witness_left", "exact hentry", "exact hs_witness_witness" ], "script_sha256": "6f195217fb9d46bc09747aea05721815f89dedc04daecbdf636cf7d789b4fb49", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_left_constant_candidate.py", "sha256": "9a7a4de30f5f389bcabc2e6267a0d2cc5dc5f061059dcea303a0a03dab58509a" }, "statement": "forall p k ab ac L. (~((p) = 1) /\\ forall pfa_factor_left_left_constant_exists_prime pfa_factor_right_left_constant_exists_prime. (p) = pfa_factor_left_left_constant_exists_prime * pfa_factor_right_left_constant_exists_prime -> pfa_factor_left_left_constant_exists_prime = 1 \\/ pfa_factor_right_left_constant_exists_prime = 1) -> (exists pfa_gap_left_constant_exists_scalar. pfa_gap_left_constant_exists_scalar + S (k) = (p)) -> (forall fom_index_pfp_left_constant_exists_input. (exists fom_gap_pfp_left_constant_exists_input_index_bound. fom_gap_pfp_left_constant_exists_input_index_bound + S (fom_index_pfp_left_constant_exists_input) = L) -> exists fom_value_pfp_left_constant_exists_input. ((((exists fom_beta_height_pfp_left_constant_exists_input_entry. fom_beta_height_pfp_left_constant_exists_input_entry + S (fom_value_pfp_left_constant_exists_input) = S ((S (fom_index_pfp_left_constant_exists_input)) * ac)) /\\ exists fom_beta_quotient_pfp_left_constant_exists_input_entry. ab = fom_beta_quotient_pfp_left_constant_exists_input_entry * S ((S (fom_index_pfp_left_constant_exists_input)) * ac) + (fom_value_pfp_left_constant_exists_input))) /\\ (exists fom_gap_pfp_left_constant_exists_input_value_bound. fom_gap_pfp_left_constant_exists_input_value_bound + S (fom_value_pfp_left_constant_exists_input) = p))) -> (exists kb kc hb hc. ((forall fom_index_pfp_left_constant_exists_result_K. (exists fom_gap_pfp_left_constant_exists_result_K_index_bound. fom_gap_pfp_left_constant_exists_result_K_index_bound + S (fom_index_pfp_left_constant_exists_result_K) = 1) -> exists fom_value_pfp_left_constant_exists_result_K. ((((exists fom_beta_height_pfp_left_constant_exists_result_K_entry. fom_beta_height_pfp_left_constant_exists_result_K_entry + S (fom_value_pfp_left_constant_exists_result_K) = S ((S (fom_index_pfp_left_constant_exists_result_K)) * kc)) /\\ exists fom_beta_quotient_pfp_left_constant_exists_result_K_entry. kb = fom_beta_quotient_pfp_left_constant_exists_result_K_entry * S ((S (fom_index_pfp_left_constant_exists_result_K)) * kc) + (fom_value_pfp_left_constant_exists_result_K))) /\\ (exists fom_gap_pfp_left_constant_exists_result_K_value_bound. fom_gap_pfp_left_constant_exists_result_K_value_bound + S (fom_value_pfp_left_constant_exists_result_K) = p))) /\\ (((((exists ff_h_pfp_left_constant_exists_result_head. ff_h_pfp_left_constant_exists_result_head + S (k) = S ((S (0)) * kc)) /\\ exists ff_q_pfp_left_constant_exists_result_head. kb = ff_q_pfp_left_constant_exists_result_head * S ((S (0)) * kc) + (k))) /\\ (((((exists pfa_gap_left_constant_exists_result_scalescalar. pfa_gap_left_constant_exists_result_scalescalar + S (k) = (p)) /\\ ((forall pfp_index_left_constant_exists_result_scale. (exists pfa_gap_left_constant_exists_result_scaleindex. pfa_gap_left_constant_exists_result_scaleindex + S (pfp_index_left_constant_exists_result_scale) = (L)) -> exists pfp_source_left_constant_exists_result_scale pfp_value_left_constant_exists_result_scale. ((((exists ff_h_pfp_left_constant_exists_result_scalesource. ff_h_pfp_left_constant_exists_result_scalesource + S (pfp_source_left_constant_exists_result_scale) = S ((S (pfp_index_left_constant_exists_result_scale)) * ac)) /\\ exists ff_q_pfp_left_constant_exists_result_scalesource. ab = ff_q_pfp_left_constant_exists_result_scalesource * S ((S (pfp_index_left_constant_exists_result_scale)) * ac) + (pfp_source_left_constant_exists_result_scale))) /\\ (((((exists ff_h_pfp_left_constant_exists_result_scaletarget. ff_h_pfp_left_constant_exists_result_scaletarget + S (pfp_value_left_constant_exists_result_scale) = S ((S (pfp_index_left_constant_exists_result_scale)) * hc)) /\\ exists ff_q_pfp_left_constant_exists_result_scaletarget. hb = ff_q_pfp_left_constant_exists_result_scaletarget * S ((S (pfp_index_left_constant_exists_result_scale)) * hc) + (pfp_value_left_constant_exists_result_scale))) /\\ ((((exists pfa_gap_left_constant_exists_result_scaleoperationleft. pfa_gap_left_constant_exists_result_scaleoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_left_constant_exists_result_scaleoperationright. pfa_gap_left_constant_exists_result_scaleoperationright + S (pfp_source_left_constant_exists_result_scale) = (p)) /\\ ((((exists pfa_gap_left_constant_exists_result_scaleoperationresultbound. pfa_gap_left_constant_exists_result_scaleoperationresultbound + S (pfp_value_left_constant_exists_result_scale) = (p)) /\\ ((exists pfa_offset_left_left_constant_exists_result_scaleoperationresultcongruence pfa_offset_right_left_constant_exists_result_scaleoperationresultcongruence. ((k) * (pfp_source_left_constant_exists_result_scale)) + (p) * pfa_offset_left_left_constant_exists_result_scaleoperationresultcongruence = (pfp_value_left_constant_exists_result_scale) + (p) * pfa_offset_right_left_constant_exists_result_scaleoperationresultcongruence))))))))))))))))) /\\ ((((forall fom_index_pfp_left_constant_exists_result_productleft. (exists fom_gap_pfp_left_constant_exists_result_productleft_index_bound. fom_gap_pfp_left_constant_exists_result_productleft_index_bound + S (fom_index_pfp_left_constant_exists_result_productleft) = 1) -> exists fom_value_pfp_left_constant_exists_result_productleft. ((((exists fom_beta_height_pfp_left_constant_exists_result_productleft_entry. fom_beta_height_pfp_left_constant_exists_result_productleft_entry + S (fom_value_pfp_left_constant_exists_result_productleft) = S ((S (fom_index_pfp_left_constant_exists_result_productleft)) * kc)) /\\ exists fom_beta_quotient_pfp_left_constant_exists_result_productleft_entry. kb = fom_beta_quotient_pfp_left_constant_exists_result_productleft_entry * S ((S (fom_index_pfp_left_constant_exists_result_productleft)) * kc) + (fom_value_pfp_left_constant_exists_result_productleft))) /\\ (exists fom_gap_pfp_left_constant_exists_result_productleft_value_bound. fom_gap_pfp_left_constant_exists_result_productleft_value_bound + S (fom_value_pfp_left_constant_exists_result_productleft) = p))) /\\ (((forall fom_index_pfp_left_constant_exists_result_productright. (exists fom_gap_pfp_left_constant_exists_result_productright_index_bound. fom_gap_pfp_left_constant_exists_result_productright_index_bound + S (fom_index_pfp_left_constant_exists_result_productright) = L) -> exists fom_value_pfp_left_constant_exists_result_productright. ((((exists fom_beta_height_pfp_left_constant_exists_result_productright_entry. fom_beta_height_pfp_left_constant_exists_result_productright_entry + S (fom_value_pfp_left_constant_exists_result_productright) = S ((S (fom_index_pfp_left_constant_exists_result_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_left_constant_exists_result_productright_entry. ab = fom_beta_quotient_pfp_left_constant_exists_result_productright_entry * S ((S (fom_index_pfp_left_constant_exists_result_productright)) * ac) + (fom_value_pfp_left_constant_exists_result_productright))) /\\ (exists fom_gap_pfp_left_constant_exists_result_productright_value_bound. fom_gap_pfp_left_constant_exists_result_productright_value_bound + S (fom_value_pfp_left_constant_exists_result_productright) = p))) /\\ (((((((1)=0 \\/ (L)=0) /\\ (((L)=0)))) \\/ (((~((1)=0)) /\\ (((~((L)=0)) /\\ (((1)+(L)=S (L)))))))) /\\ ((forall pfc_index_left_constant_exists_result_productcoefficients. (exists pfa_gap_left_constant_exists_result_productcoefficientsbound. pfa_gap_left_constant_exists_result_productcoefficientsbound + S (pfc_index_left_constant_exists_result_productcoefficients) = (L)) -> exists pfc_value_left_constant_exists_result_productcoefficients. ((((exists ff_h_pfp_left_constant_exists_result_productcoefficientsentry. ff_h_pfp_left_constant_exists_result_productcoefficientsentry + S (pfc_value_left_constant_exists_result_productcoefficients) = S ((S (pfc_index_left_constant_exists_result_productcoefficients)) * hc)) /\\ exists ff_q_pfp_left_constant_exists_result_productcoefficientsentry. hb = ff_q_pfp_left_constant_exists_result_productcoefficientsentry * S ((S (pfc_index_left_constant_exists_result_productcoefficients)) * hc) + (pfc_value_left_constant_exists_result_productcoefficients))) /\\ ((exists pfc_terms_code_left_constant_exists_result_productcoefficientscoefficient pfc_terms_scale_left_constant_exists_result_productcoefficientscoefficient pfc_natural_sum_left_constant_exists_result_productcoefficientscoefficient. ((forall pfc_index_left_constant_exists_result_productcoefficientscoefficientdiagonal. (exists pfa_gap_left_constant_exists_result_productcoefficientscoefficientdiagonalbound. pfa_gap_left_constant_exists_result_productcoefficientscoefficientdiagonalbound + S (pfc_index_left_constant_exists_result_productcoefficientscoefficientdiagonal) = (S (pfc_index_left_constant_exists_result_productcoefficients))) -> exists pfc_value_left_constant_exists_result_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_left_constant_exists_result_productcoefficientscoefficientdiagonalentry. ff_h_pfp_left_constant_exists_result_productcoefficientscoefficientdiagonalentry + S (pfc_value_left_constant_exists_result_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_left_constant_exists_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_left_constant_exists_result_productcoefficientscoefficient)) /\\ exists ff_q_pfp_left_constant_exists_result_productcoefficientscoefficientdiagonalentry. pfc_terms_code_left_constant_exists_result_productcoefficientscoefficient = ff_q_pfp_left_constant_exists_result_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_left_constant_exists_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_left_constant_exists_result_productcoefficientscoefficient) + (pfc_value_left_constant_exists_result_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_left_constant_exists_result_productcoefficientscoefficientdiagonalterm pfc_left_left_constant_exists_result_productcoefficientscoefficientdiagonalterm pfc_right_left_constant_exists_result_productcoefficientscoefficientdiagonalterm. (((pfc_index_left_constant_exists_result_productcoefficientscoefficientdiagonal)+pfc_complement_left_constant_exists_result_productcoefficientscoefficientdiagonalterm=(pfc_index_left_constant_exists_result_productcoefficients)) /\\ ((((((exists pfa_gap_left_constant_exists_result_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_left_constant_exists_result_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_left_constant_exists_result_productcoefficientscoefficientdiagonal) = (1)) /\\ ((((exists ff_h_pfp_left_constant_exists_result_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_left_constant_exists_result_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_left_constant_exists_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_left_constant_exists_result_productcoefficientscoefficientdiagonal)) * kc)) /\\ exists ff_q_pfp_left_constant_exists_result_productcoefficientscoefficientdiagonaltermleftentry. kb = ff_q_pfp_left_constant_exists_result_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_left_constant_exists_result_productcoefficientscoefficientdiagonal)) * kc) + (pfc_left_left_constant_exists_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_left_constant_exists_result_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_left_constant_exists_result_productcoefficientscoefficientdiagonaltermleftoutside+(1)=(pfc_index_left_constant_exists_result_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_left_constant_exists_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_left_constant_exists_result_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_left_constant_exists_result_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_left_constant_exists_result_productcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_left_constant_exists_result_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_left_constant_exists_result_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_left_constant_exists_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_left_constant_exists_result_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_left_constant_exists_result_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_left_constant_exists_result_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_left_constant_exists_result_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_left_constant_exists_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_left_constant_exists_result_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_left_constant_exists_result_productcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_left_constant_exists_result_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_left_constant_exists_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_left_constant_exists_result_productcoefficientscoefficientdiagonal)=pfc_left_left_constant_exists_result_productcoefficientscoefficientdiagonalterm*pfc_right_left_constant_exists_result_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_left_constant_exists_result_productcoefficientscoefficientsum fs_v_pfc_left_constant_exists_result_productcoefficientscoefficientsum. ((((exists fs_h_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_start. fs_h_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_left_constant_exists_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_start. fs_u_pfc_left_constant_exists_result_productcoefficientscoefficientsum = fs_q_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_left_constant_exists_result_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_left_constant_exists_result_productcoefficientscoefficient) = S ((S (S (pfc_index_left_constant_exists_result_productcoefficients))) * fs_v_pfc_left_constant_exists_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_left_constant_exists_result_productcoefficientscoefficientsum = fs_q_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_left_constant_exists_result_productcoefficients))) * fs_v_pfc_left_constant_exists_result_productcoefficientscoefficientsum) + (pfc_natural_sum_left_constant_exists_result_productcoefficientscoefficient))) /\\ forall fs_i_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps = S (pfc_index_left_constant_exists_result_productcoefficients)) -> exists fs_a_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps fs_r_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps fs_s_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_left_constant_exists_result_productcoefficientscoefficient)) /\\ exists fs_q_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_left_constant_exists_result_productcoefficientscoefficient = fs_q_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_left_constant_exists_result_productcoefficientscoefficient) + (fs_a_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_constant_exists_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_left_constant_exists_result_productcoefficientscoefficientsum = fs_q_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_constant_exists_result_productcoefficientscoefficientsum) + (fs_r_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_constant_exists_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_left_constant_exists_result_productcoefficientscoefficientsum = fs_q_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_constant_exists_result_productcoefficientscoefficientsum) + (fs_s_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps = fs_r_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps + fs_a_pfc_left_constant_exists_result_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_left_constant_exists_result_productcoefficientscoefficientresiduebound. pfa_gap_left_constant_exists_result_productcoefficientscoefficientresiduebound + S (pfc_value_left_constant_exists_result_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_left_constant_exists_result_productcoefficientscoefficientresiduecongruence pfa_offset_right_left_constant_exists_result_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_left_constant_exists_result_productcoefficientscoefficient) + (p) * pfa_offset_left_left_constant_exists_result_productcoefficientscoefficientresiduecongruence = (pfc_value_left_constant_exists_result_productcoefficients) + (p) * pfa_offset_right_left_constant_exists_result_productcoefficientscoefficientresiduecongruence))))))))))))))))))))))))))", "statement_sha256": "27fb809df4b4e25c993baa9a282d224aa18553949e0ab62a9f0a62b2a758dfcf", "summary": "Construct the canonical singleton and actual scalar output, then prove their genuine left-factor product using those same output codes. 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(((((R)=S (pfd_remainder_degree_normalization_descent_degree)) /\\ (((forall fom_index_pfp_normalization_descent_degreerepresentedcoefficients. (exists fom_gap_pfp_normalization_descent_degreerepresentedcoefficients_index_bound. fom_gap_pfp_normalization_descent_degreerepresentedcoefficients_index_bound + S (fom_index_pfp_normalization_descent_degreerepresentedcoefficients) = R) -> exists fom_value_pfp_normalization_descent_degreerepresentedcoefficients. ((((exists fom_beta_height_pfp_normalization_descent_degreerepresentedcoefficients_entry. fom_beta_height_pfp_normalization_descent_degreerepresentedcoefficients_entry + S (fom_value_pfp_normalization_descent_degreerepresentedcoefficients) = S ((S (fom_index_pfp_normalization_descent_degreerepresentedcoefficients)) * rc)) /\\ exists fom_beta_quotient_pfp_normalization_descent_degreerepresentedcoefficients_entry. rb = fom_beta_quotient_pfp_normalization_descent_degreerepresentedcoefficients_entry * S ((S (fom_index_pfp_normalization_descent_degreerepresentedcoefficients)) * rc) + (fom_value_pfp_normalization_descent_degreerepresentedcoefficients))) /\\ (exists fom_gap_pfp_normalization_descent_degreerepresentedcoefficients_value_bound. fom_gap_pfp_normalization_descent_degreerepresentedcoefficients_value_bound + S (fom_value_pfp_normalization_descent_degreerepresentedcoefficients) = p))) /\\ ((exists pfd_leading_normalization_descent_degreerepresented. ((((exists ff_h_pfp_normalization_descent_degreerepresentedentry. ff_h_pfp_normalization_descent_degreerepresentedentry + S (pfd_leading_normalization_descent_degreerepresented) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_normalization_descent_degreerepresentedentry. rb = ff_q_pfp_normalization_descent_degreerepresentedentry * S ((S (0)) * rc) + (pfd_leading_normalization_descent_degreerepresented))) /\\ ((~(pfd_leading_normalization_descent_degreerepresented=0)))))))))) /\\ ((exists pfa_gap_normalization_descent_degreestrict. pfa_gap_normalization_descent_degreestrict + S (pfd_remainder_degree_normalization_descent_degree) = (d)))))", "specialize prime_field_polynomial_division_remainder_degree (p)", "specialize prime_field_polynomial_division_remainder_degree (ab)", "specialize prime_field_polynomial_division_remainder_degree (ac)", "specialize prime_field_polynomial_division_remainder_degree (L)", "specialize prime_field_polynomial_division_remainder_degree (bb)", "specialize prime_field_polynomial_division_remainder_degree (bc)", "specialize prime_field_polynomial_division_remainder_degree (d)", "specialize prime_field_polynomial_division_remainder_degree (qb)", "specialize prime_field_polynomial_division_remainder_degree (qc)", "specialize prime_field_polynomial_division_remainder_degree (q)", "specialize prime_field_polynomial_division_remainder_degree (rb)", "specialize prime_field_polynomial_division_remainder_degree (rc)", "specialize prime_field_polynomial_division_remainder_degree (R)", "apply prime_field_polynomial_division_remainder_degree", "exact hp", "exact hexecution", "have hbound : exists pfc_gap_normalization_descent_bound. pfc_gap_normalization_descent_bound+(R)=(d)", "cases hdegree", "rewrite hdegree_left", "specialize zero_le (d)", "apply zero_le", "cases hdegree_right", "cases hdegree_right_witness", "cases hdegree_right_witness_left", "rewrite hdegree_right_witness_left_left", "exact hdegree_right_witness_right", "split", "exact hbound", "specialize succ_le_succ (R)", "specialize succ_le_succ (d)", "apply succ_le_succ", "exact hbound" ], "script_sha256": "9c9dbedc3923c943aef17f6dd5c9a27bbba036f10ab8fed46dbc7cd60adc7ebb", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_euclidean_normalization_candidate.py", "sha256": "d2cddfe42dc0d22104dc4e85e95116222914df11ac840d2082a4ff2e462f146f" }, "statement": "forall p ab ac L bb bc d qb qc q rb rc R. (~((p) = 1) /\\ forall pfa_factor_left_normalization_descent_prime pfa_factor_right_normalization_descent_prime. (p) = pfa_factor_left_normalization_descent_prime * pfa_factor_right_normalization_descent_prime -> pfa_factor_left_normalization_descent_prime = 1 \\/ pfa_factor_right_normalization_descent_prime = 1) -> (((forall fom_index_pfp_normalization_descent_executioninput. (exists fom_gap_pfp_normalization_descent_executioninput_index_bound. fom_gap_pfp_normalization_descent_executioninput_index_bound + S (fom_index_pfp_normalization_descent_executioninput) = L) -> exists fom_value_pfp_normalization_descent_executioninput. ((((exists fom_beta_height_pfp_normalization_descent_executioninput_entry. fom_beta_height_pfp_normalization_descent_executioninput_entry + S (fom_value_pfp_normalization_descent_executioninput) = S ((S (fom_index_pfp_normalization_descent_executioninput)) * ac)) /\\ exists fom_beta_quotient_pfp_normalization_descent_executioninput_entry. ab = fom_beta_quotient_pfp_normalization_descent_executioninput_entry * S ((S (fom_index_pfp_normalization_descent_executioninput)) * ac) + (fom_value_pfp_normalization_descent_executioninput))) /\\ (exists fom_gap_pfp_normalization_descent_executioninput_value_bound. fom_gap_pfp_normalization_descent_executioninput_value_bound + S (fom_value_pfp_normalization_descent_executioninput) = p))) /\\ (((forall fom_index_pfp_normalization_descent_executiondivisor. (exists fom_gap_pfp_normalization_descent_executiondivisor_index_bound. fom_gap_pfp_normalization_descent_executiondivisor_index_bound + S (fom_index_pfp_normalization_descent_executiondivisor) = S (d)) -> exists fom_value_pfp_normalization_descent_executiondivisor. ((((exists fom_beta_height_pfp_normalization_descent_executiondivisor_entry. fom_beta_height_pfp_normalization_descent_executiondivisor_entry + S (fom_value_pfp_normalization_descent_executiondivisor) = S ((S (fom_index_pfp_normalization_descent_executiondivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_normalization_descent_executiondivisor_entry. bb = fom_beta_quotient_pfp_normalization_descent_executiondivisor_entry * S ((S (fom_index_pfp_normalization_descent_executiondivisor)) * bc) + (fom_value_pfp_normalization_descent_executiondivisor))) /\\ (exists fom_gap_pfp_normalization_descent_executiondivisor_value_bound. fom_gap_pfp_normalization_descent_executiondivisor_value_bound + S (fom_value_pfp_normalization_descent_executiondivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_normalization_descent_executionlengthshort. pfc_gap_normalization_descent_executionlengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((exists pfd_head_normalization_descent_execution pfd_inverse_normalization_descent_execution pfd_product_code_normalization_descent_execution pfd_product_scale_normalization_descent_execution pfd_residual_code_normalization_descent_execution pfd_residual_scale_normalization_descent_execution pfd_cut_normalization_descent_execution. ((((exists ff_h_pfp_normalization_descent_executionhead. ff_h_pfp_normalization_descent_executionhead + S (pfd_head_normalization_descent_execution) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_normalization_descent_executionhead. bb = ff_q_pfp_normalization_descent_executionhead * S ((S (0)) * bc) + (pfd_head_normalization_descent_execution))) /\\ (((((~((pfd_head_normalization_descent_execution) = 0)) /\\ ((((exists pfa_gap_normalization_descent_executioninversemultiplicationleft. pfa_gap_normalization_descent_executioninversemultiplicationleft + S (pfd_head_normalization_descent_execution) = (p)) /\\ (((exists pfa_gap_normalization_descent_executioninversemultiplicationright. pfa_gap_normalization_descent_executioninversemultiplicationright + S (pfd_inverse_normalization_descent_execution) = (p)) /\\ ((((exists pfa_gap_normalization_descent_executioninversemultiplicationresultbound. pfa_gap_normalization_descent_executioninversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_normalization_descent_executioninversemultiplicationresultcongruence pfa_offset_right_normalization_descent_executioninversemultiplicationresultcongruence. ((pfd_head_normalization_descent_execution) * (pfd_inverse_normalization_descent_execution)) + (p) * pfa_offset_left_normalization_descent_executioninversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_descent_executioninversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_normalization_descent_executionquotient. (exists pfa_gap_normalization_descent_executionquotientbound. pfa_gap_normalization_descent_executionquotientbound + S (pfd_index_normalization_descent_executionquotient) = (q)) -> exists pfd_value_normalization_descent_executionquotient. ((((exists ff_h_pfp_normalization_descent_executionquotiententry. ff_h_pfp_normalization_descent_executionquotiententry + S (pfd_value_normalization_descent_executionquotient) = S ((S (pfd_index_normalization_descent_executionquotient)) * qc)) /\\ exists ff_q_pfp_normalization_descent_executionquotiententry. qb = ff_q_pfp_normalization_descent_executionquotiententry * S ((S (pfd_index_normalization_descent_executionquotient)) * qc) + (pfd_value_normalization_descent_executionquotient))) /\\ ((exists pfd_input_normalization_descent_executionquotientstep pfd_previous_normalization_descent_executionquotientstep pfd_difference_normalization_descent_executionquotientstep. ((((exists ff_h_pfp_normalization_descent_executionquotientstepinput. ff_h_pfp_normalization_descent_executionquotientstepinput + S (pfd_input_normalization_descent_executionquotientstep) = S ((S (pfd_index_normalization_descent_executionquotient)) * ac)) /\\ exists ff_q_pfp_normalization_descent_executionquotientstepinput. ab = ff_q_pfp_normalization_descent_executionquotientstepinput * S ((S (pfd_index_normalization_descent_executionquotient)) * ac) + (pfd_input_normalization_descent_executionquotientstep))) /\\ (((exists pfc_terms_code_normalization_descent_executionquotientstepprevious pfc_terms_scale_normalization_descent_executionquotientstepprevious pfc_natural_sum_normalization_descent_executionquotientstepprevious. ((forall pfc_index_normalization_descent_executionquotientsteppreviousdiagonal. (exists pfa_gap_normalization_descent_executionquotientsteppreviousdiagonalbound. pfa_gap_normalization_descent_executionquotientsteppreviousdiagonalbound + S (pfc_index_normalization_descent_executionquotientsteppreviousdiagonal) = (S (pfd_index_normalization_descent_executionquotient))) -> exists pfc_value_normalization_descent_executionquotientsteppreviousdiagonal. ((((exists ff_h_pfp_normalization_descent_executionquotientsteppreviousdiagonalentry. ff_h_pfp_normalization_descent_executionquotientsteppreviousdiagonalentry + S (pfc_value_normalization_descent_executionquotientsteppreviousdiagonal) = S ((S (pfc_index_normalization_descent_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_normalization_descent_executionquotientstepprevious)) /\\ exists ff_q_pfp_normalization_descent_executionquotientsteppreviousdiagonalentry. pfc_terms_code_normalization_descent_executionquotientstepprevious = ff_q_pfp_normalization_descent_executionquotientsteppreviousdiagonalentry * S ((S (pfc_index_normalization_descent_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_normalization_descent_executionquotientstepprevious) + (pfc_value_normalization_descent_executionquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_normalization_descent_executionquotientsteppreviousdiagonalterm pfc_left_normalization_descent_executionquotientsteppreviousdiagonalterm pfc_right_normalization_descent_executionquotientsteppreviousdiagonalterm. (((pfc_index_normalization_descent_executionquotientsteppreviousdiagonal)+pfc_complement_normalization_descent_executionquotientsteppreviousdiagonalterm=(pfd_index_normalization_descent_executionquotient)) /\\ ((((((exists pfa_gap_normalization_descent_executionquotientsteppreviousdiagonaltermleftinside. pfa_gap_normalization_descent_executionquotientsteppreviousdiagonaltermleftinside + S (pfc_index_normalization_descent_executionquotientsteppreviousdiagonal) = (pfd_index_normalization_descent_executionquotient)) /\\ ((((exists ff_h_pfp_normalization_descent_executionquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_normalization_descent_executionquotientsteppreviousdiagonaltermleftentry + S (pfc_left_normalization_descent_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_index_normalization_descent_executionquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_normalization_descent_executionquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_normalization_descent_executionquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_normalization_descent_executionquotientsteppreviousdiagonal)) * qc) + (pfc_left_normalization_descent_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_descent_executionquotientsteppreviousdiagonaltermleftoutside. pfc_gap_normalization_descent_executionquotientsteppreviousdiagonaltermleftoutside+(pfd_index_normalization_descent_executionquotient)=(pfc_index_normalization_descent_executionquotientsteppreviousdiagonal)) /\\ (((pfc_left_normalization_descent_executionquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalization_descent_executionquotientsteppreviousdiagonaltermrightinside. pfa_gap_normalization_descent_executionquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_normalization_descent_executionquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_normalization_descent_executionquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_normalization_descent_executionquotientsteppreviousdiagonaltermrightentry + S (pfc_right_normalization_descent_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_normalization_descent_executionquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_normalization_descent_executionquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_normalization_descent_executionquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_normalization_descent_executionquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_normalization_descent_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_descent_executionquotientsteppreviousdiagonaltermrightoutside. pfc_gap_normalization_descent_executionquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_normalization_descent_executionquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_normalization_descent_executionquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_normalization_descent_executionquotientsteppreviousdiagonal)=pfc_left_normalization_descent_executionquotientsteppreviousdiagonalterm*pfc_right_normalization_descent_executionquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalization_descent_executionquotientstepprevioussum fs_v_pfc_normalization_descent_executionquotientstepprevioussum. ((((exists fs_h_pfc_normalization_descent_executionquotientstepprevioussum_body_start. fs_h_pfc_normalization_descent_executionquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalization_descent_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_normalization_descent_executionquotientstepprevioussum_body_start. fs_u_pfc_normalization_descent_executionquotientstepprevioussum = fs_q_pfc_normalization_descent_executionquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_normalization_descent_executionquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_normalization_descent_executionquotientstepprevioussum_body_terminal. fs_h_pfc_normalization_descent_executionquotientstepprevioussum_body_terminal + S (pfc_natural_sum_normalization_descent_executionquotientstepprevious) = S ((S (S (pfd_index_normalization_descent_executionquotient))) * fs_v_pfc_normalization_descent_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_normalization_descent_executionquotientstepprevioussum_body_terminal. fs_u_pfc_normalization_descent_executionquotientstepprevioussum = fs_q_pfc_normalization_descent_executionquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_normalization_descent_executionquotient))) * fs_v_pfc_normalization_descent_executionquotientstepprevioussum) + (pfc_natural_sum_normalization_descent_executionquotientstepprevious))) /\\ forall fs_i_pfc_normalization_descent_executionquotientstepprevioussum_body_steps. (exists fs_lt_pfc_normalization_descent_executionquotientstepprevioussum_body_steps_bound. fs_lt_pfc_normalization_descent_executionquotientstepprevioussum_body_steps_bound + S fs_i_pfc_normalization_descent_executionquotientstepprevioussum_body_steps = S (pfd_index_normalization_descent_executionquotient)) -> exists fs_a_pfc_normalization_descent_executionquotientstepprevioussum_body_steps fs_r_pfc_normalization_descent_executionquotientstepprevioussum_body_steps fs_s_pfc_normalization_descent_executionquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_normalization_descent_executionquotientstepprevioussum_body_steps_summand. fs_h_pfc_normalization_descent_executionquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_normalization_descent_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_normalization_descent_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_normalization_descent_executionquotientstepprevious)) /\\ exists fs_q_pfc_normalization_descent_executionquotientstepprevioussum_body_steps_summand. pfc_terms_code_normalization_descent_executionquotientstepprevious = fs_q_pfc_normalization_descent_executionquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_normalization_descent_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_normalization_descent_executionquotientstepprevious) + (fs_a_pfc_normalization_descent_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_descent_executionquotientstepprevioussum_body_steps_partial. fs_h_pfc_normalization_descent_executionquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_normalization_descent_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_normalization_descent_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_normalization_descent_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_normalization_descent_executionquotientstepprevioussum_body_steps_partial. fs_u_pfc_normalization_descent_executionquotientstepprevioussum = fs_q_pfc_normalization_descent_executionquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_normalization_descent_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_normalization_descent_executionquotientstepprevioussum) + (fs_r_pfc_normalization_descent_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_descent_executionquotientstepprevioussum_body_steps_successor. fs_h_pfc_normalization_descent_executionquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_normalization_descent_executionquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_normalization_descent_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_normalization_descent_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_normalization_descent_executionquotientstepprevioussum_body_steps_successor. fs_u_pfc_normalization_descent_executionquotientstepprevioussum = fs_q_pfc_normalization_descent_executionquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_normalization_descent_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_normalization_descent_executionquotientstepprevioussum) + (fs_s_pfc_normalization_descent_executionquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_normalization_descent_executionquotientstepprevioussum_body_steps = fs_r_pfc_normalization_descent_executionquotientstepprevioussum_body_steps + fs_a_pfc_normalization_descent_executionquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_normalization_descent_executionquotientsteppreviousresiduebound. pfa_gap_normalization_descent_executionquotientsteppreviousresiduebound + S (pfd_previous_normalization_descent_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_normalization_descent_executionquotientsteppreviousresiduecongruence pfa_offset_right_normalization_descent_executionquotientsteppreviousresiduecongruence. (pfc_natural_sum_normalization_descent_executionquotientstepprevious) + (p) * pfa_offset_left_normalization_descent_executionquotientsteppreviousresiduecongruence = (pfd_previous_normalization_descent_executionquotientstep) + (p) * pfa_offset_right_normalization_descent_executionquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_normalization_descent_executionquotientstepsubtractleft. pfa_gap_normalization_descent_executionquotientstepsubtractleft + S (pfd_previous_normalization_descent_executionquotientstep) = (p)) /\\ (((exists pfa_gap_normalization_descent_executionquotientstepsubtractright. pfa_gap_normalization_descent_executionquotientstepsubtractright + S (pfd_difference_normalization_descent_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_normalization_descent_executionquotientstepsubtractresultbound. pfa_gap_normalization_descent_executionquotientstepsubtractresultbound + S (pfd_input_normalization_descent_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_normalization_descent_executionquotientstepsubtractresultcongruence pfa_offset_right_normalization_descent_executionquotientstepsubtractresultcongruence. ((pfd_previous_normalization_descent_executionquotientstep) + (pfd_difference_normalization_descent_executionquotientstep)) + (p) * pfa_offset_left_normalization_descent_executionquotientstepsubtractresultcongruence = (pfd_input_normalization_descent_executionquotientstep) + (p) * pfa_offset_right_normalization_descent_executionquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_normalization_descent_executionquotientstepmultiplyleft. pfa_gap_normalization_descent_executionquotientstepmultiplyleft + S (pfd_inverse_normalization_descent_execution) = (p)) /\\ (((exists pfa_gap_normalization_descent_executionquotientstepmultiplyright. pfa_gap_normalization_descent_executionquotientstepmultiplyright + S (pfd_difference_normalization_descent_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_normalization_descent_executionquotientstepmultiplyresultbound. pfa_gap_normalization_descent_executionquotientstepmultiplyresultbound + S (pfd_value_normalization_descent_executionquotient) = (p)) /\\ ((exists pfa_offset_left_normalization_descent_executionquotientstepmultiplyresultcongruence pfa_offset_right_normalization_descent_executionquotientstepmultiplyresultcongruence. ((pfd_inverse_normalization_descent_execution) * (pfd_difference_normalization_descent_executionquotientstep)) + (p) * pfa_offset_left_normalization_descent_executionquotientstepmultiplyresultcongruence = (pfd_value_normalization_descent_executionquotient) + (p) * pfa_offset_right_normalization_descent_executionquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_normalization_descent_executionproduct. (exists pfa_gap_normalization_descent_executionproductbound. pfa_gap_normalization_descent_executionproductbound + S (pfc_index_normalization_descent_executionproduct) = (L)) -> exists pfc_value_normalization_descent_executionproduct. ((((exists ff_h_pfp_normalization_descent_executionproductentry. ff_h_pfp_normalization_descent_executionproductentry + S (pfc_value_normalization_descent_executionproduct) = S ((S (pfc_index_normalization_descent_executionproduct)) * pfd_product_scale_normalization_descent_execution)) /\\ exists ff_q_pfp_normalization_descent_executionproductentry. pfd_product_code_normalization_descent_execution = ff_q_pfp_normalization_descent_executionproductentry * S ((S (pfc_index_normalization_descent_executionproduct)) * pfd_product_scale_normalization_descent_execution) + (pfc_value_normalization_descent_executionproduct))) /\\ ((exists pfc_terms_code_normalization_descent_executionproductcoefficient pfc_terms_scale_normalization_descent_executionproductcoefficient pfc_natural_sum_normalization_descent_executionproductcoefficient. ((forall pfc_index_normalization_descent_executionproductcoefficientdiagonal. (exists pfa_gap_normalization_descent_executionproductcoefficientdiagonalbound. pfa_gap_normalization_descent_executionproductcoefficientdiagonalbound + S (pfc_index_normalization_descent_executionproductcoefficientdiagonal) = (S (pfc_index_normalization_descent_executionproduct))) -> exists pfc_value_normalization_descent_executionproductcoefficientdiagonal. ((((exists ff_h_pfp_normalization_descent_executionproductcoefficientdiagonalentry. ff_h_pfp_normalization_descent_executionproductcoefficientdiagonalentry + S (pfc_value_normalization_descent_executionproductcoefficientdiagonal) = S ((S (pfc_index_normalization_descent_executionproductcoefficientdiagonal)) * pfc_terms_scale_normalization_descent_executionproductcoefficient)) /\\ exists ff_q_pfp_normalization_descent_executionproductcoefficientdiagonalentry. pfc_terms_code_normalization_descent_executionproductcoefficient = ff_q_pfp_normalization_descent_executionproductcoefficientdiagonalentry * S ((S (pfc_index_normalization_descent_executionproductcoefficientdiagonal)) * pfc_terms_scale_normalization_descent_executionproductcoefficient) + (pfc_value_normalization_descent_executionproductcoefficientdiagonal))) /\\ ((exists pfc_complement_normalization_descent_executionproductcoefficientdiagonalterm pfc_left_normalization_descent_executionproductcoefficientdiagonalterm pfc_right_normalization_descent_executionproductcoefficientdiagonalterm. (((pfc_index_normalization_descent_executionproductcoefficientdiagonal)+pfc_complement_normalization_descent_executionproductcoefficientdiagonalterm=(pfc_index_normalization_descent_executionproduct)) /\\ ((((((exists pfa_gap_normalization_descent_executionproductcoefficientdiagonaltermleftinside. pfa_gap_normalization_descent_executionproductcoefficientdiagonaltermleftinside + S (pfc_index_normalization_descent_executionproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_normalization_descent_executionproductcoefficientdiagonaltermleftentry. ff_h_pfp_normalization_descent_executionproductcoefficientdiagonaltermleftentry + S (pfc_left_normalization_descent_executionproductcoefficientdiagonalterm) = S ((S (pfc_index_normalization_descent_executionproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_normalization_descent_executionproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_normalization_descent_executionproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_normalization_descent_executionproductcoefficientdiagonal)) * qc) + (pfc_left_normalization_descent_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_descent_executionproductcoefficientdiagonaltermleftoutside. pfc_gap_normalization_descent_executionproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_normalization_descent_executionproductcoefficientdiagonal)) /\\ (((pfc_left_normalization_descent_executionproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalization_descent_executionproductcoefficientdiagonaltermrightinside. pfa_gap_normalization_descent_executionproductcoefficientdiagonaltermrightinside + S (pfc_complement_normalization_descent_executionproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_normalization_descent_executionproductcoefficientdiagonaltermrightentry. ff_h_pfp_normalization_descent_executionproductcoefficientdiagonaltermrightentry + S (pfc_right_normalization_descent_executionproductcoefficientdiagonalterm) = S ((S (pfc_complement_normalization_descent_executionproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_normalization_descent_executionproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_normalization_descent_executionproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalization_descent_executionproductcoefficientdiagonalterm)) * bc) + (pfc_right_normalization_descent_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_descent_executionproductcoefficientdiagonaltermrightoutside. pfc_gap_normalization_descent_executionproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_normalization_descent_executionproductcoefficientdiagonalterm)) /\\ (((pfc_right_normalization_descent_executionproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalization_descent_executionproductcoefficientdiagonal)=pfc_left_normalization_descent_executionproductcoefficientdiagonalterm*pfc_right_normalization_descent_executionproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalization_descent_executionproductcoefficientsum fs_v_pfc_normalization_descent_executionproductcoefficientsum. ((((exists fs_h_pfc_normalization_descent_executionproductcoefficientsum_body_start. fs_h_pfc_normalization_descent_executionproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalization_descent_executionproductcoefficientsum)) /\\ exists fs_q_pfc_normalization_descent_executionproductcoefficientsum_body_start. fs_u_pfc_normalization_descent_executionproductcoefficientsum = fs_q_pfc_normalization_descent_executionproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalization_descent_executionproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalization_descent_executionproductcoefficientsum_body_terminal. fs_h_pfc_normalization_descent_executionproductcoefficientsum_body_terminal + S (pfc_natural_sum_normalization_descent_executionproductcoefficient) = S ((S (S (pfc_index_normalization_descent_executionproduct))) * fs_v_pfc_normalization_descent_executionproductcoefficientsum)) /\\ exists fs_q_pfc_normalization_descent_executionproductcoefficientsum_body_terminal. fs_u_pfc_normalization_descent_executionproductcoefficientsum = fs_q_pfc_normalization_descent_executionproductcoefficientsum_body_terminal * S ((S (S (pfc_index_normalization_descent_executionproduct))) * fs_v_pfc_normalization_descent_executionproductcoefficientsum) + (pfc_natural_sum_normalization_descent_executionproductcoefficient))) /\\ forall fs_i_pfc_normalization_descent_executionproductcoefficientsum_body_steps. (exists fs_lt_pfc_normalization_descent_executionproductcoefficientsum_body_steps_bound. fs_lt_pfc_normalization_descent_executionproductcoefficientsum_body_steps_bound + S fs_i_pfc_normalization_descent_executionproductcoefficientsum_body_steps = S (pfc_index_normalization_descent_executionproduct)) -> exists fs_a_pfc_normalization_descent_executionproductcoefficientsum_body_steps fs_r_pfc_normalization_descent_executionproductcoefficientsum_body_steps fs_s_pfc_normalization_descent_executionproductcoefficientsum_body_steps. ((((exists fs_h_pfc_normalization_descent_executionproductcoefficientsum_body_steps_summand. fs_h_pfc_normalization_descent_executionproductcoefficientsum_body_steps_summand + S (fs_a_pfc_normalization_descent_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_normalization_descent_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_normalization_descent_executionproductcoefficient)) /\\ exists fs_q_pfc_normalization_descent_executionproductcoefficientsum_body_steps_summand. pfc_terms_code_normalization_descent_executionproductcoefficient = fs_q_pfc_normalization_descent_executionproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalization_descent_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_normalization_descent_executionproductcoefficient) + (fs_a_pfc_normalization_descent_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_descent_executionproductcoefficientsum_body_steps_partial. fs_h_pfc_normalization_descent_executionproductcoefficientsum_body_steps_partial + S (fs_r_pfc_normalization_descent_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_normalization_descent_executionproductcoefficientsum_body_steps)) * fs_v_pfc_normalization_descent_executionproductcoefficientsum)) /\\ exists fs_q_pfc_normalization_descent_executionproductcoefficientsum_body_steps_partial. fs_u_pfc_normalization_descent_executionproductcoefficientsum = fs_q_pfc_normalization_descent_executionproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalization_descent_executionproductcoefficientsum_body_steps)) * fs_v_pfc_normalization_descent_executionproductcoefficientsum) + (fs_r_pfc_normalization_descent_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_descent_executionproductcoefficientsum_body_steps_successor. fs_h_pfc_normalization_descent_executionproductcoefficientsum_body_steps_successor + S (fs_s_pfc_normalization_descent_executionproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalization_descent_executionproductcoefficientsum_body_steps)) * fs_v_pfc_normalization_descent_executionproductcoefficientsum)) /\\ exists fs_q_pfc_normalization_descent_executionproductcoefficientsum_body_steps_successor. fs_u_pfc_normalization_descent_executionproductcoefficientsum = fs_q_pfc_normalization_descent_executionproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalization_descent_executionproductcoefficientsum_body_steps)) * fs_v_pfc_normalization_descent_executionproductcoefficientsum) + (fs_s_pfc_normalization_descent_executionproductcoefficientsum_body_steps))) /\\ fs_s_pfc_normalization_descent_executionproductcoefficientsum_body_steps = fs_r_pfc_normalization_descent_executionproductcoefficientsum_body_steps + fs_a_pfc_normalization_descent_executionproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalization_descent_executionproductcoefficientresiduebound. pfa_gap_normalization_descent_executionproductcoefficientresiduebound + S (pfc_value_normalization_descent_executionproduct) = (p)) /\\ ((exists pfa_offset_left_normalization_descent_executionproductcoefficientresiduecongruence pfa_offset_right_normalization_descent_executionproductcoefficientresiduecongruence. (pfc_natural_sum_normalization_descent_executionproductcoefficient) + (p) * pfa_offset_left_normalization_descent_executionproductcoefficientresiduecongruence = (pfc_value_normalization_descent_executionproduct) + (p) * pfa_offset_right_normalization_descent_executionproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_normalization_descent_executiondifference. (exists pfa_gap_normalization_descent_executiondifferenceindex. pfa_gap_normalization_descent_executiondifferenceindex + S (pfs_index_normalization_descent_executiondifference) = (L)) -> exists pfs_left_normalization_descent_executiondifference pfs_right_normalization_descent_executiondifference pfs_result_normalization_descent_executiondifference. ((((exists ff_h_pfp_normalization_descent_executiondifferenceleft. ff_h_pfp_normalization_descent_executiondifferenceleft + S (pfs_left_normalization_descent_executiondifference) = S ((S (pfs_index_normalization_descent_executiondifference)) * ac)) /\\ exists ff_q_pfp_normalization_descent_executiondifferenceleft. ab = ff_q_pfp_normalization_descent_executiondifferenceleft * S ((S (pfs_index_normalization_descent_executiondifference)) * ac) + (pfs_left_normalization_descent_executiondifference))) /\\ (((((exists ff_h_pfp_normalization_descent_executiondifferenceright. ff_h_pfp_normalization_descent_executiondifferenceright + S (pfs_right_normalization_descent_executiondifference) = S ((S (pfs_index_normalization_descent_executiondifference)) * pfd_product_scale_normalization_descent_execution)) /\\ exists ff_q_pfp_normalization_descent_executiondifferenceright. pfd_product_code_normalization_descent_execution = ff_q_pfp_normalization_descent_executiondifferenceright * S ((S (pfs_index_normalization_descent_executiondifference)) * pfd_product_scale_normalization_descent_execution) + (pfs_right_normalization_descent_executiondifference))) /\\ (((((exists ff_h_pfp_normalization_descent_executiondifferenceresult. ff_h_pfp_normalization_descent_executiondifferenceresult + S (pfs_result_normalization_descent_executiondifference) = S ((S (pfs_index_normalization_descent_executiondifference)) * pfd_residual_scale_normalization_descent_execution)) /\\ exists ff_q_pfp_normalization_descent_executiondifferenceresult. pfd_residual_code_normalization_descent_execution = ff_q_pfp_normalization_descent_executiondifferenceresult * S ((S (pfs_index_normalization_descent_executiondifference)) * pfd_residual_scale_normalization_descent_execution) + (pfs_result_normalization_descent_executiondifference))) /\\ ((((exists pfa_gap_normalization_descent_executiondifferenceoperationleft. pfa_gap_normalization_descent_executiondifferenceoperationleft + S (pfs_right_normalization_descent_executiondifference) = (p)) /\\ (((exists pfa_gap_normalization_descent_executiondifferenceoperationright. pfa_gap_normalization_descent_executiondifferenceoperationright + S (pfs_result_normalization_descent_executiondifference) = (p)) /\\ ((((exists pfa_gap_normalization_descent_executiondifferenceoperationresultbound. pfa_gap_normalization_descent_executiondifferenceoperationresultbound + S (pfs_left_normalization_descent_executiondifference) = (p)) /\\ ((exists pfa_offset_left_normalization_descent_executiondifferenceoperationresultcongruence pfa_offset_right_normalization_descent_executiondifferenceoperationresultcongruence. ((pfs_right_normalization_descent_executiondifference) + (pfs_result_normalization_descent_executiondifference)) + (p) * pfa_offset_left_normalization_descent_executiondifferenceoperationresultcongruence = (pfs_left_normalization_descent_executiondifference) + (p) * pfa_offset_right_normalization_descent_executiondifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_normalization_descent_execution)+(R)) /\\ (((forall fom_index_pfp_normalization_descent_executiontriminput. (exists fom_gap_pfp_normalization_descent_executiontriminput_index_bound. fom_gap_pfp_normalization_descent_executiontriminput_index_bound + S (fom_index_pfp_normalization_descent_executiontriminput) = L) -> exists fom_value_pfp_normalization_descent_executiontriminput. ((((exists fom_beta_height_pfp_normalization_descent_executiontriminput_entry. fom_beta_height_pfp_normalization_descent_executiontriminput_entry + S (fom_value_pfp_normalization_descent_executiontriminput) = S ((S (fom_index_pfp_normalization_descent_executiontriminput)) * pfd_residual_scale_normalization_descent_execution)) /\\ exists fom_beta_quotient_pfp_normalization_descent_executiontriminput_entry. pfd_residual_code_normalization_descent_execution = fom_beta_quotient_pfp_normalization_descent_executiontriminput_entry * S ((S (fom_index_pfp_normalization_descent_executiontriminput)) * pfd_residual_scale_normalization_descent_execution) + (fom_value_pfp_normalization_descent_executiontriminput))) /\\ (exists fom_gap_pfp_normalization_descent_executiontriminput_value_bound. fom_gap_pfp_normalization_descent_executiontriminput_value_bound + S (fom_value_pfp_normalization_descent_executiontriminput) = p))) /\\ (((forall pfp_repeat_index_normalization_descent_executiontrimremoved. (exists pfa_gap_normalization_descent_executiontrimremovedindex. pfa_gap_normalization_descent_executiontrimremovedindex + S (pfp_repeat_index_normalization_descent_executiontrimremoved) = (pfd_cut_normalization_descent_execution)) -> (((exists ff_h_pfp_normalization_descent_executiontrimremovedentry. ff_h_pfp_normalization_descent_executiontrimremovedentry + S (0) = S ((S (pfp_repeat_index_normalization_descent_executiontrimremoved)) * pfd_residual_scale_normalization_descent_execution)) /\\ exists ff_q_pfp_normalization_descent_executiontrimremovedentry. pfd_residual_code_normalization_descent_execution = ff_q_pfp_normalization_descent_executiontrimremovedentry * S ((S (pfp_repeat_index_normalization_descent_executiontrimremoved)) * pfd_residual_scale_normalization_descent_execution) + (0)))) /\\ (((forall pftrim_index_normalization_descent_executiontrimsuffix pftrim_value_normalization_descent_executiontrimsuffix. (exists pfa_gap_normalization_descent_executiontrimsuffixbound. pfa_gap_normalization_descent_executiontrimsuffixbound + S (pftrim_index_normalization_descent_executiontrimsuffix) = (R)) -> (((exists ff_h_pfp_normalization_descent_executiontrimsuffixsource. ff_h_pfp_normalization_descent_executiontrimsuffixsource + S (pftrim_value_normalization_descent_executiontrimsuffix) = S ((S ((pfd_cut_normalization_descent_execution)+pftrim_index_normalization_descent_executiontrimsuffix)) * pfd_residual_scale_normalization_descent_execution)) /\\ exists ff_q_pfp_normalization_descent_executiontrimsuffixsource. pfd_residual_code_normalization_descent_execution = ff_q_pfp_normalization_descent_executiontrimsuffixsource * S ((S ((pfd_cut_normalization_descent_execution)+pftrim_index_normalization_descent_executiontrimsuffix)) * pfd_residual_scale_normalization_descent_execution) + (pftrim_value_normalization_descent_executiontrimsuffix))) -> (((exists ff_h_pfp_normalization_descent_executiontrimsuffixoutput. ff_h_pfp_normalization_descent_executiontrimsuffixoutput + S (pftrim_value_normalization_descent_executiontrimsuffix) = S ((S (pftrim_index_normalization_descent_executiontrimsuffix)) * rc)) /\\ exists ff_q_pfp_normalization_descent_executiontrimsuffixoutput. rb = ff_q_pfp_normalization_descent_executiontrimsuffixoutput * S ((S (pftrim_index_normalization_descent_executiontrimsuffix)) * rc) + (pftrim_value_normalization_descent_executiontrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_normalization_descent_executiontrimnormal. ((((exists ff_h_pfp_normalization_descent_executiontrimnormalentry. ff_h_pfp_normalization_descent_executiontrimnormalentry + S (pftrim_leading_normalization_descent_executiontrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_normalization_descent_executiontrimnormalentry. rb = ff_q_pfp_normalization_descent_executiontrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_normalization_descent_executiontrimnormal))) /\\ ((~(pftrim_leading_normalization_descent_executiontrimnormal=0))))))))))))))))))))))))))))))))) -> (((exists pfc_gap_normalization_descent_weak. pfc_gap_normalization_descent_weak+(R)=(d)) /\\ ((exists pfa_gap_normalization_descent_strict. pfa_gap_normalization_descent_strict + S (R) = (S d)))))", "statement_sha256": "0dbfd097e92907da10ba575fae7c7517cb05e8f5b8fcd8950e62f1cd809bce20", "summary": "Every actual normalized remainder has retained length at most d and strictly less than the actual divisor length S d. 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0, "name": "prime_field_polynomial_division_constant_remainder_empty", "proof_depth": 34, "proof_edges": 37, "proof_nodes": 38, "proof_objects": 38, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "prime_field_polynomial_division_remainder_length_descent", "le_zero" ], "dependencies_sha256": "395314bed3ee65a750780e85e74dcc52ff4cf351d0396fda8fa59bb8dc8c27e1", "empty_context_closure": { "body_proof_depth": 34, "body_proof_nodes": 38, "bundle_campaign": "polynomial-gcd-bezout", "bundle_dependency_edge_count": 1578, "bundle_node_count": 493, "bundle_node_id": 416, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "bundle_root_id": 492, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "closure_kind": "dependency_closed_bundle_node", "digest_kind": 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"artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "f8ec117f8eba4715b36920d4c3f724a749e7561f56de83751e70f35a63df80e6", "membership": "alpha_only", "name": "prime_field_polynomial_division_constant_remainder_empty", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro qb", "intro qc", "intro q", "intro rb", "intro rc", "intro R", "intro hp", "intro hexecution", "have hbound : ((exists pfc_gap_normalization_constant_bound. pfc_gap_normalization_constant_bound+(R)=(0)) /\\ ((exists pfa_gap_normalization_constant_strict. pfa_gap_normalization_constant_strict + S (R) = (S 0))))", "specialize prime_field_polynomial_division_remainder_length_descent (p)", "specialize prime_field_polynomial_division_remainder_length_descent (ab)", "specialize prime_field_polynomial_division_remainder_length_descent (ac)", "specialize prime_field_polynomial_division_remainder_length_descent (L)", "specialize prime_field_polynomial_division_remainder_length_descent (bb)", "specialize prime_field_polynomial_division_remainder_length_descent (bc)", "specialize prime_field_polynomial_division_remainder_length_descent (0)", "specialize prime_field_polynomial_division_remainder_length_descent (qb)", "specialize prime_field_polynomial_division_remainder_length_descent (qc)", "specialize prime_field_polynomial_division_remainder_length_descent (q)", "specialize prime_field_polynomial_division_remainder_length_descent (rb)", "specialize prime_field_polynomial_division_remainder_length_descent (rc)", "specialize prime_field_polynomial_division_remainder_length_descent (R)", "apply prime_field_polynomial_division_remainder_length_descent", "exact hp", "exact hexecution", "cases hbound", "specialize le_zero (R)", "apply le_zero", "exact hbound_left" ], "script_sha256": "0ba7d804fd54e1cfcd4121497899054ac448a2232c674106579a28c41e6d14fe", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_euclidean_normalization_candidate.py", "sha256": "d2cddfe42dc0d22104dc4e85e95116222914df11ac840d2082a4ff2e462f146f" }, "statement": "forall p ab ac L bb bc qb qc q rb rc R. (~((p) = 1) /\\ forall pfa_factor_left_normalization_constant_prime pfa_factor_right_normalization_constant_prime. (p) = pfa_factor_left_normalization_constant_prime * pfa_factor_right_normalization_constant_prime -> pfa_factor_left_normalization_constant_prime = 1 \\/ pfa_factor_right_normalization_constant_prime = 1) -> (((forall fom_index_pfp_normalization_constant_executioninput. (exists fom_gap_pfp_normalization_constant_executioninput_index_bound. fom_gap_pfp_normalization_constant_executioninput_index_bound + S (fom_index_pfp_normalization_constant_executioninput) = L) -> exists fom_value_pfp_normalization_constant_executioninput. ((((exists fom_beta_height_pfp_normalization_constant_executioninput_entry. fom_beta_height_pfp_normalization_constant_executioninput_entry + S (fom_value_pfp_normalization_constant_executioninput) = S ((S (fom_index_pfp_normalization_constant_executioninput)) * ac)) /\\ exists fom_beta_quotient_pfp_normalization_constant_executioninput_entry. ab = fom_beta_quotient_pfp_normalization_constant_executioninput_entry * S ((S (fom_index_pfp_normalization_constant_executioninput)) * ac) + (fom_value_pfp_normalization_constant_executioninput))) /\\ (exists fom_gap_pfp_normalization_constant_executioninput_value_bound. fom_gap_pfp_normalization_constant_executioninput_value_bound + S (fom_value_pfp_normalization_constant_executioninput) = p))) /\\ (((forall fom_index_pfp_normalization_constant_executiondivisor. (exists fom_gap_pfp_normalization_constant_executiondivisor_index_bound. fom_gap_pfp_normalization_constant_executiondivisor_index_bound + S (fom_index_pfp_normalization_constant_executiondivisor) = S (0)) -> exists fom_value_pfp_normalization_constant_executiondivisor. ((((exists fom_beta_height_pfp_normalization_constant_executiondivisor_entry. fom_beta_height_pfp_normalization_constant_executiondivisor_entry + S (fom_value_pfp_normalization_constant_executiondivisor) = S ((S (fom_index_pfp_normalization_constant_executiondivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_normalization_constant_executiondivisor_entry. bb = fom_beta_quotient_pfp_normalization_constant_executiondivisor_entry * S ((S (fom_index_pfp_normalization_constant_executiondivisor)) * bc) + (fom_value_pfp_normalization_constant_executiondivisor))) /\\ (exists fom_gap_pfp_normalization_constant_executiondivisor_value_bound. fom_gap_pfp_normalization_constant_executiondivisor_value_bound + S (fom_value_pfp_normalization_constant_executiondivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_normalization_constant_executionlengthshort. pfc_gap_normalization_constant_executionlengthshort+(L)=(0))))) \\/ (((~((q)=0)) /\\ (((q)+(0)=(L)))))) /\\ ((exists pfd_head_normalization_constant_execution pfd_inverse_normalization_constant_execution pfd_product_code_normalization_constant_execution pfd_product_scale_normalization_constant_execution pfd_residual_code_normalization_constant_execution pfd_residual_scale_normalization_constant_execution pfd_cut_normalization_constant_execution. ((((exists ff_h_pfp_normalization_constant_executionhead. ff_h_pfp_normalization_constant_executionhead + S (pfd_head_normalization_constant_execution) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_normalization_constant_executionhead. bb = ff_q_pfp_normalization_constant_executionhead * S ((S (0)) * bc) + (pfd_head_normalization_constant_execution))) /\\ (((((~((pfd_head_normalization_constant_execution) = 0)) /\\ ((((exists pfa_gap_normalization_constant_executioninversemultiplicationleft. pfa_gap_normalization_constant_executioninversemultiplicationleft + S (pfd_head_normalization_constant_execution) = (p)) /\\ (((exists pfa_gap_normalization_constant_executioninversemultiplicationright. pfa_gap_normalization_constant_executioninversemultiplicationright + S (pfd_inverse_normalization_constant_execution) = (p)) /\\ ((((exists pfa_gap_normalization_constant_executioninversemultiplicationresultbound. pfa_gap_normalization_constant_executioninversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_normalization_constant_executioninversemultiplicationresultcongruence pfa_offset_right_normalization_constant_executioninversemultiplicationresultcongruence. ((pfd_head_normalization_constant_execution) * (pfd_inverse_normalization_constant_execution)) + (p) * pfa_offset_left_normalization_constant_executioninversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_constant_executioninversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_normalization_constant_executionquotient. (exists pfa_gap_normalization_constant_executionquotientbound. pfa_gap_normalization_constant_executionquotientbound + S (pfd_index_normalization_constant_executionquotient) = (q)) -> exists pfd_value_normalization_constant_executionquotient. ((((exists ff_h_pfp_normalization_constant_executionquotiententry. ff_h_pfp_normalization_constant_executionquotiententry + S (pfd_value_normalization_constant_executionquotient) = S ((S (pfd_index_normalization_constant_executionquotient)) * qc)) /\\ exists ff_q_pfp_normalization_constant_executionquotiententry. qb = ff_q_pfp_normalization_constant_executionquotiententry * S ((S (pfd_index_normalization_constant_executionquotient)) * qc) + (pfd_value_normalization_constant_executionquotient))) /\\ ((exists pfd_input_normalization_constant_executionquotientstep pfd_previous_normalization_constant_executionquotientstep pfd_difference_normalization_constant_executionquotientstep. ((((exists ff_h_pfp_normalization_constant_executionquotientstepinput. ff_h_pfp_normalization_constant_executionquotientstepinput + S (pfd_input_normalization_constant_executionquotientstep) = S ((S (pfd_index_normalization_constant_executionquotient)) * ac)) /\\ exists ff_q_pfp_normalization_constant_executionquotientstepinput. ab = ff_q_pfp_normalization_constant_executionquotientstepinput * S ((S (pfd_index_normalization_constant_executionquotient)) * ac) + (pfd_input_normalization_constant_executionquotientstep))) /\\ (((exists pfc_terms_code_normalization_constant_executionquotientstepprevious pfc_terms_scale_normalization_constant_executionquotientstepprevious pfc_natural_sum_normalization_constant_executionquotientstepprevious. ((forall pfc_index_normalization_constant_executionquotientsteppreviousdiagonal. (exists pfa_gap_normalization_constant_executionquotientsteppreviousdiagonalbound. pfa_gap_normalization_constant_executionquotientsteppreviousdiagonalbound + S (pfc_index_normalization_constant_executionquotientsteppreviousdiagonal) = (S (pfd_index_normalization_constant_executionquotient))) -> exists pfc_value_normalization_constant_executionquotientsteppreviousdiagonal. ((((exists ff_h_pfp_normalization_constant_executionquotientsteppreviousdiagonalentry. ff_h_pfp_normalization_constant_executionquotientsteppreviousdiagonalentry + S (pfc_value_normalization_constant_executionquotientsteppreviousdiagonal) = S ((S (pfc_index_normalization_constant_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_normalization_constant_executionquotientstepprevious)) /\\ exists ff_q_pfp_normalization_constant_executionquotientsteppreviousdiagonalentry. pfc_terms_code_normalization_constant_executionquotientstepprevious = ff_q_pfp_normalization_constant_executionquotientsteppreviousdiagonalentry * S ((S (pfc_index_normalization_constant_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_normalization_constant_executionquotientstepprevious) + (pfc_value_normalization_constant_executionquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_normalization_constant_executionquotientsteppreviousdiagonalterm pfc_left_normalization_constant_executionquotientsteppreviousdiagonalterm pfc_right_normalization_constant_executionquotientsteppreviousdiagonalterm. (((pfc_index_normalization_constant_executionquotientsteppreviousdiagonal)+pfc_complement_normalization_constant_executionquotientsteppreviousdiagonalterm=(pfd_index_normalization_constant_executionquotient)) /\\ ((((((exists pfa_gap_normalization_constant_executionquotientsteppreviousdiagonaltermleftinside. pfa_gap_normalization_constant_executionquotientsteppreviousdiagonaltermleftinside + S (pfc_index_normalization_constant_executionquotientsteppreviousdiagonal) = (pfd_index_normalization_constant_executionquotient)) /\\ ((((exists ff_h_pfp_normalization_constant_executionquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_normalization_constant_executionquotientsteppreviousdiagonaltermleftentry + S (pfc_left_normalization_constant_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_index_normalization_constant_executionquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_normalization_constant_executionquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_normalization_constant_executionquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_normalization_constant_executionquotientsteppreviousdiagonal)) * qc) + (pfc_left_normalization_constant_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_constant_executionquotientsteppreviousdiagonaltermleftoutside. pfc_gap_normalization_constant_executionquotientsteppreviousdiagonaltermleftoutside+(pfd_index_normalization_constant_executionquotient)=(pfc_index_normalization_constant_executionquotientsteppreviousdiagonal)) /\\ (((pfc_left_normalization_constant_executionquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalization_constant_executionquotientsteppreviousdiagonaltermrightinside. pfa_gap_normalization_constant_executionquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_normalization_constant_executionquotientsteppreviousdiagonalterm) = (S (0))) /\\ ((((exists ff_h_pfp_normalization_constant_executionquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_normalization_constant_executionquotientsteppreviousdiagonaltermrightentry + S (pfc_right_normalization_constant_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_normalization_constant_executionquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_normalization_constant_executionquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_normalization_constant_executionquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_normalization_constant_executionquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_normalization_constant_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_constant_executionquotientsteppreviousdiagonaltermrightoutside. pfc_gap_normalization_constant_executionquotientsteppreviousdiagonaltermrightoutside+(S (0))=(pfc_complement_normalization_constant_executionquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_normalization_constant_executionquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_normalization_constant_executionquotientsteppreviousdiagonal)=pfc_left_normalization_constant_executionquotientsteppreviousdiagonalterm*pfc_right_normalization_constant_executionquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalization_constant_executionquotientstepprevioussum fs_v_pfc_normalization_constant_executionquotientstepprevioussum. ((((exists fs_h_pfc_normalization_constant_executionquotientstepprevioussum_body_start. fs_h_pfc_normalization_constant_executionquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalization_constant_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_normalization_constant_executionquotientstepprevioussum_body_start. fs_u_pfc_normalization_constant_executionquotientstepprevioussum = fs_q_pfc_normalization_constant_executionquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_normalization_constant_executionquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_normalization_constant_executionquotientstepprevioussum_body_terminal. fs_h_pfc_normalization_constant_executionquotientstepprevioussum_body_terminal + S (pfc_natural_sum_normalization_constant_executionquotientstepprevious) = S ((S (S (pfd_index_normalization_constant_executionquotient))) * fs_v_pfc_normalization_constant_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_normalization_constant_executionquotientstepprevioussum_body_terminal. fs_u_pfc_normalization_constant_executionquotientstepprevioussum = fs_q_pfc_normalization_constant_executionquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_normalization_constant_executionquotient))) * fs_v_pfc_normalization_constant_executionquotientstepprevioussum) + (pfc_natural_sum_normalization_constant_executionquotientstepprevious))) /\\ forall fs_i_pfc_normalization_constant_executionquotientstepprevioussum_body_steps. (exists fs_lt_pfc_normalization_constant_executionquotientstepprevioussum_body_steps_bound. fs_lt_pfc_normalization_constant_executionquotientstepprevioussum_body_steps_bound + S fs_i_pfc_normalization_constant_executionquotientstepprevioussum_body_steps = S (pfd_index_normalization_constant_executionquotient)) -> exists fs_a_pfc_normalization_constant_executionquotientstepprevioussum_body_steps fs_r_pfc_normalization_constant_executionquotientstepprevioussum_body_steps fs_s_pfc_normalization_constant_executionquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_normalization_constant_executionquotientstepprevioussum_body_steps_summand. fs_h_pfc_normalization_constant_executionquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_normalization_constant_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_normalization_constant_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_normalization_constant_executionquotientstepprevious)) /\\ exists fs_q_pfc_normalization_constant_executionquotientstepprevioussum_body_steps_summand. pfc_terms_code_normalization_constant_executionquotientstepprevious = fs_q_pfc_normalization_constant_executionquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_normalization_constant_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_normalization_constant_executionquotientstepprevious) + (fs_a_pfc_normalization_constant_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_constant_executionquotientstepprevioussum_body_steps_partial. fs_h_pfc_normalization_constant_executionquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_normalization_constant_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_normalization_constant_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_normalization_constant_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_normalization_constant_executionquotientstepprevioussum_body_steps_partial. fs_u_pfc_normalization_constant_executionquotientstepprevioussum = fs_q_pfc_normalization_constant_executionquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_normalization_constant_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_normalization_constant_executionquotientstepprevioussum) + (fs_r_pfc_normalization_constant_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_constant_executionquotientstepprevioussum_body_steps_successor. fs_h_pfc_normalization_constant_executionquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_normalization_constant_executionquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_normalization_constant_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_normalization_constant_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_normalization_constant_executionquotientstepprevioussum_body_steps_successor. fs_u_pfc_normalization_constant_executionquotientstepprevioussum = fs_q_pfc_normalization_constant_executionquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_normalization_constant_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_normalization_constant_executionquotientstepprevioussum) + (fs_s_pfc_normalization_constant_executionquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_normalization_constant_executionquotientstepprevioussum_body_steps = fs_r_pfc_normalization_constant_executionquotientstepprevioussum_body_steps + fs_a_pfc_normalization_constant_executionquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_normalization_constant_executionquotientsteppreviousresiduebound. pfa_gap_normalization_constant_executionquotientsteppreviousresiduebound + S (pfd_previous_normalization_constant_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_normalization_constant_executionquotientsteppreviousresiduecongruence pfa_offset_right_normalization_constant_executionquotientsteppreviousresiduecongruence. (pfc_natural_sum_normalization_constant_executionquotientstepprevious) + (p) * pfa_offset_left_normalization_constant_executionquotientsteppreviousresiduecongruence = (pfd_previous_normalization_constant_executionquotientstep) + (p) * pfa_offset_right_normalization_constant_executionquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_normalization_constant_executionquotientstepsubtractleft. pfa_gap_normalization_constant_executionquotientstepsubtractleft + S (pfd_previous_normalization_constant_executionquotientstep) = (p)) /\\ (((exists pfa_gap_normalization_constant_executionquotientstepsubtractright. pfa_gap_normalization_constant_executionquotientstepsubtractright + S (pfd_difference_normalization_constant_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_normalization_constant_executionquotientstepsubtractresultbound. pfa_gap_normalization_constant_executionquotientstepsubtractresultbound + S (pfd_input_normalization_constant_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_normalization_constant_executionquotientstepsubtractresultcongruence pfa_offset_right_normalization_constant_executionquotientstepsubtractresultcongruence. ((pfd_previous_normalization_constant_executionquotientstep) + (pfd_difference_normalization_constant_executionquotientstep)) + (p) * pfa_offset_left_normalization_constant_executionquotientstepsubtractresultcongruence = (pfd_input_normalization_constant_executionquotientstep) + (p) * pfa_offset_right_normalization_constant_executionquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_normalization_constant_executionquotientstepmultiplyleft. pfa_gap_normalization_constant_executionquotientstepmultiplyleft + S (pfd_inverse_normalization_constant_execution) = (p)) /\\ (((exists pfa_gap_normalization_constant_executionquotientstepmultiplyright. pfa_gap_normalization_constant_executionquotientstepmultiplyright + S (pfd_difference_normalization_constant_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_normalization_constant_executionquotientstepmultiplyresultbound. pfa_gap_normalization_constant_executionquotientstepmultiplyresultbound + S (pfd_value_normalization_constant_executionquotient) = (p)) /\\ ((exists pfa_offset_left_normalization_constant_executionquotientstepmultiplyresultcongruence pfa_offset_right_normalization_constant_executionquotientstepmultiplyresultcongruence. ((pfd_inverse_normalization_constant_execution) * (pfd_difference_normalization_constant_executionquotientstep)) + (p) * pfa_offset_left_normalization_constant_executionquotientstepmultiplyresultcongruence = (pfd_value_normalization_constant_executionquotient) + (p) * pfa_offset_right_normalization_constant_executionquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_normalization_constant_executionproduct. (exists pfa_gap_normalization_constant_executionproductbound. pfa_gap_normalization_constant_executionproductbound + S (pfc_index_normalization_constant_executionproduct) = (L)) -> exists pfc_value_normalization_constant_executionproduct. ((((exists ff_h_pfp_normalization_constant_executionproductentry. ff_h_pfp_normalization_constant_executionproductentry + S (pfc_value_normalization_constant_executionproduct) = S ((S (pfc_index_normalization_constant_executionproduct)) * pfd_product_scale_normalization_constant_execution)) /\\ exists ff_q_pfp_normalization_constant_executionproductentry. pfd_product_code_normalization_constant_execution = ff_q_pfp_normalization_constant_executionproductentry * S ((S (pfc_index_normalization_constant_executionproduct)) * pfd_product_scale_normalization_constant_execution) + (pfc_value_normalization_constant_executionproduct))) /\\ ((exists pfc_terms_code_normalization_constant_executionproductcoefficient pfc_terms_scale_normalization_constant_executionproductcoefficient pfc_natural_sum_normalization_constant_executionproductcoefficient. ((forall pfc_index_normalization_constant_executionproductcoefficientdiagonal. (exists pfa_gap_normalization_constant_executionproductcoefficientdiagonalbound. pfa_gap_normalization_constant_executionproductcoefficientdiagonalbound + S (pfc_index_normalization_constant_executionproductcoefficientdiagonal) = (S (pfc_index_normalization_constant_executionproduct))) -> exists pfc_value_normalization_constant_executionproductcoefficientdiagonal. ((((exists ff_h_pfp_normalization_constant_executionproductcoefficientdiagonalentry. ff_h_pfp_normalization_constant_executionproductcoefficientdiagonalentry + S (pfc_value_normalization_constant_executionproductcoefficientdiagonal) = S ((S (pfc_index_normalization_constant_executionproductcoefficientdiagonal)) * pfc_terms_scale_normalization_constant_executionproductcoefficient)) /\\ exists ff_q_pfp_normalization_constant_executionproductcoefficientdiagonalentry. pfc_terms_code_normalization_constant_executionproductcoefficient = ff_q_pfp_normalization_constant_executionproductcoefficientdiagonalentry * S ((S (pfc_index_normalization_constant_executionproductcoefficientdiagonal)) * pfc_terms_scale_normalization_constant_executionproductcoefficient) + (pfc_value_normalization_constant_executionproductcoefficientdiagonal))) /\\ ((exists pfc_complement_normalization_constant_executionproductcoefficientdiagonalterm pfc_left_normalization_constant_executionproductcoefficientdiagonalterm pfc_right_normalization_constant_executionproductcoefficientdiagonalterm. (((pfc_index_normalization_constant_executionproductcoefficientdiagonal)+pfc_complement_normalization_constant_executionproductcoefficientdiagonalterm=(pfc_index_normalization_constant_executionproduct)) /\\ ((((((exists pfa_gap_normalization_constant_executionproductcoefficientdiagonaltermleftinside. pfa_gap_normalization_constant_executionproductcoefficientdiagonaltermleftinside + S (pfc_index_normalization_constant_executionproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_normalization_constant_executionproductcoefficientdiagonaltermleftentry. ff_h_pfp_normalization_constant_executionproductcoefficientdiagonaltermleftentry + S (pfc_left_normalization_constant_executionproductcoefficientdiagonalterm) = S ((S (pfc_index_normalization_constant_executionproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_normalization_constant_executionproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_normalization_constant_executionproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_normalization_constant_executionproductcoefficientdiagonal)) * qc) + (pfc_left_normalization_constant_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_constant_executionproductcoefficientdiagonaltermleftoutside. pfc_gap_normalization_constant_executionproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_normalization_constant_executionproductcoefficientdiagonal)) /\\ (((pfc_left_normalization_constant_executionproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalization_constant_executionproductcoefficientdiagonaltermrightinside. pfa_gap_normalization_constant_executionproductcoefficientdiagonaltermrightinside + S (pfc_complement_normalization_constant_executionproductcoefficientdiagonalterm) = (S (0))) /\\ ((((exists ff_h_pfp_normalization_constant_executionproductcoefficientdiagonaltermrightentry. ff_h_pfp_normalization_constant_executionproductcoefficientdiagonaltermrightentry + S (pfc_right_normalization_constant_executionproductcoefficientdiagonalterm) = S ((S (pfc_complement_normalization_constant_executionproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_normalization_constant_executionproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_normalization_constant_executionproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalization_constant_executionproductcoefficientdiagonalterm)) * bc) + (pfc_right_normalization_constant_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_constant_executionproductcoefficientdiagonaltermrightoutside. pfc_gap_normalization_constant_executionproductcoefficientdiagonaltermrightoutside+(S (0))=(pfc_complement_normalization_constant_executionproductcoefficientdiagonalterm)) /\\ (((pfc_right_normalization_constant_executionproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalization_constant_executionproductcoefficientdiagonal)=pfc_left_normalization_constant_executionproductcoefficientdiagonalterm*pfc_right_normalization_constant_executionproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalization_constant_executionproductcoefficientsum fs_v_pfc_normalization_constant_executionproductcoefficientsum. ((((exists fs_h_pfc_normalization_constant_executionproductcoefficientsum_body_start. fs_h_pfc_normalization_constant_executionproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalization_constant_executionproductcoefficientsum)) /\\ exists fs_q_pfc_normalization_constant_executionproductcoefficientsum_body_start. fs_u_pfc_normalization_constant_executionproductcoefficientsum = fs_q_pfc_normalization_constant_executionproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalization_constant_executionproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalization_constant_executionproductcoefficientsum_body_terminal. fs_h_pfc_normalization_constant_executionproductcoefficientsum_body_terminal + S (pfc_natural_sum_normalization_constant_executionproductcoefficient) = S ((S (S (pfc_index_normalization_constant_executionproduct))) * fs_v_pfc_normalization_constant_executionproductcoefficientsum)) /\\ exists fs_q_pfc_normalization_constant_executionproductcoefficientsum_body_terminal. fs_u_pfc_normalization_constant_executionproductcoefficientsum = fs_q_pfc_normalization_constant_executionproductcoefficientsum_body_terminal * S ((S (S (pfc_index_normalization_constant_executionproduct))) * fs_v_pfc_normalization_constant_executionproductcoefficientsum) + (pfc_natural_sum_normalization_constant_executionproductcoefficient))) /\\ forall fs_i_pfc_normalization_constant_executionproductcoefficientsum_body_steps. (exists fs_lt_pfc_normalization_constant_executionproductcoefficientsum_body_steps_bound. fs_lt_pfc_normalization_constant_executionproductcoefficientsum_body_steps_bound + S fs_i_pfc_normalization_constant_executionproductcoefficientsum_body_steps = S (pfc_index_normalization_constant_executionproduct)) -> exists fs_a_pfc_normalization_constant_executionproductcoefficientsum_body_steps fs_r_pfc_normalization_constant_executionproductcoefficientsum_body_steps fs_s_pfc_normalization_constant_executionproductcoefficientsum_body_steps. ((((exists fs_h_pfc_normalization_constant_executionproductcoefficientsum_body_steps_summand. fs_h_pfc_normalization_constant_executionproductcoefficientsum_body_steps_summand + S (fs_a_pfc_normalization_constant_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_normalization_constant_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_normalization_constant_executionproductcoefficient)) /\\ exists fs_q_pfc_normalization_constant_executionproductcoefficientsum_body_steps_summand. pfc_terms_code_normalization_constant_executionproductcoefficient = fs_q_pfc_normalization_constant_executionproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalization_constant_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_normalization_constant_executionproductcoefficient) + (fs_a_pfc_normalization_constant_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_constant_executionproductcoefficientsum_body_steps_partial. fs_h_pfc_normalization_constant_executionproductcoefficientsum_body_steps_partial + S (fs_r_pfc_normalization_constant_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_normalization_constant_executionproductcoefficientsum_body_steps)) * fs_v_pfc_normalization_constant_executionproductcoefficientsum)) /\\ exists fs_q_pfc_normalization_constant_executionproductcoefficientsum_body_steps_partial. fs_u_pfc_normalization_constant_executionproductcoefficientsum = fs_q_pfc_normalization_constant_executionproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalization_constant_executionproductcoefficientsum_body_steps)) * fs_v_pfc_normalization_constant_executionproductcoefficientsum) + (fs_r_pfc_normalization_constant_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_constant_executionproductcoefficientsum_body_steps_successor. fs_h_pfc_normalization_constant_executionproductcoefficientsum_body_steps_successor + S (fs_s_pfc_normalization_constant_executionproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalization_constant_executionproductcoefficientsum_body_steps)) * fs_v_pfc_normalization_constant_executionproductcoefficientsum)) /\\ exists fs_q_pfc_normalization_constant_executionproductcoefficientsum_body_steps_successor. fs_u_pfc_normalization_constant_executionproductcoefficientsum = fs_q_pfc_normalization_constant_executionproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalization_constant_executionproductcoefficientsum_body_steps)) * fs_v_pfc_normalization_constant_executionproductcoefficientsum) + (fs_s_pfc_normalization_constant_executionproductcoefficientsum_body_steps))) /\\ fs_s_pfc_normalization_constant_executionproductcoefficientsum_body_steps = fs_r_pfc_normalization_constant_executionproductcoefficientsum_body_steps + fs_a_pfc_normalization_constant_executionproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalization_constant_executionproductcoefficientresiduebound. pfa_gap_normalization_constant_executionproductcoefficientresiduebound + S (pfc_value_normalization_constant_executionproduct) = (p)) /\\ ((exists pfa_offset_left_normalization_constant_executionproductcoefficientresiduecongruence pfa_offset_right_normalization_constant_executionproductcoefficientresiduecongruence. (pfc_natural_sum_normalization_constant_executionproductcoefficient) + (p) * pfa_offset_left_normalization_constant_executionproductcoefficientresiduecongruence = (pfc_value_normalization_constant_executionproduct) + (p) * pfa_offset_right_normalization_constant_executionproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_normalization_constant_executiondifference. (exists pfa_gap_normalization_constant_executiondifferenceindex. pfa_gap_normalization_constant_executiondifferenceindex + S (pfs_index_normalization_constant_executiondifference) = (L)) -> exists pfs_left_normalization_constant_executiondifference pfs_right_normalization_constant_executiondifference pfs_result_normalization_constant_executiondifference. ((((exists ff_h_pfp_normalization_constant_executiondifferenceleft. ff_h_pfp_normalization_constant_executiondifferenceleft + S (pfs_left_normalization_constant_executiondifference) = S ((S (pfs_index_normalization_constant_executiondifference)) * ac)) /\\ exists ff_q_pfp_normalization_constant_executiondifferenceleft. ab = ff_q_pfp_normalization_constant_executiondifferenceleft * S ((S (pfs_index_normalization_constant_executiondifference)) * ac) + (pfs_left_normalization_constant_executiondifference))) /\\ (((((exists ff_h_pfp_normalization_constant_executiondifferenceright. ff_h_pfp_normalization_constant_executiondifferenceright + S (pfs_right_normalization_constant_executiondifference) = S ((S (pfs_index_normalization_constant_executiondifference)) * pfd_product_scale_normalization_constant_execution)) /\\ exists ff_q_pfp_normalization_constant_executiondifferenceright. pfd_product_code_normalization_constant_execution = ff_q_pfp_normalization_constant_executiondifferenceright * S ((S (pfs_index_normalization_constant_executiondifference)) * pfd_product_scale_normalization_constant_execution) + (pfs_right_normalization_constant_executiondifference))) /\\ (((((exists ff_h_pfp_normalization_constant_executiondifferenceresult. ff_h_pfp_normalization_constant_executiondifferenceresult + S (pfs_result_normalization_constant_executiondifference) = S ((S (pfs_index_normalization_constant_executiondifference)) * pfd_residual_scale_normalization_constant_execution)) /\\ exists ff_q_pfp_normalization_constant_executiondifferenceresult. pfd_residual_code_normalization_constant_execution = ff_q_pfp_normalization_constant_executiondifferenceresult * S ((S (pfs_index_normalization_constant_executiondifference)) * pfd_residual_scale_normalization_constant_execution) + (pfs_result_normalization_constant_executiondifference))) /\\ ((((exists pfa_gap_normalization_constant_executiondifferenceoperationleft. pfa_gap_normalization_constant_executiondifferenceoperationleft + S (pfs_right_normalization_constant_executiondifference) = (p)) /\\ (((exists pfa_gap_normalization_constant_executiondifferenceoperationright. pfa_gap_normalization_constant_executiondifferenceoperationright + S (pfs_result_normalization_constant_executiondifference) = (p)) /\\ ((((exists pfa_gap_normalization_constant_executiondifferenceoperationresultbound. pfa_gap_normalization_constant_executiondifferenceoperationresultbound + S (pfs_left_normalization_constant_executiondifference) = (p)) /\\ ((exists pfa_offset_left_normalization_constant_executiondifferenceoperationresultcongruence pfa_offset_right_normalization_constant_executiondifferenceoperationresultcongruence. ((pfs_right_normalization_constant_executiondifference) + (pfs_result_normalization_constant_executiondifference)) + (p) * pfa_offset_left_normalization_constant_executiondifferenceoperationresultcongruence = (pfs_left_normalization_constant_executiondifference) + (p) * pfa_offset_right_normalization_constant_executiondifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_normalization_constant_execution)+(R)) /\\ (((forall fom_index_pfp_normalization_constant_executiontriminput. (exists fom_gap_pfp_normalization_constant_executiontriminput_index_bound. fom_gap_pfp_normalization_constant_executiontriminput_index_bound + S (fom_index_pfp_normalization_constant_executiontriminput) = L) -> exists fom_value_pfp_normalization_constant_executiontriminput. ((((exists fom_beta_height_pfp_normalization_constant_executiontriminput_entry. fom_beta_height_pfp_normalization_constant_executiontriminput_entry + S (fom_value_pfp_normalization_constant_executiontriminput) = S ((S (fom_index_pfp_normalization_constant_executiontriminput)) * pfd_residual_scale_normalization_constant_execution)) /\\ exists fom_beta_quotient_pfp_normalization_constant_executiontriminput_entry. pfd_residual_code_normalization_constant_execution = fom_beta_quotient_pfp_normalization_constant_executiontriminput_entry * S ((S (fom_index_pfp_normalization_constant_executiontriminput)) * pfd_residual_scale_normalization_constant_execution) + (fom_value_pfp_normalization_constant_executiontriminput))) /\\ (exists fom_gap_pfp_normalization_constant_executiontriminput_value_bound. fom_gap_pfp_normalization_constant_executiontriminput_value_bound + S (fom_value_pfp_normalization_constant_executiontriminput) = p))) /\\ (((forall pfp_repeat_index_normalization_constant_executiontrimremoved. (exists pfa_gap_normalization_constant_executiontrimremovedindex. pfa_gap_normalization_constant_executiontrimremovedindex + S (pfp_repeat_index_normalization_constant_executiontrimremoved) = (pfd_cut_normalization_constant_execution)) -> (((exists ff_h_pfp_normalization_constant_executiontrimremovedentry. ff_h_pfp_normalization_constant_executiontrimremovedentry + S (0) = S ((S (pfp_repeat_index_normalization_constant_executiontrimremoved)) * pfd_residual_scale_normalization_constant_execution)) /\\ exists ff_q_pfp_normalization_constant_executiontrimremovedentry. pfd_residual_code_normalization_constant_execution = ff_q_pfp_normalization_constant_executiontrimremovedentry * S ((S (pfp_repeat_index_normalization_constant_executiontrimremoved)) * pfd_residual_scale_normalization_constant_execution) + (0)))) /\\ (((forall pftrim_index_normalization_constant_executiontrimsuffix pftrim_value_normalization_constant_executiontrimsuffix. (exists pfa_gap_normalization_constant_executiontrimsuffixbound. pfa_gap_normalization_constant_executiontrimsuffixbound + S (pftrim_index_normalization_constant_executiontrimsuffix) = (R)) -> (((exists ff_h_pfp_normalization_constant_executiontrimsuffixsource. ff_h_pfp_normalization_constant_executiontrimsuffixsource + S (pftrim_value_normalization_constant_executiontrimsuffix) = S ((S ((pfd_cut_normalization_constant_execution)+pftrim_index_normalization_constant_executiontrimsuffix)) * pfd_residual_scale_normalization_constant_execution)) /\\ exists ff_q_pfp_normalization_constant_executiontrimsuffixsource. pfd_residual_code_normalization_constant_execution = ff_q_pfp_normalization_constant_executiontrimsuffixsource * S ((S ((pfd_cut_normalization_constant_execution)+pftrim_index_normalization_constant_executiontrimsuffix)) * pfd_residual_scale_normalization_constant_execution) + (pftrim_value_normalization_constant_executiontrimsuffix))) -> (((exists ff_h_pfp_normalization_constant_executiontrimsuffixoutput. ff_h_pfp_normalization_constant_executiontrimsuffixoutput + S (pftrim_value_normalization_constant_executiontrimsuffix) = S ((S (pftrim_index_normalization_constant_executiontrimsuffix)) * rc)) /\\ exists ff_q_pfp_normalization_constant_executiontrimsuffixoutput. rb = ff_q_pfp_normalization_constant_executiontrimsuffixoutput * S ((S (pftrim_index_normalization_constant_executiontrimsuffix)) * rc) + (pftrim_value_normalization_constant_executiontrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_normalization_constant_executiontrimnormal. ((((exists ff_h_pfp_normalization_constant_executiontrimnormalentry. ff_h_pfp_normalization_constant_executiontrimnormalentry + S (pftrim_leading_normalization_constant_executiontrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_normalization_constant_executiontrimnormalentry. rb = ff_q_pfp_normalization_constant_executiontrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_normalization_constant_executiontrimnormal))) /\\ ((~(pftrim_leading_normalization_constant_executiontrimnormal=0))))))))))))))))))))))))))))))))) -> (R=0)", "statement_sha256": "ac7f30f0841995aa9fe25e0546803c6bcf4aab7c09fa337a4c61eafa6f196a9b", "summary": "Actual division by the nonzero degree-zero divisor produces an empty normalized remainder, including an empty dividend. This assigns no degree to the zero polynomial.", "summary_sha256": "f076ee92ad6ee7e68edf2731b0f5325b72f1f431aecd6a1094de31f7e1f20d95" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "fdda3075499caa56446b0d85f9c15aebf045a671a934c7b4a81baa7d2d64081b", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 417, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "d2cddfe42dc0d22104dc4e85e95116222914df11ac840d2082a4ff2e462f146f", "test_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85" }, "body_checked": true, "body_receipt": { "command_count": 76, "dependency_count": 5, "dne_command_count": 0, "name": "prime_field_polynomial_scale_implies_right_divides", "proof_depth": 35, "proof_edges": 99, "proof_nodes": 100, "proof_objects": 100, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "prime_field_polynomial_scale_bounded", "prime_field_polynomial_left_constant_product_exists", "prime_field_polynomial_scale_functional", "prime_field_polynomial_right_divides_from_product", "prime_field_polynomial_equal_implies_equivalent" ], "dependencies_sha256": "a14722915dcee42b4a67d6c4972078416151146d704c71204df969339c8299ea", "empty_context_closure": { "body_proof_depth": 35, "body_proof_nodes": 100, "bundle_campaign": "polynomial-gcd-bezout", "bundle_dependency_edge_count": 1578, "bundle_node_count": 493, "bundle_node_id": 417, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "bundle_root_id": 492, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "23df9a5525ee2ad2dce1775209e3da3fd8e95dde015dd91a316bd217f7f443eb", "status": "checked" }, "enrollment_index": 4176, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "d2cddfe42dc0d22104dc4e85e95116222914df11ac840d2082a4ff2e462f146f", "kind": "alpha_v34_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_euclidean_normalization_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85", "kind": "alpha_v34_frontier_executable_audit", "path": "peano-lab/py/tests/test_campaign_research_v34_closure.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "kind": "alpha_v34_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-field-polynomial-gcd-bezout-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "kind": "alpha_v34_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=417]" }, { "document_sha256": "81943a545e7b9f667ad758c24de14bb47280e8cf3fbb3ca714efec55fb096408", "kind": "alpha_v34_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "573516ecd1cad88023f6475ec5b1595140775a1b95d6604e67dd3f51b42b4589", "membership": "alpha_only", "name": "prime_field_polynomial_scale_implies_right_divides", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro k", "intro ab", "intro ac", "intro hb", "intro hc", "intro L", "intro hp", "intro hs", "have hcopy : ((exists pfa_gap_normalization_scale_sourcescalar. pfa_gap_normalization_scale_sourcescalar + S (k) = (p)) /\\ ((forall pfp_index_normalization_scale_source. (exists pfa_gap_normalization_scale_sourceindex. pfa_gap_normalization_scale_sourceindex + S (pfp_index_normalization_scale_source) = (L)) -> exists pfp_source_normalization_scale_source pfp_value_normalization_scale_source. ((((exists ff_h_pfp_normalization_scale_sourcesource. ff_h_pfp_normalization_scale_sourcesource + S (pfp_source_normalization_scale_source) = S ((S (pfp_index_normalization_scale_source)) * ac)) /\\ exists ff_q_pfp_normalization_scale_sourcesource. ab = ff_q_pfp_normalization_scale_sourcesource * S ((S (pfp_index_normalization_scale_source)) * ac) + (pfp_source_normalization_scale_source))) /\\ (((((exists ff_h_pfp_normalization_scale_sourcetarget. ff_h_pfp_normalization_scale_sourcetarget + S (pfp_value_normalization_scale_source) = S ((S (pfp_index_normalization_scale_source)) * hc)) /\\ exists ff_q_pfp_normalization_scale_sourcetarget. hb = ff_q_pfp_normalization_scale_sourcetarget * S ((S (pfp_index_normalization_scale_source)) * hc) + (pfp_value_normalization_scale_source))) /\\ ((((exists pfa_gap_normalization_scale_sourceoperationleft. pfa_gap_normalization_scale_sourceoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_normalization_scale_sourceoperationright. pfa_gap_normalization_scale_sourceoperationright + S (pfp_source_normalization_scale_source) = (p)) /\\ ((((exists pfa_gap_normalization_scale_sourceoperationresultbound. pfa_gap_normalization_scale_sourceoperationresultbound + S (pfp_value_normalization_scale_source) = (p)) /\\ ((exists pfa_offset_left_normalization_scale_sourceoperationresultcongruence pfa_offset_right_normalization_scale_sourceoperationresultcongruence. ((k) * (pfp_source_normalization_scale_source)) + (p) * pfa_offset_left_normalization_scale_sourceoperationresultcongruence = (pfp_value_normalization_scale_source) + (p) * pfa_offset_right_normalization_scale_sourceoperationresultcongruence))))))))))))))))", "exact hs", "cases hcopy", "have hbounds : ((forall fom_index_pfp_normalization_scale_A. (exists fom_gap_pfp_normalization_scale_A_index_bound. fom_gap_pfp_normalization_scale_A_index_bound + S (fom_index_pfp_normalization_scale_A) = L) -> exists fom_value_pfp_normalization_scale_A. ((((exists fom_beta_height_pfp_normalization_scale_A_entry. fom_beta_height_pfp_normalization_scale_A_entry + S (fom_value_pfp_normalization_scale_A) = S ((S (fom_index_pfp_normalization_scale_A)) * ac)) /\\ exists fom_beta_quotient_pfp_normalization_scale_A_entry. ab = fom_beta_quotient_pfp_normalization_scale_A_entry * S ((S (fom_index_pfp_normalization_scale_A)) * ac) + (fom_value_pfp_normalization_scale_A))) /\\ (exists fom_gap_pfp_normalization_scale_A_value_bound. fom_gap_pfp_normalization_scale_A_value_bound + S (fom_value_pfp_normalization_scale_A) = p))) /\\ ((forall fom_index_pfp_normalization_scale_H. (exists fom_gap_pfp_normalization_scale_H_index_bound. fom_gap_pfp_normalization_scale_H_index_bound + S (fom_index_pfp_normalization_scale_H) = L) -> exists fom_value_pfp_normalization_scale_H. ((((exists fom_beta_height_pfp_normalization_scale_H_entry. fom_beta_height_pfp_normalization_scale_H_entry + S (fom_value_pfp_normalization_scale_H) = S ((S (fom_index_pfp_normalization_scale_H)) * hc)) /\\ exists fom_beta_quotient_pfp_normalization_scale_H_entry. hb = fom_beta_quotient_pfp_normalization_scale_H_entry * S ((S (fom_index_pfp_normalization_scale_H)) * hc) + (fom_value_pfp_normalization_scale_H))) /\\ (exists fom_gap_pfp_normalization_scale_H_value_bound. fom_gap_pfp_normalization_scale_H_value_bound + S (fom_value_pfp_normalization_scale_H) = p)))))", "specialize prime_field_polynomial_scale_bounded (p)", "specialize prime_field_polynomial_scale_bounded (k)", "specialize prime_field_polynomial_scale_bounded (ab)", "specialize prime_field_polynomial_scale_bounded (ac)", "specialize prime_field_polynomial_scale_bounded (hb)", "specialize prime_field_polynomial_scale_bounded (hc)", "specialize prime_field_polynomial_scale_bounded (L)", "apply prime_field_polynomial_scale_bounded", "exact hs", "cases hbounds", "have hactual : exists ub uc vb vc. ((forall fom_index_pfp_normalization_scale_singleton. (exists fom_gap_pfp_normalization_scale_singleton_index_bound. fom_gap_pfp_normalization_scale_singleton_index_bound + S (fom_index_pfp_normalization_scale_singleton) = 1) -> exists fom_value_pfp_normalization_scale_singleton. ((((exists fom_beta_height_pfp_normalization_scale_singleton_entry. fom_beta_height_pfp_normalization_scale_singleton_entry + S (fom_value_pfp_normalization_scale_singleton) = S ((S (fom_index_pfp_normalization_scale_singleton)) * uc)) /\\ exists fom_beta_quotient_pfp_normalization_scale_singleton_entry. ub = fom_beta_quotient_pfp_normalization_scale_singleton_entry * S ((S (fom_index_pfp_normalization_scale_singleton)) * uc) + (fom_value_pfp_normalization_scale_singleton))) /\\ (exists fom_gap_pfp_normalization_scale_singleton_value_bound. fom_gap_pfp_normalization_scale_singleton_value_bound + S (fom_value_pfp_normalization_scale_singleton) = p))) /\\ (((((exists ff_h_pfp_normalization_scale_head. ff_h_pfp_normalization_scale_head + S (k) = S ((S (0)) * uc)) /\\ exists ff_q_pfp_normalization_scale_head. ub = ff_q_pfp_normalization_scale_head * S ((S (0)) * uc) + (k))) /\\ (((((exists pfa_gap_normalization_scale_constructedscalar. pfa_gap_normalization_scale_constructedscalar + S (k) = (p)) /\\ ((forall pfp_index_normalization_scale_constructed. (exists pfa_gap_normalization_scale_constructedindex. pfa_gap_normalization_scale_constructedindex + S (pfp_index_normalization_scale_constructed) = (L)) -> exists pfp_source_normalization_scale_constructed pfp_value_normalization_scale_constructed. ((((exists ff_h_pfp_normalization_scale_constructedsource. ff_h_pfp_normalization_scale_constructedsource + S (pfp_source_normalization_scale_constructed) = S ((S (pfp_index_normalization_scale_constructed)) * ac)) /\\ exists ff_q_pfp_normalization_scale_constructedsource. ab = ff_q_pfp_normalization_scale_constructedsource * S ((S (pfp_index_normalization_scale_constructed)) * ac) + (pfp_source_normalization_scale_constructed))) /\\ (((((exists ff_h_pfp_normalization_scale_constructedtarget. ff_h_pfp_normalization_scale_constructedtarget + S (pfp_value_normalization_scale_constructed) = S ((S (pfp_index_normalization_scale_constructed)) * vc)) /\\ exists ff_q_pfp_normalization_scale_constructedtarget. vb = ff_q_pfp_normalization_scale_constructedtarget * S ((S (pfp_index_normalization_scale_constructed)) * vc) + (pfp_value_normalization_scale_constructed))) /\\ ((((exists pfa_gap_normalization_scale_constructedoperationleft. pfa_gap_normalization_scale_constructedoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_normalization_scale_constructedoperationright. pfa_gap_normalization_scale_constructedoperationright + S (pfp_source_normalization_scale_constructed) = (p)) /\\ ((((exists pfa_gap_normalization_scale_constructedoperationresultbound. pfa_gap_normalization_scale_constructedoperationresultbound + S (pfp_value_normalization_scale_constructed) = (p)) /\\ ((exists pfa_offset_left_normalization_scale_constructedoperationresultcongruence pfa_offset_right_normalization_scale_constructedoperationresultcongruence. ((k) * (pfp_source_normalization_scale_constructed)) + (p) * pfa_offset_left_normalization_scale_constructedoperationresultcongruence = (pfp_value_normalization_scale_constructed) + (p) * pfa_offset_right_normalization_scale_constructedoperationresultcongruence))))))))))))))))) /\\ ((((forall fom_index_pfp_normalization_scale_productleft. (exists fom_gap_pfp_normalization_scale_productleft_index_bound. fom_gap_pfp_normalization_scale_productleft_index_bound + S (fom_index_pfp_normalization_scale_productleft) = 1) -> exists fom_value_pfp_normalization_scale_productleft. ((((exists fom_beta_height_pfp_normalization_scale_productleft_entry. fom_beta_height_pfp_normalization_scale_productleft_entry + S (fom_value_pfp_normalization_scale_productleft) = S ((S (fom_index_pfp_normalization_scale_productleft)) * uc)) /\\ exists fom_beta_quotient_pfp_normalization_scale_productleft_entry. ub = fom_beta_quotient_pfp_normalization_scale_productleft_entry * S ((S (fom_index_pfp_normalization_scale_productleft)) * uc) + (fom_value_pfp_normalization_scale_productleft))) /\\ (exists fom_gap_pfp_normalization_scale_productleft_value_bound. fom_gap_pfp_normalization_scale_productleft_value_bound + S (fom_value_pfp_normalization_scale_productleft) = p))) /\\ (((forall fom_index_pfp_normalization_scale_productright. (exists fom_gap_pfp_normalization_scale_productright_index_bound. fom_gap_pfp_normalization_scale_productright_index_bound + S (fom_index_pfp_normalization_scale_productright) = L) -> exists fom_value_pfp_normalization_scale_productright. ((((exists fom_beta_height_pfp_normalization_scale_productright_entry. fom_beta_height_pfp_normalization_scale_productright_entry + S (fom_value_pfp_normalization_scale_productright) = S ((S (fom_index_pfp_normalization_scale_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_normalization_scale_productright_entry. ab = fom_beta_quotient_pfp_normalization_scale_productright_entry * S ((S (fom_index_pfp_normalization_scale_productright)) * ac) + (fom_value_pfp_normalization_scale_productright))) /\\ (exists fom_gap_pfp_normalization_scale_productright_value_bound. fom_gap_pfp_normalization_scale_productright_value_bound + S (fom_value_pfp_normalization_scale_productright) = p))) /\\ (((((((1)=0 \\/ (L)=0) /\\ (((L)=0)))) \\/ (((~((1)=0)) /\\ (((~((L)=0)) /\\ (((1)+(L)=S (L)))))))) /\\ ((forall pfc_index_normalization_scale_productcoefficients. (exists pfa_gap_normalization_scale_productcoefficientsbound. pfa_gap_normalization_scale_productcoefficientsbound + S (pfc_index_normalization_scale_productcoefficients) = (L)) -> exists pfc_value_normalization_scale_productcoefficients. ((((exists ff_h_pfp_normalization_scale_productcoefficientsentry. ff_h_pfp_normalization_scale_productcoefficientsentry + S (pfc_value_normalization_scale_productcoefficients) = S ((S (pfc_index_normalization_scale_productcoefficients)) * vc)) /\\ exists ff_q_pfp_normalization_scale_productcoefficientsentry. vb = ff_q_pfp_normalization_scale_productcoefficientsentry * S ((S (pfc_index_normalization_scale_productcoefficients)) * vc) + (pfc_value_normalization_scale_productcoefficients))) /\\ ((exists pfc_terms_code_normalization_scale_productcoefficientscoefficient pfc_terms_scale_normalization_scale_productcoefficientscoefficient pfc_natural_sum_normalization_scale_productcoefficientscoefficient. ((forall pfc_index_normalization_scale_productcoefficientscoefficientdiagonal. (exists pfa_gap_normalization_scale_productcoefficientscoefficientdiagonalbound. pfa_gap_normalization_scale_productcoefficientscoefficientdiagonalbound + S (pfc_index_normalization_scale_productcoefficientscoefficientdiagonal) = (S (pfc_index_normalization_scale_productcoefficients))) -> exists pfc_value_normalization_scale_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalization_scale_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normalization_scale_productcoefficientscoefficientdiagonalentry + S (pfc_value_normalization_scale_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalization_scale_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalization_scale_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normalization_scale_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normalization_scale_productcoefficientscoefficient = ff_q_pfp_normalization_scale_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalization_scale_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalization_scale_productcoefficientscoefficient) + (pfc_value_normalization_scale_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalization_scale_productcoefficientscoefficientdiagonalterm pfc_left_normalization_scale_productcoefficientscoefficientdiagonalterm pfc_right_normalization_scale_productcoefficientscoefficientdiagonalterm. (((pfc_index_normalization_scale_productcoefficientscoefficientdiagonal)+pfc_complement_normalization_scale_productcoefficientscoefficientdiagonalterm=(pfc_index_normalization_scale_productcoefficients)) /\\ ((((((exists pfa_gap_normalization_scale_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalization_scale_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalization_scale_productcoefficientscoefficientdiagonal) = (1)) /\\ ((((exists ff_h_pfp_normalization_scale_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalization_scale_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalization_scale_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalization_scale_productcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_normalization_scale_productcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_normalization_scale_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalization_scale_productcoefficientscoefficientdiagonal)) * uc) + (pfc_left_normalization_scale_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_scale_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalization_scale_productcoefficientscoefficientdiagonaltermleftoutside+(1)=(pfc_index_normalization_scale_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalization_scale_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalization_scale_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalization_scale_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalization_scale_productcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_normalization_scale_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalization_scale_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalization_scale_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalization_scale_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_normalization_scale_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_normalization_scale_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalization_scale_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_normalization_scale_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_scale_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalization_scale_productcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_normalization_scale_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalization_scale_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalization_scale_productcoefficientscoefficientdiagonal)=pfc_left_normalization_scale_productcoefficientscoefficientdiagonalterm*pfc_right_normalization_scale_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalization_scale_productcoefficientscoefficientsum fs_v_pfc_normalization_scale_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normalization_scale_productcoefficientscoefficientsum_body_start. fs_h_pfc_normalization_scale_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalization_scale_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_scale_productcoefficientscoefficientsum_body_start. fs_u_pfc_normalization_scale_productcoefficientscoefficientsum = fs_q_pfc_normalization_scale_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalization_scale_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalization_scale_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalization_scale_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalization_scale_productcoefficientscoefficient) = S ((S (S (pfc_index_normalization_scale_productcoefficients))) * fs_v_pfc_normalization_scale_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_scale_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalization_scale_productcoefficientscoefficientsum = fs_q_pfc_normalization_scale_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalization_scale_productcoefficients))) * fs_v_pfc_normalization_scale_productcoefficientscoefficientsum) + (pfc_natural_sum_normalization_scale_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps = S (pfc_index_normalization_scale_productcoefficients)) -> exists fs_a_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps fs_r_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps fs_s_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalization_scale_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalization_scale_productcoefficientscoefficient = fs_q_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalization_scale_productcoefficientscoefficient) + (fs_a_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_scale_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalization_scale_productcoefficientscoefficientsum = fs_q_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_scale_productcoefficientscoefficientsum) + (fs_r_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_scale_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalization_scale_productcoefficientscoefficientsum = fs_q_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_scale_productcoefficientscoefficientsum) + (fs_s_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normalization_scale_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalization_scale_productcoefficientscoefficientresiduebound. pfa_gap_normalization_scale_productcoefficientscoefficientresiduebound + S (pfc_value_normalization_scale_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalization_scale_productcoefficientscoefficientresiduecongruence pfa_offset_right_normalization_scale_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalization_scale_productcoefficientscoefficient) + (p) * pfa_offset_left_normalization_scale_productcoefficientscoefficientresiduecongruence = (pfc_value_normalization_scale_productcoefficients) + (p) * pfa_offset_right_normalization_scale_productcoefficientscoefficientresiduecongruence)))))))))))))))))))))))))", "specialize prime_field_polynomial_left_constant_product_exists (p)", "specialize prime_field_polynomial_left_constant_product_exists (k)", "specialize prime_field_polynomial_left_constant_product_exists (ab)", "specialize prime_field_polynomial_left_constant_product_exists (ac)", "specialize prime_field_polynomial_left_constant_product_exists (L)", "apply prime_field_polynomial_left_constant_product_exists", "exact hp", "exact hcopy_left", "exact hbounds_left", "cases hactual", "cases hactual_witness", "cases hactual_witness_witness", "cases hactual_witness_witness_witness", "cases hactual_witness_witness_witness_witness", "cases hactual_witness_witness_witness_witness_right", "cases hactual_witness_witness_witness_witness_right_right", "have hequal : forall mdr_i_pfp_normalization_scale_recode mdr_a_pfp_normalization_scale_recode. (exists mdr_gap_pfp_normalization_scale_recodeb. mdr_gap_pfp_normalization_scale_recodeb + S (mdr_i_pfp_normalization_scale_recode) = (L)) -> (((exists ff_h_mdr_pfp_normalization_scale_recodeo. ff_h_mdr_pfp_normalization_scale_recodeo + S (mdr_a_pfp_normalization_scale_recode) = S ((S (mdr_i_pfp_normalization_scale_recode)) * x3)) /\\ exists ff_q_mdr_pfp_normalization_scale_recodeo. x2 = ff_q_mdr_pfp_normalization_scale_recodeo * S ((S (mdr_i_pfp_normalization_scale_recode)) * x3) + (mdr_a_pfp_normalization_scale_recode))) -> (((exists ff_h_mdr_pfp_normalization_scale_recoden. ff_h_mdr_pfp_normalization_scale_recoden + S (mdr_a_pfp_normalization_scale_recode) = S ((S (mdr_i_pfp_normalization_scale_recode)) * hc)) /\\ exists ff_q_mdr_pfp_normalization_scale_recoden. hb = ff_q_mdr_pfp_normalization_scale_recoden * S ((S (mdr_i_pfp_normalization_scale_recode)) * hc) + (mdr_a_pfp_normalization_scale_recode)))", "specialize prime_field_polynomial_scale_functional (p)", "specialize prime_field_polynomial_scale_functional (k)", "specialize prime_field_polynomial_scale_functional (ab)", "specialize prime_field_polynomial_scale_functional (ac)", "specialize prime_field_polynomial_scale_functional (x2)", "specialize prime_field_polynomial_scale_functional (x3)", "specialize prime_field_polynomial_scale_functional (hb)", "specialize prime_field_polynomial_scale_functional (hc)", "specialize prime_field_polynomial_scale_functional (L)", "apply prime_field_polynomial_scale_functional", "exact hactual_witness_witness_witness_witness_right_right_left", "exact hs", "specialize prime_field_polynomial_right_divides_from_product (p)", "specialize prime_field_polynomial_right_divides_from_product (ab)", "specialize prime_field_polynomial_right_divides_from_product (ac)", "specialize prime_field_polynomial_right_divides_from_product (L)", "specialize prime_field_polynomial_right_divides_from_product (hb)", "specialize prime_field_polynomial_right_divides_from_product (hc)", "specialize prime_field_polynomial_right_divides_from_product (L)", "specialize prime_field_polynomial_right_divides_from_product (x)", "specialize prime_field_polynomial_right_divides_from_product (x1)", "specialize prime_field_polynomial_right_divides_from_product (1)", "specialize prime_field_polynomial_right_divides_from_product (x2)", "specialize prime_field_polynomial_right_divides_from_product (x3)", "specialize prime_field_polynomial_right_divides_from_product (L)", "apply prime_field_polynomial_right_divides_from_product", "exact hbounds_right", "exact hactual_witness_witness_witness_witness_right_right_right", "specialize prime_field_polynomial_equal_implies_equivalent (x2)", "specialize prime_field_polynomial_equal_implies_equivalent (x3)", "specialize prime_field_polynomial_equal_implies_equivalent (hb)", "specialize prime_field_polynomial_equal_implies_equivalent (hc)", "specialize prime_field_polynomial_equal_implies_equivalent (L)", "apply prime_field_polynomial_equal_implies_equivalent", "exact hequal" ], "script_sha256": "3b4bd87c12c5e48e1c45a6c068cb61140eafd4cec24c80d6efed6c458df41b61", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_euclidean_normalization_candidate.py", "sha256": "d2cddfe42dc0d22104dc4e85e95116222914df11ac840d2082a4ff2e462f146f" }, "statement": "forall p k ab ac hb hc L. (~((p) = 1) /\\ forall pfa_factor_left_normalization_scale_prime pfa_factor_right_normalization_scale_prime. (p) = pfa_factor_left_normalization_scale_prime * pfa_factor_right_normalization_scale_prime -> pfa_factor_left_normalization_scale_prime = 1 \\/ pfa_factor_right_normalization_scale_prime = 1) -> (((exists pfa_gap_normalization_scale_sourcescalar. pfa_gap_normalization_scale_sourcescalar + S (k) = (p)) /\\ ((forall pfp_index_normalization_scale_source. (exists pfa_gap_normalization_scale_sourceindex. pfa_gap_normalization_scale_sourceindex + S (pfp_index_normalization_scale_source) = (L)) -> exists pfp_source_normalization_scale_source pfp_value_normalization_scale_source. ((((exists ff_h_pfp_normalization_scale_sourcesource. ff_h_pfp_normalization_scale_sourcesource + S (pfp_source_normalization_scale_source) = S ((S (pfp_index_normalization_scale_source)) * ac)) /\\ exists ff_q_pfp_normalization_scale_sourcesource. ab = ff_q_pfp_normalization_scale_sourcesource * S ((S (pfp_index_normalization_scale_source)) * ac) + (pfp_source_normalization_scale_source))) /\\ (((((exists ff_h_pfp_normalization_scale_sourcetarget. ff_h_pfp_normalization_scale_sourcetarget + S (pfp_value_normalization_scale_source) = S ((S (pfp_index_normalization_scale_source)) * hc)) /\\ exists ff_q_pfp_normalization_scale_sourcetarget. hb = ff_q_pfp_normalization_scale_sourcetarget * S ((S (pfp_index_normalization_scale_source)) * hc) + (pfp_value_normalization_scale_source))) /\\ ((((exists pfa_gap_normalization_scale_sourceoperationleft. pfa_gap_normalization_scale_sourceoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_normalization_scale_sourceoperationright. pfa_gap_normalization_scale_sourceoperationright + S (pfp_source_normalization_scale_source) = (p)) /\\ ((((exists pfa_gap_normalization_scale_sourceoperationresultbound. pfa_gap_normalization_scale_sourceoperationresultbound + S (pfp_value_normalization_scale_source) = (p)) /\\ ((exists pfa_offset_left_normalization_scale_sourceoperationresultcongruence pfa_offset_right_normalization_scale_sourceoperationresultcongruence. ((k) * (pfp_source_normalization_scale_source)) + (p) * pfa_offset_left_normalization_scale_sourceoperationresultcongruence = (pfp_value_normalization_scale_source) + (p) * pfa_offset_right_normalization_scale_sourceoperationresultcongruence))))))))))))))))) -> (((forall fom_index_pfp_normalization_scale_divides_canonical. (exists fom_gap_pfp_normalization_scale_divides_canonical_index_bound. fom_gap_pfp_normalization_scale_divides_canonical_index_bound + S (fom_index_pfp_normalization_scale_divides_canonical) = L) -> exists fom_value_pfp_normalization_scale_divides_canonical. ((((exists fom_beta_height_pfp_normalization_scale_divides_canonical_entry. fom_beta_height_pfp_normalization_scale_divides_canonical_entry + S (fom_value_pfp_normalization_scale_divides_canonical) = S ((S (fom_index_pfp_normalization_scale_divides_canonical)) * hc)) /\\ exists fom_beta_quotient_pfp_normalization_scale_divides_canonical_entry. hb = fom_beta_quotient_pfp_normalization_scale_divides_canonical_entry * S ((S (fom_index_pfp_normalization_scale_divides_canonical)) * hc) + (fom_value_pfp_normalization_scale_divides_canonical))) /\\ (exists fom_gap_pfp_normalization_scale_divides_canonical_value_bound. fom_gap_pfp_normalization_scale_divides_canonical_value_bound + S (fom_value_pfp_normalization_scale_divides_canonical) = p))) /\\ ((exists pfen_qb_normalization_scale_divides pfen_qc_normalization_scale_divides pfen_qlen_normalization_scale_divides pfen_pb_normalization_scale_divides pfen_pc_normalization_scale_divides pfen_plen_normalization_scale_divides. ((((forall fom_index_pfp_normalization_scale_divides_productleft. (exists fom_gap_pfp_normalization_scale_divides_productleft_index_bound. fom_gap_pfp_normalization_scale_divides_productleft_index_bound + S (fom_index_pfp_normalization_scale_divides_productleft) = pfen_qlen_normalization_scale_divides) -> exists fom_value_pfp_normalization_scale_divides_productleft. ((((exists fom_beta_height_pfp_normalization_scale_divides_productleft_entry. fom_beta_height_pfp_normalization_scale_divides_productleft_entry + S (fom_value_pfp_normalization_scale_divides_productleft) = S ((S (fom_index_pfp_normalization_scale_divides_productleft)) * pfen_qc_normalization_scale_divides)) /\\ exists fom_beta_quotient_pfp_normalization_scale_divides_productleft_entry. pfen_qb_normalization_scale_divides = fom_beta_quotient_pfp_normalization_scale_divides_productleft_entry * S ((S (fom_index_pfp_normalization_scale_divides_productleft)) * pfen_qc_normalization_scale_divides) + (fom_value_pfp_normalization_scale_divides_productleft))) /\\ (exists fom_gap_pfp_normalization_scale_divides_productleft_value_bound. fom_gap_pfp_normalization_scale_divides_productleft_value_bound + S (fom_value_pfp_normalization_scale_divides_productleft) = p))) /\\ (((forall fom_index_pfp_normalization_scale_divides_productright. (exists fom_gap_pfp_normalization_scale_divides_productright_index_bound. fom_gap_pfp_normalization_scale_divides_productright_index_bound + S (fom_index_pfp_normalization_scale_divides_productright) = L) -> exists fom_value_pfp_normalization_scale_divides_productright. ((((exists fom_beta_height_pfp_normalization_scale_divides_productright_entry. fom_beta_height_pfp_normalization_scale_divides_productright_entry + S (fom_value_pfp_normalization_scale_divides_productright) = S ((S (fom_index_pfp_normalization_scale_divides_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_normalization_scale_divides_productright_entry. ab = fom_beta_quotient_pfp_normalization_scale_divides_productright_entry * S ((S (fom_index_pfp_normalization_scale_divides_productright)) * ac) + (fom_value_pfp_normalization_scale_divides_productright))) /\\ (exists fom_gap_pfp_normalization_scale_divides_productright_value_bound. fom_gap_pfp_normalization_scale_divides_productright_value_bound + S (fom_value_pfp_normalization_scale_divides_productright) = p))) /\\ (((((((pfen_qlen_normalization_scale_divides)=0 \\/ (L)=0) /\\ (((pfen_plen_normalization_scale_divides)=0)))) \\/ (((~((pfen_qlen_normalization_scale_divides)=0)) /\\ (((~((L)=0)) /\\ (((pfen_qlen_normalization_scale_divides)+(L)=S (pfen_plen_normalization_scale_divides)))))))) /\\ ((forall pfc_index_normalization_scale_divides_productcoefficients. (exists pfa_gap_normalization_scale_divides_productcoefficientsbound. pfa_gap_normalization_scale_divides_productcoefficientsbound + S (pfc_index_normalization_scale_divides_productcoefficients) = (pfen_plen_normalization_scale_divides)) -> exists pfc_value_normalization_scale_divides_productcoefficients. ((((exists ff_h_pfp_normalization_scale_divides_productcoefficientsentry. ff_h_pfp_normalization_scale_divides_productcoefficientsentry + S (pfc_value_normalization_scale_divides_productcoefficients) = S ((S (pfc_index_normalization_scale_divides_productcoefficients)) * pfen_pc_normalization_scale_divides)) /\\ exists ff_q_pfp_normalization_scale_divides_productcoefficientsentry. pfen_pb_normalization_scale_divides = ff_q_pfp_normalization_scale_divides_productcoefficientsentry * S ((S (pfc_index_normalization_scale_divides_productcoefficients)) * pfen_pc_normalization_scale_divides) + (pfc_value_normalization_scale_divides_productcoefficients))) /\\ ((exists pfc_terms_code_normalization_scale_divides_productcoefficientscoefficient pfc_terms_scale_normalization_scale_divides_productcoefficientscoefficient pfc_natural_sum_normalization_scale_divides_productcoefficientscoefficient. ((forall pfc_index_normalization_scale_divides_productcoefficientscoefficientdiagonal. (exists pfa_gap_normalization_scale_divides_productcoefficientscoefficientdiagonalbound. pfa_gap_normalization_scale_divides_productcoefficientscoefficientdiagonalbound + S (pfc_index_normalization_scale_divides_productcoefficientscoefficientdiagonal) = (S (pfc_index_normalization_scale_divides_productcoefficients))) -> exists pfc_value_normalization_scale_divides_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalization_scale_divides_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normalization_scale_divides_productcoefficientscoefficientdiagonalentry + S (pfc_value_normalization_scale_divides_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalization_scale_divides_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalization_scale_divides_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normalization_scale_divides_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normalization_scale_divides_productcoefficientscoefficient = ff_q_pfp_normalization_scale_divides_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalization_scale_divides_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalization_scale_divides_productcoefficientscoefficient) + (pfc_value_normalization_scale_divides_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalization_scale_divides_productcoefficientscoefficientdiagonalterm pfc_left_normalization_scale_divides_productcoefficientscoefficientdiagonalterm pfc_right_normalization_scale_divides_productcoefficientscoefficientdiagonalterm. (((pfc_index_normalization_scale_divides_productcoefficientscoefficientdiagonal)+pfc_complement_normalization_scale_divides_productcoefficientscoefficientdiagonalterm=(pfc_index_normalization_scale_divides_productcoefficients)) /\\ ((((((exists pfa_gap_normalization_scale_divides_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalization_scale_divides_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalization_scale_divides_productcoefficientscoefficientdiagonal) = (pfen_qlen_normalization_scale_divides)) /\\ ((((exists ff_h_pfp_normalization_scale_divides_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalization_scale_divides_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalization_scale_divides_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalization_scale_divides_productcoefficientscoefficientdiagonal)) * pfen_qc_normalization_scale_divides)) /\\ exists ff_q_pfp_normalization_scale_divides_productcoefficientscoefficientdiagonaltermleftentry. pfen_qb_normalization_scale_divides = ff_q_pfp_normalization_scale_divides_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalization_scale_divides_productcoefficientscoefficientdiagonal)) * pfen_qc_normalization_scale_divides) + (pfc_left_normalization_scale_divides_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_scale_divides_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalization_scale_divides_productcoefficientscoefficientdiagonaltermleftoutside+(pfen_qlen_normalization_scale_divides)=(pfc_index_normalization_scale_divides_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalization_scale_divides_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalization_scale_divides_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalization_scale_divides_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalization_scale_divides_productcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_normalization_scale_divides_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalization_scale_divides_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalization_scale_divides_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalization_scale_divides_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_normalization_scale_divides_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_normalization_scale_divides_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalization_scale_divides_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_normalization_scale_divides_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_scale_divides_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalization_scale_divides_productcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_normalization_scale_divides_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalization_scale_divides_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalization_scale_divides_productcoefficientscoefficientdiagonal)=pfc_left_normalization_scale_divides_productcoefficientscoefficientdiagonalterm*pfc_right_normalization_scale_divides_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalization_scale_divides_productcoefficientscoefficientsum fs_v_pfc_normalization_scale_divides_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_start. fs_h_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalization_scale_divides_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_start. fs_u_pfc_normalization_scale_divides_productcoefficientscoefficientsum = fs_q_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalization_scale_divides_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalization_scale_divides_productcoefficientscoefficient) = S ((S (S (pfc_index_normalization_scale_divides_productcoefficients))) * fs_v_pfc_normalization_scale_divides_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalization_scale_divides_productcoefficientscoefficientsum = fs_q_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalization_scale_divides_productcoefficients))) * fs_v_pfc_normalization_scale_divides_productcoefficientscoefficientsum) + (pfc_natural_sum_normalization_scale_divides_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps = S (pfc_index_normalization_scale_divides_productcoefficients)) -> exists fs_a_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps fs_r_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps fs_s_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalization_scale_divides_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalization_scale_divides_productcoefficientscoefficient = fs_q_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalization_scale_divides_productcoefficientscoefficient) + (fs_a_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_scale_divides_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalization_scale_divides_productcoefficientscoefficientsum = fs_q_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_scale_divides_productcoefficientscoefficientsum) + (fs_r_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_scale_divides_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalization_scale_divides_productcoefficientscoefficientsum = fs_q_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_scale_divides_productcoefficientscoefficientsum) + (fs_s_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normalization_scale_divides_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalization_scale_divides_productcoefficientscoefficientresiduebound. pfa_gap_normalization_scale_divides_productcoefficientscoefficientresiduebound + S (pfc_value_normalization_scale_divides_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalization_scale_divides_productcoefficientscoefficientresiduecongruence pfa_offset_right_normalization_scale_divides_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalization_scale_divides_productcoefficientscoefficient) + (p) * pfa_offset_left_normalization_scale_divides_productcoefficientscoefficientresiduecongruence = (pfc_value_normalization_scale_divides_productcoefficients) + (p) * pfa_offset_right_normalization_scale_divides_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normalization_scale_divides_target pfrep_left_normalization_scale_divides_target pfrep_right_normalization_scale_divides_target. ((exists pfrep_position_normalization_scale_divides_targetfirst. ((pfrep_position_normalization_scale_divides_targetfirst+S (pfrep_power_normalization_scale_divides_target)=(pfen_plen_normalization_scale_divides)) /\\ ((((exists ff_h_pfp_normalization_scale_divides_targetfirstentry. ff_h_pfp_normalization_scale_divides_targetfirstentry + S (pfrep_left_normalization_scale_divides_target) = S ((S (pfrep_position_normalization_scale_divides_targetfirst)) * pfen_pc_normalization_scale_divides)) /\\ exists ff_q_pfp_normalization_scale_divides_targetfirstentry. pfen_pb_normalization_scale_divides = ff_q_pfp_normalization_scale_divides_targetfirstentry * S ((S (pfrep_position_normalization_scale_divides_targetfirst)) * pfen_pc_normalization_scale_divides) + (pfrep_left_normalization_scale_divides_target)))))) \\/ (((exists pfrep_gap_normalization_scale_divides_targetfirstoutside. pfrep_gap_normalization_scale_divides_targetfirstoutside+(pfen_plen_normalization_scale_divides)=(pfrep_power_normalization_scale_divides_target)) /\\ (((pfrep_left_normalization_scale_divides_target)=0))))) -> ((exists pfrep_position_normalization_scale_divides_targetsecond. ((pfrep_position_normalization_scale_divides_targetsecond+S (pfrep_power_normalization_scale_divides_target)=(L)) /\\ ((((exists ff_h_pfp_normalization_scale_divides_targetsecondentry. ff_h_pfp_normalization_scale_divides_targetsecondentry + S (pfrep_right_normalization_scale_divides_target) = S ((S (pfrep_position_normalization_scale_divides_targetsecond)) * hc)) /\\ exists ff_q_pfp_normalization_scale_divides_targetsecondentry. hb = ff_q_pfp_normalization_scale_divides_targetsecondentry * S ((S (pfrep_position_normalization_scale_divides_targetsecond)) * hc) + (pfrep_right_normalization_scale_divides_target)))))) \\/ (((exists pfrep_gap_normalization_scale_divides_targetsecondoutside. pfrep_gap_normalization_scale_divides_targetsecondoutside+(L)=(pfrep_power_normalization_scale_divides_target)) /\\ (((pfrep_right_normalization_scale_divides_target)=0))))) -> pfrep_left_normalization_scale_divides_target=pfrep_right_normalization_scale_divides_target)))))))", "statement_sha256": "23df9a5525ee2ad2dce1775209e3da3fd8e95dde015dd91a316bd217f7f443eb", "summary": "A genuine scalar output is a right multiple of its source: construct an actual LEFT singleton quotient and actual product, then transport the independently encoded product to the supplied target by decoded-prefix equality. 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((((exists ff_h_pfp_normalization_associates_actualsource. ff_h_pfp_normalization_associates_actualsource + S (pfm_leading_normalization_associates_actual) = S ((S (0)) * gc)) /\\ exists ff_q_pfp_normalization_associates_actualsource. gb = ff_q_pfp_normalization_associates_actualsource * S ((S (0)) * gc) + (pfm_leading_normalization_associates_actual))) /\\ ((((~((pfm_leading_normalization_associates_actual) = 0)) /\\ ((((exists pfa_gap_normalization_associates_actualinversemultiplicationleft. pfa_gap_normalization_associates_actualinversemultiplicationleft + S (pfm_leading_normalization_associates_actual) = (p)) /\\ (((exists pfa_gap_normalization_associates_actualinversemultiplicationright. pfa_gap_normalization_associates_actualinversemultiplicationright + S (k) = (p)) /\\ ((((exists pfa_gap_normalization_associates_actualinversemultiplicationresultbound. pfa_gap_normalization_associates_actualinversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_normalization_associates_actualinversemultiplicationresultcongruence pfa_offset_right_normalization_associates_actualinversemultiplicationresultcongruence. ((pfm_leading_normalization_associates_actual) * (k)) + (p) * pfa_offset_left_normalization_associates_actualinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_associates_actualinversemultiplicationresultcongruence))))))))))))))) /\\ ((((exists pfa_gap_normalization_associates_actualscalescalar. pfa_gap_normalization_associates_actualscalescalar + S (k) = (p)) /\\ ((forall pfp_index_normalization_associates_actualscale. (exists pfa_gap_normalization_associates_actualscaleindex. pfa_gap_normalization_associates_actualscaleindex + S (pfp_index_normalization_associates_actualscale) = (L)) -> exists pfp_source_normalization_associates_actualscale pfp_value_normalization_associates_actualscale. ((((exists ff_h_pfp_normalization_associates_actualscalesource. ff_h_pfp_normalization_associates_actualscalesource + S (pfp_source_normalization_associates_actualscale) = S ((S (pfp_index_normalization_associates_actualscale)) * gc)) /\\ exists ff_q_pfp_normalization_associates_actualscalesource. gb = ff_q_pfp_normalization_associates_actualscalesource * S ((S (pfp_index_normalization_associates_actualscale)) * gc) + (pfp_source_normalization_associates_actualscale))) /\\ (((((exists ff_h_pfp_normalization_associates_actualscaletarget. ff_h_pfp_normalization_associates_actualscaletarget + S (pfp_value_normalization_associates_actualscale) = S ((S (pfp_index_normalization_associates_actualscale)) * hc)) /\\ exists ff_q_pfp_normalization_associates_actualscaletarget. hb = ff_q_pfp_normalization_associates_actualscaletarget * S ((S (pfp_index_normalization_associates_actualscale)) * hc) + (pfp_value_normalization_associates_actualscale))) /\\ ((((exists pfa_gap_normalization_associates_actualscaleoperationleft. pfa_gap_normalization_associates_actualscaleoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_normalization_associates_actualscaleoperationright. pfa_gap_normalization_associates_actualscaleoperationright + S (pfp_source_normalization_associates_actualscale) = (p)) /\\ ((((exists pfa_gap_normalization_associates_actualscaleoperationresultbound. pfa_gap_normalization_associates_actualscaleoperationresultbound + S (pfp_value_normalization_associates_actualscale) = (p)) /\\ ((exists pfa_offset_left_normalization_associates_actualscaleoperationresultcongruence pfa_offset_right_normalization_associates_actualscaleoperationresultcongruence. ((k) * (pfp_source_normalization_associates_actualscale)) + (p) * pfa_offset_left_normalization_associates_actualscaleoperationresultcongruence = (pfp_value_normalization_associates_actualscale) + (p) * pfa_offset_right_normalization_associates_actualscaleoperationresultcongruence)))))))))))))))))))))", "exact hnormalization", "cases hcopy", "cases hcopy_right", "split", "specialize prime_field_polynomial_scale_implies_right_divides (p)", "specialize prime_field_polynomial_scale_implies_right_divides (k)", "specialize prime_field_polynomial_scale_implies_right_divides (gb)", "specialize prime_field_polynomial_scale_implies_right_divides (gc)", "specialize prime_field_polynomial_scale_implies_right_divides (hb)", "specialize prime_field_polynomial_scale_implies_right_divides (hc)", "specialize prime_field_polynomial_scale_implies_right_divides (L)", "apply prime_field_polynomial_scale_implies_right_divides", "exact hp", "exact hcopy_right_right", "cases hcopy_right_left", "cases hcopy_right_left_witness", "have hreverse : ((exists pfa_gap_normalization_associates_reversescalar. pfa_gap_normalization_associates_reversescalar + S (x) = (p)) /\\ ((forall pfp_index_normalization_associates_reverse. (exists pfa_gap_normalization_associates_reverseindex. pfa_gap_normalization_associates_reverseindex + S (pfp_index_normalization_associates_reverse) = (L)) -> exists pfp_source_normalization_associates_reverse pfp_value_normalization_associates_reverse. ((((exists ff_h_pfp_normalization_associates_reversesource. ff_h_pfp_normalization_associates_reversesource + S (pfp_source_normalization_associates_reverse) = S ((S (pfp_index_normalization_associates_reverse)) * hc)) /\\ exists ff_q_pfp_normalization_associates_reversesource. hb = ff_q_pfp_normalization_associates_reversesource * S ((S (pfp_index_normalization_associates_reverse)) * hc) + (pfp_source_normalization_associates_reverse))) /\\ (((((exists ff_h_pfp_normalization_associates_reversetarget. ff_h_pfp_normalization_associates_reversetarget + S (pfp_value_normalization_associates_reverse) = S ((S (pfp_index_normalization_associates_reverse)) * gc)) /\\ exists ff_q_pfp_normalization_associates_reversetarget. gb = ff_q_pfp_normalization_associates_reversetarget * S ((S (pfp_index_normalization_associates_reverse)) * gc) + (pfp_value_normalization_associates_reverse))) /\\ ((((exists pfa_gap_normalization_associates_reverseoperationleft. pfa_gap_normalization_associates_reverseoperationleft + S (x) = (p)) /\\ (((exists pfa_gap_normalization_associates_reverseoperationright. pfa_gap_normalization_associates_reverseoperationright + S (pfp_source_normalization_associates_reverse) = (p)) /\\ ((((exists pfa_gap_normalization_associates_reverseoperationresultbound. pfa_gap_normalization_associates_reverseoperationresultbound + S (pfp_value_normalization_associates_reverse) = (p)) /\\ ((exists pfa_offset_left_normalization_associates_reverseoperationresultcongruence pfa_offset_right_normalization_associates_reverseoperationresultcongruence. ((x) * (pfp_source_normalization_associates_reverse)) + (p) * pfa_offset_left_normalization_associates_reverseoperationresultcongruence = (pfp_value_normalization_associates_reverse) + (p) * pfa_offset_right_normalization_associates_reverseoperationresultcongruence))))))))))))))))", "specialize prime_field_polynomial_inverse_scale (p)", "specialize prime_field_polynomial_inverse_scale (x)", "specialize prime_field_polynomial_inverse_scale (k)", "specialize prime_field_polynomial_inverse_scale (gb)", "specialize prime_field_polynomial_inverse_scale (gc)", "specialize prime_field_polynomial_inverse_scale (hb)", "specialize prime_field_polynomial_inverse_scale (hc)", "specialize prime_field_polynomial_inverse_scale (L)", "apply prime_field_polynomial_inverse_scale", "exact hp", "exact hcopy_right_left_witness_right", "exact hcopy_right_right", "specialize prime_field_polynomial_scale_implies_right_divides (p)", "specialize prime_field_polynomial_scale_implies_right_divides (x)", "specialize prime_field_polynomial_scale_implies_right_divides (hb)", "specialize prime_field_polynomial_scale_implies_right_divides (hc)", "specialize prime_field_polynomial_scale_implies_right_divides (gb)", "specialize prime_field_polynomial_scale_implies_right_divides (gc)", "specialize prime_field_polynomial_scale_implies_right_divides (L)", "apply prime_field_polynomial_scale_implies_right_divides", "exact hp", "exact hreverse" ], "script_sha256": "2e64754b693bb8f17c17bd1592c45b47504ae68c0080f1d354b1c8e6dc82c8d9", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_euclidean_normalization_candidate.py", "sha256": "d2cddfe42dc0d22104dc4e85e95116222914df11ac840d2082a4ff2e462f146f" }, "statement": "forall p k gb gc hb hc L. (~((p) = 1) /\\ forall pfa_factor_left_normalization_associates_prime pfa_factor_right_normalization_associates_prime. (p) = pfa_factor_left_normalization_associates_prime * pfa_factor_right_normalization_associates_prime -> pfa_factor_left_normalization_associates_prime = 1 \\/ pfa_factor_right_normalization_associates_prime = 1) -> (((~((L) = 0)) /\\ (((exists pfm_leading_normalization_associates_actual. ((((exists ff_h_pfp_normalization_associates_actualsource. ff_h_pfp_normalization_associates_actualsource + S (pfm_leading_normalization_associates_actual) = S ((S (0)) * gc)) /\\ exists ff_q_pfp_normalization_associates_actualsource. gb = ff_q_pfp_normalization_associates_actualsource * S ((S (0)) * gc) + (pfm_leading_normalization_associates_actual))) /\\ ((((~((pfm_leading_normalization_associates_actual) = 0)) /\\ ((((exists pfa_gap_normalization_associates_actualinversemultiplicationleft. pfa_gap_normalization_associates_actualinversemultiplicationleft + S (pfm_leading_normalization_associates_actual) = (p)) /\\ (((exists pfa_gap_normalization_associates_actualinversemultiplicationright. pfa_gap_normalization_associates_actualinversemultiplicationright + S (k) = (p)) /\\ ((((exists pfa_gap_normalization_associates_actualinversemultiplicationresultbound. pfa_gap_normalization_associates_actualinversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_normalization_associates_actualinversemultiplicationresultcongruence pfa_offset_right_normalization_associates_actualinversemultiplicationresultcongruence. ((pfm_leading_normalization_associates_actual) * (k)) + (p) * pfa_offset_left_normalization_associates_actualinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_associates_actualinversemultiplicationresultcongruence))))))))))))))) /\\ ((((exists pfa_gap_normalization_associates_actualscalescalar. pfa_gap_normalization_associates_actualscalescalar + S (k) = (p)) /\\ ((forall pfp_index_normalization_associates_actualscale. (exists pfa_gap_normalization_associates_actualscaleindex. pfa_gap_normalization_associates_actualscaleindex + S (pfp_index_normalization_associates_actualscale) = (L)) -> exists pfp_source_normalization_associates_actualscale pfp_value_normalization_associates_actualscale. ((((exists ff_h_pfp_normalization_associates_actualscalesource. ff_h_pfp_normalization_associates_actualscalesource + S (pfp_source_normalization_associates_actualscale) = S ((S (pfp_index_normalization_associates_actualscale)) * gc)) /\\ exists ff_q_pfp_normalization_associates_actualscalesource. gb = ff_q_pfp_normalization_associates_actualscalesource * S ((S (pfp_index_normalization_associates_actualscale)) * gc) + (pfp_source_normalization_associates_actualscale))) /\\ (((((exists ff_h_pfp_normalization_associates_actualscaletarget. ff_h_pfp_normalization_associates_actualscaletarget + S (pfp_value_normalization_associates_actualscale) = S ((S (pfp_index_normalization_associates_actualscale)) * hc)) /\\ exists ff_q_pfp_normalization_associates_actualscaletarget. hb = ff_q_pfp_normalization_associates_actualscaletarget * S ((S (pfp_index_normalization_associates_actualscale)) * hc) + (pfp_value_normalization_associates_actualscale))) /\\ ((((exists pfa_gap_normalization_associates_actualscaleoperationleft. pfa_gap_normalization_associates_actualscaleoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_normalization_associates_actualscaleoperationright. pfa_gap_normalization_associates_actualscaleoperationright + S (pfp_source_normalization_associates_actualscale) = (p)) /\\ ((((exists pfa_gap_normalization_associates_actualscaleoperationresultbound. pfa_gap_normalization_associates_actualscaleoperationresultbound + S (pfp_value_normalization_associates_actualscale) = (p)) /\\ ((exists pfa_offset_left_normalization_associates_actualscaleoperationresultcongruence pfa_offset_right_normalization_associates_actualscaleoperationresultcongruence. ((k) * (pfp_source_normalization_associates_actualscale)) + (p) * pfa_offset_left_normalization_associates_actualscaleoperationresultcongruence = (pfp_value_normalization_associates_actualscale) + (p) * pfa_offset_right_normalization_associates_actualscaleoperationresultcongruence)))))))))))))))))))))) -> (((((forall fom_index_pfp_normalization_associates_forward_canonical. (exists fom_gap_pfp_normalization_associates_forward_canonical_index_bound. fom_gap_pfp_normalization_associates_forward_canonical_index_bound + S (fom_index_pfp_normalization_associates_forward_canonical) = L) -> exists fom_value_pfp_normalization_associates_forward_canonical. ((((exists fom_beta_height_pfp_normalization_associates_forward_canonical_entry. fom_beta_height_pfp_normalization_associates_forward_canonical_entry + S (fom_value_pfp_normalization_associates_forward_canonical) = S ((S (fom_index_pfp_normalization_associates_forward_canonical)) * hc)) /\\ exists fom_beta_quotient_pfp_normalization_associates_forward_canonical_entry. hb = fom_beta_quotient_pfp_normalization_associates_forward_canonical_entry * S ((S (fom_index_pfp_normalization_associates_forward_canonical)) * hc) + (fom_value_pfp_normalization_associates_forward_canonical))) /\\ (exists fom_gap_pfp_normalization_associates_forward_canonical_value_bound. fom_gap_pfp_normalization_associates_forward_canonical_value_bound + S (fom_value_pfp_normalization_associates_forward_canonical) = p))) /\\ ((exists pfen_qb_normalization_associates_forward pfen_qc_normalization_associates_forward pfen_qlen_normalization_associates_forward pfen_pb_normalization_associates_forward pfen_pc_normalization_associates_forward pfen_plen_normalization_associates_forward. ((((forall fom_index_pfp_normalization_associates_forward_productleft. (exists fom_gap_pfp_normalization_associates_forward_productleft_index_bound. fom_gap_pfp_normalization_associates_forward_productleft_index_bound + S (fom_index_pfp_normalization_associates_forward_productleft) = pfen_qlen_normalization_associates_forward) -> exists fom_value_pfp_normalization_associates_forward_productleft. ((((exists fom_beta_height_pfp_normalization_associates_forward_productleft_entry. fom_beta_height_pfp_normalization_associates_forward_productleft_entry + S (fom_value_pfp_normalization_associates_forward_productleft) = S ((S (fom_index_pfp_normalization_associates_forward_productleft)) * pfen_qc_normalization_associates_forward)) /\\ exists fom_beta_quotient_pfp_normalization_associates_forward_productleft_entry. pfen_qb_normalization_associates_forward = fom_beta_quotient_pfp_normalization_associates_forward_productleft_entry * S ((S (fom_index_pfp_normalization_associates_forward_productleft)) * pfen_qc_normalization_associates_forward) + (fom_value_pfp_normalization_associates_forward_productleft))) /\\ (exists fom_gap_pfp_normalization_associates_forward_productleft_value_bound. fom_gap_pfp_normalization_associates_forward_productleft_value_bound + S (fom_value_pfp_normalization_associates_forward_productleft) = p))) /\\ (((forall fom_index_pfp_normalization_associates_forward_productright. (exists fom_gap_pfp_normalization_associates_forward_productright_index_bound. fom_gap_pfp_normalization_associates_forward_productright_index_bound + S (fom_index_pfp_normalization_associates_forward_productright) = L) -> exists fom_value_pfp_normalization_associates_forward_productright. ((((exists fom_beta_height_pfp_normalization_associates_forward_productright_entry. fom_beta_height_pfp_normalization_associates_forward_productright_entry + S (fom_value_pfp_normalization_associates_forward_productright) = S ((S (fom_index_pfp_normalization_associates_forward_productright)) * gc)) /\\ exists fom_beta_quotient_pfp_normalization_associates_forward_productright_entry. gb = fom_beta_quotient_pfp_normalization_associates_forward_productright_entry * S ((S (fom_index_pfp_normalization_associates_forward_productright)) * gc) + (fom_value_pfp_normalization_associates_forward_productright))) /\\ (exists fom_gap_pfp_normalization_associates_forward_productright_value_bound. fom_gap_pfp_normalization_associates_forward_productright_value_bound + S (fom_value_pfp_normalization_associates_forward_productright) = p))) /\\ (((((((pfen_qlen_normalization_associates_forward)=0 \\/ (L)=0) /\\ (((pfen_plen_normalization_associates_forward)=0)))) \\/ (((~((pfen_qlen_normalization_associates_forward)=0)) /\\ (((~((L)=0)) /\\ (((pfen_qlen_normalization_associates_forward)+(L)=S (pfen_plen_normalization_associates_forward)))))))) /\\ ((forall pfc_index_normalization_associates_forward_productcoefficients. (exists pfa_gap_normalization_associates_forward_productcoefficientsbound. pfa_gap_normalization_associates_forward_productcoefficientsbound + S (pfc_index_normalization_associates_forward_productcoefficients) = (pfen_plen_normalization_associates_forward)) -> exists pfc_value_normalization_associates_forward_productcoefficients. ((((exists ff_h_pfp_normalization_associates_forward_productcoefficientsentry. ff_h_pfp_normalization_associates_forward_productcoefficientsentry + S (pfc_value_normalization_associates_forward_productcoefficients) = S ((S (pfc_index_normalization_associates_forward_productcoefficients)) * pfen_pc_normalization_associates_forward)) /\\ exists ff_q_pfp_normalization_associates_forward_productcoefficientsentry. pfen_pb_normalization_associates_forward = ff_q_pfp_normalization_associates_forward_productcoefficientsentry * S ((S (pfc_index_normalization_associates_forward_productcoefficients)) * pfen_pc_normalization_associates_forward) + (pfc_value_normalization_associates_forward_productcoefficients))) /\\ ((exists pfc_terms_code_normalization_associates_forward_productcoefficientscoefficient pfc_terms_scale_normalization_associates_forward_productcoefficientscoefficient pfc_natural_sum_normalization_associates_forward_productcoefficientscoefficient. ((forall pfc_index_normalization_associates_forward_productcoefficientscoefficientdiagonal. (exists pfa_gap_normalization_associates_forward_productcoefficientscoefficientdiagonalbound. pfa_gap_normalization_associates_forward_productcoefficientscoefficientdiagonalbound + S (pfc_index_normalization_associates_forward_productcoefficientscoefficientdiagonal) = (S (pfc_index_normalization_associates_forward_productcoefficients))) -> exists pfc_value_normalization_associates_forward_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalization_associates_forward_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normalization_associates_forward_productcoefficientscoefficientdiagonalentry + S (pfc_value_normalization_associates_forward_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalization_associates_forward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalization_associates_forward_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normalization_associates_forward_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normalization_associates_forward_productcoefficientscoefficient = ff_q_pfp_normalization_associates_forward_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalization_associates_forward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalization_associates_forward_productcoefficientscoefficient) + (pfc_value_normalization_associates_forward_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalization_associates_forward_productcoefficientscoefficientdiagonalterm pfc_left_normalization_associates_forward_productcoefficientscoefficientdiagonalterm pfc_right_normalization_associates_forward_productcoefficientscoefficientdiagonalterm. (((pfc_index_normalization_associates_forward_productcoefficientscoefficientdiagonal)+pfc_complement_normalization_associates_forward_productcoefficientscoefficientdiagonalterm=(pfc_index_normalization_associates_forward_productcoefficients)) /\\ ((((((exists pfa_gap_normalization_associates_forward_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalization_associates_forward_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalization_associates_forward_productcoefficientscoefficientdiagonal) = (pfen_qlen_normalization_associates_forward)) /\\ ((((exists ff_h_pfp_normalization_associates_forward_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalization_associates_forward_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalization_associates_forward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalization_associates_forward_productcoefficientscoefficientdiagonal)) * pfen_qc_normalization_associates_forward)) /\\ exists ff_q_pfp_normalization_associates_forward_productcoefficientscoefficientdiagonaltermleftentry. pfen_qb_normalization_associates_forward = ff_q_pfp_normalization_associates_forward_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalization_associates_forward_productcoefficientscoefficientdiagonal)) * pfen_qc_normalization_associates_forward) + (pfc_left_normalization_associates_forward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_associates_forward_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalization_associates_forward_productcoefficientscoefficientdiagonaltermleftoutside+(pfen_qlen_normalization_associates_forward)=(pfc_index_normalization_associates_forward_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalization_associates_forward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalization_associates_forward_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalization_associates_forward_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalization_associates_forward_productcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_normalization_associates_forward_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalization_associates_forward_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalization_associates_forward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalization_associates_forward_productcoefficientscoefficientdiagonalterm)) * gc)) /\\ exists ff_q_pfp_normalization_associates_forward_productcoefficientscoefficientdiagonaltermrightentry. gb = ff_q_pfp_normalization_associates_forward_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalization_associates_forward_productcoefficientscoefficientdiagonalterm)) * gc) + (pfc_right_normalization_associates_forward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_associates_forward_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalization_associates_forward_productcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_normalization_associates_forward_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalization_associates_forward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalization_associates_forward_productcoefficientscoefficientdiagonal)=pfc_left_normalization_associates_forward_productcoefficientscoefficientdiagonalterm*pfc_right_normalization_associates_forward_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalization_associates_forward_productcoefficientscoefficientsum fs_v_pfc_normalization_associates_forward_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_start. fs_h_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalization_associates_forward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_start. fs_u_pfc_normalization_associates_forward_productcoefficientscoefficientsum = fs_q_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalization_associates_forward_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalization_associates_forward_productcoefficientscoefficient) = S ((S (S (pfc_index_normalization_associates_forward_productcoefficients))) * fs_v_pfc_normalization_associates_forward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalization_associates_forward_productcoefficientscoefficientsum = fs_q_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalization_associates_forward_productcoefficients))) * fs_v_pfc_normalization_associates_forward_productcoefficientscoefficientsum) + (pfc_natural_sum_normalization_associates_forward_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps = S (pfc_index_normalization_associates_forward_productcoefficients)) -> exists fs_a_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps fs_r_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps fs_s_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalization_associates_forward_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalization_associates_forward_productcoefficientscoefficient = fs_q_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalization_associates_forward_productcoefficientscoefficient) + (fs_a_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_associates_forward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalization_associates_forward_productcoefficientscoefficientsum = fs_q_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_associates_forward_productcoefficientscoefficientsum) + (fs_r_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_associates_forward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalization_associates_forward_productcoefficientscoefficientsum = fs_q_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_associates_forward_productcoefficientscoefficientsum) + (fs_s_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normalization_associates_forward_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalization_associates_forward_productcoefficientscoefficientresiduebound. pfa_gap_normalization_associates_forward_productcoefficientscoefficientresiduebound + S (pfc_value_normalization_associates_forward_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalization_associates_forward_productcoefficientscoefficientresiduecongruence pfa_offset_right_normalization_associates_forward_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalization_associates_forward_productcoefficientscoefficient) + (p) * pfa_offset_left_normalization_associates_forward_productcoefficientscoefficientresiduecongruence = (pfc_value_normalization_associates_forward_productcoefficients) + (p) * pfa_offset_right_normalization_associates_forward_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normalization_associates_forward_target pfrep_left_normalization_associates_forward_target pfrep_right_normalization_associates_forward_target. ((exists pfrep_position_normalization_associates_forward_targetfirst. ((pfrep_position_normalization_associates_forward_targetfirst+S (pfrep_power_normalization_associates_forward_target)=(pfen_plen_normalization_associates_forward)) /\\ ((((exists ff_h_pfp_normalization_associates_forward_targetfirstentry. ff_h_pfp_normalization_associates_forward_targetfirstentry + S (pfrep_left_normalization_associates_forward_target) = S ((S (pfrep_position_normalization_associates_forward_targetfirst)) * pfen_pc_normalization_associates_forward)) /\\ exists ff_q_pfp_normalization_associates_forward_targetfirstentry. pfen_pb_normalization_associates_forward = ff_q_pfp_normalization_associates_forward_targetfirstentry * S ((S (pfrep_position_normalization_associates_forward_targetfirst)) * pfen_pc_normalization_associates_forward) + (pfrep_left_normalization_associates_forward_target)))))) \\/ (((exists pfrep_gap_normalization_associates_forward_targetfirstoutside. pfrep_gap_normalization_associates_forward_targetfirstoutside+(pfen_plen_normalization_associates_forward)=(pfrep_power_normalization_associates_forward_target)) /\\ (((pfrep_left_normalization_associates_forward_target)=0))))) -> ((exists pfrep_position_normalization_associates_forward_targetsecond. ((pfrep_position_normalization_associates_forward_targetsecond+S (pfrep_power_normalization_associates_forward_target)=(L)) /\\ ((((exists ff_h_pfp_normalization_associates_forward_targetsecondentry. ff_h_pfp_normalization_associates_forward_targetsecondentry + S (pfrep_right_normalization_associates_forward_target) = S ((S (pfrep_position_normalization_associates_forward_targetsecond)) * hc)) /\\ exists ff_q_pfp_normalization_associates_forward_targetsecondentry. hb = ff_q_pfp_normalization_associates_forward_targetsecondentry * S ((S (pfrep_position_normalization_associates_forward_targetsecond)) * hc) + (pfrep_right_normalization_associates_forward_target)))))) \\/ (((exists pfrep_gap_normalization_associates_forward_targetsecondoutside. pfrep_gap_normalization_associates_forward_targetsecondoutside+(L)=(pfrep_power_normalization_associates_forward_target)) /\\ (((pfrep_right_normalization_associates_forward_target)=0))))) -> pfrep_left_normalization_associates_forward_target=pfrep_right_normalization_associates_forward_target))))))) /\\ ((((forall fom_index_pfp_normalization_associates_backward_canonical. (exists fom_gap_pfp_normalization_associates_backward_canonical_index_bound. fom_gap_pfp_normalization_associates_backward_canonical_index_bound + S (fom_index_pfp_normalization_associates_backward_canonical) = L) -> exists fom_value_pfp_normalization_associates_backward_canonical. ((((exists fom_beta_height_pfp_normalization_associates_backward_canonical_entry. fom_beta_height_pfp_normalization_associates_backward_canonical_entry + S (fom_value_pfp_normalization_associates_backward_canonical) = S ((S (fom_index_pfp_normalization_associates_backward_canonical)) * gc)) /\\ exists fom_beta_quotient_pfp_normalization_associates_backward_canonical_entry. gb = fom_beta_quotient_pfp_normalization_associates_backward_canonical_entry * S ((S (fom_index_pfp_normalization_associates_backward_canonical)) * gc) + (fom_value_pfp_normalization_associates_backward_canonical))) /\\ (exists fom_gap_pfp_normalization_associates_backward_canonical_value_bound. fom_gap_pfp_normalization_associates_backward_canonical_value_bound + S (fom_value_pfp_normalization_associates_backward_canonical) = p))) /\\ ((exists pfen_qb_normalization_associates_backward pfen_qc_normalization_associates_backward pfen_qlen_normalization_associates_backward pfen_pb_normalization_associates_backward pfen_pc_normalization_associates_backward pfen_plen_normalization_associates_backward. ((((forall fom_index_pfp_normalization_associates_backward_productleft. (exists fom_gap_pfp_normalization_associates_backward_productleft_index_bound. fom_gap_pfp_normalization_associates_backward_productleft_index_bound + S (fom_index_pfp_normalization_associates_backward_productleft) = pfen_qlen_normalization_associates_backward) -> exists fom_value_pfp_normalization_associates_backward_productleft. ((((exists fom_beta_height_pfp_normalization_associates_backward_productleft_entry. fom_beta_height_pfp_normalization_associates_backward_productleft_entry + S (fom_value_pfp_normalization_associates_backward_productleft) = S ((S (fom_index_pfp_normalization_associates_backward_productleft)) * pfen_qc_normalization_associates_backward)) /\\ exists fom_beta_quotient_pfp_normalization_associates_backward_productleft_entry. pfen_qb_normalization_associates_backward = fom_beta_quotient_pfp_normalization_associates_backward_productleft_entry * S ((S (fom_index_pfp_normalization_associates_backward_productleft)) * pfen_qc_normalization_associates_backward) + (fom_value_pfp_normalization_associates_backward_productleft))) /\\ (exists fom_gap_pfp_normalization_associates_backward_productleft_value_bound. fom_gap_pfp_normalization_associates_backward_productleft_value_bound + S (fom_value_pfp_normalization_associates_backward_productleft) = p))) /\\ (((forall fom_index_pfp_normalization_associates_backward_productright. (exists fom_gap_pfp_normalization_associates_backward_productright_index_bound. fom_gap_pfp_normalization_associates_backward_productright_index_bound + S (fom_index_pfp_normalization_associates_backward_productright) = L) -> exists fom_value_pfp_normalization_associates_backward_productright. ((((exists fom_beta_height_pfp_normalization_associates_backward_productright_entry. fom_beta_height_pfp_normalization_associates_backward_productright_entry + S (fom_value_pfp_normalization_associates_backward_productright) = S ((S (fom_index_pfp_normalization_associates_backward_productright)) * hc)) /\\ exists fom_beta_quotient_pfp_normalization_associates_backward_productright_entry. hb = fom_beta_quotient_pfp_normalization_associates_backward_productright_entry * S ((S (fom_index_pfp_normalization_associates_backward_productright)) * hc) + (fom_value_pfp_normalization_associates_backward_productright))) /\\ (exists fom_gap_pfp_normalization_associates_backward_productright_value_bound. fom_gap_pfp_normalization_associates_backward_productright_value_bound + S (fom_value_pfp_normalization_associates_backward_productright) = p))) /\\ (((((((pfen_qlen_normalization_associates_backward)=0 \\/ (L)=0) /\\ (((pfen_plen_normalization_associates_backward)=0)))) \\/ (((~((pfen_qlen_normalization_associates_backward)=0)) /\\ (((~((L)=0)) /\\ (((pfen_qlen_normalization_associates_backward)+(L)=S (pfen_plen_normalization_associates_backward)))))))) /\\ ((forall pfc_index_normalization_associates_backward_productcoefficients. (exists pfa_gap_normalization_associates_backward_productcoefficientsbound. pfa_gap_normalization_associates_backward_productcoefficientsbound + S (pfc_index_normalization_associates_backward_productcoefficients) = (pfen_plen_normalization_associates_backward)) -> exists pfc_value_normalization_associates_backward_productcoefficients. ((((exists ff_h_pfp_normalization_associates_backward_productcoefficientsentry. ff_h_pfp_normalization_associates_backward_productcoefficientsentry + S (pfc_value_normalization_associates_backward_productcoefficients) = S ((S (pfc_index_normalization_associates_backward_productcoefficients)) * pfen_pc_normalization_associates_backward)) /\\ exists ff_q_pfp_normalization_associates_backward_productcoefficientsentry. pfen_pb_normalization_associates_backward = ff_q_pfp_normalization_associates_backward_productcoefficientsentry * S ((S (pfc_index_normalization_associates_backward_productcoefficients)) * pfen_pc_normalization_associates_backward) + (pfc_value_normalization_associates_backward_productcoefficients))) /\\ ((exists pfc_terms_code_normalization_associates_backward_productcoefficientscoefficient pfc_terms_scale_normalization_associates_backward_productcoefficientscoefficient pfc_natural_sum_normalization_associates_backward_productcoefficientscoefficient. ((forall pfc_index_normalization_associates_backward_productcoefficientscoefficientdiagonal. (exists pfa_gap_normalization_associates_backward_productcoefficientscoefficientdiagonalbound. pfa_gap_normalization_associates_backward_productcoefficientscoefficientdiagonalbound + S (pfc_index_normalization_associates_backward_productcoefficientscoefficientdiagonal) = (S (pfc_index_normalization_associates_backward_productcoefficients))) -> exists pfc_value_normalization_associates_backward_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalization_associates_backward_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normalization_associates_backward_productcoefficientscoefficientdiagonalentry + S (pfc_value_normalization_associates_backward_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalization_associates_backward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalization_associates_backward_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normalization_associates_backward_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normalization_associates_backward_productcoefficientscoefficient = ff_q_pfp_normalization_associates_backward_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalization_associates_backward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalization_associates_backward_productcoefficientscoefficient) + (pfc_value_normalization_associates_backward_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalization_associates_backward_productcoefficientscoefficientdiagonalterm pfc_left_normalization_associates_backward_productcoefficientscoefficientdiagonalterm pfc_right_normalization_associates_backward_productcoefficientscoefficientdiagonalterm. (((pfc_index_normalization_associates_backward_productcoefficientscoefficientdiagonal)+pfc_complement_normalization_associates_backward_productcoefficientscoefficientdiagonalterm=(pfc_index_normalization_associates_backward_productcoefficients)) /\\ ((((((exists pfa_gap_normalization_associates_backward_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalization_associates_backward_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalization_associates_backward_productcoefficientscoefficientdiagonal) = (pfen_qlen_normalization_associates_backward)) /\\ ((((exists ff_h_pfp_normalization_associates_backward_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalization_associates_backward_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalization_associates_backward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalization_associates_backward_productcoefficientscoefficientdiagonal)) * pfen_qc_normalization_associates_backward)) /\\ exists ff_q_pfp_normalization_associates_backward_productcoefficientscoefficientdiagonaltermleftentry. pfen_qb_normalization_associates_backward = ff_q_pfp_normalization_associates_backward_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalization_associates_backward_productcoefficientscoefficientdiagonal)) * pfen_qc_normalization_associates_backward) + (pfc_left_normalization_associates_backward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_associates_backward_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalization_associates_backward_productcoefficientscoefficientdiagonaltermleftoutside+(pfen_qlen_normalization_associates_backward)=(pfc_index_normalization_associates_backward_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalization_associates_backward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalization_associates_backward_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalization_associates_backward_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalization_associates_backward_productcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_normalization_associates_backward_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalization_associates_backward_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalization_associates_backward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalization_associates_backward_productcoefficientscoefficientdiagonalterm)) * hc)) /\\ exists ff_q_pfp_normalization_associates_backward_productcoefficientscoefficientdiagonaltermrightentry. hb = ff_q_pfp_normalization_associates_backward_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalization_associates_backward_productcoefficientscoefficientdiagonalterm)) * hc) + (pfc_right_normalization_associates_backward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalization_associates_backward_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalization_associates_backward_productcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_normalization_associates_backward_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalization_associates_backward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalization_associates_backward_productcoefficientscoefficientdiagonal)=pfc_left_normalization_associates_backward_productcoefficientscoefficientdiagonalterm*pfc_right_normalization_associates_backward_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalization_associates_backward_productcoefficientscoefficientsum fs_v_pfc_normalization_associates_backward_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_start. fs_h_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalization_associates_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_start. fs_u_pfc_normalization_associates_backward_productcoefficientscoefficientsum = fs_q_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalization_associates_backward_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalization_associates_backward_productcoefficientscoefficient) = S ((S (S (pfc_index_normalization_associates_backward_productcoefficients))) * fs_v_pfc_normalization_associates_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalization_associates_backward_productcoefficientscoefficientsum = fs_q_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalization_associates_backward_productcoefficients))) * fs_v_pfc_normalization_associates_backward_productcoefficientscoefficientsum) + (pfc_natural_sum_normalization_associates_backward_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps = S (pfc_index_normalization_associates_backward_productcoefficients)) -> exists fs_a_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps fs_r_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps fs_s_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalization_associates_backward_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalization_associates_backward_productcoefficientscoefficient = fs_q_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalization_associates_backward_productcoefficientscoefficient) + (fs_a_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_associates_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalization_associates_backward_productcoefficientscoefficientsum = fs_q_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_associates_backward_productcoefficientscoefficientsum) + (fs_r_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_associates_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalization_associates_backward_productcoefficientscoefficientsum = fs_q_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalization_associates_backward_productcoefficientscoefficientsum) + (fs_s_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normalization_associates_backward_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalization_associates_backward_productcoefficientscoefficientresiduebound. pfa_gap_normalization_associates_backward_productcoefficientscoefficientresiduebound + S (pfc_value_normalization_associates_backward_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalization_associates_backward_productcoefficientscoefficientresiduecongruence pfa_offset_right_normalization_associates_backward_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalization_associates_backward_productcoefficientscoefficient) + (p) * pfa_offset_left_normalization_associates_backward_productcoefficientscoefficientresiduecongruence = (pfc_value_normalization_associates_backward_productcoefficients) + (p) * pfa_offset_right_normalization_associates_backward_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normalization_associates_backward_target pfrep_left_normalization_associates_backward_target pfrep_right_normalization_associates_backward_target. ((exists pfrep_position_normalization_associates_backward_targetfirst. ((pfrep_position_normalization_associates_backward_targetfirst+S (pfrep_power_normalization_associates_backward_target)=(pfen_plen_normalization_associates_backward)) /\\ ((((exists ff_h_pfp_normalization_associates_backward_targetfirstentry. ff_h_pfp_normalization_associates_backward_targetfirstentry + S (pfrep_left_normalization_associates_backward_target) = S ((S (pfrep_position_normalization_associates_backward_targetfirst)) * pfen_pc_normalization_associates_backward)) /\\ exists ff_q_pfp_normalization_associates_backward_targetfirstentry. pfen_pb_normalization_associates_backward = ff_q_pfp_normalization_associates_backward_targetfirstentry * S ((S (pfrep_position_normalization_associates_backward_targetfirst)) * pfen_pc_normalization_associates_backward) + (pfrep_left_normalization_associates_backward_target)))))) \\/ (((exists pfrep_gap_normalization_associates_backward_targetfirstoutside. pfrep_gap_normalization_associates_backward_targetfirstoutside+(pfen_plen_normalization_associates_backward)=(pfrep_power_normalization_associates_backward_target)) /\\ (((pfrep_left_normalization_associates_backward_target)=0))))) -> ((exists pfrep_position_normalization_associates_backward_targetsecond. ((pfrep_position_normalization_associates_backward_targetsecond+S (pfrep_power_normalization_associates_backward_target)=(L)) /\\ ((((exists ff_h_pfp_normalization_associates_backward_targetsecondentry. ff_h_pfp_normalization_associates_backward_targetsecondentry + S (pfrep_right_normalization_associates_backward_target) = S ((S (pfrep_position_normalization_associates_backward_targetsecond)) * gc)) /\\ exists ff_q_pfp_normalization_associates_backward_targetsecondentry. gb = ff_q_pfp_normalization_associates_backward_targetsecondentry * S ((S (pfrep_position_normalization_associates_backward_targetsecond)) * gc) + (pfrep_right_normalization_associates_backward_target)))))) \\/ (((exists pfrep_gap_normalization_associates_backward_targetsecondoutside. pfrep_gap_normalization_associates_backward_targetsecondoutside+(L)=(pfrep_power_normalization_associates_backward_target)) /\\ (((pfrep_right_normalization_associates_backward_target)=0))))) -> pfrep_left_normalization_associates_backward_target=pfrep_right_normalization_associates_backward_target))))))))))", "statement_sha256": "bf3f1a6b027ea739a4f9b9f12140d145f94b2571b3a06d7fe6813ddbdfe43359", "summary": "An actual monic normalization and its genuinely inverted scalar action supply actual right-divisibility witnesses in both directions. 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"intro p", "intro ab", "intro ac", "intro L", "intro hp", "intro hA", "have hp0 : ~(p=0)", "intro hpzero", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hpzero", "have htrim : exists t tb tc M. (((L)=(t)+(M)) /\\ (((forall fom_index_pfp_normalized_associate_triminput. (exists fom_gap_pfp_normalized_associate_triminput_index_bound. fom_gap_pfp_normalized_associate_triminput_index_bound + S (fom_index_pfp_normalized_associate_triminput) = L) -> exists fom_value_pfp_normalized_associate_triminput. ((((exists fom_beta_height_pfp_normalized_associate_triminput_entry. fom_beta_height_pfp_normalized_associate_triminput_entry + S (fom_value_pfp_normalized_associate_triminput) = S ((S (fom_index_pfp_normalized_associate_triminput)) * ac)) /\\ exists fom_beta_quotient_pfp_normalized_associate_triminput_entry. ab = fom_beta_quotient_pfp_normalized_associate_triminput_entry * S ((S (fom_index_pfp_normalized_associate_triminput)) * ac) + (fom_value_pfp_normalized_associate_triminput))) /\\ (exists fom_gap_pfp_normalized_associate_triminput_value_bound. fom_gap_pfp_normalized_associate_triminput_value_bound + S (fom_value_pfp_normalized_associate_triminput) = p))) /\\ (((forall pfp_repeat_index_normalized_associate_trimremoved. (exists pfa_gap_normalized_associate_trimremovedindex. pfa_gap_normalized_associate_trimremovedindex + S (pfp_repeat_index_normalized_associate_trimremoved) = (t)) -> (((exists ff_h_pfp_normalized_associate_trimremovedentry. ff_h_pfp_normalized_associate_trimremovedentry + S (0) = S ((S (pfp_repeat_index_normalized_associate_trimremoved)) * ac)) /\\ exists ff_q_pfp_normalized_associate_trimremovedentry. ab = ff_q_pfp_normalized_associate_trimremovedentry * S ((S (pfp_repeat_index_normalized_associate_trimremoved)) * ac) + (0)))) /\\ (((forall pftrim_index_normalized_associate_trimsuffix pftrim_value_normalized_associate_trimsuffix. (exists pfa_gap_normalized_associate_trimsuffixbound. pfa_gap_normalized_associate_trimsuffixbound + S (pftrim_index_normalized_associate_trimsuffix) = (M)) -> (((exists ff_h_pfp_normalized_associate_trimsuffixsource. ff_h_pfp_normalized_associate_trimsuffixsource + S (pftrim_value_normalized_associate_trimsuffix) = S ((S ((t)+pftrim_index_normalized_associate_trimsuffix)) * ac)) /\\ exists ff_q_pfp_normalized_associate_trimsuffixsource. ab = ff_q_pfp_normalized_associate_trimsuffixsource * S ((S ((t)+pftrim_index_normalized_associate_trimsuffix)) * ac) + (pftrim_value_normalized_associate_trimsuffix))) -> (((exists ff_h_pfp_normalized_associate_trimsuffixoutput. ff_h_pfp_normalized_associate_trimsuffixoutput + S (pftrim_value_normalized_associate_trimsuffix) = S ((S (pftrim_index_normalized_associate_trimsuffix)) * tc)) /\\ exists ff_q_pfp_normalized_associate_trimsuffixoutput. tb = ff_q_pfp_normalized_associate_trimsuffixoutput * S ((S (pftrim_index_normalized_associate_trimsuffix)) * tc) + (pftrim_value_normalized_associate_trimsuffix)))) /\\ (((M)=0 \\/ (exists pftrim_leading_normalized_associate_trimnormal. ((((exists ff_h_pfp_normalized_associate_trimnormalentry. ff_h_pfp_normalized_associate_trimnormalentry + S (pftrim_leading_normalized_associate_trimnormal) = S ((S (0)) * tc)) /\\ exists ff_q_pfp_normalized_associate_trimnormalentry. tb = ff_q_pfp_normalized_associate_trimnormalentry * S ((S (0)) * tc) + (pftrim_leading_normalized_associate_trimnormal))) /\\ ((~(pftrim_leading_normalized_associate_trimnormal=0))))))))))))))", "specialize prime_field_polynomial_trim_exists (p)", "specialize prime_field_polynomial_trim_exists (ab)", "specialize prime_field_polynomial_trim_exists (ac)", "specialize prime_field_polynomial_trim_exists (L)", "apply prime_field_polynomial_trim_exists", "exact hA", "cases htrim", "cases htrim_witness", "cases htrim_witness_witness", "cases htrim_witness_witness_witness", "have hT : forall fom_index_pfp_normalized_associate_trim_bound. (exists fom_gap_pfp_normalized_associate_trim_bound_index_bound. fom_gap_pfp_normalized_associate_trim_bound_index_bound + S (fom_index_pfp_normalized_associate_trim_bound) = x3) -> exists fom_value_pfp_normalized_associate_trim_bound. ((((exists fom_beta_height_pfp_normalized_associate_trim_bound_entry. fom_beta_height_pfp_normalized_associate_trim_bound_entry + S (fom_value_pfp_normalized_associate_trim_bound) = S ((S (fom_index_pfp_normalized_associate_trim_bound)) * x2)) /\\ exists fom_beta_quotient_pfp_normalized_associate_trim_bound_entry. x1 = fom_beta_quotient_pfp_normalized_associate_trim_bound_entry * S ((S (fom_index_pfp_normalized_associate_trim_bound)) * x2) + (fom_value_pfp_normalized_associate_trim_bound))) /\\ (exists fom_gap_pfp_normalized_associate_trim_bound_value_bound. fom_gap_pfp_normalized_associate_trim_bound_value_bound + S (fom_value_pfp_normalized_associate_trim_bound) = p))", "specialize prime_field_polynomial_trim_output_coefficients (p)", "specialize prime_field_polynomial_trim_output_coefficients (ab)", "specialize prime_field_polynomial_trim_output_coefficients (ac)", "specialize prime_field_polynomial_trim_output_coefficients (L)", "specialize prime_field_polynomial_trim_output_coefficients (x)", "specialize prime_field_polynomial_trim_output_coefficients (x1)", "specialize prime_field_polynomial_trim_output_coefficients (x2)", "specialize prime_field_polynomial_trim_output_coefficients (x3)", "apply prime_field_polynomial_trim_output_coefficients", "exact htrim_witness_witness_witness_witness", "have hTA : forall pfrep_power_normalized_associate_trim_equivalent pfrep_left_normalized_associate_trim_equivalent pfrep_right_normalized_associate_trim_equivalent. ((exists pfrep_position_normalized_associate_trim_equivalentfirst. ((pfrep_position_normalized_associate_trim_equivalentfirst+S (pfrep_power_normalized_associate_trim_equivalent)=(x3)) /\\ ((((exists ff_h_pfp_normalized_associate_trim_equivalentfirstentry. ff_h_pfp_normalized_associate_trim_equivalentfirstentry + S (pfrep_left_normalized_associate_trim_equivalent) = S ((S (pfrep_position_normalized_associate_trim_equivalentfirst)) * x2)) /\\ exists ff_q_pfp_normalized_associate_trim_equivalentfirstentry. x1 = ff_q_pfp_normalized_associate_trim_equivalentfirstentry * S ((S (pfrep_position_normalized_associate_trim_equivalentfirst)) * x2) + (pfrep_left_normalized_associate_trim_equivalent)))))) \\/ (((exists pfrep_gap_normalized_associate_trim_equivalentfirstoutside. pfrep_gap_normalized_associate_trim_equivalentfirstoutside+(x3)=(pfrep_power_normalized_associate_trim_equivalent)) /\\ (((pfrep_left_normalized_associate_trim_equivalent)=0))))) -> ((exists pfrep_position_normalized_associate_trim_equivalentsecond. ((pfrep_position_normalized_associate_trim_equivalentsecond+S (pfrep_power_normalized_associate_trim_equivalent)=(L)) /\\ ((((exists ff_h_pfp_normalized_associate_trim_equivalentsecondentry. ff_h_pfp_normalized_associate_trim_equivalentsecondentry + S (pfrep_right_normalized_associate_trim_equivalent) = S ((S (pfrep_position_normalized_associate_trim_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_normalized_associate_trim_equivalentsecondentry. ab = ff_q_pfp_normalized_associate_trim_equivalentsecondentry * S ((S (pfrep_position_normalized_associate_trim_equivalentsecond)) * ac) + (pfrep_right_normalized_associate_trim_equivalent)))))) \\/ (((exists pfrep_gap_normalized_associate_trim_equivalentsecondoutside. pfrep_gap_normalized_associate_trim_equivalentsecondoutside+(L)=(pfrep_power_normalized_associate_trim_equivalent)) /\\ (((pfrep_right_normalized_associate_trim_equivalent)=0))))) -> pfrep_left_normalized_associate_trim_equivalent=pfrep_right_normalized_associate_trim_equivalent", "specialize prime_field_polynomial_equivalent_symmetric (ab)", "specialize prime_field_polynomial_equivalent_symmetric (ac)", "specialize prime_field_polynomial_equivalent_symmetric (L)", "specialize prime_field_polynomial_equivalent_symmetric (x1)", "specialize prime_field_polynomial_equivalent_symmetric (x2)", "specialize prime_field_polynomial_equivalent_symmetric (x3)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_trim_equivalent (p)", "specialize prime_field_polynomial_trim_equivalent (ab)", "specialize prime_field_polynomial_trim_equivalent (ac)", "specialize prime_field_polynomial_trim_equivalent (L)", "specialize prime_field_polynomial_trim_equivalent (x)", "specialize prime_field_polynomial_trim_equivalent (x1)", "specialize prime_field_polynomial_trim_equivalent (x2)", "specialize prime_field_polynomial_trim_equivalent (x3)", "apply prime_field_polynomial_trim_equivalent", "exact htrim_witness_witness_witness_witness", "have hcase : x3=0 \\/ ~(x3=0)", "specialize eq_decidable (x3)", "specialize eq_decidable (0)", "apply eq_decidable", "cases hcase", "rewrite hcase_left at hT", "rewrite hcase_left at hTA", "rewrite hcase_left at hTA", "exists x1", "exists x2", "exists 0", "split", "left", "refl", "split", "specialize prime_field_polynomial_right_divides_empty (p)", "specialize prime_field_polynomial_right_divides_empty (ab)", "specialize prime_field_polynomial_right_divides_empty (ac)", "specialize prime_field_polynomial_right_divides_empty (L)", "specialize prime_field_polynomial_right_divides_empty (x1)", "specialize prime_field_polynomial_right_divides_empty (x2)", "apply prime_field_polynomial_right_divides_empty", "exact hA", "specialize prime_field_polynomial_right_divides_equivalent_target (p)", "specialize prime_field_polynomial_right_divides_equivalent_target (x1)", "specialize prime_field_polynomial_right_divides_equivalent_target (x2)", "specialize prime_field_polynomial_right_divides_equivalent_target (0)", "specialize prime_field_polynomial_right_divides_equivalent_target (x1)", "specialize prime_field_polynomial_right_divides_equivalent_target (x2)", "specialize prime_field_polynomial_right_divides_equivalent_target (0)", "specialize prime_field_polynomial_right_divides_equivalent_target (ab)", "specialize prime_field_polynomial_right_divides_equivalent_target (ac)", "specialize prime_field_polynomial_right_divides_equivalent_target (L)", "apply prime_field_polynomial_right_divides_equivalent_target", "exact hA", "specialize prime_field_polynomial_right_divides_empty (p)", "specialize prime_field_polynomial_right_divides_empty (x1)", "specialize prime_field_polynomial_right_divides_empty (x2)", "specialize prime_field_polynomial_right_divides_empty (0)", "specialize prime_field_polynomial_right_divides_empty (x1)", "specialize prime_field_polynomial_right_divides_empty (x2)", "apply prime_field_polynomial_right_divides_empty", "exact hT", "exact hTA", "have hdegree : exists d. (((x3)=S (d)) /\\ (((forall fom_index_pfp_normalized_associate_degreecoefficients. (exists fom_gap_pfp_normalized_associate_degreecoefficients_index_bound. fom_gap_pfp_normalized_associate_degreecoefficients_index_bound + S (fom_index_pfp_normalized_associate_degreecoefficients) = x3) -> exists fom_value_pfp_normalized_associate_degreecoefficients. ((((exists fom_beta_height_pfp_normalized_associate_degreecoefficients_entry. fom_beta_height_pfp_normalized_associate_degreecoefficients_entry + S (fom_value_pfp_normalized_associate_degreecoefficients) = S ((S (fom_index_pfp_normalized_associate_degreecoefficients)) * x2)) /\\ exists fom_beta_quotient_pfp_normalized_associate_degreecoefficients_entry. x1 = fom_beta_quotient_pfp_normalized_associate_degreecoefficients_entry * S ((S (fom_index_pfp_normalized_associate_degreecoefficients)) * x2) + (fom_value_pfp_normalized_associate_degreecoefficients))) /\\ (exists fom_gap_pfp_normalized_associate_degreecoefficients_value_bound. fom_gap_pfp_normalized_associate_degreecoefficients_value_bound + S (fom_value_pfp_normalized_associate_degreecoefficients) = p))) /\\ ((exists pfd_leading_normalized_associate_degree. ((((exists ff_h_pfp_normalized_associate_degreeentry. ff_h_pfp_normalized_associate_degreeentry + S (pfd_leading_normalized_associate_degree) = S ((S (0)) * x2)) /\\ exists ff_q_pfp_normalized_associate_degreeentry. x1 = ff_q_pfp_normalized_associate_degreeentry * S ((S (0)) * x2) + (pfd_leading_normalized_associate_degree))) /\\ ((~(pfd_leading_normalized_associate_degree=0)))))))))", "specialize prime_field_polynomial_trim_nonempty_degree_exists (p)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (ab)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (ac)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (L)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (x)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (x1)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (x2)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (x3)", "apply prime_field_polynomial_trim_nonempty_degree_exists", "exact htrim_witness_witness_witness_witness", "exact hcase_right", "cases hdegree", "have hnormalization : exists k hb hc. ((~((x3) = 0)) /\\ (((exists pfm_leading_normalized_associate_monic. ((((exists ff_h_pfp_normalized_associate_monicsource. ff_h_pfp_normalized_associate_monicsource + S (pfm_leading_normalized_associate_monic) = S ((S (0)) * x2)) /\\ exists ff_q_pfp_normalized_associate_monicsource. x1 = ff_q_pfp_normalized_associate_monicsource * S ((S (0)) * x2) + (pfm_leading_normalized_associate_monic))) /\\ ((((~((pfm_leading_normalized_associate_monic) = 0)) /\\ ((((exists pfa_gap_normalized_associate_monicinversemultiplicationleft. pfa_gap_normalized_associate_monicinversemultiplicationleft + S (pfm_leading_normalized_associate_monic) = (p)) /\\ (((exists pfa_gap_normalized_associate_monicinversemultiplicationright. pfa_gap_normalized_associate_monicinversemultiplicationright + S (k) = (p)) /\\ ((((exists pfa_gap_normalized_associate_monicinversemultiplicationresultbound. pfa_gap_normalized_associate_monicinversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_normalized_associate_monicinversemultiplicationresultcongruence pfa_offset_right_normalized_associate_monicinversemultiplicationresultcongruence. ((pfm_leading_normalized_associate_monic) * (k)) + (p) * pfa_offset_left_normalized_associate_monicinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalized_associate_monicinversemultiplicationresultcongruence))))))))))))))) /\\ ((((exists pfa_gap_normalized_associate_monicscalescalar. pfa_gap_normalized_associate_monicscalescalar + S (k) = (p)) /\\ ((forall pfp_index_normalized_associate_monicscale. (exists pfa_gap_normalized_associate_monicscaleindex. pfa_gap_normalized_associate_monicscaleindex + S (pfp_index_normalized_associate_monicscale) = (x3)) -> exists pfp_source_normalized_associate_monicscale pfp_value_normalized_associate_monicscale. ((((exists ff_h_pfp_normalized_associate_monicscalesource. ff_h_pfp_normalized_associate_monicscalesource + S (pfp_source_normalized_associate_monicscale) = S ((S (pfp_index_normalized_associate_monicscale)) * x2)) /\\ exists ff_q_pfp_normalized_associate_monicscalesource. x1 = ff_q_pfp_normalized_associate_monicscalesource * S ((S (pfp_index_normalized_associate_monicscale)) * x2) + (pfp_source_normalized_associate_monicscale))) /\\ (((((exists ff_h_pfp_normalized_associate_monicscaletarget. ff_h_pfp_normalized_associate_monicscaletarget + S (pfp_value_normalized_associate_monicscale) = S ((S (pfp_index_normalized_associate_monicscale)) * hc)) /\\ exists ff_q_pfp_normalized_associate_monicscaletarget. hb = ff_q_pfp_normalized_associate_monicscaletarget * S ((S (pfp_index_normalized_associate_monicscale)) * hc) + (pfp_value_normalized_associate_monicscale))) /\\ ((((exists pfa_gap_normalized_associate_monicscaleoperationleft. pfa_gap_normalized_associate_monicscaleoperationleft + S (k) = (p)) /\\ (((exists pfa_gap_normalized_associate_monicscaleoperationright. pfa_gap_normalized_associate_monicscaleoperationright + S (pfp_source_normalized_associate_monicscale) = (p)) /\\ ((((exists pfa_gap_normalized_associate_monicscaleoperationresultbound. pfa_gap_normalized_associate_monicscaleoperationresultbound + S (pfp_value_normalized_associate_monicscale) = (p)) /\\ ((exists pfa_offset_left_normalized_associate_monicscaleoperationresultcongruence pfa_offset_right_normalized_associate_monicscaleoperationresultcongruence. ((k) * (pfp_source_normalized_associate_monicscale)) + (p) * pfa_offset_left_normalized_associate_monicscaleoperationresultcongruence = (pfp_value_normalized_associate_monicscale) + (p) * pfa_offset_right_normalized_associate_monicscaleoperationresultcongruence)))))))))))))))))))))", "specialize prime_field_polynomial_monic_normalization_exists (p)", "specialize prime_field_polynomial_monic_normalization_exists (x1)", "specialize prime_field_polynomial_monic_normalization_exists (x2)", "specialize prime_field_polynomial_monic_normalization_exists (x3)", "specialize prime_field_polynomial_monic_normalization_exists (x4)", "apply prime_field_polynomial_monic_normalization_exists", "exact hp", "exact hdegree_witness", "cases hnormalization", "cases hnormalization_witness", "cases hnormalization_witness_witness", "have hassociates : ((((forall fom_index_pfp_normalized_associate_forward_canonical. (exists fom_gap_pfp_normalized_associate_forward_canonical_index_bound. fom_gap_pfp_normalized_associate_forward_canonical_index_bound + S (fom_index_pfp_normalized_associate_forward_canonical) = x3) -> exists fom_value_pfp_normalized_associate_forward_canonical. ((((exists fom_beta_height_pfp_normalized_associate_forward_canonical_entry. fom_beta_height_pfp_normalized_associate_forward_canonical_entry + S (fom_value_pfp_normalized_associate_forward_canonical) = S ((S (fom_index_pfp_normalized_associate_forward_canonical)) * x7)) /\\ exists fom_beta_quotient_pfp_normalized_associate_forward_canonical_entry. x6 = fom_beta_quotient_pfp_normalized_associate_forward_canonical_entry * S ((S (fom_index_pfp_normalized_associate_forward_canonical)) * x7) + (fom_value_pfp_normalized_associate_forward_canonical))) /\\ (exists fom_gap_pfp_normalized_associate_forward_canonical_value_bound. fom_gap_pfp_normalized_associate_forward_canonical_value_bound + S (fom_value_pfp_normalized_associate_forward_canonical) = p))) /\\ ((exists pfen_qb_normalized_associate_forward pfen_qc_normalized_associate_forward pfen_qlen_normalized_associate_forward pfen_pb_normalized_associate_forward pfen_pc_normalized_associate_forward pfen_plen_normalized_associate_forward. ((((forall fom_index_pfp_normalized_associate_forward_productleft. (exists fom_gap_pfp_normalized_associate_forward_productleft_index_bound. fom_gap_pfp_normalized_associate_forward_productleft_index_bound + S (fom_index_pfp_normalized_associate_forward_productleft) = pfen_qlen_normalized_associate_forward) -> exists fom_value_pfp_normalized_associate_forward_productleft. ((((exists fom_beta_height_pfp_normalized_associate_forward_productleft_entry. fom_beta_height_pfp_normalized_associate_forward_productleft_entry + S (fom_value_pfp_normalized_associate_forward_productleft) = S ((S (fom_index_pfp_normalized_associate_forward_productleft)) * pfen_qc_normalized_associate_forward)) /\\ exists fom_beta_quotient_pfp_normalized_associate_forward_productleft_entry. pfen_qb_normalized_associate_forward = fom_beta_quotient_pfp_normalized_associate_forward_productleft_entry * S ((S (fom_index_pfp_normalized_associate_forward_productleft)) * pfen_qc_normalized_associate_forward) + (fom_value_pfp_normalized_associate_forward_productleft))) /\\ (exists fom_gap_pfp_normalized_associate_forward_productleft_value_bound. fom_gap_pfp_normalized_associate_forward_productleft_value_bound + S (fom_value_pfp_normalized_associate_forward_productleft) = p))) /\\ (((forall fom_index_pfp_normalized_associate_forward_productright. (exists fom_gap_pfp_normalized_associate_forward_productright_index_bound. fom_gap_pfp_normalized_associate_forward_productright_index_bound + S (fom_index_pfp_normalized_associate_forward_productright) = x3) -> exists fom_value_pfp_normalized_associate_forward_productright. ((((exists fom_beta_height_pfp_normalized_associate_forward_productright_entry. fom_beta_height_pfp_normalized_associate_forward_productright_entry + S (fom_value_pfp_normalized_associate_forward_productright) = S ((S (fom_index_pfp_normalized_associate_forward_productright)) * x2)) /\\ exists fom_beta_quotient_pfp_normalized_associate_forward_productright_entry. x1 = fom_beta_quotient_pfp_normalized_associate_forward_productright_entry * S ((S (fom_index_pfp_normalized_associate_forward_productright)) * x2) + (fom_value_pfp_normalized_associate_forward_productright))) /\\ (exists fom_gap_pfp_normalized_associate_forward_productright_value_bound. fom_gap_pfp_normalized_associate_forward_productright_value_bound + S (fom_value_pfp_normalized_associate_forward_productright) = p))) /\\ (((((((pfen_qlen_normalized_associate_forward)=0 \\/ (x3)=0) /\\ (((pfen_plen_normalized_associate_forward)=0)))) \\/ (((~((pfen_qlen_normalized_associate_forward)=0)) /\\ (((~((x3)=0)) /\\ (((pfen_qlen_normalized_associate_forward)+(x3)=S (pfen_plen_normalized_associate_forward)))))))) /\\ ((forall pfc_index_normalized_associate_forward_productcoefficients. (exists pfa_gap_normalized_associate_forward_productcoefficientsbound. pfa_gap_normalized_associate_forward_productcoefficientsbound + S (pfc_index_normalized_associate_forward_productcoefficients) = (pfen_plen_normalized_associate_forward)) -> exists pfc_value_normalized_associate_forward_productcoefficients. ((((exists ff_h_pfp_normalized_associate_forward_productcoefficientsentry. ff_h_pfp_normalized_associate_forward_productcoefficientsentry + S (pfc_value_normalized_associate_forward_productcoefficients) = S ((S (pfc_index_normalized_associate_forward_productcoefficients)) * pfen_pc_normalized_associate_forward)) /\\ exists ff_q_pfp_normalized_associate_forward_productcoefficientsentry. pfen_pb_normalized_associate_forward = ff_q_pfp_normalized_associate_forward_productcoefficientsentry * S ((S (pfc_index_normalized_associate_forward_productcoefficients)) * pfen_pc_normalized_associate_forward) + (pfc_value_normalized_associate_forward_productcoefficients))) /\\ ((exists pfc_terms_code_normalized_associate_forward_productcoefficientscoefficient pfc_terms_scale_normalized_associate_forward_productcoefficientscoefficient pfc_natural_sum_normalized_associate_forward_productcoefficientscoefficient. ((forall pfc_index_normalized_associate_forward_productcoefficientscoefficientdiagonal. (exists pfa_gap_normalized_associate_forward_productcoefficientscoefficientdiagonalbound. pfa_gap_normalized_associate_forward_productcoefficientscoefficientdiagonalbound + S (pfc_index_normalized_associate_forward_productcoefficientscoefficientdiagonal) = (S (pfc_index_normalized_associate_forward_productcoefficients))) -> exists pfc_value_normalized_associate_forward_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalized_associate_forward_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normalized_associate_forward_productcoefficientscoefficientdiagonalentry + S (pfc_value_normalized_associate_forward_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalized_associate_forward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_associate_forward_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normalized_associate_forward_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normalized_associate_forward_productcoefficientscoefficient = ff_q_pfp_normalized_associate_forward_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalized_associate_forward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_associate_forward_productcoefficientscoefficient) + (pfc_value_normalized_associate_forward_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalized_associate_forward_productcoefficientscoefficientdiagonalterm pfc_left_normalized_associate_forward_productcoefficientscoefficientdiagonalterm pfc_right_normalized_associate_forward_productcoefficientscoefficientdiagonalterm. (((pfc_index_normalized_associate_forward_productcoefficientscoefficientdiagonal)+pfc_complement_normalized_associate_forward_productcoefficientscoefficientdiagonalterm=(pfc_index_normalized_associate_forward_productcoefficients)) /\\ ((((((exists pfa_gap_normalized_associate_forward_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalized_associate_forward_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalized_associate_forward_productcoefficientscoefficientdiagonal) = (pfen_qlen_normalized_associate_forward)) /\\ ((((exists ff_h_pfp_normalized_associate_forward_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalized_associate_forward_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalized_associate_forward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalized_associate_forward_productcoefficientscoefficientdiagonal)) * pfen_qc_normalized_associate_forward)) /\\ exists ff_q_pfp_normalized_associate_forward_productcoefficientscoefficientdiagonaltermleftentry. pfen_qb_normalized_associate_forward = ff_q_pfp_normalized_associate_forward_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalized_associate_forward_productcoefficientscoefficientdiagonal)) * pfen_qc_normalized_associate_forward) + (pfc_left_normalized_associate_forward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_associate_forward_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalized_associate_forward_productcoefficientscoefficientdiagonaltermleftoutside+(pfen_qlen_normalized_associate_forward)=(pfc_index_normalized_associate_forward_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalized_associate_forward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalized_associate_forward_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalized_associate_forward_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalized_associate_forward_productcoefficientscoefficientdiagonalterm) = (x3)) /\\ ((((exists ff_h_pfp_normalized_associate_forward_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalized_associate_forward_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalized_associate_forward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalized_associate_forward_productcoefficientscoefficientdiagonalterm)) * x2)) /\\ exists ff_q_pfp_normalized_associate_forward_productcoefficientscoefficientdiagonaltermrightentry. x1 = ff_q_pfp_normalized_associate_forward_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalized_associate_forward_productcoefficientscoefficientdiagonalterm)) * x2) + (pfc_right_normalized_associate_forward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_associate_forward_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalized_associate_forward_productcoefficientscoefficientdiagonaltermrightoutside+(x3)=(pfc_complement_normalized_associate_forward_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalized_associate_forward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalized_associate_forward_productcoefficientscoefficientdiagonal)=pfc_left_normalized_associate_forward_productcoefficientscoefficientdiagonalterm*pfc_right_normalized_associate_forward_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalized_associate_forward_productcoefficientscoefficientsum fs_v_pfc_normalized_associate_forward_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_start. fs_h_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalized_associate_forward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_start. fs_u_pfc_normalized_associate_forward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalized_associate_forward_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalized_associate_forward_productcoefficientscoefficient) = S ((S (S (pfc_index_normalized_associate_forward_productcoefficients))) * fs_v_pfc_normalized_associate_forward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalized_associate_forward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalized_associate_forward_productcoefficients))) * fs_v_pfc_normalized_associate_forward_productcoefficientscoefficientsum) + (pfc_natural_sum_normalized_associate_forward_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps = S (pfc_index_normalized_associate_forward_productcoefficients)) -> exists fs_a_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps fs_r_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps fs_s_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_associate_forward_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalized_associate_forward_productcoefficientscoefficient = fs_q_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_associate_forward_productcoefficientscoefficient) + (fs_a_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_forward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalized_associate_forward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_forward_productcoefficientscoefficientsum) + (fs_r_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_forward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalized_associate_forward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_forward_productcoefficientscoefficientsum) + (fs_s_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normalized_associate_forward_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalized_associate_forward_productcoefficientscoefficientresiduebound. pfa_gap_normalized_associate_forward_productcoefficientscoefficientresiduebound + S (pfc_value_normalized_associate_forward_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalized_associate_forward_productcoefficientscoefficientresiduecongruence pfa_offset_right_normalized_associate_forward_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalized_associate_forward_productcoefficientscoefficient) + (p) * pfa_offset_left_normalized_associate_forward_productcoefficientscoefficientresiduecongruence = (pfc_value_normalized_associate_forward_productcoefficients) + (p) * pfa_offset_right_normalized_associate_forward_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normalized_associate_forward_target pfrep_left_normalized_associate_forward_target pfrep_right_normalized_associate_forward_target. ((exists pfrep_position_normalized_associate_forward_targetfirst. ((pfrep_position_normalized_associate_forward_targetfirst+S (pfrep_power_normalized_associate_forward_target)=(pfen_plen_normalized_associate_forward)) /\\ ((((exists ff_h_pfp_normalized_associate_forward_targetfirstentry. ff_h_pfp_normalized_associate_forward_targetfirstentry + S (pfrep_left_normalized_associate_forward_target) = S ((S (pfrep_position_normalized_associate_forward_targetfirst)) * pfen_pc_normalized_associate_forward)) /\\ exists ff_q_pfp_normalized_associate_forward_targetfirstentry. pfen_pb_normalized_associate_forward = ff_q_pfp_normalized_associate_forward_targetfirstentry * S ((S (pfrep_position_normalized_associate_forward_targetfirst)) * pfen_pc_normalized_associate_forward) + (pfrep_left_normalized_associate_forward_target)))))) \\/ (((exists pfrep_gap_normalized_associate_forward_targetfirstoutside. pfrep_gap_normalized_associate_forward_targetfirstoutside+(pfen_plen_normalized_associate_forward)=(pfrep_power_normalized_associate_forward_target)) /\\ (((pfrep_left_normalized_associate_forward_target)=0))))) -> ((exists pfrep_position_normalized_associate_forward_targetsecond. ((pfrep_position_normalized_associate_forward_targetsecond+S (pfrep_power_normalized_associate_forward_target)=(x3)) /\\ ((((exists ff_h_pfp_normalized_associate_forward_targetsecondentry. ff_h_pfp_normalized_associate_forward_targetsecondentry + S (pfrep_right_normalized_associate_forward_target) = S ((S (pfrep_position_normalized_associate_forward_targetsecond)) * x7)) /\\ exists ff_q_pfp_normalized_associate_forward_targetsecondentry. x6 = ff_q_pfp_normalized_associate_forward_targetsecondentry * S ((S (pfrep_position_normalized_associate_forward_targetsecond)) * x7) + (pfrep_right_normalized_associate_forward_target)))))) \\/ (((exists pfrep_gap_normalized_associate_forward_targetsecondoutside. pfrep_gap_normalized_associate_forward_targetsecondoutside+(x3)=(pfrep_power_normalized_associate_forward_target)) /\\ (((pfrep_right_normalized_associate_forward_target)=0))))) -> pfrep_left_normalized_associate_forward_target=pfrep_right_normalized_associate_forward_target))))))) /\\ ((((forall fom_index_pfp_normalized_associate_backward_canonical. (exists fom_gap_pfp_normalized_associate_backward_canonical_index_bound. fom_gap_pfp_normalized_associate_backward_canonical_index_bound + S (fom_index_pfp_normalized_associate_backward_canonical) = x3) -> exists fom_value_pfp_normalized_associate_backward_canonical. ((((exists fom_beta_height_pfp_normalized_associate_backward_canonical_entry. fom_beta_height_pfp_normalized_associate_backward_canonical_entry + S (fom_value_pfp_normalized_associate_backward_canonical) = S ((S (fom_index_pfp_normalized_associate_backward_canonical)) * x2)) /\\ exists fom_beta_quotient_pfp_normalized_associate_backward_canonical_entry. x1 = fom_beta_quotient_pfp_normalized_associate_backward_canonical_entry * S ((S (fom_index_pfp_normalized_associate_backward_canonical)) * x2) + (fom_value_pfp_normalized_associate_backward_canonical))) /\\ (exists fom_gap_pfp_normalized_associate_backward_canonical_value_bound. fom_gap_pfp_normalized_associate_backward_canonical_value_bound + S (fom_value_pfp_normalized_associate_backward_canonical) = p))) /\\ ((exists pfen_qb_normalized_associate_backward pfen_qc_normalized_associate_backward pfen_qlen_normalized_associate_backward pfen_pb_normalized_associate_backward pfen_pc_normalized_associate_backward pfen_plen_normalized_associate_backward. ((((forall fom_index_pfp_normalized_associate_backward_productleft. (exists fom_gap_pfp_normalized_associate_backward_productleft_index_bound. fom_gap_pfp_normalized_associate_backward_productleft_index_bound + S (fom_index_pfp_normalized_associate_backward_productleft) = pfen_qlen_normalized_associate_backward) -> exists fom_value_pfp_normalized_associate_backward_productleft. ((((exists fom_beta_height_pfp_normalized_associate_backward_productleft_entry. fom_beta_height_pfp_normalized_associate_backward_productleft_entry + S (fom_value_pfp_normalized_associate_backward_productleft) = S ((S (fom_index_pfp_normalized_associate_backward_productleft)) * pfen_qc_normalized_associate_backward)) /\\ exists fom_beta_quotient_pfp_normalized_associate_backward_productleft_entry. pfen_qb_normalized_associate_backward = fom_beta_quotient_pfp_normalized_associate_backward_productleft_entry * S ((S (fom_index_pfp_normalized_associate_backward_productleft)) * pfen_qc_normalized_associate_backward) + (fom_value_pfp_normalized_associate_backward_productleft))) /\\ (exists fom_gap_pfp_normalized_associate_backward_productleft_value_bound. fom_gap_pfp_normalized_associate_backward_productleft_value_bound + S (fom_value_pfp_normalized_associate_backward_productleft) = p))) /\\ (((forall fom_index_pfp_normalized_associate_backward_productright. (exists fom_gap_pfp_normalized_associate_backward_productright_index_bound. fom_gap_pfp_normalized_associate_backward_productright_index_bound + S (fom_index_pfp_normalized_associate_backward_productright) = x3) -> exists fom_value_pfp_normalized_associate_backward_productright. ((((exists fom_beta_height_pfp_normalized_associate_backward_productright_entry. fom_beta_height_pfp_normalized_associate_backward_productright_entry + S (fom_value_pfp_normalized_associate_backward_productright) = S ((S (fom_index_pfp_normalized_associate_backward_productright)) * x7)) /\\ exists fom_beta_quotient_pfp_normalized_associate_backward_productright_entry. x6 = fom_beta_quotient_pfp_normalized_associate_backward_productright_entry * S ((S (fom_index_pfp_normalized_associate_backward_productright)) * x7) + (fom_value_pfp_normalized_associate_backward_productright))) /\\ (exists fom_gap_pfp_normalized_associate_backward_productright_value_bound. fom_gap_pfp_normalized_associate_backward_productright_value_bound + S (fom_value_pfp_normalized_associate_backward_productright) = p))) /\\ (((((((pfen_qlen_normalized_associate_backward)=0 \\/ (x3)=0) /\\ (((pfen_plen_normalized_associate_backward)=0)))) \\/ (((~((pfen_qlen_normalized_associate_backward)=0)) /\\ (((~((x3)=0)) /\\ (((pfen_qlen_normalized_associate_backward)+(x3)=S (pfen_plen_normalized_associate_backward)))))))) /\\ ((forall pfc_index_normalized_associate_backward_productcoefficients. (exists pfa_gap_normalized_associate_backward_productcoefficientsbound. pfa_gap_normalized_associate_backward_productcoefficientsbound + S (pfc_index_normalized_associate_backward_productcoefficients) = (pfen_plen_normalized_associate_backward)) -> exists pfc_value_normalized_associate_backward_productcoefficients. ((((exists ff_h_pfp_normalized_associate_backward_productcoefficientsentry. ff_h_pfp_normalized_associate_backward_productcoefficientsentry + S (pfc_value_normalized_associate_backward_productcoefficients) = S ((S (pfc_index_normalized_associate_backward_productcoefficients)) * pfen_pc_normalized_associate_backward)) /\\ exists ff_q_pfp_normalized_associate_backward_productcoefficientsentry. pfen_pb_normalized_associate_backward = ff_q_pfp_normalized_associate_backward_productcoefficientsentry * S ((S (pfc_index_normalized_associate_backward_productcoefficients)) * pfen_pc_normalized_associate_backward) + (pfc_value_normalized_associate_backward_productcoefficients))) /\\ ((exists pfc_terms_code_normalized_associate_backward_productcoefficientscoefficient pfc_terms_scale_normalized_associate_backward_productcoefficientscoefficient pfc_natural_sum_normalized_associate_backward_productcoefficientscoefficient. ((forall pfc_index_normalized_associate_backward_productcoefficientscoefficientdiagonal. (exists pfa_gap_normalized_associate_backward_productcoefficientscoefficientdiagonalbound. pfa_gap_normalized_associate_backward_productcoefficientscoefficientdiagonalbound + S (pfc_index_normalized_associate_backward_productcoefficientscoefficientdiagonal) = (S (pfc_index_normalized_associate_backward_productcoefficients))) -> exists pfc_value_normalized_associate_backward_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalized_associate_backward_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normalized_associate_backward_productcoefficientscoefficientdiagonalentry + S (pfc_value_normalized_associate_backward_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalized_associate_backward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_associate_backward_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normalized_associate_backward_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normalized_associate_backward_productcoefficientscoefficient = ff_q_pfp_normalized_associate_backward_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalized_associate_backward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_associate_backward_productcoefficientscoefficient) + (pfc_value_normalized_associate_backward_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalized_associate_backward_productcoefficientscoefficientdiagonalterm pfc_left_normalized_associate_backward_productcoefficientscoefficientdiagonalterm pfc_right_normalized_associate_backward_productcoefficientscoefficientdiagonalterm. (((pfc_index_normalized_associate_backward_productcoefficientscoefficientdiagonal)+pfc_complement_normalized_associate_backward_productcoefficientscoefficientdiagonalterm=(pfc_index_normalized_associate_backward_productcoefficients)) /\\ ((((((exists pfa_gap_normalized_associate_backward_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalized_associate_backward_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalized_associate_backward_productcoefficientscoefficientdiagonal) = (pfen_qlen_normalized_associate_backward)) /\\ ((((exists ff_h_pfp_normalized_associate_backward_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalized_associate_backward_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalized_associate_backward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalized_associate_backward_productcoefficientscoefficientdiagonal)) * pfen_qc_normalized_associate_backward)) /\\ exists ff_q_pfp_normalized_associate_backward_productcoefficientscoefficientdiagonaltermleftentry. pfen_qb_normalized_associate_backward = ff_q_pfp_normalized_associate_backward_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalized_associate_backward_productcoefficientscoefficientdiagonal)) * pfen_qc_normalized_associate_backward) + (pfc_left_normalized_associate_backward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_associate_backward_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalized_associate_backward_productcoefficientscoefficientdiagonaltermleftoutside+(pfen_qlen_normalized_associate_backward)=(pfc_index_normalized_associate_backward_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalized_associate_backward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalized_associate_backward_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalized_associate_backward_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalized_associate_backward_productcoefficientscoefficientdiagonalterm) = (x3)) /\\ ((((exists ff_h_pfp_normalized_associate_backward_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalized_associate_backward_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalized_associate_backward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalized_associate_backward_productcoefficientscoefficientdiagonalterm)) * x7)) /\\ exists ff_q_pfp_normalized_associate_backward_productcoefficientscoefficientdiagonaltermrightentry. x6 = ff_q_pfp_normalized_associate_backward_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalized_associate_backward_productcoefficientscoefficientdiagonalterm)) * x7) + (pfc_right_normalized_associate_backward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_associate_backward_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalized_associate_backward_productcoefficientscoefficientdiagonaltermrightoutside+(x3)=(pfc_complement_normalized_associate_backward_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalized_associate_backward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalized_associate_backward_productcoefficientscoefficientdiagonal)=pfc_left_normalized_associate_backward_productcoefficientscoefficientdiagonalterm*pfc_right_normalized_associate_backward_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalized_associate_backward_productcoefficientscoefficientsum fs_v_pfc_normalized_associate_backward_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_start. fs_h_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalized_associate_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_start. fs_u_pfc_normalized_associate_backward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalized_associate_backward_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalized_associate_backward_productcoefficientscoefficient) = S ((S (S (pfc_index_normalized_associate_backward_productcoefficients))) * fs_v_pfc_normalized_associate_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalized_associate_backward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalized_associate_backward_productcoefficients))) * fs_v_pfc_normalized_associate_backward_productcoefficientscoefficientsum) + (pfc_natural_sum_normalized_associate_backward_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps = S (pfc_index_normalized_associate_backward_productcoefficients)) -> exists fs_a_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps fs_r_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps fs_s_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_associate_backward_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalized_associate_backward_productcoefficientscoefficient = fs_q_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_associate_backward_productcoefficientscoefficient) + (fs_a_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalized_associate_backward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_backward_productcoefficientscoefficientsum) + (fs_r_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalized_associate_backward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_backward_productcoefficientscoefficientsum) + (fs_s_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normalized_associate_backward_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalized_associate_backward_productcoefficientscoefficientresiduebound. pfa_gap_normalized_associate_backward_productcoefficientscoefficientresiduebound + S (pfc_value_normalized_associate_backward_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalized_associate_backward_productcoefficientscoefficientresiduecongruence pfa_offset_right_normalized_associate_backward_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalized_associate_backward_productcoefficientscoefficient) + (p) * pfa_offset_left_normalized_associate_backward_productcoefficientscoefficientresiduecongruence = (pfc_value_normalized_associate_backward_productcoefficients) + (p) * pfa_offset_right_normalized_associate_backward_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normalized_associate_backward_target pfrep_left_normalized_associate_backward_target pfrep_right_normalized_associate_backward_target. ((exists pfrep_position_normalized_associate_backward_targetfirst. ((pfrep_position_normalized_associate_backward_targetfirst+S (pfrep_power_normalized_associate_backward_target)=(pfen_plen_normalized_associate_backward)) /\\ ((((exists ff_h_pfp_normalized_associate_backward_targetfirstentry. ff_h_pfp_normalized_associate_backward_targetfirstentry + S (pfrep_left_normalized_associate_backward_target) = S ((S (pfrep_position_normalized_associate_backward_targetfirst)) * pfen_pc_normalized_associate_backward)) /\\ exists ff_q_pfp_normalized_associate_backward_targetfirstentry. pfen_pb_normalized_associate_backward = ff_q_pfp_normalized_associate_backward_targetfirstentry * S ((S (pfrep_position_normalized_associate_backward_targetfirst)) * pfen_pc_normalized_associate_backward) + (pfrep_left_normalized_associate_backward_target)))))) \\/ (((exists pfrep_gap_normalized_associate_backward_targetfirstoutside. pfrep_gap_normalized_associate_backward_targetfirstoutside+(pfen_plen_normalized_associate_backward)=(pfrep_power_normalized_associate_backward_target)) /\\ (((pfrep_left_normalized_associate_backward_target)=0))))) -> ((exists pfrep_position_normalized_associate_backward_targetsecond. ((pfrep_position_normalized_associate_backward_targetsecond+S (pfrep_power_normalized_associate_backward_target)=(x3)) /\\ ((((exists ff_h_pfp_normalized_associate_backward_targetsecondentry. ff_h_pfp_normalized_associate_backward_targetsecondentry + S (pfrep_right_normalized_associate_backward_target) = S ((S (pfrep_position_normalized_associate_backward_targetsecond)) * x2)) /\\ exists ff_q_pfp_normalized_associate_backward_targetsecondentry. x1 = ff_q_pfp_normalized_associate_backward_targetsecondentry * S ((S (pfrep_position_normalized_associate_backward_targetsecond)) * x2) + (pfrep_right_normalized_associate_backward_target)))))) \\/ (((exists pfrep_gap_normalized_associate_backward_targetsecondoutside. pfrep_gap_normalized_associate_backward_targetsecondoutside+(x3)=(pfrep_power_normalized_associate_backward_target)) /\\ (((pfrep_right_normalized_associate_backward_target)=0))))) -> pfrep_left_normalized_associate_backward_target=pfrep_right_normalized_associate_backward_target)))))))))", "specialize prime_field_polynomial_monic_normalization_right_associates (p)", "specialize prime_field_polynomial_monic_normalization_right_associates (x5)", "specialize prime_field_polynomial_monic_normalization_right_associates (x1)", "specialize prime_field_polynomial_monic_normalization_right_associates (x2)", "specialize prime_field_polynomial_monic_normalization_right_associates (x6)", "specialize prime_field_polynomial_monic_normalization_right_associates (x7)", "specialize prime_field_polynomial_monic_normalization_right_associates (x3)", "apply prime_field_polynomial_monic_normalization_right_associates", "exact hp", "exact hnormalization_witness_witness_witness", "cases hassociates", "exists x6", "exists x7", "exists x3", "split", "right", "specialize prime_field_polynomial_monic_normalization_monic (p)", "specialize prime_field_polynomial_monic_normalization_monic (x5)", "specialize prime_field_polynomial_monic_normalization_monic (x1)", "specialize prime_field_polynomial_monic_normalization_monic (x2)", "specialize prime_field_polynomial_monic_normalization_monic (x6)", "specialize prime_field_polynomial_monic_normalization_monic (x7)", "specialize prime_field_polynomial_monic_normalization_monic (x3)", "apply prime_field_polynomial_monic_normalization_monic", "exact hnormalization_witness_witness_witness", "split", "specialize prime_field_polynomial_right_divides_equivalent_divisor (p)", "specialize prime_field_polynomial_right_divides_equivalent_divisor (x1)", "specialize prime_field_polynomial_right_divides_equivalent_divisor (x2)", "specialize prime_field_polynomial_right_divides_equivalent_divisor (x3)", "specialize prime_field_polynomial_right_divides_equivalent_divisor (x6)", "specialize prime_field_polynomial_right_divides_equivalent_divisor (x7)", "specialize prime_field_polynomial_right_divides_equivalent_divisor (x3)", "specialize prime_field_polynomial_right_divides_equivalent_divisor (ab)", "specialize prime_field_polynomial_right_divides_equivalent_divisor (ac)", "specialize prime_field_polynomial_right_divides_equivalent_divisor (L)", "apply prime_field_polynomial_right_divides_equivalent_divisor", "exact hp0", "exact hA", "exact hTA", "exact hassociates_left", "specialize prime_field_polynomial_right_divides_equivalent_target (p)", "specialize prime_field_polynomial_right_divides_equivalent_target (x6)", "specialize prime_field_polynomial_right_divides_equivalent_target (x7)", "specialize prime_field_polynomial_right_divides_equivalent_target (x3)", "specialize prime_field_polynomial_right_divides_equivalent_target (x1)", "specialize prime_field_polynomial_right_divides_equivalent_target (x2)", "specialize prime_field_polynomial_right_divides_equivalent_target (x3)", "specialize prime_field_polynomial_right_divides_equivalent_target (ab)", "specialize prime_field_polynomial_right_divides_equivalent_target (ac)", "specialize prime_field_polynomial_right_divides_equivalent_target (L)", "apply prime_field_polynomial_right_divides_equivalent_target", "exact hA", "exact hassociates_right", "exact hTA" ], "script_sha256": "3906a4a1282eb3f86af0e4ef0c2e9041a5c396b519775c46dbdcc013efdae54a", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_euclidean_normalization_candidate.py", "sha256": "d2cddfe42dc0d22104dc4e85e95116222914df11ac840d2082a4ff2e462f146f" }, "statement": "forall p ab ac L. (~((p) = 1) /\\ forall pfa_factor_left_normalized_associate_prime pfa_factor_right_normalized_associate_prime. (p) = pfa_factor_left_normalized_associate_prime * pfa_factor_right_normalized_associate_prime -> pfa_factor_left_normalized_associate_prime = 1 \\/ pfa_factor_right_normalized_associate_prime = 1) -> (forall fom_index_pfp_normalized_associate_input. (exists fom_gap_pfp_normalized_associate_input_index_bound. fom_gap_pfp_normalized_associate_input_index_bound + S (fom_index_pfp_normalized_associate_input) = L) -> exists fom_value_pfp_normalized_associate_input. ((((exists fom_beta_height_pfp_normalized_associate_input_entry. fom_beta_height_pfp_normalized_associate_input_entry + S (fom_value_pfp_normalized_associate_input) = S ((S (fom_index_pfp_normalized_associate_input)) * ac)) /\\ exists fom_beta_quotient_pfp_normalized_associate_input_entry. ab = fom_beta_quotient_pfp_normalized_associate_input_entry * S ((S (fom_index_pfp_normalized_associate_input)) * ac) + (fom_value_pfp_normalized_associate_input))) /\\ (exists fom_gap_pfp_normalized_associate_input_value_bound. fom_gap_pfp_normalized_associate_input_value_bound + S (fom_value_pfp_normalized_associate_input) = p))) -> (exists hb hc H. ((H=0 \\/ (((~((H) = 0)) /\\ (((forall fom_index_pfp_normalized_associate_result_moniccoefficients. (exists fom_gap_pfp_normalized_associate_result_moniccoefficients_index_bound. fom_gap_pfp_normalized_associate_result_moniccoefficients_index_bound + S (fom_index_pfp_normalized_associate_result_moniccoefficients) = H) -> exists fom_value_pfp_normalized_associate_result_moniccoefficients. ((((exists fom_beta_height_pfp_normalized_associate_result_moniccoefficients_entry. fom_beta_height_pfp_normalized_associate_result_moniccoefficients_entry + S (fom_value_pfp_normalized_associate_result_moniccoefficients) = S ((S (fom_index_pfp_normalized_associate_result_moniccoefficients)) * hc)) /\\ exists fom_beta_quotient_pfp_normalized_associate_result_moniccoefficients_entry. hb = fom_beta_quotient_pfp_normalized_associate_result_moniccoefficients_entry * S ((S (fom_index_pfp_normalized_associate_result_moniccoefficients)) * hc) + (fom_value_pfp_normalized_associate_result_moniccoefficients))) /\\ (exists fom_gap_pfp_normalized_associate_result_moniccoefficients_value_bound. fom_gap_pfp_normalized_associate_result_moniccoefficients_value_bound + S (fom_value_pfp_normalized_associate_result_moniccoefficients) = p))) /\\ ((((exists ff_h_pfp_normalized_associate_result_monicleading. ff_h_pfp_normalized_associate_result_monicleading + S (1) = S ((S (0)) * hc)) /\\ exists ff_q_pfp_normalized_associate_result_monicleading. hb = ff_q_pfp_normalized_associate_result_monicleading * S ((S (0)) * hc) + (1))))))))) /\\ (((((forall fom_index_pfp_normalized_associate_result_forward_canonical. (exists fom_gap_pfp_normalized_associate_result_forward_canonical_index_bound. fom_gap_pfp_normalized_associate_result_forward_canonical_index_bound + S (fom_index_pfp_normalized_associate_result_forward_canonical) = H) -> exists fom_value_pfp_normalized_associate_result_forward_canonical. ((((exists fom_beta_height_pfp_normalized_associate_result_forward_canonical_entry. fom_beta_height_pfp_normalized_associate_result_forward_canonical_entry + S (fom_value_pfp_normalized_associate_result_forward_canonical) = S ((S (fom_index_pfp_normalized_associate_result_forward_canonical)) * hc)) /\\ exists fom_beta_quotient_pfp_normalized_associate_result_forward_canonical_entry. hb = fom_beta_quotient_pfp_normalized_associate_result_forward_canonical_entry * S ((S (fom_index_pfp_normalized_associate_result_forward_canonical)) * hc) + (fom_value_pfp_normalized_associate_result_forward_canonical))) /\\ (exists fom_gap_pfp_normalized_associate_result_forward_canonical_value_bound. fom_gap_pfp_normalized_associate_result_forward_canonical_value_bound + S (fom_value_pfp_normalized_associate_result_forward_canonical) = p))) /\\ ((exists pfen_qb_normalized_associate_result_forward pfen_qc_normalized_associate_result_forward pfen_qlen_normalized_associate_result_forward pfen_pb_normalized_associate_result_forward pfen_pc_normalized_associate_result_forward pfen_plen_normalized_associate_result_forward. ((((forall fom_index_pfp_normalized_associate_result_forward_productleft. (exists fom_gap_pfp_normalized_associate_result_forward_productleft_index_bound. fom_gap_pfp_normalized_associate_result_forward_productleft_index_bound + S (fom_index_pfp_normalized_associate_result_forward_productleft) = pfen_qlen_normalized_associate_result_forward) -> exists fom_value_pfp_normalized_associate_result_forward_productleft. ((((exists fom_beta_height_pfp_normalized_associate_result_forward_productleft_entry. fom_beta_height_pfp_normalized_associate_result_forward_productleft_entry + S (fom_value_pfp_normalized_associate_result_forward_productleft) = S ((S (fom_index_pfp_normalized_associate_result_forward_productleft)) * pfen_qc_normalized_associate_result_forward)) /\\ exists fom_beta_quotient_pfp_normalized_associate_result_forward_productleft_entry. pfen_qb_normalized_associate_result_forward = fom_beta_quotient_pfp_normalized_associate_result_forward_productleft_entry * S ((S (fom_index_pfp_normalized_associate_result_forward_productleft)) * pfen_qc_normalized_associate_result_forward) + (fom_value_pfp_normalized_associate_result_forward_productleft))) /\\ (exists fom_gap_pfp_normalized_associate_result_forward_productleft_value_bound. fom_gap_pfp_normalized_associate_result_forward_productleft_value_bound + S (fom_value_pfp_normalized_associate_result_forward_productleft) = p))) /\\ (((forall fom_index_pfp_normalized_associate_result_forward_productright. (exists fom_gap_pfp_normalized_associate_result_forward_productright_index_bound. fom_gap_pfp_normalized_associate_result_forward_productright_index_bound + S (fom_index_pfp_normalized_associate_result_forward_productright) = L) -> exists fom_value_pfp_normalized_associate_result_forward_productright. ((((exists fom_beta_height_pfp_normalized_associate_result_forward_productright_entry. fom_beta_height_pfp_normalized_associate_result_forward_productright_entry + S (fom_value_pfp_normalized_associate_result_forward_productright) = S ((S (fom_index_pfp_normalized_associate_result_forward_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_normalized_associate_result_forward_productright_entry. ab = fom_beta_quotient_pfp_normalized_associate_result_forward_productright_entry * S ((S (fom_index_pfp_normalized_associate_result_forward_productright)) * ac) + (fom_value_pfp_normalized_associate_result_forward_productright))) /\\ (exists fom_gap_pfp_normalized_associate_result_forward_productright_value_bound. fom_gap_pfp_normalized_associate_result_forward_productright_value_bound + S (fom_value_pfp_normalized_associate_result_forward_productright) = p))) /\\ (((((((pfen_qlen_normalized_associate_result_forward)=0 \\/ (L)=0) /\\ (((pfen_plen_normalized_associate_result_forward)=0)))) \\/ (((~((pfen_qlen_normalized_associate_result_forward)=0)) /\\ (((~((L)=0)) /\\ (((pfen_qlen_normalized_associate_result_forward)+(L)=S (pfen_plen_normalized_associate_result_forward)))))))) /\\ ((forall pfc_index_normalized_associate_result_forward_productcoefficients. (exists pfa_gap_normalized_associate_result_forward_productcoefficientsbound. pfa_gap_normalized_associate_result_forward_productcoefficientsbound + S (pfc_index_normalized_associate_result_forward_productcoefficients) = (pfen_plen_normalized_associate_result_forward)) -> exists pfc_value_normalized_associate_result_forward_productcoefficients. ((((exists ff_h_pfp_normalized_associate_result_forward_productcoefficientsentry. ff_h_pfp_normalized_associate_result_forward_productcoefficientsentry + S (pfc_value_normalized_associate_result_forward_productcoefficients) = S ((S (pfc_index_normalized_associate_result_forward_productcoefficients)) * pfen_pc_normalized_associate_result_forward)) /\\ exists ff_q_pfp_normalized_associate_result_forward_productcoefficientsentry. pfen_pb_normalized_associate_result_forward = ff_q_pfp_normalized_associate_result_forward_productcoefficientsentry * S ((S (pfc_index_normalized_associate_result_forward_productcoefficients)) * pfen_pc_normalized_associate_result_forward) + (pfc_value_normalized_associate_result_forward_productcoefficients))) /\\ ((exists pfc_terms_code_normalized_associate_result_forward_productcoefficientscoefficient pfc_terms_scale_normalized_associate_result_forward_productcoefficientscoefficient pfc_natural_sum_normalized_associate_result_forward_productcoefficientscoefficient. ((forall pfc_index_normalized_associate_result_forward_productcoefficientscoefficientdiagonal. (exists pfa_gap_normalized_associate_result_forward_productcoefficientscoefficientdiagonalbound. pfa_gap_normalized_associate_result_forward_productcoefficientscoefficientdiagonalbound + S (pfc_index_normalized_associate_result_forward_productcoefficientscoefficientdiagonal) = (S (pfc_index_normalized_associate_result_forward_productcoefficients))) -> exists pfc_value_normalized_associate_result_forward_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalized_associate_result_forward_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normalized_associate_result_forward_productcoefficientscoefficientdiagonalentry + S (pfc_value_normalized_associate_result_forward_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalized_associate_result_forward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_associate_result_forward_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normalized_associate_result_forward_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normalized_associate_result_forward_productcoefficientscoefficient = ff_q_pfp_normalized_associate_result_forward_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalized_associate_result_forward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_associate_result_forward_productcoefficientscoefficient) + (pfc_value_normalized_associate_result_forward_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm pfc_left_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm pfc_right_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm. (((pfc_index_normalized_associate_result_forward_productcoefficientscoefficientdiagonal)+pfc_complement_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm=(pfc_index_normalized_associate_result_forward_productcoefficients)) /\\ ((((((exists pfa_gap_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalized_associate_result_forward_productcoefficientscoefficientdiagonal) = (pfen_qlen_normalized_associate_result_forward)) /\\ ((((exists ff_h_pfp_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalized_associate_result_forward_productcoefficientscoefficientdiagonal)) * pfen_qc_normalized_associate_result_forward)) /\\ exists ff_q_pfp_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermleftentry. pfen_qb_normalized_associate_result_forward = ff_q_pfp_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalized_associate_result_forward_productcoefficientscoefficientdiagonal)) * pfen_qc_normalized_associate_result_forward) + (pfc_left_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermleftoutside+(pfen_qlen_normalized_associate_result_forward)=(pfc_index_normalized_associate_result_forward_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalized_associate_result_forward_productcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalized_associate_result_forward_productcoefficientscoefficientdiagonal)=pfc_left_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm*pfc_right_normalized_associate_result_forward_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum fs_v_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_start. fs_h_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_start. fs_u_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalized_associate_result_forward_productcoefficientscoefficient) = S ((S (S (pfc_index_normalized_associate_result_forward_productcoefficients))) * fs_v_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalized_associate_result_forward_productcoefficients))) * fs_v_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum) + (pfc_natural_sum_normalized_associate_result_forward_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps = S (pfc_index_normalized_associate_result_forward_productcoefficients)) -> exists fs_a_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps fs_r_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps fs_s_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_associate_result_forward_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalized_associate_result_forward_productcoefficientscoefficient = fs_q_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_associate_result_forward_productcoefficientscoefficient) + (fs_a_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum) + (fs_r_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum) + (fs_s_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normalized_associate_result_forward_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalized_associate_result_forward_productcoefficientscoefficientresiduebound. pfa_gap_normalized_associate_result_forward_productcoefficientscoefficientresiduebound + S (pfc_value_normalized_associate_result_forward_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalized_associate_result_forward_productcoefficientscoefficientresiduecongruence pfa_offset_right_normalized_associate_result_forward_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalized_associate_result_forward_productcoefficientscoefficient) + (p) * pfa_offset_left_normalized_associate_result_forward_productcoefficientscoefficientresiduecongruence = (pfc_value_normalized_associate_result_forward_productcoefficients) + (p) * pfa_offset_right_normalized_associate_result_forward_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normalized_associate_result_forward_target pfrep_left_normalized_associate_result_forward_target pfrep_right_normalized_associate_result_forward_target. ((exists pfrep_position_normalized_associate_result_forward_targetfirst. ((pfrep_position_normalized_associate_result_forward_targetfirst+S (pfrep_power_normalized_associate_result_forward_target)=(pfen_plen_normalized_associate_result_forward)) /\\ ((((exists ff_h_pfp_normalized_associate_result_forward_targetfirstentry. ff_h_pfp_normalized_associate_result_forward_targetfirstentry + S (pfrep_left_normalized_associate_result_forward_target) = S ((S (pfrep_position_normalized_associate_result_forward_targetfirst)) * pfen_pc_normalized_associate_result_forward)) /\\ exists ff_q_pfp_normalized_associate_result_forward_targetfirstentry. pfen_pb_normalized_associate_result_forward = ff_q_pfp_normalized_associate_result_forward_targetfirstentry * S ((S (pfrep_position_normalized_associate_result_forward_targetfirst)) * pfen_pc_normalized_associate_result_forward) + (pfrep_left_normalized_associate_result_forward_target)))))) \\/ (((exists pfrep_gap_normalized_associate_result_forward_targetfirstoutside. pfrep_gap_normalized_associate_result_forward_targetfirstoutside+(pfen_plen_normalized_associate_result_forward)=(pfrep_power_normalized_associate_result_forward_target)) /\\ (((pfrep_left_normalized_associate_result_forward_target)=0))))) -> ((exists pfrep_position_normalized_associate_result_forward_targetsecond. ((pfrep_position_normalized_associate_result_forward_targetsecond+S (pfrep_power_normalized_associate_result_forward_target)=(H)) /\\ ((((exists ff_h_pfp_normalized_associate_result_forward_targetsecondentry. ff_h_pfp_normalized_associate_result_forward_targetsecondentry + S (pfrep_right_normalized_associate_result_forward_target) = S ((S (pfrep_position_normalized_associate_result_forward_targetsecond)) * hc)) /\\ exists ff_q_pfp_normalized_associate_result_forward_targetsecondentry. hb = ff_q_pfp_normalized_associate_result_forward_targetsecondentry * S ((S (pfrep_position_normalized_associate_result_forward_targetsecond)) * hc) + (pfrep_right_normalized_associate_result_forward_target)))))) \\/ (((exists pfrep_gap_normalized_associate_result_forward_targetsecondoutside. pfrep_gap_normalized_associate_result_forward_targetsecondoutside+(H)=(pfrep_power_normalized_associate_result_forward_target)) /\\ (((pfrep_right_normalized_associate_result_forward_target)=0))))) -> pfrep_left_normalized_associate_result_forward_target=pfrep_right_normalized_associate_result_forward_target))))))) /\\ ((((forall fom_index_pfp_normalized_associate_result_backward_canonical. (exists fom_gap_pfp_normalized_associate_result_backward_canonical_index_bound. fom_gap_pfp_normalized_associate_result_backward_canonical_index_bound + S (fom_index_pfp_normalized_associate_result_backward_canonical) = L) -> exists fom_value_pfp_normalized_associate_result_backward_canonical. ((((exists fom_beta_height_pfp_normalized_associate_result_backward_canonical_entry. fom_beta_height_pfp_normalized_associate_result_backward_canonical_entry + S (fom_value_pfp_normalized_associate_result_backward_canonical) = S ((S (fom_index_pfp_normalized_associate_result_backward_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_normalized_associate_result_backward_canonical_entry. ab = fom_beta_quotient_pfp_normalized_associate_result_backward_canonical_entry * S ((S (fom_index_pfp_normalized_associate_result_backward_canonical)) * ac) + (fom_value_pfp_normalized_associate_result_backward_canonical))) /\\ (exists fom_gap_pfp_normalized_associate_result_backward_canonical_value_bound. fom_gap_pfp_normalized_associate_result_backward_canonical_value_bound + S (fom_value_pfp_normalized_associate_result_backward_canonical) = p))) /\\ ((exists pfen_qb_normalized_associate_result_backward pfen_qc_normalized_associate_result_backward pfen_qlen_normalized_associate_result_backward pfen_pb_normalized_associate_result_backward pfen_pc_normalized_associate_result_backward pfen_plen_normalized_associate_result_backward. ((((forall fom_index_pfp_normalized_associate_result_backward_productleft. (exists fom_gap_pfp_normalized_associate_result_backward_productleft_index_bound. fom_gap_pfp_normalized_associate_result_backward_productleft_index_bound + S (fom_index_pfp_normalized_associate_result_backward_productleft) = pfen_qlen_normalized_associate_result_backward) -> exists fom_value_pfp_normalized_associate_result_backward_productleft. ((((exists fom_beta_height_pfp_normalized_associate_result_backward_productleft_entry. fom_beta_height_pfp_normalized_associate_result_backward_productleft_entry + S (fom_value_pfp_normalized_associate_result_backward_productleft) = S ((S (fom_index_pfp_normalized_associate_result_backward_productleft)) * pfen_qc_normalized_associate_result_backward)) /\\ exists fom_beta_quotient_pfp_normalized_associate_result_backward_productleft_entry. pfen_qb_normalized_associate_result_backward = fom_beta_quotient_pfp_normalized_associate_result_backward_productleft_entry * S ((S (fom_index_pfp_normalized_associate_result_backward_productleft)) * pfen_qc_normalized_associate_result_backward) + (fom_value_pfp_normalized_associate_result_backward_productleft))) /\\ (exists fom_gap_pfp_normalized_associate_result_backward_productleft_value_bound. fom_gap_pfp_normalized_associate_result_backward_productleft_value_bound + S (fom_value_pfp_normalized_associate_result_backward_productleft) = p))) /\\ (((forall fom_index_pfp_normalized_associate_result_backward_productright. (exists fom_gap_pfp_normalized_associate_result_backward_productright_index_bound. fom_gap_pfp_normalized_associate_result_backward_productright_index_bound + S (fom_index_pfp_normalized_associate_result_backward_productright) = H) -> exists fom_value_pfp_normalized_associate_result_backward_productright. ((((exists fom_beta_height_pfp_normalized_associate_result_backward_productright_entry. fom_beta_height_pfp_normalized_associate_result_backward_productright_entry + S (fom_value_pfp_normalized_associate_result_backward_productright) = S ((S (fom_index_pfp_normalized_associate_result_backward_productright)) * hc)) /\\ exists fom_beta_quotient_pfp_normalized_associate_result_backward_productright_entry. hb = fom_beta_quotient_pfp_normalized_associate_result_backward_productright_entry * S ((S (fom_index_pfp_normalized_associate_result_backward_productright)) * hc) + (fom_value_pfp_normalized_associate_result_backward_productright))) /\\ (exists fom_gap_pfp_normalized_associate_result_backward_productright_value_bound. fom_gap_pfp_normalized_associate_result_backward_productright_value_bound + S (fom_value_pfp_normalized_associate_result_backward_productright) = p))) /\\ (((((((pfen_qlen_normalized_associate_result_backward)=0 \\/ (H)=0) /\\ (((pfen_plen_normalized_associate_result_backward)=0)))) \\/ (((~((pfen_qlen_normalized_associate_result_backward)=0)) /\\ (((~((H)=0)) /\\ (((pfen_qlen_normalized_associate_result_backward)+(H)=S (pfen_plen_normalized_associate_result_backward)))))))) /\\ ((forall pfc_index_normalized_associate_result_backward_productcoefficients. (exists pfa_gap_normalized_associate_result_backward_productcoefficientsbound. pfa_gap_normalized_associate_result_backward_productcoefficientsbound + S (pfc_index_normalized_associate_result_backward_productcoefficients) = (pfen_plen_normalized_associate_result_backward)) -> exists pfc_value_normalized_associate_result_backward_productcoefficients. ((((exists ff_h_pfp_normalized_associate_result_backward_productcoefficientsentry. ff_h_pfp_normalized_associate_result_backward_productcoefficientsentry + S (pfc_value_normalized_associate_result_backward_productcoefficients) = S ((S (pfc_index_normalized_associate_result_backward_productcoefficients)) * pfen_pc_normalized_associate_result_backward)) /\\ exists ff_q_pfp_normalized_associate_result_backward_productcoefficientsentry. pfen_pb_normalized_associate_result_backward = ff_q_pfp_normalized_associate_result_backward_productcoefficientsentry * S ((S (pfc_index_normalized_associate_result_backward_productcoefficients)) * pfen_pc_normalized_associate_result_backward) + (pfc_value_normalized_associate_result_backward_productcoefficients))) /\\ ((exists pfc_terms_code_normalized_associate_result_backward_productcoefficientscoefficient pfc_terms_scale_normalized_associate_result_backward_productcoefficientscoefficient pfc_natural_sum_normalized_associate_result_backward_productcoefficientscoefficient. ((forall pfc_index_normalized_associate_result_backward_productcoefficientscoefficientdiagonal. (exists pfa_gap_normalized_associate_result_backward_productcoefficientscoefficientdiagonalbound. pfa_gap_normalized_associate_result_backward_productcoefficientscoefficientdiagonalbound + S (pfc_index_normalized_associate_result_backward_productcoefficientscoefficientdiagonal) = (S (pfc_index_normalized_associate_result_backward_productcoefficients))) -> exists pfc_value_normalized_associate_result_backward_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalized_associate_result_backward_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normalized_associate_result_backward_productcoefficientscoefficientdiagonalentry + S (pfc_value_normalized_associate_result_backward_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalized_associate_result_backward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_associate_result_backward_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normalized_associate_result_backward_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normalized_associate_result_backward_productcoefficientscoefficient = ff_q_pfp_normalized_associate_result_backward_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalized_associate_result_backward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_associate_result_backward_productcoefficientscoefficient) + (pfc_value_normalized_associate_result_backward_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm pfc_left_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm pfc_right_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm. (((pfc_index_normalized_associate_result_backward_productcoefficientscoefficientdiagonal)+pfc_complement_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm=(pfc_index_normalized_associate_result_backward_productcoefficients)) /\\ ((((((exists pfa_gap_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalized_associate_result_backward_productcoefficientscoefficientdiagonal) = (pfen_qlen_normalized_associate_result_backward)) /\\ ((((exists ff_h_pfp_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalized_associate_result_backward_productcoefficientscoefficientdiagonal)) * pfen_qc_normalized_associate_result_backward)) /\\ exists ff_q_pfp_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermleftentry. pfen_qb_normalized_associate_result_backward = ff_q_pfp_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalized_associate_result_backward_productcoefficientscoefficientdiagonal)) * pfen_qc_normalized_associate_result_backward) + (pfc_left_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermleftoutside+(pfen_qlen_normalized_associate_result_backward)=(pfc_index_normalized_associate_result_backward_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm) = (H)) /\\ ((((exists ff_h_pfp_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm)) * hc)) /\\ exists ff_q_pfp_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermrightentry. hb = ff_q_pfp_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm)) * hc) + (pfc_right_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalized_associate_result_backward_productcoefficientscoefficientdiagonaltermrightoutside+(H)=(pfc_complement_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalized_associate_result_backward_productcoefficientscoefficientdiagonal)=pfc_left_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm*pfc_right_normalized_associate_result_backward_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum fs_v_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_start. fs_h_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_start. fs_u_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalized_associate_result_backward_productcoefficientscoefficient) = S ((S (S (pfc_index_normalized_associate_result_backward_productcoefficients))) * fs_v_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalized_associate_result_backward_productcoefficients))) * fs_v_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum) + (pfc_natural_sum_normalized_associate_result_backward_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps = S (pfc_index_normalized_associate_result_backward_productcoefficients)) -> exists fs_a_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps fs_r_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps fs_s_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_associate_result_backward_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalized_associate_result_backward_productcoefficientscoefficient = fs_q_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_associate_result_backward_productcoefficientscoefficient) + (fs_a_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum) + (fs_r_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum = fs_q_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum) + (fs_s_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normalized_associate_result_backward_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalized_associate_result_backward_productcoefficientscoefficientresiduebound. pfa_gap_normalized_associate_result_backward_productcoefficientscoefficientresiduebound + S (pfc_value_normalized_associate_result_backward_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalized_associate_result_backward_productcoefficientscoefficientresiduecongruence pfa_offset_right_normalized_associate_result_backward_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalized_associate_result_backward_productcoefficientscoefficient) + (p) * pfa_offset_left_normalized_associate_result_backward_productcoefficientscoefficientresiduecongruence = (pfc_value_normalized_associate_result_backward_productcoefficients) + (p) * pfa_offset_right_normalized_associate_result_backward_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normalized_associate_result_backward_target pfrep_left_normalized_associate_result_backward_target pfrep_right_normalized_associate_result_backward_target. ((exists pfrep_position_normalized_associate_result_backward_targetfirst. ((pfrep_position_normalized_associate_result_backward_targetfirst+S (pfrep_power_normalized_associate_result_backward_target)=(pfen_plen_normalized_associate_result_backward)) /\\ ((((exists ff_h_pfp_normalized_associate_result_backward_targetfirstentry. ff_h_pfp_normalized_associate_result_backward_targetfirstentry + S (pfrep_left_normalized_associate_result_backward_target) = S ((S (pfrep_position_normalized_associate_result_backward_targetfirst)) * pfen_pc_normalized_associate_result_backward)) /\\ exists ff_q_pfp_normalized_associate_result_backward_targetfirstentry. pfen_pb_normalized_associate_result_backward = ff_q_pfp_normalized_associate_result_backward_targetfirstentry * S ((S (pfrep_position_normalized_associate_result_backward_targetfirst)) * pfen_pc_normalized_associate_result_backward) + (pfrep_left_normalized_associate_result_backward_target)))))) \\/ (((exists pfrep_gap_normalized_associate_result_backward_targetfirstoutside. pfrep_gap_normalized_associate_result_backward_targetfirstoutside+(pfen_plen_normalized_associate_result_backward)=(pfrep_power_normalized_associate_result_backward_target)) /\\ (((pfrep_left_normalized_associate_result_backward_target)=0))))) -> ((exists pfrep_position_normalized_associate_result_backward_targetsecond. ((pfrep_position_normalized_associate_result_backward_targetsecond+S (pfrep_power_normalized_associate_result_backward_target)=(L)) /\\ ((((exists ff_h_pfp_normalized_associate_result_backward_targetsecondentry. ff_h_pfp_normalized_associate_result_backward_targetsecondentry + S (pfrep_right_normalized_associate_result_backward_target) = S ((S (pfrep_position_normalized_associate_result_backward_targetsecond)) * ac)) /\\ exists ff_q_pfp_normalized_associate_result_backward_targetsecondentry. ab = ff_q_pfp_normalized_associate_result_backward_targetsecondentry * S ((S (pfrep_position_normalized_associate_result_backward_targetsecond)) * ac) + (pfrep_right_normalized_associate_result_backward_target)))))) \\/ (((exists pfrep_gap_normalized_associate_result_backward_targetsecondoutside. pfrep_gap_normalized_associate_result_backward_targetsecondoutside+(L)=(pfrep_power_normalized_associate_result_backward_target)) /\\ (((pfrep_right_normalized_associate_result_backward_target)=0))))) -> pfrep_left_normalized_associate_result_backward_target=pfrep_right_normalized_associate_result_backward_target))))))))))))", "statement_sha256": "21d68a815be43fce3458f0b51fa22b3080e0f67ae51702d3bb09a7e9a7a6711c", "summary": "Construct a zero-or-monic right associate of every canonical polynomial, first trimming its actual leading zeros and then normalizing only a nonempty trim. Both divisibility directions have real product witnesses and are transported to the original representation. 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hop", "have hp0 : ~(p=0)", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hz", "cases hDA", "cases hDA_right", "cases hDA_right_witness", "cases hDA_right_witness_witness", "cases hDA_right_witness_witness_witness", "cases hDA_right_witness_witness_witness_witness", "cases hDA_right_witness_witness_witness_witness_witness", "cases hDA_right_witness_witness_witness_witness_witness_witness", "cases hDB", "cases hDB_right", "cases hDB_right_witness", "cases hDB_right_witness_witness", "cases hDB_right_witness_witness_witness", "cases hDB_right_witness_witness_witness_witness", "cases hDB_right_witness_witness_witness_witness_witness", "cases hDB_right_witness_witness_witness_witness_witness_witness", "have hfirst : ((forall fom_index_pfp_add_hfirstleft. (exists fom_gap_pfp_add_hfirstleft_index_bound. fom_gap_pfp_add_hfirstleft_index_bound + S (fom_index_pfp_add_hfirstleft) = x2) -> exists fom_value_pfp_add_hfirstleft. ((((exists fom_beta_height_pfp_add_hfirstleft_entry. fom_beta_height_pfp_add_hfirstleft_entry + S (fom_value_pfp_add_hfirstleft) = S ((S (fom_index_pfp_add_hfirstleft)) * x1)) /\\ exists fom_beta_quotient_pfp_add_hfirstleft_entry. x = fom_beta_quotient_pfp_add_hfirstleft_entry * S ((S (fom_index_pfp_add_hfirstleft)) * x1) + (fom_value_pfp_add_hfirstleft))) /\\ (exists fom_gap_pfp_add_hfirstleft_value_bound. fom_gap_pfp_add_hfirstleft_value_bound + S (fom_value_pfp_add_hfirstleft) = p))) /\\ (((forall fom_index_pfp_add_hfirstright. (exists fom_gap_pfp_add_hfirstright_index_bound. fom_gap_pfp_add_hfirstright_index_bound + S (fom_index_pfp_add_hfirstright) = J) -> exists fom_value_pfp_add_hfirstright. ((((exists fom_beta_height_pfp_add_hfirstright_entry. fom_beta_height_pfp_add_hfirstright_entry + S (fom_value_pfp_add_hfirstright) = S ((S (fom_index_pfp_add_hfirstright)) * dc)) /\\ exists fom_beta_quotient_pfp_add_hfirstright_entry. db = fom_beta_quotient_pfp_add_hfirstright_entry * S ((S (fom_index_pfp_add_hfirstright)) * dc) + (fom_value_pfp_add_hfirstright))) /\\ (exists fom_gap_pfp_add_hfirstright_value_bound. fom_gap_pfp_add_hfirstright_value_bound + S (fom_value_pfp_add_hfirstright) = p))) /\\ (((((((x2)=0 \\/ (J)=0) /\\ (((x5)=0)))) \\/ (((~((x2)=0)) /\\ (((~((J)=0)) /\\ (((x2)+(J)=S (x5)))))))) /\\ ((forall pfc_index_add_hfirstcoefficients. (exists pfa_gap_add_hfirstcoefficientsbound. pfa_gap_add_hfirstcoefficientsbound + S (pfc_index_add_hfirstcoefficients) = (x5)) -> exists pfc_value_add_hfirstcoefficients. ((((exists ff_h_pfp_add_hfirstcoefficientsentry. ff_h_pfp_add_hfirstcoefficientsentry + S (pfc_value_add_hfirstcoefficients) = S ((S (pfc_index_add_hfirstcoefficients)) * x4)) /\\ exists ff_q_pfp_add_hfirstcoefficientsentry. x3 = ff_q_pfp_add_hfirstcoefficientsentry * S ((S (pfc_index_add_hfirstcoefficients)) * x4) + (pfc_value_add_hfirstcoefficients))) /\\ ((exists pfc_terms_code_add_hfirstcoefficientscoefficient pfc_terms_scale_add_hfirstcoefficientscoefficient pfc_natural_sum_add_hfirstcoefficientscoefficient. ((forall pfc_index_add_hfirstcoefficientscoefficientdiagonal. (exists pfa_gap_add_hfirstcoefficientscoefficientdiagonalbound. pfa_gap_add_hfirstcoefficientscoefficientdiagonalbound + S (pfc_index_add_hfirstcoefficientscoefficientdiagonal) = (S (pfc_index_add_hfirstcoefficients))) -> exists pfc_value_add_hfirstcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_add_hfirstcoefficientscoefficientdiagonalentry. ff_h_pfp_add_hfirstcoefficientscoefficientdiagonalentry + S (pfc_value_add_hfirstcoefficientscoefficientdiagonal) = S ((S (pfc_index_add_hfirstcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_hfirstcoefficientscoefficient)) /\\ exists ff_q_pfp_add_hfirstcoefficientscoefficientdiagonalentry. pfc_terms_code_add_hfirstcoefficientscoefficient = ff_q_pfp_add_hfirstcoefficientscoefficientdiagonalentry * S ((S (pfc_index_add_hfirstcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_hfirstcoefficientscoefficient) + (pfc_value_add_hfirstcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_add_hfirstcoefficientscoefficientdiagonalterm pfc_left_add_hfirstcoefficientscoefficientdiagonalterm pfc_right_add_hfirstcoefficientscoefficientdiagonalterm. (((pfc_index_add_hfirstcoefficientscoefficientdiagonal)+pfc_complement_add_hfirstcoefficientscoefficientdiagonalterm=(pfc_index_add_hfirstcoefficients)) /\\ ((((((exists pfa_gap_add_hfirstcoefficientscoefficientdiagonaltermleftinside. pfa_gap_add_hfirstcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_add_hfirstcoefficientscoefficientdiagonal) = (x2)) /\\ ((((exists ff_h_pfp_add_hfirstcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_add_hfirstcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_add_hfirstcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_add_hfirstcoefficientscoefficientdiagonal)) * x1)) /\\ exists ff_q_pfp_add_hfirstcoefficientscoefficientdiagonaltermleftentry. x = ff_q_pfp_add_hfirstcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_add_hfirstcoefficientscoefficientdiagonal)) * x1) + (pfc_left_add_hfirstcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_add_hfirstcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_add_hfirstcoefficientscoefficientdiagonaltermleftoutside+(x2)=(pfc_index_add_hfirstcoefficientscoefficientdiagonal)) /\\ (((pfc_left_add_hfirstcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_add_hfirstcoefficientscoefficientdiagonaltermrightinside. pfa_gap_add_hfirstcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_add_hfirstcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_add_hfirstcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_add_hfirstcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_add_hfirstcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_add_hfirstcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_add_hfirstcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_add_hfirstcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_add_hfirstcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_add_hfirstcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_add_hfirstcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_add_hfirstcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_add_hfirstcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_add_hfirstcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_add_hfirstcoefficientscoefficientdiagonal)=pfc_left_add_hfirstcoefficientscoefficientdiagonalterm*pfc_right_add_hfirstcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_add_hfirstcoefficientscoefficientsum fs_v_pfc_add_hfirstcoefficientscoefficientsum. ((((exists fs_h_pfc_add_hfirstcoefficientscoefficientsum_body_start. fs_h_pfc_add_hfirstcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_add_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_hfirstcoefficientscoefficientsum_body_start. fs_u_pfc_add_hfirstcoefficientscoefficientsum = fs_q_pfc_add_hfirstcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_add_hfirstcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_add_hfirstcoefficientscoefficientsum_body_terminal. fs_h_pfc_add_hfirstcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_add_hfirstcoefficientscoefficient) = S ((S (S (pfc_index_add_hfirstcoefficients))) * fs_v_pfc_add_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_hfirstcoefficientscoefficientsum_body_terminal. fs_u_pfc_add_hfirstcoefficientscoefficientsum = fs_q_pfc_add_hfirstcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_add_hfirstcoefficients))) * fs_v_pfc_add_hfirstcoefficientscoefficientsum) + (pfc_natural_sum_add_hfirstcoefficientscoefficient))) /\\ forall fs_i_pfc_add_hfirstcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_add_hfirstcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_add_hfirstcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_add_hfirstcoefficientscoefficientsum_body_steps = S (pfc_index_add_hfirstcoefficients)) -> exists fs_a_pfc_add_hfirstcoefficientscoefficientsum_body_steps fs_r_pfc_add_hfirstcoefficientscoefficientsum_body_steps fs_s_pfc_add_hfirstcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_add_hfirstcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_add_hfirstcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_add_hfirstcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_hfirstcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_hfirstcoefficientscoefficient)) /\\ exists fs_q_pfc_add_hfirstcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_add_hfirstcoefficientscoefficient = fs_q_pfc_add_hfirstcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_add_hfirstcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_hfirstcoefficientscoefficient) + (fs_a_pfc_add_hfirstcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_add_hfirstcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_add_hfirstcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_add_hfirstcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_hfirstcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_add_hfirstcoefficientscoefficientsum = fs_q_pfc_add_hfirstcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_add_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_hfirstcoefficientscoefficientsum) + (fs_r_pfc_add_hfirstcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_add_hfirstcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_add_hfirstcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_add_hfirstcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_add_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_hfirstcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_add_hfirstcoefficientscoefficientsum = fs_q_pfc_add_hfirstcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_add_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_hfirstcoefficientscoefficientsum) + (fs_s_pfc_add_hfirstcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_add_hfirstcoefficientscoefficientsum_body_steps = fs_r_pfc_add_hfirstcoefficientscoefficientsum_body_steps + fs_a_pfc_add_hfirstcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_add_hfirstcoefficientscoefficientresiduebound. pfa_gap_add_hfirstcoefficientscoefficientresiduebound + S (pfc_value_add_hfirstcoefficients) = (p)) /\\ ((exists pfa_offset_left_add_hfirstcoefficientscoefficientresiduecongruence pfa_offset_right_add_hfirstcoefficientscoefficientresiduecongruence. (pfc_natural_sum_add_hfirstcoefficientscoefficient) + (p) * pfa_offset_left_add_hfirstcoefficientscoefficientresiduecongruence = (pfc_value_add_hfirstcoefficients) + (p) * pfa_offset_right_add_hfirstcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hDA_right_witness_witness_witness_witness_witness_witness_left", "cases hfirst", "cases hfirst_right", "cases hfirst_right_right", "have hfirst_bounded : forall fom_index_pfp_add_hfirst_bounded. (exists fom_gap_pfp_add_hfirst_bounded_index_bound. fom_gap_pfp_add_hfirst_bounded_index_bound + S (fom_index_pfp_add_hfirst_bounded) = x5) -> exists fom_value_pfp_add_hfirst_bounded. ((((exists fom_beta_height_pfp_add_hfirst_bounded_entry. fom_beta_height_pfp_add_hfirst_bounded_entry + S (fom_value_pfp_add_hfirst_bounded) = S ((S (fom_index_pfp_add_hfirst_bounded)) * x4)) /\\ exists fom_beta_quotient_pfp_add_hfirst_bounded_entry. x3 = fom_beta_quotient_pfp_add_hfirst_bounded_entry * S ((S (fom_index_pfp_add_hfirst_bounded)) * x4) + (fom_value_pfp_add_hfirst_bounded))) /\\ (exists fom_gap_pfp_add_hfirst_bounded_value_bound. fom_gap_pfp_add_hfirst_bounded_value_bound + S (fom_value_pfp_add_hfirst_bounded) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (x)", "specialize prime_field_polynomial_convolution_bounded (x1)", "specialize prime_field_polynomial_convolution_bounded (x2)", "specialize prime_field_polynomial_convolution_bounded (db)", "specialize prime_field_polynomial_convolution_bounded (dc)", "specialize prime_field_polynomial_convolution_bounded (J)", "specialize prime_field_polynomial_convolution_bounded (x3)", "specialize prime_field_polynomial_convolution_bounded (x4)", "specialize prime_field_polynomial_convolution_bounded (x5)", "apply prime_field_polynomial_convolution_bounded", "exact hDA_right_witness_witness_witness_witness_witness_witness_left", "have hsecond : ((forall fom_index_pfp_add_hsecondleft. (exists fom_gap_pfp_add_hsecondleft_index_bound. fom_gap_pfp_add_hsecondleft_index_bound + S (fom_index_pfp_add_hsecondleft) = x8) -> exists fom_value_pfp_add_hsecondleft. ((((exists fom_beta_height_pfp_add_hsecondleft_entry. fom_beta_height_pfp_add_hsecondleft_entry + S (fom_value_pfp_add_hsecondleft) = S ((S (fom_index_pfp_add_hsecondleft)) * x7)) /\\ exists fom_beta_quotient_pfp_add_hsecondleft_entry. x6 = fom_beta_quotient_pfp_add_hsecondleft_entry * S ((S (fom_index_pfp_add_hsecondleft)) * x7) + (fom_value_pfp_add_hsecondleft))) /\\ (exists fom_gap_pfp_add_hsecondleft_value_bound. fom_gap_pfp_add_hsecondleft_value_bound + S (fom_value_pfp_add_hsecondleft) = p))) /\\ (((forall fom_index_pfp_add_hsecondright. (exists fom_gap_pfp_add_hsecondright_index_bound. fom_gap_pfp_add_hsecondright_index_bound + S (fom_index_pfp_add_hsecondright) = J) -> exists fom_value_pfp_add_hsecondright. ((((exists fom_beta_height_pfp_add_hsecondright_entry. fom_beta_height_pfp_add_hsecondright_entry + S (fom_value_pfp_add_hsecondright) = S ((S (fom_index_pfp_add_hsecondright)) * dc)) /\\ exists fom_beta_quotient_pfp_add_hsecondright_entry. db = fom_beta_quotient_pfp_add_hsecondright_entry * S ((S (fom_index_pfp_add_hsecondright)) * dc) + (fom_value_pfp_add_hsecondright))) /\\ (exists fom_gap_pfp_add_hsecondright_value_bound. fom_gap_pfp_add_hsecondright_value_bound + S (fom_value_pfp_add_hsecondright) = p))) /\\ (((((((x8)=0 \\/ (J)=0) /\\ (((x11)=0)))) \\/ (((~((x8)=0)) /\\ (((~((J)=0)) /\\ (((x8)+(J)=S (x11)))))))) /\\ ((forall pfc_index_add_hsecondcoefficients. (exists pfa_gap_add_hsecondcoefficientsbound. pfa_gap_add_hsecondcoefficientsbound + S (pfc_index_add_hsecondcoefficients) = (x11)) -> exists pfc_value_add_hsecondcoefficients. ((((exists ff_h_pfp_add_hsecondcoefficientsentry. ff_h_pfp_add_hsecondcoefficientsentry + S (pfc_value_add_hsecondcoefficients) = S ((S (pfc_index_add_hsecondcoefficients)) * x10)) /\\ exists ff_q_pfp_add_hsecondcoefficientsentry. x9 = ff_q_pfp_add_hsecondcoefficientsentry * S ((S (pfc_index_add_hsecondcoefficients)) * x10) + (pfc_value_add_hsecondcoefficients))) /\\ ((exists pfc_terms_code_add_hsecondcoefficientscoefficient pfc_terms_scale_add_hsecondcoefficientscoefficient pfc_natural_sum_add_hsecondcoefficientscoefficient. ((forall pfc_index_add_hsecondcoefficientscoefficientdiagonal. (exists pfa_gap_add_hsecondcoefficientscoefficientdiagonalbound. pfa_gap_add_hsecondcoefficientscoefficientdiagonalbound + S (pfc_index_add_hsecondcoefficientscoefficientdiagonal) = (S (pfc_index_add_hsecondcoefficients))) -> exists pfc_value_add_hsecondcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_add_hsecondcoefficientscoefficientdiagonalentry. ff_h_pfp_add_hsecondcoefficientscoefficientdiagonalentry + S (pfc_value_add_hsecondcoefficientscoefficientdiagonal) = S ((S (pfc_index_add_hsecondcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_hsecondcoefficientscoefficient)) /\\ exists ff_q_pfp_add_hsecondcoefficientscoefficientdiagonalentry. pfc_terms_code_add_hsecondcoefficientscoefficient = ff_q_pfp_add_hsecondcoefficientscoefficientdiagonalentry * S ((S (pfc_index_add_hsecondcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_hsecondcoefficientscoefficient) + (pfc_value_add_hsecondcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_add_hsecondcoefficientscoefficientdiagonalterm pfc_left_add_hsecondcoefficientscoefficientdiagonalterm pfc_right_add_hsecondcoefficientscoefficientdiagonalterm. (((pfc_index_add_hsecondcoefficientscoefficientdiagonal)+pfc_complement_add_hsecondcoefficientscoefficientdiagonalterm=(pfc_index_add_hsecondcoefficients)) /\\ ((((((exists pfa_gap_add_hsecondcoefficientscoefficientdiagonaltermleftinside. pfa_gap_add_hsecondcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_add_hsecondcoefficientscoefficientdiagonal) = (x8)) /\\ ((((exists ff_h_pfp_add_hsecondcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_add_hsecondcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_add_hsecondcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_add_hsecondcoefficientscoefficientdiagonal)) * x7)) /\\ exists ff_q_pfp_add_hsecondcoefficientscoefficientdiagonaltermleftentry. x6 = ff_q_pfp_add_hsecondcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_add_hsecondcoefficientscoefficientdiagonal)) * x7) + (pfc_left_add_hsecondcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_add_hsecondcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_add_hsecondcoefficientscoefficientdiagonaltermleftoutside+(x8)=(pfc_index_add_hsecondcoefficientscoefficientdiagonal)) /\\ (((pfc_left_add_hsecondcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_add_hsecondcoefficientscoefficientdiagonaltermrightinside. pfa_gap_add_hsecondcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_add_hsecondcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_add_hsecondcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_add_hsecondcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_add_hsecondcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_add_hsecondcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_add_hsecondcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_add_hsecondcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_add_hsecondcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_add_hsecondcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_add_hsecondcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_add_hsecondcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_add_hsecondcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_add_hsecondcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_add_hsecondcoefficientscoefficientdiagonal)=pfc_left_add_hsecondcoefficientscoefficientdiagonalterm*pfc_right_add_hsecondcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_add_hsecondcoefficientscoefficientsum fs_v_pfc_add_hsecondcoefficientscoefficientsum. ((((exists fs_h_pfc_add_hsecondcoefficientscoefficientsum_body_start. fs_h_pfc_add_hsecondcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_add_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_hsecondcoefficientscoefficientsum_body_start. fs_u_pfc_add_hsecondcoefficientscoefficientsum = fs_q_pfc_add_hsecondcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_add_hsecondcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_add_hsecondcoefficientscoefficientsum_body_terminal. fs_h_pfc_add_hsecondcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_add_hsecondcoefficientscoefficient) = S ((S (S (pfc_index_add_hsecondcoefficients))) * fs_v_pfc_add_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_hsecondcoefficientscoefficientsum_body_terminal. fs_u_pfc_add_hsecondcoefficientscoefficientsum = fs_q_pfc_add_hsecondcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_add_hsecondcoefficients))) * fs_v_pfc_add_hsecondcoefficientscoefficientsum) + (pfc_natural_sum_add_hsecondcoefficientscoefficient))) /\\ forall fs_i_pfc_add_hsecondcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_add_hsecondcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_add_hsecondcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_add_hsecondcoefficientscoefficientsum_body_steps = S (pfc_index_add_hsecondcoefficients)) -> exists fs_a_pfc_add_hsecondcoefficientscoefficientsum_body_steps fs_r_pfc_add_hsecondcoefficientscoefficientsum_body_steps fs_s_pfc_add_hsecondcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_add_hsecondcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_add_hsecondcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_add_hsecondcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_hsecondcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_hsecondcoefficientscoefficient)) /\\ exists fs_q_pfc_add_hsecondcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_add_hsecondcoefficientscoefficient = fs_q_pfc_add_hsecondcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_add_hsecondcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_hsecondcoefficientscoefficient) + (fs_a_pfc_add_hsecondcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_add_hsecondcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_add_hsecondcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_add_hsecondcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_hsecondcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_add_hsecondcoefficientscoefficientsum = fs_q_pfc_add_hsecondcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_add_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_hsecondcoefficientscoefficientsum) + (fs_r_pfc_add_hsecondcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_add_hsecondcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_add_hsecondcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_add_hsecondcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_add_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_hsecondcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_add_hsecondcoefficientscoefficientsum = fs_q_pfc_add_hsecondcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_add_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_hsecondcoefficientscoefficientsum) + (fs_s_pfc_add_hsecondcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_add_hsecondcoefficientscoefficientsum_body_steps = fs_r_pfc_add_hsecondcoefficientscoefficientsum_body_steps + fs_a_pfc_add_hsecondcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_add_hsecondcoefficientscoefficientresiduebound. pfa_gap_add_hsecondcoefficientscoefficientresiduebound + S (pfc_value_add_hsecondcoefficients) = (p)) /\\ ((exists pfa_offset_left_add_hsecondcoefficientscoefficientresiduecongruence pfa_offset_right_add_hsecondcoefficientscoefficientresiduecongruence. (pfc_natural_sum_add_hsecondcoefficientscoefficient) + (p) * pfa_offset_left_add_hsecondcoefficientscoefficientresiduecongruence = (pfc_value_add_hsecondcoefficients) + (p) * pfa_offset_right_add_hsecondcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hDB_right_witness_witness_witness_witness_witness_witness_left", "cases hsecond", "cases hsecond_right", "cases hsecond_right_right", "have hsecond_bounded : forall fom_index_pfp_add_hsecond_bounded. (exists fom_gap_pfp_add_hsecond_bounded_index_bound. fom_gap_pfp_add_hsecond_bounded_index_bound + S (fom_index_pfp_add_hsecond_bounded) = x11) -> exists fom_value_pfp_add_hsecond_bounded. ((((exists fom_beta_height_pfp_add_hsecond_bounded_entry. fom_beta_height_pfp_add_hsecond_bounded_entry + S (fom_value_pfp_add_hsecond_bounded) = S ((S (fom_index_pfp_add_hsecond_bounded)) * x10)) /\\ exists fom_beta_quotient_pfp_add_hsecond_bounded_entry. x9 = fom_beta_quotient_pfp_add_hsecond_bounded_entry * S ((S (fom_index_pfp_add_hsecond_bounded)) * x10) + (fom_value_pfp_add_hsecond_bounded))) /\\ (exists fom_gap_pfp_add_hsecond_bounded_value_bound. fom_gap_pfp_add_hsecond_bounded_value_bound + S (fom_value_pfp_add_hsecond_bounded) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (x6)", "specialize prime_field_polynomial_convolution_bounded (x7)", "specialize prime_field_polynomial_convolution_bounded (x8)", "specialize prime_field_polynomial_convolution_bounded (db)", "specialize prime_field_polynomial_convolution_bounded (dc)", "specialize prime_field_polynomial_convolution_bounded (J)", "specialize prime_field_polynomial_convolution_bounded (x9)", "specialize prime_field_polynomial_convolution_bounded (x10)", "specialize prime_field_polynomial_convolution_bounded (x11)", "apply prime_field_polynomial_convolution_bounded", "exact hDB_right_witness_witness_witness_witness_witness_witness_left", "have hw : exists wb wc. ((forall fom_index_pfp_add_quotient_operation_left_bounded. (exists fom_gap_pfp_add_quotient_operation_left_bounded_index_bound. fom_gap_pfp_add_quotient_operation_left_bounded_index_bound + S (fom_index_pfp_add_quotient_operation_left_bounded) = x2) -> exists fom_value_pfp_add_quotient_operation_left_bounded. ((((exists fom_beta_height_pfp_add_quotient_operation_left_bounded_entry. fom_beta_height_pfp_add_quotient_operation_left_bounded_entry + S (fom_value_pfp_add_quotient_operation_left_bounded) = S ((S (fom_index_pfp_add_quotient_operation_left_bounded)) * x1)) /\\ exists fom_beta_quotient_pfp_add_quotient_operation_left_bounded_entry. x = fom_beta_quotient_pfp_add_quotient_operation_left_bounded_entry * S ((S (fom_index_pfp_add_quotient_operation_left_bounded)) * x1) + (fom_value_pfp_add_quotient_operation_left_bounded))) /\\ (exists fom_gap_pfp_add_quotient_operation_left_bounded_value_bound. fom_gap_pfp_add_quotient_operation_left_bounded_value_bound + S (fom_value_pfp_add_quotient_operation_left_bounded) = p))) /\\ (((forall fom_index_pfp_add_quotient_operation_right_bounded. (exists fom_gap_pfp_add_quotient_operation_right_bounded_index_bound. fom_gap_pfp_add_quotient_operation_right_bounded_index_bound + S (fom_index_pfp_add_quotient_operation_right_bounded) = x8) -> exists fom_value_pfp_add_quotient_operation_right_bounded. ((((exists fom_beta_height_pfp_add_quotient_operation_right_bounded_entry. fom_beta_height_pfp_add_quotient_operation_right_bounded_entry + S (fom_value_pfp_add_quotient_operation_right_bounded) = S ((S (fom_index_pfp_add_quotient_operation_right_bounded)) * x7)) /\\ exists fom_beta_quotient_pfp_add_quotient_operation_right_bounded_entry. x6 = fom_beta_quotient_pfp_add_quotient_operation_right_bounded_entry * S ((S (fom_index_pfp_add_quotient_operation_right_bounded)) * x7) + (fom_value_pfp_add_quotient_operation_right_bounded))) /\\ (exists fom_gap_pfp_add_quotient_operation_right_bounded_value_bound. fom_gap_pfp_add_quotient_operation_right_bounded_value_bound + S (fom_value_pfp_add_quotient_operation_right_bounded) = p))) /\\ (((forall fom_index_pfp_add_quotient_operation_result_bounded. (exists fom_gap_pfp_add_quotient_operation_result_bounded_index_bound. fom_gap_pfp_add_quotient_operation_result_bounded_index_bound + S (fom_index_pfp_add_quotient_operation_result_bounded) = (x2)+(x8)) -> exists fom_value_pfp_add_quotient_operation_result_bounded. ((((exists fom_beta_height_pfp_add_quotient_operation_result_bounded_entry. fom_beta_height_pfp_add_quotient_operation_result_bounded_entry + S (fom_value_pfp_add_quotient_operation_result_bounded) = S ((S (fom_index_pfp_add_quotient_operation_result_bounded)) * wc)) /\\ exists fom_beta_quotient_pfp_add_quotient_operation_result_bounded_entry. wb = fom_beta_quotient_pfp_add_quotient_operation_result_bounded_entry * S ((S (fom_index_pfp_add_quotient_operation_result_bounded)) * wc) + (fom_value_pfp_add_quotient_operation_result_bounded))) /\\ (exists fom_gap_pfp_add_quotient_operation_result_bounded_value_bound. fom_gap_pfp_add_quotient_operation_result_bounded_value_bound + S (fom_value_pfp_add_quotient_operation_result_bounded) = p))) /\\ ((exists pfaa_left_b_add_quotient_operation pfaa_left_c_add_quotient_operation pfaa_right_b_add_quotient_operation pfaa_right_c_add_quotient_operation pfaa_sum_b_add_quotient_operation pfaa_sum_c_add_quotient_operation pfaa_length_add_quotient_operation. ((((forall pfrep_power_add_quotient_operation_witness_common_left pfrep_left_add_quotient_operation_witness_common_left pfrep_right_add_quotient_operation_witness_common_left. ((exists pfrep_position_add_quotient_operation_witness_common_leftfirst. ((pfrep_position_add_quotient_operation_witness_common_leftfirst+S (pfrep_power_add_quotient_operation_witness_common_left)=(x2)) /\\ ((((exists ff_h_pfp_add_quotient_operation_witness_common_leftfirstentry. ff_h_pfp_add_quotient_operation_witness_common_leftfirstentry + S (pfrep_left_add_quotient_operation_witness_common_left) = S ((S (pfrep_position_add_quotient_operation_witness_common_leftfirst)) * x1)) /\\ exists ff_q_pfp_add_quotient_operation_witness_common_leftfirstentry. x = ff_q_pfp_add_quotient_operation_witness_common_leftfirstentry * S ((S (pfrep_position_add_quotient_operation_witness_common_leftfirst)) * x1) + (pfrep_left_add_quotient_operation_witness_common_left)))))) \\/ (((exists pfrep_gap_add_quotient_operation_witness_common_leftfirstoutside. pfrep_gap_add_quotient_operation_witness_common_leftfirstoutside+(x2)=(pfrep_power_add_quotient_operation_witness_common_left)) /\\ (((pfrep_left_add_quotient_operation_witness_common_left)=0))))) -> ((exists pfrep_position_add_quotient_operation_witness_common_leftsecond. ((pfrep_position_add_quotient_operation_witness_common_leftsecond+S (pfrep_power_add_quotient_operation_witness_common_left)=(pfaa_length_add_quotient_operation)) /\\ ((((exists ff_h_pfp_add_quotient_operation_witness_common_leftsecondentry. ff_h_pfp_add_quotient_operation_witness_common_leftsecondentry + S (pfrep_right_add_quotient_operation_witness_common_left) = S ((S (pfrep_position_add_quotient_operation_witness_common_leftsecond)) * pfaa_left_c_add_quotient_operation)) /\\ exists ff_q_pfp_add_quotient_operation_witness_common_leftsecondentry. pfaa_left_b_add_quotient_operation = ff_q_pfp_add_quotient_operation_witness_common_leftsecondentry * S ((S (pfrep_position_add_quotient_operation_witness_common_leftsecond)) * pfaa_left_c_add_quotient_operation) + (pfrep_right_add_quotient_operation_witness_common_left)))))) \\/ (((exists pfrep_gap_add_quotient_operation_witness_common_leftsecondoutside. pfrep_gap_add_quotient_operation_witness_common_leftsecondoutside+(pfaa_length_add_quotient_operation)=(pfrep_power_add_quotient_operation_witness_common_left)) /\\ (((pfrep_right_add_quotient_operation_witness_common_left)=0))))) -> pfrep_left_add_quotient_operation_witness_common_left=pfrep_right_add_quotient_operation_witness_common_left) /\\ ((forall pfrep_power_add_quotient_operation_witness_common_right pfrep_left_add_quotient_operation_witness_common_right pfrep_right_add_quotient_operation_witness_common_right. ((exists pfrep_position_add_quotient_operation_witness_common_rightfirst. ((pfrep_position_add_quotient_operation_witness_common_rightfirst+S (pfrep_power_add_quotient_operation_witness_common_right)=(x8)) /\\ ((((exists ff_h_pfp_add_quotient_operation_witness_common_rightfirstentry. ff_h_pfp_add_quotient_operation_witness_common_rightfirstentry + S (pfrep_left_add_quotient_operation_witness_common_right) = S ((S (pfrep_position_add_quotient_operation_witness_common_rightfirst)) * x7)) /\\ exists ff_q_pfp_add_quotient_operation_witness_common_rightfirstentry. x6 = ff_q_pfp_add_quotient_operation_witness_common_rightfirstentry * S ((S (pfrep_position_add_quotient_operation_witness_common_rightfirst)) * x7) + (pfrep_left_add_quotient_operation_witness_common_right)))))) \\/ (((exists pfrep_gap_add_quotient_operation_witness_common_rightfirstoutside. pfrep_gap_add_quotient_operation_witness_common_rightfirstoutside+(x8)=(pfrep_power_add_quotient_operation_witness_common_right)) /\\ (((pfrep_left_add_quotient_operation_witness_common_right)=0))))) -> ((exists pfrep_position_add_quotient_operation_witness_common_rightsecond. ((pfrep_position_add_quotient_operation_witness_common_rightsecond+S (pfrep_power_add_quotient_operation_witness_common_right)=(pfaa_length_add_quotient_operation)) /\\ ((((exists ff_h_pfp_add_quotient_operation_witness_common_rightsecondentry. ff_h_pfp_add_quotient_operation_witness_common_rightsecondentry + S (pfrep_right_add_quotient_operation_witness_common_right) = S ((S (pfrep_position_add_quotient_operation_witness_common_rightsecond)) * pfaa_right_c_add_quotient_operation)) /\\ exists ff_q_pfp_add_quotient_operation_witness_common_rightsecondentry. pfaa_right_b_add_quotient_operation = ff_q_pfp_add_quotient_operation_witness_common_rightsecondentry * S ((S (pfrep_position_add_quotient_operation_witness_common_rightsecond)) * pfaa_right_c_add_quotient_operation) + (pfrep_right_add_quotient_operation_witness_common_right)))))) \\/ (((exists pfrep_gap_add_quotient_operation_witness_common_rightsecondoutside. pfrep_gap_add_quotient_operation_witness_common_rightsecondoutside+(pfaa_length_add_quotient_operation)=(pfrep_power_add_quotient_operation_witness_common_right)) /\\ (((pfrep_right_add_quotient_operation_witness_common_right)=0))))) -> pfrep_left_add_quotient_operation_witness_common_right=pfrep_right_add_quotient_operation_witness_common_right)))) /\\ (((forall pfp_index_add_quotient_operation_witness_operation. (exists pfa_gap_add_quotient_operation_witness_operationindex. pfa_gap_add_quotient_operation_witness_operationindex + S (pfp_index_add_quotient_operation_witness_operation) = (pfaa_length_add_quotient_operation)) -> exists pfp_left_add_quotient_operation_witness_operation pfp_right_add_quotient_operation_witness_operation pfp_value_add_quotient_operation_witness_operation. ((((exists ff_h_pfp_add_quotient_operation_witness_operationleft. ff_h_pfp_add_quotient_operation_witness_operationleft + S (pfp_left_add_quotient_operation_witness_operation) = S ((S (pfp_index_add_quotient_operation_witness_operation)) * pfaa_left_c_add_quotient_operation)) /\\ exists ff_q_pfp_add_quotient_operation_witness_operationleft. pfaa_left_b_add_quotient_operation = ff_q_pfp_add_quotient_operation_witness_operationleft * S ((S (pfp_index_add_quotient_operation_witness_operation)) * pfaa_left_c_add_quotient_operation) + (pfp_left_add_quotient_operation_witness_operation))) /\\ (((((exists ff_h_pfp_add_quotient_operation_witness_operationright. ff_h_pfp_add_quotient_operation_witness_operationright + S (pfp_right_add_quotient_operation_witness_operation) = S ((S (pfp_index_add_quotient_operation_witness_operation)) * pfaa_right_c_add_quotient_operation)) /\\ exists ff_q_pfp_add_quotient_operation_witness_operationright. pfaa_right_b_add_quotient_operation = ff_q_pfp_add_quotient_operation_witness_operationright * S ((S (pfp_index_add_quotient_operation_witness_operation)) * pfaa_right_c_add_quotient_operation) + (pfp_right_add_quotient_operation_witness_operation))) /\\ (((((exists ff_h_pfp_add_quotient_operation_witness_operationtarget. ff_h_pfp_add_quotient_operation_witness_operationtarget + S (pfp_value_add_quotient_operation_witness_operation) = S ((S (pfp_index_add_quotient_operation_witness_operation)) * pfaa_sum_c_add_quotient_operation)) /\\ exists ff_q_pfp_add_quotient_operation_witness_operationtarget. pfaa_sum_b_add_quotient_operation = ff_q_pfp_add_quotient_operation_witness_operationtarget * S ((S (pfp_index_add_quotient_operation_witness_operation)) * pfaa_sum_c_add_quotient_operation) + (pfp_value_add_quotient_operation_witness_operation))) /\\ ((((exists pfa_gap_add_quotient_operation_witness_operationoperationleft. pfa_gap_add_quotient_operation_witness_operationoperationleft + S (pfp_left_add_quotient_operation_witness_operation) = (p)) /\\ (((exists pfa_gap_add_quotient_operation_witness_operationoperationright. pfa_gap_add_quotient_operation_witness_operationoperationright + S (pfp_right_add_quotient_operation_witness_operation) = (p)) /\\ ((((exists pfa_gap_add_quotient_operation_witness_operationoperationresultbound. pfa_gap_add_quotient_operation_witness_operationoperationresultbound + S (pfp_value_add_quotient_operation_witness_operation) = (p)) /\\ ((exists pfa_offset_left_add_quotient_operation_witness_operationoperationresultcongruence pfa_offset_right_add_quotient_operation_witness_operationoperationresultcongruence. ((pfp_left_add_quotient_operation_witness_operation) + (pfp_right_add_quotient_operation_witness_operation)) + (p) * pfa_offset_left_add_quotient_operation_witness_operationoperationresultcongruence = (pfp_value_add_quotient_operation_witness_operation) + (p) * pfa_offset_right_add_quotient_operation_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_add_quotient_operation_witness_output pfrep_left_add_quotient_operation_witness_output pfrep_right_add_quotient_operation_witness_output. ((exists pfrep_position_add_quotient_operation_witness_outputfirst. ((pfrep_position_add_quotient_operation_witness_outputfirst+S (pfrep_power_add_quotient_operation_witness_output)=(pfaa_length_add_quotient_operation)) /\\ ((((exists ff_h_pfp_add_quotient_operation_witness_outputfirstentry. ff_h_pfp_add_quotient_operation_witness_outputfirstentry + S (pfrep_left_add_quotient_operation_witness_output) = S ((S (pfrep_position_add_quotient_operation_witness_outputfirst)) * pfaa_sum_c_add_quotient_operation)) /\\ exists ff_q_pfp_add_quotient_operation_witness_outputfirstentry. pfaa_sum_b_add_quotient_operation = ff_q_pfp_add_quotient_operation_witness_outputfirstentry * S ((S (pfrep_position_add_quotient_operation_witness_outputfirst)) * pfaa_sum_c_add_quotient_operation) + (pfrep_left_add_quotient_operation_witness_output)))))) \\/ (((exists pfrep_gap_add_quotient_operation_witness_outputfirstoutside. pfrep_gap_add_quotient_operation_witness_outputfirstoutside+(pfaa_length_add_quotient_operation)=(pfrep_power_add_quotient_operation_witness_output)) /\\ (((pfrep_left_add_quotient_operation_witness_output)=0))))) -> ((exists pfrep_position_add_quotient_operation_witness_outputsecond. ((pfrep_position_add_quotient_operation_witness_outputsecond+S (pfrep_power_add_quotient_operation_witness_output)=((x2)+(x8))) /\\ ((((exists ff_h_pfp_add_quotient_operation_witness_outputsecondentry. ff_h_pfp_add_quotient_operation_witness_outputsecondentry + S (pfrep_right_add_quotient_operation_witness_output) = S ((S (pfrep_position_add_quotient_operation_witness_outputsecond)) * wc)) /\\ exists ff_q_pfp_add_quotient_operation_witness_outputsecondentry. wb = ff_q_pfp_add_quotient_operation_witness_outputsecondentry * S ((S (pfrep_position_add_quotient_operation_witness_outputsecond)) * wc) + (pfrep_right_add_quotient_operation_witness_output)))))) \\/ (((exists pfrep_gap_add_quotient_operation_witness_outputsecondoutside. pfrep_gap_add_quotient_operation_witness_outputsecondoutside+((x2)+(x8))=(pfrep_power_add_quotient_operation_witness_output)) /\\ (((pfrep_right_add_quotient_operation_witness_output)=0))))) -> pfrep_left_add_quotient_operation_witness_output=pfrep_right_add_quotient_operation_witness_output))))))))))))", "specialize prime_field_polynomial_aligned_add_exists (p)", "specialize prime_field_polynomial_aligned_add_exists (x)", "specialize prime_field_polynomial_aligned_add_exists (x1)", "specialize prime_field_polynomial_aligned_add_exists (x2)", "specialize prime_field_polynomial_aligned_add_exists (x6)", "specialize prime_field_polynomial_aligned_add_exists (x7)", "specialize prime_field_polynomial_aligned_add_exists (x8)", "apply prime_field_polynomial_aligned_add_exists", "exact hp", "exact hfirst_left", "exact hsecond_left", "cases hw", "cases hw_witness", "have hwbound : ((forall fom_index_pfp_add_quotient_bound_0. (exists fom_gap_pfp_add_quotient_bound_0_index_bound. fom_gap_pfp_add_quotient_bound_0_index_bound + S (fom_index_pfp_add_quotient_bound_0) = x2) -> exists fom_value_pfp_add_quotient_bound_0. ((((exists fom_beta_height_pfp_add_quotient_bound_0_entry. fom_beta_height_pfp_add_quotient_bound_0_entry + S (fom_value_pfp_add_quotient_bound_0) = S ((S (fom_index_pfp_add_quotient_bound_0)) * x1)) /\\ exists fom_beta_quotient_pfp_add_quotient_bound_0_entry. x = fom_beta_quotient_pfp_add_quotient_bound_0_entry * S ((S (fom_index_pfp_add_quotient_bound_0)) * x1) + (fom_value_pfp_add_quotient_bound_0))) /\\ (exists fom_gap_pfp_add_quotient_bound_0_value_bound. fom_gap_pfp_add_quotient_bound_0_value_bound + S (fom_value_pfp_add_quotient_bound_0) = p))) /\\ (((forall fom_index_pfp_add_quotient_bound_1. (exists fom_gap_pfp_add_quotient_bound_1_index_bound. fom_gap_pfp_add_quotient_bound_1_index_bound + S (fom_index_pfp_add_quotient_bound_1) = x8) -> exists fom_value_pfp_add_quotient_bound_1. ((((exists fom_beta_height_pfp_add_quotient_bound_1_entry. fom_beta_height_pfp_add_quotient_bound_1_entry + S (fom_value_pfp_add_quotient_bound_1) = S ((S (fom_index_pfp_add_quotient_bound_1)) * x7)) /\\ exists fom_beta_quotient_pfp_add_quotient_bound_1_entry. x6 = fom_beta_quotient_pfp_add_quotient_bound_1_entry * S ((S (fom_index_pfp_add_quotient_bound_1)) * x7) + (fom_value_pfp_add_quotient_bound_1))) /\\ (exists fom_gap_pfp_add_quotient_bound_1_value_bound. fom_gap_pfp_add_quotient_bound_1_value_bound + S (fom_value_pfp_add_quotient_bound_1) = p))) /\\ ((forall fom_index_pfp_add_quotient_bound_2. (exists fom_gap_pfp_add_quotient_bound_2_index_bound. fom_gap_pfp_add_quotient_bound_2_index_bound + S (fom_index_pfp_add_quotient_bound_2) = (x2)+(x8)) -> exists fom_value_pfp_add_quotient_bound_2. ((((exists fom_beta_height_pfp_add_quotient_bound_2_entry. fom_beta_height_pfp_add_quotient_bound_2_entry + S (fom_value_pfp_add_quotient_bound_2) = S ((S (fom_index_pfp_add_quotient_bound_2)) * x13)) /\\ exists fom_beta_quotient_pfp_add_quotient_bound_2_entry. x12 = fom_beta_quotient_pfp_add_quotient_bound_2_entry * S ((S (fom_index_pfp_add_quotient_bound_2)) * x13) + (fom_value_pfp_add_quotient_bound_2))) /\\ (exists fom_gap_pfp_add_quotient_bound_2_value_bound. fom_gap_pfp_add_quotient_bound_2_value_bound + S (fom_value_pfp_add_quotient_bound_2) = p)))))))", "specialize prime_field_polynomial_aligned_add_bounded (p)", "specialize prime_field_polynomial_aligned_add_bounded (x)", "specialize prime_field_polynomial_aligned_add_bounded (x1)", "specialize prime_field_polynomial_aligned_add_bounded (x2)", "specialize prime_field_polynomial_aligned_add_bounded (x6)", "specialize prime_field_polynomial_aligned_add_bounded (x7)", "specialize prime_field_polynomial_aligned_add_bounded (x8)", "specialize prime_field_polynomial_aligned_add_bounded (x12)", "specialize prime_field_polynomial_aligned_add_bounded (x13)", "specialize prime_field_polynomial_aligned_add_bounded ((x2)+(x8))", "apply prime_field_polynomial_aligned_add_bounded", "exact hw_witness_witness", "cases hwbound", "cases hwbound_right", "have hresult_length : exists n. (((((x2)+(x8))=0 \\/ (J)=0) /\\ (((n)=0)))) \\/ (((~(((x2)+(x8))=0)) /\\ (((~((J)=0)) /\\ ((((x2)+(x8))+(J)=S (n)))))))", "specialize polynomial_product_length_exists ((x2)+(x8))", "specialize polynomial_product_length_exists (J)", "apply polynomial_product_length_exists", "cases hresult_length", "have hresult_product : exists b c. ((forall fom_index_pfp_hresult_productleft. (exists fom_gap_pfp_hresult_productleft_index_bound. fom_gap_pfp_hresult_productleft_index_bound + S (fom_index_pfp_hresult_productleft) = (x2)+(x8)) -> exists fom_value_pfp_hresult_productleft. ((((exists fom_beta_height_pfp_hresult_productleft_entry. fom_beta_height_pfp_hresult_productleft_entry + S (fom_value_pfp_hresult_productleft) = S ((S (fom_index_pfp_hresult_productleft)) * x13)) /\\ exists fom_beta_quotient_pfp_hresult_productleft_entry. x12 = fom_beta_quotient_pfp_hresult_productleft_entry * S ((S (fom_index_pfp_hresult_productleft)) * x13) + (fom_value_pfp_hresult_productleft))) /\\ (exists fom_gap_pfp_hresult_productleft_value_bound. fom_gap_pfp_hresult_productleft_value_bound + S (fom_value_pfp_hresult_productleft) = p))) /\\ (((forall fom_index_pfp_hresult_productright. (exists fom_gap_pfp_hresult_productright_index_bound. fom_gap_pfp_hresult_productright_index_bound + S (fom_index_pfp_hresult_productright) = J) -> exists fom_value_pfp_hresult_productright. ((((exists fom_beta_height_pfp_hresult_productright_entry. fom_beta_height_pfp_hresult_productright_entry + S (fom_value_pfp_hresult_productright) = S ((S (fom_index_pfp_hresult_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_hresult_productright_entry. db = fom_beta_quotient_pfp_hresult_productright_entry * S ((S (fom_index_pfp_hresult_productright)) * dc) + (fom_value_pfp_hresult_productright))) /\\ (exists fom_gap_pfp_hresult_productright_value_bound. fom_gap_pfp_hresult_productright_value_bound + S (fom_value_pfp_hresult_productright) = p))) /\\ ((((((((x2)+(x8))=0 \\/ (J)=0) /\\ (((x14)=0)))) \\/ (((~(((x2)+(x8))=0)) /\\ (((~((J)=0)) /\\ ((((x2)+(x8))+(J)=S (x14)))))))) /\\ ((forall pfc_index_hresult_productcoefficients. (exists pfa_gap_hresult_productcoefficientsbound. pfa_gap_hresult_productcoefficientsbound + S (pfc_index_hresult_productcoefficients) = (x14)) -> exists pfc_value_hresult_productcoefficients. ((((exists ff_h_pfp_hresult_productcoefficientsentry. ff_h_pfp_hresult_productcoefficientsentry + S (pfc_value_hresult_productcoefficients) = S ((S (pfc_index_hresult_productcoefficients)) * c)) /\\ exists ff_q_pfp_hresult_productcoefficientsentry. b = ff_q_pfp_hresult_productcoefficientsentry * S ((S (pfc_index_hresult_productcoefficients)) * c) + (pfc_value_hresult_productcoefficients))) /\\ ((exists pfc_terms_code_hresult_productcoefficientscoefficient pfc_terms_scale_hresult_productcoefficientscoefficient pfc_natural_sum_hresult_productcoefficientscoefficient. ((forall pfc_index_hresult_productcoefficientscoefficientdiagonal. (exists pfa_gap_hresult_productcoefficientscoefficientdiagonalbound. pfa_gap_hresult_productcoefficientscoefficientdiagonalbound + S (pfc_index_hresult_productcoefficientscoefficientdiagonal) = (S (pfc_index_hresult_productcoefficients))) -> exists pfc_value_hresult_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_hresult_productcoefficientscoefficientdiagonalentry. ff_h_pfp_hresult_productcoefficientscoefficientdiagonalentry + S (pfc_value_hresult_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_hresult_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hresult_productcoefficientscoefficient)) /\\ exists ff_q_pfp_hresult_productcoefficientscoefficientdiagonalentry. pfc_terms_code_hresult_productcoefficientscoefficient = ff_q_pfp_hresult_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_hresult_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hresult_productcoefficientscoefficient) + (pfc_value_hresult_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_hresult_productcoefficientscoefficientdiagonalterm pfc_left_hresult_productcoefficientscoefficientdiagonalterm pfc_right_hresult_productcoefficientscoefficientdiagonalterm. (((pfc_index_hresult_productcoefficientscoefficientdiagonal)+pfc_complement_hresult_productcoefficientscoefficientdiagonalterm=(pfc_index_hresult_productcoefficients)) /\\ ((((((exists pfa_gap_hresult_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_hresult_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_hresult_productcoefficientscoefficientdiagonal) = ((x2)+(x8))) /\\ ((((exists ff_h_pfp_hresult_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_hresult_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_hresult_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_hresult_productcoefficientscoefficientdiagonal)) * x13)) /\\ exists ff_q_pfp_hresult_productcoefficientscoefficientdiagonaltermleftentry. x12 = ff_q_pfp_hresult_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_hresult_productcoefficientscoefficientdiagonal)) * x13) + (pfc_left_hresult_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hresult_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_hresult_productcoefficientscoefficientdiagonaltermleftoutside+((x2)+(x8))=(pfc_index_hresult_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_hresult_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_hresult_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_hresult_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_hresult_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_hresult_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_hresult_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_hresult_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_hresult_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_hresult_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_hresult_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_hresult_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_hresult_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hresult_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_hresult_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_hresult_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_hresult_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_hresult_productcoefficientscoefficientdiagonal)=pfc_left_hresult_productcoefficientscoefficientdiagonalterm*pfc_right_hresult_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_hresult_productcoefficientscoefficientsum fs_v_pfc_hresult_productcoefficientscoefficientsum. ((((exists fs_h_pfc_hresult_productcoefficientscoefficientsum_body_start. fs_h_pfc_hresult_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_hresult_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hresult_productcoefficientscoefficientsum_body_start. fs_u_pfc_hresult_productcoefficientscoefficientsum = fs_q_pfc_hresult_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_hresult_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_hresult_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_hresult_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_hresult_productcoefficientscoefficient) = S ((S (S (pfc_index_hresult_productcoefficients))) * fs_v_pfc_hresult_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hresult_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_hresult_productcoefficientscoefficientsum = fs_q_pfc_hresult_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_hresult_productcoefficients))) * fs_v_pfc_hresult_productcoefficientscoefficientsum) + (pfc_natural_sum_hresult_productcoefficientscoefficient))) /\\ forall fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_hresult_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_hresult_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps = S (pfc_index_hresult_productcoefficients)) -> exists fs_a_pfc_hresult_productcoefficientscoefficientsum_body_steps fs_r_pfc_hresult_productcoefficientscoefficientsum_body_steps fs_s_pfc_hresult_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_hresult_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_hresult_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_hresult_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hresult_productcoefficientscoefficient)) /\\ exists fs_q_pfc_hresult_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_hresult_productcoefficientscoefficient = fs_q_pfc_hresult_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hresult_productcoefficientscoefficient) + (fs_a_pfc_hresult_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hresult_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_hresult_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_hresult_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hresult_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hresult_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_hresult_productcoefficientscoefficientsum = fs_q_pfc_hresult_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hresult_productcoefficientscoefficientsum) + (fs_r_pfc_hresult_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hresult_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_hresult_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_hresult_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hresult_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hresult_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_hresult_productcoefficientscoefficientsum = fs_q_pfc_hresult_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hresult_productcoefficientscoefficientsum) + (fs_s_pfc_hresult_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_hresult_productcoefficientscoefficientsum_body_steps = fs_r_pfc_hresult_productcoefficientscoefficientsum_body_steps + fs_a_pfc_hresult_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_hresult_productcoefficientscoefficientresiduebound. pfa_gap_hresult_productcoefficientscoefficientresiduebound + S (pfc_value_hresult_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_hresult_productcoefficientscoefficientresiduecongruence pfa_offset_right_hresult_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_hresult_productcoefficientscoefficient) + (p) * pfa_offset_left_hresult_productcoefficientscoefficientresiduecongruence = (pfc_value_hresult_productcoefficients) + (p) * pfa_offset_right_hresult_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (x12)", "specialize prime_field_polynomial_convolution_at_length_exists (x13)", "specialize prime_field_polynomial_convolution_at_length_exists ((x2)+(x8))", "specialize prime_field_polynomial_convolution_at_length_exists (db)", "specialize prime_field_polynomial_convolution_at_length_exists (dc)", "specialize prime_field_polynomial_convolution_at_length_exists (J)", "specialize prime_field_polynomial_convolution_at_length_exists (x14)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hwbound_right_right", "exact hfirst_right_left", "exact hresult_length_witness", "cases hresult_product", "cases hresult_product_witness", "have htbound : forall fom_index_pfp_add_result_bound. (exists fom_gap_pfp_add_result_bound_index_bound. fom_gap_pfp_add_result_bound_index_bound + S (fom_index_pfp_add_result_bound) = x14) -> exists fom_value_pfp_add_result_bound. ((((exists fom_beta_height_pfp_add_result_bound_entry. fom_beta_height_pfp_add_result_bound_entry + S (fom_value_pfp_add_result_bound) = S ((S (fom_index_pfp_add_result_bound)) * x16)) /\\ exists fom_beta_quotient_pfp_add_result_bound_entry. x15 = fom_beta_quotient_pfp_add_result_bound_entry * S ((S (fom_index_pfp_add_result_bound)) * x16) + (fom_value_pfp_add_result_bound))) /\\ (exists fom_gap_pfp_add_result_bound_value_bound. fom_gap_pfp_add_result_bound_value_bound + S (fom_value_pfp_add_result_bound) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (x12)", "specialize prime_field_polynomial_convolution_bounded (x13)", "specialize prime_field_polynomial_convolution_bounded ((x2)+(x8))", "specialize prime_field_polynomial_convolution_bounded (db)", "specialize prime_field_polynomial_convolution_bounded (dc)", "specialize prime_field_polynomial_convolution_bounded (J)", "specialize prime_field_polynomial_convolution_bounded (x15)", "specialize prime_field_polynomial_convolution_bounded (x16)", "specialize prime_field_polynomial_convolution_bounded (x14)", "apply prime_field_polynomial_convolution_bounded", "exact hresult_product_witness_witness", "have hdistr : ((forall fom_index_pfp_add_distribution_left_bounded. (exists fom_gap_pfp_add_distribution_left_bounded_index_bound. fom_gap_pfp_add_distribution_left_bounded_index_bound + S (fom_index_pfp_add_distribution_left_bounded) = x5) -> exists fom_value_pfp_add_distribution_left_bounded. ((((exists fom_beta_height_pfp_add_distribution_left_bounded_entry. fom_beta_height_pfp_add_distribution_left_bounded_entry + S (fom_value_pfp_add_distribution_left_bounded) = S ((S (fom_index_pfp_add_distribution_left_bounded)) * x4)) /\\ exists fom_beta_quotient_pfp_add_distribution_left_bounded_entry. x3 = fom_beta_quotient_pfp_add_distribution_left_bounded_entry * S ((S (fom_index_pfp_add_distribution_left_bounded)) * x4) + (fom_value_pfp_add_distribution_left_bounded))) /\\ (exists fom_gap_pfp_add_distribution_left_bounded_value_bound. fom_gap_pfp_add_distribution_left_bounded_value_bound + S (fom_value_pfp_add_distribution_left_bounded) = p))) /\\ (((forall fom_index_pfp_add_distribution_right_bounded. (exists fom_gap_pfp_add_distribution_right_bounded_index_bound. fom_gap_pfp_add_distribution_right_bounded_index_bound + S (fom_index_pfp_add_distribution_right_bounded) = x11) -> exists fom_value_pfp_add_distribution_right_bounded. ((((exists fom_beta_height_pfp_add_distribution_right_bounded_entry. fom_beta_height_pfp_add_distribution_right_bounded_entry + S (fom_value_pfp_add_distribution_right_bounded) = S ((S (fom_index_pfp_add_distribution_right_bounded)) * x10)) /\\ exists fom_beta_quotient_pfp_add_distribution_right_bounded_entry. x9 = fom_beta_quotient_pfp_add_distribution_right_bounded_entry * S ((S (fom_index_pfp_add_distribution_right_bounded)) * x10) + (fom_value_pfp_add_distribution_right_bounded))) /\\ (exists fom_gap_pfp_add_distribution_right_bounded_value_bound. fom_gap_pfp_add_distribution_right_bounded_value_bound + S (fom_value_pfp_add_distribution_right_bounded) = p))) /\\ (((forall fom_index_pfp_add_distribution_result_bounded. (exists fom_gap_pfp_add_distribution_result_bounded_index_bound. fom_gap_pfp_add_distribution_result_bounded_index_bound + S (fom_index_pfp_add_distribution_result_bounded) = x14) -> exists fom_value_pfp_add_distribution_result_bounded. ((((exists fom_beta_height_pfp_add_distribution_result_bounded_entry. fom_beta_height_pfp_add_distribution_result_bounded_entry + S (fom_value_pfp_add_distribution_result_bounded) = S ((S (fom_index_pfp_add_distribution_result_bounded)) * x16)) /\\ exists fom_beta_quotient_pfp_add_distribution_result_bounded_entry. x15 = fom_beta_quotient_pfp_add_distribution_result_bounded_entry * S ((S (fom_index_pfp_add_distribution_result_bounded)) * x16) + (fom_value_pfp_add_distribution_result_bounded))) /\\ (exists fom_gap_pfp_add_distribution_result_bounded_value_bound. fom_gap_pfp_add_distribution_result_bounded_value_bound + S (fom_value_pfp_add_distribution_result_bounded) = p))) /\\ ((exists pfaa_left_b_add_distribution pfaa_left_c_add_distribution pfaa_right_b_add_distribution pfaa_right_c_add_distribution pfaa_sum_b_add_distribution pfaa_sum_c_add_distribution pfaa_length_add_distribution. ((((forall pfrep_power_add_distribution_witness_common_left pfrep_left_add_distribution_witness_common_left pfrep_right_add_distribution_witness_common_left. ((exists pfrep_position_add_distribution_witness_common_leftfirst. ((pfrep_position_add_distribution_witness_common_leftfirst+S (pfrep_power_add_distribution_witness_common_left)=(x5)) /\\ ((((exists ff_h_pfp_add_distribution_witness_common_leftfirstentry. ff_h_pfp_add_distribution_witness_common_leftfirstentry + S (pfrep_left_add_distribution_witness_common_left) = S ((S (pfrep_position_add_distribution_witness_common_leftfirst)) * x4)) /\\ exists ff_q_pfp_add_distribution_witness_common_leftfirstentry. x3 = ff_q_pfp_add_distribution_witness_common_leftfirstentry * S ((S (pfrep_position_add_distribution_witness_common_leftfirst)) * x4) + (pfrep_left_add_distribution_witness_common_left)))))) \\/ (((exists pfrep_gap_add_distribution_witness_common_leftfirstoutside. pfrep_gap_add_distribution_witness_common_leftfirstoutside+(x5)=(pfrep_power_add_distribution_witness_common_left)) /\\ (((pfrep_left_add_distribution_witness_common_left)=0))))) -> ((exists pfrep_position_add_distribution_witness_common_leftsecond. ((pfrep_position_add_distribution_witness_common_leftsecond+S (pfrep_power_add_distribution_witness_common_left)=(pfaa_length_add_distribution)) /\\ ((((exists ff_h_pfp_add_distribution_witness_common_leftsecondentry. ff_h_pfp_add_distribution_witness_common_leftsecondentry + S (pfrep_right_add_distribution_witness_common_left) = S ((S (pfrep_position_add_distribution_witness_common_leftsecond)) * pfaa_left_c_add_distribution)) /\\ exists ff_q_pfp_add_distribution_witness_common_leftsecondentry. pfaa_left_b_add_distribution = ff_q_pfp_add_distribution_witness_common_leftsecondentry * S ((S (pfrep_position_add_distribution_witness_common_leftsecond)) * pfaa_left_c_add_distribution) + (pfrep_right_add_distribution_witness_common_left)))))) \\/ (((exists pfrep_gap_add_distribution_witness_common_leftsecondoutside. pfrep_gap_add_distribution_witness_common_leftsecondoutside+(pfaa_length_add_distribution)=(pfrep_power_add_distribution_witness_common_left)) /\\ (((pfrep_right_add_distribution_witness_common_left)=0))))) -> pfrep_left_add_distribution_witness_common_left=pfrep_right_add_distribution_witness_common_left) /\\ ((forall pfrep_power_add_distribution_witness_common_right pfrep_left_add_distribution_witness_common_right pfrep_right_add_distribution_witness_common_right. ((exists pfrep_position_add_distribution_witness_common_rightfirst. ((pfrep_position_add_distribution_witness_common_rightfirst+S (pfrep_power_add_distribution_witness_common_right)=(x11)) /\\ ((((exists ff_h_pfp_add_distribution_witness_common_rightfirstentry. ff_h_pfp_add_distribution_witness_common_rightfirstentry + S (pfrep_left_add_distribution_witness_common_right) = S ((S (pfrep_position_add_distribution_witness_common_rightfirst)) * x10)) /\\ exists ff_q_pfp_add_distribution_witness_common_rightfirstentry. x9 = ff_q_pfp_add_distribution_witness_common_rightfirstentry * S ((S (pfrep_position_add_distribution_witness_common_rightfirst)) * x10) + (pfrep_left_add_distribution_witness_common_right)))))) \\/ (((exists pfrep_gap_add_distribution_witness_common_rightfirstoutside. pfrep_gap_add_distribution_witness_common_rightfirstoutside+(x11)=(pfrep_power_add_distribution_witness_common_right)) /\\ (((pfrep_left_add_distribution_witness_common_right)=0))))) -> ((exists pfrep_position_add_distribution_witness_common_rightsecond. ((pfrep_position_add_distribution_witness_common_rightsecond+S (pfrep_power_add_distribution_witness_common_right)=(pfaa_length_add_distribution)) /\\ ((((exists ff_h_pfp_add_distribution_witness_common_rightsecondentry. ff_h_pfp_add_distribution_witness_common_rightsecondentry + S (pfrep_right_add_distribution_witness_common_right) = S ((S (pfrep_position_add_distribution_witness_common_rightsecond)) * pfaa_right_c_add_distribution)) /\\ exists ff_q_pfp_add_distribution_witness_common_rightsecondentry. pfaa_right_b_add_distribution = ff_q_pfp_add_distribution_witness_common_rightsecondentry * S ((S (pfrep_position_add_distribution_witness_common_rightsecond)) * pfaa_right_c_add_distribution) + (pfrep_right_add_distribution_witness_common_right)))))) \\/ (((exists pfrep_gap_add_distribution_witness_common_rightsecondoutside. pfrep_gap_add_distribution_witness_common_rightsecondoutside+(pfaa_length_add_distribution)=(pfrep_power_add_distribution_witness_common_right)) /\\ (((pfrep_right_add_distribution_witness_common_right)=0))))) -> pfrep_left_add_distribution_witness_common_right=pfrep_right_add_distribution_witness_common_right)))) /\\ (((forall pfp_index_add_distribution_witness_operation. (exists pfa_gap_add_distribution_witness_operationindex. pfa_gap_add_distribution_witness_operationindex + S (pfp_index_add_distribution_witness_operation) = (pfaa_length_add_distribution)) -> exists pfp_left_add_distribution_witness_operation pfp_right_add_distribution_witness_operation pfp_value_add_distribution_witness_operation. ((((exists ff_h_pfp_add_distribution_witness_operationleft. ff_h_pfp_add_distribution_witness_operationleft + S (pfp_left_add_distribution_witness_operation) = S ((S (pfp_index_add_distribution_witness_operation)) * pfaa_left_c_add_distribution)) /\\ exists ff_q_pfp_add_distribution_witness_operationleft. pfaa_left_b_add_distribution = ff_q_pfp_add_distribution_witness_operationleft * S ((S (pfp_index_add_distribution_witness_operation)) * pfaa_left_c_add_distribution) + (pfp_left_add_distribution_witness_operation))) /\\ (((((exists ff_h_pfp_add_distribution_witness_operationright. ff_h_pfp_add_distribution_witness_operationright + S (pfp_right_add_distribution_witness_operation) = S ((S (pfp_index_add_distribution_witness_operation)) * pfaa_right_c_add_distribution)) /\\ exists ff_q_pfp_add_distribution_witness_operationright. pfaa_right_b_add_distribution = ff_q_pfp_add_distribution_witness_operationright * S ((S (pfp_index_add_distribution_witness_operation)) * pfaa_right_c_add_distribution) + (pfp_right_add_distribution_witness_operation))) /\\ (((((exists ff_h_pfp_add_distribution_witness_operationtarget. ff_h_pfp_add_distribution_witness_operationtarget + S (pfp_value_add_distribution_witness_operation) = S ((S (pfp_index_add_distribution_witness_operation)) * pfaa_sum_c_add_distribution)) /\\ exists ff_q_pfp_add_distribution_witness_operationtarget. pfaa_sum_b_add_distribution = ff_q_pfp_add_distribution_witness_operationtarget * S ((S (pfp_index_add_distribution_witness_operation)) * pfaa_sum_c_add_distribution) + (pfp_value_add_distribution_witness_operation))) /\\ ((((exists pfa_gap_add_distribution_witness_operationoperationleft. pfa_gap_add_distribution_witness_operationoperationleft + S (pfp_left_add_distribution_witness_operation) = (p)) /\\ (((exists pfa_gap_add_distribution_witness_operationoperationright. pfa_gap_add_distribution_witness_operationoperationright + S (pfp_right_add_distribution_witness_operation) = (p)) /\\ ((((exists pfa_gap_add_distribution_witness_operationoperationresultbound. pfa_gap_add_distribution_witness_operationoperationresultbound + S (pfp_value_add_distribution_witness_operation) = (p)) /\\ ((exists pfa_offset_left_add_distribution_witness_operationoperationresultcongruence pfa_offset_right_add_distribution_witness_operationoperationresultcongruence. ((pfp_left_add_distribution_witness_operation) + (pfp_right_add_distribution_witness_operation)) + (p) * pfa_offset_left_add_distribution_witness_operationoperationresultcongruence = (pfp_value_add_distribution_witness_operation) + (p) * pfa_offset_right_add_distribution_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_add_distribution_witness_output pfrep_left_add_distribution_witness_output pfrep_right_add_distribution_witness_output. ((exists pfrep_position_add_distribution_witness_outputfirst. ((pfrep_position_add_distribution_witness_outputfirst+S (pfrep_power_add_distribution_witness_output)=(pfaa_length_add_distribution)) /\\ ((((exists ff_h_pfp_add_distribution_witness_outputfirstentry. ff_h_pfp_add_distribution_witness_outputfirstentry + S (pfrep_left_add_distribution_witness_output) = S ((S (pfrep_position_add_distribution_witness_outputfirst)) * pfaa_sum_c_add_distribution)) /\\ exists ff_q_pfp_add_distribution_witness_outputfirstentry. pfaa_sum_b_add_distribution = ff_q_pfp_add_distribution_witness_outputfirstentry * S ((S (pfrep_position_add_distribution_witness_outputfirst)) * pfaa_sum_c_add_distribution) + (pfrep_left_add_distribution_witness_output)))))) \\/ (((exists pfrep_gap_add_distribution_witness_outputfirstoutside. pfrep_gap_add_distribution_witness_outputfirstoutside+(pfaa_length_add_distribution)=(pfrep_power_add_distribution_witness_output)) /\\ (((pfrep_left_add_distribution_witness_output)=0))))) -> ((exists pfrep_position_add_distribution_witness_outputsecond. ((pfrep_position_add_distribution_witness_outputsecond+S (pfrep_power_add_distribution_witness_output)=(x14)) /\\ ((((exists ff_h_pfp_add_distribution_witness_outputsecondentry. ff_h_pfp_add_distribution_witness_outputsecondentry + S (pfrep_right_add_distribution_witness_output) = S ((S (pfrep_position_add_distribution_witness_outputsecond)) * x16)) /\\ exists ff_q_pfp_add_distribution_witness_outputsecondentry. x15 = ff_q_pfp_add_distribution_witness_outputsecondentry * S ((S (pfrep_position_add_distribution_witness_outputsecond)) * x16) + (pfrep_right_add_distribution_witness_output)))))) \\/ (((exists pfrep_gap_add_distribution_witness_outputsecondoutside. pfrep_gap_add_distribution_witness_outputsecondoutside+(x14)=(pfrep_power_add_distribution_witness_output)) /\\ (((pfrep_right_add_distribution_witness_output)=0))))) -> pfrep_left_add_distribution_witness_output=pfrep_right_add_distribution_witness_output))))))))))))", "specialize prime_field_polynomial_aligned_convolution_right_add (p)", "specialize prime_field_polynomial_aligned_convolution_right_add (x)", "specialize prime_field_polynomial_aligned_convolution_right_add (x1)", "specialize prime_field_polynomial_aligned_convolution_right_add (x2)", "specialize prime_field_polynomial_aligned_convolution_right_add (x6)", "specialize prime_field_polynomial_aligned_convolution_right_add (x7)", "specialize prime_field_polynomial_aligned_convolution_right_add (x8)", "specialize prime_field_polynomial_aligned_convolution_right_add (x12)", "specialize prime_field_polynomial_aligned_convolution_right_add (x13)", "specialize prime_field_polynomial_aligned_convolution_right_add ((x2)+(x8))", "specialize prime_field_polynomial_aligned_convolution_right_add (db)", "specialize prime_field_polynomial_aligned_convolution_right_add (dc)", "specialize prime_field_polynomial_aligned_convolution_right_add (J)", "specialize prime_field_polynomial_aligned_convolution_right_add (x3)", "specialize prime_field_polynomial_aligned_convolution_right_add (x4)", "specialize prime_field_polynomial_aligned_convolution_right_add (x5)", "specialize prime_field_polynomial_aligned_convolution_right_add (x9)", "specialize prime_field_polynomial_aligned_convolution_right_add (x10)", "specialize prime_field_polynomial_aligned_convolution_right_add (x11)", "specialize prime_field_polynomial_aligned_convolution_right_add (x15)", "specialize prime_field_polynomial_aligned_convolution_right_add (x16)", "specialize prime_field_polynomial_aligned_convolution_right_add (x14)", "apply prime_field_polynomial_aligned_convolution_right_add", "exact hp", "exact hw_witness_witness", "exact hDA_right_witness_witness_witness_witness_witness_witness_left", "exact hDB_right_witness_witness_witness_witness_witness_witness_left", "exact hresult_product_witness_witness", "have hcompare : ((forall fom_index_pfp_add_comparison_left_bounded. (exists fom_gap_pfp_add_comparison_left_bounded_index_bound. fom_gap_pfp_add_comparison_left_bounded_index_bound + S (fom_index_pfp_add_comparison_left_bounded) = x5) -> exists fom_value_pfp_add_comparison_left_bounded. ((((exists fom_beta_height_pfp_add_comparison_left_bounded_entry. fom_beta_height_pfp_add_comparison_left_bounded_entry + S (fom_value_pfp_add_comparison_left_bounded) = S ((S (fom_index_pfp_add_comparison_left_bounded)) * x4)) /\\ exists fom_beta_quotient_pfp_add_comparison_left_bounded_entry. x3 = fom_beta_quotient_pfp_add_comparison_left_bounded_entry * S ((S (fom_index_pfp_add_comparison_left_bounded)) * x4) + (fom_value_pfp_add_comparison_left_bounded))) /\\ (exists fom_gap_pfp_add_comparison_left_bounded_value_bound. fom_gap_pfp_add_comparison_left_bounded_value_bound + S (fom_value_pfp_add_comparison_left_bounded) = p))) /\\ (((forall fom_index_pfp_add_comparison_right_bounded. (exists fom_gap_pfp_add_comparison_right_bounded_index_bound. fom_gap_pfp_add_comparison_right_bounded_index_bound + S (fom_index_pfp_add_comparison_right_bounded) = x11) -> exists fom_value_pfp_add_comparison_right_bounded. ((((exists fom_beta_height_pfp_add_comparison_right_bounded_entry. fom_beta_height_pfp_add_comparison_right_bounded_entry + S (fom_value_pfp_add_comparison_right_bounded) = S ((S (fom_index_pfp_add_comparison_right_bounded)) * x10)) /\\ exists fom_beta_quotient_pfp_add_comparison_right_bounded_entry. x9 = fom_beta_quotient_pfp_add_comparison_right_bounded_entry * S ((S (fom_index_pfp_add_comparison_right_bounded)) * x10) + (fom_value_pfp_add_comparison_right_bounded))) /\\ (exists fom_gap_pfp_add_comparison_right_bounded_value_bound. fom_gap_pfp_add_comparison_right_bounded_value_bound + S (fom_value_pfp_add_comparison_right_bounded) = p))) /\\ (((forall fom_index_pfp_add_comparison_result_bounded. (exists fom_gap_pfp_add_comparison_result_bounded_index_bound. fom_gap_pfp_add_comparison_result_bounded_index_bound + S (fom_index_pfp_add_comparison_result_bounded) = N) -> exists fom_value_pfp_add_comparison_result_bounded. ((((exists fom_beta_height_pfp_add_comparison_result_bounded_entry. fom_beta_height_pfp_add_comparison_result_bounded_entry + S (fom_value_pfp_add_comparison_result_bounded) = S ((S (fom_index_pfp_add_comparison_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_add_comparison_result_bounded_entry. rb = fom_beta_quotient_pfp_add_comparison_result_bounded_entry * S ((S (fom_index_pfp_add_comparison_result_bounded)) * rc) + (fom_value_pfp_add_comparison_result_bounded))) /\\ (exists fom_gap_pfp_add_comparison_result_bounded_value_bound. fom_gap_pfp_add_comparison_result_bounded_value_bound + S (fom_value_pfp_add_comparison_result_bounded) = p))) /\\ ((exists pfaa_left_b_add_comparison pfaa_left_c_add_comparison pfaa_right_b_add_comparison pfaa_right_c_add_comparison pfaa_sum_b_add_comparison pfaa_sum_c_add_comparison pfaa_length_add_comparison. ((((forall pfrep_power_add_comparison_witness_common_left pfrep_left_add_comparison_witness_common_left pfrep_right_add_comparison_witness_common_left. ((exists pfrep_position_add_comparison_witness_common_leftfirst. ((pfrep_position_add_comparison_witness_common_leftfirst+S (pfrep_power_add_comparison_witness_common_left)=(x5)) /\\ ((((exists ff_h_pfp_add_comparison_witness_common_leftfirstentry. ff_h_pfp_add_comparison_witness_common_leftfirstentry + S (pfrep_left_add_comparison_witness_common_left) = S ((S (pfrep_position_add_comparison_witness_common_leftfirst)) * x4)) /\\ exists ff_q_pfp_add_comparison_witness_common_leftfirstentry. x3 = ff_q_pfp_add_comparison_witness_common_leftfirstentry * S ((S (pfrep_position_add_comparison_witness_common_leftfirst)) * x4) + (pfrep_left_add_comparison_witness_common_left)))))) \\/ (((exists pfrep_gap_add_comparison_witness_common_leftfirstoutside. pfrep_gap_add_comparison_witness_common_leftfirstoutside+(x5)=(pfrep_power_add_comparison_witness_common_left)) /\\ (((pfrep_left_add_comparison_witness_common_left)=0))))) -> ((exists pfrep_position_add_comparison_witness_common_leftsecond. ((pfrep_position_add_comparison_witness_common_leftsecond+S (pfrep_power_add_comparison_witness_common_left)=(pfaa_length_add_comparison)) /\\ ((((exists ff_h_pfp_add_comparison_witness_common_leftsecondentry. ff_h_pfp_add_comparison_witness_common_leftsecondentry + S (pfrep_right_add_comparison_witness_common_left) = S ((S (pfrep_position_add_comparison_witness_common_leftsecond)) * pfaa_left_c_add_comparison)) /\\ exists ff_q_pfp_add_comparison_witness_common_leftsecondentry. pfaa_left_b_add_comparison = ff_q_pfp_add_comparison_witness_common_leftsecondentry * S ((S (pfrep_position_add_comparison_witness_common_leftsecond)) * pfaa_left_c_add_comparison) + (pfrep_right_add_comparison_witness_common_left)))))) \\/ (((exists pfrep_gap_add_comparison_witness_common_leftsecondoutside. pfrep_gap_add_comparison_witness_common_leftsecondoutside+(pfaa_length_add_comparison)=(pfrep_power_add_comparison_witness_common_left)) /\\ (((pfrep_right_add_comparison_witness_common_left)=0))))) -> pfrep_left_add_comparison_witness_common_left=pfrep_right_add_comparison_witness_common_left) /\\ ((forall pfrep_power_add_comparison_witness_common_right pfrep_left_add_comparison_witness_common_right pfrep_right_add_comparison_witness_common_right. ((exists pfrep_position_add_comparison_witness_common_rightfirst. ((pfrep_position_add_comparison_witness_common_rightfirst+S (pfrep_power_add_comparison_witness_common_right)=(x11)) /\\ ((((exists ff_h_pfp_add_comparison_witness_common_rightfirstentry. ff_h_pfp_add_comparison_witness_common_rightfirstentry + S (pfrep_left_add_comparison_witness_common_right) = S ((S (pfrep_position_add_comparison_witness_common_rightfirst)) * x10)) /\\ exists ff_q_pfp_add_comparison_witness_common_rightfirstentry. x9 = ff_q_pfp_add_comparison_witness_common_rightfirstentry * S ((S (pfrep_position_add_comparison_witness_common_rightfirst)) * x10) + (pfrep_left_add_comparison_witness_common_right)))))) \\/ (((exists pfrep_gap_add_comparison_witness_common_rightfirstoutside. pfrep_gap_add_comparison_witness_common_rightfirstoutside+(x11)=(pfrep_power_add_comparison_witness_common_right)) /\\ (((pfrep_left_add_comparison_witness_common_right)=0))))) -> ((exists pfrep_position_add_comparison_witness_common_rightsecond. ((pfrep_position_add_comparison_witness_common_rightsecond+S (pfrep_power_add_comparison_witness_common_right)=(pfaa_length_add_comparison)) /\\ ((((exists ff_h_pfp_add_comparison_witness_common_rightsecondentry. ff_h_pfp_add_comparison_witness_common_rightsecondentry + S (pfrep_right_add_comparison_witness_common_right) = S ((S (pfrep_position_add_comparison_witness_common_rightsecond)) * pfaa_right_c_add_comparison)) /\\ exists ff_q_pfp_add_comparison_witness_common_rightsecondentry. pfaa_right_b_add_comparison = ff_q_pfp_add_comparison_witness_common_rightsecondentry * S ((S (pfrep_position_add_comparison_witness_common_rightsecond)) * pfaa_right_c_add_comparison) + (pfrep_right_add_comparison_witness_common_right)))))) \\/ (((exists pfrep_gap_add_comparison_witness_common_rightsecondoutside. pfrep_gap_add_comparison_witness_common_rightsecondoutside+(pfaa_length_add_comparison)=(pfrep_power_add_comparison_witness_common_right)) /\\ (((pfrep_right_add_comparison_witness_common_right)=0))))) -> pfrep_left_add_comparison_witness_common_right=pfrep_right_add_comparison_witness_common_right)))) /\\ (((forall pfp_index_add_comparison_witness_operation. (exists pfa_gap_add_comparison_witness_operationindex. pfa_gap_add_comparison_witness_operationindex + S (pfp_index_add_comparison_witness_operation) = (pfaa_length_add_comparison)) -> exists pfp_left_add_comparison_witness_operation pfp_right_add_comparison_witness_operation pfp_value_add_comparison_witness_operation. ((((exists ff_h_pfp_add_comparison_witness_operationleft. ff_h_pfp_add_comparison_witness_operationleft + S (pfp_left_add_comparison_witness_operation) = S ((S (pfp_index_add_comparison_witness_operation)) * pfaa_left_c_add_comparison)) /\\ exists ff_q_pfp_add_comparison_witness_operationleft. pfaa_left_b_add_comparison = ff_q_pfp_add_comparison_witness_operationleft * S ((S (pfp_index_add_comparison_witness_operation)) * pfaa_left_c_add_comparison) + (pfp_left_add_comparison_witness_operation))) /\\ (((((exists ff_h_pfp_add_comparison_witness_operationright. ff_h_pfp_add_comparison_witness_operationright + S (pfp_right_add_comparison_witness_operation) = S ((S (pfp_index_add_comparison_witness_operation)) * pfaa_right_c_add_comparison)) /\\ exists ff_q_pfp_add_comparison_witness_operationright. pfaa_right_b_add_comparison = ff_q_pfp_add_comparison_witness_operationright * S ((S (pfp_index_add_comparison_witness_operation)) * pfaa_right_c_add_comparison) + (pfp_right_add_comparison_witness_operation))) /\\ (((((exists ff_h_pfp_add_comparison_witness_operationtarget. ff_h_pfp_add_comparison_witness_operationtarget + S (pfp_value_add_comparison_witness_operation) = S ((S (pfp_index_add_comparison_witness_operation)) * pfaa_sum_c_add_comparison)) /\\ exists ff_q_pfp_add_comparison_witness_operationtarget. pfaa_sum_b_add_comparison = ff_q_pfp_add_comparison_witness_operationtarget * S ((S (pfp_index_add_comparison_witness_operation)) * pfaa_sum_c_add_comparison) + (pfp_value_add_comparison_witness_operation))) /\\ ((((exists pfa_gap_add_comparison_witness_operationoperationleft. pfa_gap_add_comparison_witness_operationoperationleft + S (pfp_left_add_comparison_witness_operation) = (p)) /\\ (((exists pfa_gap_add_comparison_witness_operationoperationright. pfa_gap_add_comparison_witness_operationoperationright + S (pfp_right_add_comparison_witness_operation) = (p)) /\\ ((((exists pfa_gap_add_comparison_witness_operationoperationresultbound. pfa_gap_add_comparison_witness_operationoperationresultbound + S (pfp_value_add_comparison_witness_operation) = (p)) /\\ ((exists pfa_offset_left_add_comparison_witness_operationoperationresultcongruence pfa_offset_right_add_comparison_witness_operationoperationresultcongruence. ((pfp_left_add_comparison_witness_operation) + (pfp_right_add_comparison_witness_operation)) + (p) * pfa_offset_left_add_comparison_witness_operationoperationresultcongruence = (pfp_value_add_comparison_witness_operation) + (p) * pfa_offset_right_add_comparison_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_add_comparison_witness_output pfrep_left_add_comparison_witness_output pfrep_right_add_comparison_witness_output. ((exists pfrep_position_add_comparison_witness_outputfirst. ((pfrep_position_add_comparison_witness_outputfirst+S (pfrep_power_add_comparison_witness_output)=(pfaa_length_add_comparison)) /\\ ((((exists ff_h_pfp_add_comparison_witness_outputfirstentry. ff_h_pfp_add_comparison_witness_outputfirstentry + S (pfrep_left_add_comparison_witness_output) = S ((S (pfrep_position_add_comparison_witness_outputfirst)) * pfaa_sum_c_add_comparison)) /\\ exists ff_q_pfp_add_comparison_witness_outputfirstentry. pfaa_sum_b_add_comparison = ff_q_pfp_add_comparison_witness_outputfirstentry * S ((S (pfrep_position_add_comparison_witness_outputfirst)) * pfaa_sum_c_add_comparison) + (pfrep_left_add_comparison_witness_output)))))) \\/ (((exists pfrep_gap_add_comparison_witness_outputfirstoutside. pfrep_gap_add_comparison_witness_outputfirstoutside+(pfaa_length_add_comparison)=(pfrep_power_add_comparison_witness_output)) /\\ (((pfrep_left_add_comparison_witness_output)=0))))) -> ((exists pfrep_position_add_comparison_witness_outputsecond. ((pfrep_position_add_comparison_witness_outputsecond+S (pfrep_power_add_comparison_witness_output)=(N)) /\\ ((((exists ff_h_pfp_add_comparison_witness_outputsecondentry. ff_h_pfp_add_comparison_witness_outputsecondentry + S (pfrep_right_add_comparison_witness_output) = S ((S (pfrep_position_add_comparison_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_add_comparison_witness_outputsecondentry. rb = ff_q_pfp_add_comparison_witness_outputsecondentry * S ((S (pfrep_position_add_comparison_witness_outputsecond)) * rc) + (pfrep_right_add_comparison_witness_output)))))) \\/ (((exists pfrep_gap_add_comparison_witness_outputsecondoutside. pfrep_gap_add_comparison_witness_outputsecondoutside+(N)=(pfrep_power_add_comparison_witness_output)) /\\ (((pfrep_right_add_comparison_witness_output)=0))))) -> pfrep_left_add_comparison_witness_output=pfrep_right_add_comparison_witness_output))))))))))))", "specialize prime_field_polynomial_aligned_add_transport (p)", "specialize prime_field_polynomial_aligned_add_transport (ab)", "specialize prime_field_polynomial_aligned_add_transport (ac)", "specialize prime_field_polynomial_aligned_add_transport (L)", "specialize prime_field_polynomial_aligned_add_transport (bb)", "specialize prime_field_polynomial_aligned_add_transport (bc)", "specialize prime_field_polynomial_aligned_add_transport (M)", "specialize prime_field_polynomial_aligned_add_transport (rb)", "specialize prime_field_polynomial_aligned_add_transport (rc)", "specialize prime_field_polynomial_aligned_add_transport (N)", "specialize prime_field_polynomial_aligned_add_transport (x3)", "specialize prime_field_polynomial_aligned_add_transport (x4)", "specialize prime_field_polynomial_aligned_add_transport (x5)", "specialize prime_field_polynomial_aligned_add_transport (x9)", "specialize prime_field_polynomial_aligned_add_transport (x10)", "specialize prime_field_polynomial_aligned_add_transport (x11)", "specialize prime_field_polynomial_aligned_add_transport (rb)", "specialize prime_field_polynomial_aligned_add_transport (rc)", "specialize prime_field_polynomial_aligned_add_transport (N)", "apply prime_field_polynomial_aligned_add_transport", "exact hfirst_bounded", "exact hsecond_bounded", "have hrbound : ((forall fom_index_pfp_add_input_A. (exists fom_gap_pfp_add_input_A_index_bound. fom_gap_pfp_add_input_A_index_bound + S (fom_index_pfp_add_input_A) = L) -> exists fom_value_pfp_add_input_A. ((((exists fom_beta_height_pfp_add_input_A_entry. fom_beta_height_pfp_add_input_A_entry + S (fom_value_pfp_add_input_A) = S ((S (fom_index_pfp_add_input_A)) * ac)) /\\ exists fom_beta_quotient_pfp_add_input_A_entry. ab = fom_beta_quotient_pfp_add_input_A_entry * S ((S (fom_index_pfp_add_input_A)) * ac) + (fom_value_pfp_add_input_A))) /\\ (exists fom_gap_pfp_add_input_A_value_bound. fom_gap_pfp_add_input_A_value_bound + S (fom_value_pfp_add_input_A) = p))) /\\ (((forall fom_index_pfp_add_input_B. (exists fom_gap_pfp_add_input_B_index_bound. fom_gap_pfp_add_input_B_index_bound + S (fom_index_pfp_add_input_B) = M) -> exists fom_value_pfp_add_input_B. ((((exists fom_beta_height_pfp_add_input_B_entry. fom_beta_height_pfp_add_input_B_entry + S (fom_value_pfp_add_input_B) = S ((S (fom_index_pfp_add_input_B)) * bc)) /\\ exists fom_beta_quotient_pfp_add_input_B_entry. bb = fom_beta_quotient_pfp_add_input_B_entry * S ((S (fom_index_pfp_add_input_B)) * bc) + (fom_value_pfp_add_input_B))) /\\ (exists fom_gap_pfp_add_input_B_value_bound. fom_gap_pfp_add_input_B_value_bound + S (fom_value_pfp_add_input_B) = p))) /\\ ((forall fom_index_pfp_add_input_R. (exists fom_gap_pfp_add_input_R_index_bound. fom_gap_pfp_add_input_R_index_bound + S (fom_index_pfp_add_input_R) = N) -> exists fom_value_pfp_add_input_R. ((((exists fom_beta_height_pfp_add_input_R_entry. fom_beta_height_pfp_add_input_R_entry + S (fom_value_pfp_add_input_R) = S ((S (fom_index_pfp_add_input_R)) * rc)) /\\ exists fom_beta_quotient_pfp_add_input_R_entry. rb = fom_beta_quotient_pfp_add_input_R_entry * S ((S (fom_index_pfp_add_input_R)) * rc) + (fom_value_pfp_add_input_R))) /\\ (exists fom_gap_pfp_add_input_R_value_bound. fom_gap_pfp_add_input_R_value_bound + S (fom_value_pfp_add_input_R) = p)))))))", "specialize prime_field_polynomial_aligned_add_bounded (p)", "specialize prime_field_polynomial_aligned_add_bounded (ab)", "specialize prime_field_polynomial_aligned_add_bounded (ac)", "specialize prime_field_polynomial_aligned_add_bounded (L)", "specialize prime_field_polynomial_aligned_add_bounded (bb)", "specialize prime_field_polynomial_aligned_add_bounded (bc)", "specialize prime_field_polynomial_aligned_add_bounded (M)", "specialize prime_field_polynomial_aligned_add_bounded (rb)", "specialize prime_field_polynomial_aligned_add_bounded (rc)", "specialize prime_field_polynomial_aligned_add_bounded (N)", "apply prime_field_polynomial_aligned_add_bounded", "exact hop", "cases hrbound", "cases hrbound_right", "exact hrbound_right_right", "exact hDA_right_witness_witness_witness_witness_witness_witness_right", "exact hDB_right_witness_witness_witness_witness_witness_witness_right", "specialize prime_field_polynomial_power_coefficient_functional (rb)", "specialize prime_field_polynomial_power_coefficient_functional (rc)", "specialize prime_field_polynomial_power_coefficient_functional (N)", "apply prime_field_polynomial_power_coefficient_functional", "exact hop", "have heq : forall pfrep_power_add_output_equivalent pfrep_left_add_output_equivalent pfrep_right_add_output_equivalent. ((exists pfrep_position_add_output_equivalentfirst. ((pfrep_position_add_output_equivalentfirst+S (pfrep_power_add_output_equivalent)=(x14)) /\\ ((((exists ff_h_pfp_add_output_equivalentfirstentry. ff_h_pfp_add_output_equivalentfirstentry + S (pfrep_left_add_output_equivalent) = S ((S (pfrep_position_add_output_equivalentfirst)) * x16)) /\\ exists ff_q_pfp_add_output_equivalentfirstentry. x15 = ff_q_pfp_add_output_equivalentfirstentry * S ((S (pfrep_position_add_output_equivalentfirst)) * x16) + (pfrep_left_add_output_equivalent)))))) \\/ (((exists pfrep_gap_add_output_equivalentfirstoutside. pfrep_gap_add_output_equivalentfirstoutside+(x14)=(pfrep_power_add_output_equivalent)) /\\ (((pfrep_left_add_output_equivalent)=0))))) -> ((exists pfrep_position_add_output_equivalentsecond. ((pfrep_position_add_output_equivalentsecond+S (pfrep_power_add_output_equivalent)=(N)) /\\ ((((exists ff_h_pfp_add_output_equivalentsecondentry. ff_h_pfp_add_output_equivalentsecondentry + S (pfrep_right_add_output_equivalent) = S ((S (pfrep_position_add_output_equivalentsecond)) * rc)) /\\ exists ff_q_pfp_add_output_equivalentsecondentry. rb = ff_q_pfp_add_output_equivalentsecondentry * S ((S (pfrep_position_add_output_equivalentsecond)) * rc) + (pfrep_right_add_output_equivalent)))))) \\/ (((exists pfrep_gap_add_output_equivalentsecondoutside. pfrep_gap_add_output_equivalentsecondoutside+(N)=(pfrep_power_add_output_equivalent)) /\\ (((pfrep_right_add_output_equivalent)=0))))) -> pfrep_left_add_output_equivalent=pfrep_right_add_output_equivalent", "specialize prime_field_polynomial_aligned_add_functional (p)", "specialize prime_field_polynomial_aligned_add_functional (x3)", "specialize prime_field_polynomial_aligned_add_functional (x4)", "specialize prime_field_polynomial_aligned_add_functional (x5)", "specialize prime_field_polynomial_aligned_add_functional (x9)", "specialize prime_field_polynomial_aligned_add_functional (x10)", "specialize prime_field_polynomial_aligned_add_functional (x11)", "specialize prime_field_polynomial_aligned_add_functional (x15)", "specialize prime_field_polynomial_aligned_add_functional (x16)", "specialize prime_field_polynomial_aligned_add_functional (x14)", "specialize prime_field_polynomial_aligned_add_functional (rb)", "specialize prime_field_polynomial_aligned_add_functional (rc)", "specialize prime_field_polynomial_aligned_add_functional (N)", "apply prime_field_polynomial_aligned_add_functional", "exact hp", "exact hdistr", "exact hcompare", "have hopbound : ((forall fom_index_pfp_add_input_bound_0. (exists fom_gap_pfp_add_input_bound_0_index_bound. fom_gap_pfp_add_input_bound_0_index_bound + S (fom_index_pfp_add_input_bound_0) = L) -> exists fom_value_pfp_add_input_bound_0. ((((exists fom_beta_height_pfp_add_input_bound_0_entry. fom_beta_height_pfp_add_input_bound_0_entry + S (fom_value_pfp_add_input_bound_0) = S ((S (fom_index_pfp_add_input_bound_0)) * ac)) /\\ exists fom_beta_quotient_pfp_add_input_bound_0_entry. ab = fom_beta_quotient_pfp_add_input_bound_0_entry * S ((S (fom_index_pfp_add_input_bound_0)) * ac) + (fom_value_pfp_add_input_bound_0))) /\\ (exists fom_gap_pfp_add_input_bound_0_value_bound. fom_gap_pfp_add_input_bound_0_value_bound + S (fom_value_pfp_add_input_bound_0) = p))) /\\ (((forall fom_index_pfp_add_input_bound_1. (exists fom_gap_pfp_add_input_bound_1_index_bound. fom_gap_pfp_add_input_bound_1_index_bound + S (fom_index_pfp_add_input_bound_1) = M) -> exists fom_value_pfp_add_input_bound_1. ((((exists fom_beta_height_pfp_add_input_bound_1_entry. fom_beta_height_pfp_add_input_bound_1_entry + S (fom_value_pfp_add_input_bound_1) = S ((S (fom_index_pfp_add_input_bound_1)) * bc)) /\\ exists fom_beta_quotient_pfp_add_input_bound_1_entry. bb = fom_beta_quotient_pfp_add_input_bound_1_entry * S ((S (fom_index_pfp_add_input_bound_1)) * bc) + (fom_value_pfp_add_input_bound_1))) /\\ (exists fom_gap_pfp_add_input_bound_1_value_bound. fom_gap_pfp_add_input_bound_1_value_bound + S (fom_value_pfp_add_input_bound_1) = p))) /\\ ((forall fom_index_pfp_add_input_bound_2. (exists fom_gap_pfp_add_input_bound_2_index_bound. fom_gap_pfp_add_input_bound_2_index_bound + S (fom_index_pfp_add_input_bound_2) = N) -> exists fom_value_pfp_add_input_bound_2. ((((exists fom_beta_height_pfp_add_input_bound_2_entry. fom_beta_height_pfp_add_input_bound_2_entry + S (fom_value_pfp_add_input_bound_2) = S ((S (fom_index_pfp_add_input_bound_2)) * rc)) /\\ exists fom_beta_quotient_pfp_add_input_bound_2_entry. rb = fom_beta_quotient_pfp_add_input_bound_2_entry * S ((S (fom_index_pfp_add_input_bound_2)) * rc) + (fom_value_pfp_add_input_bound_2))) /\\ (exists fom_gap_pfp_add_input_bound_2_value_bound. fom_gap_pfp_add_input_bound_2_value_bound + S (fom_value_pfp_add_input_bound_2) = p)))))))", "specialize prime_field_polynomial_aligned_add_bounded (p)", "specialize prime_field_polynomial_aligned_add_bounded (ab)", "specialize prime_field_polynomial_aligned_add_bounded (ac)", "specialize prime_field_polynomial_aligned_add_bounded (L)", "specialize prime_field_polynomial_aligned_add_bounded (bb)", "specialize prime_field_polynomial_aligned_add_bounded (bc)", "specialize prime_field_polynomial_aligned_add_bounded (M)", "specialize prime_field_polynomial_aligned_add_bounded (rb)", "specialize prime_field_polynomial_aligned_add_bounded (rc)", "specialize prime_field_polynomial_aligned_add_bounded (N)", "apply prime_field_polynomial_aligned_add_bounded", "exact hop", "cases hopbound", "cases hopbound_right", "specialize prime_field_polynomial_right_divides_from_product (p)", "specialize prime_field_polynomial_right_divides_from_product (db)", "specialize prime_field_polynomial_right_divides_from_product (dc)", "specialize prime_field_polynomial_right_divides_from_product (J)", "specialize prime_field_polynomial_right_divides_from_product (rb)", "specialize prime_field_polynomial_right_divides_from_product (rc)", "specialize prime_field_polynomial_right_divides_from_product (N)", "specialize prime_field_polynomial_right_divides_from_product (x12)", "specialize prime_field_polynomial_right_divides_from_product (x13)", "specialize prime_field_polynomial_right_divides_from_product ((x2)+(x8))", "specialize prime_field_polynomial_right_divides_from_product (x15)", "specialize prime_field_polynomial_right_divides_from_product (x16)", "specialize prime_field_polynomial_right_divides_from_product (x14)", "apply prime_field_polynomial_right_divides_from_product", "exact hopbound_right_right", "exact hresult_product_witness_witness", "exact heq" ], "script_sha256": "7e59340e19d153bba51dcf7ff87bc783e1b93bb9794736ea403482b2ce7d3536", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_euclidean_transport_candidate.py", "sha256": "9a589d1749eb38d30d1a24364bc4d66f7df0efb59247527f7831f97557da9c30" }, "statement": "forall p db dc J ab ac L bb bc M rb rc N. (~((p) = 1) /\\ forall pfa_factor_left_add_divides_prime pfa_factor_right_add_divides_prime. (p) = pfa_factor_left_add_divides_prime * pfa_factor_right_add_divides_prime -> pfa_factor_left_add_divides_prime = 1 \\/ pfa_factor_right_add_divides_prime = 1) -> (((forall fom_index_pfp_add_divides_left_canonical. (exists fom_gap_pfp_add_divides_left_canonical_index_bound. fom_gap_pfp_add_divides_left_canonical_index_bound + S (fom_index_pfp_add_divides_left_canonical) = L) -> exists fom_value_pfp_add_divides_left_canonical. ((((exists fom_beta_height_pfp_add_divides_left_canonical_entry. fom_beta_height_pfp_add_divides_left_canonical_entry + S (fom_value_pfp_add_divides_left_canonical) = S ((S (fom_index_pfp_add_divides_left_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_add_divides_left_canonical_entry. ab = fom_beta_quotient_pfp_add_divides_left_canonical_entry * S ((S (fom_index_pfp_add_divides_left_canonical)) * ac) + (fom_value_pfp_add_divides_left_canonical))) /\\ (exists fom_gap_pfp_add_divides_left_canonical_value_bound. fom_gap_pfp_add_divides_left_canonical_value_bound + S (fom_value_pfp_add_divides_left_canonical) = p))) /\\ ((exists pfrd_qb_add_divides_left pfrd_qc_add_divides_left pfrd_qlen_add_divides_left pfrd_pb_add_divides_left pfrd_pc_add_divides_left pfrd_plen_add_divides_left. ((((forall fom_index_pfp_add_divides_left_productleft. (exists fom_gap_pfp_add_divides_left_productleft_index_bound. fom_gap_pfp_add_divides_left_productleft_index_bound + S (fom_index_pfp_add_divides_left_productleft) = pfrd_qlen_add_divides_left) -> exists fom_value_pfp_add_divides_left_productleft. ((((exists fom_beta_height_pfp_add_divides_left_productleft_entry. fom_beta_height_pfp_add_divides_left_productleft_entry + S (fom_value_pfp_add_divides_left_productleft) = S ((S (fom_index_pfp_add_divides_left_productleft)) * pfrd_qc_add_divides_left)) /\\ exists fom_beta_quotient_pfp_add_divides_left_productleft_entry. pfrd_qb_add_divides_left = fom_beta_quotient_pfp_add_divides_left_productleft_entry * S ((S (fom_index_pfp_add_divides_left_productleft)) * pfrd_qc_add_divides_left) + (fom_value_pfp_add_divides_left_productleft))) /\\ (exists fom_gap_pfp_add_divides_left_productleft_value_bound. fom_gap_pfp_add_divides_left_productleft_value_bound + S (fom_value_pfp_add_divides_left_productleft) = p))) /\\ (((forall fom_index_pfp_add_divides_left_productright. (exists fom_gap_pfp_add_divides_left_productright_index_bound. fom_gap_pfp_add_divides_left_productright_index_bound + S (fom_index_pfp_add_divides_left_productright) = J) -> exists fom_value_pfp_add_divides_left_productright. ((((exists fom_beta_height_pfp_add_divides_left_productright_entry. fom_beta_height_pfp_add_divides_left_productright_entry + S (fom_value_pfp_add_divides_left_productright) = S ((S (fom_index_pfp_add_divides_left_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_add_divides_left_productright_entry. db = fom_beta_quotient_pfp_add_divides_left_productright_entry * S ((S (fom_index_pfp_add_divides_left_productright)) * dc) + (fom_value_pfp_add_divides_left_productright))) /\\ (exists fom_gap_pfp_add_divides_left_productright_value_bound. fom_gap_pfp_add_divides_left_productright_value_bound + S (fom_value_pfp_add_divides_left_productright) = p))) /\\ (((((((pfrd_qlen_add_divides_left)=0 \\/ (J)=0) /\\ (((pfrd_plen_add_divides_left)=0)))) \\/ (((~((pfrd_qlen_add_divides_left)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_add_divides_left)+(J)=S (pfrd_plen_add_divides_left)))))))) /\\ ((forall pfc_index_add_divides_left_productcoefficients. (exists pfa_gap_add_divides_left_productcoefficientsbound. pfa_gap_add_divides_left_productcoefficientsbound + S (pfc_index_add_divides_left_productcoefficients) = (pfrd_plen_add_divides_left)) -> exists pfc_value_add_divides_left_productcoefficients. ((((exists ff_h_pfp_add_divides_left_productcoefficientsentry. ff_h_pfp_add_divides_left_productcoefficientsentry + S (pfc_value_add_divides_left_productcoefficients) = S ((S (pfc_index_add_divides_left_productcoefficients)) * pfrd_pc_add_divides_left)) /\\ exists ff_q_pfp_add_divides_left_productcoefficientsentry. pfrd_pb_add_divides_left = ff_q_pfp_add_divides_left_productcoefficientsentry * S ((S (pfc_index_add_divides_left_productcoefficients)) * pfrd_pc_add_divides_left) + (pfc_value_add_divides_left_productcoefficients))) /\\ ((exists pfc_terms_code_add_divides_left_productcoefficientscoefficient pfc_terms_scale_add_divides_left_productcoefficientscoefficient pfc_natural_sum_add_divides_left_productcoefficientscoefficient. ((forall pfc_index_add_divides_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_add_divides_left_productcoefficientscoefficientdiagonalbound. pfa_gap_add_divides_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_add_divides_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_add_divides_left_productcoefficients))) -> exists pfc_value_add_divides_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_add_divides_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_add_divides_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_add_divides_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_add_divides_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_divides_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_add_divides_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_add_divides_left_productcoefficientscoefficient = ff_q_pfp_add_divides_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_add_divides_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_divides_left_productcoefficientscoefficient) + (pfc_value_add_divides_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_add_divides_left_productcoefficientscoefficientdiagonalterm pfc_left_add_divides_left_productcoefficientscoefficientdiagonalterm pfc_right_add_divides_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_add_divides_left_productcoefficientscoefficientdiagonal)+pfc_complement_add_divides_left_productcoefficientscoefficientdiagonalterm=(pfc_index_add_divides_left_productcoefficients)) /\\ ((((((exists pfa_gap_add_divides_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_add_divides_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_add_divides_left_productcoefficientscoefficientdiagonal) = (pfrd_qlen_add_divides_left)) /\\ ((((exists ff_h_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_add_divides_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_add_divides_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_add_divides_left)) /\\ exists ff_q_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_add_divides_left = ff_q_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_add_divides_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_add_divides_left) + (pfc_left_add_divides_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_add_divides_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_add_divides_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_add_divides_left)=(pfc_index_add_divides_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_add_divides_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_add_divides_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_add_divides_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_add_divides_left_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_add_divides_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_add_divides_left_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_add_divides_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_add_divides_left_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_add_divides_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_add_divides_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_add_divides_left_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_add_divides_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_add_divides_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_add_divides_left_productcoefficientscoefficientdiagonal)=pfc_left_add_divides_left_productcoefficientscoefficientdiagonalterm*pfc_right_add_divides_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_add_divides_left_productcoefficientscoefficientsum fs_v_pfc_add_divides_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_add_divides_left_productcoefficientscoefficientsum = fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_add_divides_left_productcoefficientscoefficient) = S ((S (S (pfc_index_add_divides_left_productcoefficients))) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_add_divides_left_productcoefficientscoefficientsum = fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_add_divides_left_productcoefficients))) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum) + (pfc_natural_sum_add_divides_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_add_divides_left_productcoefficients)) -> exists fs_a_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_divides_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_add_divides_left_productcoefficientscoefficient = fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_divides_left_productcoefficientscoefficient) + (fs_a_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_add_divides_left_productcoefficientscoefficientsum = fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum) + (fs_r_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_add_divides_left_productcoefficientscoefficientsum = fs_q_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_left_productcoefficientscoefficientsum) + (fs_s_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_add_divides_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_add_divides_left_productcoefficientscoefficientresiduebound. pfa_gap_add_divides_left_productcoefficientscoefficientresiduebound + S (pfc_value_add_divides_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_add_divides_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_add_divides_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_add_divides_left_productcoefficientscoefficient) + (p) * pfa_offset_left_add_divides_left_productcoefficientscoefficientresiduecongruence = (pfc_value_add_divides_left_productcoefficients) + (p) * pfa_offset_right_add_divides_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_add_divides_left_target pfrep_left_add_divides_left_target pfrep_right_add_divides_left_target. ((exists pfrep_position_add_divides_left_targetfirst. ((pfrep_position_add_divides_left_targetfirst+S (pfrep_power_add_divides_left_target)=(pfrd_plen_add_divides_left)) /\\ ((((exists ff_h_pfp_add_divides_left_targetfirstentry. ff_h_pfp_add_divides_left_targetfirstentry + S (pfrep_left_add_divides_left_target) = S ((S (pfrep_position_add_divides_left_targetfirst)) * pfrd_pc_add_divides_left)) /\\ exists ff_q_pfp_add_divides_left_targetfirstentry. pfrd_pb_add_divides_left = ff_q_pfp_add_divides_left_targetfirstentry * S ((S (pfrep_position_add_divides_left_targetfirst)) * pfrd_pc_add_divides_left) + (pfrep_left_add_divides_left_target)))))) \\/ (((exists pfrep_gap_add_divides_left_targetfirstoutside. pfrep_gap_add_divides_left_targetfirstoutside+(pfrd_plen_add_divides_left)=(pfrep_power_add_divides_left_target)) /\\ (((pfrep_left_add_divides_left_target)=0))))) -> ((exists pfrep_position_add_divides_left_targetsecond. ((pfrep_position_add_divides_left_targetsecond+S (pfrep_power_add_divides_left_target)=(L)) /\\ ((((exists ff_h_pfp_add_divides_left_targetsecondentry. ff_h_pfp_add_divides_left_targetsecondentry + S (pfrep_right_add_divides_left_target) = S ((S (pfrep_position_add_divides_left_targetsecond)) * ac)) /\\ exists ff_q_pfp_add_divides_left_targetsecondentry. ab = ff_q_pfp_add_divides_left_targetsecondentry * S ((S (pfrep_position_add_divides_left_targetsecond)) * ac) + (pfrep_right_add_divides_left_target)))))) \\/ (((exists pfrep_gap_add_divides_left_targetsecondoutside. pfrep_gap_add_divides_left_targetsecondoutside+(L)=(pfrep_power_add_divides_left_target)) /\\ (((pfrep_right_add_divides_left_target)=0))))) -> pfrep_left_add_divides_left_target=pfrep_right_add_divides_left_target))))))) -> (((forall fom_index_pfp_add_divides_right_canonical. (exists fom_gap_pfp_add_divides_right_canonical_index_bound. fom_gap_pfp_add_divides_right_canonical_index_bound + S (fom_index_pfp_add_divides_right_canonical) = M) -> exists fom_value_pfp_add_divides_right_canonical. ((((exists fom_beta_height_pfp_add_divides_right_canonical_entry. fom_beta_height_pfp_add_divides_right_canonical_entry + S (fom_value_pfp_add_divides_right_canonical) = S ((S (fom_index_pfp_add_divides_right_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_add_divides_right_canonical_entry. bb = fom_beta_quotient_pfp_add_divides_right_canonical_entry * S ((S (fom_index_pfp_add_divides_right_canonical)) * bc) + (fom_value_pfp_add_divides_right_canonical))) /\\ (exists fom_gap_pfp_add_divides_right_canonical_value_bound. fom_gap_pfp_add_divides_right_canonical_value_bound + S (fom_value_pfp_add_divides_right_canonical) = p))) /\\ ((exists pfrd_qb_add_divides_right pfrd_qc_add_divides_right pfrd_qlen_add_divides_right pfrd_pb_add_divides_right pfrd_pc_add_divides_right pfrd_plen_add_divides_right. ((((forall fom_index_pfp_add_divides_right_productleft. (exists fom_gap_pfp_add_divides_right_productleft_index_bound. fom_gap_pfp_add_divides_right_productleft_index_bound + S (fom_index_pfp_add_divides_right_productleft) = pfrd_qlen_add_divides_right) -> exists fom_value_pfp_add_divides_right_productleft. ((((exists fom_beta_height_pfp_add_divides_right_productleft_entry. fom_beta_height_pfp_add_divides_right_productleft_entry + S (fom_value_pfp_add_divides_right_productleft) = S ((S (fom_index_pfp_add_divides_right_productleft)) * pfrd_qc_add_divides_right)) /\\ exists fom_beta_quotient_pfp_add_divides_right_productleft_entry. pfrd_qb_add_divides_right = fom_beta_quotient_pfp_add_divides_right_productleft_entry * S ((S (fom_index_pfp_add_divides_right_productleft)) * pfrd_qc_add_divides_right) + (fom_value_pfp_add_divides_right_productleft))) /\\ (exists fom_gap_pfp_add_divides_right_productleft_value_bound. fom_gap_pfp_add_divides_right_productleft_value_bound + S (fom_value_pfp_add_divides_right_productleft) = p))) /\\ (((forall fom_index_pfp_add_divides_right_productright. (exists fom_gap_pfp_add_divides_right_productright_index_bound. fom_gap_pfp_add_divides_right_productright_index_bound + S (fom_index_pfp_add_divides_right_productright) = J) -> exists fom_value_pfp_add_divides_right_productright. ((((exists fom_beta_height_pfp_add_divides_right_productright_entry. fom_beta_height_pfp_add_divides_right_productright_entry + S (fom_value_pfp_add_divides_right_productright) = S ((S (fom_index_pfp_add_divides_right_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_add_divides_right_productright_entry. db = fom_beta_quotient_pfp_add_divides_right_productright_entry * S ((S (fom_index_pfp_add_divides_right_productright)) * dc) + (fom_value_pfp_add_divides_right_productright))) /\\ (exists fom_gap_pfp_add_divides_right_productright_value_bound. fom_gap_pfp_add_divides_right_productright_value_bound + S (fom_value_pfp_add_divides_right_productright) = p))) /\\ (((((((pfrd_qlen_add_divides_right)=0 \\/ (J)=0) /\\ (((pfrd_plen_add_divides_right)=0)))) \\/ (((~((pfrd_qlen_add_divides_right)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_add_divides_right)+(J)=S (pfrd_plen_add_divides_right)))))))) /\\ ((forall pfc_index_add_divides_right_productcoefficients. (exists pfa_gap_add_divides_right_productcoefficientsbound. pfa_gap_add_divides_right_productcoefficientsbound + S (pfc_index_add_divides_right_productcoefficients) = (pfrd_plen_add_divides_right)) -> exists pfc_value_add_divides_right_productcoefficients. ((((exists ff_h_pfp_add_divides_right_productcoefficientsentry. ff_h_pfp_add_divides_right_productcoefficientsentry + S (pfc_value_add_divides_right_productcoefficients) = S ((S (pfc_index_add_divides_right_productcoefficients)) * pfrd_pc_add_divides_right)) /\\ exists ff_q_pfp_add_divides_right_productcoefficientsentry. pfrd_pb_add_divides_right = ff_q_pfp_add_divides_right_productcoefficientsentry * S ((S (pfc_index_add_divides_right_productcoefficients)) * pfrd_pc_add_divides_right) + (pfc_value_add_divides_right_productcoefficients))) /\\ ((exists pfc_terms_code_add_divides_right_productcoefficientscoefficient pfc_terms_scale_add_divides_right_productcoefficientscoefficient pfc_natural_sum_add_divides_right_productcoefficientscoefficient. ((forall pfc_index_add_divides_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_add_divides_right_productcoefficientscoefficientdiagonalbound. pfa_gap_add_divides_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_add_divides_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_add_divides_right_productcoefficients))) -> exists pfc_value_add_divides_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_add_divides_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_add_divides_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_add_divides_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_add_divides_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_divides_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_add_divides_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_add_divides_right_productcoefficientscoefficient = ff_q_pfp_add_divides_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_add_divides_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_divides_right_productcoefficientscoefficient) + (pfc_value_add_divides_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_add_divides_right_productcoefficientscoefficientdiagonalterm pfc_left_add_divides_right_productcoefficientscoefficientdiagonalterm pfc_right_add_divides_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_add_divides_right_productcoefficientscoefficientdiagonal)+pfc_complement_add_divides_right_productcoefficientscoefficientdiagonalterm=(pfc_index_add_divides_right_productcoefficients)) /\\ ((((((exists pfa_gap_add_divides_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_add_divides_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_add_divides_right_productcoefficientscoefficientdiagonal) = (pfrd_qlen_add_divides_right)) /\\ ((((exists ff_h_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_add_divides_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_add_divides_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_add_divides_right)) /\\ exists ff_q_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_add_divides_right = ff_q_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_add_divides_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_add_divides_right) + (pfc_left_add_divides_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_add_divides_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_add_divides_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_add_divides_right)=(pfc_index_add_divides_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_add_divides_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_add_divides_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_add_divides_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_add_divides_right_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_add_divides_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_add_divides_right_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_add_divides_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_add_divides_right_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_add_divides_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_add_divides_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_add_divides_right_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_add_divides_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_add_divides_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_add_divides_right_productcoefficientscoefficientdiagonal)=pfc_left_add_divides_right_productcoefficientscoefficientdiagonalterm*pfc_right_add_divides_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_add_divides_right_productcoefficientscoefficientsum fs_v_pfc_add_divides_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_add_divides_right_productcoefficientscoefficientsum = fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_add_divides_right_productcoefficientscoefficient) = S ((S (S (pfc_index_add_divides_right_productcoefficients))) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_add_divides_right_productcoefficientscoefficientsum = fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_add_divides_right_productcoefficients))) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum) + (pfc_natural_sum_add_divides_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_add_divides_right_productcoefficients)) -> exists fs_a_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_divides_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_add_divides_right_productcoefficientscoefficient = fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_divides_right_productcoefficientscoefficient) + (fs_a_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_add_divides_right_productcoefficientscoefficientsum = fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum) + (fs_r_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_add_divides_right_productcoefficientscoefficientsum = fs_q_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_right_productcoefficientscoefficientsum) + (fs_s_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_add_divides_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_add_divides_right_productcoefficientscoefficientresiduebound. pfa_gap_add_divides_right_productcoefficientscoefficientresiduebound + S (pfc_value_add_divides_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_add_divides_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_add_divides_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_add_divides_right_productcoefficientscoefficient) + (p) * pfa_offset_left_add_divides_right_productcoefficientscoefficientresiduecongruence = (pfc_value_add_divides_right_productcoefficients) + (p) * pfa_offset_right_add_divides_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_add_divides_right_target pfrep_left_add_divides_right_target pfrep_right_add_divides_right_target. ((exists pfrep_position_add_divides_right_targetfirst. ((pfrep_position_add_divides_right_targetfirst+S (pfrep_power_add_divides_right_target)=(pfrd_plen_add_divides_right)) /\\ ((((exists ff_h_pfp_add_divides_right_targetfirstentry. ff_h_pfp_add_divides_right_targetfirstentry + S (pfrep_left_add_divides_right_target) = S ((S (pfrep_position_add_divides_right_targetfirst)) * pfrd_pc_add_divides_right)) /\\ exists ff_q_pfp_add_divides_right_targetfirstentry. pfrd_pb_add_divides_right = ff_q_pfp_add_divides_right_targetfirstentry * S ((S (pfrep_position_add_divides_right_targetfirst)) * pfrd_pc_add_divides_right) + (pfrep_left_add_divides_right_target)))))) \\/ (((exists pfrep_gap_add_divides_right_targetfirstoutside. pfrep_gap_add_divides_right_targetfirstoutside+(pfrd_plen_add_divides_right)=(pfrep_power_add_divides_right_target)) /\\ (((pfrep_left_add_divides_right_target)=0))))) -> ((exists pfrep_position_add_divides_right_targetsecond. ((pfrep_position_add_divides_right_targetsecond+S (pfrep_power_add_divides_right_target)=(M)) /\\ ((((exists ff_h_pfp_add_divides_right_targetsecondentry. ff_h_pfp_add_divides_right_targetsecondentry + S (pfrep_right_add_divides_right_target) = S ((S (pfrep_position_add_divides_right_targetsecond)) * bc)) /\\ exists ff_q_pfp_add_divides_right_targetsecondentry. bb = ff_q_pfp_add_divides_right_targetsecondentry * S ((S (pfrep_position_add_divides_right_targetsecond)) * bc) + (pfrep_right_add_divides_right_target)))))) \\/ (((exists pfrep_gap_add_divides_right_targetsecondoutside. pfrep_gap_add_divides_right_targetsecondoutside+(M)=(pfrep_power_add_divides_right_target)) /\\ (((pfrep_right_add_divides_right_target)=0))))) -> pfrep_left_add_divides_right_target=pfrep_right_add_divides_right_target))))))) -> (((forall fom_index_pfp_add_divides_operation_left_bounded. (exists fom_gap_pfp_add_divides_operation_left_bounded_index_bound. fom_gap_pfp_add_divides_operation_left_bounded_index_bound + S (fom_index_pfp_add_divides_operation_left_bounded) = L) -> exists fom_value_pfp_add_divides_operation_left_bounded. ((((exists fom_beta_height_pfp_add_divides_operation_left_bounded_entry. fom_beta_height_pfp_add_divides_operation_left_bounded_entry + S (fom_value_pfp_add_divides_operation_left_bounded) = S ((S (fom_index_pfp_add_divides_operation_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_add_divides_operation_left_bounded_entry. ab = fom_beta_quotient_pfp_add_divides_operation_left_bounded_entry * S ((S (fom_index_pfp_add_divides_operation_left_bounded)) * ac) + (fom_value_pfp_add_divides_operation_left_bounded))) /\\ (exists fom_gap_pfp_add_divides_operation_left_bounded_value_bound. fom_gap_pfp_add_divides_operation_left_bounded_value_bound + S (fom_value_pfp_add_divides_operation_left_bounded) = p))) /\\ (((forall fom_index_pfp_add_divides_operation_right_bounded. (exists fom_gap_pfp_add_divides_operation_right_bounded_index_bound. fom_gap_pfp_add_divides_operation_right_bounded_index_bound + S (fom_index_pfp_add_divides_operation_right_bounded) = M) -> exists fom_value_pfp_add_divides_operation_right_bounded. ((((exists fom_beta_height_pfp_add_divides_operation_right_bounded_entry. fom_beta_height_pfp_add_divides_operation_right_bounded_entry + S (fom_value_pfp_add_divides_operation_right_bounded) = S ((S (fom_index_pfp_add_divides_operation_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_add_divides_operation_right_bounded_entry. bb = fom_beta_quotient_pfp_add_divides_operation_right_bounded_entry * S ((S (fom_index_pfp_add_divides_operation_right_bounded)) * bc) + (fom_value_pfp_add_divides_operation_right_bounded))) /\\ (exists fom_gap_pfp_add_divides_operation_right_bounded_value_bound. fom_gap_pfp_add_divides_operation_right_bounded_value_bound + S (fom_value_pfp_add_divides_operation_right_bounded) = p))) /\\ (((forall fom_index_pfp_add_divides_operation_result_bounded. (exists fom_gap_pfp_add_divides_operation_result_bounded_index_bound. fom_gap_pfp_add_divides_operation_result_bounded_index_bound + S (fom_index_pfp_add_divides_operation_result_bounded) = N) -> exists fom_value_pfp_add_divides_operation_result_bounded. ((((exists fom_beta_height_pfp_add_divides_operation_result_bounded_entry. fom_beta_height_pfp_add_divides_operation_result_bounded_entry + S (fom_value_pfp_add_divides_operation_result_bounded) = S ((S (fom_index_pfp_add_divides_operation_result_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_add_divides_operation_result_bounded_entry. rb = fom_beta_quotient_pfp_add_divides_operation_result_bounded_entry * S ((S (fom_index_pfp_add_divides_operation_result_bounded)) * rc) + (fom_value_pfp_add_divides_operation_result_bounded))) /\\ (exists fom_gap_pfp_add_divides_operation_result_bounded_value_bound. fom_gap_pfp_add_divides_operation_result_bounded_value_bound + S (fom_value_pfp_add_divides_operation_result_bounded) = p))) /\\ ((exists pfaa_left_b_add_divides_operation pfaa_left_c_add_divides_operation pfaa_right_b_add_divides_operation pfaa_right_c_add_divides_operation pfaa_sum_b_add_divides_operation pfaa_sum_c_add_divides_operation pfaa_length_add_divides_operation. ((((forall pfrep_power_add_divides_operation_witness_common_left pfrep_left_add_divides_operation_witness_common_left pfrep_right_add_divides_operation_witness_common_left. ((exists pfrep_position_add_divides_operation_witness_common_leftfirst. ((pfrep_position_add_divides_operation_witness_common_leftfirst+S (pfrep_power_add_divides_operation_witness_common_left)=(L)) /\\ ((((exists ff_h_pfp_add_divides_operation_witness_common_leftfirstentry. ff_h_pfp_add_divides_operation_witness_common_leftfirstentry + S (pfrep_left_add_divides_operation_witness_common_left) = S ((S (pfrep_position_add_divides_operation_witness_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_add_divides_operation_witness_common_leftfirstentry. ab = ff_q_pfp_add_divides_operation_witness_common_leftfirstentry * S ((S (pfrep_position_add_divides_operation_witness_common_leftfirst)) * ac) + (pfrep_left_add_divides_operation_witness_common_left)))))) \\/ (((exists pfrep_gap_add_divides_operation_witness_common_leftfirstoutside. pfrep_gap_add_divides_operation_witness_common_leftfirstoutside+(L)=(pfrep_power_add_divides_operation_witness_common_left)) /\\ (((pfrep_left_add_divides_operation_witness_common_left)=0))))) -> ((exists pfrep_position_add_divides_operation_witness_common_leftsecond. ((pfrep_position_add_divides_operation_witness_common_leftsecond+S (pfrep_power_add_divides_operation_witness_common_left)=(pfaa_length_add_divides_operation)) /\\ ((((exists ff_h_pfp_add_divides_operation_witness_common_leftsecondentry. ff_h_pfp_add_divides_operation_witness_common_leftsecondentry + S (pfrep_right_add_divides_operation_witness_common_left) = S ((S (pfrep_position_add_divides_operation_witness_common_leftsecond)) * pfaa_left_c_add_divides_operation)) /\\ exists ff_q_pfp_add_divides_operation_witness_common_leftsecondentry. pfaa_left_b_add_divides_operation = ff_q_pfp_add_divides_operation_witness_common_leftsecondentry * S ((S (pfrep_position_add_divides_operation_witness_common_leftsecond)) * pfaa_left_c_add_divides_operation) + (pfrep_right_add_divides_operation_witness_common_left)))))) \\/ (((exists pfrep_gap_add_divides_operation_witness_common_leftsecondoutside. pfrep_gap_add_divides_operation_witness_common_leftsecondoutside+(pfaa_length_add_divides_operation)=(pfrep_power_add_divides_operation_witness_common_left)) /\\ (((pfrep_right_add_divides_operation_witness_common_left)=0))))) -> pfrep_left_add_divides_operation_witness_common_left=pfrep_right_add_divides_operation_witness_common_left) /\\ ((forall pfrep_power_add_divides_operation_witness_common_right pfrep_left_add_divides_operation_witness_common_right pfrep_right_add_divides_operation_witness_common_right. ((exists pfrep_position_add_divides_operation_witness_common_rightfirst. ((pfrep_position_add_divides_operation_witness_common_rightfirst+S (pfrep_power_add_divides_operation_witness_common_right)=(M)) /\\ ((((exists ff_h_pfp_add_divides_operation_witness_common_rightfirstentry. ff_h_pfp_add_divides_operation_witness_common_rightfirstentry + S (pfrep_left_add_divides_operation_witness_common_right) = S ((S (pfrep_position_add_divides_operation_witness_common_rightfirst)) * bc)) /\\ exists ff_q_pfp_add_divides_operation_witness_common_rightfirstentry. bb = ff_q_pfp_add_divides_operation_witness_common_rightfirstentry * S ((S (pfrep_position_add_divides_operation_witness_common_rightfirst)) * bc) + (pfrep_left_add_divides_operation_witness_common_right)))))) \\/ (((exists pfrep_gap_add_divides_operation_witness_common_rightfirstoutside. pfrep_gap_add_divides_operation_witness_common_rightfirstoutside+(M)=(pfrep_power_add_divides_operation_witness_common_right)) /\\ (((pfrep_left_add_divides_operation_witness_common_right)=0))))) -> ((exists pfrep_position_add_divides_operation_witness_common_rightsecond. ((pfrep_position_add_divides_operation_witness_common_rightsecond+S (pfrep_power_add_divides_operation_witness_common_right)=(pfaa_length_add_divides_operation)) /\\ ((((exists ff_h_pfp_add_divides_operation_witness_common_rightsecondentry. ff_h_pfp_add_divides_operation_witness_common_rightsecondentry + S (pfrep_right_add_divides_operation_witness_common_right) = S ((S (pfrep_position_add_divides_operation_witness_common_rightsecond)) * pfaa_right_c_add_divides_operation)) /\\ exists ff_q_pfp_add_divides_operation_witness_common_rightsecondentry. pfaa_right_b_add_divides_operation = ff_q_pfp_add_divides_operation_witness_common_rightsecondentry * S ((S (pfrep_position_add_divides_operation_witness_common_rightsecond)) * pfaa_right_c_add_divides_operation) + (pfrep_right_add_divides_operation_witness_common_right)))))) \\/ (((exists pfrep_gap_add_divides_operation_witness_common_rightsecondoutside. pfrep_gap_add_divides_operation_witness_common_rightsecondoutside+(pfaa_length_add_divides_operation)=(pfrep_power_add_divides_operation_witness_common_right)) /\\ (((pfrep_right_add_divides_operation_witness_common_right)=0))))) -> pfrep_left_add_divides_operation_witness_common_right=pfrep_right_add_divides_operation_witness_common_right)))) /\\ (((forall pfp_index_add_divides_operation_witness_operation. (exists pfa_gap_add_divides_operation_witness_operationindex. pfa_gap_add_divides_operation_witness_operationindex + S (pfp_index_add_divides_operation_witness_operation) = (pfaa_length_add_divides_operation)) -> exists pfp_left_add_divides_operation_witness_operation pfp_right_add_divides_operation_witness_operation pfp_value_add_divides_operation_witness_operation. ((((exists ff_h_pfp_add_divides_operation_witness_operationleft. ff_h_pfp_add_divides_operation_witness_operationleft + S (pfp_left_add_divides_operation_witness_operation) = S ((S (pfp_index_add_divides_operation_witness_operation)) * pfaa_left_c_add_divides_operation)) /\\ exists ff_q_pfp_add_divides_operation_witness_operationleft. pfaa_left_b_add_divides_operation = ff_q_pfp_add_divides_operation_witness_operationleft * S ((S (pfp_index_add_divides_operation_witness_operation)) * pfaa_left_c_add_divides_operation) + (pfp_left_add_divides_operation_witness_operation))) /\\ (((((exists ff_h_pfp_add_divides_operation_witness_operationright. ff_h_pfp_add_divides_operation_witness_operationright + S (pfp_right_add_divides_operation_witness_operation) = S ((S (pfp_index_add_divides_operation_witness_operation)) * pfaa_right_c_add_divides_operation)) /\\ exists ff_q_pfp_add_divides_operation_witness_operationright. pfaa_right_b_add_divides_operation = ff_q_pfp_add_divides_operation_witness_operationright * S ((S (pfp_index_add_divides_operation_witness_operation)) * pfaa_right_c_add_divides_operation) + (pfp_right_add_divides_operation_witness_operation))) /\\ (((((exists ff_h_pfp_add_divides_operation_witness_operationtarget. ff_h_pfp_add_divides_operation_witness_operationtarget + S (pfp_value_add_divides_operation_witness_operation) = S ((S (pfp_index_add_divides_operation_witness_operation)) * pfaa_sum_c_add_divides_operation)) /\\ exists ff_q_pfp_add_divides_operation_witness_operationtarget. pfaa_sum_b_add_divides_operation = ff_q_pfp_add_divides_operation_witness_operationtarget * S ((S (pfp_index_add_divides_operation_witness_operation)) * pfaa_sum_c_add_divides_operation) + (pfp_value_add_divides_operation_witness_operation))) /\\ ((((exists pfa_gap_add_divides_operation_witness_operationoperationleft. pfa_gap_add_divides_operation_witness_operationoperationleft + S (pfp_left_add_divides_operation_witness_operation) = (p)) /\\ (((exists pfa_gap_add_divides_operation_witness_operationoperationright. pfa_gap_add_divides_operation_witness_operationoperationright + S (pfp_right_add_divides_operation_witness_operation) = (p)) /\\ ((((exists pfa_gap_add_divides_operation_witness_operationoperationresultbound. pfa_gap_add_divides_operation_witness_operationoperationresultbound + S (pfp_value_add_divides_operation_witness_operation) = (p)) /\\ ((exists pfa_offset_left_add_divides_operation_witness_operationoperationresultcongruence pfa_offset_right_add_divides_operation_witness_operationoperationresultcongruence. ((pfp_left_add_divides_operation_witness_operation) + (pfp_right_add_divides_operation_witness_operation)) + (p) * pfa_offset_left_add_divides_operation_witness_operationoperationresultcongruence = (pfp_value_add_divides_operation_witness_operation) + (p) * pfa_offset_right_add_divides_operation_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_add_divides_operation_witness_output pfrep_left_add_divides_operation_witness_output pfrep_right_add_divides_operation_witness_output. ((exists pfrep_position_add_divides_operation_witness_outputfirst. ((pfrep_position_add_divides_operation_witness_outputfirst+S (pfrep_power_add_divides_operation_witness_output)=(pfaa_length_add_divides_operation)) /\\ ((((exists ff_h_pfp_add_divides_operation_witness_outputfirstentry. ff_h_pfp_add_divides_operation_witness_outputfirstentry + S (pfrep_left_add_divides_operation_witness_output) = S ((S (pfrep_position_add_divides_operation_witness_outputfirst)) * pfaa_sum_c_add_divides_operation)) /\\ exists ff_q_pfp_add_divides_operation_witness_outputfirstentry. pfaa_sum_b_add_divides_operation = ff_q_pfp_add_divides_operation_witness_outputfirstentry * S ((S (pfrep_position_add_divides_operation_witness_outputfirst)) * pfaa_sum_c_add_divides_operation) + (pfrep_left_add_divides_operation_witness_output)))))) \\/ (((exists pfrep_gap_add_divides_operation_witness_outputfirstoutside. pfrep_gap_add_divides_operation_witness_outputfirstoutside+(pfaa_length_add_divides_operation)=(pfrep_power_add_divides_operation_witness_output)) /\\ (((pfrep_left_add_divides_operation_witness_output)=0))))) -> ((exists pfrep_position_add_divides_operation_witness_outputsecond. ((pfrep_position_add_divides_operation_witness_outputsecond+S (pfrep_power_add_divides_operation_witness_output)=(N)) /\\ ((((exists ff_h_pfp_add_divides_operation_witness_outputsecondentry. ff_h_pfp_add_divides_operation_witness_outputsecondentry + S (pfrep_right_add_divides_operation_witness_output) = S ((S (pfrep_position_add_divides_operation_witness_outputsecond)) * rc)) /\\ exists ff_q_pfp_add_divides_operation_witness_outputsecondentry. rb = ff_q_pfp_add_divides_operation_witness_outputsecondentry * S ((S (pfrep_position_add_divides_operation_witness_outputsecond)) * rc) + (pfrep_right_add_divides_operation_witness_output)))))) \\/ (((exists pfrep_gap_add_divides_operation_witness_outputsecondoutside. pfrep_gap_add_divides_operation_witness_outputsecondoutside+(N)=(pfrep_power_add_divides_operation_witness_output)) /\\ (((pfrep_right_add_divides_operation_witness_output)=0))))) -> pfrep_left_add_divides_operation_witness_output=pfrep_right_add_divides_operation_witness_output))))))))))))) -> (((forall fom_index_pfp_add_divides_result_canonical. (exists fom_gap_pfp_add_divides_result_canonical_index_bound. fom_gap_pfp_add_divides_result_canonical_index_bound + S (fom_index_pfp_add_divides_result_canonical) = N) -> exists fom_value_pfp_add_divides_result_canonical. ((((exists fom_beta_height_pfp_add_divides_result_canonical_entry. fom_beta_height_pfp_add_divides_result_canonical_entry + S (fom_value_pfp_add_divides_result_canonical) = S ((S (fom_index_pfp_add_divides_result_canonical)) * rc)) /\\ exists fom_beta_quotient_pfp_add_divides_result_canonical_entry. rb = fom_beta_quotient_pfp_add_divides_result_canonical_entry * S ((S (fom_index_pfp_add_divides_result_canonical)) * rc) + (fom_value_pfp_add_divides_result_canonical))) /\\ (exists fom_gap_pfp_add_divides_result_canonical_value_bound. fom_gap_pfp_add_divides_result_canonical_value_bound + S (fom_value_pfp_add_divides_result_canonical) = p))) /\\ ((exists pfrd_qb_add_divides_result pfrd_qc_add_divides_result pfrd_qlen_add_divides_result pfrd_pb_add_divides_result pfrd_pc_add_divides_result pfrd_plen_add_divides_result. ((((forall fom_index_pfp_add_divides_result_productleft. (exists fom_gap_pfp_add_divides_result_productleft_index_bound. fom_gap_pfp_add_divides_result_productleft_index_bound + S (fom_index_pfp_add_divides_result_productleft) = pfrd_qlen_add_divides_result) -> exists fom_value_pfp_add_divides_result_productleft. ((((exists fom_beta_height_pfp_add_divides_result_productleft_entry. fom_beta_height_pfp_add_divides_result_productleft_entry + S (fom_value_pfp_add_divides_result_productleft) = S ((S (fom_index_pfp_add_divides_result_productleft)) * pfrd_qc_add_divides_result)) /\\ exists fom_beta_quotient_pfp_add_divides_result_productleft_entry. pfrd_qb_add_divides_result = fom_beta_quotient_pfp_add_divides_result_productleft_entry * S ((S (fom_index_pfp_add_divides_result_productleft)) * pfrd_qc_add_divides_result) + (fom_value_pfp_add_divides_result_productleft))) /\\ (exists fom_gap_pfp_add_divides_result_productleft_value_bound. fom_gap_pfp_add_divides_result_productleft_value_bound + S (fom_value_pfp_add_divides_result_productleft) = p))) /\\ (((forall fom_index_pfp_add_divides_result_productright. (exists fom_gap_pfp_add_divides_result_productright_index_bound. fom_gap_pfp_add_divides_result_productright_index_bound + S (fom_index_pfp_add_divides_result_productright) = J) -> exists fom_value_pfp_add_divides_result_productright. ((((exists fom_beta_height_pfp_add_divides_result_productright_entry. fom_beta_height_pfp_add_divides_result_productright_entry + S (fom_value_pfp_add_divides_result_productright) = S ((S (fom_index_pfp_add_divides_result_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_add_divides_result_productright_entry. db = fom_beta_quotient_pfp_add_divides_result_productright_entry * S ((S (fom_index_pfp_add_divides_result_productright)) * dc) + (fom_value_pfp_add_divides_result_productright))) /\\ (exists fom_gap_pfp_add_divides_result_productright_value_bound. fom_gap_pfp_add_divides_result_productright_value_bound + S (fom_value_pfp_add_divides_result_productright) = p))) /\\ (((((((pfrd_qlen_add_divides_result)=0 \\/ (J)=0) /\\ (((pfrd_plen_add_divides_result)=0)))) \\/ (((~((pfrd_qlen_add_divides_result)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_add_divides_result)+(J)=S (pfrd_plen_add_divides_result)))))))) /\\ ((forall pfc_index_add_divides_result_productcoefficients. (exists pfa_gap_add_divides_result_productcoefficientsbound. pfa_gap_add_divides_result_productcoefficientsbound + S (pfc_index_add_divides_result_productcoefficients) = (pfrd_plen_add_divides_result)) -> exists pfc_value_add_divides_result_productcoefficients. ((((exists ff_h_pfp_add_divides_result_productcoefficientsentry. ff_h_pfp_add_divides_result_productcoefficientsentry + S (pfc_value_add_divides_result_productcoefficients) = S ((S (pfc_index_add_divides_result_productcoefficients)) * pfrd_pc_add_divides_result)) /\\ exists ff_q_pfp_add_divides_result_productcoefficientsentry. pfrd_pb_add_divides_result = ff_q_pfp_add_divides_result_productcoefficientsentry * S ((S (pfc_index_add_divides_result_productcoefficients)) * pfrd_pc_add_divides_result) + (pfc_value_add_divides_result_productcoefficients))) /\\ ((exists pfc_terms_code_add_divides_result_productcoefficientscoefficient pfc_terms_scale_add_divides_result_productcoefficientscoefficient pfc_natural_sum_add_divides_result_productcoefficientscoefficient. ((forall pfc_index_add_divides_result_productcoefficientscoefficientdiagonal. (exists pfa_gap_add_divides_result_productcoefficientscoefficientdiagonalbound. pfa_gap_add_divides_result_productcoefficientscoefficientdiagonalbound + S (pfc_index_add_divides_result_productcoefficientscoefficientdiagonal) = (S (pfc_index_add_divides_result_productcoefficients))) -> exists pfc_value_add_divides_result_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_add_divides_result_productcoefficientscoefficientdiagonalentry. ff_h_pfp_add_divides_result_productcoefficientscoefficientdiagonalentry + S (pfc_value_add_divides_result_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_add_divides_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_divides_result_productcoefficientscoefficient)) /\\ exists ff_q_pfp_add_divides_result_productcoefficientscoefficientdiagonalentry. pfc_terms_code_add_divides_result_productcoefficientscoefficient = ff_q_pfp_add_divides_result_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_add_divides_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_add_divides_result_productcoefficientscoefficient) + (pfc_value_add_divides_result_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_add_divides_result_productcoefficientscoefficientdiagonalterm pfc_left_add_divides_result_productcoefficientscoefficientdiagonalterm pfc_right_add_divides_result_productcoefficientscoefficientdiagonalterm. (((pfc_index_add_divides_result_productcoefficientscoefficientdiagonal)+pfc_complement_add_divides_result_productcoefficientscoefficientdiagonalterm=(pfc_index_add_divides_result_productcoefficients)) /\\ ((((((exists pfa_gap_add_divides_result_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_add_divides_result_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_add_divides_result_productcoefficientscoefficientdiagonal) = (pfrd_qlen_add_divides_result)) /\\ ((((exists ff_h_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_add_divides_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_add_divides_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_add_divides_result)) /\\ exists ff_q_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_add_divides_result = ff_q_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_add_divides_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_add_divides_result) + (pfc_left_add_divides_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_add_divides_result_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_add_divides_result_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_add_divides_result)=(pfc_index_add_divides_result_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_add_divides_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_add_divides_result_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_add_divides_result_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_add_divides_result_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_add_divides_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_add_divides_result_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_add_divides_result_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_add_divides_result_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_add_divides_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_add_divides_result_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_add_divides_result_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_add_divides_result_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_add_divides_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_add_divides_result_productcoefficientscoefficientdiagonal)=pfc_left_add_divides_result_productcoefficientscoefficientdiagonalterm*pfc_right_add_divides_result_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_add_divides_result_productcoefficientscoefficientsum fs_v_pfc_add_divides_result_productcoefficientscoefficientsum. ((((exists fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_start. fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_start. fs_u_pfc_add_divides_result_productcoefficientscoefficientsum = fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_add_divides_result_productcoefficientscoefficient) = S ((S (S (pfc_index_add_divides_result_productcoefficients))) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_add_divides_result_productcoefficientscoefficientsum = fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_add_divides_result_productcoefficients))) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum) + (pfc_natural_sum_add_divides_result_productcoefficientscoefficient))) /\\ forall fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps = S (pfc_index_add_divides_result_productcoefficients)) -> exists fs_a_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps fs_r_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps fs_s_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_divides_result_productcoefficientscoefficient)) /\\ exists fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_add_divides_result_productcoefficientscoefficient = fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_add_divides_result_productcoefficientscoefficient) + (fs_a_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_add_divides_result_productcoefficientscoefficientsum = fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum) + (fs_r_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_add_divides_result_productcoefficientscoefficientsum = fs_q_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_add_divides_result_productcoefficientscoefficientsum) + (fs_s_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps = fs_r_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps + fs_a_pfc_add_divides_result_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_add_divides_result_productcoefficientscoefficientresiduebound. pfa_gap_add_divides_result_productcoefficientscoefficientresiduebound + S (pfc_value_add_divides_result_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_add_divides_result_productcoefficientscoefficientresiduecongruence pfa_offset_right_add_divides_result_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_add_divides_result_productcoefficientscoefficient) + (p) * pfa_offset_left_add_divides_result_productcoefficientscoefficientresiduecongruence = (pfc_value_add_divides_result_productcoefficients) + (p) * pfa_offset_right_add_divides_result_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_add_divides_result_target pfrep_left_add_divides_result_target pfrep_right_add_divides_result_target. ((exists pfrep_position_add_divides_result_targetfirst. ((pfrep_position_add_divides_result_targetfirst+S (pfrep_power_add_divides_result_target)=(pfrd_plen_add_divides_result)) /\\ ((((exists ff_h_pfp_add_divides_result_targetfirstentry. ff_h_pfp_add_divides_result_targetfirstentry + S (pfrep_left_add_divides_result_target) = S ((S (pfrep_position_add_divides_result_targetfirst)) * pfrd_pc_add_divides_result)) /\\ exists ff_q_pfp_add_divides_result_targetfirstentry. pfrd_pb_add_divides_result = ff_q_pfp_add_divides_result_targetfirstentry * S ((S (pfrep_position_add_divides_result_targetfirst)) * pfrd_pc_add_divides_result) + (pfrep_left_add_divides_result_target)))))) \\/ (((exists pfrep_gap_add_divides_result_targetfirstoutside. pfrep_gap_add_divides_result_targetfirstoutside+(pfrd_plen_add_divides_result)=(pfrep_power_add_divides_result_target)) /\\ (((pfrep_left_add_divides_result_target)=0))))) -> ((exists pfrep_position_add_divides_result_targetsecond. 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p", "intro db", "intro dc", "intro J", "intro ab", "intro ac", "intro L", "intro bb", "intro bc", "intro M", "intro rb", "intro rc", "intro N", "intro hp", "intro hDA", "intro hDB", "intro hop", "have hp0 : ~(p=0)", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hz", "cases hDA", "cases hDA_right", "cases hDA_right_witness", "cases hDA_right_witness_witness", "cases hDA_right_witness_witness_witness", "cases hDA_right_witness_witness_witness_witness", "cases hDA_right_witness_witness_witness_witness_witness", "cases hDA_right_witness_witness_witness_witness_witness_witness", "cases hDB", "cases hDB_right", "cases hDB_right_witness", "cases hDB_right_witness_witness", "cases hDB_right_witness_witness_witness", "cases hDB_right_witness_witness_witness_witness", "cases hDB_right_witness_witness_witness_witness_witness", "cases hDB_right_witness_witness_witness_witness_witness_witness", "have hfirst : ((forall fom_index_pfp_subtract_hfirstleft. (exists fom_gap_pfp_subtract_hfirstleft_index_bound. fom_gap_pfp_subtract_hfirstleft_index_bound + S (fom_index_pfp_subtract_hfirstleft) = x2) -> exists fom_value_pfp_subtract_hfirstleft. ((((exists fom_beta_height_pfp_subtract_hfirstleft_entry. fom_beta_height_pfp_subtract_hfirstleft_entry + S (fom_value_pfp_subtract_hfirstleft) = S ((S (fom_index_pfp_subtract_hfirstleft)) * x1)) /\\ exists fom_beta_quotient_pfp_subtract_hfirstleft_entry. x = fom_beta_quotient_pfp_subtract_hfirstleft_entry * S ((S (fom_index_pfp_subtract_hfirstleft)) * x1) + (fom_value_pfp_subtract_hfirstleft))) /\\ (exists fom_gap_pfp_subtract_hfirstleft_value_bound. fom_gap_pfp_subtract_hfirstleft_value_bound + S (fom_value_pfp_subtract_hfirstleft) = p))) /\\ (((forall fom_index_pfp_subtract_hfirstright. (exists fom_gap_pfp_subtract_hfirstright_index_bound. fom_gap_pfp_subtract_hfirstright_index_bound + S (fom_index_pfp_subtract_hfirstright) = J) -> exists fom_value_pfp_subtract_hfirstright. ((((exists fom_beta_height_pfp_subtract_hfirstright_entry. fom_beta_height_pfp_subtract_hfirstright_entry + S (fom_value_pfp_subtract_hfirstright) = S ((S (fom_index_pfp_subtract_hfirstright)) * dc)) /\\ exists fom_beta_quotient_pfp_subtract_hfirstright_entry. db = fom_beta_quotient_pfp_subtract_hfirstright_entry * S ((S (fom_index_pfp_subtract_hfirstright)) * dc) + (fom_value_pfp_subtract_hfirstright))) /\\ (exists fom_gap_pfp_subtract_hfirstright_value_bound. fom_gap_pfp_subtract_hfirstright_value_bound + S (fom_value_pfp_subtract_hfirstright) = p))) /\\ (((((((x2)=0 \\/ (J)=0) /\\ (((x5)=0)))) \\/ (((~((x2)=0)) /\\ (((~((J)=0)) /\\ (((x2)+(J)=S (x5)))))))) /\\ ((forall pfc_index_subtract_hfirstcoefficients. (exists pfa_gap_subtract_hfirstcoefficientsbound. pfa_gap_subtract_hfirstcoefficientsbound + S (pfc_index_subtract_hfirstcoefficients) = (x5)) -> exists pfc_value_subtract_hfirstcoefficients. ((((exists ff_h_pfp_subtract_hfirstcoefficientsentry. ff_h_pfp_subtract_hfirstcoefficientsentry + S (pfc_value_subtract_hfirstcoefficients) = S ((S (pfc_index_subtract_hfirstcoefficients)) * x4)) /\\ exists ff_q_pfp_subtract_hfirstcoefficientsentry. x3 = ff_q_pfp_subtract_hfirstcoefficientsentry * S ((S (pfc_index_subtract_hfirstcoefficients)) * x4) + (pfc_value_subtract_hfirstcoefficients))) /\\ ((exists pfc_terms_code_subtract_hfirstcoefficientscoefficient pfc_terms_scale_subtract_hfirstcoefficientscoefficient pfc_natural_sum_subtract_hfirstcoefficientscoefficient. ((forall pfc_index_subtract_hfirstcoefficientscoefficientdiagonal. (exists pfa_gap_subtract_hfirstcoefficientscoefficientdiagonalbound. pfa_gap_subtract_hfirstcoefficientscoefficientdiagonalbound + S (pfc_index_subtract_hfirstcoefficientscoefficientdiagonal) = (S (pfc_index_subtract_hfirstcoefficients))) -> exists pfc_value_subtract_hfirstcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_subtract_hfirstcoefficientscoefficientdiagonalentry. ff_h_pfp_subtract_hfirstcoefficientscoefficientdiagonalentry + S (pfc_value_subtract_hfirstcoefficientscoefficientdiagonal) = S ((S (pfc_index_subtract_hfirstcoefficientscoefficientdiagonal)) * pfc_terms_scale_subtract_hfirstcoefficientscoefficient)) /\\ exists ff_q_pfp_subtract_hfirstcoefficientscoefficientdiagonalentry. pfc_terms_code_subtract_hfirstcoefficientscoefficient = ff_q_pfp_subtract_hfirstcoefficientscoefficientdiagonalentry * S ((S (pfc_index_subtract_hfirstcoefficientscoefficientdiagonal)) * pfc_terms_scale_subtract_hfirstcoefficientscoefficient) + (pfc_value_subtract_hfirstcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_subtract_hfirstcoefficientscoefficientdiagonalterm pfc_left_subtract_hfirstcoefficientscoefficientdiagonalterm pfc_right_subtract_hfirstcoefficientscoefficientdiagonalterm. (((pfc_index_subtract_hfirstcoefficientscoefficientdiagonal)+pfc_complement_subtract_hfirstcoefficientscoefficientdiagonalterm=(pfc_index_subtract_hfirstcoefficients)) /\\ ((((((exists pfa_gap_subtract_hfirstcoefficientscoefficientdiagonaltermleftinside. pfa_gap_subtract_hfirstcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_subtract_hfirstcoefficientscoefficientdiagonal) = (x2)) /\\ ((((exists ff_h_pfp_subtract_hfirstcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_subtract_hfirstcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_subtract_hfirstcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_subtract_hfirstcoefficientscoefficientdiagonal)) * x1)) /\\ exists ff_q_pfp_subtract_hfirstcoefficientscoefficientdiagonaltermleftentry. x = ff_q_pfp_subtract_hfirstcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_subtract_hfirstcoefficientscoefficientdiagonal)) * x1) + (pfc_left_subtract_hfirstcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_subtract_hfirstcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_subtract_hfirstcoefficientscoefficientdiagonaltermleftoutside+(x2)=(pfc_index_subtract_hfirstcoefficientscoefficientdiagonal)) /\\ (((pfc_left_subtract_hfirstcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_subtract_hfirstcoefficientscoefficientdiagonaltermrightinside. pfa_gap_subtract_hfirstcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_subtract_hfirstcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_subtract_hfirstcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_subtract_hfirstcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_subtract_hfirstcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_subtract_hfirstcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_subtract_hfirstcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_subtract_hfirstcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_subtract_hfirstcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_subtract_hfirstcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_subtract_hfirstcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_subtract_hfirstcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_subtract_hfirstcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_subtract_hfirstcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_subtract_hfirstcoefficientscoefficientdiagonal)=pfc_left_subtract_hfirstcoefficientscoefficientdiagonalterm*pfc_right_subtract_hfirstcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_subtract_hfirstcoefficientscoefficientsum fs_v_pfc_subtract_hfirstcoefficientscoefficientsum. ((((exists fs_h_pfc_subtract_hfirstcoefficientscoefficientsum_body_start. fs_h_pfc_subtract_hfirstcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_subtract_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_hfirstcoefficientscoefficientsum_body_start. fs_u_pfc_subtract_hfirstcoefficientscoefficientsum = fs_q_pfc_subtract_hfirstcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_subtract_hfirstcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_subtract_hfirstcoefficientscoefficientsum_body_terminal. fs_h_pfc_subtract_hfirstcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_subtract_hfirstcoefficientscoefficient) = S ((S (S (pfc_index_subtract_hfirstcoefficients))) * fs_v_pfc_subtract_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_hfirstcoefficientscoefficientsum_body_terminal. fs_u_pfc_subtract_hfirstcoefficientscoefficientsum = fs_q_pfc_subtract_hfirstcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_subtract_hfirstcoefficients))) * fs_v_pfc_subtract_hfirstcoefficientscoefficientsum) + (pfc_natural_sum_subtract_hfirstcoefficientscoefficient))) /\\ forall fs_i_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps = S (pfc_index_subtract_hfirstcoefficients)) -> exists fs_a_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps fs_r_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps fs_s_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_subtract_hfirstcoefficientscoefficient)) /\\ exists fs_q_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_subtract_hfirstcoefficientscoefficient = fs_q_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_subtract_hfirstcoefficientscoefficient) + (fs_a_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_subtract_hfirstcoefficientscoefficientsum = fs_q_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_hfirstcoefficientscoefficientsum) + (fs_r_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_hfirstcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_subtract_hfirstcoefficientscoefficientsum = fs_q_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_hfirstcoefficientscoefficientsum) + (fs_s_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps = fs_r_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps + fs_a_pfc_subtract_hfirstcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_subtract_hfirstcoefficientscoefficientresiduebound. pfa_gap_subtract_hfirstcoefficientscoefficientresiduebound + S (pfc_value_subtract_hfirstcoefficients) = (p)) /\\ ((exists pfa_offset_left_subtract_hfirstcoefficientscoefficientresiduecongruence pfa_offset_right_subtract_hfirstcoefficientscoefficientresiduecongruence. (pfc_natural_sum_subtract_hfirstcoefficientscoefficient) + (p) * pfa_offset_left_subtract_hfirstcoefficientscoefficientresiduecongruence = (pfc_value_subtract_hfirstcoefficients) + (p) * pfa_offset_right_subtract_hfirstcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hDA_right_witness_witness_witness_witness_witness_witness_left", "cases hfirst", "cases hfirst_right", "cases hfirst_right_right", "have hfirst_bounded : forall fom_index_pfp_subtract_hfirst_bounded. (exists fom_gap_pfp_subtract_hfirst_bounded_index_bound. fom_gap_pfp_subtract_hfirst_bounded_index_bound + S (fom_index_pfp_subtract_hfirst_bounded) = x5) -> exists fom_value_pfp_subtract_hfirst_bounded. ((((exists fom_beta_height_pfp_subtract_hfirst_bounded_entry. fom_beta_height_pfp_subtract_hfirst_bounded_entry + S (fom_value_pfp_subtract_hfirst_bounded) = S ((S (fom_index_pfp_subtract_hfirst_bounded)) * x4)) /\\ exists fom_beta_quotient_pfp_subtract_hfirst_bounded_entry. x3 = fom_beta_quotient_pfp_subtract_hfirst_bounded_entry * S ((S (fom_index_pfp_subtract_hfirst_bounded)) * x4) + (fom_value_pfp_subtract_hfirst_bounded))) /\\ (exists fom_gap_pfp_subtract_hfirst_bounded_value_bound. fom_gap_pfp_subtract_hfirst_bounded_value_bound + S (fom_value_pfp_subtract_hfirst_bounded) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (x)", "specialize prime_field_polynomial_convolution_bounded (x1)", "specialize prime_field_polynomial_convolution_bounded (x2)", "specialize prime_field_polynomial_convolution_bounded (db)", "specialize prime_field_polynomial_convolution_bounded (dc)", "specialize prime_field_polynomial_convolution_bounded (J)", "specialize prime_field_polynomial_convolution_bounded (x3)", "specialize prime_field_polynomial_convolution_bounded (x4)", "specialize prime_field_polynomial_convolution_bounded (x5)", "apply prime_field_polynomial_convolution_bounded", "exact hDA_right_witness_witness_witness_witness_witness_witness_left", "have hsecond : ((forall fom_index_pfp_subtract_hsecondleft. (exists fom_gap_pfp_subtract_hsecondleft_index_bound. fom_gap_pfp_subtract_hsecondleft_index_bound + S (fom_index_pfp_subtract_hsecondleft) = x8) -> exists fom_value_pfp_subtract_hsecondleft. ((((exists fom_beta_height_pfp_subtract_hsecondleft_entry. fom_beta_height_pfp_subtract_hsecondleft_entry + S (fom_value_pfp_subtract_hsecondleft) = S ((S (fom_index_pfp_subtract_hsecondleft)) * x7)) /\\ exists fom_beta_quotient_pfp_subtract_hsecondleft_entry. x6 = fom_beta_quotient_pfp_subtract_hsecondleft_entry * S ((S (fom_index_pfp_subtract_hsecondleft)) * x7) + (fom_value_pfp_subtract_hsecondleft))) /\\ (exists fom_gap_pfp_subtract_hsecondleft_value_bound. fom_gap_pfp_subtract_hsecondleft_value_bound + S (fom_value_pfp_subtract_hsecondleft) = p))) /\\ (((forall fom_index_pfp_subtract_hsecondright. (exists fom_gap_pfp_subtract_hsecondright_index_bound. fom_gap_pfp_subtract_hsecondright_index_bound + S (fom_index_pfp_subtract_hsecondright) = J) -> exists fom_value_pfp_subtract_hsecondright. ((((exists fom_beta_height_pfp_subtract_hsecondright_entry. fom_beta_height_pfp_subtract_hsecondright_entry + S (fom_value_pfp_subtract_hsecondright) = S ((S (fom_index_pfp_subtract_hsecondright)) * dc)) /\\ exists fom_beta_quotient_pfp_subtract_hsecondright_entry. db = fom_beta_quotient_pfp_subtract_hsecondright_entry * S ((S (fom_index_pfp_subtract_hsecondright)) * dc) + (fom_value_pfp_subtract_hsecondright))) /\\ (exists fom_gap_pfp_subtract_hsecondright_value_bound. fom_gap_pfp_subtract_hsecondright_value_bound + S (fom_value_pfp_subtract_hsecondright) = p))) /\\ (((((((x8)=0 \\/ (J)=0) /\\ (((x11)=0)))) \\/ (((~((x8)=0)) /\\ (((~((J)=0)) /\\ (((x8)+(J)=S (x11)))))))) /\\ ((forall pfc_index_subtract_hsecondcoefficients. (exists pfa_gap_subtract_hsecondcoefficientsbound. pfa_gap_subtract_hsecondcoefficientsbound + S (pfc_index_subtract_hsecondcoefficients) = (x11)) -> exists pfc_value_subtract_hsecondcoefficients. ((((exists ff_h_pfp_subtract_hsecondcoefficientsentry. ff_h_pfp_subtract_hsecondcoefficientsentry + S (pfc_value_subtract_hsecondcoefficients) = S ((S (pfc_index_subtract_hsecondcoefficients)) * x10)) /\\ exists ff_q_pfp_subtract_hsecondcoefficientsentry. x9 = ff_q_pfp_subtract_hsecondcoefficientsentry * S ((S (pfc_index_subtract_hsecondcoefficients)) * x10) + (pfc_value_subtract_hsecondcoefficients))) /\\ ((exists pfc_terms_code_subtract_hsecondcoefficientscoefficient pfc_terms_scale_subtract_hsecondcoefficientscoefficient pfc_natural_sum_subtract_hsecondcoefficientscoefficient. ((forall pfc_index_subtract_hsecondcoefficientscoefficientdiagonal. (exists pfa_gap_subtract_hsecondcoefficientscoefficientdiagonalbound. pfa_gap_subtract_hsecondcoefficientscoefficientdiagonalbound + S (pfc_index_subtract_hsecondcoefficientscoefficientdiagonal) = (S (pfc_index_subtract_hsecondcoefficients))) -> exists pfc_value_subtract_hsecondcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_subtract_hsecondcoefficientscoefficientdiagonalentry. ff_h_pfp_subtract_hsecondcoefficientscoefficientdiagonalentry + S (pfc_value_subtract_hsecondcoefficientscoefficientdiagonal) = S ((S (pfc_index_subtract_hsecondcoefficientscoefficientdiagonal)) * pfc_terms_scale_subtract_hsecondcoefficientscoefficient)) /\\ exists ff_q_pfp_subtract_hsecondcoefficientscoefficientdiagonalentry. pfc_terms_code_subtract_hsecondcoefficientscoefficient = ff_q_pfp_subtract_hsecondcoefficientscoefficientdiagonalentry * S ((S (pfc_index_subtract_hsecondcoefficientscoefficientdiagonal)) * pfc_terms_scale_subtract_hsecondcoefficientscoefficient) + (pfc_value_subtract_hsecondcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_subtract_hsecondcoefficientscoefficientdiagonalterm pfc_left_subtract_hsecondcoefficientscoefficientdiagonalterm pfc_right_subtract_hsecondcoefficientscoefficientdiagonalterm. (((pfc_index_subtract_hsecondcoefficientscoefficientdiagonal)+pfc_complement_subtract_hsecondcoefficientscoefficientdiagonalterm=(pfc_index_subtract_hsecondcoefficients)) /\\ ((((((exists pfa_gap_subtract_hsecondcoefficientscoefficientdiagonaltermleftinside. pfa_gap_subtract_hsecondcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_subtract_hsecondcoefficientscoefficientdiagonal) = (x8)) /\\ ((((exists ff_h_pfp_subtract_hsecondcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_subtract_hsecondcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_subtract_hsecondcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_subtract_hsecondcoefficientscoefficientdiagonal)) * x7)) /\\ exists ff_q_pfp_subtract_hsecondcoefficientscoefficientdiagonaltermleftentry. x6 = ff_q_pfp_subtract_hsecondcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_subtract_hsecondcoefficientscoefficientdiagonal)) * x7) + (pfc_left_subtract_hsecondcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_subtract_hsecondcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_subtract_hsecondcoefficientscoefficientdiagonaltermleftoutside+(x8)=(pfc_index_subtract_hsecondcoefficientscoefficientdiagonal)) /\\ (((pfc_left_subtract_hsecondcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_subtract_hsecondcoefficientscoefficientdiagonaltermrightinside. pfa_gap_subtract_hsecondcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_subtract_hsecondcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_subtract_hsecondcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_subtract_hsecondcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_subtract_hsecondcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_subtract_hsecondcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_subtract_hsecondcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_subtract_hsecondcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_subtract_hsecondcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_subtract_hsecondcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_subtract_hsecondcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_subtract_hsecondcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_subtract_hsecondcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_subtract_hsecondcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_subtract_hsecondcoefficientscoefficientdiagonal)=pfc_left_subtract_hsecondcoefficientscoefficientdiagonalterm*pfc_right_subtract_hsecondcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_subtract_hsecondcoefficientscoefficientsum fs_v_pfc_subtract_hsecondcoefficientscoefficientsum. ((((exists fs_h_pfc_subtract_hsecondcoefficientscoefficientsum_body_start. fs_h_pfc_subtract_hsecondcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_subtract_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_hsecondcoefficientscoefficientsum_body_start. fs_u_pfc_subtract_hsecondcoefficientscoefficientsum = fs_q_pfc_subtract_hsecondcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_subtract_hsecondcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_subtract_hsecondcoefficientscoefficientsum_body_terminal. fs_h_pfc_subtract_hsecondcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_subtract_hsecondcoefficientscoefficient) = S ((S (S (pfc_index_subtract_hsecondcoefficients))) * fs_v_pfc_subtract_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_hsecondcoefficientscoefficientsum_body_terminal. fs_u_pfc_subtract_hsecondcoefficientscoefficientsum = fs_q_pfc_subtract_hsecondcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_subtract_hsecondcoefficients))) * fs_v_pfc_subtract_hsecondcoefficientscoefficientsum) + (pfc_natural_sum_subtract_hsecondcoefficientscoefficient))) /\\ forall fs_i_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps = S (pfc_index_subtract_hsecondcoefficients)) -> exists fs_a_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps fs_r_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps fs_s_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_subtract_hsecondcoefficientscoefficient)) /\\ exists fs_q_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_subtract_hsecondcoefficientscoefficient = fs_q_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_subtract_hsecondcoefficientscoefficient) + (fs_a_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_subtract_hsecondcoefficientscoefficientsum = fs_q_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_hsecondcoefficientscoefficientsum) + (fs_r_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_hsecondcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_subtract_hsecondcoefficientscoefficientsum = fs_q_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_hsecondcoefficientscoefficientsum) + (fs_s_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps = fs_r_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps + fs_a_pfc_subtract_hsecondcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_subtract_hsecondcoefficientscoefficientresiduebound. pfa_gap_subtract_hsecondcoefficientscoefficientresiduebound + S (pfc_value_subtract_hsecondcoefficients) = (p)) /\\ ((exists pfa_offset_left_subtract_hsecondcoefficientscoefficientresiduecongruence pfa_offset_right_subtract_hsecondcoefficientscoefficientresiduecongruence. (pfc_natural_sum_subtract_hsecondcoefficientscoefficient) + (p) * pfa_offset_left_subtract_hsecondcoefficientscoefficientresiduecongruence = (pfc_value_subtract_hsecondcoefficients) + (p) * pfa_offset_right_subtract_hsecondcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hDB_right_witness_witness_witness_witness_witness_witness_left", "cases hsecond", "cases hsecond_right", "cases hsecond_right_right", "have hsecond_bounded : forall fom_index_pfp_subtract_hsecond_bounded. (exists fom_gap_pfp_subtract_hsecond_bounded_index_bound. fom_gap_pfp_subtract_hsecond_bounded_index_bound + S (fom_index_pfp_subtract_hsecond_bounded) = x11) -> exists fom_value_pfp_subtract_hsecond_bounded. ((((exists fom_beta_height_pfp_subtract_hsecond_bounded_entry. fom_beta_height_pfp_subtract_hsecond_bounded_entry + S (fom_value_pfp_subtract_hsecond_bounded) = S ((S (fom_index_pfp_subtract_hsecond_bounded)) * x10)) /\\ exists fom_beta_quotient_pfp_subtract_hsecond_bounded_entry. x9 = fom_beta_quotient_pfp_subtract_hsecond_bounded_entry * S ((S (fom_index_pfp_subtract_hsecond_bounded)) * x10) + (fom_value_pfp_subtract_hsecond_bounded))) /\\ (exists fom_gap_pfp_subtract_hsecond_bounded_value_bound. fom_gap_pfp_subtract_hsecond_bounded_value_bound + S (fom_value_pfp_subtract_hsecond_bounded) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (x6)", "specialize prime_field_polynomial_convolution_bounded (x7)", "specialize prime_field_polynomial_convolution_bounded (x8)", "specialize prime_field_polynomial_convolution_bounded (db)", "specialize prime_field_polynomial_convolution_bounded (dc)", "specialize prime_field_polynomial_convolution_bounded (J)", "specialize prime_field_polynomial_convolution_bounded (x9)", "specialize prime_field_polynomial_convolution_bounded (x10)", "specialize prime_field_polynomial_convolution_bounded (x11)", "apply prime_field_polynomial_convolution_bounded", "exact hDB_right_witness_witness_witness_witness_witness_witness_left", "have hw : exists wb wc. ((forall fom_index_pfp_subtract_quotient_operation_left_bounded. (exists fom_gap_pfp_subtract_quotient_operation_left_bounded_index_bound. fom_gap_pfp_subtract_quotient_operation_left_bounded_index_bound + S (fom_index_pfp_subtract_quotient_operation_left_bounded) = x8) -> exists fom_value_pfp_subtract_quotient_operation_left_bounded. ((((exists fom_beta_height_pfp_subtract_quotient_operation_left_bounded_entry. fom_beta_height_pfp_subtract_quotient_operation_left_bounded_entry + S (fom_value_pfp_subtract_quotient_operation_left_bounded) = S ((S (fom_index_pfp_subtract_quotient_operation_left_bounded)) * x7)) /\\ exists fom_beta_quotient_pfp_subtract_quotient_operation_left_bounded_entry. x6 = fom_beta_quotient_pfp_subtract_quotient_operation_left_bounded_entry * S ((S (fom_index_pfp_subtract_quotient_operation_left_bounded)) * x7) + (fom_value_pfp_subtract_quotient_operation_left_bounded))) /\\ (exists fom_gap_pfp_subtract_quotient_operation_left_bounded_value_bound. fom_gap_pfp_subtract_quotient_operation_left_bounded_value_bound + S (fom_value_pfp_subtract_quotient_operation_left_bounded) = p))) /\\ (((forall fom_index_pfp_subtract_quotient_operation_right_bounded. (exists fom_gap_pfp_subtract_quotient_operation_right_bounded_index_bound. fom_gap_pfp_subtract_quotient_operation_right_bounded_index_bound + S (fom_index_pfp_subtract_quotient_operation_right_bounded) = (x2)+(x8)) -> exists fom_value_pfp_subtract_quotient_operation_right_bounded. ((((exists fom_beta_height_pfp_subtract_quotient_operation_right_bounded_entry. fom_beta_height_pfp_subtract_quotient_operation_right_bounded_entry + S (fom_value_pfp_subtract_quotient_operation_right_bounded) = S ((S (fom_index_pfp_subtract_quotient_operation_right_bounded)) * wc)) /\\ exists fom_beta_quotient_pfp_subtract_quotient_operation_right_bounded_entry. wb = fom_beta_quotient_pfp_subtract_quotient_operation_right_bounded_entry * S ((S (fom_index_pfp_subtract_quotient_operation_right_bounded)) * wc) + (fom_value_pfp_subtract_quotient_operation_right_bounded))) /\\ (exists fom_gap_pfp_subtract_quotient_operation_right_bounded_value_bound. fom_gap_pfp_subtract_quotient_operation_right_bounded_value_bound + S (fom_value_pfp_subtract_quotient_operation_right_bounded) = p))) /\\ (((forall fom_index_pfp_subtract_quotient_operation_result_bounded. (exists fom_gap_pfp_subtract_quotient_operation_result_bounded_index_bound. fom_gap_pfp_subtract_quotient_operation_result_bounded_index_bound + S (fom_index_pfp_subtract_quotient_operation_result_bounded) = x2) -> exists fom_value_pfp_subtract_quotient_operation_result_bounded. ((((exists fom_beta_height_pfp_subtract_quotient_operation_result_bounded_entry. fom_beta_height_pfp_subtract_quotient_operation_result_bounded_entry + S (fom_value_pfp_subtract_quotient_operation_result_bounded) = S ((S (fom_index_pfp_subtract_quotient_operation_result_bounded)) * x1)) /\\ exists fom_beta_quotient_pfp_subtract_quotient_operation_result_bounded_entry. x = fom_beta_quotient_pfp_subtract_quotient_operation_result_bounded_entry * S ((S (fom_index_pfp_subtract_quotient_operation_result_bounded)) * x1) + (fom_value_pfp_subtract_quotient_operation_result_bounded))) /\\ (exists fom_gap_pfp_subtract_quotient_operation_result_bounded_value_bound. fom_gap_pfp_subtract_quotient_operation_result_bounded_value_bound + S (fom_value_pfp_subtract_quotient_operation_result_bounded) = p))) /\\ ((exists pfaa_left_b_subtract_quotient_operation pfaa_left_c_subtract_quotient_operation pfaa_right_b_subtract_quotient_operation pfaa_right_c_subtract_quotient_operation pfaa_sum_b_subtract_quotient_operation pfaa_sum_c_subtract_quotient_operation pfaa_length_subtract_quotient_operation. ((((forall pfrep_power_subtract_quotient_operation_witness_common_left pfrep_left_subtract_quotient_operation_witness_common_left pfrep_right_subtract_quotient_operation_witness_common_left. ((exists pfrep_position_subtract_quotient_operation_witness_common_leftfirst. ((pfrep_position_subtract_quotient_operation_witness_common_leftfirst+S (pfrep_power_subtract_quotient_operation_witness_common_left)=(x8)) /\\ ((((exists ff_h_pfp_subtract_quotient_operation_witness_common_leftfirstentry. ff_h_pfp_subtract_quotient_operation_witness_common_leftfirstentry + S (pfrep_left_subtract_quotient_operation_witness_common_left) = S ((S (pfrep_position_subtract_quotient_operation_witness_common_leftfirst)) * x7)) /\\ exists ff_q_pfp_subtract_quotient_operation_witness_common_leftfirstentry. x6 = ff_q_pfp_subtract_quotient_operation_witness_common_leftfirstentry * S ((S (pfrep_position_subtract_quotient_operation_witness_common_leftfirst)) * x7) + (pfrep_left_subtract_quotient_operation_witness_common_left)))))) \\/ (((exists pfrep_gap_subtract_quotient_operation_witness_common_leftfirstoutside. pfrep_gap_subtract_quotient_operation_witness_common_leftfirstoutside+(x8)=(pfrep_power_subtract_quotient_operation_witness_common_left)) /\\ (((pfrep_left_subtract_quotient_operation_witness_common_left)=0))))) -> ((exists pfrep_position_subtract_quotient_operation_witness_common_leftsecond. ((pfrep_position_subtract_quotient_operation_witness_common_leftsecond+S (pfrep_power_subtract_quotient_operation_witness_common_left)=(pfaa_length_subtract_quotient_operation)) /\\ ((((exists ff_h_pfp_subtract_quotient_operation_witness_common_leftsecondentry. ff_h_pfp_subtract_quotient_operation_witness_common_leftsecondentry + S (pfrep_right_subtract_quotient_operation_witness_common_left) = S ((S (pfrep_position_subtract_quotient_operation_witness_common_leftsecond)) * pfaa_left_c_subtract_quotient_operation)) /\\ exists ff_q_pfp_subtract_quotient_operation_witness_common_leftsecondentry. pfaa_left_b_subtract_quotient_operation = ff_q_pfp_subtract_quotient_operation_witness_common_leftsecondentry * S ((S (pfrep_position_subtract_quotient_operation_witness_common_leftsecond)) * pfaa_left_c_subtract_quotient_operation) + (pfrep_right_subtract_quotient_operation_witness_common_left)))))) \\/ (((exists pfrep_gap_subtract_quotient_operation_witness_common_leftsecondoutside. pfrep_gap_subtract_quotient_operation_witness_common_leftsecondoutside+(pfaa_length_subtract_quotient_operation)=(pfrep_power_subtract_quotient_operation_witness_common_left)) /\\ (((pfrep_right_subtract_quotient_operation_witness_common_left)=0))))) -> pfrep_left_subtract_quotient_operation_witness_common_left=pfrep_right_subtract_quotient_operation_witness_common_left) /\\ ((forall pfrep_power_subtract_quotient_operation_witness_common_right pfrep_left_subtract_quotient_operation_witness_common_right pfrep_right_subtract_quotient_operation_witness_common_right. ((exists pfrep_position_subtract_quotient_operation_witness_common_rightfirst. ((pfrep_position_subtract_quotient_operation_witness_common_rightfirst+S (pfrep_power_subtract_quotient_operation_witness_common_right)=((x2)+(x8))) /\\ ((((exists ff_h_pfp_subtract_quotient_operation_witness_common_rightfirstentry. ff_h_pfp_subtract_quotient_operation_witness_common_rightfirstentry + S (pfrep_left_subtract_quotient_operation_witness_common_right) = S ((S (pfrep_position_subtract_quotient_operation_witness_common_rightfirst)) * wc)) /\\ exists ff_q_pfp_subtract_quotient_operation_witness_common_rightfirstentry. wb = ff_q_pfp_subtract_quotient_operation_witness_common_rightfirstentry * S ((S (pfrep_position_subtract_quotient_operation_witness_common_rightfirst)) * wc) + (pfrep_left_subtract_quotient_operation_witness_common_right)))))) \\/ (((exists pfrep_gap_subtract_quotient_operation_witness_common_rightfirstoutside. pfrep_gap_subtract_quotient_operation_witness_common_rightfirstoutside+((x2)+(x8))=(pfrep_power_subtract_quotient_operation_witness_common_right)) /\\ (((pfrep_left_subtract_quotient_operation_witness_common_right)=0))))) -> ((exists pfrep_position_subtract_quotient_operation_witness_common_rightsecond. ((pfrep_position_subtract_quotient_operation_witness_common_rightsecond+S (pfrep_power_subtract_quotient_operation_witness_common_right)=(pfaa_length_subtract_quotient_operation)) /\\ ((((exists ff_h_pfp_subtract_quotient_operation_witness_common_rightsecondentry. ff_h_pfp_subtract_quotient_operation_witness_common_rightsecondentry + S (pfrep_right_subtract_quotient_operation_witness_common_right) = S ((S (pfrep_position_subtract_quotient_operation_witness_common_rightsecond)) * pfaa_right_c_subtract_quotient_operation)) /\\ exists ff_q_pfp_subtract_quotient_operation_witness_common_rightsecondentry. pfaa_right_b_subtract_quotient_operation = ff_q_pfp_subtract_quotient_operation_witness_common_rightsecondentry * S ((S (pfrep_position_subtract_quotient_operation_witness_common_rightsecond)) * pfaa_right_c_subtract_quotient_operation) + (pfrep_right_subtract_quotient_operation_witness_common_right)))))) \\/ (((exists pfrep_gap_subtract_quotient_operation_witness_common_rightsecondoutside. pfrep_gap_subtract_quotient_operation_witness_common_rightsecondoutside+(pfaa_length_subtract_quotient_operation)=(pfrep_power_subtract_quotient_operation_witness_common_right)) /\\ (((pfrep_right_subtract_quotient_operation_witness_common_right)=0))))) -> pfrep_left_subtract_quotient_operation_witness_common_right=pfrep_right_subtract_quotient_operation_witness_common_right)))) /\\ (((forall pfp_index_subtract_quotient_operation_witness_operation. (exists pfa_gap_subtract_quotient_operation_witness_operationindex. pfa_gap_subtract_quotient_operation_witness_operationindex + S (pfp_index_subtract_quotient_operation_witness_operation) = (pfaa_length_subtract_quotient_operation)) -> exists pfp_left_subtract_quotient_operation_witness_operation pfp_right_subtract_quotient_operation_witness_operation pfp_value_subtract_quotient_operation_witness_operation. ((((exists ff_h_pfp_subtract_quotient_operation_witness_operationleft. ff_h_pfp_subtract_quotient_operation_witness_operationleft + S (pfp_left_subtract_quotient_operation_witness_operation) = S ((S (pfp_index_subtract_quotient_operation_witness_operation)) * pfaa_left_c_subtract_quotient_operation)) /\\ exists ff_q_pfp_subtract_quotient_operation_witness_operationleft. pfaa_left_b_subtract_quotient_operation = ff_q_pfp_subtract_quotient_operation_witness_operationleft * S ((S (pfp_index_subtract_quotient_operation_witness_operation)) * pfaa_left_c_subtract_quotient_operation) + (pfp_left_subtract_quotient_operation_witness_operation))) /\\ (((((exists ff_h_pfp_subtract_quotient_operation_witness_operationright. ff_h_pfp_subtract_quotient_operation_witness_operationright + S (pfp_right_subtract_quotient_operation_witness_operation) = S ((S (pfp_index_subtract_quotient_operation_witness_operation)) * pfaa_right_c_subtract_quotient_operation)) /\\ exists ff_q_pfp_subtract_quotient_operation_witness_operationright. pfaa_right_b_subtract_quotient_operation = ff_q_pfp_subtract_quotient_operation_witness_operationright * S ((S (pfp_index_subtract_quotient_operation_witness_operation)) * pfaa_right_c_subtract_quotient_operation) + (pfp_right_subtract_quotient_operation_witness_operation))) /\\ (((((exists ff_h_pfp_subtract_quotient_operation_witness_operationtarget. ff_h_pfp_subtract_quotient_operation_witness_operationtarget + S (pfp_value_subtract_quotient_operation_witness_operation) = S ((S (pfp_index_subtract_quotient_operation_witness_operation)) * pfaa_sum_c_subtract_quotient_operation)) /\\ exists ff_q_pfp_subtract_quotient_operation_witness_operationtarget. pfaa_sum_b_subtract_quotient_operation = ff_q_pfp_subtract_quotient_operation_witness_operationtarget * S ((S (pfp_index_subtract_quotient_operation_witness_operation)) * pfaa_sum_c_subtract_quotient_operation) + (pfp_value_subtract_quotient_operation_witness_operation))) /\\ ((((exists pfa_gap_subtract_quotient_operation_witness_operationoperationleft. pfa_gap_subtract_quotient_operation_witness_operationoperationleft + S (pfp_left_subtract_quotient_operation_witness_operation) = (p)) /\\ (((exists pfa_gap_subtract_quotient_operation_witness_operationoperationright. pfa_gap_subtract_quotient_operation_witness_operationoperationright + S (pfp_right_subtract_quotient_operation_witness_operation) = (p)) /\\ ((((exists pfa_gap_subtract_quotient_operation_witness_operationoperationresultbound. pfa_gap_subtract_quotient_operation_witness_operationoperationresultbound + S (pfp_value_subtract_quotient_operation_witness_operation) = (p)) /\\ ((exists pfa_offset_left_subtract_quotient_operation_witness_operationoperationresultcongruence pfa_offset_right_subtract_quotient_operation_witness_operationoperationresultcongruence. ((pfp_left_subtract_quotient_operation_witness_operation) + (pfp_right_subtract_quotient_operation_witness_operation)) + (p) * pfa_offset_left_subtract_quotient_operation_witness_operationoperationresultcongruence = (pfp_value_subtract_quotient_operation_witness_operation) + (p) * pfa_offset_right_subtract_quotient_operation_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_subtract_quotient_operation_witness_output pfrep_left_subtract_quotient_operation_witness_output pfrep_right_subtract_quotient_operation_witness_output. ((exists pfrep_position_subtract_quotient_operation_witness_outputfirst. ((pfrep_position_subtract_quotient_operation_witness_outputfirst+S (pfrep_power_subtract_quotient_operation_witness_output)=(pfaa_length_subtract_quotient_operation)) /\\ ((((exists ff_h_pfp_subtract_quotient_operation_witness_outputfirstentry. ff_h_pfp_subtract_quotient_operation_witness_outputfirstentry + S (pfrep_left_subtract_quotient_operation_witness_output) = S ((S (pfrep_position_subtract_quotient_operation_witness_outputfirst)) * pfaa_sum_c_subtract_quotient_operation)) /\\ exists ff_q_pfp_subtract_quotient_operation_witness_outputfirstentry. pfaa_sum_b_subtract_quotient_operation = ff_q_pfp_subtract_quotient_operation_witness_outputfirstentry * S ((S (pfrep_position_subtract_quotient_operation_witness_outputfirst)) * pfaa_sum_c_subtract_quotient_operation) + (pfrep_left_subtract_quotient_operation_witness_output)))))) \\/ (((exists pfrep_gap_subtract_quotient_operation_witness_outputfirstoutside. pfrep_gap_subtract_quotient_operation_witness_outputfirstoutside+(pfaa_length_subtract_quotient_operation)=(pfrep_power_subtract_quotient_operation_witness_output)) /\\ (((pfrep_left_subtract_quotient_operation_witness_output)=0))))) -> ((exists pfrep_position_subtract_quotient_operation_witness_outputsecond. ((pfrep_position_subtract_quotient_operation_witness_outputsecond+S (pfrep_power_subtract_quotient_operation_witness_output)=(x2)) /\\ ((((exists ff_h_pfp_subtract_quotient_operation_witness_outputsecondentry. ff_h_pfp_subtract_quotient_operation_witness_outputsecondentry + S (pfrep_right_subtract_quotient_operation_witness_output) = S ((S (pfrep_position_subtract_quotient_operation_witness_outputsecond)) * x1)) /\\ exists ff_q_pfp_subtract_quotient_operation_witness_outputsecondentry. x = ff_q_pfp_subtract_quotient_operation_witness_outputsecondentry * S ((S (pfrep_position_subtract_quotient_operation_witness_outputsecond)) * x1) + (pfrep_right_subtract_quotient_operation_witness_output)))))) \\/ (((exists pfrep_gap_subtract_quotient_operation_witness_outputsecondoutside. pfrep_gap_subtract_quotient_operation_witness_outputsecondoutside+(x2)=(pfrep_power_subtract_quotient_operation_witness_output)) /\\ (((pfrep_right_subtract_quotient_operation_witness_output)=0))))) -> pfrep_left_subtract_quotient_operation_witness_output=pfrep_right_subtract_quotient_operation_witness_output))))))))))))", "specialize prime_field_polynomial_aligned_subtract_exists (p)", "specialize prime_field_polynomial_aligned_subtract_exists (x)", "specialize prime_field_polynomial_aligned_subtract_exists (x1)", "specialize prime_field_polynomial_aligned_subtract_exists (x2)", "specialize prime_field_polynomial_aligned_subtract_exists (x6)", "specialize prime_field_polynomial_aligned_subtract_exists (x7)", "specialize prime_field_polynomial_aligned_subtract_exists (x8)", "apply prime_field_polynomial_aligned_subtract_exists", "exact hp", "exact hfirst_left", "exact hsecond_left", "cases hw", "cases hw_witness", "have hwbound : ((forall fom_index_pfp_subtract_quotient_bound_0. (exists fom_gap_pfp_subtract_quotient_bound_0_index_bound. fom_gap_pfp_subtract_quotient_bound_0_index_bound + S (fom_index_pfp_subtract_quotient_bound_0) = x8) -> exists fom_value_pfp_subtract_quotient_bound_0. ((((exists fom_beta_height_pfp_subtract_quotient_bound_0_entry. fom_beta_height_pfp_subtract_quotient_bound_0_entry + S (fom_value_pfp_subtract_quotient_bound_0) = S ((S (fom_index_pfp_subtract_quotient_bound_0)) * x7)) /\\ exists fom_beta_quotient_pfp_subtract_quotient_bound_0_entry. x6 = fom_beta_quotient_pfp_subtract_quotient_bound_0_entry * S ((S (fom_index_pfp_subtract_quotient_bound_0)) * x7) + (fom_value_pfp_subtract_quotient_bound_0))) /\\ (exists fom_gap_pfp_subtract_quotient_bound_0_value_bound. fom_gap_pfp_subtract_quotient_bound_0_value_bound + S (fom_value_pfp_subtract_quotient_bound_0) = p))) /\\ (((forall fom_index_pfp_subtract_quotient_bound_1. (exists fom_gap_pfp_subtract_quotient_bound_1_index_bound. fom_gap_pfp_subtract_quotient_bound_1_index_bound + S (fom_index_pfp_subtract_quotient_bound_1) = (x2)+(x8)) -> exists fom_value_pfp_subtract_quotient_bound_1. ((((exists fom_beta_height_pfp_subtract_quotient_bound_1_entry. fom_beta_height_pfp_subtract_quotient_bound_1_entry + S (fom_value_pfp_subtract_quotient_bound_1) = S ((S (fom_index_pfp_subtract_quotient_bound_1)) * x13)) /\\ exists fom_beta_quotient_pfp_subtract_quotient_bound_1_entry. x12 = fom_beta_quotient_pfp_subtract_quotient_bound_1_entry * S ((S (fom_index_pfp_subtract_quotient_bound_1)) * x13) + (fom_value_pfp_subtract_quotient_bound_1))) /\\ (exists fom_gap_pfp_subtract_quotient_bound_1_value_bound. fom_gap_pfp_subtract_quotient_bound_1_value_bound + S (fom_value_pfp_subtract_quotient_bound_1) = p))) /\\ ((forall fom_index_pfp_subtract_quotient_bound_2. (exists fom_gap_pfp_subtract_quotient_bound_2_index_bound. fom_gap_pfp_subtract_quotient_bound_2_index_bound + S (fom_index_pfp_subtract_quotient_bound_2) = x2) -> exists fom_value_pfp_subtract_quotient_bound_2. ((((exists fom_beta_height_pfp_subtract_quotient_bound_2_entry. fom_beta_height_pfp_subtract_quotient_bound_2_entry + S (fom_value_pfp_subtract_quotient_bound_2) = S ((S (fom_index_pfp_subtract_quotient_bound_2)) * x1)) /\\ exists fom_beta_quotient_pfp_subtract_quotient_bound_2_entry. x = fom_beta_quotient_pfp_subtract_quotient_bound_2_entry * S ((S (fom_index_pfp_subtract_quotient_bound_2)) * x1) + (fom_value_pfp_subtract_quotient_bound_2))) /\\ (exists fom_gap_pfp_subtract_quotient_bound_2_value_bound. fom_gap_pfp_subtract_quotient_bound_2_value_bound + S (fom_value_pfp_subtract_quotient_bound_2) = p)))))))", "specialize prime_field_polynomial_aligned_add_bounded (p)", "specialize prime_field_polynomial_aligned_add_bounded (x6)", "specialize prime_field_polynomial_aligned_add_bounded (x7)", "specialize prime_field_polynomial_aligned_add_bounded (x8)", "specialize prime_field_polynomial_aligned_add_bounded (x12)", "specialize prime_field_polynomial_aligned_add_bounded (x13)", "specialize prime_field_polynomial_aligned_add_bounded ((x2)+(x8))", "specialize prime_field_polynomial_aligned_add_bounded (x)", "specialize prime_field_polynomial_aligned_add_bounded (x1)", "specialize prime_field_polynomial_aligned_add_bounded (x2)", "apply prime_field_polynomial_aligned_add_bounded", "exact hw_witness_witness", "cases hwbound", "cases hwbound_right", "have hresult_length : exists n. (((((x2)+(x8))=0 \\/ (J)=0) /\\ (((n)=0)))) \\/ (((~(((x2)+(x8))=0)) /\\ (((~((J)=0)) /\\ ((((x2)+(x8))+(J)=S (n)))))))", "specialize polynomial_product_length_exists ((x2)+(x8))", "specialize polynomial_product_length_exists (J)", "apply polynomial_product_length_exists", "cases hresult_length", "have hresult_product : exists b c. ((forall fom_index_pfp_hresult_productleft. (exists fom_gap_pfp_hresult_productleft_index_bound. fom_gap_pfp_hresult_productleft_index_bound + S (fom_index_pfp_hresult_productleft) = (x2)+(x8)) -> exists fom_value_pfp_hresult_productleft. ((((exists fom_beta_height_pfp_hresult_productleft_entry. fom_beta_height_pfp_hresult_productleft_entry + S (fom_value_pfp_hresult_productleft) = S ((S (fom_index_pfp_hresult_productleft)) * x13)) /\\ exists fom_beta_quotient_pfp_hresult_productleft_entry. x12 = fom_beta_quotient_pfp_hresult_productleft_entry * S ((S (fom_index_pfp_hresult_productleft)) * x13) + (fom_value_pfp_hresult_productleft))) /\\ (exists fom_gap_pfp_hresult_productleft_value_bound. fom_gap_pfp_hresult_productleft_value_bound + S (fom_value_pfp_hresult_productleft) = p))) /\\ (((forall fom_index_pfp_hresult_productright. (exists fom_gap_pfp_hresult_productright_index_bound. fom_gap_pfp_hresult_productright_index_bound + S (fom_index_pfp_hresult_productright) = J) -> exists fom_value_pfp_hresult_productright. ((((exists fom_beta_height_pfp_hresult_productright_entry. fom_beta_height_pfp_hresult_productright_entry + S (fom_value_pfp_hresult_productright) = S ((S (fom_index_pfp_hresult_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_hresult_productright_entry. db = fom_beta_quotient_pfp_hresult_productright_entry * S ((S (fom_index_pfp_hresult_productright)) * dc) + (fom_value_pfp_hresult_productright))) /\\ (exists fom_gap_pfp_hresult_productright_value_bound. fom_gap_pfp_hresult_productright_value_bound + S (fom_value_pfp_hresult_productright) = p))) /\\ ((((((((x2)+(x8))=0 \\/ (J)=0) /\\ (((x14)=0)))) \\/ (((~(((x2)+(x8))=0)) /\\ (((~((J)=0)) /\\ ((((x2)+(x8))+(J)=S (x14)))))))) /\\ ((forall pfc_index_hresult_productcoefficients. (exists pfa_gap_hresult_productcoefficientsbound. pfa_gap_hresult_productcoefficientsbound + S (pfc_index_hresult_productcoefficients) = (x14)) -> exists pfc_value_hresult_productcoefficients. ((((exists ff_h_pfp_hresult_productcoefficientsentry. ff_h_pfp_hresult_productcoefficientsentry + S (pfc_value_hresult_productcoefficients) = S ((S (pfc_index_hresult_productcoefficients)) * c)) /\\ exists ff_q_pfp_hresult_productcoefficientsentry. b = ff_q_pfp_hresult_productcoefficientsentry * S ((S (pfc_index_hresult_productcoefficients)) * c) + (pfc_value_hresult_productcoefficients))) /\\ ((exists pfc_terms_code_hresult_productcoefficientscoefficient pfc_terms_scale_hresult_productcoefficientscoefficient pfc_natural_sum_hresult_productcoefficientscoefficient. ((forall pfc_index_hresult_productcoefficientscoefficientdiagonal. (exists pfa_gap_hresult_productcoefficientscoefficientdiagonalbound. pfa_gap_hresult_productcoefficientscoefficientdiagonalbound + S (pfc_index_hresult_productcoefficientscoefficientdiagonal) = (S (pfc_index_hresult_productcoefficients))) -> exists pfc_value_hresult_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_hresult_productcoefficientscoefficientdiagonalentry. ff_h_pfp_hresult_productcoefficientscoefficientdiagonalentry + S (pfc_value_hresult_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_hresult_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hresult_productcoefficientscoefficient)) /\\ exists ff_q_pfp_hresult_productcoefficientscoefficientdiagonalentry. pfc_terms_code_hresult_productcoefficientscoefficient = ff_q_pfp_hresult_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_hresult_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hresult_productcoefficientscoefficient) + (pfc_value_hresult_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_hresult_productcoefficientscoefficientdiagonalterm pfc_left_hresult_productcoefficientscoefficientdiagonalterm pfc_right_hresult_productcoefficientscoefficientdiagonalterm. (((pfc_index_hresult_productcoefficientscoefficientdiagonal)+pfc_complement_hresult_productcoefficientscoefficientdiagonalterm=(pfc_index_hresult_productcoefficients)) /\\ ((((((exists pfa_gap_hresult_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_hresult_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_hresult_productcoefficientscoefficientdiagonal) = ((x2)+(x8))) /\\ ((((exists ff_h_pfp_hresult_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_hresult_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_hresult_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_hresult_productcoefficientscoefficientdiagonal)) * x13)) /\\ exists ff_q_pfp_hresult_productcoefficientscoefficientdiagonaltermleftentry. x12 = ff_q_pfp_hresult_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_hresult_productcoefficientscoefficientdiagonal)) * x13) + (pfc_left_hresult_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hresult_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_hresult_productcoefficientscoefficientdiagonaltermleftoutside+((x2)+(x8))=(pfc_index_hresult_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_hresult_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_hresult_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_hresult_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_hresult_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_hresult_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_hresult_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_hresult_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_hresult_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_hresult_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_hresult_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_hresult_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_hresult_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hresult_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_hresult_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_hresult_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_hresult_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_hresult_productcoefficientscoefficientdiagonal)=pfc_left_hresult_productcoefficientscoefficientdiagonalterm*pfc_right_hresult_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_hresult_productcoefficientscoefficientsum fs_v_pfc_hresult_productcoefficientscoefficientsum. ((((exists fs_h_pfc_hresult_productcoefficientscoefficientsum_body_start. fs_h_pfc_hresult_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_hresult_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hresult_productcoefficientscoefficientsum_body_start. fs_u_pfc_hresult_productcoefficientscoefficientsum = fs_q_pfc_hresult_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_hresult_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_hresult_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_hresult_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_hresult_productcoefficientscoefficient) = S ((S (S (pfc_index_hresult_productcoefficients))) * fs_v_pfc_hresult_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hresult_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_hresult_productcoefficientscoefficientsum = fs_q_pfc_hresult_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_hresult_productcoefficients))) * fs_v_pfc_hresult_productcoefficientscoefficientsum) + (pfc_natural_sum_hresult_productcoefficientscoefficient))) /\\ forall fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_hresult_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_hresult_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps = S (pfc_index_hresult_productcoefficients)) -> exists fs_a_pfc_hresult_productcoefficientscoefficientsum_body_steps fs_r_pfc_hresult_productcoefficientscoefficientsum_body_steps fs_s_pfc_hresult_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_hresult_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_hresult_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_hresult_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hresult_productcoefficientscoefficient)) /\\ exists fs_q_pfc_hresult_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_hresult_productcoefficientscoefficient = fs_q_pfc_hresult_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hresult_productcoefficientscoefficient) + (fs_a_pfc_hresult_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hresult_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_hresult_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_hresult_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hresult_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hresult_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_hresult_productcoefficientscoefficientsum = fs_q_pfc_hresult_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hresult_productcoefficientscoefficientsum) + (fs_r_pfc_hresult_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hresult_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_hresult_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_hresult_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hresult_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hresult_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_hresult_productcoefficientscoefficientsum = fs_q_pfc_hresult_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_hresult_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hresult_productcoefficientscoefficientsum) + (fs_s_pfc_hresult_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_hresult_productcoefficientscoefficientsum_body_steps = fs_r_pfc_hresult_productcoefficientscoefficientsum_body_steps + fs_a_pfc_hresult_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_hresult_productcoefficientscoefficientresiduebound. pfa_gap_hresult_productcoefficientscoefficientresiduebound + S (pfc_value_hresult_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_hresult_productcoefficientscoefficientresiduecongruence pfa_offset_right_hresult_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_hresult_productcoefficientscoefficient) + (p) * pfa_offset_left_hresult_productcoefficientscoefficientresiduecongruence = (pfc_value_hresult_productcoefficients) + (p) * pfa_offset_right_hresult_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (x12)", "specialize prime_field_polynomial_convolution_at_length_exists (x13)", "specialize prime_field_polynomial_convolution_at_length_exists ((x2)+(x8))", "specialize prime_field_polynomial_convolution_at_length_exists (db)", "specialize prime_field_polynomial_convolution_at_length_exists (dc)", "specialize prime_field_polynomial_convolution_at_length_exists (J)", "specialize prime_field_polynomial_convolution_at_length_exists (x14)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hwbound_right_left", "exact hfirst_right_left", "exact hresult_length_witness", "cases hresult_product", "cases hresult_product_witness", "have htbound : forall fom_index_pfp_subtract_result_bound. (exists fom_gap_pfp_subtract_result_bound_index_bound. fom_gap_pfp_subtract_result_bound_index_bound + S (fom_index_pfp_subtract_result_bound) = x14) -> exists fom_value_pfp_subtract_result_bound. ((((exists fom_beta_height_pfp_subtract_result_bound_entry. fom_beta_height_pfp_subtract_result_bound_entry + S (fom_value_pfp_subtract_result_bound) = S ((S (fom_index_pfp_subtract_result_bound)) * x16)) /\\ exists fom_beta_quotient_pfp_subtract_result_bound_entry. x15 = fom_beta_quotient_pfp_subtract_result_bound_entry * S ((S (fom_index_pfp_subtract_result_bound)) * x16) + (fom_value_pfp_subtract_result_bound))) /\\ (exists fom_gap_pfp_subtract_result_bound_value_bound. fom_gap_pfp_subtract_result_bound_value_bound + S (fom_value_pfp_subtract_result_bound) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (x12)", "specialize prime_field_polynomial_convolution_bounded (x13)", "specialize prime_field_polynomial_convolution_bounded ((x2)+(x8))", "specialize prime_field_polynomial_convolution_bounded (db)", "specialize prime_field_polynomial_convolution_bounded (dc)", "specialize prime_field_polynomial_convolution_bounded (J)", "specialize prime_field_polynomial_convolution_bounded (x15)", "specialize prime_field_polynomial_convolution_bounded (x16)", "specialize prime_field_polynomial_convolution_bounded (x14)", "apply prime_field_polynomial_convolution_bounded", "exact hresult_product_witness_witness", "have hdistr : ((forall fom_index_pfp_subtract_distribution_left_bounded. (exists fom_gap_pfp_subtract_distribution_left_bounded_index_bound. fom_gap_pfp_subtract_distribution_left_bounded_index_bound + S (fom_index_pfp_subtract_distribution_left_bounded) = x11) -> exists fom_value_pfp_subtract_distribution_left_bounded. ((((exists fom_beta_height_pfp_subtract_distribution_left_bounded_entry. fom_beta_height_pfp_subtract_distribution_left_bounded_entry + S (fom_value_pfp_subtract_distribution_left_bounded) = S ((S (fom_index_pfp_subtract_distribution_left_bounded)) * x10)) /\\ exists fom_beta_quotient_pfp_subtract_distribution_left_bounded_entry. x9 = fom_beta_quotient_pfp_subtract_distribution_left_bounded_entry * S ((S (fom_index_pfp_subtract_distribution_left_bounded)) * x10) + (fom_value_pfp_subtract_distribution_left_bounded))) /\\ (exists fom_gap_pfp_subtract_distribution_left_bounded_value_bound. fom_gap_pfp_subtract_distribution_left_bounded_value_bound + S (fom_value_pfp_subtract_distribution_left_bounded) = p))) /\\ (((forall fom_index_pfp_subtract_distribution_right_bounded. (exists fom_gap_pfp_subtract_distribution_right_bounded_index_bound. fom_gap_pfp_subtract_distribution_right_bounded_index_bound + S (fom_index_pfp_subtract_distribution_right_bounded) = x14) -> exists fom_value_pfp_subtract_distribution_right_bounded. ((((exists fom_beta_height_pfp_subtract_distribution_right_bounded_entry. fom_beta_height_pfp_subtract_distribution_right_bounded_entry + S (fom_value_pfp_subtract_distribution_right_bounded) = S ((S (fom_index_pfp_subtract_distribution_right_bounded)) * x16)) /\\ exists fom_beta_quotient_pfp_subtract_distribution_right_bounded_entry. x15 = fom_beta_quotient_pfp_subtract_distribution_right_bounded_entry * S ((S (fom_index_pfp_subtract_distribution_right_bounded)) * x16) + (fom_value_pfp_subtract_distribution_right_bounded))) /\\ (exists fom_gap_pfp_subtract_distribution_right_bounded_value_bound. fom_gap_pfp_subtract_distribution_right_bounded_value_bound + S (fom_value_pfp_subtract_distribution_right_bounded) = p))) /\\ (((forall fom_index_pfp_subtract_distribution_result_bounded. (exists fom_gap_pfp_subtract_distribution_result_bounded_index_bound. fom_gap_pfp_subtract_distribution_result_bounded_index_bound + S (fom_index_pfp_subtract_distribution_result_bounded) = x5) -> exists fom_value_pfp_subtract_distribution_result_bounded. ((((exists fom_beta_height_pfp_subtract_distribution_result_bounded_entry. fom_beta_height_pfp_subtract_distribution_result_bounded_entry + S (fom_value_pfp_subtract_distribution_result_bounded) = S ((S (fom_index_pfp_subtract_distribution_result_bounded)) * x4)) /\\ exists fom_beta_quotient_pfp_subtract_distribution_result_bounded_entry. x3 = fom_beta_quotient_pfp_subtract_distribution_result_bounded_entry * S ((S (fom_index_pfp_subtract_distribution_result_bounded)) * x4) + (fom_value_pfp_subtract_distribution_result_bounded))) /\\ (exists fom_gap_pfp_subtract_distribution_result_bounded_value_bound. fom_gap_pfp_subtract_distribution_result_bounded_value_bound + S (fom_value_pfp_subtract_distribution_result_bounded) = p))) /\\ ((exists pfaa_left_b_subtract_distribution pfaa_left_c_subtract_distribution pfaa_right_b_subtract_distribution pfaa_right_c_subtract_distribution pfaa_sum_b_subtract_distribution pfaa_sum_c_subtract_distribution pfaa_length_subtract_distribution. ((((forall pfrep_power_subtract_distribution_witness_common_left pfrep_left_subtract_distribution_witness_common_left pfrep_right_subtract_distribution_witness_common_left. ((exists pfrep_position_subtract_distribution_witness_common_leftfirst. ((pfrep_position_subtract_distribution_witness_common_leftfirst+S (pfrep_power_subtract_distribution_witness_common_left)=(x11)) /\\ ((((exists ff_h_pfp_subtract_distribution_witness_common_leftfirstentry. ff_h_pfp_subtract_distribution_witness_common_leftfirstentry + S (pfrep_left_subtract_distribution_witness_common_left) = S ((S (pfrep_position_subtract_distribution_witness_common_leftfirst)) * x10)) /\\ exists ff_q_pfp_subtract_distribution_witness_common_leftfirstentry. x9 = ff_q_pfp_subtract_distribution_witness_common_leftfirstentry * S ((S (pfrep_position_subtract_distribution_witness_common_leftfirst)) * x10) + (pfrep_left_subtract_distribution_witness_common_left)))))) \\/ (((exists pfrep_gap_subtract_distribution_witness_common_leftfirstoutside. pfrep_gap_subtract_distribution_witness_common_leftfirstoutside+(x11)=(pfrep_power_subtract_distribution_witness_common_left)) /\\ (((pfrep_left_subtract_distribution_witness_common_left)=0))))) -> ((exists pfrep_position_subtract_distribution_witness_common_leftsecond. ((pfrep_position_subtract_distribution_witness_common_leftsecond+S (pfrep_power_subtract_distribution_witness_common_left)=(pfaa_length_subtract_distribution)) /\\ ((((exists ff_h_pfp_subtract_distribution_witness_common_leftsecondentry. ff_h_pfp_subtract_distribution_witness_common_leftsecondentry + S (pfrep_right_subtract_distribution_witness_common_left) = S ((S (pfrep_position_subtract_distribution_witness_common_leftsecond)) * pfaa_left_c_subtract_distribution)) /\\ exists ff_q_pfp_subtract_distribution_witness_common_leftsecondentry. pfaa_left_b_subtract_distribution = ff_q_pfp_subtract_distribution_witness_common_leftsecondentry * S ((S (pfrep_position_subtract_distribution_witness_common_leftsecond)) * pfaa_left_c_subtract_distribution) + (pfrep_right_subtract_distribution_witness_common_left)))))) \\/ (((exists pfrep_gap_subtract_distribution_witness_common_leftsecondoutside. pfrep_gap_subtract_distribution_witness_common_leftsecondoutside+(pfaa_length_subtract_distribution)=(pfrep_power_subtract_distribution_witness_common_left)) /\\ (((pfrep_right_subtract_distribution_witness_common_left)=0))))) -> pfrep_left_subtract_distribution_witness_common_left=pfrep_right_subtract_distribution_witness_common_left) /\\ ((forall pfrep_power_subtract_distribution_witness_common_right pfrep_left_subtract_distribution_witness_common_right pfrep_right_subtract_distribution_witness_common_right. ((exists pfrep_position_subtract_distribution_witness_common_rightfirst. ((pfrep_position_subtract_distribution_witness_common_rightfirst+S (pfrep_power_subtract_distribution_witness_common_right)=(x14)) /\\ ((((exists ff_h_pfp_subtract_distribution_witness_common_rightfirstentry. ff_h_pfp_subtract_distribution_witness_common_rightfirstentry + S (pfrep_left_subtract_distribution_witness_common_right) = S ((S (pfrep_position_subtract_distribution_witness_common_rightfirst)) * x16)) /\\ exists ff_q_pfp_subtract_distribution_witness_common_rightfirstentry. x15 = ff_q_pfp_subtract_distribution_witness_common_rightfirstentry * S ((S (pfrep_position_subtract_distribution_witness_common_rightfirst)) * x16) + (pfrep_left_subtract_distribution_witness_common_right)))))) \\/ (((exists pfrep_gap_subtract_distribution_witness_common_rightfirstoutside. pfrep_gap_subtract_distribution_witness_common_rightfirstoutside+(x14)=(pfrep_power_subtract_distribution_witness_common_right)) /\\ (((pfrep_left_subtract_distribution_witness_common_right)=0))))) -> ((exists pfrep_position_subtract_distribution_witness_common_rightsecond. ((pfrep_position_subtract_distribution_witness_common_rightsecond+S (pfrep_power_subtract_distribution_witness_common_right)=(pfaa_length_subtract_distribution)) /\\ ((((exists ff_h_pfp_subtract_distribution_witness_common_rightsecondentry. ff_h_pfp_subtract_distribution_witness_common_rightsecondentry + S (pfrep_right_subtract_distribution_witness_common_right) = S ((S (pfrep_position_subtract_distribution_witness_common_rightsecond)) * pfaa_right_c_subtract_distribution)) /\\ exists ff_q_pfp_subtract_distribution_witness_common_rightsecondentry. pfaa_right_b_subtract_distribution = ff_q_pfp_subtract_distribution_witness_common_rightsecondentry * S ((S (pfrep_position_subtract_distribution_witness_common_rightsecond)) * pfaa_right_c_subtract_distribution) + (pfrep_right_subtract_distribution_witness_common_right)))))) \\/ (((exists pfrep_gap_subtract_distribution_witness_common_rightsecondoutside. pfrep_gap_subtract_distribution_witness_common_rightsecondoutside+(pfaa_length_subtract_distribution)=(pfrep_power_subtract_distribution_witness_common_right)) /\\ (((pfrep_right_subtract_distribution_witness_common_right)=0))))) -> pfrep_left_subtract_distribution_witness_common_right=pfrep_right_subtract_distribution_witness_common_right)))) /\\ (((forall pfp_index_subtract_distribution_witness_operation. (exists pfa_gap_subtract_distribution_witness_operationindex. pfa_gap_subtract_distribution_witness_operationindex + S (pfp_index_subtract_distribution_witness_operation) = (pfaa_length_subtract_distribution)) -> exists pfp_left_subtract_distribution_witness_operation pfp_right_subtract_distribution_witness_operation pfp_value_subtract_distribution_witness_operation. ((((exists ff_h_pfp_subtract_distribution_witness_operationleft. ff_h_pfp_subtract_distribution_witness_operationleft + S (pfp_left_subtract_distribution_witness_operation) = S ((S (pfp_index_subtract_distribution_witness_operation)) * pfaa_left_c_subtract_distribution)) /\\ exists ff_q_pfp_subtract_distribution_witness_operationleft. pfaa_left_b_subtract_distribution = ff_q_pfp_subtract_distribution_witness_operationleft * S ((S (pfp_index_subtract_distribution_witness_operation)) * pfaa_left_c_subtract_distribution) + (pfp_left_subtract_distribution_witness_operation))) /\\ (((((exists ff_h_pfp_subtract_distribution_witness_operationright. ff_h_pfp_subtract_distribution_witness_operationright + S (pfp_right_subtract_distribution_witness_operation) = S ((S (pfp_index_subtract_distribution_witness_operation)) * pfaa_right_c_subtract_distribution)) /\\ exists ff_q_pfp_subtract_distribution_witness_operationright. pfaa_right_b_subtract_distribution = ff_q_pfp_subtract_distribution_witness_operationright * S ((S (pfp_index_subtract_distribution_witness_operation)) * pfaa_right_c_subtract_distribution) + (pfp_right_subtract_distribution_witness_operation))) /\\ (((((exists ff_h_pfp_subtract_distribution_witness_operationtarget. ff_h_pfp_subtract_distribution_witness_operationtarget + S (pfp_value_subtract_distribution_witness_operation) = S ((S (pfp_index_subtract_distribution_witness_operation)) * pfaa_sum_c_subtract_distribution)) /\\ exists ff_q_pfp_subtract_distribution_witness_operationtarget. pfaa_sum_b_subtract_distribution = ff_q_pfp_subtract_distribution_witness_operationtarget * S ((S (pfp_index_subtract_distribution_witness_operation)) * pfaa_sum_c_subtract_distribution) + (pfp_value_subtract_distribution_witness_operation))) /\\ ((((exists pfa_gap_subtract_distribution_witness_operationoperationleft. pfa_gap_subtract_distribution_witness_operationoperationleft + S (pfp_left_subtract_distribution_witness_operation) = (p)) /\\ (((exists pfa_gap_subtract_distribution_witness_operationoperationright. pfa_gap_subtract_distribution_witness_operationoperationright + S (pfp_right_subtract_distribution_witness_operation) = (p)) /\\ ((((exists pfa_gap_subtract_distribution_witness_operationoperationresultbound. pfa_gap_subtract_distribution_witness_operationoperationresultbound + S (pfp_value_subtract_distribution_witness_operation) = (p)) /\\ ((exists pfa_offset_left_subtract_distribution_witness_operationoperationresultcongruence pfa_offset_right_subtract_distribution_witness_operationoperationresultcongruence. ((pfp_left_subtract_distribution_witness_operation) + (pfp_right_subtract_distribution_witness_operation)) + (p) * pfa_offset_left_subtract_distribution_witness_operationoperationresultcongruence = (pfp_value_subtract_distribution_witness_operation) + (p) * pfa_offset_right_subtract_distribution_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_subtract_distribution_witness_output pfrep_left_subtract_distribution_witness_output pfrep_right_subtract_distribution_witness_output. ((exists pfrep_position_subtract_distribution_witness_outputfirst. ((pfrep_position_subtract_distribution_witness_outputfirst+S (pfrep_power_subtract_distribution_witness_output)=(pfaa_length_subtract_distribution)) /\\ ((((exists ff_h_pfp_subtract_distribution_witness_outputfirstentry. ff_h_pfp_subtract_distribution_witness_outputfirstentry + S (pfrep_left_subtract_distribution_witness_output) = S ((S (pfrep_position_subtract_distribution_witness_outputfirst)) * pfaa_sum_c_subtract_distribution)) /\\ exists ff_q_pfp_subtract_distribution_witness_outputfirstentry. pfaa_sum_b_subtract_distribution = ff_q_pfp_subtract_distribution_witness_outputfirstentry * S ((S (pfrep_position_subtract_distribution_witness_outputfirst)) * pfaa_sum_c_subtract_distribution) + (pfrep_left_subtract_distribution_witness_output)))))) \\/ (((exists pfrep_gap_subtract_distribution_witness_outputfirstoutside. pfrep_gap_subtract_distribution_witness_outputfirstoutside+(pfaa_length_subtract_distribution)=(pfrep_power_subtract_distribution_witness_output)) /\\ (((pfrep_left_subtract_distribution_witness_output)=0))))) -> ((exists pfrep_position_subtract_distribution_witness_outputsecond. ((pfrep_position_subtract_distribution_witness_outputsecond+S (pfrep_power_subtract_distribution_witness_output)=(x5)) /\\ ((((exists ff_h_pfp_subtract_distribution_witness_outputsecondentry. ff_h_pfp_subtract_distribution_witness_outputsecondentry + S (pfrep_right_subtract_distribution_witness_output) = S ((S (pfrep_position_subtract_distribution_witness_outputsecond)) * x4)) /\\ exists ff_q_pfp_subtract_distribution_witness_outputsecondentry. x3 = ff_q_pfp_subtract_distribution_witness_outputsecondentry * S ((S (pfrep_position_subtract_distribution_witness_outputsecond)) * x4) + (pfrep_right_subtract_distribution_witness_output)))))) \\/ (((exists pfrep_gap_subtract_distribution_witness_outputsecondoutside. pfrep_gap_subtract_distribution_witness_outputsecondoutside+(x5)=(pfrep_power_subtract_distribution_witness_output)) /\\ (((pfrep_right_subtract_distribution_witness_output)=0))))) -> pfrep_left_subtract_distribution_witness_output=pfrep_right_subtract_distribution_witness_output))))))))))))", "specialize prime_field_polynomial_aligned_convolution_right_add (p)", "specialize prime_field_polynomial_aligned_convolution_right_add (x6)", "specialize prime_field_polynomial_aligned_convolution_right_add (x7)", "specialize prime_field_polynomial_aligned_convolution_right_add (x8)", "specialize prime_field_polynomial_aligned_convolution_right_add (x12)", "specialize prime_field_polynomial_aligned_convolution_right_add (x13)", "specialize prime_field_polynomial_aligned_convolution_right_add ((x2)+(x8))", "specialize prime_field_polynomial_aligned_convolution_right_add (x)", "specialize prime_field_polynomial_aligned_convolution_right_add (x1)", "specialize prime_field_polynomial_aligned_convolution_right_add (x2)", "specialize prime_field_polynomial_aligned_convolution_right_add (db)", "specialize prime_field_polynomial_aligned_convolution_right_add (dc)", "specialize prime_field_polynomial_aligned_convolution_right_add (J)", "specialize prime_field_polynomial_aligned_convolution_right_add (x9)", "specialize prime_field_polynomial_aligned_convolution_right_add (x10)", "specialize prime_field_polynomial_aligned_convolution_right_add (x11)", "specialize prime_field_polynomial_aligned_convolution_right_add (x15)", "specialize prime_field_polynomial_aligned_convolution_right_add (x16)", "specialize prime_field_polynomial_aligned_convolution_right_add (x14)", "specialize prime_field_polynomial_aligned_convolution_right_add (x3)", "specialize prime_field_polynomial_aligned_convolution_right_add (x4)", "specialize prime_field_polynomial_aligned_convolution_right_add (x5)", "apply prime_field_polynomial_aligned_convolution_right_add", "exact hp", "exact hw_witness_witness", "exact hDB_right_witness_witness_witness_witness_witness_witness_left", "exact hresult_product_witness_witness", "exact hDA_right_witness_witness_witness_witness_witness_witness_left", "have hcompare : ((forall fom_index_pfp_subtract_comparison_left_bounded. (exists fom_gap_pfp_subtract_comparison_left_bounded_index_bound. fom_gap_pfp_subtract_comparison_left_bounded_index_bound + S (fom_index_pfp_subtract_comparison_left_bounded) = M) -> exists fom_value_pfp_subtract_comparison_left_bounded. ((((exists fom_beta_height_pfp_subtract_comparison_left_bounded_entry. fom_beta_height_pfp_subtract_comparison_left_bounded_entry + S (fom_value_pfp_subtract_comparison_left_bounded) = S ((S (fom_index_pfp_subtract_comparison_left_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_subtract_comparison_left_bounded_entry. bb = fom_beta_quotient_pfp_subtract_comparison_left_bounded_entry * S ((S (fom_index_pfp_subtract_comparison_left_bounded)) * bc) + (fom_value_pfp_subtract_comparison_left_bounded))) /\\ (exists fom_gap_pfp_subtract_comparison_left_bounded_value_bound. fom_gap_pfp_subtract_comparison_left_bounded_value_bound + S (fom_value_pfp_subtract_comparison_left_bounded) = p))) /\\ (((forall fom_index_pfp_subtract_comparison_right_bounded. (exists fom_gap_pfp_subtract_comparison_right_bounded_index_bound. fom_gap_pfp_subtract_comparison_right_bounded_index_bound + S (fom_index_pfp_subtract_comparison_right_bounded) = x14) -> exists fom_value_pfp_subtract_comparison_right_bounded. ((((exists fom_beta_height_pfp_subtract_comparison_right_bounded_entry. fom_beta_height_pfp_subtract_comparison_right_bounded_entry + S (fom_value_pfp_subtract_comparison_right_bounded) = S ((S (fom_index_pfp_subtract_comparison_right_bounded)) * x16)) /\\ exists fom_beta_quotient_pfp_subtract_comparison_right_bounded_entry. x15 = fom_beta_quotient_pfp_subtract_comparison_right_bounded_entry * S ((S (fom_index_pfp_subtract_comparison_right_bounded)) * x16) + (fom_value_pfp_subtract_comparison_right_bounded))) /\\ (exists fom_gap_pfp_subtract_comparison_right_bounded_value_bound. fom_gap_pfp_subtract_comparison_right_bounded_value_bound + S (fom_value_pfp_subtract_comparison_right_bounded) = p))) /\\ (((forall fom_index_pfp_subtract_comparison_result_bounded. (exists fom_gap_pfp_subtract_comparison_result_bounded_index_bound. fom_gap_pfp_subtract_comparison_result_bounded_index_bound + S (fom_index_pfp_subtract_comparison_result_bounded) = L) -> exists fom_value_pfp_subtract_comparison_result_bounded. ((((exists fom_beta_height_pfp_subtract_comparison_result_bounded_entry. fom_beta_height_pfp_subtract_comparison_result_bounded_entry + S (fom_value_pfp_subtract_comparison_result_bounded) = S ((S (fom_index_pfp_subtract_comparison_result_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_subtract_comparison_result_bounded_entry. ab = fom_beta_quotient_pfp_subtract_comparison_result_bounded_entry * S ((S (fom_index_pfp_subtract_comparison_result_bounded)) * ac) + (fom_value_pfp_subtract_comparison_result_bounded))) /\\ (exists fom_gap_pfp_subtract_comparison_result_bounded_value_bound. fom_gap_pfp_subtract_comparison_result_bounded_value_bound + S (fom_value_pfp_subtract_comparison_result_bounded) = p))) /\\ ((exists pfaa_left_b_subtract_comparison pfaa_left_c_subtract_comparison pfaa_right_b_subtract_comparison pfaa_right_c_subtract_comparison pfaa_sum_b_subtract_comparison pfaa_sum_c_subtract_comparison pfaa_length_subtract_comparison. ((((forall pfrep_power_subtract_comparison_witness_common_left pfrep_left_subtract_comparison_witness_common_left pfrep_right_subtract_comparison_witness_common_left. ((exists pfrep_position_subtract_comparison_witness_common_leftfirst. ((pfrep_position_subtract_comparison_witness_common_leftfirst+S (pfrep_power_subtract_comparison_witness_common_left)=(M)) /\\ ((((exists ff_h_pfp_subtract_comparison_witness_common_leftfirstentry. ff_h_pfp_subtract_comparison_witness_common_leftfirstentry + S (pfrep_left_subtract_comparison_witness_common_left) = S ((S (pfrep_position_subtract_comparison_witness_common_leftfirst)) * bc)) /\\ exists ff_q_pfp_subtract_comparison_witness_common_leftfirstentry. bb = ff_q_pfp_subtract_comparison_witness_common_leftfirstentry * S ((S (pfrep_position_subtract_comparison_witness_common_leftfirst)) * bc) + (pfrep_left_subtract_comparison_witness_common_left)))))) \\/ (((exists pfrep_gap_subtract_comparison_witness_common_leftfirstoutside. pfrep_gap_subtract_comparison_witness_common_leftfirstoutside+(M)=(pfrep_power_subtract_comparison_witness_common_left)) /\\ (((pfrep_left_subtract_comparison_witness_common_left)=0))))) -> ((exists pfrep_position_subtract_comparison_witness_common_leftsecond. ((pfrep_position_subtract_comparison_witness_common_leftsecond+S (pfrep_power_subtract_comparison_witness_common_left)=(pfaa_length_subtract_comparison)) /\\ ((((exists ff_h_pfp_subtract_comparison_witness_common_leftsecondentry. ff_h_pfp_subtract_comparison_witness_common_leftsecondentry + S (pfrep_right_subtract_comparison_witness_common_left) = S ((S (pfrep_position_subtract_comparison_witness_common_leftsecond)) * pfaa_left_c_subtract_comparison)) /\\ exists ff_q_pfp_subtract_comparison_witness_common_leftsecondentry. pfaa_left_b_subtract_comparison = ff_q_pfp_subtract_comparison_witness_common_leftsecondentry * S ((S (pfrep_position_subtract_comparison_witness_common_leftsecond)) * pfaa_left_c_subtract_comparison) + (pfrep_right_subtract_comparison_witness_common_left)))))) \\/ (((exists pfrep_gap_subtract_comparison_witness_common_leftsecondoutside. pfrep_gap_subtract_comparison_witness_common_leftsecondoutside+(pfaa_length_subtract_comparison)=(pfrep_power_subtract_comparison_witness_common_left)) /\\ (((pfrep_right_subtract_comparison_witness_common_left)=0))))) -> pfrep_left_subtract_comparison_witness_common_left=pfrep_right_subtract_comparison_witness_common_left) /\\ ((forall pfrep_power_subtract_comparison_witness_common_right pfrep_left_subtract_comparison_witness_common_right pfrep_right_subtract_comparison_witness_common_right. ((exists pfrep_position_subtract_comparison_witness_common_rightfirst. ((pfrep_position_subtract_comparison_witness_common_rightfirst+S (pfrep_power_subtract_comparison_witness_common_right)=(x14)) /\\ ((((exists ff_h_pfp_subtract_comparison_witness_common_rightfirstentry. ff_h_pfp_subtract_comparison_witness_common_rightfirstentry + S (pfrep_left_subtract_comparison_witness_common_right) = S ((S (pfrep_position_subtract_comparison_witness_common_rightfirst)) * x16)) /\\ exists ff_q_pfp_subtract_comparison_witness_common_rightfirstentry. x15 = ff_q_pfp_subtract_comparison_witness_common_rightfirstentry * S ((S (pfrep_position_subtract_comparison_witness_common_rightfirst)) * x16) + (pfrep_left_subtract_comparison_witness_common_right)))))) \\/ (((exists pfrep_gap_subtract_comparison_witness_common_rightfirstoutside. pfrep_gap_subtract_comparison_witness_common_rightfirstoutside+(x14)=(pfrep_power_subtract_comparison_witness_common_right)) /\\ (((pfrep_left_subtract_comparison_witness_common_right)=0))))) -> ((exists pfrep_position_subtract_comparison_witness_common_rightsecond. ((pfrep_position_subtract_comparison_witness_common_rightsecond+S (pfrep_power_subtract_comparison_witness_common_right)=(pfaa_length_subtract_comparison)) /\\ ((((exists ff_h_pfp_subtract_comparison_witness_common_rightsecondentry. ff_h_pfp_subtract_comparison_witness_common_rightsecondentry + S (pfrep_right_subtract_comparison_witness_common_right) = S ((S (pfrep_position_subtract_comparison_witness_common_rightsecond)) * pfaa_right_c_subtract_comparison)) /\\ exists ff_q_pfp_subtract_comparison_witness_common_rightsecondentry. pfaa_right_b_subtract_comparison = ff_q_pfp_subtract_comparison_witness_common_rightsecondentry * S ((S (pfrep_position_subtract_comparison_witness_common_rightsecond)) * pfaa_right_c_subtract_comparison) + (pfrep_right_subtract_comparison_witness_common_right)))))) \\/ (((exists pfrep_gap_subtract_comparison_witness_common_rightsecondoutside. pfrep_gap_subtract_comparison_witness_common_rightsecondoutside+(pfaa_length_subtract_comparison)=(pfrep_power_subtract_comparison_witness_common_right)) /\\ (((pfrep_right_subtract_comparison_witness_common_right)=0))))) -> pfrep_left_subtract_comparison_witness_common_right=pfrep_right_subtract_comparison_witness_common_right)))) /\\ (((forall pfp_index_subtract_comparison_witness_operation. (exists pfa_gap_subtract_comparison_witness_operationindex. pfa_gap_subtract_comparison_witness_operationindex + S (pfp_index_subtract_comparison_witness_operation) = (pfaa_length_subtract_comparison)) -> exists pfp_left_subtract_comparison_witness_operation pfp_right_subtract_comparison_witness_operation pfp_value_subtract_comparison_witness_operation. ((((exists ff_h_pfp_subtract_comparison_witness_operationleft. ff_h_pfp_subtract_comparison_witness_operationleft + S (pfp_left_subtract_comparison_witness_operation) = S ((S (pfp_index_subtract_comparison_witness_operation)) * pfaa_left_c_subtract_comparison)) /\\ exists ff_q_pfp_subtract_comparison_witness_operationleft. pfaa_left_b_subtract_comparison = ff_q_pfp_subtract_comparison_witness_operationleft * S ((S (pfp_index_subtract_comparison_witness_operation)) * pfaa_left_c_subtract_comparison) + (pfp_left_subtract_comparison_witness_operation))) /\\ (((((exists ff_h_pfp_subtract_comparison_witness_operationright. ff_h_pfp_subtract_comparison_witness_operationright + S (pfp_right_subtract_comparison_witness_operation) = S ((S (pfp_index_subtract_comparison_witness_operation)) * pfaa_right_c_subtract_comparison)) /\\ exists ff_q_pfp_subtract_comparison_witness_operationright. pfaa_right_b_subtract_comparison = ff_q_pfp_subtract_comparison_witness_operationright * S ((S (pfp_index_subtract_comparison_witness_operation)) * pfaa_right_c_subtract_comparison) + (pfp_right_subtract_comparison_witness_operation))) /\\ (((((exists ff_h_pfp_subtract_comparison_witness_operationtarget. ff_h_pfp_subtract_comparison_witness_operationtarget + S (pfp_value_subtract_comparison_witness_operation) = S ((S (pfp_index_subtract_comparison_witness_operation)) * pfaa_sum_c_subtract_comparison)) /\\ exists ff_q_pfp_subtract_comparison_witness_operationtarget. pfaa_sum_b_subtract_comparison = ff_q_pfp_subtract_comparison_witness_operationtarget * S ((S (pfp_index_subtract_comparison_witness_operation)) * pfaa_sum_c_subtract_comparison) + (pfp_value_subtract_comparison_witness_operation))) /\\ ((((exists pfa_gap_subtract_comparison_witness_operationoperationleft. pfa_gap_subtract_comparison_witness_operationoperationleft + S (pfp_left_subtract_comparison_witness_operation) = (p)) /\\ (((exists pfa_gap_subtract_comparison_witness_operationoperationright. pfa_gap_subtract_comparison_witness_operationoperationright + S (pfp_right_subtract_comparison_witness_operation) = (p)) /\\ ((((exists pfa_gap_subtract_comparison_witness_operationoperationresultbound. pfa_gap_subtract_comparison_witness_operationoperationresultbound + S (pfp_value_subtract_comparison_witness_operation) = (p)) /\\ ((exists pfa_offset_left_subtract_comparison_witness_operationoperationresultcongruence pfa_offset_right_subtract_comparison_witness_operationoperationresultcongruence. ((pfp_left_subtract_comparison_witness_operation) + (pfp_right_subtract_comparison_witness_operation)) + (p) * pfa_offset_left_subtract_comparison_witness_operationoperationresultcongruence = (pfp_value_subtract_comparison_witness_operation) + (p) * pfa_offset_right_subtract_comparison_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_subtract_comparison_witness_output pfrep_left_subtract_comparison_witness_output pfrep_right_subtract_comparison_witness_output. ((exists pfrep_position_subtract_comparison_witness_outputfirst. ((pfrep_position_subtract_comparison_witness_outputfirst+S (pfrep_power_subtract_comparison_witness_output)=(pfaa_length_subtract_comparison)) /\\ ((((exists ff_h_pfp_subtract_comparison_witness_outputfirstentry. ff_h_pfp_subtract_comparison_witness_outputfirstentry + S (pfrep_left_subtract_comparison_witness_output) = S ((S (pfrep_position_subtract_comparison_witness_outputfirst)) * pfaa_sum_c_subtract_comparison)) /\\ exists ff_q_pfp_subtract_comparison_witness_outputfirstentry. pfaa_sum_b_subtract_comparison = ff_q_pfp_subtract_comparison_witness_outputfirstentry * S ((S (pfrep_position_subtract_comparison_witness_outputfirst)) * pfaa_sum_c_subtract_comparison) + (pfrep_left_subtract_comparison_witness_output)))))) \\/ (((exists pfrep_gap_subtract_comparison_witness_outputfirstoutside. pfrep_gap_subtract_comparison_witness_outputfirstoutside+(pfaa_length_subtract_comparison)=(pfrep_power_subtract_comparison_witness_output)) /\\ (((pfrep_left_subtract_comparison_witness_output)=0))))) -> ((exists pfrep_position_subtract_comparison_witness_outputsecond. ((pfrep_position_subtract_comparison_witness_outputsecond+S (pfrep_power_subtract_comparison_witness_output)=(L)) /\\ ((((exists ff_h_pfp_subtract_comparison_witness_outputsecondentry. ff_h_pfp_subtract_comparison_witness_outputsecondentry + S (pfrep_right_subtract_comparison_witness_output) = S ((S (pfrep_position_subtract_comparison_witness_outputsecond)) * ac)) /\\ exists ff_q_pfp_subtract_comparison_witness_outputsecondentry. ab = ff_q_pfp_subtract_comparison_witness_outputsecondentry * S ((S (pfrep_position_subtract_comparison_witness_outputsecond)) * ac) + (pfrep_right_subtract_comparison_witness_output)))))) \\/ (((exists pfrep_gap_subtract_comparison_witness_outputsecondoutside. pfrep_gap_subtract_comparison_witness_outputsecondoutside+(L)=(pfrep_power_subtract_comparison_witness_output)) /\\ (((pfrep_right_subtract_comparison_witness_output)=0))))) -> pfrep_left_subtract_comparison_witness_output=pfrep_right_subtract_comparison_witness_output))))))))))))", "specialize prime_field_polynomial_aligned_add_transport (p)", "specialize prime_field_polynomial_aligned_add_transport (x9)", "specialize prime_field_polynomial_aligned_add_transport (x10)", "specialize prime_field_polynomial_aligned_add_transport (x11)", "specialize prime_field_polynomial_aligned_add_transport (x15)", "specialize prime_field_polynomial_aligned_add_transport (x16)", "specialize prime_field_polynomial_aligned_add_transport (x14)", "specialize prime_field_polynomial_aligned_add_transport (x3)", "specialize prime_field_polynomial_aligned_add_transport (x4)", "specialize prime_field_polynomial_aligned_add_transport (x5)", "specialize prime_field_polynomial_aligned_add_transport (bb)", "specialize prime_field_polynomial_aligned_add_transport (bc)", "specialize prime_field_polynomial_aligned_add_transport (M)", "specialize prime_field_polynomial_aligned_add_transport (x15)", "specialize prime_field_polynomial_aligned_add_transport (x16)", "specialize prime_field_polynomial_aligned_add_transport (x14)", "specialize prime_field_polynomial_aligned_add_transport (ab)", "specialize prime_field_polynomial_aligned_add_transport (ac)", "specialize prime_field_polynomial_aligned_add_transport (L)", "apply prime_field_polynomial_aligned_add_transport", "exact hDB_left", "exact htbound", "exact hDA_left", "specialize prime_field_polynomial_equivalent_symmetric (x9)", "specialize prime_field_polynomial_equivalent_symmetric (x10)", "specialize prime_field_polynomial_equivalent_symmetric (x11)", "specialize prime_field_polynomial_equivalent_symmetric (bb)", "specialize prime_field_polynomial_equivalent_symmetric (bc)", "specialize prime_field_polynomial_equivalent_symmetric (M)", "apply prime_field_polynomial_equivalent_symmetric", "exact hDB_right_witness_witness_witness_witness_witness_witness_right", "specialize prime_field_polynomial_power_coefficient_functional (x15)", "specialize prime_field_polynomial_power_coefficient_functional (x16)", "specialize prime_field_polynomial_power_coefficient_functional (x14)", "apply prime_field_polynomial_power_coefficient_functional", "exact hDA_right_witness_witness_witness_witness_witness_witness_right", "exact hdistr", "have heq : forall pfrep_power_subtract_output_equivalent pfrep_left_subtract_output_equivalent pfrep_right_subtract_output_equivalent. ((exists pfrep_position_subtract_output_equivalentfirst. ((pfrep_position_subtract_output_equivalentfirst+S (pfrep_power_subtract_output_equivalent)=(x14)) /\\ ((((exists ff_h_pfp_subtract_output_equivalentfirstentry. ff_h_pfp_subtract_output_equivalentfirstentry + S (pfrep_left_subtract_output_equivalent) = S ((S (pfrep_position_subtract_output_equivalentfirst)) * x16)) /\\ exists ff_q_pfp_subtract_output_equivalentfirstentry. x15 = ff_q_pfp_subtract_output_equivalentfirstentry * S ((S (pfrep_position_subtract_output_equivalentfirst)) * x16) + (pfrep_left_subtract_output_equivalent)))))) \\/ (((exists pfrep_gap_subtract_output_equivalentfirstoutside. pfrep_gap_subtract_output_equivalentfirstoutside+(x14)=(pfrep_power_subtract_output_equivalent)) /\\ (((pfrep_left_subtract_output_equivalent)=0))))) -> ((exists pfrep_position_subtract_output_equivalentsecond. ((pfrep_position_subtract_output_equivalentsecond+S (pfrep_power_subtract_output_equivalent)=(N)) /\\ ((((exists ff_h_pfp_subtract_output_equivalentsecondentry. ff_h_pfp_subtract_output_equivalentsecondentry + S (pfrep_right_subtract_output_equivalent) = S ((S (pfrep_position_subtract_output_equivalentsecond)) * rc)) /\\ exists ff_q_pfp_subtract_output_equivalentsecondentry. rb = ff_q_pfp_subtract_output_equivalentsecondentry * S ((S (pfrep_position_subtract_output_equivalentsecond)) * rc) + (pfrep_right_subtract_output_equivalent)))))) \\/ (((exists pfrep_gap_subtract_output_equivalentsecondoutside. pfrep_gap_subtract_output_equivalentsecondoutside+(N)=(pfrep_power_subtract_output_equivalent)) /\\ (((pfrep_right_subtract_output_equivalent)=0))))) -> pfrep_left_subtract_output_equivalent=pfrep_right_subtract_output_equivalent", "specialize prime_field_polynomial_aligned_add_cancel_left (p)", "specialize prime_field_polynomial_aligned_add_cancel_left (bb)", "specialize prime_field_polynomial_aligned_add_cancel_left (bc)", "specialize prime_field_polynomial_aligned_add_cancel_left (M)", "specialize prime_field_polynomial_aligned_add_cancel_left (x15)", "specialize prime_field_polynomial_aligned_add_cancel_left (x16)", "specialize prime_field_polynomial_aligned_add_cancel_left (x14)", "specialize prime_field_polynomial_aligned_add_cancel_left (rb)", "specialize prime_field_polynomial_aligned_add_cancel_left (rc)", "specialize prime_field_polynomial_aligned_add_cancel_left (N)", "specialize prime_field_polynomial_aligned_add_cancel_left (ab)", "specialize prime_field_polynomial_aligned_add_cancel_left (ac)", "specialize prime_field_polynomial_aligned_add_cancel_left (L)", "apply prime_field_polynomial_aligned_add_cancel_left", "exact hp", "exact hcompare", "exact hop", "have hopbound : ((forall fom_index_pfp_subtract_input_bound_0. (exists fom_gap_pfp_subtract_input_bound_0_index_bound. fom_gap_pfp_subtract_input_bound_0_index_bound + S (fom_index_pfp_subtract_input_bound_0) = M) -> exists fom_value_pfp_subtract_input_bound_0. ((((exists fom_beta_height_pfp_subtract_input_bound_0_entry. fom_beta_height_pfp_subtract_input_bound_0_entry + S (fom_value_pfp_subtract_input_bound_0) = S ((S (fom_index_pfp_subtract_input_bound_0)) * bc)) /\\ exists fom_beta_quotient_pfp_subtract_input_bound_0_entry. bb = fom_beta_quotient_pfp_subtract_input_bound_0_entry * S ((S (fom_index_pfp_subtract_input_bound_0)) * bc) + (fom_value_pfp_subtract_input_bound_0))) /\\ (exists fom_gap_pfp_subtract_input_bound_0_value_bound. fom_gap_pfp_subtract_input_bound_0_value_bound + S (fom_value_pfp_subtract_input_bound_0) = p))) /\\ (((forall fom_index_pfp_subtract_input_bound_1. (exists fom_gap_pfp_subtract_input_bound_1_index_bound. fom_gap_pfp_subtract_input_bound_1_index_bound + S (fom_index_pfp_subtract_input_bound_1) = N) -> exists fom_value_pfp_subtract_input_bound_1. ((((exists fom_beta_height_pfp_subtract_input_bound_1_entry. fom_beta_height_pfp_subtract_input_bound_1_entry + S (fom_value_pfp_subtract_input_bound_1) = S ((S (fom_index_pfp_subtract_input_bound_1)) * rc)) /\\ exists fom_beta_quotient_pfp_subtract_input_bound_1_entry. rb = fom_beta_quotient_pfp_subtract_input_bound_1_entry * S ((S (fom_index_pfp_subtract_input_bound_1)) * rc) + (fom_value_pfp_subtract_input_bound_1))) /\\ (exists fom_gap_pfp_subtract_input_bound_1_value_bound. fom_gap_pfp_subtract_input_bound_1_value_bound + S (fom_value_pfp_subtract_input_bound_1) = p))) /\\ ((forall fom_index_pfp_subtract_input_bound_2. (exists fom_gap_pfp_subtract_input_bound_2_index_bound. fom_gap_pfp_subtract_input_bound_2_index_bound + S (fom_index_pfp_subtract_input_bound_2) = L) -> exists fom_value_pfp_subtract_input_bound_2. ((((exists fom_beta_height_pfp_subtract_input_bound_2_entry. fom_beta_height_pfp_subtract_input_bound_2_entry + S (fom_value_pfp_subtract_input_bound_2) = S ((S (fom_index_pfp_subtract_input_bound_2)) * ac)) /\\ exists fom_beta_quotient_pfp_subtract_input_bound_2_entry. ab = fom_beta_quotient_pfp_subtract_input_bound_2_entry * S ((S (fom_index_pfp_subtract_input_bound_2)) * ac) + (fom_value_pfp_subtract_input_bound_2))) /\\ (exists fom_gap_pfp_subtract_input_bound_2_value_bound. fom_gap_pfp_subtract_input_bound_2_value_bound + S (fom_value_pfp_subtract_input_bound_2) = p)))))))", "specialize prime_field_polynomial_aligned_add_bounded (p)", "specialize prime_field_polynomial_aligned_add_bounded (bb)", "specialize prime_field_polynomial_aligned_add_bounded (bc)", "specialize prime_field_polynomial_aligned_add_bounded (M)", "specialize prime_field_polynomial_aligned_add_bounded (rb)", "specialize prime_field_polynomial_aligned_add_bounded (rc)", "specialize prime_field_polynomial_aligned_add_bounded (N)", "specialize prime_field_polynomial_aligned_add_bounded (ab)", "specialize prime_field_polynomial_aligned_add_bounded (ac)", "specialize prime_field_polynomial_aligned_add_bounded (L)", "apply prime_field_polynomial_aligned_add_bounded", "exact hop", "cases hopbound", "cases hopbound_right", "specialize prime_field_polynomial_right_divides_from_product (p)", "specialize prime_field_polynomial_right_divides_from_product (db)", "specialize prime_field_polynomial_right_divides_from_product (dc)", "specialize prime_field_polynomial_right_divides_from_product (J)", "specialize prime_field_polynomial_right_divides_from_product (rb)", "specialize prime_field_polynomial_right_divides_from_product (rc)", "specialize prime_field_polynomial_right_divides_from_product (N)", "specialize prime_field_polynomial_right_divides_from_product (x12)", "specialize prime_field_polynomial_right_divides_from_product (x13)", "specialize prime_field_polynomial_right_divides_from_product ((x2)+(x8))", "specialize prime_field_polynomial_right_divides_from_product (x15)", "specialize prime_field_polynomial_right_divides_from_product (x16)", "specialize prime_field_polynomial_right_divides_from_product (x14)", "apply prime_field_polynomial_right_divides_from_product", "exact hopbound_right_left", "exact hresult_product_witness_witness", "exact heq" ], "script_sha256": "b110e2e4e65f332e9b0d9b042ff67ca6d133511779181dac4661a6006200843f", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_euclidean_transport_candidate.py", "sha256": "9a589d1749eb38d30d1a24364bc4d66f7df0efb59247527f7831f97557da9c30" }, "statement": "forall p db dc J ab ac L bb bc M rb rc N. (~((p) = 1) /\\ forall pfa_factor_left_subtract_divides_prime pfa_factor_right_subtract_divides_prime. (p) = pfa_factor_left_subtract_divides_prime * pfa_factor_right_subtract_divides_prime -> pfa_factor_left_subtract_divides_prime = 1 \\/ pfa_factor_right_subtract_divides_prime = 1) -> (((forall fom_index_pfp_subtract_divides_left_canonical. (exists fom_gap_pfp_subtract_divides_left_canonical_index_bound. fom_gap_pfp_subtract_divides_left_canonical_index_bound + S (fom_index_pfp_subtract_divides_left_canonical) = L) -> exists fom_value_pfp_subtract_divides_left_canonical. ((((exists fom_beta_height_pfp_subtract_divides_left_canonical_entry. fom_beta_height_pfp_subtract_divides_left_canonical_entry + S (fom_value_pfp_subtract_divides_left_canonical) = S ((S (fom_index_pfp_subtract_divides_left_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_subtract_divides_left_canonical_entry. ab = fom_beta_quotient_pfp_subtract_divides_left_canonical_entry * S ((S (fom_index_pfp_subtract_divides_left_canonical)) * ac) + (fom_value_pfp_subtract_divides_left_canonical))) /\\ (exists fom_gap_pfp_subtract_divides_left_canonical_value_bound. fom_gap_pfp_subtract_divides_left_canonical_value_bound + S (fom_value_pfp_subtract_divides_left_canonical) = p))) /\\ ((exists pfrd_qb_subtract_divides_left pfrd_qc_subtract_divides_left pfrd_qlen_subtract_divides_left pfrd_pb_subtract_divides_left pfrd_pc_subtract_divides_left pfrd_plen_subtract_divides_left. ((((forall fom_index_pfp_subtract_divides_left_productleft. (exists fom_gap_pfp_subtract_divides_left_productleft_index_bound. fom_gap_pfp_subtract_divides_left_productleft_index_bound + S (fom_index_pfp_subtract_divides_left_productleft) = pfrd_qlen_subtract_divides_left) -> exists fom_value_pfp_subtract_divides_left_productleft. ((((exists fom_beta_height_pfp_subtract_divides_left_productleft_entry. fom_beta_height_pfp_subtract_divides_left_productleft_entry + S (fom_value_pfp_subtract_divides_left_productleft) = S ((S (fom_index_pfp_subtract_divides_left_productleft)) * pfrd_qc_subtract_divides_left)) /\\ exists fom_beta_quotient_pfp_subtract_divides_left_productleft_entry. pfrd_qb_subtract_divides_left = fom_beta_quotient_pfp_subtract_divides_left_productleft_entry * S ((S (fom_index_pfp_subtract_divides_left_productleft)) * pfrd_qc_subtract_divides_left) + (fom_value_pfp_subtract_divides_left_productleft))) /\\ (exists fom_gap_pfp_subtract_divides_left_productleft_value_bound. fom_gap_pfp_subtract_divides_left_productleft_value_bound + S (fom_value_pfp_subtract_divides_left_productleft) = p))) /\\ (((forall fom_index_pfp_subtract_divides_left_productright. (exists fom_gap_pfp_subtract_divides_left_productright_index_bound. fom_gap_pfp_subtract_divides_left_productright_index_bound + S (fom_index_pfp_subtract_divides_left_productright) = J) -> exists fom_value_pfp_subtract_divides_left_productright. ((((exists fom_beta_height_pfp_subtract_divides_left_productright_entry. fom_beta_height_pfp_subtract_divides_left_productright_entry + S (fom_value_pfp_subtract_divides_left_productright) = S ((S (fom_index_pfp_subtract_divides_left_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_subtract_divides_left_productright_entry. db = fom_beta_quotient_pfp_subtract_divides_left_productright_entry * S ((S (fom_index_pfp_subtract_divides_left_productright)) * dc) + (fom_value_pfp_subtract_divides_left_productright))) /\\ (exists fom_gap_pfp_subtract_divides_left_productright_value_bound. fom_gap_pfp_subtract_divides_left_productright_value_bound + S (fom_value_pfp_subtract_divides_left_productright) = p))) /\\ (((((((pfrd_qlen_subtract_divides_left)=0 \\/ (J)=0) /\\ (((pfrd_plen_subtract_divides_left)=0)))) \\/ (((~((pfrd_qlen_subtract_divides_left)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_subtract_divides_left)+(J)=S (pfrd_plen_subtract_divides_left)))))))) /\\ ((forall pfc_index_subtract_divides_left_productcoefficients. (exists pfa_gap_subtract_divides_left_productcoefficientsbound. pfa_gap_subtract_divides_left_productcoefficientsbound + S (pfc_index_subtract_divides_left_productcoefficients) = (pfrd_plen_subtract_divides_left)) -> exists pfc_value_subtract_divides_left_productcoefficients. ((((exists ff_h_pfp_subtract_divides_left_productcoefficientsentry. ff_h_pfp_subtract_divides_left_productcoefficientsentry + S (pfc_value_subtract_divides_left_productcoefficients) = S ((S (pfc_index_subtract_divides_left_productcoefficients)) * pfrd_pc_subtract_divides_left)) /\\ exists ff_q_pfp_subtract_divides_left_productcoefficientsentry. pfrd_pb_subtract_divides_left = ff_q_pfp_subtract_divides_left_productcoefficientsentry * S ((S (pfc_index_subtract_divides_left_productcoefficients)) * pfrd_pc_subtract_divides_left) + (pfc_value_subtract_divides_left_productcoefficients))) /\\ ((exists pfc_terms_code_subtract_divides_left_productcoefficientscoefficient pfc_terms_scale_subtract_divides_left_productcoefficientscoefficient pfc_natural_sum_subtract_divides_left_productcoefficientscoefficient. ((forall pfc_index_subtract_divides_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_subtract_divides_left_productcoefficientscoefficientdiagonalbound. pfa_gap_subtract_divides_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_subtract_divides_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_subtract_divides_left_productcoefficients))) -> exists pfc_value_subtract_divides_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_subtract_divides_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_subtract_divides_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_subtract_divides_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_subtract_divides_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_subtract_divides_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_subtract_divides_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_subtract_divides_left_productcoefficientscoefficient = ff_q_pfp_subtract_divides_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_subtract_divides_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_subtract_divides_left_productcoefficientscoefficient) + (pfc_value_subtract_divides_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_subtract_divides_left_productcoefficientscoefficientdiagonalterm pfc_left_subtract_divides_left_productcoefficientscoefficientdiagonalterm pfc_right_subtract_divides_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_subtract_divides_left_productcoefficientscoefficientdiagonal)+pfc_complement_subtract_divides_left_productcoefficientscoefficientdiagonalterm=(pfc_index_subtract_divides_left_productcoefficients)) /\\ ((((((exists pfa_gap_subtract_divides_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_subtract_divides_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_subtract_divides_left_productcoefficientscoefficientdiagonal) = (pfrd_qlen_subtract_divides_left)) /\\ ((((exists ff_h_pfp_subtract_divides_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_subtract_divides_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_subtract_divides_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_subtract_divides_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_subtract_divides_left)) /\\ exists ff_q_pfp_subtract_divides_left_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_subtract_divides_left = ff_q_pfp_subtract_divides_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_subtract_divides_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_subtract_divides_left) + (pfc_left_subtract_divides_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_subtract_divides_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_subtract_divides_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_subtract_divides_left)=(pfc_index_subtract_divides_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_subtract_divides_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_subtract_divides_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_subtract_divides_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_subtract_divides_left_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_subtract_divides_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_subtract_divides_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_subtract_divides_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_subtract_divides_left_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_subtract_divides_left_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_subtract_divides_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_subtract_divides_left_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_subtract_divides_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_subtract_divides_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_subtract_divides_left_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_subtract_divides_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_subtract_divides_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_subtract_divides_left_productcoefficientscoefficientdiagonal)=pfc_left_subtract_divides_left_productcoefficientscoefficientdiagonalterm*pfc_right_subtract_divides_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_subtract_divides_left_productcoefficientscoefficientsum fs_v_pfc_subtract_divides_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_subtract_divides_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_subtract_divides_left_productcoefficientscoefficientsum = fs_q_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_subtract_divides_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_subtract_divides_left_productcoefficientscoefficient) = S ((S (S (pfc_index_subtract_divides_left_productcoefficients))) * fs_v_pfc_subtract_divides_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_subtract_divides_left_productcoefficientscoefficientsum = fs_q_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_subtract_divides_left_productcoefficients))) * fs_v_pfc_subtract_divides_left_productcoefficientscoefficientsum) + (pfc_natural_sum_subtract_divides_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_subtract_divides_left_productcoefficients)) -> exists fs_a_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_subtract_divides_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_subtract_divides_left_productcoefficientscoefficient = fs_q_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_subtract_divides_left_productcoefficientscoefficient) + (fs_a_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_divides_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_subtract_divides_left_productcoefficientscoefficientsum = fs_q_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_divides_left_productcoefficientscoefficientsum) + (fs_r_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_divides_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_subtract_divides_left_productcoefficientscoefficientsum = fs_q_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_divides_left_productcoefficientscoefficientsum) + (fs_s_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_subtract_divides_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_subtract_divides_left_productcoefficientscoefficientresiduebound. pfa_gap_subtract_divides_left_productcoefficientscoefficientresiduebound + S (pfc_value_subtract_divides_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_subtract_divides_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_subtract_divides_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_subtract_divides_left_productcoefficientscoefficient) + (p) * pfa_offset_left_subtract_divides_left_productcoefficientscoefficientresiduecongruence = (pfc_value_subtract_divides_left_productcoefficients) + (p) * pfa_offset_right_subtract_divides_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_subtract_divides_left_target pfrep_left_subtract_divides_left_target pfrep_right_subtract_divides_left_target. ((exists pfrep_position_subtract_divides_left_targetfirst. ((pfrep_position_subtract_divides_left_targetfirst+S (pfrep_power_subtract_divides_left_target)=(pfrd_plen_subtract_divides_left)) /\\ ((((exists ff_h_pfp_subtract_divides_left_targetfirstentry. ff_h_pfp_subtract_divides_left_targetfirstentry + S (pfrep_left_subtract_divides_left_target) = S ((S (pfrep_position_subtract_divides_left_targetfirst)) * pfrd_pc_subtract_divides_left)) /\\ exists ff_q_pfp_subtract_divides_left_targetfirstentry. pfrd_pb_subtract_divides_left = ff_q_pfp_subtract_divides_left_targetfirstentry * S ((S (pfrep_position_subtract_divides_left_targetfirst)) * pfrd_pc_subtract_divides_left) + (pfrep_left_subtract_divides_left_target)))))) \\/ (((exists pfrep_gap_subtract_divides_left_targetfirstoutside. pfrep_gap_subtract_divides_left_targetfirstoutside+(pfrd_plen_subtract_divides_left)=(pfrep_power_subtract_divides_left_target)) /\\ (((pfrep_left_subtract_divides_left_target)=0))))) -> ((exists pfrep_position_subtract_divides_left_targetsecond. ((pfrep_position_subtract_divides_left_targetsecond+S (pfrep_power_subtract_divides_left_target)=(L)) /\\ ((((exists ff_h_pfp_subtract_divides_left_targetsecondentry. ff_h_pfp_subtract_divides_left_targetsecondentry + S (pfrep_right_subtract_divides_left_target) = S ((S (pfrep_position_subtract_divides_left_targetsecond)) * ac)) /\\ exists ff_q_pfp_subtract_divides_left_targetsecondentry. ab = ff_q_pfp_subtract_divides_left_targetsecondentry * S ((S (pfrep_position_subtract_divides_left_targetsecond)) * ac) + (pfrep_right_subtract_divides_left_target)))))) \\/ (((exists pfrep_gap_subtract_divides_left_targetsecondoutside. pfrep_gap_subtract_divides_left_targetsecondoutside+(L)=(pfrep_power_subtract_divides_left_target)) /\\ (((pfrep_right_subtract_divides_left_target)=0))))) -> pfrep_left_subtract_divides_left_target=pfrep_right_subtract_divides_left_target))))))) -> (((forall fom_index_pfp_subtract_divides_right_canonical. (exists fom_gap_pfp_subtract_divides_right_canonical_index_bound. fom_gap_pfp_subtract_divides_right_canonical_index_bound + S (fom_index_pfp_subtract_divides_right_canonical) = M) -> exists fom_value_pfp_subtract_divides_right_canonical. ((((exists fom_beta_height_pfp_subtract_divides_right_canonical_entry. fom_beta_height_pfp_subtract_divides_right_canonical_entry + S (fom_value_pfp_subtract_divides_right_canonical) = S ((S (fom_index_pfp_subtract_divides_right_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_subtract_divides_right_canonical_entry. bb = fom_beta_quotient_pfp_subtract_divides_right_canonical_entry * S ((S (fom_index_pfp_subtract_divides_right_canonical)) * bc) + (fom_value_pfp_subtract_divides_right_canonical))) /\\ (exists fom_gap_pfp_subtract_divides_right_canonical_value_bound. fom_gap_pfp_subtract_divides_right_canonical_value_bound + S (fom_value_pfp_subtract_divides_right_canonical) = p))) /\\ ((exists pfrd_qb_subtract_divides_right pfrd_qc_subtract_divides_right pfrd_qlen_subtract_divides_right pfrd_pb_subtract_divides_right pfrd_pc_subtract_divides_right pfrd_plen_subtract_divides_right. ((((forall fom_index_pfp_subtract_divides_right_productleft. (exists fom_gap_pfp_subtract_divides_right_productleft_index_bound. fom_gap_pfp_subtract_divides_right_productleft_index_bound + S (fom_index_pfp_subtract_divides_right_productleft) = pfrd_qlen_subtract_divides_right) -> exists fom_value_pfp_subtract_divides_right_productleft. ((((exists fom_beta_height_pfp_subtract_divides_right_productleft_entry. fom_beta_height_pfp_subtract_divides_right_productleft_entry + S (fom_value_pfp_subtract_divides_right_productleft) = S ((S (fom_index_pfp_subtract_divides_right_productleft)) * pfrd_qc_subtract_divides_right)) /\\ exists fom_beta_quotient_pfp_subtract_divides_right_productleft_entry. pfrd_qb_subtract_divides_right = fom_beta_quotient_pfp_subtract_divides_right_productleft_entry * S ((S (fom_index_pfp_subtract_divides_right_productleft)) * pfrd_qc_subtract_divides_right) + (fom_value_pfp_subtract_divides_right_productleft))) /\\ (exists fom_gap_pfp_subtract_divides_right_productleft_value_bound. fom_gap_pfp_subtract_divides_right_productleft_value_bound + S (fom_value_pfp_subtract_divides_right_productleft) = p))) /\\ (((forall fom_index_pfp_subtract_divides_right_productright. (exists fom_gap_pfp_subtract_divides_right_productright_index_bound. fom_gap_pfp_subtract_divides_right_productright_index_bound + S (fom_index_pfp_subtract_divides_right_productright) = J) -> exists fom_value_pfp_subtract_divides_right_productright. ((((exists fom_beta_height_pfp_subtract_divides_right_productright_entry. fom_beta_height_pfp_subtract_divides_right_productright_entry + S (fom_value_pfp_subtract_divides_right_productright) = S ((S (fom_index_pfp_subtract_divides_right_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_subtract_divides_right_productright_entry. db = fom_beta_quotient_pfp_subtract_divides_right_productright_entry * S ((S (fom_index_pfp_subtract_divides_right_productright)) * dc) + (fom_value_pfp_subtract_divides_right_productright))) /\\ (exists fom_gap_pfp_subtract_divides_right_productright_value_bound. fom_gap_pfp_subtract_divides_right_productright_value_bound + S (fom_value_pfp_subtract_divides_right_productright) = p))) /\\ (((((((pfrd_qlen_subtract_divides_right)=0 \\/ (J)=0) /\\ (((pfrd_plen_subtract_divides_right)=0)))) \\/ (((~((pfrd_qlen_subtract_divides_right)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_subtract_divides_right)+(J)=S (pfrd_plen_subtract_divides_right)))))))) /\\ ((forall pfc_index_subtract_divides_right_productcoefficients. (exists pfa_gap_subtract_divides_right_productcoefficientsbound. pfa_gap_subtract_divides_right_productcoefficientsbound + S (pfc_index_subtract_divides_right_productcoefficients) = (pfrd_plen_subtract_divides_right)) -> exists pfc_value_subtract_divides_right_productcoefficients. ((((exists ff_h_pfp_subtract_divides_right_productcoefficientsentry. ff_h_pfp_subtract_divides_right_productcoefficientsentry + S (pfc_value_subtract_divides_right_productcoefficients) = S ((S (pfc_index_subtract_divides_right_productcoefficients)) * pfrd_pc_subtract_divides_right)) /\\ exists ff_q_pfp_subtract_divides_right_productcoefficientsentry. pfrd_pb_subtract_divides_right = ff_q_pfp_subtract_divides_right_productcoefficientsentry * S ((S (pfc_index_subtract_divides_right_productcoefficients)) * pfrd_pc_subtract_divides_right) + (pfc_value_subtract_divides_right_productcoefficients))) /\\ ((exists pfc_terms_code_subtract_divides_right_productcoefficientscoefficient pfc_terms_scale_subtract_divides_right_productcoefficientscoefficient pfc_natural_sum_subtract_divides_right_productcoefficientscoefficient. ((forall pfc_index_subtract_divides_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_subtract_divides_right_productcoefficientscoefficientdiagonalbound. pfa_gap_subtract_divides_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_subtract_divides_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_subtract_divides_right_productcoefficients))) -> exists pfc_value_subtract_divides_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_subtract_divides_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_subtract_divides_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_subtract_divides_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_subtract_divides_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_subtract_divides_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_subtract_divides_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_subtract_divides_right_productcoefficientscoefficient = ff_q_pfp_subtract_divides_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_subtract_divides_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_subtract_divides_right_productcoefficientscoefficient) + (pfc_value_subtract_divides_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_subtract_divides_right_productcoefficientscoefficientdiagonalterm pfc_left_subtract_divides_right_productcoefficientscoefficientdiagonalterm pfc_right_subtract_divides_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_subtract_divides_right_productcoefficientscoefficientdiagonal)+pfc_complement_subtract_divides_right_productcoefficientscoefficientdiagonalterm=(pfc_index_subtract_divides_right_productcoefficients)) /\\ ((((((exists pfa_gap_subtract_divides_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_subtract_divides_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_subtract_divides_right_productcoefficientscoefficientdiagonal) = (pfrd_qlen_subtract_divides_right)) /\\ ((((exists ff_h_pfp_subtract_divides_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_subtract_divides_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_subtract_divides_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_subtract_divides_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_subtract_divides_right)) /\\ exists ff_q_pfp_subtract_divides_right_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_subtract_divides_right = ff_q_pfp_subtract_divides_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_subtract_divides_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_subtract_divides_right) + (pfc_left_subtract_divides_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_subtract_divides_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_subtract_divides_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_subtract_divides_right)=(pfc_index_subtract_divides_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_subtract_divides_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_subtract_divides_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_subtract_divides_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_subtract_divides_right_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_subtract_divides_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_subtract_divides_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_subtract_divides_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_subtract_divides_right_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_subtract_divides_right_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_subtract_divides_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_subtract_divides_right_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_subtract_divides_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_subtract_divides_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_subtract_divides_right_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_subtract_divides_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_subtract_divides_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_subtract_divides_right_productcoefficientscoefficientdiagonal)=pfc_left_subtract_divides_right_productcoefficientscoefficientdiagonalterm*pfc_right_subtract_divides_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_subtract_divides_right_productcoefficientscoefficientsum fs_v_pfc_subtract_divides_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_subtract_divides_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_subtract_divides_right_productcoefficientscoefficientsum = fs_q_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_subtract_divides_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_subtract_divides_right_productcoefficientscoefficient) = S ((S (S (pfc_index_subtract_divides_right_productcoefficients))) * fs_v_pfc_subtract_divides_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_subtract_divides_right_productcoefficientscoefficientsum = fs_q_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_subtract_divides_right_productcoefficients))) * fs_v_pfc_subtract_divides_right_productcoefficientscoefficientsum) + (pfc_natural_sum_subtract_divides_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_subtract_divides_right_productcoefficients)) -> exists fs_a_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_subtract_divides_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_subtract_divides_right_productcoefficientscoefficient = fs_q_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_subtract_divides_right_productcoefficientscoefficient) + (fs_a_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_divides_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_subtract_divides_right_productcoefficientscoefficientsum = fs_q_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_divides_right_productcoefficientscoefficientsum) + (fs_r_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_divides_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_subtract_divides_right_productcoefficientscoefficientsum = fs_q_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_divides_right_productcoefficientscoefficientsum) + (fs_s_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_subtract_divides_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_subtract_divides_right_productcoefficientscoefficientresiduebound. pfa_gap_subtract_divides_right_productcoefficientscoefficientresiduebound + S (pfc_value_subtract_divides_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_subtract_divides_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_subtract_divides_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_subtract_divides_right_productcoefficientscoefficient) + (p) * pfa_offset_left_subtract_divides_right_productcoefficientscoefficientresiduecongruence = (pfc_value_subtract_divides_right_productcoefficients) + (p) * pfa_offset_right_subtract_divides_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_subtract_divides_right_target pfrep_left_subtract_divides_right_target pfrep_right_subtract_divides_right_target. ((exists pfrep_position_subtract_divides_right_targetfirst. ((pfrep_position_subtract_divides_right_targetfirst+S (pfrep_power_subtract_divides_right_target)=(pfrd_plen_subtract_divides_right)) /\\ ((((exists ff_h_pfp_subtract_divides_right_targetfirstentry. ff_h_pfp_subtract_divides_right_targetfirstentry + S (pfrep_left_subtract_divides_right_target) = S ((S (pfrep_position_subtract_divides_right_targetfirst)) * pfrd_pc_subtract_divides_right)) /\\ exists ff_q_pfp_subtract_divides_right_targetfirstentry. pfrd_pb_subtract_divides_right = ff_q_pfp_subtract_divides_right_targetfirstentry * S ((S (pfrep_position_subtract_divides_right_targetfirst)) * pfrd_pc_subtract_divides_right) + (pfrep_left_subtract_divides_right_target)))))) \\/ (((exists pfrep_gap_subtract_divides_right_targetfirstoutside. pfrep_gap_subtract_divides_right_targetfirstoutside+(pfrd_plen_subtract_divides_right)=(pfrep_power_subtract_divides_right_target)) /\\ (((pfrep_left_subtract_divides_right_target)=0))))) -> ((exists pfrep_position_subtract_divides_right_targetsecond. ((pfrep_position_subtract_divides_right_targetsecond+S (pfrep_power_subtract_divides_right_target)=(M)) /\\ ((((exists ff_h_pfp_subtract_divides_right_targetsecondentry. ff_h_pfp_subtract_divides_right_targetsecondentry + S (pfrep_right_subtract_divides_right_target) = S ((S (pfrep_position_subtract_divides_right_targetsecond)) * bc)) /\\ exists ff_q_pfp_subtract_divides_right_targetsecondentry. bb = ff_q_pfp_subtract_divides_right_targetsecondentry * S ((S (pfrep_position_subtract_divides_right_targetsecond)) * bc) + (pfrep_right_subtract_divides_right_target)))))) \\/ (((exists pfrep_gap_subtract_divides_right_targetsecondoutside. pfrep_gap_subtract_divides_right_targetsecondoutside+(M)=(pfrep_power_subtract_divides_right_target)) /\\ (((pfrep_right_subtract_divides_right_target)=0))))) -> pfrep_left_subtract_divides_right_target=pfrep_right_subtract_divides_right_target))))))) -> (((forall fom_index_pfp_subtract_divides_operation_left_bounded. (exists fom_gap_pfp_subtract_divides_operation_left_bounded_index_bound. fom_gap_pfp_subtract_divides_operation_left_bounded_index_bound + S (fom_index_pfp_subtract_divides_operation_left_bounded) = M) -> exists fom_value_pfp_subtract_divides_operation_left_bounded. ((((exists fom_beta_height_pfp_subtract_divides_operation_left_bounded_entry. fom_beta_height_pfp_subtract_divides_operation_left_bounded_entry + S (fom_value_pfp_subtract_divides_operation_left_bounded) = S ((S (fom_index_pfp_subtract_divides_operation_left_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_subtract_divides_operation_left_bounded_entry. bb = fom_beta_quotient_pfp_subtract_divides_operation_left_bounded_entry * S ((S (fom_index_pfp_subtract_divides_operation_left_bounded)) * bc) + (fom_value_pfp_subtract_divides_operation_left_bounded))) /\\ (exists fom_gap_pfp_subtract_divides_operation_left_bounded_value_bound. fom_gap_pfp_subtract_divides_operation_left_bounded_value_bound + S (fom_value_pfp_subtract_divides_operation_left_bounded) = p))) /\\ (((forall fom_index_pfp_subtract_divides_operation_right_bounded. (exists fom_gap_pfp_subtract_divides_operation_right_bounded_index_bound. fom_gap_pfp_subtract_divides_operation_right_bounded_index_bound + S (fom_index_pfp_subtract_divides_operation_right_bounded) = N) -> exists fom_value_pfp_subtract_divides_operation_right_bounded. ((((exists fom_beta_height_pfp_subtract_divides_operation_right_bounded_entry. fom_beta_height_pfp_subtract_divides_operation_right_bounded_entry + S (fom_value_pfp_subtract_divides_operation_right_bounded) = S ((S (fom_index_pfp_subtract_divides_operation_right_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_subtract_divides_operation_right_bounded_entry. rb = fom_beta_quotient_pfp_subtract_divides_operation_right_bounded_entry * S ((S (fom_index_pfp_subtract_divides_operation_right_bounded)) * rc) + (fom_value_pfp_subtract_divides_operation_right_bounded))) /\\ (exists fom_gap_pfp_subtract_divides_operation_right_bounded_value_bound. fom_gap_pfp_subtract_divides_operation_right_bounded_value_bound + S (fom_value_pfp_subtract_divides_operation_right_bounded) = p))) /\\ (((forall fom_index_pfp_subtract_divides_operation_result_bounded. (exists fom_gap_pfp_subtract_divides_operation_result_bounded_index_bound. fom_gap_pfp_subtract_divides_operation_result_bounded_index_bound + S (fom_index_pfp_subtract_divides_operation_result_bounded) = L) -> exists fom_value_pfp_subtract_divides_operation_result_bounded. ((((exists fom_beta_height_pfp_subtract_divides_operation_result_bounded_entry. fom_beta_height_pfp_subtract_divides_operation_result_bounded_entry + S (fom_value_pfp_subtract_divides_operation_result_bounded) = S ((S (fom_index_pfp_subtract_divides_operation_result_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_subtract_divides_operation_result_bounded_entry. ab = fom_beta_quotient_pfp_subtract_divides_operation_result_bounded_entry * S ((S (fom_index_pfp_subtract_divides_operation_result_bounded)) * ac) + (fom_value_pfp_subtract_divides_operation_result_bounded))) /\\ (exists fom_gap_pfp_subtract_divides_operation_result_bounded_value_bound. fom_gap_pfp_subtract_divides_operation_result_bounded_value_bound + S (fom_value_pfp_subtract_divides_operation_result_bounded) = p))) /\\ ((exists pfaa_left_b_subtract_divides_operation pfaa_left_c_subtract_divides_operation pfaa_right_b_subtract_divides_operation pfaa_right_c_subtract_divides_operation pfaa_sum_b_subtract_divides_operation pfaa_sum_c_subtract_divides_operation pfaa_length_subtract_divides_operation. ((((forall pfrep_power_subtract_divides_operation_witness_common_left pfrep_left_subtract_divides_operation_witness_common_left pfrep_right_subtract_divides_operation_witness_common_left. ((exists pfrep_position_subtract_divides_operation_witness_common_leftfirst. ((pfrep_position_subtract_divides_operation_witness_common_leftfirst+S (pfrep_power_subtract_divides_operation_witness_common_left)=(M)) /\\ ((((exists ff_h_pfp_subtract_divides_operation_witness_common_leftfirstentry. ff_h_pfp_subtract_divides_operation_witness_common_leftfirstentry + S (pfrep_left_subtract_divides_operation_witness_common_left) = S ((S (pfrep_position_subtract_divides_operation_witness_common_leftfirst)) * bc)) /\\ exists ff_q_pfp_subtract_divides_operation_witness_common_leftfirstentry. bb = ff_q_pfp_subtract_divides_operation_witness_common_leftfirstentry * S ((S (pfrep_position_subtract_divides_operation_witness_common_leftfirst)) * bc) + (pfrep_left_subtract_divides_operation_witness_common_left)))))) \\/ (((exists pfrep_gap_subtract_divides_operation_witness_common_leftfirstoutside. pfrep_gap_subtract_divides_operation_witness_common_leftfirstoutside+(M)=(pfrep_power_subtract_divides_operation_witness_common_left)) /\\ (((pfrep_left_subtract_divides_operation_witness_common_left)=0))))) -> ((exists pfrep_position_subtract_divides_operation_witness_common_leftsecond. ((pfrep_position_subtract_divides_operation_witness_common_leftsecond+S (pfrep_power_subtract_divides_operation_witness_common_left)=(pfaa_length_subtract_divides_operation)) /\\ ((((exists ff_h_pfp_subtract_divides_operation_witness_common_leftsecondentry. ff_h_pfp_subtract_divides_operation_witness_common_leftsecondentry + S (pfrep_right_subtract_divides_operation_witness_common_left) = S ((S (pfrep_position_subtract_divides_operation_witness_common_leftsecond)) * pfaa_left_c_subtract_divides_operation)) /\\ exists ff_q_pfp_subtract_divides_operation_witness_common_leftsecondentry. pfaa_left_b_subtract_divides_operation = ff_q_pfp_subtract_divides_operation_witness_common_leftsecondentry * S ((S (pfrep_position_subtract_divides_operation_witness_common_leftsecond)) * pfaa_left_c_subtract_divides_operation) + (pfrep_right_subtract_divides_operation_witness_common_left)))))) \\/ (((exists pfrep_gap_subtract_divides_operation_witness_common_leftsecondoutside. pfrep_gap_subtract_divides_operation_witness_common_leftsecondoutside+(pfaa_length_subtract_divides_operation)=(pfrep_power_subtract_divides_operation_witness_common_left)) /\\ (((pfrep_right_subtract_divides_operation_witness_common_left)=0))))) -> pfrep_left_subtract_divides_operation_witness_common_left=pfrep_right_subtract_divides_operation_witness_common_left) /\\ ((forall pfrep_power_subtract_divides_operation_witness_common_right pfrep_left_subtract_divides_operation_witness_common_right pfrep_right_subtract_divides_operation_witness_common_right. ((exists pfrep_position_subtract_divides_operation_witness_common_rightfirst. ((pfrep_position_subtract_divides_operation_witness_common_rightfirst+S (pfrep_power_subtract_divides_operation_witness_common_right)=(N)) /\\ ((((exists ff_h_pfp_subtract_divides_operation_witness_common_rightfirstentry. ff_h_pfp_subtract_divides_operation_witness_common_rightfirstentry + S (pfrep_left_subtract_divides_operation_witness_common_right) = S ((S (pfrep_position_subtract_divides_operation_witness_common_rightfirst)) * rc)) /\\ exists ff_q_pfp_subtract_divides_operation_witness_common_rightfirstentry. rb = ff_q_pfp_subtract_divides_operation_witness_common_rightfirstentry * S ((S (pfrep_position_subtract_divides_operation_witness_common_rightfirst)) * rc) + (pfrep_left_subtract_divides_operation_witness_common_right)))))) \\/ (((exists pfrep_gap_subtract_divides_operation_witness_common_rightfirstoutside. pfrep_gap_subtract_divides_operation_witness_common_rightfirstoutside+(N)=(pfrep_power_subtract_divides_operation_witness_common_right)) /\\ (((pfrep_left_subtract_divides_operation_witness_common_right)=0))))) -> ((exists pfrep_position_subtract_divides_operation_witness_common_rightsecond. ((pfrep_position_subtract_divides_operation_witness_common_rightsecond+S (pfrep_power_subtract_divides_operation_witness_common_right)=(pfaa_length_subtract_divides_operation)) /\\ ((((exists ff_h_pfp_subtract_divides_operation_witness_common_rightsecondentry. ff_h_pfp_subtract_divides_operation_witness_common_rightsecondentry + S (pfrep_right_subtract_divides_operation_witness_common_right) = S ((S (pfrep_position_subtract_divides_operation_witness_common_rightsecond)) * pfaa_right_c_subtract_divides_operation)) /\\ exists ff_q_pfp_subtract_divides_operation_witness_common_rightsecondentry. pfaa_right_b_subtract_divides_operation = ff_q_pfp_subtract_divides_operation_witness_common_rightsecondentry * S ((S (pfrep_position_subtract_divides_operation_witness_common_rightsecond)) * pfaa_right_c_subtract_divides_operation) + (pfrep_right_subtract_divides_operation_witness_common_right)))))) \\/ (((exists pfrep_gap_subtract_divides_operation_witness_common_rightsecondoutside. pfrep_gap_subtract_divides_operation_witness_common_rightsecondoutside+(pfaa_length_subtract_divides_operation)=(pfrep_power_subtract_divides_operation_witness_common_right)) /\\ (((pfrep_right_subtract_divides_operation_witness_common_right)=0))))) -> pfrep_left_subtract_divides_operation_witness_common_right=pfrep_right_subtract_divides_operation_witness_common_right)))) /\\ (((forall pfp_index_subtract_divides_operation_witness_operation. (exists pfa_gap_subtract_divides_operation_witness_operationindex. pfa_gap_subtract_divides_operation_witness_operationindex + S (pfp_index_subtract_divides_operation_witness_operation) = (pfaa_length_subtract_divides_operation)) -> exists pfp_left_subtract_divides_operation_witness_operation pfp_right_subtract_divides_operation_witness_operation pfp_value_subtract_divides_operation_witness_operation. ((((exists ff_h_pfp_subtract_divides_operation_witness_operationleft. ff_h_pfp_subtract_divides_operation_witness_operationleft + S (pfp_left_subtract_divides_operation_witness_operation) = S ((S (pfp_index_subtract_divides_operation_witness_operation)) * pfaa_left_c_subtract_divides_operation)) /\\ exists ff_q_pfp_subtract_divides_operation_witness_operationleft. pfaa_left_b_subtract_divides_operation = ff_q_pfp_subtract_divides_operation_witness_operationleft * S ((S (pfp_index_subtract_divides_operation_witness_operation)) * pfaa_left_c_subtract_divides_operation) + (pfp_left_subtract_divides_operation_witness_operation))) /\\ (((((exists ff_h_pfp_subtract_divides_operation_witness_operationright. ff_h_pfp_subtract_divides_operation_witness_operationright + S (pfp_right_subtract_divides_operation_witness_operation) = S ((S (pfp_index_subtract_divides_operation_witness_operation)) * pfaa_right_c_subtract_divides_operation)) /\\ exists ff_q_pfp_subtract_divides_operation_witness_operationright. pfaa_right_b_subtract_divides_operation = ff_q_pfp_subtract_divides_operation_witness_operationright * S ((S (pfp_index_subtract_divides_operation_witness_operation)) * pfaa_right_c_subtract_divides_operation) + (pfp_right_subtract_divides_operation_witness_operation))) /\\ (((((exists ff_h_pfp_subtract_divides_operation_witness_operationtarget. ff_h_pfp_subtract_divides_operation_witness_operationtarget + S (pfp_value_subtract_divides_operation_witness_operation) = S ((S (pfp_index_subtract_divides_operation_witness_operation)) * pfaa_sum_c_subtract_divides_operation)) /\\ exists ff_q_pfp_subtract_divides_operation_witness_operationtarget. pfaa_sum_b_subtract_divides_operation = ff_q_pfp_subtract_divides_operation_witness_operationtarget * S ((S (pfp_index_subtract_divides_operation_witness_operation)) * pfaa_sum_c_subtract_divides_operation) + (pfp_value_subtract_divides_operation_witness_operation))) /\\ ((((exists pfa_gap_subtract_divides_operation_witness_operationoperationleft. pfa_gap_subtract_divides_operation_witness_operationoperationleft + S (pfp_left_subtract_divides_operation_witness_operation) = (p)) /\\ (((exists pfa_gap_subtract_divides_operation_witness_operationoperationright. pfa_gap_subtract_divides_operation_witness_operationoperationright + S (pfp_right_subtract_divides_operation_witness_operation) = (p)) /\\ ((((exists pfa_gap_subtract_divides_operation_witness_operationoperationresultbound. pfa_gap_subtract_divides_operation_witness_operationoperationresultbound + S (pfp_value_subtract_divides_operation_witness_operation) = (p)) /\\ ((exists pfa_offset_left_subtract_divides_operation_witness_operationoperationresultcongruence pfa_offset_right_subtract_divides_operation_witness_operationoperationresultcongruence. ((pfp_left_subtract_divides_operation_witness_operation) + (pfp_right_subtract_divides_operation_witness_operation)) + (p) * pfa_offset_left_subtract_divides_operation_witness_operationoperationresultcongruence = (pfp_value_subtract_divides_operation_witness_operation) + (p) * pfa_offset_right_subtract_divides_operation_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_subtract_divides_operation_witness_output pfrep_left_subtract_divides_operation_witness_output pfrep_right_subtract_divides_operation_witness_output. ((exists pfrep_position_subtract_divides_operation_witness_outputfirst. ((pfrep_position_subtract_divides_operation_witness_outputfirst+S (pfrep_power_subtract_divides_operation_witness_output)=(pfaa_length_subtract_divides_operation)) /\\ ((((exists ff_h_pfp_subtract_divides_operation_witness_outputfirstentry. ff_h_pfp_subtract_divides_operation_witness_outputfirstentry + S (pfrep_left_subtract_divides_operation_witness_output) = S ((S (pfrep_position_subtract_divides_operation_witness_outputfirst)) * pfaa_sum_c_subtract_divides_operation)) /\\ exists ff_q_pfp_subtract_divides_operation_witness_outputfirstentry. pfaa_sum_b_subtract_divides_operation = ff_q_pfp_subtract_divides_operation_witness_outputfirstentry * S ((S (pfrep_position_subtract_divides_operation_witness_outputfirst)) * pfaa_sum_c_subtract_divides_operation) + (pfrep_left_subtract_divides_operation_witness_output)))))) \\/ (((exists pfrep_gap_subtract_divides_operation_witness_outputfirstoutside. pfrep_gap_subtract_divides_operation_witness_outputfirstoutside+(pfaa_length_subtract_divides_operation)=(pfrep_power_subtract_divides_operation_witness_output)) /\\ (((pfrep_left_subtract_divides_operation_witness_output)=0))))) -> ((exists pfrep_position_subtract_divides_operation_witness_outputsecond. ((pfrep_position_subtract_divides_operation_witness_outputsecond+S (pfrep_power_subtract_divides_operation_witness_output)=(L)) /\\ ((((exists ff_h_pfp_subtract_divides_operation_witness_outputsecondentry. ff_h_pfp_subtract_divides_operation_witness_outputsecondentry + S (pfrep_right_subtract_divides_operation_witness_output) = S ((S (pfrep_position_subtract_divides_operation_witness_outputsecond)) * ac)) /\\ exists ff_q_pfp_subtract_divides_operation_witness_outputsecondentry. ab = ff_q_pfp_subtract_divides_operation_witness_outputsecondentry * S ((S (pfrep_position_subtract_divides_operation_witness_outputsecond)) * ac) + (pfrep_right_subtract_divides_operation_witness_output)))))) \\/ (((exists pfrep_gap_subtract_divides_operation_witness_outputsecondoutside. pfrep_gap_subtract_divides_operation_witness_outputsecondoutside+(L)=(pfrep_power_subtract_divides_operation_witness_output)) /\\ (((pfrep_right_subtract_divides_operation_witness_output)=0))))) -> pfrep_left_subtract_divides_operation_witness_output=pfrep_right_subtract_divides_operation_witness_output))))))))))))) -> (((forall fom_index_pfp_subtract_divides_result_canonical. (exists fom_gap_pfp_subtract_divides_result_canonical_index_bound. fom_gap_pfp_subtract_divides_result_canonical_index_bound + S (fom_index_pfp_subtract_divides_result_canonical) = N) -> exists fom_value_pfp_subtract_divides_result_canonical. ((((exists fom_beta_height_pfp_subtract_divides_result_canonical_entry. fom_beta_height_pfp_subtract_divides_result_canonical_entry + S (fom_value_pfp_subtract_divides_result_canonical) = S ((S (fom_index_pfp_subtract_divides_result_canonical)) * rc)) /\\ exists fom_beta_quotient_pfp_subtract_divides_result_canonical_entry. rb = fom_beta_quotient_pfp_subtract_divides_result_canonical_entry * S ((S (fom_index_pfp_subtract_divides_result_canonical)) * rc) + (fom_value_pfp_subtract_divides_result_canonical))) /\\ (exists fom_gap_pfp_subtract_divides_result_canonical_value_bound. fom_gap_pfp_subtract_divides_result_canonical_value_bound + S (fom_value_pfp_subtract_divides_result_canonical) = p))) /\\ ((exists pfrd_qb_subtract_divides_result pfrd_qc_subtract_divides_result pfrd_qlen_subtract_divides_result pfrd_pb_subtract_divides_result pfrd_pc_subtract_divides_result pfrd_plen_subtract_divides_result. ((((forall fom_index_pfp_subtract_divides_result_productleft. (exists fom_gap_pfp_subtract_divides_result_productleft_index_bound. fom_gap_pfp_subtract_divides_result_productleft_index_bound + S (fom_index_pfp_subtract_divides_result_productleft) = pfrd_qlen_subtract_divides_result) -> exists fom_value_pfp_subtract_divides_result_productleft. ((((exists fom_beta_height_pfp_subtract_divides_result_productleft_entry. fom_beta_height_pfp_subtract_divides_result_productleft_entry + S (fom_value_pfp_subtract_divides_result_productleft) = S ((S (fom_index_pfp_subtract_divides_result_productleft)) * pfrd_qc_subtract_divides_result)) /\\ exists fom_beta_quotient_pfp_subtract_divides_result_productleft_entry. pfrd_qb_subtract_divides_result = fom_beta_quotient_pfp_subtract_divides_result_productleft_entry * S ((S (fom_index_pfp_subtract_divides_result_productleft)) * pfrd_qc_subtract_divides_result) + (fom_value_pfp_subtract_divides_result_productleft))) /\\ (exists fom_gap_pfp_subtract_divides_result_productleft_value_bound. fom_gap_pfp_subtract_divides_result_productleft_value_bound + S (fom_value_pfp_subtract_divides_result_productleft) = p))) /\\ (((forall fom_index_pfp_subtract_divides_result_productright. (exists fom_gap_pfp_subtract_divides_result_productright_index_bound. fom_gap_pfp_subtract_divides_result_productright_index_bound + S (fom_index_pfp_subtract_divides_result_productright) = J) -> exists fom_value_pfp_subtract_divides_result_productright. ((((exists fom_beta_height_pfp_subtract_divides_result_productright_entry. fom_beta_height_pfp_subtract_divides_result_productright_entry + S (fom_value_pfp_subtract_divides_result_productright) = S ((S (fom_index_pfp_subtract_divides_result_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_subtract_divides_result_productright_entry. db = fom_beta_quotient_pfp_subtract_divides_result_productright_entry * S ((S (fom_index_pfp_subtract_divides_result_productright)) * dc) + (fom_value_pfp_subtract_divides_result_productright))) /\\ (exists fom_gap_pfp_subtract_divides_result_productright_value_bound. fom_gap_pfp_subtract_divides_result_productright_value_bound + S (fom_value_pfp_subtract_divides_result_productright) = p))) /\\ (((((((pfrd_qlen_subtract_divides_result)=0 \\/ (J)=0) /\\ (((pfrd_plen_subtract_divides_result)=0)))) \\/ (((~((pfrd_qlen_subtract_divides_result)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_subtract_divides_result)+(J)=S (pfrd_plen_subtract_divides_result)))))))) /\\ ((forall pfc_index_subtract_divides_result_productcoefficients. (exists pfa_gap_subtract_divides_result_productcoefficientsbound. pfa_gap_subtract_divides_result_productcoefficientsbound + S (pfc_index_subtract_divides_result_productcoefficients) = (pfrd_plen_subtract_divides_result)) -> exists pfc_value_subtract_divides_result_productcoefficients. ((((exists ff_h_pfp_subtract_divides_result_productcoefficientsentry. ff_h_pfp_subtract_divides_result_productcoefficientsentry + S (pfc_value_subtract_divides_result_productcoefficients) = S ((S (pfc_index_subtract_divides_result_productcoefficients)) * pfrd_pc_subtract_divides_result)) /\\ exists ff_q_pfp_subtract_divides_result_productcoefficientsentry. pfrd_pb_subtract_divides_result = ff_q_pfp_subtract_divides_result_productcoefficientsentry * S ((S (pfc_index_subtract_divides_result_productcoefficients)) * pfrd_pc_subtract_divides_result) + (pfc_value_subtract_divides_result_productcoefficients))) /\\ ((exists pfc_terms_code_subtract_divides_result_productcoefficientscoefficient pfc_terms_scale_subtract_divides_result_productcoefficientscoefficient pfc_natural_sum_subtract_divides_result_productcoefficientscoefficient. ((forall pfc_index_subtract_divides_result_productcoefficientscoefficientdiagonal. (exists pfa_gap_subtract_divides_result_productcoefficientscoefficientdiagonalbound. pfa_gap_subtract_divides_result_productcoefficientscoefficientdiagonalbound + S (pfc_index_subtract_divides_result_productcoefficientscoefficientdiagonal) = (S (pfc_index_subtract_divides_result_productcoefficients))) -> exists pfc_value_subtract_divides_result_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_subtract_divides_result_productcoefficientscoefficientdiagonalentry. ff_h_pfp_subtract_divides_result_productcoefficientscoefficientdiagonalentry + S (pfc_value_subtract_divides_result_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_subtract_divides_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_subtract_divides_result_productcoefficientscoefficient)) /\\ exists ff_q_pfp_subtract_divides_result_productcoefficientscoefficientdiagonalentry. pfc_terms_code_subtract_divides_result_productcoefficientscoefficient = ff_q_pfp_subtract_divides_result_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_subtract_divides_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_subtract_divides_result_productcoefficientscoefficient) + (pfc_value_subtract_divides_result_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_subtract_divides_result_productcoefficientscoefficientdiagonalterm pfc_left_subtract_divides_result_productcoefficientscoefficientdiagonalterm pfc_right_subtract_divides_result_productcoefficientscoefficientdiagonalterm. (((pfc_index_subtract_divides_result_productcoefficientscoefficientdiagonal)+pfc_complement_subtract_divides_result_productcoefficientscoefficientdiagonalterm=(pfc_index_subtract_divides_result_productcoefficients)) /\\ ((((((exists pfa_gap_subtract_divides_result_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_subtract_divides_result_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_subtract_divides_result_productcoefficientscoefficientdiagonal) = (pfrd_qlen_subtract_divides_result)) /\\ ((((exists ff_h_pfp_subtract_divides_result_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_subtract_divides_result_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_subtract_divides_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_subtract_divides_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_subtract_divides_result)) /\\ exists ff_q_pfp_subtract_divides_result_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_subtract_divides_result = ff_q_pfp_subtract_divides_result_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_subtract_divides_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_subtract_divides_result) + (pfc_left_subtract_divides_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_subtract_divides_result_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_subtract_divides_result_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_subtract_divides_result)=(pfc_index_subtract_divides_result_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_subtract_divides_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_subtract_divides_result_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_subtract_divides_result_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_subtract_divides_result_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_subtract_divides_result_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_subtract_divides_result_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_subtract_divides_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_subtract_divides_result_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_subtract_divides_result_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_subtract_divides_result_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_subtract_divides_result_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_subtract_divides_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_subtract_divides_result_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_subtract_divides_result_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_subtract_divides_result_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_subtract_divides_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_subtract_divides_result_productcoefficientscoefficientdiagonal)=pfc_left_subtract_divides_result_productcoefficientscoefficientdiagonalterm*pfc_right_subtract_divides_result_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_subtract_divides_result_productcoefficientscoefficientsum fs_v_pfc_subtract_divides_result_productcoefficientscoefficientsum. ((((exists fs_h_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_start. fs_h_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_subtract_divides_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_start. fs_u_pfc_subtract_divides_result_productcoefficientscoefficientsum = fs_q_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_subtract_divides_result_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_subtract_divides_result_productcoefficientscoefficient) = S ((S (S (pfc_index_subtract_divides_result_productcoefficients))) * fs_v_pfc_subtract_divides_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_subtract_divides_result_productcoefficientscoefficientsum = fs_q_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_subtract_divides_result_productcoefficients))) * fs_v_pfc_subtract_divides_result_productcoefficientscoefficientsum) + (pfc_natural_sum_subtract_divides_result_productcoefficientscoefficient))) /\\ forall fs_i_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps = S (pfc_index_subtract_divides_result_productcoefficients)) -> exists fs_a_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps fs_r_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps fs_s_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_subtract_divides_result_productcoefficientscoefficient)) /\\ exists fs_q_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_subtract_divides_result_productcoefficientscoefficient = fs_q_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_subtract_divides_result_productcoefficientscoefficient) + (fs_a_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_divides_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_subtract_divides_result_productcoefficientscoefficientsum = fs_q_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_divides_result_productcoefficientscoefficientsum) + (fs_r_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_divides_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_subtract_divides_result_productcoefficientscoefficientsum = fs_q_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_subtract_divides_result_productcoefficientscoefficientsum) + (fs_s_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps = fs_r_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps + fs_a_pfc_subtract_divides_result_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_subtract_divides_result_productcoefficientscoefficientresiduebound. pfa_gap_subtract_divides_result_productcoefficientscoefficientresiduebound + S (pfc_value_subtract_divides_result_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_subtract_divides_result_productcoefficientscoefficientresiduecongruence pfa_offset_right_subtract_divides_result_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_subtract_divides_result_productcoefficientscoefficient) + (p) * pfa_offset_left_subtract_divides_result_productcoefficientscoefficientresiduecongruence = (pfc_value_subtract_divides_result_productcoefficients) + (p) * pfa_offset_right_subtract_divides_result_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_subtract_divides_result_target pfrep_left_subtract_divides_result_target pfrep_right_subtract_divides_result_target. ((exists pfrep_position_subtract_divides_result_targetfirst. ((pfrep_position_subtract_divides_result_targetfirst+S (pfrep_power_subtract_divides_result_target)=(pfrd_plen_subtract_divides_result)) /\\ ((((exists ff_h_pfp_subtract_divides_result_targetfirstentry. ff_h_pfp_subtract_divides_result_targetfirstentry + S (pfrep_left_subtract_divides_result_target) = S ((S (pfrep_position_subtract_divides_result_targetfirst)) * pfrd_pc_subtract_divides_result)) /\\ exists ff_q_pfp_subtract_divides_result_targetfirstentry. pfrd_pb_subtract_divides_result = ff_q_pfp_subtract_divides_result_targetfirstentry * S ((S (pfrep_position_subtract_divides_result_targetfirst)) * pfrd_pc_subtract_divides_result) + (pfrep_left_subtract_divides_result_target)))))) \\/ (((exists pfrep_gap_subtract_divides_result_targetfirstoutside. pfrep_gap_subtract_divides_result_targetfirstoutside+(pfrd_plen_subtract_divides_result)=(pfrep_power_subtract_divides_result_target)) /\\ (((pfrep_left_subtract_divides_result_target)=0))))) -> ((exists pfrep_position_subtract_divides_result_targetsecond. 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prime_field_polynomial_right_divides_from_product (I)", "specialize prime_field_polynomial_right_divides_from_product (qb)", "specialize prime_field_polynomial_right_divides_from_product (qc)", "specialize prime_field_polynomial_right_divides_from_product (H)", "specialize prime_field_polynomial_right_divides_from_product (pb)", "specialize prime_field_polynomial_right_divides_from_product (pc)", "specialize prime_field_polynomial_right_divides_from_product (I)", "apply prime_field_polynomial_right_divides_from_product", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (qb)", "specialize prime_field_polynomial_convolution_bounded (qc)", "specialize prime_field_polynomial_convolution_bounded (H)", "specialize prime_field_polynomial_convolution_bounded (bb)", "specialize prime_field_polynomial_convolution_bounded (bc)", "specialize prime_field_polynomial_convolution_bounded (M)", "specialize prime_field_polynomial_convolution_bounded (pb)", "specialize prime_field_polynomial_convolution_bounded (pc)", "specialize prime_field_polynomial_convolution_bounded (I)", "apply prime_field_polynomial_convolution_bounded", "exact hc", "exact hc", "specialize prime_field_polynomial_power_coefficient_functional (pb)", "specialize prime_field_polynomial_power_coefficient_functional (pc)", "specialize prime_field_polynomial_power_coefficient_functional (I)", "apply prime_field_polynomial_power_coefficient_functional" ], "script_sha256": "cb46ec58e051f31a1706b41ed32ed01bab34bef4ae9d380d6aec411298e20f07", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_euclidean_transport_candidate.py", "sha256": "9a589d1749eb38d30d1a24364bc4d66f7df0efb59247527f7831f97557da9c30" }, "statement": "forall p db dc J bb bc M qb qc H pb pc I. (~((p) = 1) /\\ forall pfa_factor_left_left_product_prime pfa_factor_right_left_product_prime. (p) = pfa_factor_left_left_product_prime * pfa_factor_right_left_product_prime -> pfa_factor_left_left_product_prime = 1 \\/ pfa_factor_right_left_product_prime = 1) -> (((forall fom_index_pfp_left_product_divisor_canonical. (exists fom_gap_pfp_left_product_divisor_canonical_index_bound. fom_gap_pfp_left_product_divisor_canonical_index_bound + S (fom_index_pfp_left_product_divisor_canonical) = M) -> exists fom_value_pfp_left_product_divisor_canonical. ((((exists fom_beta_height_pfp_left_product_divisor_canonical_entry. fom_beta_height_pfp_left_product_divisor_canonical_entry + S (fom_value_pfp_left_product_divisor_canonical) = S ((S (fom_index_pfp_left_product_divisor_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_left_product_divisor_canonical_entry. bb = fom_beta_quotient_pfp_left_product_divisor_canonical_entry * S ((S (fom_index_pfp_left_product_divisor_canonical)) * bc) + (fom_value_pfp_left_product_divisor_canonical))) /\\ (exists fom_gap_pfp_left_product_divisor_canonical_value_bound. fom_gap_pfp_left_product_divisor_canonical_value_bound + S (fom_value_pfp_left_product_divisor_canonical) = p))) /\\ ((exists pfrd_qb_left_product_divisor pfrd_qc_left_product_divisor pfrd_qlen_left_product_divisor pfrd_pb_left_product_divisor pfrd_pc_left_product_divisor pfrd_plen_left_product_divisor. ((((forall fom_index_pfp_left_product_divisor_productleft. (exists fom_gap_pfp_left_product_divisor_productleft_index_bound. fom_gap_pfp_left_product_divisor_productleft_index_bound + S (fom_index_pfp_left_product_divisor_productleft) = pfrd_qlen_left_product_divisor) -> exists fom_value_pfp_left_product_divisor_productleft. ((((exists fom_beta_height_pfp_left_product_divisor_productleft_entry. fom_beta_height_pfp_left_product_divisor_productleft_entry + S (fom_value_pfp_left_product_divisor_productleft) = S ((S (fom_index_pfp_left_product_divisor_productleft)) * pfrd_qc_left_product_divisor)) /\\ exists fom_beta_quotient_pfp_left_product_divisor_productleft_entry. pfrd_qb_left_product_divisor = fom_beta_quotient_pfp_left_product_divisor_productleft_entry * S ((S (fom_index_pfp_left_product_divisor_productleft)) * pfrd_qc_left_product_divisor) + (fom_value_pfp_left_product_divisor_productleft))) /\\ (exists fom_gap_pfp_left_product_divisor_productleft_value_bound. fom_gap_pfp_left_product_divisor_productleft_value_bound + S (fom_value_pfp_left_product_divisor_productleft) = p))) /\\ (((forall fom_index_pfp_left_product_divisor_productright. (exists fom_gap_pfp_left_product_divisor_productright_index_bound. fom_gap_pfp_left_product_divisor_productright_index_bound + S (fom_index_pfp_left_product_divisor_productright) = J) -> exists fom_value_pfp_left_product_divisor_productright. ((((exists fom_beta_height_pfp_left_product_divisor_productright_entry. fom_beta_height_pfp_left_product_divisor_productright_entry + S (fom_value_pfp_left_product_divisor_productright) = S ((S (fom_index_pfp_left_product_divisor_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_left_product_divisor_productright_entry. db = fom_beta_quotient_pfp_left_product_divisor_productright_entry * S ((S (fom_index_pfp_left_product_divisor_productright)) * dc) + (fom_value_pfp_left_product_divisor_productright))) /\\ (exists fom_gap_pfp_left_product_divisor_productright_value_bound. fom_gap_pfp_left_product_divisor_productright_value_bound + S (fom_value_pfp_left_product_divisor_productright) = p))) /\\ (((((((pfrd_qlen_left_product_divisor)=0 \\/ (J)=0) /\\ (((pfrd_plen_left_product_divisor)=0)))) \\/ (((~((pfrd_qlen_left_product_divisor)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_left_product_divisor)+(J)=S (pfrd_plen_left_product_divisor)))))))) /\\ ((forall pfc_index_left_product_divisor_productcoefficients. (exists pfa_gap_left_product_divisor_productcoefficientsbound. pfa_gap_left_product_divisor_productcoefficientsbound + S (pfc_index_left_product_divisor_productcoefficients) = (pfrd_plen_left_product_divisor)) -> exists pfc_value_left_product_divisor_productcoefficients. ((((exists ff_h_pfp_left_product_divisor_productcoefficientsentry. ff_h_pfp_left_product_divisor_productcoefficientsentry + S (pfc_value_left_product_divisor_productcoefficients) = S ((S (pfc_index_left_product_divisor_productcoefficients)) * pfrd_pc_left_product_divisor)) /\\ exists ff_q_pfp_left_product_divisor_productcoefficientsentry. pfrd_pb_left_product_divisor = ff_q_pfp_left_product_divisor_productcoefficientsentry * S ((S (pfc_index_left_product_divisor_productcoefficients)) * pfrd_pc_left_product_divisor) + (pfc_value_left_product_divisor_productcoefficients))) /\\ ((exists pfc_terms_code_left_product_divisor_productcoefficientscoefficient pfc_terms_scale_left_product_divisor_productcoefficientscoefficient pfc_natural_sum_left_product_divisor_productcoefficientscoefficient. ((forall pfc_index_left_product_divisor_productcoefficientscoefficientdiagonal. (exists pfa_gap_left_product_divisor_productcoefficientscoefficientdiagonalbound. pfa_gap_left_product_divisor_productcoefficientscoefficientdiagonalbound + S (pfc_index_left_product_divisor_productcoefficientscoefficientdiagonal) = (S (pfc_index_left_product_divisor_productcoefficients))) -> exists pfc_value_left_product_divisor_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_left_product_divisor_productcoefficientscoefficientdiagonalentry. ff_h_pfp_left_product_divisor_productcoefficientscoefficientdiagonalentry + S (pfc_value_left_product_divisor_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_left_product_divisor_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_left_product_divisor_productcoefficientscoefficient)) /\\ exists ff_q_pfp_left_product_divisor_productcoefficientscoefficientdiagonalentry. pfc_terms_code_left_product_divisor_productcoefficientscoefficient = ff_q_pfp_left_product_divisor_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_left_product_divisor_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_left_product_divisor_productcoefficientscoefficient) + (pfc_value_left_product_divisor_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_left_product_divisor_productcoefficientscoefficientdiagonalterm pfc_left_left_product_divisor_productcoefficientscoefficientdiagonalterm pfc_right_left_product_divisor_productcoefficientscoefficientdiagonalterm. (((pfc_index_left_product_divisor_productcoefficientscoefficientdiagonal)+pfc_complement_left_product_divisor_productcoefficientscoefficientdiagonalterm=(pfc_index_left_product_divisor_productcoefficients)) /\\ ((((((exists pfa_gap_left_product_divisor_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_left_product_divisor_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_left_product_divisor_productcoefficientscoefficientdiagonal) = (pfrd_qlen_left_product_divisor)) /\\ ((((exists ff_h_pfp_left_product_divisor_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_left_product_divisor_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_left_product_divisor_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_left_product_divisor_productcoefficientscoefficientdiagonal)) * pfrd_qc_left_product_divisor)) /\\ exists ff_q_pfp_left_product_divisor_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_left_product_divisor = ff_q_pfp_left_product_divisor_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_left_product_divisor_productcoefficientscoefficientdiagonal)) * pfrd_qc_left_product_divisor) + (pfc_left_left_product_divisor_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_left_product_divisor_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_left_product_divisor_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_left_product_divisor)=(pfc_index_left_product_divisor_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_left_product_divisor_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_left_product_divisor_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_left_product_divisor_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_left_product_divisor_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_left_product_divisor_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_left_product_divisor_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_left_product_divisor_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_left_product_divisor_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_left_product_divisor_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_left_product_divisor_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_left_product_divisor_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_left_product_divisor_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_left_product_divisor_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_left_product_divisor_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_left_product_divisor_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_left_product_divisor_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_left_product_divisor_productcoefficientscoefficientdiagonal)=pfc_left_left_product_divisor_productcoefficientscoefficientdiagonalterm*pfc_right_left_product_divisor_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_left_product_divisor_productcoefficientscoefficientsum fs_v_pfc_left_product_divisor_productcoefficientscoefficientsum. ((((exists fs_h_pfc_left_product_divisor_productcoefficientscoefficientsum_body_start. fs_h_pfc_left_product_divisor_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_left_product_divisor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_product_divisor_productcoefficientscoefficientsum_body_start. fs_u_pfc_left_product_divisor_productcoefficientscoefficientsum = fs_q_pfc_left_product_divisor_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_left_product_divisor_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_left_product_divisor_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_left_product_divisor_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_left_product_divisor_productcoefficientscoefficient) = S ((S (S (pfc_index_left_product_divisor_productcoefficients))) * fs_v_pfc_left_product_divisor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_product_divisor_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_left_product_divisor_productcoefficientscoefficientsum = fs_q_pfc_left_product_divisor_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_left_product_divisor_productcoefficients))) * fs_v_pfc_left_product_divisor_productcoefficientscoefficientsum) + (pfc_natural_sum_left_product_divisor_productcoefficientscoefficient))) /\\ forall fs_i_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps = S (pfc_index_left_product_divisor_productcoefficients)) -> exists fs_a_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps fs_r_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps fs_s_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_left_product_divisor_productcoefficientscoefficient)) /\\ exists fs_q_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_left_product_divisor_productcoefficientscoefficient = fs_q_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_left_product_divisor_productcoefficientscoefficient) + (fs_a_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_product_divisor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_left_product_divisor_productcoefficientscoefficientsum = fs_q_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_product_divisor_productcoefficientscoefficientsum) + (fs_r_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_product_divisor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_left_product_divisor_productcoefficientscoefficientsum = fs_q_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_product_divisor_productcoefficientscoefficientsum) + (fs_s_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps = fs_r_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps + fs_a_pfc_left_product_divisor_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_left_product_divisor_productcoefficientscoefficientresiduebound. pfa_gap_left_product_divisor_productcoefficientscoefficientresiduebound + S (pfc_value_left_product_divisor_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_left_product_divisor_productcoefficientscoefficientresiduecongruence pfa_offset_right_left_product_divisor_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_left_product_divisor_productcoefficientscoefficient) + (p) * pfa_offset_left_left_product_divisor_productcoefficientscoefficientresiduecongruence = (pfc_value_left_product_divisor_productcoefficients) + (p) * pfa_offset_right_left_product_divisor_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_left_product_divisor_target pfrep_left_left_product_divisor_target pfrep_right_left_product_divisor_target. ((exists pfrep_position_left_product_divisor_targetfirst. ((pfrep_position_left_product_divisor_targetfirst+S (pfrep_power_left_product_divisor_target)=(pfrd_plen_left_product_divisor)) /\\ ((((exists ff_h_pfp_left_product_divisor_targetfirstentry. ff_h_pfp_left_product_divisor_targetfirstentry + S (pfrep_left_left_product_divisor_target) = S ((S (pfrep_position_left_product_divisor_targetfirst)) * pfrd_pc_left_product_divisor)) /\\ exists ff_q_pfp_left_product_divisor_targetfirstentry. pfrd_pb_left_product_divisor = ff_q_pfp_left_product_divisor_targetfirstentry * S ((S (pfrep_position_left_product_divisor_targetfirst)) * pfrd_pc_left_product_divisor) + (pfrep_left_left_product_divisor_target)))))) \\/ (((exists pfrep_gap_left_product_divisor_targetfirstoutside. pfrep_gap_left_product_divisor_targetfirstoutside+(pfrd_plen_left_product_divisor)=(pfrep_power_left_product_divisor_target)) /\\ (((pfrep_left_left_product_divisor_target)=0))))) -> ((exists pfrep_position_left_product_divisor_targetsecond. ((pfrep_position_left_product_divisor_targetsecond+S (pfrep_power_left_product_divisor_target)=(M)) /\\ ((((exists ff_h_pfp_left_product_divisor_targetsecondentry. ff_h_pfp_left_product_divisor_targetsecondentry + S (pfrep_right_left_product_divisor_target) = S ((S (pfrep_position_left_product_divisor_targetsecond)) * bc)) /\\ exists ff_q_pfp_left_product_divisor_targetsecondentry. bb = ff_q_pfp_left_product_divisor_targetsecondentry * S ((S (pfrep_position_left_product_divisor_targetsecond)) * bc) + (pfrep_right_left_product_divisor_target)))))) \\/ (((exists pfrep_gap_left_product_divisor_targetsecondoutside. pfrep_gap_left_product_divisor_targetsecondoutside+(M)=(pfrep_power_left_product_divisor_target)) /\\ (((pfrep_right_left_product_divisor_target)=0))))) -> pfrep_left_left_product_divisor_target=pfrep_right_left_product_divisor_target))))))) -> (((forall fom_index_pfp_left_product_actualleft. (exists fom_gap_pfp_left_product_actualleft_index_bound. fom_gap_pfp_left_product_actualleft_index_bound + S (fom_index_pfp_left_product_actualleft) = H) -> exists fom_value_pfp_left_product_actualleft. ((((exists fom_beta_height_pfp_left_product_actualleft_entry. fom_beta_height_pfp_left_product_actualleft_entry + S (fom_value_pfp_left_product_actualleft) = S ((S (fom_index_pfp_left_product_actualleft)) * qc)) /\\ exists fom_beta_quotient_pfp_left_product_actualleft_entry. qb = fom_beta_quotient_pfp_left_product_actualleft_entry * S ((S (fom_index_pfp_left_product_actualleft)) * qc) + (fom_value_pfp_left_product_actualleft))) /\\ (exists fom_gap_pfp_left_product_actualleft_value_bound. fom_gap_pfp_left_product_actualleft_value_bound + S (fom_value_pfp_left_product_actualleft) = p))) /\\ (((forall fom_index_pfp_left_product_actualright. (exists fom_gap_pfp_left_product_actualright_index_bound. fom_gap_pfp_left_product_actualright_index_bound + S (fom_index_pfp_left_product_actualright) = M) -> exists fom_value_pfp_left_product_actualright. ((((exists fom_beta_height_pfp_left_product_actualright_entry. fom_beta_height_pfp_left_product_actualright_entry + S (fom_value_pfp_left_product_actualright) = S ((S (fom_index_pfp_left_product_actualright)) * bc)) /\\ exists fom_beta_quotient_pfp_left_product_actualright_entry. bb = fom_beta_quotient_pfp_left_product_actualright_entry * S ((S (fom_index_pfp_left_product_actualright)) * bc) + (fom_value_pfp_left_product_actualright))) /\\ (exists fom_gap_pfp_left_product_actualright_value_bound. fom_gap_pfp_left_product_actualright_value_bound + S (fom_value_pfp_left_product_actualright) = p))) /\\ (((((((H)=0 \\/ (M)=0) /\\ (((I)=0)))) \\/ (((~((H)=0)) /\\ (((~((M)=0)) /\\ (((H)+(M)=S (I)))))))) /\\ ((forall pfc_index_left_product_actualcoefficients. (exists pfa_gap_left_product_actualcoefficientsbound. pfa_gap_left_product_actualcoefficientsbound + S (pfc_index_left_product_actualcoefficients) = (I)) -> exists pfc_value_left_product_actualcoefficients. ((((exists ff_h_pfp_left_product_actualcoefficientsentry. ff_h_pfp_left_product_actualcoefficientsentry + S (pfc_value_left_product_actualcoefficients) = S ((S (pfc_index_left_product_actualcoefficients)) * pc)) /\\ exists ff_q_pfp_left_product_actualcoefficientsentry. pb = ff_q_pfp_left_product_actualcoefficientsentry * S ((S (pfc_index_left_product_actualcoefficients)) * pc) + (pfc_value_left_product_actualcoefficients))) /\\ ((exists pfc_terms_code_left_product_actualcoefficientscoefficient pfc_terms_scale_left_product_actualcoefficientscoefficient pfc_natural_sum_left_product_actualcoefficientscoefficient. ((forall pfc_index_left_product_actualcoefficientscoefficientdiagonal. (exists pfa_gap_left_product_actualcoefficientscoefficientdiagonalbound. pfa_gap_left_product_actualcoefficientscoefficientdiagonalbound + S (pfc_index_left_product_actualcoefficientscoefficientdiagonal) = (S (pfc_index_left_product_actualcoefficients))) -> exists pfc_value_left_product_actualcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_left_product_actualcoefficientscoefficientdiagonalentry. ff_h_pfp_left_product_actualcoefficientscoefficientdiagonalentry + S (pfc_value_left_product_actualcoefficientscoefficientdiagonal) = S ((S (pfc_index_left_product_actualcoefficientscoefficientdiagonal)) * pfc_terms_scale_left_product_actualcoefficientscoefficient)) /\\ exists ff_q_pfp_left_product_actualcoefficientscoefficientdiagonalentry. pfc_terms_code_left_product_actualcoefficientscoefficient = ff_q_pfp_left_product_actualcoefficientscoefficientdiagonalentry * S ((S (pfc_index_left_product_actualcoefficientscoefficientdiagonal)) * pfc_terms_scale_left_product_actualcoefficientscoefficient) + (pfc_value_left_product_actualcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_left_product_actualcoefficientscoefficientdiagonalterm pfc_left_left_product_actualcoefficientscoefficientdiagonalterm pfc_right_left_product_actualcoefficientscoefficientdiagonalterm. (((pfc_index_left_product_actualcoefficientscoefficientdiagonal)+pfc_complement_left_product_actualcoefficientscoefficientdiagonalterm=(pfc_index_left_product_actualcoefficients)) /\\ ((((((exists pfa_gap_left_product_actualcoefficientscoefficientdiagonaltermleftinside. pfa_gap_left_product_actualcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_left_product_actualcoefficientscoefficientdiagonal) = (H)) /\\ ((((exists ff_h_pfp_left_product_actualcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_left_product_actualcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_left_product_actualcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_left_product_actualcoefficientscoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_left_product_actualcoefficientscoefficientdiagonaltermleftentry. qb = ff_q_pfp_left_product_actualcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_left_product_actualcoefficientscoefficientdiagonal)) * qc) + (pfc_left_left_product_actualcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_left_product_actualcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_left_product_actualcoefficientscoefficientdiagonaltermleftoutside+(H)=(pfc_index_left_product_actualcoefficientscoefficientdiagonal)) /\\ (((pfc_left_left_product_actualcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_left_product_actualcoefficientscoefficientdiagonaltermrightinside. pfa_gap_left_product_actualcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_left_product_actualcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_left_product_actualcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_left_product_actualcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_left_product_actualcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_left_product_actualcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_left_product_actualcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_left_product_actualcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_left_product_actualcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_left_product_actualcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_left_product_actualcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_left_product_actualcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_left_product_actualcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_left_product_actualcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_left_product_actualcoefficientscoefficientdiagonal)=pfc_left_left_product_actualcoefficientscoefficientdiagonalterm*pfc_right_left_product_actualcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_left_product_actualcoefficientscoefficientsum fs_v_pfc_left_product_actualcoefficientscoefficientsum. ((((exists fs_h_pfc_left_product_actualcoefficientscoefficientsum_body_start. fs_h_pfc_left_product_actualcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_left_product_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_product_actualcoefficientscoefficientsum_body_start. fs_u_pfc_left_product_actualcoefficientscoefficientsum = fs_q_pfc_left_product_actualcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_left_product_actualcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_left_product_actualcoefficientscoefficientsum_body_terminal. fs_h_pfc_left_product_actualcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_left_product_actualcoefficientscoefficient) = S ((S (S (pfc_index_left_product_actualcoefficients))) * fs_v_pfc_left_product_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_product_actualcoefficientscoefficientsum_body_terminal. fs_u_pfc_left_product_actualcoefficientscoefficientsum = fs_q_pfc_left_product_actualcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_left_product_actualcoefficients))) * fs_v_pfc_left_product_actualcoefficientscoefficientsum) + (pfc_natural_sum_left_product_actualcoefficientscoefficient))) /\\ forall fs_i_pfc_left_product_actualcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_left_product_actualcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_left_product_actualcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_left_product_actualcoefficientscoefficientsum_body_steps = S (pfc_index_left_product_actualcoefficients)) -> exists fs_a_pfc_left_product_actualcoefficientscoefficientsum_body_steps fs_r_pfc_left_product_actualcoefficientscoefficientsum_body_steps fs_s_pfc_left_product_actualcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_left_product_actualcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_left_product_actualcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_left_product_actualcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_left_product_actualcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_left_product_actualcoefficientscoefficient)) /\\ exists fs_q_pfc_left_product_actualcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_left_product_actualcoefficientscoefficient = fs_q_pfc_left_product_actualcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_left_product_actualcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_left_product_actualcoefficientscoefficient) + (fs_a_pfc_left_product_actualcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_left_product_actualcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_left_product_actualcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_left_product_actualcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_left_product_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_product_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_product_actualcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_left_product_actualcoefficientscoefficientsum = fs_q_pfc_left_product_actualcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_left_product_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_product_actualcoefficientscoefficientsum) + (fs_r_pfc_left_product_actualcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_left_product_actualcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_left_product_actualcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_left_product_actualcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_left_product_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_product_actualcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_product_actualcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_left_product_actualcoefficientscoefficientsum = fs_q_pfc_left_product_actualcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_left_product_actualcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_product_actualcoefficientscoefficientsum) + (fs_s_pfc_left_product_actualcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_left_product_actualcoefficientscoefficientsum_body_steps = fs_r_pfc_left_product_actualcoefficientscoefficientsum_body_steps + fs_a_pfc_left_product_actualcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_left_product_actualcoefficientscoefficientresiduebound. pfa_gap_left_product_actualcoefficientscoefficientresiduebound + S (pfc_value_left_product_actualcoefficients) = (p)) /\\ ((exists pfa_offset_left_left_product_actualcoefficientscoefficientresiduecongruence pfa_offset_right_left_product_actualcoefficientscoefficientresiduecongruence. (pfc_natural_sum_left_product_actualcoefficientscoefficient) + (p) * pfa_offset_left_left_product_actualcoefficientscoefficientresiduecongruence = (pfc_value_left_product_actualcoefficients) + (p) * pfa_offset_right_left_product_actualcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_left_product_result_canonical. (exists fom_gap_pfp_left_product_result_canonical_index_bound. fom_gap_pfp_left_product_result_canonical_index_bound + S (fom_index_pfp_left_product_result_canonical) = I) -> exists fom_value_pfp_left_product_result_canonical. ((((exists fom_beta_height_pfp_left_product_result_canonical_entry. fom_beta_height_pfp_left_product_result_canonical_entry + S (fom_value_pfp_left_product_result_canonical) = S ((S (fom_index_pfp_left_product_result_canonical)) * pc)) /\\ exists fom_beta_quotient_pfp_left_product_result_canonical_entry. pb = fom_beta_quotient_pfp_left_product_result_canonical_entry * S ((S (fom_index_pfp_left_product_result_canonical)) * pc) + (fom_value_pfp_left_product_result_canonical))) /\\ (exists fom_gap_pfp_left_product_result_canonical_value_bound. fom_gap_pfp_left_product_result_canonical_value_bound + S (fom_value_pfp_left_product_result_canonical) = p))) /\\ ((exists pfrd_qb_left_product_result pfrd_qc_left_product_result pfrd_qlen_left_product_result pfrd_pb_left_product_result pfrd_pc_left_product_result pfrd_plen_left_product_result. ((((forall fom_index_pfp_left_product_result_productleft. (exists fom_gap_pfp_left_product_result_productleft_index_bound. fom_gap_pfp_left_product_result_productleft_index_bound + S (fom_index_pfp_left_product_result_productleft) = pfrd_qlen_left_product_result) -> exists fom_value_pfp_left_product_result_productleft. ((((exists fom_beta_height_pfp_left_product_result_productleft_entry. fom_beta_height_pfp_left_product_result_productleft_entry + S (fom_value_pfp_left_product_result_productleft) = S ((S (fom_index_pfp_left_product_result_productleft)) * pfrd_qc_left_product_result)) /\\ exists fom_beta_quotient_pfp_left_product_result_productleft_entry. pfrd_qb_left_product_result = fom_beta_quotient_pfp_left_product_result_productleft_entry * S ((S (fom_index_pfp_left_product_result_productleft)) * pfrd_qc_left_product_result) + (fom_value_pfp_left_product_result_productleft))) /\\ (exists fom_gap_pfp_left_product_result_productleft_value_bound. fom_gap_pfp_left_product_result_productleft_value_bound + S (fom_value_pfp_left_product_result_productleft) = p))) /\\ (((forall fom_index_pfp_left_product_result_productright. (exists fom_gap_pfp_left_product_result_productright_index_bound. fom_gap_pfp_left_product_result_productright_index_bound + S (fom_index_pfp_left_product_result_productright) = J) -> exists fom_value_pfp_left_product_result_productright. ((((exists fom_beta_height_pfp_left_product_result_productright_entry. fom_beta_height_pfp_left_product_result_productright_entry + S (fom_value_pfp_left_product_result_productright) = S ((S (fom_index_pfp_left_product_result_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_left_product_result_productright_entry. db = fom_beta_quotient_pfp_left_product_result_productright_entry * S ((S (fom_index_pfp_left_product_result_productright)) * dc) + (fom_value_pfp_left_product_result_productright))) /\\ (exists fom_gap_pfp_left_product_result_productright_value_bound. fom_gap_pfp_left_product_result_productright_value_bound + S (fom_value_pfp_left_product_result_productright) = p))) /\\ (((((((pfrd_qlen_left_product_result)=0 \\/ (J)=0) /\\ (((pfrd_plen_left_product_result)=0)))) \\/ (((~((pfrd_qlen_left_product_result)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_left_product_result)+(J)=S (pfrd_plen_left_product_result)))))))) /\\ ((forall pfc_index_left_product_result_productcoefficients. (exists pfa_gap_left_product_result_productcoefficientsbound. pfa_gap_left_product_result_productcoefficientsbound + S (pfc_index_left_product_result_productcoefficients) = (pfrd_plen_left_product_result)) -> exists pfc_value_left_product_result_productcoefficients. ((((exists ff_h_pfp_left_product_result_productcoefficientsentry. ff_h_pfp_left_product_result_productcoefficientsentry + S (pfc_value_left_product_result_productcoefficients) = S ((S (pfc_index_left_product_result_productcoefficients)) * pfrd_pc_left_product_result)) /\\ exists ff_q_pfp_left_product_result_productcoefficientsentry. pfrd_pb_left_product_result = ff_q_pfp_left_product_result_productcoefficientsentry * S ((S (pfc_index_left_product_result_productcoefficients)) * pfrd_pc_left_product_result) + (pfc_value_left_product_result_productcoefficients))) /\\ ((exists pfc_terms_code_left_product_result_productcoefficientscoefficient pfc_terms_scale_left_product_result_productcoefficientscoefficient pfc_natural_sum_left_product_result_productcoefficientscoefficient. ((forall pfc_index_left_product_result_productcoefficientscoefficientdiagonal. (exists pfa_gap_left_product_result_productcoefficientscoefficientdiagonalbound. pfa_gap_left_product_result_productcoefficientscoefficientdiagonalbound + S (pfc_index_left_product_result_productcoefficientscoefficientdiagonal) = (S (pfc_index_left_product_result_productcoefficients))) -> exists pfc_value_left_product_result_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_left_product_result_productcoefficientscoefficientdiagonalentry. ff_h_pfp_left_product_result_productcoefficientscoefficientdiagonalentry + S (pfc_value_left_product_result_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_left_product_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_left_product_result_productcoefficientscoefficient)) /\\ exists ff_q_pfp_left_product_result_productcoefficientscoefficientdiagonalentry. pfc_terms_code_left_product_result_productcoefficientscoefficient = ff_q_pfp_left_product_result_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_left_product_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_left_product_result_productcoefficientscoefficient) + (pfc_value_left_product_result_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_left_product_result_productcoefficientscoefficientdiagonalterm pfc_left_left_product_result_productcoefficientscoefficientdiagonalterm pfc_right_left_product_result_productcoefficientscoefficientdiagonalterm. (((pfc_index_left_product_result_productcoefficientscoefficientdiagonal)+pfc_complement_left_product_result_productcoefficientscoefficientdiagonalterm=(pfc_index_left_product_result_productcoefficients)) /\\ ((((((exists pfa_gap_left_product_result_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_left_product_result_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_left_product_result_productcoefficientscoefficientdiagonal) = (pfrd_qlen_left_product_result)) /\\ ((((exists ff_h_pfp_left_product_result_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_left_product_result_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_left_product_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_left_product_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_left_product_result)) /\\ exists ff_q_pfp_left_product_result_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_left_product_result = ff_q_pfp_left_product_result_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_left_product_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_left_product_result) + (pfc_left_left_product_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_left_product_result_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_left_product_result_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_left_product_result)=(pfc_index_left_product_result_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_left_product_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_left_product_result_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_left_product_result_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_left_product_result_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_left_product_result_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_left_product_result_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_left_product_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_left_product_result_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_left_product_result_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_left_product_result_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_left_product_result_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_left_product_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_left_product_result_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_left_product_result_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_left_product_result_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_left_product_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_left_product_result_productcoefficientscoefficientdiagonal)=pfc_left_left_product_result_productcoefficientscoefficientdiagonalterm*pfc_right_left_product_result_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_left_product_result_productcoefficientscoefficientsum fs_v_pfc_left_product_result_productcoefficientscoefficientsum. ((((exists fs_h_pfc_left_product_result_productcoefficientscoefficientsum_body_start. fs_h_pfc_left_product_result_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_left_product_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_product_result_productcoefficientscoefficientsum_body_start. fs_u_pfc_left_product_result_productcoefficientscoefficientsum = fs_q_pfc_left_product_result_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_left_product_result_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_left_product_result_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_left_product_result_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_left_product_result_productcoefficientscoefficient) = S ((S (S (pfc_index_left_product_result_productcoefficients))) * fs_v_pfc_left_product_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_product_result_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_left_product_result_productcoefficientscoefficientsum = fs_q_pfc_left_product_result_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_left_product_result_productcoefficients))) * fs_v_pfc_left_product_result_productcoefficientscoefficientsum) + (pfc_natural_sum_left_product_result_productcoefficientscoefficient))) /\\ forall fs_i_pfc_left_product_result_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_left_product_result_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_left_product_result_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_left_product_result_productcoefficientscoefficientsum_body_steps = S (pfc_index_left_product_result_productcoefficients)) -> exists fs_a_pfc_left_product_result_productcoefficientscoefficientsum_body_steps fs_r_pfc_left_product_result_productcoefficientscoefficientsum_body_steps fs_s_pfc_left_product_result_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_left_product_result_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_left_product_result_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_left_product_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_left_product_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_left_product_result_productcoefficientscoefficient)) /\\ exists fs_q_pfc_left_product_result_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_left_product_result_productcoefficientscoefficient = fs_q_pfc_left_product_result_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_left_product_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_left_product_result_productcoefficientscoefficient) + (fs_a_pfc_left_product_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_left_product_result_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_left_product_result_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_left_product_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_left_product_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_product_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_product_result_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_left_product_result_productcoefficientscoefficientsum = fs_q_pfc_left_product_result_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_left_product_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_product_result_productcoefficientscoefficientsum) + (fs_r_pfc_left_product_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_left_product_result_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_left_product_result_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_left_product_result_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_left_product_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_product_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_left_product_result_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_left_product_result_productcoefficientscoefficientsum = fs_q_pfc_left_product_result_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_left_product_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_left_product_result_productcoefficientscoefficientsum) + (fs_s_pfc_left_product_result_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_left_product_result_productcoefficientscoefficientsum_body_steps = fs_r_pfc_left_product_result_productcoefficientscoefficientsum_body_steps + fs_a_pfc_left_product_result_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_left_product_result_productcoefficientscoefficientresiduebound. pfa_gap_left_product_result_productcoefficientscoefficientresiduebound + S (pfc_value_left_product_result_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_left_product_result_productcoefficientscoefficientresiduecongruence pfa_offset_right_left_product_result_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_left_product_result_productcoefficientscoefficient) + (p) * pfa_offset_left_left_product_result_productcoefficientscoefficientresiduecongruence = (pfc_value_left_product_result_productcoefficients) + (p) * pfa_offset_right_left_product_result_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_left_product_result_target pfrep_left_left_product_result_target pfrep_right_left_product_result_target. ((exists pfrep_position_left_product_result_targetfirst. ((pfrep_position_left_product_result_targetfirst+S (pfrep_power_left_product_result_target)=(pfrd_plen_left_product_result)) /\\ ((((exists ff_h_pfp_left_product_result_targetfirstentry. ff_h_pfp_left_product_result_targetfirstentry + S (pfrep_left_left_product_result_target) = S ((S (pfrep_position_left_product_result_targetfirst)) * pfrd_pc_left_product_result)) /\\ exists ff_q_pfp_left_product_result_targetfirstentry. pfrd_pb_left_product_result = ff_q_pfp_left_product_result_targetfirstentry * S ((S (pfrep_position_left_product_result_targetfirst)) * pfrd_pc_left_product_result) + (pfrep_left_left_product_result_target)))))) \\/ (((exists pfrep_gap_left_product_result_targetfirstoutside. pfrep_gap_left_product_result_targetfirstoutside+(pfrd_plen_left_product_result)=(pfrep_power_left_product_result_target)) /\\ (((pfrep_left_left_product_result_target)=0))))) -> ((exists pfrep_position_left_product_result_targetsecond. 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(~((p) = 1) /\\ forall pfa_factor_left_euclidean_common_prime pfa_factor_right_euclidean_common_prime. (p) = pfa_factor_left_euclidean_common_prime * pfa_factor_right_euclidean_common_prime -> pfa_factor_left_euclidean_common_prime = 1 \\/ pfa_factor_right_euclidean_common_prime = 1) -> (((forall fom_index_pfp_euclidean_common_productleft. (exists fom_gap_pfp_euclidean_common_productleft_index_bound. fom_gap_pfp_euclidean_common_productleft_index_bound + S (fom_index_pfp_euclidean_common_productleft) = H) -> exists fom_value_pfp_euclidean_common_productleft. ((((exists fom_beta_height_pfp_euclidean_common_productleft_entry. fom_beta_height_pfp_euclidean_common_productleft_entry + S (fom_value_pfp_euclidean_common_productleft) = S ((S (fom_index_pfp_euclidean_common_productleft)) * qc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_productleft_entry. qb = fom_beta_quotient_pfp_euclidean_common_productleft_entry * S ((S (fom_index_pfp_euclidean_common_productleft)) * qc) + (fom_value_pfp_euclidean_common_productleft))) /\\ (exists fom_gap_pfp_euclidean_common_productleft_value_bound. fom_gap_pfp_euclidean_common_productleft_value_bound + S (fom_value_pfp_euclidean_common_productleft) = p))) /\\ (((forall fom_index_pfp_euclidean_common_productright. (exists fom_gap_pfp_euclidean_common_productright_index_bound. fom_gap_pfp_euclidean_common_productright_index_bound + S (fom_index_pfp_euclidean_common_productright) = M) -> exists fom_value_pfp_euclidean_common_productright. ((((exists fom_beta_height_pfp_euclidean_common_productright_entry. fom_beta_height_pfp_euclidean_common_productright_entry + S (fom_value_pfp_euclidean_common_productright) = S ((S (fom_index_pfp_euclidean_common_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_productright_entry. bb = fom_beta_quotient_pfp_euclidean_common_productright_entry * S ((S (fom_index_pfp_euclidean_common_productright)) * bc) + (fom_value_pfp_euclidean_common_productright))) /\\ (exists fom_gap_pfp_euclidean_common_productright_value_bound. fom_gap_pfp_euclidean_common_productright_value_bound + S (fom_value_pfp_euclidean_common_productright) = p))) /\\ (((((((H)=0 \\/ (M)=0) /\\ (((I)=0)))) \\/ (((~((H)=0)) /\\ (((~((M)=0)) /\\ (((H)+(M)=S (I)))))))) /\\ ((forall pfc_index_euclidean_common_productcoefficients. (exists pfa_gap_euclidean_common_productcoefficientsbound. pfa_gap_euclidean_common_productcoefficientsbound + S (pfc_index_euclidean_common_productcoefficients) = (I)) -> exists pfc_value_euclidean_common_productcoefficients. ((((exists ff_h_pfp_euclidean_common_productcoefficientsentry. ff_h_pfp_euclidean_common_productcoefficientsentry + S (pfc_value_euclidean_common_productcoefficients) = S ((S (pfc_index_euclidean_common_productcoefficients)) * pc)) /\\ exists ff_q_pfp_euclidean_common_productcoefficientsentry. pb = ff_q_pfp_euclidean_common_productcoefficientsentry * S ((S (pfc_index_euclidean_common_productcoefficients)) * pc) + (pfc_value_euclidean_common_productcoefficients))) /\\ ((exists pfc_terms_code_euclidean_common_productcoefficientscoefficient pfc_terms_scale_euclidean_common_productcoefficientscoefficient pfc_natural_sum_euclidean_common_productcoefficientscoefficient. ((forall pfc_index_euclidean_common_productcoefficientscoefficientdiagonal. (exists pfa_gap_euclidean_common_productcoefficientscoefficientdiagonalbound. pfa_gap_euclidean_common_productcoefficientscoefficientdiagonalbound + S (pfc_index_euclidean_common_productcoefficientscoefficientdiagonal) = (S (pfc_index_euclidean_common_productcoefficients))) -> exists pfc_value_euclidean_common_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_euclidean_common_productcoefficientscoefficientdiagonalentry. ff_h_pfp_euclidean_common_productcoefficientscoefficientdiagonalentry + S (pfc_value_euclidean_common_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_euclidean_common_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_productcoefficientscoefficient)) /\\ exists ff_q_pfp_euclidean_common_productcoefficientscoefficientdiagonalentry. pfc_terms_code_euclidean_common_productcoefficientscoefficient = ff_q_pfp_euclidean_common_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_euclidean_common_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_productcoefficientscoefficient) + (pfc_value_euclidean_common_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_euclidean_common_productcoefficientscoefficientdiagonalterm pfc_left_euclidean_common_productcoefficientscoefficientdiagonalterm pfc_right_euclidean_common_productcoefficientscoefficientdiagonalterm. (((pfc_index_euclidean_common_productcoefficientscoefficientdiagonal)+pfc_complement_euclidean_common_productcoefficientscoefficientdiagonalterm=(pfc_index_euclidean_common_productcoefficients)) /\\ ((((((exists pfa_gap_euclidean_common_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_euclidean_common_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_euclidean_common_productcoefficientscoefficientdiagonal) = (H)) /\\ ((((exists ff_h_pfp_euclidean_common_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_euclidean_common_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_euclidean_common_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_euclidean_common_productcoefficientscoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_euclidean_common_productcoefficientscoefficientdiagonaltermleftentry. qb = ff_q_pfp_euclidean_common_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_euclidean_common_productcoefficientscoefficientdiagonal)) * qc) + (pfc_left_euclidean_common_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_euclidean_common_productcoefficientscoefficientdiagonaltermleftoutside+(H)=(pfc_index_euclidean_common_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_euclidean_common_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_euclidean_common_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_euclidean_common_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_euclidean_common_productcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_euclidean_common_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_euclidean_common_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_euclidean_common_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_euclidean_common_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_euclidean_common_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_euclidean_common_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_euclidean_common_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_euclidean_common_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_euclidean_common_productcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_euclidean_common_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_euclidean_common_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_euclidean_common_productcoefficientscoefficientdiagonal)=pfc_left_euclidean_common_productcoefficientscoefficientdiagonalterm*pfc_right_euclidean_common_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_euclidean_common_productcoefficientscoefficientsum fs_v_pfc_euclidean_common_productcoefficientscoefficientsum. ((((exists fs_h_pfc_euclidean_common_productcoefficientscoefficientsum_body_start. fs_h_pfc_euclidean_common_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_euclidean_common_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_productcoefficientscoefficientsum_body_start. fs_u_pfc_euclidean_common_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_euclidean_common_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_euclidean_common_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_euclidean_common_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_euclidean_common_productcoefficientscoefficient) = S ((S (S (pfc_index_euclidean_common_productcoefficients))) * fs_v_pfc_euclidean_common_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_euclidean_common_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_euclidean_common_productcoefficients))) * fs_v_pfc_euclidean_common_productcoefficientscoefficientsum) + (pfc_natural_sum_euclidean_common_productcoefficientscoefficient))) /\\ forall fs_i_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps = S (pfc_index_euclidean_common_productcoefficients)) -> exists fs_a_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps fs_r_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps fs_s_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_productcoefficientscoefficient)) /\\ exists fs_q_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_euclidean_common_productcoefficientscoefficient = fs_q_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_productcoefficientscoefficient) + (fs_a_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_euclidean_common_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_productcoefficientscoefficientsum) + (fs_r_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_euclidean_common_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_productcoefficientscoefficientsum) + (fs_s_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps = fs_r_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps + fs_a_pfc_euclidean_common_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_euclidean_common_productcoefficientscoefficientresiduebound. pfa_gap_euclidean_common_productcoefficientscoefficientresiduebound + S (pfc_value_euclidean_common_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_euclidean_common_productcoefficientscoefficientresiduecongruence pfa_offset_right_euclidean_common_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_euclidean_common_productcoefficientscoefficient) + (p) * pfa_offset_left_euclidean_common_productcoefficientscoefficientresiduecongruence = (pfc_value_euclidean_common_productcoefficients) + (p) * pfa_offset_right_euclidean_common_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_euclidean_common_identity_left_bounded. (exists fom_gap_pfp_euclidean_common_identity_left_bounded_index_bound. fom_gap_pfp_euclidean_common_identity_left_bounded_index_bound + S (fom_index_pfp_euclidean_common_identity_left_bounded) = I) -> exists fom_value_pfp_euclidean_common_identity_left_bounded. ((((exists fom_beta_height_pfp_euclidean_common_identity_left_bounded_entry. fom_beta_height_pfp_euclidean_common_identity_left_bounded_entry + S (fom_value_pfp_euclidean_common_identity_left_bounded) = S ((S (fom_index_pfp_euclidean_common_identity_left_bounded)) * pc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_identity_left_bounded_entry. pb = fom_beta_quotient_pfp_euclidean_common_identity_left_bounded_entry * S ((S (fom_index_pfp_euclidean_common_identity_left_bounded)) * pc) + (fom_value_pfp_euclidean_common_identity_left_bounded))) /\\ (exists fom_gap_pfp_euclidean_common_identity_left_bounded_value_bound. fom_gap_pfp_euclidean_common_identity_left_bounded_value_bound + S (fom_value_pfp_euclidean_common_identity_left_bounded) = p))) /\\ (((forall fom_index_pfp_euclidean_common_identity_right_bounded. (exists fom_gap_pfp_euclidean_common_identity_right_bounded_index_bound. fom_gap_pfp_euclidean_common_identity_right_bounded_index_bound + S (fom_index_pfp_euclidean_common_identity_right_bounded) = N) -> exists fom_value_pfp_euclidean_common_identity_right_bounded. ((((exists fom_beta_height_pfp_euclidean_common_identity_right_bounded_entry. fom_beta_height_pfp_euclidean_common_identity_right_bounded_entry + S (fom_value_pfp_euclidean_common_identity_right_bounded) = S ((S (fom_index_pfp_euclidean_common_identity_right_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_identity_right_bounded_entry. rb = fom_beta_quotient_pfp_euclidean_common_identity_right_bounded_entry * S ((S (fom_index_pfp_euclidean_common_identity_right_bounded)) * rc) + (fom_value_pfp_euclidean_common_identity_right_bounded))) /\\ (exists fom_gap_pfp_euclidean_common_identity_right_bounded_value_bound. fom_gap_pfp_euclidean_common_identity_right_bounded_value_bound + S (fom_value_pfp_euclidean_common_identity_right_bounded) = p))) /\\ (((forall fom_index_pfp_euclidean_common_identity_result_bounded. (exists fom_gap_pfp_euclidean_common_identity_result_bounded_index_bound. fom_gap_pfp_euclidean_common_identity_result_bounded_index_bound + S (fom_index_pfp_euclidean_common_identity_result_bounded) = L) -> exists fom_value_pfp_euclidean_common_identity_result_bounded. ((((exists fom_beta_height_pfp_euclidean_common_identity_result_bounded_entry. fom_beta_height_pfp_euclidean_common_identity_result_bounded_entry + S (fom_value_pfp_euclidean_common_identity_result_bounded) = S ((S (fom_index_pfp_euclidean_common_identity_result_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_euclidean_common_identity_result_bounded_entry. ab = fom_beta_quotient_pfp_euclidean_common_identity_result_bounded_entry * S ((S (fom_index_pfp_euclidean_common_identity_result_bounded)) * ac) + (fom_value_pfp_euclidean_common_identity_result_bounded))) /\\ (exists fom_gap_pfp_euclidean_common_identity_result_bounded_value_bound. fom_gap_pfp_euclidean_common_identity_result_bounded_value_bound + S (fom_value_pfp_euclidean_common_identity_result_bounded) = p))) /\\ ((exists pfaa_left_b_euclidean_common_identity pfaa_left_c_euclidean_common_identity pfaa_right_b_euclidean_common_identity pfaa_right_c_euclidean_common_identity pfaa_sum_b_euclidean_common_identity pfaa_sum_c_euclidean_common_identity pfaa_length_euclidean_common_identity. ((((forall pfrep_power_euclidean_common_identity_witness_common_left pfrep_left_euclidean_common_identity_witness_common_left pfrep_right_euclidean_common_identity_witness_common_left. ((exists pfrep_position_euclidean_common_identity_witness_common_leftfirst. ((pfrep_position_euclidean_common_identity_witness_common_leftfirst+S (pfrep_power_euclidean_common_identity_witness_common_left)=(I)) /\\ ((((exists ff_h_pfp_euclidean_common_identity_witness_common_leftfirstentry. ff_h_pfp_euclidean_common_identity_witness_common_leftfirstentry + S (pfrep_left_euclidean_common_identity_witness_common_left) = S ((S (pfrep_position_euclidean_common_identity_witness_common_leftfirst)) * pc)) /\\ exists ff_q_pfp_euclidean_common_identity_witness_common_leftfirstentry. pb = ff_q_pfp_euclidean_common_identity_witness_common_leftfirstentry * S ((S (pfrep_position_euclidean_common_identity_witness_common_leftfirst)) * pc) + (pfrep_left_euclidean_common_identity_witness_common_left)))))) \\/ (((exists pfrep_gap_euclidean_common_identity_witness_common_leftfirstoutside. pfrep_gap_euclidean_common_identity_witness_common_leftfirstoutside+(I)=(pfrep_power_euclidean_common_identity_witness_common_left)) /\\ (((pfrep_left_euclidean_common_identity_witness_common_left)=0))))) -> ((exists pfrep_position_euclidean_common_identity_witness_common_leftsecond. ((pfrep_position_euclidean_common_identity_witness_common_leftsecond+S (pfrep_power_euclidean_common_identity_witness_common_left)=(pfaa_length_euclidean_common_identity)) /\\ ((((exists ff_h_pfp_euclidean_common_identity_witness_common_leftsecondentry. ff_h_pfp_euclidean_common_identity_witness_common_leftsecondentry + S (pfrep_right_euclidean_common_identity_witness_common_left) = S ((S (pfrep_position_euclidean_common_identity_witness_common_leftsecond)) * pfaa_left_c_euclidean_common_identity)) /\\ exists ff_q_pfp_euclidean_common_identity_witness_common_leftsecondentry. pfaa_left_b_euclidean_common_identity = ff_q_pfp_euclidean_common_identity_witness_common_leftsecondentry * S ((S (pfrep_position_euclidean_common_identity_witness_common_leftsecond)) * pfaa_left_c_euclidean_common_identity) + (pfrep_right_euclidean_common_identity_witness_common_left)))))) \\/ (((exists pfrep_gap_euclidean_common_identity_witness_common_leftsecondoutside. pfrep_gap_euclidean_common_identity_witness_common_leftsecondoutside+(pfaa_length_euclidean_common_identity)=(pfrep_power_euclidean_common_identity_witness_common_left)) /\\ (((pfrep_right_euclidean_common_identity_witness_common_left)=0))))) -> pfrep_left_euclidean_common_identity_witness_common_left=pfrep_right_euclidean_common_identity_witness_common_left) /\\ ((forall pfrep_power_euclidean_common_identity_witness_common_right pfrep_left_euclidean_common_identity_witness_common_right pfrep_right_euclidean_common_identity_witness_common_right. ((exists pfrep_position_euclidean_common_identity_witness_common_rightfirst. ((pfrep_position_euclidean_common_identity_witness_common_rightfirst+S (pfrep_power_euclidean_common_identity_witness_common_right)=(N)) /\\ ((((exists ff_h_pfp_euclidean_common_identity_witness_common_rightfirstentry. ff_h_pfp_euclidean_common_identity_witness_common_rightfirstentry + S (pfrep_left_euclidean_common_identity_witness_common_right) = S ((S (pfrep_position_euclidean_common_identity_witness_common_rightfirst)) * rc)) /\\ exists ff_q_pfp_euclidean_common_identity_witness_common_rightfirstentry. rb = ff_q_pfp_euclidean_common_identity_witness_common_rightfirstentry * S ((S (pfrep_position_euclidean_common_identity_witness_common_rightfirst)) * rc) + (pfrep_left_euclidean_common_identity_witness_common_right)))))) \\/ (((exists pfrep_gap_euclidean_common_identity_witness_common_rightfirstoutside. pfrep_gap_euclidean_common_identity_witness_common_rightfirstoutside+(N)=(pfrep_power_euclidean_common_identity_witness_common_right)) /\\ (((pfrep_left_euclidean_common_identity_witness_common_right)=0))))) -> ((exists pfrep_position_euclidean_common_identity_witness_common_rightsecond. ((pfrep_position_euclidean_common_identity_witness_common_rightsecond+S (pfrep_power_euclidean_common_identity_witness_common_right)=(pfaa_length_euclidean_common_identity)) /\\ ((((exists ff_h_pfp_euclidean_common_identity_witness_common_rightsecondentry. ff_h_pfp_euclidean_common_identity_witness_common_rightsecondentry + S (pfrep_right_euclidean_common_identity_witness_common_right) = S ((S (pfrep_position_euclidean_common_identity_witness_common_rightsecond)) * pfaa_right_c_euclidean_common_identity)) /\\ exists ff_q_pfp_euclidean_common_identity_witness_common_rightsecondentry. pfaa_right_b_euclidean_common_identity = ff_q_pfp_euclidean_common_identity_witness_common_rightsecondentry * S ((S (pfrep_position_euclidean_common_identity_witness_common_rightsecond)) * pfaa_right_c_euclidean_common_identity) + (pfrep_right_euclidean_common_identity_witness_common_right)))))) \\/ (((exists pfrep_gap_euclidean_common_identity_witness_common_rightsecondoutside. pfrep_gap_euclidean_common_identity_witness_common_rightsecondoutside+(pfaa_length_euclidean_common_identity)=(pfrep_power_euclidean_common_identity_witness_common_right)) /\\ (((pfrep_right_euclidean_common_identity_witness_common_right)=0))))) -> pfrep_left_euclidean_common_identity_witness_common_right=pfrep_right_euclidean_common_identity_witness_common_right)))) /\\ (((forall pfp_index_euclidean_common_identity_witness_operation. (exists pfa_gap_euclidean_common_identity_witness_operationindex. pfa_gap_euclidean_common_identity_witness_operationindex + S (pfp_index_euclidean_common_identity_witness_operation) = (pfaa_length_euclidean_common_identity)) -> exists pfp_left_euclidean_common_identity_witness_operation pfp_right_euclidean_common_identity_witness_operation pfp_value_euclidean_common_identity_witness_operation. ((((exists ff_h_pfp_euclidean_common_identity_witness_operationleft. ff_h_pfp_euclidean_common_identity_witness_operationleft + S (pfp_left_euclidean_common_identity_witness_operation) = S ((S (pfp_index_euclidean_common_identity_witness_operation)) * pfaa_left_c_euclidean_common_identity)) /\\ exists ff_q_pfp_euclidean_common_identity_witness_operationleft. pfaa_left_b_euclidean_common_identity = ff_q_pfp_euclidean_common_identity_witness_operationleft * S ((S (pfp_index_euclidean_common_identity_witness_operation)) * pfaa_left_c_euclidean_common_identity) + (pfp_left_euclidean_common_identity_witness_operation))) /\\ (((((exists ff_h_pfp_euclidean_common_identity_witness_operationright. ff_h_pfp_euclidean_common_identity_witness_operationright + S (pfp_right_euclidean_common_identity_witness_operation) = S ((S (pfp_index_euclidean_common_identity_witness_operation)) * pfaa_right_c_euclidean_common_identity)) /\\ exists ff_q_pfp_euclidean_common_identity_witness_operationright. pfaa_right_b_euclidean_common_identity = ff_q_pfp_euclidean_common_identity_witness_operationright * S ((S (pfp_index_euclidean_common_identity_witness_operation)) * pfaa_right_c_euclidean_common_identity) + (pfp_right_euclidean_common_identity_witness_operation))) /\\ (((((exists ff_h_pfp_euclidean_common_identity_witness_operationtarget. ff_h_pfp_euclidean_common_identity_witness_operationtarget + S (pfp_value_euclidean_common_identity_witness_operation) = S ((S (pfp_index_euclidean_common_identity_witness_operation)) * pfaa_sum_c_euclidean_common_identity)) /\\ exists ff_q_pfp_euclidean_common_identity_witness_operationtarget. pfaa_sum_b_euclidean_common_identity = ff_q_pfp_euclidean_common_identity_witness_operationtarget * S ((S (pfp_index_euclidean_common_identity_witness_operation)) * pfaa_sum_c_euclidean_common_identity) + (pfp_value_euclidean_common_identity_witness_operation))) /\\ ((((exists pfa_gap_euclidean_common_identity_witness_operationoperationleft. pfa_gap_euclidean_common_identity_witness_operationoperationleft + S (pfp_left_euclidean_common_identity_witness_operation) = (p)) /\\ (((exists pfa_gap_euclidean_common_identity_witness_operationoperationright. pfa_gap_euclidean_common_identity_witness_operationoperationright + S (pfp_right_euclidean_common_identity_witness_operation) = (p)) /\\ ((((exists pfa_gap_euclidean_common_identity_witness_operationoperationresultbound. pfa_gap_euclidean_common_identity_witness_operationoperationresultbound + S (pfp_value_euclidean_common_identity_witness_operation) = (p)) /\\ ((exists pfa_offset_left_euclidean_common_identity_witness_operationoperationresultcongruence pfa_offset_right_euclidean_common_identity_witness_operationoperationresultcongruence. ((pfp_left_euclidean_common_identity_witness_operation) + (pfp_right_euclidean_common_identity_witness_operation)) + (p) * pfa_offset_left_euclidean_common_identity_witness_operationoperationresultcongruence = (pfp_value_euclidean_common_identity_witness_operation) + (p) * pfa_offset_right_euclidean_common_identity_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_euclidean_common_identity_witness_output pfrep_left_euclidean_common_identity_witness_output pfrep_right_euclidean_common_identity_witness_output. ((exists pfrep_position_euclidean_common_identity_witness_outputfirst. ((pfrep_position_euclidean_common_identity_witness_outputfirst+S (pfrep_power_euclidean_common_identity_witness_output)=(pfaa_length_euclidean_common_identity)) /\\ ((((exists ff_h_pfp_euclidean_common_identity_witness_outputfirstentry. ff_h_pfp_euclidean_common_identity_witness_outputfirstentry + S (pfrep_left_euclidean_common_identity_witness_output) = S ((S (pfrep_position_euclidean_common_identity_witness_outputfirst)) * pfaa_sum_c_euclidean_common_identity)) /\\ exists ff_q_pfp_euclidean_common_identity_witness_outputfirstentry. pfaa_sum_b_euclidean_common_identity = ff_q_pfp_euclidean_common_identity_witness_outputfirstentry * S ((S (pfrep_position_euclidean_common_identity_witness_outputfirst)) * pfaa_sum_c_euclidean_common_identity) + (pfrep_left_euclidean_common_identity_witness_output)))))) \\/ (((exists pfrep_gap_euclidean_common_identity_witness_outputfirstoutside. pfrep_gap_euclidean_common_identity_witness_outputfirstoutside+(pfaa_length_euclidean_common_identity)=(pfrep_power_euclidean_common_identity_witness_output)) /\\ (((pfrep_left_euclidean_common_identity_witness_output)=0))))) -> ((exists pfrep_position_euclidean_common_identity_witness_outputsecond. ((pfrep_position_euclidean_common_identity_witness_outputsecond+S (pfrep_power_euclidean_common_identity_witness_output)=(L)) /\\ ((((exists ff_h_pfp_euclidean_common_identity_witness_outputsecondentry. ff_h_pfp_euclidean_common_identity_witness_outputsecondentry + S (pfrep_right_euclidean_common_identity_witness_output) = S ((S (pfrep_position_euclidean_common_identity_witness_outputsecond)) * ac)) /\\ exists ff_q_pfp_euclidean_common_identity_witness_outputsecondentry. ab = ff_q_pfp_euclidean_common_identity_witness_outputsecondentry * S ((S (pfrep_position_euclidean_common_identity_witness_outputsecond)) * ac) + (pfrep_right_euclidean_common_identity_witness_output)))))) \\/ (((exists pfrep_gap_euclidean_common_identity_witness_outputsecondoutside. pfrep_gap_euclidean_common_identity_witness_outputsecondoutside+(L)=(pfrep_power_euclidean_common_identity_witness_output)) /\\ (((pfrep_right_euclidean_common_identity_witness_output)=0))))) -> pfrep_left_euclidean_common_identity_witness_output=pfrep_right_euclidean_common_identity_witness_output))))))))))))) -> ((((((((forall fom_index_pfp_euclidean_common_source_left_canonical. (exists fom_gap_pfp_euclidean_common_source_left_canonical_index_bound. fom_gap_pfp_euclidean_common_source_left_canonical_index_bound + S (fom_index_pfp_euclidean_common_source_left_canonical) = L) -> exists fom_value_pfp_euclidean_common_source_left_canonical. ((((exists fom_beta_height_pfp_euclidean_common_source_left_canonical_entry. fom_beta_height_pfp_euclidean_common_source_left_canonical_entry + S (fom_value_pfp_euclidean_common_source_left_canonical) = S ((S (fom_index_pfp_euclidean_common_source_left_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_euclidean_common_source_left_canonical_entry. ab = fom_beta_quotient_pfp_euclidean_common_source_left_canonical_entry * S ((S (fom_index_pfp_euclidean_common_source_left_canonical)) * ac) + (fom_value_pfp_euclidean_common_source_left_canonical))) /\\ (exists fom_gap_pfp_euclidean_common_source_left_canonical_value_bound. fom_gap_pfp_euclidean_common_source_left_canonical_value_bound + S (fom_value_pfp_euclidean_common_source_left_canonical) = p))) /\\ ((exists pfrd_qb_euclidean_common_source_left pfrd_qc_euclidean_common_source_left pfrd_qlen_euclidean_common_source_left pfrd_pb_euclidean_common_source_left pfrd_pc_euclidean_common_source_left pfrd_plen_euclidean_common_source_left. ((((forall fom_index_pfp_euclidean_common_source_left_productleft. (exists fom_gap_pfp_euclidean_common_source_left_productleft_index_bound. fom_gap_pfp_euclidean_common_source_left_productleft_index_bound + S (fom_index_pfp_euclidean_common_source_left_productleft) = pfrd_qlen_euclidean_common_source_left) -> exists fom_value_pfp_euclidean_common_source_left_productleft. ((((exists fom_beta_height_pfp_euclidean_common_source_left_productleft_entry. fom_beta_height_pfp_euclidean_common_source_left_productleft_entry + S (fom_value_pfp_euclidean_common_source_left_productleft) = S ((S (fom_index_pfp_euclidean_common_source_left_productleft)) * pfrd_qc_euclidean_common_source_left)) /\\ exists fom_beta_quotient_pfp_euclidean_common_source_left_productleft_entry. pfrd_qb_euclidean_common_source_left = fom_beta_quotient_pfp_euclidean_common_source_left_productleft_entry * S ((S (fom_index_pfp_euclidean_common_source_left_productleft)) * pfrd_qc_euclidean_common_source_left) + (fom_value_pfp_euclidean_common_source_left_productleft))) /\\ (exists fom_gap_pfp_euclidean_common_source_left_productleft_value_bound. fom_gap_pfp_euclidean_common_source_left_productleft_value_bound + S (fom_value_pfp_euclidean_common_source_left_productleft) = p))) /\\ (((forall fom_index_pfp_euclidean_common_source_left_productright. (exists fom_gap_pfp_euclidean_common_source_left_productright_index_bound. fom_gap_pfp_euclidean_common_source_left_productright_index_bound + S (fom_index_pfp_euclidean_common_source_left_productright) = J) -> exists fom_value_pfp_euclidean_common_source_left_productright. ((((exists fom_beta_height_pfp_euclidean_common_source_left_productright_entry. fom_beta_height_pfp_euclidean_common_source_left_productright_entry + S (fom_value_pfp_euclidean_common_source_left_productright) = S ((S (fom_index_pfp_euclidean_common_source_left_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_source_left_productright_entry. db = fom_beta_quotient_pfp_euclidean_common_source_left_productright_entry * S ((S (fom_index_pfp_euclidean_common_source_left_productright)) * dc) + (fom_value_pfp_euclidean_common_source_left_productright))) /\\ (exists fom_gap_pfp_euclidean_common_source_left_productright_value_bound. fom_gap_pfp_euclidean_common_source_left_productright_value_bound + S (fom_value_pfp_euclidean_common_source_left_productright) = p))) /\\ (((((((pfrd_qlen_euclidean_common_source_left)=0 \\/ (J)=0) /\\ (((pfrd_plen_euclidean_common_source_left)=0)))) \\/ (((~((pfrd_qlen_euclidean_common_source_left)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_euclidean_common_source_left)+(J)=S (pfrd_plen_euclidean_common_source_left)))))))) /\\ ((forall pfc_index_euclidean_common_source_left_productcoefficients. (exists pfa_gap_euclidean_common_source_left_productcoefficientsbound. pfa_gap_euclidean_common_source_left_productcoefficientsbound + S (pfc_index_euclidean_common_source_left_productcoefficients) = (pfrd_plen_euclidean_common_source_left)) -> exists pfc_value_euclidean_common_source_left_productcoefficients. ((((exists ff_h_pfp_euclidean_common_source_left_productcoefficientsentry. ff_h_pfp_euclidean_common_source_left_productcoefficientsentry + S (pfc_value_euclidean_common_source_left_productcoefficients) = S ((S (pfc_index_euclidean_common_source_left_productcoefficients)) * pfrd_pc_euclidean_common_source_left)) /\\ exists ff_q_pfp_euclidean_common_source_left_productcoefficientsentry. pfrd_pb_euclidean_common_source_left = ff_q_pfp_euclidean_common_source_left_productcoefficientsentry * S ((S (pfc_index_euclidean_common_source_left_productcoefficients)) * pfrd_pc_euclidean_common_source_left) + (pfc_value_euclidean_common_source_left_productcoefficients))) /\\ ((exists pfc_terms_code_euclidean_common_source_left_productcoefficientscoefficient pfc_terms_scale_euclidean_common_source_left_productcoefficientscoefficient pfc_natural_sum_euclidean_common_source_left_productcoefficientscoefficient. ((forall pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonalbound. pfa_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_euclidean_common_source_left_productcoefficients))) -> exists pfc_value_euclidean_common_source_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_euclidean_common_source_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_source_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_euclidean_common_source_left_productcoefficientscoefficient = ff_q_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_source_left_productcoefficientscoefficient) + (pfc_value_euclidean_common_source_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm pfc_left_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm pfc_right_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal)+pfc_complement_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm=(pfc_index_euclidean_common_source_left_productcoefficients)) /\\ ((((((exists pfa_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal) = (pfrd_qlen_euclidean_common_source_left)) /\\ ((((exists ff_h_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_source_left)) /\\ exists ff_q_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_euclidean_common_source_left = ff_q_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_source_left) + (pfc_left_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_euclidean_common_source_left)=(pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_euclidean_common_source_left_productcoefficientscoefficientdiagonal)=pfc_left_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm*pfc_right_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_euclidean_common_source_left_productcoefficientscoefficientsum fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_euclidean_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_euclidean_common_source_left_productcoefficientscoefficient) = S ((S (S (pfc_index_euclidean_common_source_left_productcoefficients))) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_euclidean_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_euclidean_common_source_left_productcoefficients))) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum) + (pfc_natural_sum_euclidean_common_source_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_euclidean_common_source_left_productcoefficients)) -> exists fs_a_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_source_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_euclidean_common_source_left_productcoefficientscoefficient = fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_source_left_productcoefficientscoefficient) + (fs_a_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_euclidean_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum) + (fs_r_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_euclidean_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum) + (fs_s_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_euclidean_common_source_left_productcoefficientscoefficientresiduebound. pfa_gap_euclidean_common_source_left_productcoefficientscoefficientresiduebound + S (pfc_value_euclidean_common_source_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_euclidean_common_source_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_euclidean_common_source_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_euclidean_common_source_left_productcoefficientscoefficient) + (p) * pfa_offset_left_euclidean_common_source_left_productcoefficientscoefficientresiduecongruence = (pfc_value_euclidean_common_source_left_productcoefficients) + (p) * pfa_offset_right_euclidean_common_source_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_euclidean_common_source_left_target pfrep_left_euclidean_common_source_left_target pfrep_right_euclidean_common_source_left_target. ((exists pfrep_position_euclidean_common_source_left_targetfirst. ((pfrep_position_euclidean_common_source_left_targetfirst+S (pfrep_power_euclidean_common_source_left_target)=(pfrd_plen_euclidean_common_source_left)) /\\ ((((exists ff_h_pfp_euclidean_common_source_left_targetfirstentry. ff_h_pfp_euclidean_common_source_left_targetfirstentry + S (pfrep_left_euclidean_common_source_left_target) = S ((S (pfrep_position_euclidean_common_source_left_targetfirst)) * pfrd_pc_euclidean_common_source_left)) /\\ exists ff_q_pfp_euclidean_common_source_left_targetfirstentry. pfrd_pb_euclidean_common_source_left = ff_q_pfp_euclidean_common_source_left_targetfirstentry * S ((S (pfrep_position_euclidean_common_source_left_targetfirst)) * pfrd_pc_euclidean_common_source_left) + (pfrep_left_euclidean_common_source_left_target)))))) \\/ (((exists pfrep_gap_euclidean_common_source_left_targetfirstoutside. pfrep_gap_euclidean_common_source_left_targetfirstoutside+(pfrd_plen_euclidean_common_source_left)=(pfrep_power_euclidean_common_source_left_target)) /\\ (((pfrep_left_euclidean_common_source_left_target)=0))))) -> ((exists pfrep_position_euclidean_common_source_left_targetsecond. ((pfrep_position_euclidean_common_source_left_targetsecond+S (pfrep_power_euclidean_common_source_left_target)=(L)) /\\ ((((exists ff_h_pfp_euclidean_common_source_left_targetsecondentry. ff_h_pfp_euclidean_common_source_left_targetsecondentry + S (pfrep_right_euclidean_common_source_left_target) = S ((S (pfrep_position_euclidean_common_source_left_targetsecond)) * ac)) /\\ exists ff_q_pfp_euclidean_common_source_left_targetsecondentry. ab = ff_q_pfp_euclidean_common_source_left_targetsecondentry * S ((S (pfrep_position_euclidean_common_source_left_targetsecond)) * ac) + (pfrep_right_euclidean_common_source_left_target)))))) \\/ (((exists pfrep_gap_euclidean_common_source_left_targetsecondoutside. pfrep_gap_euclidean_common_source_left_targetsecondoutside+(L)=(pfrep_power_euclidean_common_source_left_target)) /\\ (((pfrep_right_euclidean_common_source_left_target)=0))))) -> pfrep_left_euclidean_common_source_left_target=pfrep_right_euclidean_common_source_left_target))))))) /\\ ((((forall fom_index_pfp_euclidean_common_source_right_canonical. (exists fom_gap_pfp_euclidean_common_source_right_canonical_index_bound. fom_gap_pfp_euclidean_common_source_right_canonical_index_bound + S (fom_index_pfp_euclidean_common_source_right_canonical) = M) -> exists fom_value_pfp_euclidean_common_source_right_canonical. ((((exists fom_beta_height_pfp_euclidean_common_source_right_canonical_entry. fom_beta_height_pfp_euclidean_common_source_right_canonical_entry + S (fom_value_pfp_euclidean_common_source_right_canonical) = S ((S (fom_index_pfp_euclidean_common_source_right_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_source_right_canonical_entry. bb = fom_beta_quotient_pfp_euclidean_common_source_right_canonical_entry * S ((S (fom_index_pfp_euclidean_common_source_right_canonical)) * bc) + (fom_value_pfp_euclidean_common_source_right_canonical))) /\\ (exists fom_gap_pfp_euclidean_common_source_right_canonical_value_bound. fom_gap_pfp_euclidean_common_source_right_canonical_value_bound + S (fom_value_pfp_euclidean_common_source_right_canonical) = p))) /\\ ((exists pfrd_qb_euclidean_common_source_right pfrd_qc_euclidean_common_source_right pfrd_qlen_euclidean_common_source_right pfrd_pb_euclidean_common_source_right pfrd_pc_euclidean_common_source_right pfrd_plen_euclidean_common_source_right. ((((forall fom_index_pfp_euclidean_common_source_right_productleft. (exists fom_gap_pfp_euclidean_common_source_right_productleft_index_bound. fom_gap_pfp_euclidean_common_source_right_productleft_index_bound + S (fom_index_pfp_euclidean_common_source_right_productleft) = pfrd_qlen_euclidean_common_source_right) -> exists fom_value_pfp_euclidean_common_source_right_productleft. ((((exists fom_beta_height_pfp_euclidean_common_source_right_productleft_entry. fom_beta_height_pfp_euclidean_common_source_right_productleft_entry + S (fom_value_pfp_euclidean_common_source_right_productleft) = S ((S (fom_index_pfp_euclidean_common_source_right_productleft)) * pfrd_qc_euclidean_common_source_right)) /\\ exists fom_beta_quotient_pfp_euclidean_common_source_right_productleft_entry. pfrd_qb_euclidean_common_source_right = fom_beta_quotient_pfp_euclidean_common_source_right_productleft_entry * S ((S (fom_index_pfp_euclidean_common_source_right_productleft)) * pfrd_qc_euclidean_common_source_right) + (fom_value_pfp_euclidean_common_source_right_productleft))) /\\ (exists fom_gap_pfp_euclidean_common_source_right_productleft_value_bound. fom_gap_pfp_euclidean_common_source_right_productleft_value_bound + S (fom_value_pfp_euclidean_common_source_right_productleft) = p))) /\\ (((forall fom_index_pfp_euclidean_common_source_right_productright. (exists fom_gap_pfp_euclidean_common_source_right_productright_index_bound. fom_gap_pfp_euclidean_common_source_right_productright_index_bound + S (fom_index_pfp_euclidean_common_source_right_productright) = J) -> exists fom_value_pfp_euclidean_common_source_right_productright. ((((exists fom_beta_height_pfp_euclidean_common_source_right_productright_entry. fom_beta_height_pfp_euclidean_common_source_right_productright_entry + S (fom_value_pfp_euclidean_common_source_right_productright) = S ((S (fom_index_pfp_euclidean_common_source_right_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_source_right_productright_entry. db = fom_beta_quotient_pfp_euclidean_common_source_right_productright_entry * S ((S (fom_index_pfp_euclidean_common_source_right_productright)) * dc) + (fom_value_pfp_euclidean_common_source_right_productright))) /\\ (exists fom_gap_pfp_euclidean_common_source_right_productright_value_bound. fom_gap_pfp_euclidean_common_source_right_productright_value_bound + S (fom_value_pfp_euclidean_common_source_right_productright) = p))) /\\ (((((((pfrd_qlen_euclidean_common_source_right)=0 \\/ (J)=0) /\\ (((pfrd_plen_euclidean_common_source_right)=0)))) \\/ (((~((pfrd_qlen_euclidean_common_source_right)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_euclidean_common_source_right)+(J)=S (pfrd_plen_euclidean_common_source_right)))))))) /\\ ((forall pfc_index_euclidean_common_source_right_productcoefficients. (exists pfa_gap_euclidean_common_source_right_productcoefficientsbound. pfa_gap_euclidean_common_source_right_productcoefficientsbound + S (pfc_index_euclidean_common_source_right_productcoefficients) = (pfrd_plen_euclidean_common_source_right)) -> exists pfc_value_euclidean_common_source_right_productcoefficients. ((((exists ff_h_pfp_euclidean_common_source_right_productcoefficientsentry. ff_h_pfp_euclidean_common_source_right_productcoefficientsentry + S (pfc_value_euclidean_common_source_right_productcoefficients) = S ((S (pfc_index_euclidean_common_source_right_productcoefficients)) * pfrd_pc_euclidean_common_source_right)) /\\ exists ff_q_pfp_euclidean_common_source_right_productcoefficientsentry. pfrd_pb_euclidean_common_source_right = ff_q_pfp_euclidean_common_source_right_productcoefficientsentry * S ((S (pfc_index_euclidean_common_source_right_productcoefficients)) * pfrd_pc_euclidean_common_source_right) + (pfc_value_euclidean_common_source_right_productcoefficients))) /\\ ((exists pfc_terms_code_euclidean_common_source_right_productcoefficientscoefficient pfc_terms_scale_euclidean_common_source_right_productcoefficientscoefficient pfc_natural_sum_euclidean_common_source_right_productcoefficientscoefficient. ((forall pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonalbound. pfa_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_euclidean_common_source_right_productcoefficients))) -> exists pfc_value_euclidean_common_source_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_euclidean_common_source_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_source_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_euclidean_common_source_right_productcoefficientscoefficient = ff_q_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_source_right_productcoefficientscoefficient) + (pfc_value_euclidean_common_source_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm pfc_left_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm pfc_right_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal)+pfc_complement_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm=(pfc_index_euclidean_common_source_right_productcoefficients)) /\\ ((((((exists pfa_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal) = (pfrd_qlen_euclidean_common_source_right)) /\\ ((((exists ff_h_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_source_right)) /\\ exists ff_q_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_euclidean_common_source_right = ff_q_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_source_right) + (pfc_left_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_euclidean_common_source_right)=(pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_euclidean_common_source_right_productcoefficientscoefficientdiagonal)=pfc_left_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm*pfc_right_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_euclidean_common_source_right_productcoefficientscoefficientsum fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_euclidean_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_euclidean_common_source_right_productcoefficientscoefficient) = S ((S (S (pfc_index_euclidean_common_source_right_productcoefficients))) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_euclidean_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_euclidean_common_source_right_productcoefficients))) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum) + (pfc_natural_sum_euclidean_common_source_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_euclidean_common_source_right_productcoefficients)) -> exists fs_a_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_source_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_euclidean_common_source_right_productcoefficientscoefficient = fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_source_right_productcoefficientscoefficient) + (fs_a_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_euclidean_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum) + (fs_r_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_euclidean_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum) + (fs_s_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_euclidean_common_source_right_productcoefficientscoefficientresiduebound. pfa_gap_euclidean_common_source_right_productcoefficientscoefficientresiduebound + S (pfc_value_euclidean_common_source_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_euclidean_common_source_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_euclidean_common_source_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_euclidean_common_source_right_productcoefficientscoefficient) + (p) * pfa_offset_left_euclidean_common_source_right_productcoefficientscoefficientresiduecongruence = (pfc_value_euclidean_common_source_right_productcoefficients) + (p) * pfa_offset_right_euclidean_common_source_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_euclidean_common_source_right_target pfrep_left_euclidean_common_source_right_target pfrep_right_euclidean_common_source_right_target. ((exists pfrep_position_euclidean_common_source_right_targetfirst. ((pfrep_position_euclidean_common_source_right_targetfirst+S (pfrep_power_euclidean_common_source_right_target)=(pfrd_plen_euclidean_common_source_right)) /\\ ((((exists ff_h_pfp_euclidean_common_source_right_targetfirstentry. ff_h_pfp_euclidean_common_source_right_targetfirstentry + S (pfrep_left_euclidean_common_source_right_target) = S ((S (pfrep_position_euclidean_common_source_right_targetfirst)) * pfrd_pc_euclidean_common_source_right)) /\\ exists ff_q_pfp_euclidean_common_source_right_targetfirstentry. pfrd_pb_euclidean_common_source_right = ff_q_pfp_euclidean_common_source_right_targetfirstentry * S ((S (pfrep_position_euclidean_common_source_right_targetfirst)) * pfrd_pc_euclidean_common_source_right) + (pfrep_left_euclidean_common_source_right_target)))))) \\/ (((exists pfrep_gap_euclidean_common_source_right_targetfirstoutside. pfrep_gap_euclidean_common_source_right_targetfirstoutside+(pfrd_plen_euclidean_common_source_right)=(pfrep_power_euclidean_common_source_right_target)) /\\ (((pfrep_left_euclidean_common_source_right_target)=0))))) -> ((exists pfrep_position_euclidean_common_source_right_targetsecond. ((pfrep_position_euclidean_common_source_right_targetsecond+S (pfrep_power_euclidean_common_source_right_target)=(M)) /\\ ((((exists ff_h_pfp_euclidean_common_source_right_targetsecondentry. ff_h_pfp_euclidean_common_source_right_targetsecondentry + S (pfrep_right_euclidean_common_source_right_target) = S ((S (pfrep_position_euclidean_common_source_right_targetsecond)) * bc)) /\\ exists ff_q_pfp_euclidean_common_source_right_targetsecondentry. bb = ff_q_pfp_euclidean_common_source_right_targetsecondentry * S ((S (pfrep_position_euclidean_common_source_right_targetsecond)) * bc) + (pfrep_right_euclidean_common_source_right_target)))))) \\/ (((exists pfrep_gap_euclidean_common_source_right_targetsecondoutside. pfrep_gap_euclidean_common_source_right_targetsecondoutside+(M)=(pfrep_power_euclidean_common_source_right_target)) /\\ (((pfrep_right_euclidean_common_source_right_target)=0))))) -> pfrep_left_euclidean_common_source_right_target=pfrep_right_euclidean_common_source_right_target)))))))))) -> (((((forall fom_index_pfp_euclidean_common_target_left_canonical. (exists fom_gap_pfp_euclidean_common_target_left_canonical_index_bound. fom_gap_pfp_euclidean_common_target_left_canonical_index_bound + S (fom_index_pfp_euclidean_common_target_left_canonical) = M) -> exists fom_value_pfp_euclidean_common_target_left_canonical. ((((exists fom_beta_height_pfp_euclidean_common_target_left_canonical_entry. fom_beta_height_pfp_euclidean_common_target_left_canonical_entry + S (fom_value_pfp_euclidean_common_target_left_canonical) = S ((S (fom_index_pfp_euclidean_common_target_left_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_target_left_canonical_entry. bb = fom_beta_quotient_pfp_euclidean_common_target_left_canonical_entry * S ((S (fom_index_pfp_euclidean_common_target_left_canonical)) * bc) + (fom_value_pfp_euclidean_common_target_left_canonical))) /\\ (exists fom_gap_pfp_euclidean_common_target_left_canonical_value_bound. fom_gap_pfp_euclidean_common_target_left_canonical_value_bound + S (fom_value_pfp_euclidean_common_target_left_canonical) = p))) /\\ ((exists pfrd_qb_euclidean_common_target_left pfrd_qc_euclidean_common_target_left pfrd_qlen_euclidean_common_target_left pfrd_pb_euclidean_common_target_left pfrd_pc_euclidean_common_target_left pfrd_plen_euclidean_common_target_left. ((((forall fom_index_pfp_euclidean_common_target_left_productleft. (exists fom_gap_pfp_euclidean_common_target_left_productleft_index_bound. fom_gap_pfp_euclidean_common_target_left_productleft_index_bound + S (fom_index_pfp_euclidean_common_target_left_productleft) = pfrd_qlen_euclidean_common_target_left) -> exists fom_value_pfp_euclidean_common_target_left_productleft. ((((exists fom_beta_height_pfp_euclidean_common_target_left_productleft_entry. fom_beta_height_pfp_euclidean_common_target_left_productleft_entry + S (fom_value_pfp_euclidean_common_target_left_productleft) = S ((S (fom_index_pfp_euclidean_common_target_left_productleft)) * pfrd_qc_euclidean_common_target_left)) /\\ exists fom_beta_quotient_pfp_euclidean_common_target_left_productleft_entry. pfrd_qb_euclidean_common_target_left = fom_beta_quotient_pfp_euclidean_common_target_left_productleft_entry * S ((S (fom_index_pfp_euclidean_common_target_left_productleft)) * pfrd_qc_euclidean_common_target_left) + (fom_value_pfp_euclidean_common_target_left_productleft))) /\\ (exists fom_gap_pfp_euclidean_common_target_left_productleft_value_bound. fom_gap_pfp_euclidean_common_target_left_productleft_value_bound + S (fom_value_pfp_euclidean_common_target_left_productleft) = p))) /\\ (((forall fom_index_pfp_euclidean_common_target_left_productright. (exists fom_gap_pfp_euclidean_common_target_left_productright_index_bound. fom_gap_pfp_euclidean_common_target_left_productright_index_bound + S (fom_index_pfp_euclidean_common_target_left_productright) = J) -> exists fom_value_pfp_euclidean_common_target_left_productright. ((((exists fom_beta_height_pfp_euclidean_common_target_left_productright_entry. fom_beta_height_pfp_euclidean_common_target_left_productright_entry + S (fom_value_pfp_euclidean_common_target_left_productright) = S ((S (fom_index_pfp_euclidean_common_target_left_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_target_left_productright_entry. db = fom_beta_quotient_pfp_euclidean_common_target_left_productright_entry * S ((S (fom_index_pfp_euclidean_common_target_left_productright)) * dc) + (fom_value_pfp_euclidean_common_target_left_productright))) /\\ (exists fom_gap_pfp_euclidean_common_target_left_productright_value_bound. fom_gap_pfp_euclidean_common_target_left_productright_value_bound + S (fom_value_pfp_euclidean_common_target_left_productright) = p))) /\\ (((((((pfrd_qlen_euclidean_common_target_left)=0 \\/ (J)=0) /\\ (((pfrd_plen_euclidean_common_target_left)=0)))) \\/ (((~((pfrd_qlen_euclidean_common_target_left)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_euclidean_common_target_left)+(J)=S (pfrd_plen_euclidean_common_target_left)))))))) /\\ ((forall pfc_index_euclidean_common_target_left_productcoefficients. (exists pfa_gap_euclidean_common_target_left_productcoefficientsbound. pfa_gap_euclidean_common_target_left_productcoefficientsbound + S (pfc_index_euclidean_common_target_left_productcoefficients) = (pfrd_plen_euclidean_common_target_left)) -> exists pfc_value_euclidean_common_target_left_productcoefficients. ((((exists ff_h_pfp_euclidean_common_target_left_productcoefficientsentry. ff_h_pfp_euclidean_common_target_left_productcoefficientsentry + S (pfc_value_euclidean_common_target_left_productcoefficients) = S ((S (pfc_index_euclidean_common_target_left_productcoefficients)) * pfrd_pc_euclidean_common_target_left)) /\\ exists ff_q_pfp_euclidean_common_target_left_productcoefficientsentry. pfrd_pb_euclidean_common_target_left = ff_q_pfp_euclidean_common_target_left_productcoefficientsentry * S ((S (pfc_index_euclidean_common_target_left_productcoefficients)) * pfrd_pc_euclidean_common_target_left) + (pfc_value_euclidean_common_target_left_productcoefficients))) /\\ ((exists pfc_terms_code_euclidean_common_target_left_productcoefficientscoefficient pfc_terms_scale_euclidean_common_target_left_productcoefficientscoefficient pfc_natural_sum_euclidean_common_target_left_productcoefficientscoefficient. ((forall pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonalbound. pfa_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_euclidean_common_target_left_productcoefficients))) -> exists pfc_value_euclidean_common_target_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_euclidean_common_target_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_target_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_euclidean_common_target_left_productcoefficientscoefficient = ff_q_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_target_left_productcoefficientscoefficient) + (pfc_value_euclidean_common_target_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm pfc_left_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm pfc_right_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal)+pfc_complement_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm=(pfc_index_euclidean_common_target_left_productcoefficients)) /\\ ((((((exists pfa_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal) = (pfrd_qlen_euclidean_common_target_left)) /\\ ((((exists ff_h_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_target_left)) /\\ exists ff_q_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_euclidean_common_target_left = ff_q_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_target_left) + (pfc_left_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_euclidean_common_target_left)=(pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_euclidean_common_target_left_productcoefficientscoefficientdiagonal)=pfc_left_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm*pfc_right_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_euclidean_common_target_left_productcoefficientscoefficientsum fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_euclidean_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_euclidean_common_target_left_productcoefficientscoefficient) = S ((S (S (pfc_index_euclidean_common_target_left_productcoefficients))) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_euclidean_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_euclidean_common_target_left_productcoefficients))) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum) + (pfc_natural_sum_euclidean_common_target_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_euclidean_common_target_left_productcoefficients)) -> exists fs_a_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_target_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_euclidean_common_target_left_productcoefficientscoefficient = fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_target_left_productcoefficientscoefficient) + (fs_a_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_euclidean_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum) + (fs_r_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_euclidean_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum) + (fs_s_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_euclidean_common_target_left_productcoefficientscoefficientresiduebound. pfa_gap_euclidean_common_target_left_productcoefficientscoefficientresiduebound + S (pfc_value_euclidean_common_target_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_euclidean_common_target_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_euclidean_common_target_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_euclidean_common_target_left_productcoefficientscoefficient) + (p) * pfa_offset_left_euclidean_common_target_left_productcoefficientscoefficientresiduecongruence = (pfc_value_euclidean_common_target_left_productcoefficients) + (p) * pfa_offset_right_euclidean_common_target_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_euclidean_common_target_left_target pfrep_left_euclidean_common_target_left_target pfrep_right_euclidean_common_target_left_target. ((exists pfrep_position_euclidean_common_target_left_targetfirst. ((pfrep_position_euclidean_common_target_left_targetfirst+S (pfrep_power_euclidean_common_target_left_target)=(pfrd_plen_euclidean_common_target_left)) /\\ ((((exists ff_h_pfp_euclidean_common_target_left_targetfirstentry. ff_h_pfp_euclidean_common_target_left_targetfirstentry + S (pfrep_left_euclidean_common_target_left_target) = S ((S (pfrep_position_euclidean_common_target_left_targetfirst)) * pfrd_pc_euclidean_common_target_left)) /\\ exists ff_q_pfp_euclidean_common_target_left_targetfirstentry. pfrd_pb_euclidean_common_target_left = ff_q_pfp_euclidean_common_target_left_targetfirstentry * S ((S (pfrep_position_euclidean_common_target_left_targetfirst)) * pfrd_pc_euclidean_common_target_left) + (pfrep_left_euclidean_common_target_left_target)))))) \\/ (((exists pfrep_gap_euclidean_common_target_left_targetfirstoutside. pfrep_gap_euclidean_common_target_left_targetfirstoutside+(pfrd_plen_euclidean_common_target_left)=(pfrep_power_euclidean_common_target_left_target)) /\\ (((pfrep_left_euclidean_common_target_left_target)=0))))) -> ((exists pfrep_position_euclidean_common_target_left_targetsecond. ((pfrep_position_euclidean_common_target_left_targetsecond+S (pfrep_power_euclidean_common_target_left_target)=(M)) /\\ ((((exists ff_h_pfp_euclidean_common_target_left_targetsecondentry. ff_h_pfp_euclidean_common_target_left_targetsecondentry + S (pfrep_right_euclidean_common_target_left_target) = S ((S (pfrep_position_euclidean_common_target_left_targetsecond)) * bc)) /\\ exists ff_q_pfp_euclidean_common_target_left_targetsecondentry. bb = ff_q_pfp_euclidean_common_target_left_targetsecondentry * S ((S (pfrep_position_euclidean_common_target_left_targetsecond)) * bc) + (pfrep_right_euclidean_common_target_left_target)))))) \\/ (((exists pfrep_gap_euclidean_common_target_left_targetsecondoutside. pfrep_gap_euclidean_common_target_left_targetsecondoutside+(M)=(pfrep_power_euclidean_common_target_left_target)) /\\ (((pfrep_right_euclidean_common_target_left_target)=0))))) -> pfrep_left_euclidean_common_target_left_target=pfrep_right_euclidean_common_target_left_target))))))) /\\ ((((forall fom_index_pfp_euclidean_common_target_right_canonical. (exists fom_gap_pfp_euclidean_common_target_right_canonical_index_bound. fom_gap_pfp_euclidean_common_target_right_canonical_index_bound + S (fom_index_pfp_euclidean_common_target_right_canonical) = N) -> exists fom_value_pfp_euclidean_common_target_right_canonical. ((((exists fom_beta_height_pfp_euclidean_common_target_right_canonical_entry. fom_beta_height_pfp_euclidean_common_target_right_canonical_entry + S (fom_value_pfp_euclidean_common_target_right_canonical) = S ((S (fom_index_pfp_euclidean_common_target_right_canonical)) * rc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_target_right_canonical_entry. rb = fom_beta_quotient_pfp_euclidean_common_target_right_canonical_entry * S ((S (fom_index_pfp_euclidean_common_target_right_canonical)) * rc) + (fom_value_pfp_euclidean_common_target_right_canonical))) /\\ (exists fom_gap_pfp_euclidean_common_target_right_canonical_value_bound. fom_gap_pfp_euclidean_common_target_right_canonical_value_bound + S (fom_value_pfp_euclidean_common_target_right_canonical) = p))) /\\ ((exists pfrd_qb_euclidean_common_target_right pfrd_qc_euclidean_common_target_right pfrd_qlen_euclidean_common_target_right pfrd_pb_euclidean_common_target_right pfrd_pc_euclidean_common_target_right pfrd_plen_euclidean_common_target_right. ((((forall fom_index_pfp_euclidean_common_target_right_productleft. (exists fom_gap_pfp_euclidean_common_target_right_productleft_index_bound. fom_gap_pfp_euclidean_common_target_right_productleft_index_bound + S (fom_index_pfp_euclidean_common_target_right_productleft) = pfrd_qlen_euclidean_common_target_right) -> exists fom_value_pfp_euclidean_common_target_right_productleft. ((((exists fom_beta_height_pfp_euclidean_common_target_right_productleft_entry. fom_beta_height_pfp_euclidean_common_target_right_productleft_entry + S (fom_value_pfp_euclidean_common_target_right_productleft) = S ((S (fom_index_pfp_euclidean_common_target_right_productleft)) * pfrd_qc_euclidean_common_target_right)) /\\ exists fom_beta_quotient_pfp_euclidean_common_target_right_productleft_entry. pfrd_qb_euclidean_common_target_right = fom_beta_quotient_pfp_euclidean_common_target_right_productleft_entry * S ((S (fom_index_pfp_euclidean_common_target_right_productleft)) * pfrd_qc_euclidean_common_target_right) + (fom_value_pfp_euclidean_common_target_right_productleft))) /\\ (exists fom_gap_pfp_euclidean_common_target_right_productleft_value_bound. fom_gap_pfp_euclidean_common_target_right_productleft_value_bound + S (fom_value_pfp_euclidean_common_target_right_productleft) = p))) /\\ (((forall fom_index_pfp_euclidean_common_target_right_productright. (exists fom_gap_pfp_euclidean_common_target_right_productright_index_bound. fom_gap_pfp_euclidean_common_target_right_productright_index_bound + S (fom_index_pfp_euclidean_common_target_right_productright) = J) -> exists fom_value_pfp_euclidean_common_target_right_productright. ((((exists fom_beta_height_pfp_euclidean_common_target_right_productright_entry. fom_beta_height_pfp_euclidean_common_target_right_productright_entry + S (fom_value_pfp_euclidean_common_target_right_productright) = S ((S (fom_index_pfp_euclidean_common_target_right_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_target_right_productright_entry. db = fom_beta_quotient_pfp_euclidean_common_target_right_productright_entry * S ((S (fom_index_pfp_euclidean_common_target_right_productright)) * dc) + (fom_value_pfp_euclidean_common_target_right_productright))) /\\ (exists fom_gap_pfp_euclidean_common_target_right_productright_value_bound. fom_gap_pfp_euclidean_common_target_right_productright_value_bound + S (fom_value_pfp_euclidean_common_target_right_productright) = p))) /\\ (((((((pfrd_qlen_euclidean_common_target_right)=0 \\/ (J)=0) /\\ (((pfrd_plen_euclidean_common_target_right)=0)))) \\/ (((~((pfrd_qlen_euclidean_common_target_right)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_euclidean_common_target_right)+(J)=S (pfrd_plen_euclidean_common_target_right)))))))) /\\ ((forall pfc_index_euclidean_common_target_right_productcoefficients. (exists pfa_gap_euclidean_common_target_right_productcoefficientsbound. pfa_gap_euclidean_common_target_right_productcoefficientsbound + S (pfc_index_euclidean_common_target_right_productcoefficients) = (pfrd_plen_euclidean_common_target_right)) -> exists pfc_value_euclidean_common_target_right_productcoefficients. ((((exists ff_h_pfp_euclidean_common_target_right_productcoefficientsentry. ff_h_pfp_euclidean_common_target_right_productcoefficientsentry + S (pfc_value_euclidean_common_target_right_productcoefficients) = S ((S (pfc_index_euclidean_common_target_right_productcoefficients)) * pfrd_pc_euclidean_common_target_right)) /\\ exists ff_q_pfp_euclidean_common_target_right_productcoefficientsentry. pfrd_pb_euclidean_common_target_right = ff_q_pfp_euclidean_common_target_right_productcoefficientsentry * S ((S (pfc_index_euclidean_common_target_right_productcoefficients)) * pfrd_pc_euclidean_common_target_right) + (pfc_value_euclidean_common_target_right_productcoefficients))) /\\ ((exists pfc_terms_code_euclidean_common_target_right_productcoefficientscoefficient pfc_terms_scale_euclidean_common_target_right_productcoefficientscoefficient pfc_natural_sum_euclidean_common_target_right_productcoefficientscoefficient. ((forall pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonalbound. pfa_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_euclidean_common_target_right_productcoefficients))) -> exists pfc_value_euclidean_common_target_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_euclidean_common_target_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_target_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_euclidean_common_target_right_productcoefficientscoefficient = ff_q_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_target_right_productcoefficientscoefficient) + (pfc_value_euclidean_common_target_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm pfc_left_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm pfc_right_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal)+pfc_complement_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm=(pfc_index_euclidean_common_target_right_productcoefficients)) /\\ ((((((exists pfa_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal) = (pfrd_qlen_euclidean_common_target_right)) /\\ ((((exists ff_h_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_target_right)) /\\ exists ff_q_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_euclidean_common_target_right = ff_q_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_target_right) + (pfc_left_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_euclidean_common_target_right)=(pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_euclidean_common_target_right_productcoefficientscoefficientdiagonal)=pfc_left_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm*pfc_right_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_euclidean_common_target_right_productcoefficientscoefficientsum fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_euclidean_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_euclidean_common_target_right_productcoefficientscoefficient) = S ((S (S (pfc_index_euclidean_common_target_right_productcoefficients))) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_euclidean_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_euclidean_common_target_right_productcoefficients))) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum) + (pfc_natural_sum_euclidean_common_target_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_euclidean_common_target_right_productcoefficients)) -> exists fs_a_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_target_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_euclidean_common_target_right_productcoefficientscoefficient = fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_target_right_productcoefficientscoefficient) + (fs_a_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_euclidean_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum) + (fs_r_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_euclidean_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum) + (fs_s_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_euclidean_common_target_right_productcoefficientscoefficientresiduebound. pfa_gap_euclidean_common_target_right_productcoefficientscoefficientresiduebound + S (pfc_value_euclidean_common_target_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_euclidean_common_target_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_euclidean_common_target_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_euclidean_common_target_right_productcoefficientscoefficient) + (p) * pfa_offset_left_euclidean_common_target_right_productcoefficientscoefficientresiduecongruence = (pfc_value_euclidean_common_target_right_productcoefficients) + (p) * pfa_offset_right_euclidean_common_target_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_euclidean_common_target_right_target pfrep_left_euclidean_common_target_right_target pfrep_right_euclidean_common_target_right_target. ((exists pfrep_position_euclidean_common_target_right_targetfirst. ((pfrep_position_euclidean_common_target_right_targetfirst+S (pfrep_power_euclidean_common_target_right_target)=(pfrd_plen_euclidean_common_target_right)) /\\ ((((exists ff_h_pfp_euclidean_common_target_right_targetfirstentry. ff_h_pfp_euclidean_common_target_right_targetfirstentry + S (pfrep_left_euclidean_common_target_right_target) = S ((S (pfrep_position_euclidean_common_target_right_targetfirst)) * pfrd_pc_euclidean_common_target_right)) /\\ exists ff_q_pfp_euclidean_common_target_right_targetfirstentry. pfrd_pb_euclidean_common_target_right = ff_q_pfp_euclidean_common_target_right_targetfirstentry * S ((S (pfrep_position_euclidean_common_target_right_targetfirst)) * pfrd_pc_euclidean_common_target_right) + (pfrep_left_euclidean_common_target_right_target)))))) \\/ (((exists pfrep_gap_euclidean_common_target_right_targetfirstoutside. pfrep_gap_euclidean_common_target_right_targetfirstoutside+(pfrd_plen_euclidean_common_target_right)=(pfrep_power_euclidean_common_target_right_target)) /\\ (((pfrep_left_euclidean_common_target_right_target)=0))))) -> ((exists pfrep_position_euclidean_common_target_right_targetsecond. ((pfrep_position_euclidean_common_target_right_targetsecond+S (pfrep_power_euclidean_common_target_right_target)=(N)) /\\ ((((exists ff_h_pfp_euclidean_common_target_right_targetsecondentry. ff_h_pfp_euclidean_common_target_right_targetsecondentry + S (pfrep_right_euclidean_common_target_right_target) = S ((S (pfrep_position_euclidean_common_target_right_targetsecond)) * rc)) /\\ exists ff_q_pfp_euclidean_common_target_right_targetsecondentry. rb = ff_q_pfp_euclidean_common_target_right_targetsecondentry * S ((S (pfrep_position_euclidean_common_target_right_targetsecond)) * rc) + (pfrep_right_euclidean_common_target_right_target)))))) \\/ (((exists pfrep_gap_euclidean_common_target_right_targetsecondoutside. pfrep_gap_euclidean_common_target_right_targetsecondoutside+(N)=(pfrep_power_euclidean_common_target_right_target)) /\\ (((pfrep_right_euclidean_common_target_right_target)=0))))) -> pfrep_left_euclidean_common_target_right_target=pfrep_right_euclidean_common_target_right_target))))))))))) /\\ (((((((forall fom_index_pfp_euclidean_common_target_left_canonical. (exists fom_gap_pfp_euclidean_common_target_left_canonical_index_bound. fom_gap_pfp_euclidean_common_target_left_canonical_index_bound + S (fom_index_pfp_euclidean_common_target_left_canonical) = M) -> exists fom_value_pfp_euclidean_common_target_left_canonical. ((((exists fom_beta_height_pfp_euclidean_common_target_left_canonical_entry. fom_beta_height_pfp_euclidean_common_target_left_canonical_entry + S (fom_value_pfp_euclidean_common_target_left_canonical) = S ((S (fom_index_pfp_euclidean_common_target_left_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_target_left_canonical_entry. bb = fom_beta_quotient_pfp_euclidean_common_target_left_canonical_entry * S ((S (fom_index_pfp_euclidean_common_target_left_canonical)) * bc) + (fom_value_pfp_euclidean_common_target_left_canonical))) /\\ (exists fom_gap_pfp_euclidean_common_target_left_canonical_value_bound. fom_gap_pfp_euclidean_common_target_left_canonical_value_bound + S (fom_value_pfp_euclidean_common_target_left_canonical) = p))) /\\ ((exists pfrd_qb_euclidean_common_target_left pfrd_qc_euclidean_common_target_left pfrd_qlen_euclidean_common_target_left pfrd_pb_euclidean_common_target_left pfrd_pc_euclidean_common_target_left pfrd_plen_euclidean_common_target_left. ((((forall fom_index_pfp_euclidean_common_target_left_productleft. (exists fom_gap_pfp_euclidean_common_target_left_productleft_index_bound. fom_gap_pfp_euclidean_common_target_left_productleft_index_bound + S (fom_index_pfp_euclidean_common_target_left_productleft) = pfrd_qlen_euclidean_common_target_left) -> exists fom_value_pfp_euclidean_common_target_left_productleft. ((((exists fom_beta_height_pfp_euclidean_common_target_left_productleft_entry. fom_beta_height_pfp_euclidean_common_target_left_productleft_entry + S (fom_value_pfp_euclidean_common_target_left_productleft) = S ((S (fom_index_pfp_euclidean_common_target_left_productleft)) * pfrd_qc_euclidean_common_target_left)) /\\ exists fom_beta_quotient_pfp_euclidean_common_target_left_productleft_entry. pfrd_qb_euclidean_common_target_left = fom_beta_quotient_pfp_euclidean_common_target_left_productleft_entry * S ((S (fom_index_pfp_euclidean_common_target_left_productleft)) * pfrd_qc_euclidean_common_target_left) + (fom_value_pfp_euclidean_common_target_left_productleft))) /\\ (exists fom_gap_pfp_euclidean_common_target_left_productleft_value_bound. fom_gap_pfp_euclidean_common_target_left_productleft_value_bound + S (fom_value_pfp_euclidean_common_target_left_productleft) = p))) /\\ (((forall fom_index_pfp_euclidean_common_target_left_productright. (exists fom_gap_pfp_euclidean_common_target_left_productright_index_bound. fom_gap_pfp_euclidean_common_target_left_productright_index_bound + S (fom_index_pfp_euclidean_common_target_left_productright) = J) -> exists fom_value_pfp_euclidean_common_target_left_productright. ((((exists fom_beta_height_pfp_euclidean_common_target_left_productright_entry. fom_beta_height_pfp_euclidean_common_target_left_productright_entry + S (fom_value_pfp_euclidean_common_target_left_productright) = S ((S (fom_index_pfp_euclidean_common_target_left_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_target_left_productright_entry. db = fom_beta_quotient_pfp_euclidean_common_target_left_productright_entry * S ((S (fom_index_pfp_euclidean_common_target_left_productright)) * dc) + (fom_value_pfp_euclidean_common_target_left_productright))) /\\ (exists fom_gap_pfp_euclidean_common_target_left_productright_value_bound. fom_gap_pfp_euclidean_common_target_left_productright_value_bound + S (fom_value_pfp_euclidean_common_target_left_productright) = p))) /\\ (((((((pfrd_qlen_euclidean_common_target_left)=0 \\/ (J)=0) /\\ (((pfrd_plen_euclidean_common_target_left)=0)))) \\/ (((~((pfrd_qlen_euclidean_common_target_left)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_euclidean_common_target_left)+(J)=S (pfrd_plen_euclidean_common_target_left)))))))) /\\ ((forall pfc_index_euclidean_common_target_left_productcoefficients. (exists pfa_gap_euclidean_common_target_left_productcoefficientsbound. pfa_gap_euclidean_common_target_left_productcoefficientsbound + S (pfc_index_euclidean_common_target_left_productcoefficients) = (pfrd_plen_euclidean_common_target_left)) -> exists pfc_value_euclidean_common_target_left_productcoefficients. ((((exists ff_h_pfp_euclidean_common_target_left_productcoefficientsentry. ff_h_pfp_euclidean_common_target_left_productcoefficientsentry + S (pfc_value_euclidean_common_target_left_productcoefficients) = S ((S (pfc_index_euclidean_common_target_left_productcoefficients)) * pfrd_pc_euclidean_common_target_left)) /\\ exists ff_q_pfp_euclidean_common_target_left_productcoefficientsentry. pfrd_pb_euclidean_common_target_left = ff_q_pfp_euclidean_common_target_left_productcoefficientsentry * S ((S (pfc_index_euclidean_common_target_left_productcoefficients)) * pfrd_pc_euclidean_common_target_left) + (pfc_value_euclidean_common_target_left_productcoefficients))) /\\ ((exists pfc_terms_code_euclidean_common_target_left_productcoefficientscoefficient pfc_terms_scale_euclidean_common_target_left_productcoefficientscoefficient pfc_natural_sum_euclidean_common_target_left_productcoefficientscoefficient. ((forall pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonalbound. pfa_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_euclidean_common_target_left_productcoefficients))) -> exists pfc_value_euclidean_common_target_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_euclidean_common_target_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_target_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_euclidean_common_target_left_productcoefficientscoefficient = ff_q_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_target_left_productcoefficientscoefficient) + (pfc_value_euclidean_common_target_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm pfc_left_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm pfc_right_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal)+pfc_complement_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm=(pfc_index_euclidean_common_target_left_productcoefficients)) /\\ ((((((exists pfa_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal) = (pfrd_qlen_euclidean_common_target_left)) /\\ ((((exists ff_h_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_target_left)) /\\ exists ff_q_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_euclidean_common_target_left = ff_q_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_target_left) + (pfc_left_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_euclidean_common_target_left)=(pfc_index_euclidean_common_target_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_euclidean_common_target_left_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_euclidean_common_target_left_productcoefficientscoefficientdiagonal)=pfc_left_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm*pfc_right_euclidean_common_target_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_euclidean_common_target_left_productcoefficientscoefficientsum fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_euclidean_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_euclidean_common_target_left_productcoefficientscoefficient) = S ((S (S (pfc_index_euclidean_common_target_left_productcoefficients))) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_euclidean_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_euclidean_common_target_left_productcoefficients))) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum) + (pfc_natural_sum_euclidean_common_target_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_euclidean_common_target_left_productcoefficients)) -> exists fs_a_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_target_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_euclidean_common_target_left_productcoefficientscoefficient = fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_target_left_productcoefficientscoefficient) + (fs_a_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_euclidean_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum) + (fs_r_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_euclidean_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_left_productcoefficientscoefficientsum) + (fs_s_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_euclidean_common_target_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_euclidean_common_target_left_productcoefficientscoefficientresiduebound. pfa_gap_euclidean_common_target_left_productcoefficientscoefficientresiduebound + S (pfc_value_euclidean_common_target_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_euclidean_common_target_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_euclidean_common_target_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_euclidean_common_target_left_productcoefficientscoefficient) + (p) * pfa_offset_left_euclidean_common_target_left_productcoefficientscoefficientresiduecongruence = (pfc_value_euclidean_common_target_left_productcoefficients) + (p) * pfa_offset_right_euclidean_common_target_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_euclidean_common_target_left_target pfrep_left_euclidean_common_target_left_target pfrep_right_euclidean_common_target_left_target. ((exists pfrep_position_euclidean_common_target_left_targetfirst. ((pfrep_position_euclidean_common_target_left_targetfirst+S (pfrep_power_euclidean_common_target_left_target)=(pfrd_plen_euclidean_common_target_left)) /\\ ((((exists ff_h_pfp_euclidean_common_target_left_targetfirstentry. ff_h_pfp_euclidean_common_target_left_targetfirstentry + S (pfrep_left_euclidean_common_target_left_target) = S ((S (pfrep_position_euclidean_common_target_left_targetfirst)) * pfrd_pc_euclidean_common_target_left)) /\\ exists ff_q_pfp_euclidean_common_target_left_targetfirstentry. pfrd_pb_euclidean_common_target_left = ff_q_pfp_euclidean_common_target_left_targetfirstentry * S ((S (pfrep_position_euclidean_common_target_left_targetfirst)) * pfrd_pc_euclidean_common_target_left) + (pfrep_left_euclidean_common_target_left_target)))))) \\/ (((exists pfrep_gap_euclidean_common_target_left_targetfirstoutside. pfrep_gap_euclidean_common_target_left_targetfirstoutside+(pfrd_plen_euclidean_common_target_left)=(pfrep_power_euclidean_common_target_left_target)) /\\ (((pfrep_left_euclidean_common_target_left_target)=0))))) -> ((exists pfrep_position_euclidean_common_target_left_targetsecond. ((pfrep_position_euclidean_common_target_left_targetsecond+S (pfrep_power_euclidean_common_target_left_target)=(M)) /\\ ((((exists ff_h_pfp_euclidean_common_target_left_targetsecondentry. ff_h_pfp_euclidean_common_target_left_targetsecondentry + S (pfrep_right_euclidean_common_target_left_target) = S ((S (pfrep_position_euclidean_common_target_left_targetsecond)) * bc)) /\\ exists ff_q_pfp_euclidean_common_target_left_targetsecondentry. bb = ff_q_pfp_euclidean_common_target_left_targetsecondentry * S ((S (pfrep_position_euclidean_common_target_left_targetsecond)) * bc) + (pfrep_right_euclidean_common_target_left_target)))))) \\/ (((exists pfrep_gap_euclidean_common_target_left_targetsecondoutside. pfrep_gap_euclidean_common_target_left_targetsecondoutside+(M)=(pfrep_power_euclidean_common_target_left_target)) /\\ (((pfrep_right_euclidean_common_target_left_target)=0))))) -> pfrep_left_euclidean_common_target_left_target=pfrep_right_euclidean_common_target_left_target))))))) /\\ ((((forall fom_index_pfp_euclidean_common_target_right_canonical. (exists fom_gap_pfp_euclidean_common_target_right_canonical_index_bound. fom_gap_pfp_euclidean_common_target_right_canonical_index_bound + S (fom_index_pfp_euclidean_common_target_right_canonical) = N) -> exists fom_value_pfp_euclidean_common_target_right_canonical. ((((exists fom_beta_height_pfp_euclidean_common_target_right_canonical_entry. fom_beta_height_pfp_euclidean_common_target_right_canonical_entry + S (fom_value_pfp_euclidean_common_target_right_canonical) = S ((S (fom_index_pfp_euclidean_common_target_right_canonical)) * rc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_target_right_canonical_entry. rb = fom_beta_quotient_pfp_euclidean_common_target_right_canonical_entry * S ((S (fom_index_pfp_euclidean_common_target_right_canonical)) * rc) + (fom_value_pfp_euclidean_common_target_right_canonical))) /\\ (exists fom_gap_pfp_euclidean_common_target_right_canonical_value_bound. fom_gap_pfp_euclidean_common_target_right_canonical_value_bound + S (fom_value_pfp_euclidean_common_target_right_canonical) = p))) /\\ ((exists pfrd_qb_euclidean_common_target_right pfrd_qc_euclidean_common_target_right pfrd_qlen_euclidean_common_target_right pfrd_pb_euclidean_common_target_right pfrd_pc_euclidean_common_target_right pfrd_plen_euclidean_common_target_right. ((((forall fom_index_pfp_euclidean_common_target_right_productleft. (exists fom_gap_pfp_euclidean_common_target_right_productleft_index_bound. fom_gap_pfp_euclidean_common_target_right_productleft_index_bound + S (fom_index_pfp_euclidean_common_target_right_productleft) = pfrd_qlen_euclidean_common_target_right) -> exists fom_value_pfp_euclidean_common_target_right_productleft. ((((exists fom_beta_height_pfp_euclidean_common_target_right_productleft_entry. fom_beta_height_pfp_euclidean_common_target_right_productleft_entry + S (fom_value_pfp_euclidean_common_target_right_productleft) = S ((S (fom_index_pfp_euclidean_common_target_right_productleft)) * pfrd_qc_euclidean_common_target_right)) /\\ exists fom_beta_quotient_pfp_euclidean_common_target_right_productleft_entry. pfrd_qb_euclidean_common_target_right = fom_beta_quotient_pfp_euclidean_common_target_right_productleft_entry * S ((S (fom_index_pfp_euclidean_common_target_right_productleft)) * pfrd_qc_euclidean_common_target_right) + (fom_value_pfp_euclidean_common_target_right_productleft))) /\\ (exists fom_gap_pfp_euclidean_common_target_right_productleft_value_bound. fom_gap_pfp_euclidean_common_target_right_productleft_value_bound + S (fom_value_pfp_euclidean_common_target_right_productleft) = p))) /\\ (((forall fom_index_pfp_euclidean_common_target_right_productright. (exists fom_gap_pfp_euclidean_common_target_right_productright_index_bound. fom_gap_pfp_euclidean_common_target_right_productright_index_bound + S (fom_index_pfp_euclidean_common_target_right_productright) = J) -> exists fom_value_pfp_euclidean_common_target_right_productright. ((((exists fom_beta_height_pfp_euclidean_common_target_right_productright_entry. fom_beta_height_pfp_euclidean_common_target_right_productright_entry + S (fom_value_pfp_euclidean_common_target_right_productright) = S ((S (fom_index_pfp_euclidean_common_target_right_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_target_right_productright_entry. db = fom_beta_quotient_pfp_euclidean_common_target_right_productright_entry * S ((S (fom_index_pfp_euclidean_common_target_right_productright)) * dc) + (fom_value_pfp_euclidean_common_target_right_productright))) /\\ (exists fom_gap_pfp_euclidean_common_target_right_productright_value_bound. fom_gap_pfp_euclidean_common_target_right_productright_value_bound + S (fom_value_pfp_euclidean_common_target_right_productright) = p))) /\\ (((((((pfrd_qlen_euclidean_common_target_right)=0 \\/ (J)=0) /\\ (((pfrd_plen_euclidean_common_target_right)=0)))) \\/ (((~((pfrd_qlen_euclidean_common_target_right)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_euclidean_common_target_right)+(J)=S (pfrd_plen_euclidean_common_target_right)))))))) /\\ ((forall pfc_index_euclidean_common_target_right_productcoefficients. (exists pfa_gap_euclidean_common_target_right_productcoefficientsbound. pfa_gap_euclidean_common_target_right_productcoefficientsbound + S (pfc_index_euclidean_common_target_right_productcoefficients) = (pfrd_plen_euclidean_common_target_right)) -> exists pfc_value_euclidean_common_target_right_productcoefficients. ((((exists ff_h_pfp_euclidean_common_target_right_productcoefficientsentry. ff_h_pfp_euclidean_common_target_right_productcoefficientsentry + S (pfc_value_euclidean_common_target_right_productcoefficients) = S ((S (pfc_index_euclidean_common_target_right_productcoefficients)) * pfrd_pc_euclidean_common_target_right)) /\\ exists ff_q_pfp_euclidean_common_target_right_productcoefficientsentry. pfrd_pb_euclidean_common_target_right = ff_q_pfp_euclidean_common_target_right_productcoefficientsentry * S ((S (pfc_index_euclidean_common_target_right_productcoefficients)) * pfrd_pc_euclidean_common_target_right) + (pfc_value_euclidean_common_target_right_productcoefficients))) /\\ ((exists pfc_terms_code_euclidean_common_target_right_productcoefficientscoefficient pfc_terms_scale_euclidean_common_target_right_productcoefficientscoefficient pfc_natural_sum_euclidean_common_target_right_productcoefficientscoefficient. ((forall pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonalbound. pfa_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_euclidean_common_target_right_productcoefficients))) -> exists pfc_value_euclidean_common_target_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_euclidean_common_target_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_target_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_euclidean_common_target_right_productcoefficientscoefficient = ff_q_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_target_right_productcoefficientscoefficient) + (pfc_value_euclidean_common_target_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm pfc_left_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm pfc_right_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal)+pfc_complement_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm=(pfc_index_euclidean_common_target_right_productcoefficients)) /\\ ((((((exists pfa_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal) = (pfrd_qlen_euclidean_common_target_right)) /\\ ((((exists ff_h_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_target_right)) /\\ exists ff_q_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_euclidean_common_target_right = ff_q_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_target_right) + (pfc_left_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_euclidean_common_target_right)=(pfc_index_euclidean_common_target_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_euclidean_common_target_right_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_euclidean_common_target_right_productcoefficientscoefficientdiagonal)=pfc_left_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm*pfc_right_euclidean_common_target_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_euclidean_common_target_right_productcoefficientscoefficientsum fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_euclidean_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_euclidean_common_target_right_productcoefficientscoefficient) = S ((S (S (pfc_index_euclidean_common_target_right_productcoefficients))) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_euclidean_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_euclidean_common_target_right_productcoefficients))) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum) + (pfc_natural_sum_euclidean_common_target_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_euclidean_common_target_right_productcoefficients)) -> exists fs_a_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_target_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_euclidean_common_target_right_productcoefficientscoefficient = fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_target_right_productcoefficientscoefficient) + (fs_a_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_euclidean_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum) + (fs_r_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_euclidean_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_target_right_productcoefficientscoefficientsum) + (fs_s_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_euclidean_common_target_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_euclidean_common_target_right_productcoefficientscoefficientresiduebound. pfa_gap_euclidean_common_target_right_productcoefficientscoefficientresiduebound + S (pfc_value_euclidean_common_target_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_euclidean_common_target_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_euclidean_common_target_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_euclidean_common_target_right_productcoefficientscoefficient) + (p) * pfa_offset_left_euclidean_common_target_right_productcoefficientscoefficientresiduecongruence = (pfc_value_euclidean_common_target_right_productcoefficients) + (p) * pfa_offset_right_euclidean_common_target_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_euclidean_common_target_right_target pfrep_left_euclidean_common_target_right_target pfrep_right_euclidean_common_target_right_target. ((exists pfrep_position_euclidean_common_target_right_targetfirst. ((pfrep_position_euclidean_common_target_right_targetfirst+S (pfrep_power_euclidean_common_target_right_target)=(pfrd_plen_euclidean_common_target_right)) /\\ ((((exists ff_h_pfp_euclidean_common_target_right_targetfirstentry. ff_h_pfp_euclidean_common_target_right_targetfirstentry + S (pfrep_left_euclidean_common_target_right_target) = S ((S (pfrep_position_euclidean_common_target_right_targetfirst)) * pfrd_pc_euclidean_common_target_right)) /\\ exists ff_q_pfp_euclidean_common_target_right_targetfirstentry. pfrd_pb_euclidean_common_target_right = ff_q_pfp_euclidean_common_target_right_targetfirstentry * S ((S (pfrep_position_euclidean_common_target_right_targetfirst)) * pfrd_pc_euclidean_common_target_right) + (pfrep_left_euclidean_common_target_right_target)))))) \\/ (((exists pfrep_gap_euclidean_common_target_right_targetfirstoutside. pfrep_gap_euclidean_common_target_right_targetfirstoutside+(pfrd_plen_euclidean_common_target_right)=(pfrep_power_euclidean_common_target_right_target)) /\\ (((pfrep_left_euclidean_common_target_right_target)=0))))) -> ((exists pfrep_position_euclidean_common_target_right_targetsecond. ((pfrep_position_euclidean_common_target_right_targetsecond+S (pfrep_power_euclidean_common_target_right_target)=(N)) /\\ ((((exists ff_h_pfp_euclidean_common_target_right_targetsecondentry. ff_h_pfp_euclidean_common_target_right_targetsecondentry + S (pfrep_right_euclidean_common_target_right_target) = S ((S (pfrep_position_euclidean_common_target_right_targetsecond)) * rc)) /\\ exists ff_q_pfp_euclidean_common_target_right_targetsecondentry. rb = ff_q_pfp_euclidean_common_target_right_targetsecondentry * S ((S (pfrep_position_euclidean_common_target_right_targetsecond)) * rc) + (pfrep_right_euclidean_common_target_right_target)))))) \\/ (((exists pfrep_gap_euclidean_common_target_right_targetsecondoutside. pfrep_gap_euclidean_common_target_right_targetsecondoutside+(N)=(pfrep_power_euclidean_common_target_right_target)) /\\ (((pfrep_right_euclidean_common_target_right_target)=0))))) -> pfrep_left_euclidean_common_target_right_target=pfrep_right_euclidean_common_target_right_target)))))))))) -> (((((forall fom_index_pfp_euclidean_common_source_left_canonical. (exists fom_gap_pfp_euclidean_common_source_left_canonical_index_bound. fom_gap_pfp_euclidean_common_source_left_canonical_index_bound + S (fom_index_pfp_euclidean_common_source_left_canonical) = L) -> exists fom_value_pfp_euclidean_common_source_left_canonical. ((((exists fom_beta_height_pfp_euclidean_common_source_left_canonical_entry. fom_beta_height_pfp_euclidean_common_source_left_canonical_entry + S (fom_value_pfp_euclidean_common_source_left_canonical) = S ((S (fom_index_pfp_euclidean_common_source_left_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_euclidean_common_source_left_canonical_entry. ab = fom_beta_quotient_pfp_euclidean_common_source_left_canonical_entry * S ((S (fom_index_pfp_euclidean_common_source_left_canonical)) * ac) + (fom_value_pfp_euclidean_common_source_left_canonical))) /\\ (exists fom_gap_pfp_euclidean_common_source_left_canonical_value_bound. fom_gap_pfp_euclidean_common_source_left_canonical_value_bound + S (fom_value_pfp_euclidean_common_source_left_canonical) = p))) /\\ ((exists pfrd_qb_euclidean_common_source_left pfrd_qc_euclidean_common_source_left pfrd_qlen_euclidean_common_source_left pfrd_pb_euclidean_common_source_left pfrd_pc_euclidean_common_source_left pfrd_plen_euclidean_common_source_left. ((((forall fom_index_pfp_euclidean_common_source_left_productleft. (exists fom_gap_pfp_euclidean_common_source_left_productleft_index_bound. fom_gap_pfp_euclidean_common_source_left_productleft_index_bound + S (fom_index_pfp_euclidean_common_source_left_productleft) = pfrd_qlen_euclidean_common_source_left) -> exists fom_value_pfp_euclidean_common_source_left_productleft. ((((exists fom_beta_height_pfp_euclidean_common_source_left_productleft_entry. fom_beta_height_pfp_euclidean_common_source_left_productleft_entry + S (fom_value_pfp_euclidean_common_source_left_productleft) = S ((S (fom_index_pfp_euclidean_common_source_left_productleft)) * pfrd_qc_euclidean_common_source_left)) /\\ exists fom_beta_quotient_pfp_euclidean_common_source_left_productleft_entry. pfrd_qb_euclidean_common_source_left = fom_beta_quotient_pfp_euclidean_common_source_left_productleft_entry * S ((S (fom_index_pfp_euclidean_common_source_left_productleft)) * pfrd_qc_euclidean_common_source_left) + (fom_value_pfp_euclidean_common_source_left_productleft))) /\\ (exists fom_gap_pfp_euclidean_common_source_left_productleft_value_bound. fom_gap_pfp_euclidean_common_source_left_productleft_value_bound + S (fom_value_pfp_euclidean_common_source_left_productleft) = p))) /\\ (((forall fom_index_pfp_euclidean_common_source_left_productright. (exists fom_gap_pfp_euclidean_common_source_left_productright_index_bound. fom_gap_pfp_euclidean_common_source_left_productright_index_bound + S (fom_index_pfp_euclidean_common_source_left_productright) = J) -> exists fom_value_pfp_euclidean_common_source_left_productright. ((((exists fom_beta_height_pfp_euclidean_common_source_left_productright_entry. fom_beta_height_pfp_euclidean_common_source_left_productright_entry + S (fom_value_pfp_euclidean_common_source_left_productright) = S ((S (fom_index_pfp_euclidean_common_source_left_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_source_left_productright_entry. db = fom_beta_quotient_pfp_euclidean_common_source_left_productright_entry * S ((S (fom_index_pfp_euclidean_common_source_left_productright)) * dc) + (fom_value_pfp_euclidean_common_source_left_productright))) /\\ (exists fom_gap_pfp_euclidean_common_source_left_productright_value_bound. fom_gap_pfp_euclidean_common_source_left_productright_value_bound + S (fom_value_pfp_euclidean_common_source_left_productright) = p))) /\\ (((((((pfrd_qlen_euclidean_common_source_left)=0 \\/ (J)=0) /\\ (((pfrd_plen_euclidean_common_source_left)=0)))) \\/ (((~((pfrd_qlen_euclidean_common_source_left)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_euclidean_common_source_left)+(J)=S (pfrd_plen_euclidean_common_source_left)))))))) /\\ ((forall pfc_index_euclidean_common_source_left_productcoefficients. (exists pfa_gap_euclidean_common_source_left_productcoefficientsbound. pfa_gap_euclidean_common_source_left_productcoefficientsbound + S (pfc_index_euclidean_common_source_left_productcoefficients) = (pfrd_plen_euclidean_common_source_left)) -> exists pfc_value_euclidean_common_source_left_productcoefficients. ((((exists ff_h_pfp_euclidean_common_source_left_productcoefficientsentry. ff_h_pfp_euclidean_common_source_left_productcoefficientsentry + S (pfc_value_euclidean_common_source_left_productcoefficients) = S ((S (pfc_index_euclidean_common_source_left_productcoefficients)) * pfrd_pc_euclidean_common_source_left)) /\\ exists ff_q_pfp_euclidean_common_source_left_productcoefficientsentry. pfrd_pb_euclidean_common_source_left = ff_q_pfp_euclidean_common_source_left_productcoefficientsentry * S ((S (pfc_index_euclidean_common_source_left_productcoefficients)) * pfrd_pc_euclidean_common_source_left) + (pfc_value_euclidean_common_source_left_productcoefficients))) /\\ ((exists pfc_terms_code_euclidean_common_source_left_productcoefficientscoefficient pfc_terms_scale_euclidean_common_source_left_productcoefficientscoefficient pfc_natural_sum_euclidean_common_source_left_productcoefficientscoefficient. ((forall pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonalbound. pfa_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_euclidean_common_source_left_productcoefficients))) -> exists pfc_value_euclidean_common_source_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_euclidean_common_source_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_source_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_euclidean_common_source_left_productcoefficientscoefficient = ff_q_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_source_left_productcoefficientscoefficient) + (pfc_value_euclidean_common_source_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm pfc_left_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm pfc_right_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal)+pfc_complement_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm=(pfc_index_euclidean_common_source_left_productcoefficients)) /\\ ((((((exists pfa_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal) = (pfrd_qlen_euclidean_common_source_left)) /\\ ((((exists ff_h_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_source_left)) /\\ exists ff_q_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_euclidean_common_source_left = ff_q_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_source_left) + (pfc_left_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_euclidean_common_source_left)=(pfc_index_euclidean_common_source_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_euclidean_common_source_left_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_euclidean_common_source_left_productcoefficientscoefficientdiagonal)=pfc_left_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm*pfc_right_euclidean_common_source_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_euclidean_common_source_left_productcoefficientscoefficientsum fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_euclidean_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_euclidean_common_source_left_productcoefficientscoefficient) = S ((S (S (pfc_index_euclidean_common_source_left_productcoefficients))) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_euclidean_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_euclidean_common_source_left_productcoefficients))) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum) + (pfc_natural_sum_euclidean_common_source_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_euclidean_common_source_left_productcoefficients)) -> exists fs_a_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_source_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_euclidean_common_source_left_productcoefficientscoefficient = fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_source_left_productcoefficientscoefficient) + (fs_a_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_euclidean_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum) + (fs_r_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_euclidean_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_left_productcoefficientscoefficientsum) + (fs_s_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_euclidean_common_source_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_euclidean_common_source_left_productcoefficientscoefficientresiduebound. pfa_gap_euclidean_common_source_left_productcoefficientscoefficientresiduebound + S (pfc_value_euclidean_common_source_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_euclidean_common_source_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_euclidean_common_source_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_euclidean_common_source_left_productcoefficientscoefficient) + (p) * pfa_offset_left_euclidean_common_source_left_productcoefficientscoefficientresiduecongruence = (pfc_value_euclidean_common_source_left_productcoefficients) + (p) * pfa_offset_right_euclidean_common_source_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_euclidean_common_source_left_target pfrep_left_euclidean_common_source_left_target pfrep_right_euclidean_common_source_left_target. ((exists pfrep_position_euclidean_common_source_left_targetfirst. ((pfrep_position_euclidean_common_source_left_targetfirst+S (pfrep_power_euclidean_common_source_left_target)=(pfrd_plen_euclidean_common_source_left)) /\\ ((((exists ff_h_pfp_euclidean_common_source_left_targetfirstentry. ff_h_pfp_euclidean_common_source_left_targetfirstentry + S (pfrep_left_euclidean_common_source_left_target) = S ((S (pfrep_position_euclidean_common_source_left_targetfirst)) * pfrd_pc_euclidean_common_source_left)) /\\ exists ff_q_pfp_euclidean_common_source_left_targetfirstentry. pfrd_pb_euclidean_common_source_left = ff_q_pfp_euclidean_common_source_left_targetfirstentry * S ((S (pfrep_position_euclidean_common_source_left_targetfirst)) * pfrd_pc_euclidean_common_source_left) + (pfrep_left_euclidean_common_source_left_target)))))) \\/ (((exists pfrep_gap_euclidean_common_source_left_targetfirstoutside. pfrep_gap_euclidean_common_source_left_targetfirstoutside+(pfrd_plen_euclidean_common_source_left)=(pfrep_power_euclidean_common_source_left_target)) /\\ (((pfrep_left_euclidean_common_source_left_target)=0))))) -> ((exists pfrep_position_euclidean_common_source_left_targetsecond. ((pfrep_position_euclidean_common_source_left_targetsecond+S (pfrep_power_euclidean_common_source_left_target)=(L)) /\\ ((((exists ff_h_pfp_euclidean_common_source_left_targetsecondentry. ff_h_pfp_euclidean_common_source_left_targetsecondentry + S (pfrep_right_euclidean_common_source_left_target) = S ((S (pfrep_position_euclidean_common_source_left_targetsecond)) * ac)) /\\ exists ff_q_pfp_euclidean_common_source_left_targetsecondentry. ab = ff_q_pfp_euclidean_common_source_left_targetsecondentry * S ((S (pfrep_position_euclidean_common_source_left_targetsecond)) * ac) + (pfrep_right_euclidean_common_source_left_target)))))) \\/ (((exists pfrep_gap_euclidean_common_source_left_targetsecondoutside. pfrep_gap_euclidean_common_source_left_targetsecondoutside+(L)=(pfrep_power_euclidean_common_source_left_target)) /\\ (((pfrep_right_euclidean_common_source_left_target)=0))))) -> pfrep_left_euclidean_common_source_left_target=pfrep_right_euclidean_common_source_left_target))))))) /\\ ((((forall fom_index_pfp_euclidean_common_source_right_canonical. (exists fom_gap_pfp_euclidean_common_source_right_canonical_index_bound. fom_gap_pfp_euclidean_common_source_right_canonical_index_bound + S (fom_index_pfp_euclidean_common_source_right_canonical) = M) -> exists fom_value_pfp_euclidean_common_source_right_canonical. ((((exists fom_beta_height_pfp_euclidean_common_source_right_canonical_entry. fom_beta_height_pfp_euclidean_common_source_right_canonical_entry + S (fom_value_pfp_euclidean_common_source_right_canonical) = S ((S (fom_index_pfp_euclidean_common_source_right_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_source_right_canonical_entry. bb = fom_beta_quotient_pfp_euclidean_common_source_right_canonical_entry * S ((S (fom_index_pfp_euclidean_common_source_right_canonical)) * bc) + (fom_value_pfp_euclidean_common_source_right_canonical))) /\\ (exists fom_gap_pfp_euclidean_common_source_right_canonical_value_bound. fom_gap_pfp_euclidean_common_source_right_canonical_value_bound + S (fom_value_pfp_euclidean_common_source_right_canonical) = p))) /\\ ((exists pfrd_qb_euclidean_common_source_right pfrd_qc_euclidean_common_source_right pfrd_qlen_euclidean_common_source_right pfrd_pb_euclidean_common_source_right pfrd_pc_euclidean_common_source_right pfrd_plen_euclidean_common_source_right. ((((forall fom_index_pfp_euclidean_common_source_right_productleft. (exists fom_gap_pfp_euclidean_common_source_right_productleft_index_bound. fom_gap_pfp_euclidean_common_source_right_productleft_index_bound + S (fom_index_pfp_euclidean_common_source_right_productleft) = pfrd_qlen_euclidean_common_source_right) -> exists fom_value_pfp_euclidean_common_source_right_productleft. ((((exists fom_beta_height_pfp_euclidean_common_source_right_productleft_entry. fom_beta_height_pfp_euclidean_common_source_right_productleft_entry + S (fom_value_pfp_euclidean_common_source_right_productleft) = S ((S (fom_index_pfp_euclidean_common_source_right_productleft)) * pfrd_qc_euclidean_common_source_right)) /\\ exists fom_beta_quotient_pfp_euclidean_common_source_right_productleft_entry. pfrd_qb_euclidean_common_source_right = fom_beta_quotient_pfp_euclidean_common_source_right_productleft_entry * S ((S (fom_index_pfp_euclidean_common_source_right_productleft)) * pfrd_qc_euclidean_common_source_right) + (fom_value_pfp_euclidean_common_source_right_productleft))) /\\ (exists fom_gap_pfp_euclidean_common_source_right_productleft_value_bound. fom_gap_pfp_euclidean_common_source_right_productleft_value_bound + S (fom_value_pfp_euclidean_common_source_right_productleft) = p))) /\\ (((forall fom_index_pfp_euclidean_common_source_right_productright. (exists fom_gap_pfp_euclidean_common_source_right_productright_index_bound. fom_gap_pfp_euclidean_common_source_right_productright_index_bound + S (fom_index_pfp_euclidean_common_source_right_productright) = J) -> exists fom_value_pfp_euclidean_common_source_right_productright. ((((exists fom_beta_height_pfp_euclidean_common_source_right_productright_entry. fom_beta_height_pfp_euclidean_common_source_right_productright_entry + S (fom_value_pfp_euclidean_common_source_right_productright) = S ((S (fom_index_pfp_euclidean_common_source_right_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_euclidean_common_source_right_productright_entry. db = fom_beta_quotient_pfp_euclidean_common_source_right_productright_entry * S ((S (fom_index_pfp_euclidean_common_source_right_productright)) * dc) + (fom_value_pfp_euclidean_common_source_right_productright))) /\\ (exists fom_gap_pfp_euclidean_common_source_right_productright_value_bound. fom_gap_pfp_euclidean_common_source_right_productright_value_bound + S (fom_value_pfp_euclidean_common_source_right_productright) = p))) /\\ (((((((pfrd_qlen_euclidean_common_source_right)=0 \\/ (J)=0) /\\ (((pfrd_plen_euclidean_common_source_right)=0)))) \\/ (((~((pfrd_qlen_euclidean_common_source_right)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_euclidean_common_source_right)+(J)=S (pfrd_plen_euclidean_common_source_right)))))))) /\\ ((forall pfc_index_euclidean_common_source_right_productcoefficients. (exists pfa_gap_euclidean_common_source_right_productcoefficientsbound. pfa_gap_euclidean_common_source_right_productcoefficientsbound + S (pfc_index_euclidean_common_source_right_productcoefficients) = (pfrd_plen_euclidean_common_source_right)) -> exists pfc_value_euclidean_common_source_right_productcoefficients. ((((exists ff_h_pfp_euclidean_common_source_right_productcoefficientsentry. ff_h_pfp_euclidean_common_source_right_productcoefficientsentry + S (pfc_value_euclidean_common_source_right_productcoefficients) = S ((S (pfc_index_euclidean_common_source_right_productcoefficients)) * pfrd_pc_euclidean_common_source_right)) /\\ exists ff_q_pfp_euclidean_common_source_right_productcoefficientsentry. pfrd_pb_euclidean_common_source_right = ff_q_pfp_euclidean_common_source_right_productcoefficientsentry * S ((S (pfc_index_euclidean_common_source_right_productcoefficients)) * pfrd_pc_euclidean_common_source_right) + (pfc_value_euclidean_common_source_right_productcoefficients))) /\\ ((exists pfc_terms_code_euclidean_common_source_right_productcoefficientscoefficient pfc_terms_scale_euclidean_common_source_right_productcoefficientscoefficient pfc_natural_sum_euclidean_common_source_right_productcoefficientscoefficient. ((forall pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonalbound. pfa_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_euclidean_common_source_right_productcoefficients))) -> exists pfc_value_euclidean_common_source_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_euclidean_common_source_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_source_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_euclidean_common_source_right_productcoefficientscoefficient = ff_q_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_euclidean_common_source_right_productcoefficientscoefficient) + (pfc_value_euclidean_common_source_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm pfc_left_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm pfc_right_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal)+pfc_complement_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm=(pfc_index_euclidean_common_source_right_productcoefficients)) /\\ ((((((exists pfa_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal) = (pfrd_qlen_euclidean_common_source_right)) /\\ ((((exists ff_h_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_source_right)) /\\ exists ff_q_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_euclidean_common_source_right = ff_q_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_euclidean_common_source_right) + (pfc_left_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_euclidean_common_source_right)=(pfc_index_euclidean_common_source_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_euclidean_common_source_right_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_euclidean_common_source_right_productcoefficientscoefficientdiagonal)=pfc_left_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm*pfc_right_euclidean_common_source_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_euclidean_common_source_right_productcoefficientscoefficientsum fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_euclidean_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_euclidean_common_source_right_productcoefficientscoefficient) = S ((S (S (pfc_index_euclidean_common_source_right_productcoefficients))) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_euclidean_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_euclidean_common_source_right_productcoefficients))) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum) + (pfc_natural_sum_euclidean_common_source_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_euclidean_common_source_right_productcoefficients)) -> exists fs_a_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_source_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_euclidean_common_source_right_productcoefficientscoefficient = fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_euclidean_common_source_right_productcoefficientscoefficient) + (fs_a_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_euclidean_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum) + (fs_r_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_euclidean_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_euclidean_common_source_right_productcoefficientscoefficientsum) + (fs_s_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_euclidean_common_source_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_euclidean_common_source_right_productcoefficientscoefficientresiduebound. pfa_gap_euclidean_common_source_right_productcoefficientscoefficientresiduebound + S (pfc_value_euclidean_common_source_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_euclidean_common_source_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_euclidean_common_source_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_euclidean_common_source_right_productcoefficientscoefficient) + (p) * pfa_offset_left_euclidean_common_source_right_productcoefficientscoefficientresiduecongruence = (pfc_value_euclidean_common_source_right_productcoefficients) + (p) * pfa_offset_right_euclidean_common_source_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_euclidean_common_source_right_target pfrep_left_euclidean_common_source_right_target pfrep_right_euclidean_common_source_right_target. ((exists pfrep_position_euclidean_common_source_right_targetfirst. ((pfrep_position_euclidean_common_source_right_targetfirst+S (pfrep_power_euclidean_common_source_right_target)=(pfrd_plen_euclidean_common_source_right)) /\\ ((((exists ff_h_pfp_euclidean_common_source_right_targetfirstentry. ff_h_pfp_euclidean_common_source_right_targetfirstentry + S (pfrep_left_euclidean_common_source_right_target) = S ((S (pfrep_position_euclidean_common_source_right_targetfirst)) * pfrd_pc_euclidean_common_source_right)) /\\ exists ff_q_pfp_euclidean_common_source_right_targetfirstentry. pfrd_pb_euclidean_common_source_right = ff_q_pfp_euclidean_common_source_right_targetfirstentry * S ((S (pfrep_position_euclidean_common_source_right_targetfirst)) * pfrd_pc_euclidean_common_source_right) + (pfrep_left_euclidean_common_source_right_target)))))) \\/ (((exists pfrep_gap_euclidean_common_source_right_targetfirstoutside. pfrep_gap_euclidean_common_source_right_targetfirstoutside+(pfrd_plen_euclidean_common_source_right)=(pfrep_power_euclidean_common_source_right_target)) /\\ (((pfrep_left_euclidean_common_source_right_target)=0))))) -> ((exists pfrep_position_euclidean_common_source_right_targetsecond. ((pfrep_position_euclidean_common_source_right_targetsecond+S (pfrep_power_euclidean_common_source_right_target)=(M)) /\\ ((((exists ff_h_pfp_euclidean_common_source_right_targetsecondentry. ff_h_pfp_euclidean_common_source_right_targetsecondentry + S (pfrep_right_euclidean_common_source_right_target) = S ((S (pfrep_position_euclidean_common_source_right_targetsecond)) * bc)) /\\ exists ff_q_pfp_euclidean_common_source_right_targetsecondentry. bb = ff_q_pfp_euclidean_common_source_right_targetsecondentry * S ((S (pfrep_position_euclidean_common_source_right_targetsecond)) * bc) + (pfrep_right_euclidean_common_source_right_target)))))) \\/ (((exists pfrep_gap_euclidean_common_source_right_targetsecondoutside. pfrep_gap_euclidean_common_source_right_targetsecondoutside+(M)=(pfrep_power_euclidean_common_source_right_target)) /\\ (((pfrep_right_euclidean_common_source_right_target)=0))))) -> 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((((forall fom_index_pfp_execution_common_productleft. (exists fom_gap_pfp_execution_common_productleft_index_bound. fom_gap_pfp_execution_common_productleft_index_bound + S (fom_index_pfp_execution_common_productleft) = q) -> exists fom_value_pfp_execution_common_productleft. ((((exists fom_beta_height_pfp_execution_common_productleft_entry. fom_beta_height_pfp_execution_common_productleft_entry + S (fom_value_pfp_execution_common_productleft) = S ((S (fom_index_pfp_execution_common_productleft)) * qc)) /\\ exists fom_beta_quotient_pfp_execution_common_productleft_entry. qb = fom_beta_quotient_pfp_execution_common_productleft_entry * S ((S (fom_index_pfp_execution_common_productleft)) * qc) + (fom_value_pfp_execution_common_productleft))) /\\ (exists fom_gap_pfp_execution_common_productleft_value_bound. fom_gap_pfp_execution_common_productleft_value_bound + S (fom_value_pfp_execution_common_productleft) = p))) /\\ (((forall fom_index_pfp_execution_common_productright. (exists fom_gap_pfp_execution_common_productright_index_bound. fom_gap_pfp_execution_common_productright_index_bound + S (fom_index_pfp_execution_common_productright) = S d) -> exists fom_value_pfp_execution_common_productright. ((((exists fom_beta_height_pfp_execution_common_productright_entry. fom_beta_height_pfp_execution_common_productright_entry + S (fom_value_pfp_execution_common_productright) = S ((S (fom_index_pfp_execution_common_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_common_productright_entry. bb = fom_beta_quotient_pfp_execution_common_productright_entry * S ((S (fom_index_pfp_execution_common_productright)) * bc) + (fom_value_pfp_execution_common_productright))) /\\ (exists fom_gap_pfp_execution_common_productright_value_bound. fom_gap_pfp_execution_common_productright_value_bound + S (fom_value_pfp_execution_common_productright) = p))) /\\ (((((((q)=0 \\/ (S d)=0) /\\ (((I)=0)))) \\/ (((~((q)=0)) /\\ (((~((S d)=0)) /\\ (((q)+(S d)=S (I)))))))) /\\ ((forall pfc_index_execution_common_productcoefficients. (exists pfa_gap_execution_common_productcoefficientsbound. pfa_gap_execution_common_productcoefficientsbound + S (pfc_index_execution_common_productcoefficients) = (I)) -> exists pfc_value_execution_common_productcoefficients. ((((exists ff_h_pfp_execution_common_productcoefficientsentry. ff_h_pfp_execution_common_productcoefficientsentry + S (pfc_value_execution_common_productcoefficients) = S ((S (pfc_index_execution_common_productcoefficients)) * pc)) /\\ exists ff_q_pfp_execution_common_productcoefficientsentry. pb = ff_q_pfp_execution_common_productcoefficientsentry * S ((S (pfc_index_execution_common_productcoefficients)) * pc) + (pfc_value_execution_common_productcoefficients))) /\\ ((exists pfc_terms_code_execution_common_productcoefficientscoefficient pfc_terms_scale_execution_common_productcoefficientscoefficient pfc_natural_sum_execution_common_productcoefficientscoefficient. ((forall pfc_index_execution_common_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_common_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_common_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_common_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_common_productcoefficients))) -> exists pfc_value_execution_common_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_common_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_common_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_common_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_common_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_common_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_common_productcoefficientscoefficient = ff_q_pfp_execution_common_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_common_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_productcoefficientscoefficient) + (pfc_value_execution_common_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_common_productcoefficientscoefficientdiagonalterm pfc_left_execution_common_productcoefficientscoefficientdiagonalterm pfc_right_execution_common_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_common_productcoefficientscoefficientdiagonal)+pfc_complement_execution_common_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_common_productcoefficients)) /\\ ((((((exists pfa_gap_execution_common_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_common_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_common_productcoefficientscoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_execution_common_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_common_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_common_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_common_productcoefficientscoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_common_productcoefficientscoefficientdiagonaltermleftentry. qb = ff_q_pfp_execution_common_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_common_productcoefficientscoefficientdiagonal)) * qc) + (pfc_left_execution_common_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_common_productcoefficientscoefficientdiagonaltermleftoutside+(q)=(pfc_index_execution_common_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_common_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_common_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_common_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_common_productcoefficientscoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_execution_common_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_common_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_common_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_common_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_common_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_common_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_common_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_execution_common_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_common_productcoefficientscoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_execution_common_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_common_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_common_productcoefficientscoefficientdiagonal)=pfc_left_execution_common_productcoefficientscoefficientdiagonalterm*pfc_right_execution_common_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_common_productcoefficientscoefficientsum fs_v_pfc_execution_common_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_common_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_common_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_common_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_common_productcoefficientscoefficientsum = fs_q_pfc_execution_common_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_common_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_common_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_common_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_common_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_common_productcoefficients))) * fs_v_pfc_execution_common_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_common_productcoefficientscoefficientsum = fs_q_pfc_execution_common_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_common_productcoefficients))) * fs_v_pfc_execution_common_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_common_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_common_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_common_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_common_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_common_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_common_productcoefficients)) -> exists fs_a_pfc_execution_common_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_common_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_common_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_common_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_common_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_common_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_common_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_common_productcoefficientscoefficient = fs_q_pfc_execution_common_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_common_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_productcoefficientscoefficient) + (fs_a_pfc_execution_common_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_common_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_common_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_common_productcoefficientscoefficientsum = fs_q_pfc_execution_common_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_common_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_productcoefficientscoefficientsum) + (fs_r_pfc_execution_common_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_common_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_common_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_common_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_common_productcoefficientscoefficientsum = fs_q_pfc_execution_common_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_common_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_productcoefficientscoefficientsum) + (fs_s_pfc_execution_common_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_common_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_common_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_common_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_common_productcoefficientscoefficientresiduebound. pfa_gap_execution_common_productcoefficientscoefficientresiduebound + S (pfc_value_execution_common_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_common_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_common_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_common_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_common_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_common_productcoefficients) + (p) * pfa_offset_right_execution_common_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_execution_common_sum_left_bounded. (exists fom_gap_pfp_execution_common_sum_left_bounded_index_bound. fom_gap_pfp_execution_common_sum_left_bounded_index_bound + S (fom_index_pfp_execution_common_sum_left_bounded) = I) -> exists fom_value_pfp_execution_common_sum_left_bounded. ((((exists fom_beta_height_pfp_execution_common_sum_left_bounded_entry. fom_beta_height_pfp_execution_common_sum_left_bounded_entry + S (fom_value_pfp_execution_common_sum_left_bounded) = S ((S (fom_index_pfp_execution_common_sum_left_bounded)) * pc)) /\\ exists fom_beta_quotient_pfp_execution_common_sum_left_bounded_entry. pb = fom_beta_quotient_pfp_execution_common_sum_left_bounded_entry * S ((S (fom_index_pfp_execution_common_sum_left_bounded)) * pc) + (fom_value_pfp_execution_common_sum_left_bounded))) /\\ (exists fom_gap_pfp_execution_common_sum_left_bounded_value_bound. fom_gap_pfp_execution_common_sum_left_bounded_value_bound + S (fom_value_pfp_execution_common_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_execution_common_sum_right_bounded. (exists fom_gap_pfp_execution_common_sum_right_bounded_index_bound. fom_gap_pfp_execution_common_sum_right_bounded_index_bound + S (fom_index_pfp_execution_common_sum_right_bounded) = N) -> exists fom_value_pfp_execution_common_sum_right_bounded. ((((exists fom_beta_height_pfp_execution_common_sum_right_bounded_entry. fom_beta_height_pfp_execution_common_sum_right_bounded_entry + S (fom_value_pfp_execution_common_sum_right_bounded) = S ((S (fom_index_pfp_execution_common_sum_right_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_execution_common_sum_right_bounded_entry. rb = fom_beta_quotient_pfp_execution_common_sum_right_bounded_entry * S ((S (fom_index_pfp_execution_common_sum_right_bounded)) * rc) + (fom_value_pfp_execution_common_sum_right_bounded))) /\\ (exists fom_gap_pfp_execution_common_sum_right_bounded_value_bound. fom_gap_pfp_execution_common_sum_right_bounded_value_bound + S (fom_value_pfp_execution_common_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_execution_common_sum_result_bounded. (exists fom_gap_pfp_execution_common_sum_result_bounded_index_bound. fom_gap_pfp_execution_common_sum_result_bounded_index_bound + S (fom_index_pfp_execution_common_sum_result_bounded) = L) -> exists fom_value_pfp_execution_common_sum_result_bounded. ((((exists fom_beta_height_pfp_execution_common_sum_result_bounded_entry. fom_beta_height_pfp_execution_common_sum_result_bounded_entry + S (fom_value_pfp_execution_common_sum_result_bounded) = S ((S (fom_index_pfp_execution_common_sum_result_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_common_sum_result_bounded_entry. ab = fom_beta_quotient_pfp_execution_common_sum_result_bounded_entry * S ((S (fom_index_pfp_execution_common_sum_result_bounded)) * ac) + (fom_value_pfp_execution_common_sum_result_bounded))) /\\ (exists fom_gap_pfp_execution_common_sum_result_bounded_value_bound. fom_gap_pfp_execution_common_sum_result_bounded_value_bound + S (fom_value_pfp_execution_common_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_execution_common_sum pfaa_left_c_execution_common_sum pfaa_right_b_execution_common_sum pfaa_right_c_execution_common_sum pfaa_sum_b_execution_common_sum pfaa_sum_c_execution_common_sum pfaa_length_execution_common_sum. ((((forall pfrep_power_execution_common_sum_witness_common_left pfrep_left_execution_common_sum_witness_common_left pfrep_right_execution_common_sum_witness_common_left. ((exists pfrep_position_execution_common_sum_witness_common_leftfirst. ((pfrep_position_execution_common_sum_witness_common_leftfirst+S (pfrep_power_execution_common_sum_witness_common_left)=(I)) /\\ ((((exists ff_h_pfp_execution_common_sum_witness_common_leftfirstentry. ff_h_pfp_execution_common_sum_witness_common_leftfirstentry + S (pfrep_left_execution_common_sum_witness_common_left) = S ((S (pfrep_position_execution_common_sum_witness_common_leftfirst)) * pc)) /\\ exists ff_q_pfp_execution_common_sum_witness_common_leftfirstentry. pb = ff_q_pfp_execution_common_sum_witness_common_leftfirstentry * S ((S (pfrep_position_execution_common_sum_witness_common_leftfirst)) * pc) + (pfrep_left_execution_common_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_execution_common_sum_witness_common_leftfirstoutside. pfrep_gap_execution_common_sum_witness_common_leftfirstoutside+(I)=(pfrep_power_execution_common_sum_witness_common_left)) /\\ (((pfrep_left_execution_common_sum_witness_common_left)=0))))) -> ((exists pfrep_position_execution_common_sum_witness_common_leftsecond. ((pfrep_position_execution_common_sum_witness_common_leftsecond+S (pfrep_power_execution_common_sum_witness_common_left)=(pfaa_length_execution_common_sum)) /\\ ((((exists ff_h_pfp_execution_common_sum_witness_common_leftsecondentry. ff_h_pfp_execution_common_sum_witness_common_leftsecondentry + S (pfrep_right_execution_common_sum_witness_common_left) = S ((S (pfrep_position_execution_common_sum_witness_common_leftsecond)) * pfaa_left_c_execution_common_sum)) /\\ exists ff_q_pfp_execution_common_sum_witness_common_leftsecondentry. pfaa_left_b_execution_common_sum = ff_q_pfp_execution_common_sum_witness_common_leftsecondentry * S ((S (pfrep_position_execution_common_sum_witness_common_leftsecond)) * pfaa_left_c_execution_common_sum) + (pfrep_right_execution_common_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_execution_common_sum_witness_common_leftsecondoutside. pfrep_gap_execution_common_sum_witness_common_leftsecondoutside+(pfaa_length_execution_common_sum)=(pfrep_power_execution_common_sum_witness_common_left)) /\\ (((pfrep_right_execution_common_sum_witness_common_left)=0))))) -> pfrep_left_execution_common_sum_witness_common_left=pfrep_right_execution_common_sum_witness_common_left) /\\ ((forall pfrep_power_execution_common_sum_witness_common_right pfrep_left_execution_common_sum_witness_common_right pfrep_right_execution_common_sum_witness_common_right. ((exists pfrep_position_execution_common_sum_witness_common_rightfirst. ((pfrep_position_execution_common_sum_witness_common_rightfirst+S (pfrep_power_execution_common_sum_witness_common_right)=(N)) /\\ ((((exists ff_h_pfp_execution_common_sum_witness_common_rightfirstentry. ff_h_pfp_execution_common_sum_witness_common_rightfirstentry + S (pfrep_left_execution_common_sum_witness_common_right) = S ((S (pfrep_position_execution_common_sum_witness_common_rightfirst)) * rc)) /\\ exists ff_q_pfp_execution_common_sum_witness_common_rightfirstentry. rb = ff_q_pfp_execution_common_sum_witness_common_rightfirstentry * S ((S (pfrep_position_execution_common_sum_witness_common_rightfirst)) * rc) + (pfrep_left_execution_common_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_execution_common_sum_witness_common_rightfirstoutside. pfrep_gap_execution_common_sum_witness_common_rightfirstoutside+(N)=(pfrep_power_execution_common_sum_witness_common_right)) /\\ (((pfrep_left_execution_common_sum_witness_common_right)=0))))) -> ((exists pfrep_position_execution_common_sum_witness_common_rightsecond. ((pfrep_position_execution_common_sum_witness_common_rightsecond+S (pfrep_power_execution_common_sum_witness_common_right)=(pfaa_length_execution_common_sum)) /\\ ((((exists ff_h_pfp_execution_common_sum_witness_common_rightsecondentry. ff_h_pfp_execution_common_sum_witness_common_rightsecondentry + S (pfrep_right_execution_common_sum_witness_common_right) = S ((S (pfrep_position_execution_common_sum_witness_common_rightsecond)) * pfaa_right_c_execution_common_sum)) /\\ exists ff_q_pfp_execution_common_sum_witness_common_rightsecondentry. pfaa_right_b_execution_common_sum = ff_q_pfp_execution_common_sum_witness_common_rightsecondentry * S ((S (pfrep_position_execution_common_sum_witness_common_rightsecond)) * pfaa_right_c_execution_common_sum) + (pfrep_right_execution_common_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_execution_common_sum_witness_common_rightsecondoutside. pfrep_gap_execution_common_sum_witness_common_rightsecondoutside+(pfaa_length_execution_common_sum)=(pfrep_power_execution_common_sum_witness_common_right)) /\\ (((pfrep_right_execution_common_sum_witness_common_right)=0))))) -> pfrep_left_execution_common_sum_witness_common_right=pfrep_right_execution_common_sum_witness_common_right)))) /\\ (((forall pfp_index_execution_common_sum_witness_operation. (exists pfa_gap_execution_common_sum_witness_operationindex. pfa_gap_execution_common_sum_witness_operationindex + S (pfp_index_execution_common_sum_witness_operation) = (pfaa_length_execution_common_sum)) -> exists pfp_left_execution_common_sum_witness_operation pfp_right_execution_common_sum_witness_operation pfp_value_execution_common_sum_witness_operation. ((((exists ff_h_pfp_execution_common_sum_witness_operationleft. ff_h_pfp_execution_common_sum_witness_operationleft + S (pfp_left_execution_common_sum_witness_operation) = S ((S (pfp_index_execution_common_sum_witness_operation)) * pfaa_left_c_execution_common_sum)) /\\ exists ff_q_pfp_execution_common_sum_witness_operationleft. pfaa_left_b_execution_common_sum = ff_q_pfp_execution_common_sum_witness_operationleft * S ((S (pfp_index_execution_common_sum_witness_operation)) * pfaa_left_c_execution_common_sum) + (pfp_left_execution_common_sum_witness_operation))) /\\ (((((exists ff_h_pfp_execution_common_sum_witness_operationright. ff_h_pfp_execution_common_sum_witness_operationright + S (pfp_right_execution_common_sum_witness_operation) = S ((S (pfp_index_execution_common_sum_witness_operation)) * pfaa_right_c_execution_common_sum)) /\\ exists ff_q_pfp_execution_common_sum_witness_operationright. pfaa_right_b_execution_common_sum = ff_q_pfp_execution_common_sum_witness_operationright * S ((S (pfp_index_execution_common_sum_witness_operation)) * pfaa_right_c_execution_common_sum) + (pfp_right_execution_common_sum_witness_operation))) /\\ (((((exists ff_h_pfp_execution_common_sum_witness_operationtarget. ff_h_pfp_execution_common_sum_witness_operationtarget + S (pfp_value_execution_common_sum_witness_operation) = S ((S (pfp_index_execution_common_sum_witness_operation)) * pfaa_sum_c_execution_common_sum)) /\\ exists ff_q_pfp_execution_common_sum_witness_operationtarget. pfaa_sum_b_execution_common_sum = ff_q_pfp_execution_common_sum_witness_operationtarget * S ((S (pfp_index_execution_common_sum_witness_operation)) * pfaa_sum_c_execution_common_sum) + (pfp_value_execution_common_sum_witness_operation))) /\\ ((((exists pfa_gap_execution_common_sum_witness_operationoperationleft. pfa_gap_execution_common_sum_witness_operationoperationleft + S (pfp_left_execution_common_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_execution_common_sum_witness_operationoperationright. pfa_gap_execution_common_sum_witness_operationoperationright + S (pfp_right_execution_common_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_execution_common_sum_witness_operationoperationresultbound. pfa_gap_execution_common_sum_witness_operationoperationresultbound + S (pfp_value_execution_common_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_execution_common_sum_witness_operationoperationresultcongruence pfa_offset_right_execution_common_sum_witness_operationoperationresultcongruence. ((pfp_left_execution_common_sum_witness_operation) + (pfp_right_execution_common_sum_witness_operation)) + (p) * pfa_offset_left_execution_common_sum_witness_operationoperationresultcongruence = (pfp_value_execution_common_sum_witness_operation) + (p) * pfa_offset_right_execution_common_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_execution_common_sum_witness_output pfrep_left_execution_common_sum_witness_output pfrep_right_execution_common_sum_witness_output. ((exists pfrep_position_execution_common_sum_witness_outputfirst. ((pfrep_position_execution_common_sum_witness_outputfirst+S (pfrep_power_execution_common_sum_witness_output)=(pfaa_length_execution_common_sum)) /\\ ((((exists ff_h_pfp_execution_common_sum_witness_outputfirstentry. ff_h_pfp_execution_common_sum_witness_outputfirstentry + S (pfrep_left_execution_common_sum_witness_output) = S ((S (pfrep_position_execution_common_sum_witness_outputfirst)) * pfaa_sum_c_execution_common_sum)) /\\ exists ff_q_pfp_execution_common_sum_witness_outputfirstentry. pfaa_sum_b_execution_common_sum = ff_q_pfp_execution_common_sum_witness_outputfirstentry * S ((S (pfrep_position_execution_common_sum_witness_outputfirst)) * pfaa_sum_c_execution_common_sum) + (pfrep_left_execution_common_sum_witness_output)))))) \\/ (((exists pfrep_gap_execution_common_sum_witness_outputfirstoutside. pfrep_gap_execution_common_sum_witness_outputfirstoutside+(pfaa_length_execution_common_sum)=(pfrep_power_execution_common_sum_witness_output)) /\\ (((pfrep_left_execution_common_sum_witness_output)=0))))) -> ((exists pfrep_position_execution_common_sum_witness_outputsecond. ((pfrep_position_execution_common_sum_witness_outputsecond+S (pfrep_power_execution_common_sum_witness_output)=(L)) /\\ ((((exists ff_h_pfp_execution_common_sum_witness_outputsecondentry. ff_h_pfp_execution_common_sum_witness_outputsecondentry + S (pfrep_right_execution_common_sum_witness_output) = S ((S (pfrep_position_execution_common_sum_witness_outputsecond)) * ac)) /\\ exists ff_q_pfp_execution_common_sum_witness_outputsecondentry. ab = ff_q_pfp_execution_common_sum_witness_outputsecondentry * S ((S (pfrep_position_execution_common_sum_witness_outputsecond)) * ac) + (pfrep_right_execution_common_sum_witness_output)))))) \\/ (((exists pfrep_gap_execution_common_sum_witness_outputsecondoutside. pfrep_gap_execution_common_sum_witness_outputsecondoutside+(L)=(pfrep_power_execution_common_sum_witness_output)) /\\ (((pfrep_right_execution_common_sum_witness_output)=0))))) -> pfrep_left_execution_common_sum_witness_output=pfrep_right_execution_common_sum_witness_output)))))))))))))))", "specialize prime_field_polynomial_division_execution_aligned_identity (p)", "specialize prime_field_polynomial_division_execution_aligned_identity (ab)", "specialize prime_field_polynomial_division_execution_aligned_identity (ac)", "specialize prime_field_polynomial_division_execution_aligned_identity (L)", "specialize prime_field_polynomial_division_execution_aligned_identity (bb)", "specialize prime_field_polynomial_division_execution_aligned_identity (bc)", "specialize prime_field_polynomial_division_execution_aligned_identity (d)", "specialize prime_field_polynomial_division_execution_aligned_identity (qb)", "specialize prime_field_polynomial_division_execution_aligned_identity (qc)", "specialize prime_field_polynomial_division_execution_aligned_identity (q)", "specialize prime_field_polynomial_division_execution_aligned_identity (rb)", "specialize prime_field_polynomial_division_execution_aligned_identity (rc)", "specialize prime_field_polynomial_division_execution_aligned_identity (N)", "apply prime_field_polynomial_division_execution_aligned_identity", "exact hp", "exact he", "cases hi", "cases hi_witness", "cases hi_witness_witness", "cases hi_witness_witness_witness", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (p)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (db)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (dc)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (J)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (ab)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (ac)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (L)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (bb)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (bc)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (S d)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (qb)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (qc)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (q)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (x)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (x1)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (x2)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (rb)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (rc)", "specialize prime_field_polynomial_common_right_divisor_euclidean_transport (N)", "apply prime_field_polynomial_common_right_divisor_euclidean_transport", "exact hp", "exact hi_witness_witness_witness_left", "exact hi_witness_witness_witness_right" ], "script_sha256": "cb03253ffd3ececb71d858b55548b7cadee5b4f1b22734d9b2d7b461cd1c0829", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_euclidean_transport_candidate.py", "sha256": "9a589d1749eb38d30d1a24364bc4d66f7df0efb59247527f7831f97557da9c30" }, "statement": "forall p ab ac L bb bc d qb qc q rb rc N db dc J. (~((p) = 1) /\\ forall pfa_factor_left_execution_common_prime pfa_factor_right_execution_common_prime. (p) = pfa_factor_left_execution_common_prime * pfa_factor_right_execution_common_prime -> pfa_factor_left_execution_common_prime = 1 \\/ pfa_factor_right_execution_common_prime = 1) -> (((forall fom_index_pfp_execution_common_actualinput. (exists fom_gap_pfp_execution_common_actualinput_index_bound. fom_gap_pfp_execution_common_actualinput_index_bound + S (fom_index_pfp_execution_common_actualinput) = L) -> exists fom_value_pfp_execution_common_actualinput. ((((exists fom_beta_height_pfp_execution_common_actualinput_entry. fom_beta_height_pfp_execution_common_actualinput_entry + S (fom_value_pfp_execution_common_actualinput) = S ((S (fom_index_pfp_execution_common_actualinput)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_common_actualinput_entry. ab = fom_beta_quotient_pfp_execution_common_actualinput_entry * S ((S (fom_index_pfp_execution_common_actualinput)) * ac) + (fom_value_pfp_execution_common_actualinput))) /\\ (exists fom_gap_pfp_execution_common_actualinput_value_bound. fom_gap_pfp_execution_common_actualinput_value_bound + S (fom_value_pfp_execution_common_actualinput) = p))) /\\ (((forall fom_index_pfp_execution_common_actualdivisor. (exists fom_gap_pfp_execution_common_actualdivisor_index_bound. fom_gap_pfp_execution_common_actualdivisor_index_bound + S (fom_index_pfp_execution_common_actualdivisor) = S (d)) -> exists fom_value_pfp_execution_common_actualdivisor. ((((exists fom_beta_height_pfp_execution_common_actualdivisor_entry. fom_beta_height_pfp_execution_common_actualdivisor_entry + S (fom_value_pfp_execution_common_actualdivisor) = S ((S (fom_index_pfp_execution_common_actualdivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_common_actualdivisor_entry. bb = fom_beta_quotient_pfp_execution_common_actualdivisor_entry * S ((S (fom_index_pfp_execution_common_actualdivisor)) * bc) + (fom_value_pfp_execution_common_actualdivisor))) /\\ (exists fom_gap_pfp_execution_common_actualdivisor_value_bound. fom_gap_pfp_execution_common_actualdivisor_value_bound + S (fom_value_pfp_execution_common_actualdivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_execution_common_actuallengthshort. pfc_gap_execution_common_actuallengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((exists pfd_head_execution_common_actual pfd_inverse_execution_common_actual pfd_product_code_execution_common_actual pfd_product_scale_execution_common_actual pfd_residual_code_execution_common_actual pfd_residual_scale_execution_common_actual pfd_cut_execution_common_actual. ((((exists ff_h_pfp_execution_common_actualhead. ff_h_pfp_execution_common_actualhead + S (pfd_head_execution_common_actual) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_execution_common_actualhead. bb = ff_q_pfp_execution_common_actualhead * S ((S (0)) * bc) + (pfd_head_execution_common_actual))) /\\ (((((~((pfd_head_execution_common_actual) = 0)) /\\ ((((exists pfa_gap_execution_common_actualinversemultiplicationleft. pfa_gap_execution_common_actualinversemultiplicationleft + S (pfd_head_execution_common_actual) = (p)) /\\ (((exists pfa_gap_execution_common_actualinversemultiplicationright. pfa_gap_execution_common_actualinversemultiplicationright + S (pfd_inverse_execution_common_actual) = (p)) /\\ ((((exists pfa_gap_execution_common_actualinversemultiplicationresultbound. pfa_gap_execution_common_actualinversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_execution_common_actualinversemultiplicationresultcongruence pfa_offset_right_execution_common_actualinversemultiplicationresultcongruence. ((pfd_head_execution_common_actual) * (pfd_inverse_execution_common_actual)) + (p) * pfa_offset_left_execution_common_actualinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_execution_common_actualinversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_execution_common_actualquotient. (exists pfa_gap_execution_common_actualquotientbound. pfa_gap_execution_common_actualquotientbound + S (pfd_index_execution_common_actualquotient) = (q)) -> exists pfd_value_execution_common_actualquotient. ((((exists ff_h_pfp_execution_common_actualquotiententry. ff_h_pfp_execution_common_actualquotiententry + S (pfd_value_execution_common_actualquotient) = S ((S (pfd_index_execution_common_actualquotient)) * qc)) /\\ exists ff_q_pfp_execution_common_actualquotiententry. qb = ff_q_pfp_execution_common_actualquotiententry * S ((S (pfd_index_execution_common_actualquotient)) * qc) + (pfd_value_execution_common_actualquotient))) /\\ ((exists pfd_input_execution_common_actualquotientstep pfd_previous_execution_common_actualquotientstep pfd_difference_execution_common_actualquotientstep. ((((exists ff_h_pfp_execution_common_actualquotientstepinput. ff_h_pfp_execution_common_actualquotientstepinput + S (pfd_input_execution_common_actualquotientstep) = S ((S (pfd_index_execution_common_actualquotient)) * ac)) /\\ exists ff_q_pfp_execution_common_actualquotientstepinput. ab = ff_q_pfp_execution_common_actualquotientstepinput * S ((S (pfd_index_execution_common_actualquotient)) * ac) + (pfd_input_execution_common_actualquotientstep))) /\\ (((exists pfc_terms_code_execution_common_actualquotientstepprevious pfc_terms_scale_execution_common_actualquotientstepprevious pfc_natural_sum_execution_common_actualquotientstepprevious. ((forall pfc_index_execution_common_actualquotientsteppreviousdiagonal. (exists pfa_gap_execution_common_actualquotientsteppreviousdiagonalbound. pfa_gap_execution_common_actualquotientsteppreviousdiagonalbound + S (pfc_index_execution_common_actualquotientsteppreviousdiagonal) = (S (pfd_index_execution_common_actualquotient))) -> exists pfc_value_execution_common_actualquotientsteppreviousdiagonal. ((((exists ff_h_pfp_execution_common_actualquotientsteppreviousdiagonalentry. ff_h_pfp_execution_common_actualquotientsteppreviousdiagonalentry + S (pfc_value_execution_common_actualquotientsteppreviousdiagonal) = S ((S (pfc_index_execution_common_actualquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_common_actualquotientstepprevious)) /\\ exists ff_q_pfp_execution_common_actualquotientsteppreviousdiagonalentry. pfc_terms_code_execution_common_actualquotientstepprevious = ff_q_pfp_execution_common_actualquotientsteppreviousdiagonalentry * S ((S (pfc_index_execution_common_actualquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_common_actualquotientstepprevious) + (pfc_value_execution_common_actualquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_execution_common_actualquotientsteppreviousdiagonalterm pfc_left_execution_common_actualquotientsteppreviousdiagonalterm pfc_right_execution_common_actualquotientsteppreviousdiagonalterm. (((pfc_index_execution_common_actualquotientsteppreviousdiagonal)+pfc_complement_execution_common_actualquotientsteppreviousdiagonalterm=(pfd_index_execution_common_actualquotient)) /\\ ((((((exists pfa_gap_execution_common_actualquotientsteppreviousdiagonaltermleftinside. pfa_gap_execution_common_actualquotientsteppreviousdiagonaltermleftinside + S (pfc_index_execution_common_actualquotientsteppreviousdiagonal) = (pfd_index_execution_common_actualquotient)) /\\ ((((exists ff_h_pfp_execution_common_actualquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_execution_common_actualquotientsteppreviousdiagonaltermleftentry + S (pfc_left_execution_common_actualquotientsteppreviousdiagonalterm) = S ((S (pfc_index_execution_common_actualquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_common_actualquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_execution_common_actualquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_execution_common_actualquotientsteppreviousdiagonal)) * qc) + (pfc_left_execution_common_actualquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_actualquotientsteppreviousdiagonaltermleftoutside. pfc_gap_execution_common_actualquotientsteppreviousdiagonaltermleftoutside+(pfd_index_execution_common_actualquotient)=(pfc_index_execution_common_actualquotientsteppreviousdiagonal)) /\\ (((pfc_left_execution_common_actualquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_common_actualquotientsteppreviousdiagonaltermrightinside. pfa_gap_execution_common_actualquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_execution_common_actualquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_common_actualquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_execution_common_actualquotientsteppreviousdiagonaltermrightentry + S (pfc_right_execution_common_actualquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_execution_common_actualquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_common_actualquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_execution_common_actualquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_execution_common_actualquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_execution_common_actualquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_actualquotientsteppreviousdiagonaltermrightoutside. pfc_gap_execution_common_actualquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_common_actualquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_execution_common_actualquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_execution_common_actualquotientsteppreviousdiagonal)=pfc_left_execution_common_actualquotientsteppreviousdiagonalterm*pfc_right_execution_common_actualquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_common_actualquotientstepprevioussum fs_v_pfc_execution_common_actualquotientstepprevioussum. ((((exists fs_h_pfc_execution_common_actualquotientstepprevioussum_body_start. fs_h_pfc_execution_common_actualquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_common_actualquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_common_actualquotientstepprevioussum_body_start. fs_u_pfc_execution_common_actualquotientstepprevioussum = fs_q_pfc_execution_common_actualquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_execution_common_actualquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_execution_common_actualquotientstepprevioussum_body_terminal. fs_h_pfc_execution_common_actualquotientstepprevioussum_body_terminal + S (pfc_natural_sum_execution_common_actualquotientstepprevious) = S ((S (S (pfd_index_execution_common_actualquotient))) * fs_v_pfc_execution_common_actualquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_common_actualquotientstepprevioussum_body_terminal. fs_u_pfc_execution_common_actualquotientstepprevioussum = fs_q_pfc_execution_common_actualquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_execution_common_actualquotient))) * fs_v_pfc_execution_common_actualquotientstepprevioussum) + (pfc_natural_sum_execution_common_actualquotientstepprevious))) /\\ forall fs_i_pfc_execution_common_actualquotientstepprevioussum_body_steps. (exists fs_lt_pfc_execution_common_actualquotientstepprevioussum_body_steps_bound. fs_lt_pfc_execution_common_actualquotientstepprevioussum_body_steps_bound + S fs_i_pfc_execution_common_actualquotientstepprevioussum_body_steps = S (pfd_index_execution_common_actualquotient)) -> exists fs_a_pfc_execution_common_actualquotientstepprevioussum_body_steps fs_r_pfc_execution_common_actualquotientstepprevioussum_body_steps fs_s_pfc_execution_common_actualquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_execution_common_actualquotientstepprevioussum_body_steps_summand. fs_h_pfc_execution_common_actualquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_execution_common_actualquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_common_actualquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_common_actualquotientstepprevious)) /\\ exists fs_q_pfc_execution_common_actualquotientstepprevioussum_body_steps_summand. pfc_terms_code_execution_common_actualquotientstepprevious = fs_q_pfc_execution_common_actualquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_execution_common_actualquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_common_actualquotientstepprevious) + (fs_a_pfc_execution_common_actualquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_actualquotientstepprevioussum_body_steps_partial. fs_h_pfc_execution_common_actualquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_execution_common_actualquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_common_actualquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_common_actualquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_common_actualquotientstepprevioussum_body_steps_partial. fs_u_pfc_execution_common_actualquotientstepprevioussum = fs_q_pfc_execution_common_actualquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_execution_common_actualquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_common_actualquotientstepprevioussum) + (fs_r_pfc_execution_common_actualquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_actualquotientstepprevioussum_body_steps_successor. fs_h_pfc_execution_common_actualquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_execution_common_actualquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_execution_common_actualquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_common_actualquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_common_actualquotientstepprevioussum_body_steps_successor. fs_u_pfc_execution_common_actualquotientstepprevioussum = fs_q_pfc_execution_common_actualquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_execution_common_actualquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_common_actualquotientstepprevioussum) + (fs_s_pfc_execution_common_actualquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_execution_common_actualquotientstepprevioussum_body_steps = fs_r_pfc_execution_common_actualquotientstepprevioussum_body_steps + fs_a_pfc_execution_common_actualquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_execution_common_actualquotientsteppreviousresiduebound. pfa_gap_execution_common_actualquotientsteppreviousresiduebound + S (pfd_previous_execution_common_actualquotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_common_actualquotientsteppreviousresiduecongruence pfa_offset_right_execution_common_actualquotientsteppreviousresiduecongruence. (pfc_natural_sum_execution_common_actualquotientstepprevious) + (p) * pfa_offset_left_execution_common_actualquotientsteppreviousresiduecongruence = (pfd_previous_execution_common_actualquotientstep) + (p) * pfa_offset_right_execution_common_actualquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_execution_common_actualquotientstepsubtractleft. pfa_gap_execution_common_actualquotientstepsubtractleft + S (pfd_previous_execution_common_actualquotientstep) = (p)) /\\ (((exists pfa_gap_execution_common_actualquotientstepsubtractright. pfa_gap_execution_common_actualquotientstepsubtractright + S (pfd_difference_execution_common_actualquotientstep) = (p)) /\\ ((((exists pfa_gap_execution_common_actualquotientstepsubtractresultbound. pfa_gap_execution_common_actualquotientstepsubtractresultbound + S (pfd_input_execution_common_actualquotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_common_actualquotientstepsubtractresultcongruence pfa_offset_right_execution_common_actualquotientstepsubtractresultcongruence. ((pfd_previous_execution_common_actualquotientstep) + (pfd_difference_execution_common_actualquotientstep)) + (p) * pfa_offset_left_execution_common_actualquotientstepsubtractresultcongruence = (pfd_input_execution_common_actualquotientstep) + (p) * pfa_offset_right_execution_common_actualquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_execution_common_actualquotientstepmultiplyleft. pfa_gap_execution_common_actualquotientstepmultiplyleft + S (pfd_inverse_execution_common_actual) = (p)) /\\ (((exists pfa_gap_execution_common_actualquotientstepmultiplyright. pfa_gap_execution_common_actualquotientstepmultiplyright + S (pfd_difference_execution_common_actualquotientstep) = (p)) /\\ ((((exists pfa_gap_execution_common_actualquotientstepmultiplyresultbound. pfa_gap_execution_common_actualquotientstepmultiplyresultbound + S (pfd_value_execution_common_actualquotient) = (p)) /\\ ((exists pfa_offset_left_execution_common_actualquotientstepmultiplyresultcongruence pfa_offset_right_execution_common_actualquotientstepmultiplyresultcongruence. ((pfd_inverse_execution_common_actual) * (pfd_difference_execution_common_actualquotientstep)) + (p) * pfa_offset_left_execution_common_actualquotientstepmultiplyresultcongruence = (pfd_value_execution_common_actualquotient) + (p) * pfa_offset_right_execution_common_actualquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_execution_common_actualproduct. (exists pfa_gap_execution_common_actualproductbound. pfa_gap_execution_common_actualproductbound + S (pfc_index_execution_common_actualproduct) = (L)) -> exists pfc_value_execution_common_actualproduct. ((((exists ff_h_pfp_execution_common_actualproductentry. ff_h_pfp_execution_common_actualproductentry + S (pfc_value_execution_common_actualproduct) = S ((S (pfc_index_execution_common_actualproduct)) * pfd_product_scale_execution_common_actual)) /\\ exists ff_q_pfp_execution_common_actualproductentry. pfd_product_code_execution_common_actual = ff_q_pfp_execution_common_actualproductentry * S ((S (pfc_index_execution_common_actualproduct)) * pfd_product_scale_execution_common_actual) + (pfc_value_execution_common_actualproduct))) /\\ ((exists pfc_terms_code_execution_common_actualproductcoefficient pfc_terms_scale_execution_common_actualproductcoefficient pfc_natural_sum_execution_common_actualproductcoefficient. ((forall pfc_index_execution_common_actualproductcoefficientdiagonal. (exists pfa_gap_execution_common_actualproductcoefficientdiagonalbound. pfa_gap_execution_common_actualproductcoefficientdiagonalbound + S (pfc_index_execution_common_actualproductcoefficientdiagonal) = (S (pfc_index_execution_common_actualproduct))) -> exists pfc_value_execution_common_actualproductcoefficientdiagonal. ((((exists ff_h_pfp_execution_common_actualproductcoefficientdiagonalentry. ff_h_pfp_execution_common_actualproductcoefficientdiagonalentry + S (pfc_value_execution_common_actualproductcoefficientdiagonal) = S ((S (pfc_index_execution_common_actualproductcoefficientdiagonal)) * pfc_terms_scale_execution_common_actualproductcoefficient)) /\\ exists ff_q_pfp_execution_common_actualproductcoefficientdiagonalentry. pfc_terms_code_execution_common_actualproductcoefficient = ff_q_pfp_execution_common_actualproductcoefficientdiagonalentry * S ((S (pfc_index_execution_common_actualproductcoefficientdiagonal)) * pfc_terms_scale_execution_common_actualproductcoefficient) + (pfc_value_execution_common_actualproductcoefficientdiagonal))) /\\ ((exists pfc_complement_execution_common_actualproductcoefficientdiagonalterm pfc_left_execution_common_actualproductcoefficientdiagonalterm pfc_right_execution_common_actualproductcoefficientdiagonalterm. (((pfc_index_execution_common_actualproductcoefficientdiagonal)+pfc_complement_execution_common_actualproductcoefficientdiagonalterm=(pfc_index_execution_common_actualproduct)) /\\ ((((((exists pfa_gap_execution_common_actualproductcoefficientdiagonaltermleftinside. pfa_gap_execution_common_actualproductcoefficientdiagonaltermleftinside + S (pfc_index_execution_common_actualproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_execution_common_actualproductcoefficientdiagonaltermleftentry. ff_h_pfp_execution_common_actualproductcoefficientdiagonaltermleftentry + S (pfc_left_execution_common_actualproductcoefficientdiagonalterm) = S ((S (pfc_index_execution_common_actualproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_common_actualproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_execution_common_actualproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_common_actualproductcoefficientdiagonal)) * qc) + (pfc_left_execution_common_actualproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_actualproductcoefficientdiagonaltermleftoutside. pfc_gap_execution_common_actualproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_execution_common_actualproductcoefficientdiagonal)) /\\ (((pfc_left_execution_common_actualproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_common_actualproductcoefficientdiagonaltermrightinside. pfa_gap_execution_common_actualproductcoefficientdiagonaltermrightinside + S (pfc_complement_execution_common_actualproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_common_actualproductcoefficientdiagonaltermrightentry. ff_h_pfp_execution_common_actualproductcoefficientdiagonaltermrightentry + S (pfc_right_execution_common_actualproductcoefficientdiagonalterm) = S ((S (pfc_complement_execution_common_actualproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_common_actualproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_common_actualproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_common_actualproductcoefficientdiagonalterm)) * bc) + (pfc_right_execution_common_actualproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_actualproductcoefficientdiagonaltermrightoutside. pfc_gap_execution_common_actualproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_common_actualproductcoefficientdiagonalterm)) /\\ (((pfc_right_execution_common_actualproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_common_actualproductcoefficientdiagonal)=pfc_left_execution_common_actualproductcoefficientdiagonalterm*pfc_right_execution_common_actualproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_common_actualproductcoefficientsum fs_v_pfc_execution_common_actualproductcoefficientsum. ((((exists fs_h_pfc_execution_common_actualproductcoefficientsum_body_start. fs_h_pfc_execution_common_actualproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_common_actualproductcoefficientsum)) /\\ exists fs_q_pfc_execution_common_actualproductcoefficientsum_body_start. fs_u_pfc_execution_common_actualproductcoefficientsum = fs_q_pfc_execution_common_actualproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_common_actualproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_common_actualproductcoefficientsum_body_terminal. fs_h_pfc_execution_common_actualproductcoefficientsum_body_terminal + S (pfc_natural_sum_execution_common_actualproductcoefficient) = S ((S (S (pfc_index_execution_common_actualproduct))) * fs_v_pfc_execution_common_actualproductcoefficientsum)) /\\ exists fs_q_pfc_execution_common_actualproductcoefficientsum_body_terminal. fs_u_pfc_execution_common_actualproductcoefficientsum = fs_q_pfc_execution_common_actualproductcoefficientsum_body_terminal * S ((S (S (pfc_index_execution_common_actualproduct))) * fs_v_pfc_execution_common_actualproductcoefficientsum) + (pfc_natural_sum_execution_common_actualproductcoefficient))) /\\ forall fs_i_pfc_execution_common_actualproductcoefficientsum_body_steps. (exists fs_lt_pfc_execution_common_actualproductcoefficientsum_body_steps_bound. fs_lt_pfc_execution_common_actualproductcoefficientsum_body_steps_bound + S fs_i_pfc_execution_common_actualproductcoefficientsum_body_steps = S (pfc_index_execution_common_actualproduct)) -> exists fs_a_pfc_execution_common_actualproductcoefficientsum_body_steps fs_r_pfc_execution_common_actualproductcoefficientsum_body_steps fs_s_pfc_execution_common_actualproductcoefficientsum_body_steps. ((((exists fs_h_pfc_execution_common_actualproductcoefficientsum_body_steps_summand. fs_h_pfc_execution_common_actualproductcoefficientsum_body_steps_summand + S (fs_a_pfc_execution_common_actualproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_actualproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_common_actualproductcoefficient)) /\\ exists fs_q_pfc_execution_common_actualproductcoefficientsum_body_steps_summand. pfc_terms_code_execution_common_actualproductcoefficient = fs_q_pfc_execution_common_actualproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_common_actualproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_common_actualproductcoefficient) + (fs_a_pfc_execution_common_actualproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_actualproductcoefficientsum_body_steps_partial. fs_h_pfc_execution_common_actualproductcoefficientsum_body_steps_partial + S (fs_r_pfc_execution_common_actualproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_actualproductcoefficientsum_body_steps)) * fs_v_pfc_execution_common_actualproductcoefficientsum)) /\\ exists fs_q_pfc_execution_common_actualproductcoefficientsum_body_steps_partial. fs_u_pfc_execution_common_actualproductcoefficientsum = fs_q_pfc_execution_common_actualproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_common_actualproductcoefficientsum_body_steps)) * fs_v_pfc_execution_common_actualproductcoefficientsum) + (fs_r_pfc_execution_common_actualproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_actualproductcoefficientsum_body_steps_successor. fs_h_pfc_execution_common_actualproductcoefficientsum_body_steps_successor + S (fs_s_pfc_execution_common_actualproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_common_actualproductcoefficientsum_body_steps)) * fs_v_pfc_execution_common_actualproductcoefficientsum)) /\\ exists fs_q_pfc_execution_common_actualproductcoefficientsum_body_steps_successor. fs_u_pfc_execution_common_actualproductcoefficientsum = fs_q_pfc_execution_common_actualproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_common_actualproductcoefficientsum_body_steps)) * fs_v_pfc_execution_common_actualproductcoefficientsum) + (fs_s_pfc_execution_common_actualproductcoefficientsum_body_steps))) /\\ fs_s_pfc_execution_common_actualproductcoefficientsum_body_steps = fs_r_pfc_execution_common_actualproductcoefficientsum_body_steps + fs_a_pfc_execution_common_actualproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_common_actualproductcoefficientresiduebound. pfa_gap_execution_common_actualproductcoefficientresiduebound + S (pfc_value_execution_common_actualproduct) = (p)) /\\ ((exists pfa_offset_left_execution_common_actualproductcoefficientresiduecongruence pfa_offset_right_execution_common_actualproductcoefficientresiduecongruence. (pfc_natural_sum_execution_common_actualproductcoefficient) + (p) * pfa_offset_left_execution_common_actualproductcoefficientresiduecongruence = (pfc_value_execution_common_actualproduct) + (p) * pfa_offset_right_execution_common_actualproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_execution_common_actualdifference. (exists pfa_gap_execution_common_actualdifferenceindex. pfa_gap_execution_common_actualdifferenceindex + S (pfs_index_execution_common_actualdifference) = (L)) -> exists pfs_left_execution_common_actualdifference pfs_right_execution_common_actualdifference pfs_result_execution_common_actualdifference. ((((exists ff_h_pfp_execution_common_actualdifferenceleft. ff_h_pfp_execution_common_actualdifferenceleft + S (pfs_left_execution_common_actualdifference) = S ((S (pfs_index_execution_common_actualdifference)) * ac)) /\\ exists ff_q_pfp_execution_common_actualdifferenceleft. ab = ff_q_pfp_execution_common_actualdifferenceleft * S ((S (pfs_index_execution_common_actualdifference)) * ac) + (pfs_left_execution_common_actualdifference))) /\\ (((((exists ff_h_pfp_execution_common_actualdifferenceright. ff_h_pfp_execution_common_actualdifferenceright + S (pfs_right_execution_common_actualdifference) = S ((S (pfs_index_execution_common_actualdifference)) * pfd_product_scale_execution_common_actual)) /\\ exists ff_q_pfp_execution_common_actualdifferenceright. pfd_product_code_execution_common_actual = ff_q_pfp_execution_common_actualdifferenceright * S ((S (pfs_index_execution_common_actualdifference)) * pfd_product_scale_execution_common_actual) + (pfs_right_execution_common_actualdifference))) /\\ (((((exists ff_h_pfp_execution_common_actualdifferenceresult. ff_h_pfp_execution_common_actualdifferenceresult + S (pfs_result_execution_common_actualdifference) = S ((S (pfs_index_execution_common_actualdifference)) * pfd_residual_scale_execution_common_actual)) /\\ exists ff_q_pfp_execution_common_actualdifferenceresult. pfd_residual_code_execution_common_actual = ff_q_pfp_execution_common_actualdifferenceresult * S ((S (pfs_index_execution_common_actualdifference)) * pfd_residual_scale_execution_common_actual) + (pfs_result_execution_common_actualdifference))) /\\ ((((exists pfa_gap_execution_common_actualdifferenceoperationleft. pfa_gap_execution_common_actualdifferenceoperationleft + S (pfs_right_execution_common_actualdifference) = (p)) /\\ (((exists pfa_gap_execution_common_actualdifferenceoperationright. pfa_gap_execution_common_actualdifferenceoperationright + S (pfs_result_execution_common_actualdifference) = (p)) /\\ ((((exists pfa_gap_execution_common_actualdifferenceoperationresultbound. pfa_gap_execution_common_actualdifferenceoperationresultbound + S (pfs_left_execution_common_actualdifference) = (p)) /\\ ((exists pfa_offset_left_execution_common_actualdifferenceoperationresultcongruence pfa_offset_right_execution_common_actualdifferenceoperationresultcongruence. ((pfs_right_execution_common_actualdifference) + (pfs_result_execution_common_actualdifference)) + (p) * pfa_offset_left_execution_common_actualdifferenceoperationresultcongruence = (pfs_left_execution_common_actualdifference) + (p) * pfa_offset_right_execution_common_actualdifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_execution_common_actual)+(N)) /\\ (((forall fom_index_pfp_execution_common_actualtriminput. (exists fom_gap_pfp_execution_common_actualtriminput_index_bound. fom_gap_pfp_execution_common_actualtriminput_index_bound + S (fom_index_pfp_execution_common_actualtriminput) = L) -> exists fom_value_pfp_execution_common_actualtriminput. ((((exists fom_beta_height_pfp_execution_common_actualtriminput_entry. fom_beta_height_pfp_execution_common_actualtriminput_entry + S (fom_value_pfp_execution_common_actualtriminput) = S ((S (fom_index_pfp_execution_common_actualtriminput)) * pfd_residual_scale_execution_common_actual)) /\\ exists fom_beta_quotient_pfp_execution_common_actualtriminput_entry. pfd_residual_code_execution_common_actual = fom_beta_quotient_pfp_execution_common_actualtriminput_entry * S ((S (fom_index_pfp_execution_common_actualtriminput)) * pfd_residual_scale_execution_common_actual) + (fom_value_pfp_execution_common_actualtriminput))) /\\ (exists fom_gap_pfp_execution_common_actualtriminput_value_bound. fom_gap_pfp_execution_common_actualtriminput_value_bound + S (fom_value_pfp_execution_common_actualtriminput) = p))) /\\ (((forall pfp_repeat_index_execution_common_actualtrimremoved. (exists pfa_gap_execution_common_actualtrimremovedindex. pfa_gap_execution_common_actualtrimremovedindex + S (pfp_repeat_index_execution_common_actualtrimremoved) = (pfd_cut_execution_common_actual)) -> (((exists ff_h_pfp_execution_common_actualtrimremovedentry. ff_h_pfp_execution_common_actualtrimremovedentry + S (0) = S ((S (pfp_repeat_index_execution_common_actualtrimremoved)) * pfd_residual_scale_execution_common_actual)) /\\ exists ff_q_pfp_execution_common_actualtrimremovedentry. pfd_residual_code_execution_common_actual = ff_q_pfp_execution_common_actualtrimremovedentry * S ((S (pfp_repeat_index_execution_common_actualtrimremoved)) * pfd_residual_scale_execution_common_actual) + (0)))) /\\ (((forall pftrim_index_execution_common_actualtrimsuffix pftrim_value_execution_common_actualtrimsuffix. (exists pfa_gap_execution_common_actualtrimsuffixbound. pfa_gap_execution_common_actualtrimsuffixbound + S (pftrim_index_execution_common_actualtrimsuffix) = (N)) -> (((exists ff_h_pfp_execution_common_actualtrimsuffixsource. ff_h_pfp_execution_common_actualtrimsuffixsource + S (pftrim_value_execution_common_actualtrimsuffix) = S ((S ((pfd_cut_execution_common_actual)+pftrim_index_execution_common_actualtrimsuffix)) * pfd_residual_scale_execution_common_actual)) /\\ exists ff_q_pfp_execution_common_actualtrimsuffixsource. pfd_residual_code_execution_common_actual = ff_q_pfp_execution_common_actualtrimsuffixsource * S ((S ((pfd_cut_execution_common_actual)+pftrim_index_execution_common_actualtrimsuffix)) * pfd_residual_scale_execution_common_actual) + (pftrim_value_execution_common_actualtrimsuffix))) -> (((exists ff_h_pfp_execution_common_actualtrimsuffixoutput. ff_h_pfp_execution_common_actualtrimsuffixoutput + S (pftrim_value_execution_common_actualtrimsuffix) = S ((S (pftrim_index_execution_common_actualtrimsuffix)) * rc)) /\\ exists ff_q_pfp_execution_common_actualtrimsuffixoutput. rb = ff_q_pfp_execution_common_actualtrimsuffixoutput * S ((S (pftrim_index_execution_common_actualtrimsuffix)) * rc) + (pftrim_value_execution_common_actualtrimsuffix)))) /\\ (((N)=0 \\/ (exists pftrim_leading_execution_common_actualtrimnormal. ((((exists ff_h_pfp_execution_common_actualtrimnormalentry. ff_h_pfp_execution_common_actualtrimnormalentry + S (pftrim_leading_execution_common_actualtrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_execution_common_actualtrimnormalentry. rb = ff_q_pfp_execution_common_actualtrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_execution_common_actualtrimnormal))) /\\ ((~(pftrim_leading_execution_common_actualtrimnormal=0))))))))))))))))))))))))))))))))) -> ((((((((forall fom_index_pfp_execution_common_source_left_canonical. (exists fom_gap_pfp_execution_common_source_left_canonical_index_bound. fom_gap_pfp_execution_common_source_left_canonical_index_bound + S (fom_index_pfp_execution_common_source_left_canonical) = L) -> exists fom_value_pfp_execution_common_source_left_canonical. ((((exists fom_beta_height_pfp_execution_common_source_left_canonical_entry. fom_beta_height_pfp_execution_common_source_left_canonical_entry + S (fom_value_pfp_execution_common_source_left_canonical) = S ((S (fom_index_pfp_execution_common_source_left_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_common_source_left_canonical_entry. ab = fom_beta_quotient_pfp_execution_common_source_left_canonical_entry * S ((S (fom_index_pfp_execution_common_source_left_canonical)) * ac) + (fom_value_pfp_execution_common_source_left_canonical))) /\\ (exists fom_gap_pfp_execution_common_source_left_canonical_value_bound. fom_gap_pfp_execution_common_source_left_canonical_value_bound + S (fom_value_pfp_execution_common_source_left_canonical) = p))) /\\ ((exists pfrd_qb_execution_common_source_left pfrd_qc_execution_common_source_left pfrd_qlen_execution_common_source_left pfrd_pb_execution_common_source_left pfrd_pc_execution_common_source_left pfrd_plen_execution_common_source_left. ((((forall fom_index_pfp_execution_common_source_left_productleft. (exists fom_gap_pfp_execution_common_source_left_productleft_index_bound. fom_gap_pfp_execution_common_source_left_productleft_index_bound + S (fom_index_pfp_execution_common_source_left_productleft) = pfrd_qlen_execution_common_source_left) -> exists fom_value_pfp_execution_common_source_left_productleft. ((((exists fom_beta_height_pfp_execution_common_source_left_productleft_entry. fom_beta_height_pfp_execution_common_source_left_productleft_entry + S (fom_value_pfp_execution_common_source_left_productleft) = S ((S (fom_index_pfp_execution_common_source_left_productleft)) * pfrd_qc_execution_common_source_left)) /\\ exists fom_beta_quotient_pfp_execution_common_source_left_productleft_entry. pfrd_qb_execution_common_source_left = fom_beta_quotient_pfp_execution_common_source_left_productleft_entry * S ((S (fom_index_pfp_execution_common_source_left_productleft)) * pfrd_qc_execution_common_source_left) + (fom_value_pfp_execution_common_source_left_productleft))) /\\ (exists fom_gap_pfp_execution_common_source_left_productleft_value_bound. fom_gap_pfp_execution_common_source_left_productleft_value_bound + S (fom_value_pfp_execution_common_source_left_productleft) = p))) /\\ (((forall fom_index_pfp_execution_common_source_left_productright. (exists fom_gap_pfp_execution_common_source_left_productright_index_bound. fom_gap_pfp_execution_common_source_left_productright_index_bound + S (fom_index_pfp_execution_common_source_left_productright) = J) -> exists fom_value_pfp_execution_common_source_left_productright. ((((exists fom_beta_height_pfp_execution_common_source_left_productright_entry. fom_beta_height_pfp_execution_common_source_left_productright_entry + S (fom_value_pfp_execution_common_source_left_productright) = S ((S (fom_index_pfp_execution_common_source_left_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_execution_common_source_left_productright_entry. db = fom_beta_quotient_pfp_execution_common_source_left_productright_entry * S ((S (fom_index_pfp_execution_common_source_left_productright)) * dc) + (fom_value_pfp_execution_common_source_left_productright))) /\\ (exists fom_gap_pfp_execution_common_source_left_productright_value_bound. fom_gap_pfp_execution_common_source_left_productright_value_bound + S (fom_value_pfp_execution_common_source_left_productright) = p))) /\\ (((((((pfrd_qlen_execution_common_source_left)=0 \\/ (J)=0) /\\ (((pfrd_plen_execution_common_source_left)=0)))) \\/ (((~((pfrd_qlen_execution_common_source_left)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_execution_common_source_left)+(J)=S (pfrd_plen_execution_common_source_left)))))))) /\\ ((forall pfc_index_execution_common_source_left_productcoefficients. (exists pfa_gap_execution_common_source_left_productcoefficientsbound. pfa_gap_execution_common_source_left_productcoefficientsbound + S (pfc_index_execution_common_source_left_productcoefficients) = (pfrd_plen_execution_common_source_left)) -> exists pfc_value_execution_common_source_left_productcoefficients. ((((exists ff_h_pfp_execution_common_source_left_productcoefficientsentry. ff_h_pfp_execution_common_source_left_productcoefficientsentry + S (pfc_value_execution_common_source_left_productcoefficients) = S ((S (pfc_index_execution_common_source_left_productcoefficients)) * pfrd_pc_execution_common_source_left)) /\\ exists ff_q_pfp_execution_common_source_left_productcoefficientsentry. pfrd_pb_execution_common_source_left = ff_q_pfp_execution_common_source_left_productcoefficientsentry * S ((S (pfc_index_execution_common_source_left_productcoefficients)) * pfrd_pc_execution_common_source_left) + (pfc_value_execution_common_source_left_productcoefficients))) /\\ ((exists pfc_terms_code_execution_common_source_left_productcoefficientscoefficient pfc_terms_scale_execution_common_source_left_productcoefficientscoefficient pfc_natural_sum_execution_common_source_left_productcoefficientscoefficient. ((forall pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_common_source_left_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_common_source_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_common_source_left_productcoefficients))) -> exists pfc_value_execution_common_source_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_common_source_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_common_source_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_common_source_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_source_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_common_source_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_common_source_left_productcoefficientscoefficient = ff_q_pfp_execution_common_source_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_source_left_productcoefficientscoefficient) + (pfc_value_execution_common_source_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_common_source_left_productcoefficientscoefficientdiagonalterm pfc_left_execution_common_source_left_productcoefficientscoefficientdiagonalterm pfc_right_execution_common_source_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal)+pfc_complement_execution_common_source_left_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_common_source_left_productcoefficients)) /\\ ((((((exists pfa_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal) = (pfrd_qlen_execution_common_source_left)) /\\ ((((exists ff_h_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_common_source_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_source_left)) /\\ exists ff_q_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_execution_common_source_left = ff_q_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_source_left) + (pfc_left_execution_common_source_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_execution_common_source_left)=(pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_common_source_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_common_source_left_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_common_source_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_common_source_left_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_common_source_left_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_execution_common_source_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_execution_common_source_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_common_source_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_common_source_left_productcoefficientscoefficientdiagonal)=pfc_left_execution_common_source_left_productcoefficientscoefficientdiagonalterm*pfc_right_execution_common_source_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_common_source_left_productcoefficientscoefficientsum fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_common_source_left_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_common_source_left_productcoefficients))) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_common_source_left_productcoefficients))) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_common_source_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_common_source_left_productcoefficients)) -> exists fs_a_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_source_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_common_source_left_productcoefficientscoefficient = fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_source_left_productcoefficientscoefficient) + (fs_a_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum) + (fs_r_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum) + (fs_s_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_common_source_left_productcoefficientscoefficientresiduebound. pfa_gap_execution_common_source_left_productcoefficientscoefficientresiduebound + S (pfc_value_execution_common_source_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_common_source_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_common_source_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_common_source_left_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_common_source_left_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_common_source_left_productcoefficients) + (p) * pfa_offset_right_execution_common_source_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_execution_common_source_left_target pfrep_left_execution_common_source_left_target pfrep_right_execution_common_source_left_target. ((exists pfrep_position_execution_common_source_left_targetfirst. ((pfrep_position_execution_common_source_left_targetfirst+S (pfrep_power_execution_common_source_left_target)=(pfrd_plen_execution_common_source_left)) /\\ ((((exists ff_h_pfp_execution_common_source_left_targetfirstentry. ff_h_pfp_execution_common_source_left_targetfirstentry + S (pfrep_left_execution_common_source_left_target) = S ((S (pfrep_position_execution_common_source_left_targetfirst)) * pfrd_pc_execution_common_source_left)) /\\ exists ff_q_pfp_execution_common_source_left_targetfirstentry. pfrd_pb_execution_common_source_left = ff_q_pfp_execution_common_source_left_targetfirstentry * S ((S (pfrep_position_execution_common_source_left_targetfirst)) * pfrd_pc_execution_common_source_left) + (pfrep_left_execution_common_source_left_target)))))) \\/ (((exists pfrep_gap_execution_common_source_left_targetfirstoutside. pfrep_gap_execution_common_source_left_targetfirstoutside+(pfrd_plen_execution_common_source_left)=(pfrep_power_execution_common_source_left_target)) /\\ (((pfrep_left_execution_common_source_left_target)=0))))) -> ((exists pfrep_position_execution_common_source_left_targetsecond. ((pfrep_position_execution_common_source_left_targetsecond+S (pfrep_power_execution_common_source_left_target)=(L)) /\\ ((((exists ff_h_pfp_execution_common_source_left_targetsecondentry. ff_h_pfp_execution_common_source_left_targetsecondentry + S (pfrep_right_execution_common_source_left_target) = S ((S (pfrep_position_execution_common_source_left_targetsecond)) * ac)) /\\ exists ff_q_pfp_execution_common_source_left_targetsecondentry. ab = ff_q_pfp_execution_common_source_left_targetsecondentry * S ((S (pfrep_position_execution_common_source_left_targetsecond)) * ac) + (pfrep_right_execution_common_source_left_target)))))) \\/ (((exists pfrep_gap_execution_common_source_left_targetsecondoutside. pfrep_gap_execution_common_source_left_targetsecondoutside+(L)=(pfrep_power_execution_common_source_left_target)) /\\ (((pfrep_right_execution_common_source_left_target)=0))))) -> pfrep_left_execution_common_source_left_target=pfrep_right_execution_common_source_left_target))))))) /\\ ((((forall fom_index_pfp_execution_common_source_right_canonical. (exists fom_gap_pfp_execution_common_source_right_canonical_index_bound. fom_gap_pfp_execution_common_source_right_canonical_index_bound + S (fom_index_pfp_execution_common_source_right_canonical) = S d) -> exists fom_value_pfp_execution_common_source_right_canonical. ((((exists fom_beta_height_pfp_execution_common_source_right_canonical_entry. fom_beta_height_pfp_execution_common_source_right_canonical_entry + S (fom_value_pfp_execution_common_source_right_canonical) = S ((S (fom_index_pfp_execution_common_source_right_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_common_source_right_canonical_entry. bb = fom_beta_quotient_pfp_execution_common_source_right_canonical_entry * S ((S (fom_index_pfp_execution_common_source_right_canonical)) * bc) + (fom_value_pfp_execution_common_source_right_canonical))) /\\ (exists fom_gap_pfp_execution_common_source_right_canonical_value_bound. fom_gap_pfp_execution_common_source_right_canonical_value_bound + S (fom_value_pfp_execution_common_source_right_canonical) = p))) /\\ ((exists pfrd_qb_execution_common_source_right pfrd_qc_execution_common_source_right pfrd_qlen_execution_common_source_right pfrd_pb_execution_common_source_right pfrd_pc_execution_common_source_right pfrd_plen_execution_common_source_right. ((((forall fom_index_pfp_execution_common_source_right_productleft. (exists fom_gap_pfp_execution_common_source_right_productleft_index_bound. fom_gap_pfp_execution_common_source_right_productleft_index_bound + S (fom_index_pfp_execution_common_source_right_productleft) = pfrd_qlen_execution_common_source_right) -> exists fom_value_pfp_execution_common_source_right_productleft. ((((exists fom_beta_height_pfp_execution_common_source_right_productleft_entry. fom_beta_height_pfp_execution_common_source_right_productleft_entry + S (fom_value_pfp_execution_common_source_right_productleft) = S ((S (fom_index_pfp_execution_common_source_right_productleft)) * pfrd_qc_execution_common_source_right)) /\\ exists fom_beta_quotient_pfp_execution_common_source_right_productleft_entry. pfrd_qb_execution_common_source_right = fom_beta_quotient_pfp_execution_common_source_right_productleft_entry * S ((S (fom_index_pfp_execution_common_source_right_productleft)) * pfrd_qc_execution_common_source_right) + (fom_value_pfp_execution_common_source_right_productleft))) /\\ (exists fom_gap_pfp_execution_common_source_right_productleft_value_bound. fom_gap_pfp_execution_common_source_right_productleft_value_bound + S (fom_value_pfp_execution_common_source_right_productleft) = p))) /\\ (((forall fom_index_pfp_execution_common_source_right_productright. (exists fom_gap_pfp_execution_common_source_right_productright_index_bound. fom_gap_pfp_execution_common_source_right_productright_index_bound + S (fom_index_pfp_execution_common_source_right_productright) = J) -> exists fom_value_pfp_execution_common_source_right_productright. ((((exists fom_beta_height_pfp_execution_common_source_right_productright_entry. fom_beta_height_pfp_execution_common_source_right_productright_entry + S (fom_value_pfp_execution_common_source_right_productright) = S ((S (fom_index_pfp_execution_common_source_right_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_execution_common_source_right_productright_entry. db = fom_beta_quotient_pfp_execution_common_source_right_productright_entry * S ((S (fom_index_pfp_execution_common_source_right_productright)) * dc) + (fom_value_pfp_execution_common_source_right_productright))) /\\ (exists fom_gap_pfp_execution_common_source_right_productright_value_bound. fom_gap_pfp_execution_common_source_right_productright_value_bound + S (fom_value_pfp_execution_common_source_right_productright) = p))) /\\ (((((((pfrd_qlen_execution_common_source_right)=0 \\/ (J)=0) /\\ (((pfrd_plen_execution_common_source_right)=0)))) \\/ (((~((pfrd_qlen_execution_common_source_right)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_execution_common_source_right)+(J)=S (pfrd_plen_execution_common_source_right)))))))) /\\ ((forall pfc_index_execution_common_source_right_productcoefficients. (exists pfa_gap_execution_common_source_right_productcoefficientsbound. pfa_gap_execution_common_source_right_productcoefficientsbound + S (pfc_index_execution_common_source_right_productcoefficients) = (pfrd_plen_execution_common_source_right)) -> exists pfc_value_execution_common_source_right_productcoefficients. ((((exists ff_h_pfp_execution_common_source_right_productcoefficientsentry. ff_h_pfp_execution_common_source_right_productcoefficientsentry + S (pfc_value_execution_common_source_right_productcoefficients) = S ((S (pfc_index_execution_common_source_right_productcoefficients)) * pfrd_pc_execution_common_source_right)) /\\ exists ff_q_pfp_execution_common_source_right_productcoefficientsentry. pfrd_pb_execution_common_source_right = ff_q_pfp_execution_common_source_right_productcoefficientsentry * S ((S (pfc_index_execution_common_source_right_productcoefficients)) * pfrd_pc_execution_common_source_right) + (pfc_value_execution_common_source_right_productcoefficients))) /\\ ((exists pfc_terms_code_execution_common_source_right_productcoefficientscoefficient pfc_terms_scale_execution_common_source_right_productcoefficientscoefficient pfc_natural_sum_execution_common_source_right_productcoefficientscoefficient. ((forall pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_common_source_right_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_common_source_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_common_source_right_productcoefficients))) -> exists pfc_value_execution_common_source_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_common_source_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_common_source_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_common_source_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_source_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_common_source_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_common_source_right_productcoefficientscoefficient = ff_q_pfp_execution_common_source_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_source_right_productcoefficientscoefficient) + (pfc_value_execution_common_source_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_common_source_right_productcoefficientscoefficientdiagonalterm pfc_left_execution_common_source_right_productcoefficientscoefficientdiagonalterm pfc_right_execution_common_source_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal)+pfc_complement_execution_common_source_right_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_common_source_right_productcoefficients)) /\\ ((((((exists pfa_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal) = (pfrd_qlen_execution_common_source_right)) /\\ ((((exists ff_h_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_common_source_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_source_right)) /\\ exists ff_q_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_execution_common_source_right = ff_q_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_source_right) + (pfc_left_execution_common_source_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_execution_common_source_right)=(pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_common_source_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_common_source_right_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_common_source_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_common_source_right_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_common_source_right_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_execution_common_source_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_execution_common_source_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_common_source_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_common_source_right_productcoefficientscoefficientdiagonal)=pfc_left_execution_common_source_right_productcoefficientscoefficientdiagonalterm*pfc_right_execution_common_source_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_common_source_right_productcoefficientscoefficientsum fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_common_source_right_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_common_source_right_productcoefficients))) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_common_source_right_productcoefficients))) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_common_source_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_common_source_right_productcoefficients)) -> exists fs_a_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_source_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_common_source_right_productcoefficientscoefficient = fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_source_right_productcoefficientscoefficient) + (fs_a_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum) + (fs_r_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum) + (fs_s_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_common_source_right_productcoefficientscoefficientresiduebound. pfa_gap_execution_common_source_right_productcoefficientscoefficientresiduebound + S (pfc_value_execution_common_source_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_common_source_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_common_source_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_common_source_right_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_common_source_right_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_common_source_right_productcoefficients) + (p) * pfa_offset_right_execution_common_source_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_execution_common_source_right_target pfrep_left_execution_common_source_right_target pfrep_right_execution_common_source_right_target. ((exists pfrep_position_execution_common_source_right_targetfirst. ((pfrep_position_execution_common_source_right_targetfirst+S (pfrep_power_execution_common_source_right_target)=(pfrd_plen_execution_common_source_right)) /\\ ((((exists ff_h_pfp_execution_common_source_right_targetfirstentry. ff_h_pfp_execution_common_source_right_targetfirstentry + S (pfrep_left_execution_common_source_right_target) = S ((S (pfrep_position_execution_common_source_right_targetfirst)) * pfrd_pc_execution_common_source_right)) /\\ exists ff_q_pfp_execution_common_source_right_targetfirstentry. pfrd_pb_execution_common_source_right = ff_q_pfp_execution_common_source_right_targetfirstentry * S ((S (pfrep_position_execution_common_source_right_targetfirst)) * pfrd_pc_execution_common_source_right) + (pfrep_left_execution_common_source_right_target)))))) \\/ (((exists pfrep_gap_execution_common_source_right_targetfirstoutside. pfrep_gap_execution_common_source_right_targetfirstoutside+(pfrd_plen_execution_common_source_right)=(pfrep_power_execution_common_source_right_target)) /\\ (((pfrep_left_execution_common_source_right_target)=0))))) -> ((exists pfrep_position_execution_common_source_right_targetsecond. ((pfrep_position_execution_common_source_right_targetsecond+S (pfrep_power_execution_common_source_right_target)=(S d)) /\\ ((((exists ff_h_pfp_execution_common_source_right_targetsecondentry. ff_h_pfp_execution_common_source_right_targetsecondentry + S (pfrep_right_execution_common_source_right_target) = S ((S (pfrep_position_execution_common_source_right_targetsecond)) * bc)) /\\ exists ff_q_pfp_execution_common_source_right_targetsecondentry. bb = ff_q_pfp_execution_common_source_right_targetsecondentry * S ((S (pfrep_position_execution_common_source_right_targetsecond)) * bc) + (pfrep_right_execution_common_source_right_target)))))) \\/ (((exists pfrep_gap_execution_common_source_right_targetsecondoutside. pfrep_gap_execution_common_source_right_targetsecondoutside+(S d)=(pfrep_power_execution_common_source_right_target)) /\\ (((pfrep_right_execution_common_source_right_target)=0))))) -> pfrep_left_execution_common_source_right_target=pfrep_right_execution_common_source_right_target)))))))))) -> (((((forall fom_index_pfp_execution_common_target_left_canonical. (exists fom_gap_pfp_execution_common_target_left_canonical_index_bound. fom_gap_pfp_execution_common_target_left_canonical_index_bound + S (fom_index_pfp_execution_common_target_left_canonical) = S d) -> exists fom_value_pfp_execution_common_target_left_canonical. ((((exists fom_beta_height_pfp_execution_common_target_left_canonical_entry. fom_beta_height_pfp_execution_common_target_left_canonical_entry + S (fom_value_pfp_execution_common_target_left_canonical) = S ((S (fom_index_pfp_execution_common_target_left_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_common_target_left_canonical_entry. bb = fom_beta_quotient_pfp_execution_common_target_left_canonical_entry * S ((S (fom_index_pfp_execution_common_target_left_canonical)) * bc) + (fom_value_pfp_execution_common_target_left_canonical))) /\\ (exists fom_gap_pfp_execution_common_target_left_canonical_value_bound. fom_gap_pfp_execution_common_target_left_canonical_value_bound + S (fom_value_pfp_execution_common_target_left_canonical) = p))) /\\ ((exists pfrd_qb_execution_common_target_left pfrd_qc_execution_common_target_left pfrd_qlen_execution_common_target_left pfrd_pb_execution_common_target_left pfrd_pc_execution_common_target_left pfrd_plen_execution_common_target_left. ((((forall fom_index_pfp_execution_common_target_left_productleft. (exists fom_gap_pfp_execution_common_target_left_productleft_index_bound. fom_gap_pfp_execution_common_target_left_productleft_index_bound + S (fom_index_pfp_execution_common_target_left_productleft) = pfrd_qlen_execution_common_target_left) -> exists fom_value_pfp_execution_common_target_left_productleft. ((((exists fom_beta_height_pfp_execution_common_target_left_productleft_entry. fom_beta_height_pfp_execution_common_target_left_productleft_entry + S (fom_value_pfp_execution_common_target_left_productleft) = S ((S (fom_index_pfp_execution_common_target_left_productleft)) * pfrd_qc_execution_common_target_left)) /\\ exists fom_beta_quotient_pfp_execution_common_target_left_productleft_entry. pfrd_qb_execution_common_target_left = fom_beta_quotient_pfp_execution_common_target_left_productleft_entry * S ((S (fom_index_pfp_execution_common_target_left_productleft)) * pfrd_qc_execution_common_target_left) + (fom_value_pfp_execution_common_target_left_productleft))) /\\ (exists fom_gap_pfp_execution_common_target_left_productleft_value_bound. fom_gap_pfp_execution_common_target_left_productleft_value_bound + S (fom_value_pfp_execution_common_target_left_productleft) = p))) /\\ (((forall fom_index_pfp_execution_common_target_left_productright. (exists fom_gap_pfp_execution_common_target_left_productright_index_bound. fom_gap_pfp_execution_common_target_left_productright_index_bound + S (fom_index_pfp_execution_common_target_left_productright) = J) -> exists fom_value_pfp_execution_common_target_left_productright. ((((exists fom_beta_height_pfp_execution_common_target_left_productright_entry. fom_beta_height_pfp_execution_common_target_left_productright_entry + S (fom_value_pfp_execution_common_target_left_productright) = S ((S (fom_index_pfp_execution_common_target_left_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_execution_common_target_left_productright_entry. db = fom_beta_quotient_pfp_execution_common_target_left_productright_entry * S ((S (fom_index_pfp_execution_common_target_left_productright)) * dc) + (fom_value_pfp_execution_common_target_left_productright))) /\\ (exists fom_gap_pfp_execution_common_target_left_productright_value_bound. fom_gap_pfp_execution_common_target_left_productright_value_bound + S (fom_value_pfp_execution_common_target_left_productright) = p))) /\\ (((((((pfrd_qlen_execution_common_target_left)=0 \\/ (J)=0) /\\ (((pfrd_plen_execution_common_target_left)=0)))) \\/ (((~((pfrd_qlen_execution_common_target_left)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_execution_common_target_left)+(J)=S (pfrd_plen_execution_common_target_left)))))))) /\\ ((forall pfc_index_execution_common_target_left_productcoefficients. (exists pfa_gap_execution_common_target_left_productcoefficientsbound. pfa_gap_execution_common_target_left_productcoefficientsbound + S (pfc_index_execution_common_target_left_productcoefficients) = (pfrd_plen_execution_common_target_left)) -> exists pfc_value_execution_common_target_left_productcoefficients. ((((exists ff_h_pfp_execution_common_target_left_productcoefficientsentry. ff_h_pfp_execution_common_target_left_productcoefficientsentry + S (pfc_value_execution_common_target_left_productcoefficients) = S ((S (pfc_index_execution_common_target_left_productcoefficients)) * pfrd_pc_execution_common_target_left)) /\\ exists ff_q_pfp_execution_common_target_left_productcoefficientsentry. pfrd_pb_execution_common_target_left = ff_q_pfp_execution_common_target_left_productcoefficientsentry * S ((S (pfc_index_execution_common_target_left_productcoefficients)) * pfrd_pc_execution_common_target_left) + (pfc_value_execution_common_target_left_productcoefficients))) /\\ ((exists pfc_terms_code_execution_common_target_left_productcoefficientscoefficient pfc_terms_scale_execution_common_target_left_productcoefficientscoefficient pfc_natural_sum_execution_common_target_left_productcoefficientscoefficient. ((forall pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_common_target_left_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_common_target_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_common_target_left_productcoefficients))) -> exists pfc_value_execution_common_target_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_common_target_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_common_target_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_common_target_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_target_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_common_target_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_common_target_left_productcoefficientscoefficient = ff_q_pfp_execution_common_target_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_target_left_productcoefficientscoefficient) + (pfc_value_execution_common_target_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_common_target_left_productcoefficientscoefficientdiagonalterm pfc_left_execution_common_target_left_productcoefficientscoefficientdiagonalterm pfc_right_execution_common_target_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal)+pfc_complement_execution_common_target_left_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_common_target_left_productcoefficients)) /\\ ((((((exists pfa_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal) = (pfrd_qlen_execution_common_target_left)) /\\ ((((exists ff_h_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_common_target_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_target_left)) /\\ exists ff_q_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_execution_common_target_left = ff_q_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_target_left) + (pfc_left_execution_common_target_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_execution_common_target_left)=(pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_common_target_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_common_target_left_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_common_target_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_common_target_left_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_common_target_left_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_execution_common_target_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_execution_common_target_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_common_target_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_common_target_left_productcoefficientscoefficientdiagonal)=pfc_left_execution_common_target_left_productcoefficientscoefficientdiagonalterm*pfc_right_execution_common_target_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_common_target_left_productcoefficientscoefficientsum fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_common_target_left_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_common_target_left_productcoefficients))) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_common_target_left_productcoefficients))) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_common_target_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_common_target_left_productcoefficients)) -> exists fs_a_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_target_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_common_target_left_productcoefficientscoefficient = fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_target_left_productcoefficientscoefficient) + (fs_a_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum) + (fs_r_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum) + (fs_s_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_common_target_left_productcoefficientscoefficientresiduebound. pfa_gap_execution_common_target_left_productcoefficientscoefficientresiduebound + S (pfc_value_execution_common_target_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_common_target_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_common_target_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_common_target_left_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_common_target_left_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_common_target_left_productcoefficients) + (p) * pfa_offset_right_execution_common_target_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_execution_common_target_left_target pfrep_left_execution_common_target_left_target pfrep_right_execution_common_target_left_target. ((exists pfrep_position_execution_common_target_left_targetfirst. ((pfrep_position_execution_common_target_left_targetfirst+S (pfrep_power_execution_common_target_left_target)=(pfrd_plen_execution_common_target_left)) /\\ ((((exists ff_h_pfp_execution_common_target_left_targetfirstentry. ff_h_pfp_execution_common_target_left_targetfirstentry + S (pfrep_left_execution_common_target_left_target) = S ((S (pfrep_position_execution_common_target_left_targetfirst)) * pfrd_pc_execution_common_target_left)) /\\ exists ff_q_pfp_execution_common_target_left_targetfirstentry. pfrd_pb_execution_common_target_left = ff_q_pfp_execution_common_target_left_targetfirstentry * S ((S (pfrep_position_execution_common_target_left_targetfirst)) * pfrd_pc_execution_common_target_left) + (pfrep_left_execution_common_target_left_target)))))) \\/ (((exists pfrep_gap_execution_common_target_left_targetfirstoutside. pfrep_gap_execution_common_target_left_targetfirstoutside+(pfrd_plen_execution_common_target_left)=(pfrep_power_execution_common_target_left_target)) /\\ (((pfrep_left_execution_common_target_left_target)=0))))) -> ((exists pfrep_position_execution_common_target_left_targetsecond. ((pfrep_position_execution_common_target_left_targetsecond+S (pfrep_power_execution_common_target_left_target)=(S d)) /\\ ((((exists ff_h_pfp_execution_common_target_left_targetsecondentry. ff_h_pfp_execution_common_target_left_targetsecondentry + S (pfrep_right_execution_common_target_left_target) = S ((S (pfrep_position_execution_common_target_left_targetsecond)) * bc)) /\\ exists ff_q_pfp_execution_common_target_left_targetsecondentry. bb = ff_q_pfp_execution_common_target_left_targetsecondentry * S ((S (pfrep_position_execution_common_target_left_targetsecond)) * bc) + (pfrep_right_execution_common_target_left_target)))))) \\/ (((exists pfrep_gap_execution_common_target_left_targetsecondoutside. pfrep_gap_execution_common_target_left_targetsecondoutside+(S d)=(pfrep_power_execution_common_target_left_target)) /\\ (((pfrep_right_execution_common_target_left_target)=0))))) -> pfrep_left_execution_common_target_left_target=pfrep_right_execution_common_target_left_target))))))) /\\ ((((forall fom_index_pfp_execution_common_target_right_canonical. (exists fom_gap_pfp_execution_common_target_right_canonical_index_bound. fom_gap_pfp_execution_common_target_right_canonical_index_bound + S (fom_index_pfp_execution_common_target_right_canonical) = N) -> exists fom_value_pfp_execution_common_target_right_canonical. ((((exists fom_beta_height_pfp_execution_common_target_right_canonical_entry. fom_beta_height_pfp_execution_common_target_right_canonical_entry + S (fom_value_pfp_execution_common_target_right_canonical) = S ((S (fom_index_pfp_execution_common_target_right_canonical)) * rc)) /\\ exists fom_beta_quotient_pfp_execution_common_target_right_canonical_entry. rb = fom_beta_quotient_pfp_execution_common_target_right_canonical_entry * S ((S (fom_index_pfp_execution_common_target_right_canonical)) * rc) + (fom_value_pfp_execution_common_target_right_canonical))) /\\ (exists fom_gap_pfp_execution_common_target_right_canonical_value_bound. fom_gap_pfp_execution_common_target_right_canonical_value_bound + S (fom_value_pfp_execution_common_target_right_canonical) = p))) /\\ ((exists pfrd_qb_execution_common_target_right pfrd_qc_execution_common_target_right pfrd_qlen_execution_common_target_right pfrd_pb_execution_common_target_right pfrd_pc_execution_common_target_right pfrd_plen_execution_common_target_right. ((((forall fom_index_pfp_execution_common_target_right_productleft. (exists fom_gap_pfp_execution_common_target_right_productleft_index_bound. fom_gap_pfp_execution_common_target_right_productleft_index_bound + S (fom_index_pfp_execution_common_target_right_productleft) = pfrd_qlen_execution_common_target_right) -> exists fom_value_pfp_execution_common_target_right_productleft. ((((exists fom_beta_height_pfp_execution_common_target_right_productleft_entry. fom_beta_height_pfp_execution_common_target_right_productleft_entry + S (fom_value_pfp_execution_common_target_right_productleft) = S ((S (fom_index_pfp_execution_common_target_right_productleft)) * pfrd_qc_execution_common_target_right)) /\\ exists fom_beta_quotient_pfp_execution_common_target_right_productleft_entry. pfrd_qb_execution_common_target_right = fom_beta_quotient_pfp_execution_common_target_right_productleft_entry * S ((S (fom_index_pfp_execution_common_target_right_productleft)) * pfrd_qc_execution_common_target_right) + (fom_value_pfp_execution_common_target_right_productleft))) /\\ (exists fom_gap_pfp_execution_common_target_right_productleft_value_bound. fom_gap_pfp_execution_common_target_right_productleft_value_bound + S (fom_value_pfp_execution_common_target_right_productleft) = p))) /\\ (((forall fom_index_pfp_execution_common_target_right_productright. (exists fom_gap_pfp_execution_common_target_right_productright_index_bound. fom_gap_pfp_execution_common_target_right_productright_index_bound + S (fom_index_pfp_execution_common_target_right_productright) = J) -> exists fom_value_pfp_execution_common_target_right_productright. ((((exists fom_beta_height_pfp_execution_common_target_right_productright_entry. fom_beta_height_pfp_execution_common_target_right_productright_entry + S (fom_value_pfp_execution_common_target_right_productright) = S ((S (fom_index_pfp_execution_common_target_right_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_execution_common_target_right_productright_entry. db = fom_beta_quotient_pfp_execution_common_target_right_productright_entry * S ((S (fom_index_pfp_execution_common_target_right_productright)) * dc) + (fom_value_pfp_execution_common_target_right_productright))) /\\ (exists fom_gap_pfp_execution_common_target_right_productright_value_bound. fom_gap_pfp_execution_common_target_right_productright_value_bound + S (fom_value_pfp_execution_common_target_right_productright) = p))) /\\ (((((((pfrd_qlen_execution_common_target_right)=0 \\/ (J)=0) /\\ (((pfrd_plen_execution_common_target_right)=0)))) \\/ (((~((pfrd_qlen_execution_common_target_right)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_execution_common_target_right)+(J)=S (pfrd_plen_execution_common_target_right)))))))) /\\ ((forall pfc_index_execution_common_target_right_productcoefficients. (exists pfa_gap_execution_common_target_right_productcoefficientsbound. pfa_gap_execution_common_target_right_productcoefficientsbound + S (pfc_index_execution_common_target_right_productcoefficients) = (pfrd_plen_execution_common_target_right)) -> exists pfc_value_execution_common_target_right_productcoefficients. ((((exists ff_h_pfp_execution_common_target_right_productcoefficientsentry. ff_h_pfp_execution_common_target_right_productcoefficientsentry + S (pfc_value_execution_common_target_right_productcoefficients) = S ((S (pfc_index_execution_common_target_right_productcoefficients)) * pfrd_pc_execution_common_target_right)) /\\ exists ff_q_pfp_execution_common_target_right_productcoefficientsentry. pfrd_pb_execution_common_target_right = ff_q_pfp_execution_common_target_right_productcoefficientsentry * S ((S (pfc_index_execution_common_target_right_productcoefficients)) * pfrd_pc_execution_common_target_right) + (pfc_value_execution_common_target_right_productcoefficients))) /\\ ((exists pfc_terms_code_execution_common_target_right_productcoefficientscoefficient pfc_terms_scale_execution_common_target_right_productcoefficientscoefficient pfc_natural_sum_execution_common_target_right_productcoefficientscoefficient. ((forall pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_common_target_right_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_common_target_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_common_target_right_productcoefficients))) -> exists pfc_value_execution_common_target_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_common_target_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_common_target_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_common_target_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_target_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_common_target_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_common_target_right_productcoefficientscoefficient = ff_q_pfp_execution_common_target_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_target_right_productcoefficientscoefficient) + (pfc_value_execution_common_target_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_common_target_right_productcoefficientscoefficientdiagonalterm pfc_left_execution_common_target_right_productcoefficientscoefficientdiagonalterm pfc_right_execution_common_target_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal)+pfc_complement_execution_common_target_right_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_common_target_right_productcoefficients)) /\\ ((((((exists pfa_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal) = (pfrd_qlen_execution_common_target_right)) /\\ ((((exists ff_h_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_common_target_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_target_right)) /\\ exists ff_q_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_execution_common_target_right = ff_q_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_target_right) + (pfc_left_execution_common_target_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_execution_common_target_right)=(pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_common_target_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_common_target_right_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_common_target_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_common_target_right_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_common_target_right_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_execution_common_target_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_execution_common_target_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_common_target_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_common_target_right_productcoefficientscoefficientdiagonal)=pfc_left_execution_common_target_right_productcoefficientscoefficientdiagonalterm*pfc_right_execution_common_target_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_common_target_right_productcoefficientscoefficientsum fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_common_target_right_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_common_target_right_productcoefficients))) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_common_target_right_productcoefficients))) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_common_target_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_common_target_right_productcoefficients)) -> exists fs_a_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_target_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_common_target_right_productcoefficientscoefficient = fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_target_right_productcoefficientscoefficient) + (fs_a_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum) + (fs_r_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum) + (fs_s_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_common_target_right_productcoefficientscoefficientresiduebound. pfa_gap_execution_common_target_right_productcoefficientscoefficientresiduebound + S (pfc_value_execution_common_target_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_common_target_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_common_target_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_common_target_right_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_common_target_right_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_common_target_right_productcoefficients) + (p) * pfa_offset_right_execution_common_target_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_execution_common_target_right_target pfrep_left_execution_common_target_right_target pfrep_right_execution_common_target_right_target. ((exists pfrep_position_execution_common_target_right_targetfirst. ((pfrep_position_execution_common_target_right_targetfirst+S (pfrep_power_execution_common_target_right_target)=(pfrd_plen_execution_common_target_right)) /\\ ((((exists ff_h_pfp_execution_common_target_right_targetfirstentry. ff_h_pfp_execution_common_target_right_targetfirstentry + S (pfrep_left_execution_common_target_right_target) = S ((S (pfrep_position_execution_common_target_right_targetfirst)) * pfrd_pc_execution_common_target_right)) /\\ exists ff_q_pfp_execution_common_target_right_targetfirstentry. pfrd_pb_execution_common_target_right = ff_q_pfp_execution_common_target_right_targetfirstentry * S ((S (pfrep_position_execution_common_target_right_targetfirst)) * pfrd_pc_execution_common_target_right) + (pfrep_left_execution_common_target_right_target)))))) \\/ (((exists pfrep_gap_execution_common_target_right_targetfirstoutside. pfrep_gap_execution_common_target_right_targetfirstoutside+(pfrd_plen_execution_common_target_right)=(pfrep_power_execution_common_target_right_target)) /\\ (((pfrep_left_execution_common_target_right_target)=0))))) -> ((exists pfrep_position_execution_common_target_right_targetsecond. ((pfrep_position_execution_common_target_right_targetsecond+S (pfrep_power_execution_common_target_right_target)=(N)) /\\ ((((exists ff_h_pfp_execution_common_target_right_targetsecondentry. ff_h_pfp_execution_common_target_right_targetsecondentry + S (pfrep_right_execution_common_target_right_target) = S ((S (pfrep_position_execution_common_target_right_targetsecond)) * rc)) /\\ exists ff_q_pfp_execution_common_target_right_targetsecondentry. rb = ff_q_pfp_execution_common_target_right_targetsecondentry * S ((S (pfrep_position_execution_common_target_right_targetsecond)) * rc) + (pfrep_right_execution_common_target_right_target)))))) \\/ (((exists pfrep_gap_execution_common_target_right_targetsecondoutside. pfrep_gap_execution_common_target_right_targetsecondoutside+(N)=(pfrep_power_execution_common_target_right_target)) /\\ (((pfrep_right_execution_common_target_right_target)=0))))) -> pfrep_left_execution_common_target_right_target=pfrep_right_execution_common_target_right_target))))))))))) /\\ (((((((forall fom_index_pfp_execution_common_target_left_canonical. (exists fom_gap_pfp_execution_common_target_left_canonical_index_bound. fom_gap_pfp_execution_common_target_left_canonical_index_bound + S (fom_index_pfp_execution_common_target_left_canonical) = S d) -> exists fom_value_pfp_execution_common_target_left_canonical. ((((exists fom_beta_height_pfp_execution_common_target_left_canonical_entry. fom_beta_height_pfp_execution_common_target_left_canonical_entry + S (fom_value_pfp_execution_common_target_left_canonical) = S ((S (fom_index_pfp_execution_common_target_left_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_common_target_left_canonical_entry. bb = fom_beta_quotient_pfp_execution_common_target_left_canonical_entry * S ((S (fom_index_pfp_execution_common_target_left_canonical)) * bc) + (fom_value_pfp_execution_common_target_left_canonical))) /\\ (exists fom_gap_pfp_execution_common_target_left_canonical_value_bound. fom_gap_pfp_execution_common_target_left_canonical_value_bound + S (fom_value_pfp_execution_common_target_left_canonical) = p))) /\\ ((exists pfrd_qb_execution_common_target_left pfrd_qc_execution_common_target_left pfrd_qlen_execution_common_target_left pfrd_pb_execution_common_target_left pfrd_pc_execution_common_target_left pfrd_plen_execution_common_target_left. ((((forall fom_index_pfp_execution_common_target_left_productleft. (exists fom_gap_pfp_execution_common_target_left_productleft_index_bound. fom_gap_pfp_execution_common_target_left_productleft_index_bound + S (fom_index_pfp_execution_common_target_left_productleft) = pfrd_qlen_execution_common_target_left) -> exists fom_value_pfp_execution_common_target_left_productleft. ((((exists fom_beta_height_pfp_execution_common_target_left_productleft_entry. fom_beta_height_pfp_execution_common_target_left_productleft_entry + S (fom_value_pfp_execution_common_target_left_productleft) = S ((S (fom_index_pfp_execution_common_target_left_productleft)) * pfrd_qc_execution_common_target_left)) /\\ exists fom_beta_quotient_pfp_execution_common_target_left_productleft_entry. pfrd_qb_execution_common_target_left = fom_beta_quotient_pfp_execution_common_target_left_productleft_entry * S ((S (fom_index_pfp_execution_common_target_left_productleft)) * pfrd_qc_execution_common_target_left) + (fom_value_pfp_execution_common_target_left_productleft))) /\\ (exists fom_gap_pfp_execution_common_target_left_productleft_value_bound. fom_gap_pfp_execution_common_target_left_productleft_value_bound + S (fom_value_pfp_execution_common_target_left_productleft) = p))) /\\ (((forall fom_index_pfp_execution_common_target_left_productright. (exists fom_gap_pfp_execution_common_target_left_productright_index_bound. fom_gap_pfp_execution_common_target_left_productright_index_bound + S (fom_index_pfp_execution_common_target_left_productright) = J) -> exists fom_value_pfp_execution_common_target_left_productright. ((((exists fom_beta_height_pfp_execution_common_target_left_productright_entry. fom_beta_height_pfp_execution_common_target_left_productright_entry + S (fom_value_pfp_execution_common_target_left_productright) = S ((S (fom_index_pfp_execution_common_target_left_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_execution_common_target_left_productright_entry. db = fom_beta_quotient_pfp_execution_common_target_left_productright_entry * S ((S (fom_index_pfp_execution_common_target_left_productright)) * dc) + (fom_value_pfp_execution_common_target_left_productright))) /\\ (exists fom_gap_pfp_execution_common_target_left_productright_value_bound. fom_gap_pfp_execution_common_target_left_productright_value_bound + S (fom_value_pfp_execution_common_target_left_productright) = p))) /\\ (((((((pfrd_qlen_execution_common_target_left)=0 \\/ (J)=0) /\\ (((pfrd_plen_execution_common_target_left)=0)))) \\/ (((~((pfrd_qlen_execution_common_target_left)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_execution_common_target_left)+(J)=S (pfrd_plen_execution_common_target_left)))))))) /\\ ((forall pfc_index_execution_common_target_left_productcoefficients. (exists pfa_gap_execution_common_target_left_productcoefficientsbound. pfa_gap_execution_common_target_left_productcoefficientsbound + S (pfc_index_execution_common_target_left_productcoefficients) = (pfrd_plen_execution_common_target_left)) -> exists pfc_value_execution_common_target_left_productcoefficients. ((((exists ff_h_pfp_execution_common_target_left_productcoefficientsentry. ff_h_pfp_execution_common_target_left_productcoefficientsentry + S (pfc_value_execution_common_target_left_productcoefficients) = S ((S (pfc_index_execution_common_target_left_productcoefficients)) * pfrd_pc_execution_common_target_left)) /\\ exists ff_q_pfp_execution_common_target_left_productcoefficientsentry. pfrd_pb_execution_common_target_left = ff_q_pfp_execution_common_target_left_productcoefficientsentry * S ((S (pfc_index_execution_common_target_left_productcoefficients)) * pfrd_pc_execution_common_target_left) + (pfc_value_execution_common_target_left_productcoefficients))) /\\ ((exists pfc_terms_code_execution_common_target_left_productcoefficientscoefficient pfc_terms_scale_execution_common_target_left_productcoefficientscoefficient pfc_natural_sum_execution_common_target_left_productcoefficientscoefficient. ((forall pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_common_target_left_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_common_target_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_common_target_left_productcoefficients))) -> exists pfc_value_execution_common_target_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_common_target_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_common_target_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_common_target_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_target_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_common_target_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_common_target_left_productcoefficientscoefficient = ff_q_pfp_execution_common_target_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_target_left_productcoefficientscoefficient) + (pfc_value_execution_common_target_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_common_target_left_productcoefficientscoefficientdiagonalterm pfc_left_execution_common_target_left_productcoefficientscoefficientdiagonalterm pfc_right_execution_common_target_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal)+pfc_complement_execution_common_target_left_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_common_target_left_productcoefficients)) /\\ ((((((exists pfa_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal) = (pfrd_qlen_execution_common_target_left)) /\\ ((((exists ff_h_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_common_target_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_target_left)) /\\ exists ff_q_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_execution_common_target_left = ff_q_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_target_left) + (pfc_left_execution_common_target_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_execution_common_target_left)=(pfc_index_execution_common_target_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_common_target_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_common_target_left_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_common_target_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_common_target_left_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_common_target_left_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_execution_common_target_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_common_target_left_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_execution_common_target_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_common_target_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_common_target_left_productcoefficientscoefficientdiagonal)=pfc_left_execution_common_target_left_productcoefficientscoefficientdiagonalterm*pfc_right_execution_common_target_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_common_target_left_productcoefficientscoefficientsum fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_common_target_left_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_common_target_left_productcoefficients))) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_common_target_left_productcoefficients))) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_common_target_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_common_target_left_productcoefficients)) -> exists fs_a_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_target_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_common_target_left_productcoefficientscoefficient = fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_target_left_productcoefficientscoefficient) + (fs_a_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum) + (fs_r_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_common_target_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_left_productcoefficientscoefficientsum) + (fs_s_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_common_target_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_common_target_left_productcoefficientscoefficientresiduebound. pfa_gap_execution_common_target_left_productcoefficientscoefficientresiduebound + S (pfc_value_execution_common_target_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_common_target_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_common_target_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_common_target_left_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_common_target_left_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_common_target_left_productcoefficients) + (p) * pfa_offset_right_execution_common_target_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_execution_common_target_left_target pfrep_left_execution_common_target_left_target pfrep_right_execution_common_target_left_target. ((exists pfrep_position_execution_common_target_left_targetfirst. ((pfrep_position_execution_common_target_left_targetfirst+S (pfrep_power_execution_common_target_left_target)=(pfrd_plen_execution_common_target_left)) /\\ ((((exists ff_h_pfp_execution_common_target_left_targetfirstentry. ff_h_pfp_execution_common_target_left_targetfirstentry + S (pfrep_left_execution_common_target_left_target) = S ((S (pfrep_position_execution_common_target_left_targetfirst)) * pfrd_pc_execution_common_target_left)) /\\ exists ff_q_pfp_execution_common_target_left_targetfirstentry. pfrd_pb_execution_common_target_left = ff_q_pfp_execution_common_target_left_targetfirstentry * S ((S (pfrep_position_execution_common_target_left_targetfirst)) * pfrd_pc_execution_common_target_left) + (pfrep_left_execution_common_target_left_target)))))) \\/ (((exists pfrep_gap_execution_common_target_left_targetfirstoutside. pfrep_gap_execution_common_target_left_targetfirstoutside+(pfrd_plen_execution_common_target_left)=(pfrep_power_execution_common_target_left_target)) /\\ (((pfrep_left_execution_common_target_left_target)=0))))) -> ((exists pfrep_position_execution_common_target_left_targetsecond. ((pfrep_position_execution_common_target_left_targetsecond+S (pfrep_power_execution_common_target_left_target)=(S d)) /\\ ((((exists ff_h_pfp_execution_common_target_left_targetsecondentry. ff_h_pfp_execution_common_target_left_targetsecondentry + S (pfrep_right_execution_common_target_left_target) = S ((S (pfrep_position_execution_common_target_left_targetsecond)) * bc)) /\\ exists ff_q_pfp_execution_common_target_left_targetsecondentry. bb = ff_q_pfp_execution_common_target_left_targetsecondentry * S ((S (pfrep_position_execution_common_target_left_targetsecond)) * bc) + (pfrep_right_execution_common_target_left_target)))))) \\/ (((exists pfrep_gap_execution_common_target_left_targetsecondoutside. pfrep_gap_execution_common_target_left_targetsecondoutside+(S d)=(pfrep_power_execution_common_target_left_target)) /\\ (((pfrep_right_execution_common_target_left_target)=0))))) -> pfrep_left_execution_common_target_left_target=pfrep_right_execution_common_target_left_target))))))) /\\ ((((forall fom_index_pfp_execution_common_target_right_canonical. (exists fom_gap_pfp_execution_common_target_right_canonical_index_bound. fom_gap_pfp_execution_common_target_right_canonical_index_bound + S (fom_index_pfp_execution_common_target_right_canonical) = N) -> exists fom_value_pfp_execution_common_target_right_canonical. ((((exists fom_beta_height_pfp_execution_common_target_right_canonical_entry. fom_beta_height_pfp_execution_common_target_right_canonical_entry + S (fom_value_pfp_execution_common_target_right_canonical) = S ((S (fom_index_pfp_execution_common_target_right_canonical)) * rc)) /\\ exists fom_beta_quotient_pfp_execution_common_target_right_canonical_entry. rb = fom_beta_quotient_pfp_execution_common_target_right_canonical_entry * S ((S (fom_index_pfp_execution_common_target_right_canonical)) * rc) + (fom_value_pfp_execution_common_target_right_canonical))) /\\ (exists fom_gap_pfp_execution_common_target_right_canonical_value_bound. fom_gap_pfp_execution_common_target_right_canonical_value_bound + S (fom_value_pfp_execution_common_target_right_canonical) = p))) /\\ ((exists pfrd_qb_execution_common_target_right pfrd_qc_execution_common_target_right pfrd_qlen_execution_common_target_right pfrd_pb_execution_common_target_right pfrd_pc_execution_common_target_right pfrd_plen_execution_common_target_right. ((((forall fom_index_pfp_execution_common_target_right_productleft. (exists fom_gap_pfp_execution_common_target_right_productleft_index_bound. fom_gap_pfp_execution_common_target_right_productleft_index_bound + S (fom_index_pfp_execution_common_target_right_productleft) = pfrd_qlen_execution_common_target_right) -> exists fom_value_pfp_execution_common_target_right_productleft. ((((exists fom_beta_height_pfp_execution_common_target_right_productleft_entry. fom_beta_height_pfp_execution_common_target_right_productleft_entry + S (fom_value_pfp_execution_common_target_right_productleft) = S ((S (fom_index_pfp_execution_common_target_right_productleft)) * pfrd_qc_execution_common_target_right)) /\\ exists fom_beta_quotient_pfp_execution_common_target_right_productleft_entry. pfrd_qb_execution_common_target_right = fom_beta_quotient_pfp_execution_common_target_right_productleft_entry * S ((S (fom_index_pfp_execution_common_target_right_productleft)) * pfrd_qc_execution_common_target_right) + (fom_value_pfp_execution_common_target_right_productleft))) /\\ (exists fom_gap_pfp_execution_common_target_right_productleft_value_bound. fom_gap_pfp_execution_common_target_right_productleft_value_bound + S (fom_value_pfp_execution_common_target_right_productleft) = p))) /\\ (((forall fom_index_pfp_execution_common_target_right_productright. (exists fom_gap_pfp_execution_common_target_right_productright_index_bound. fom_gap_pfp_execution_common_target_right_productright_index_bound + S (fom_index_pfp_execution_common_target_right_productright) = J) -> exists fom_value_pfp_execution_common_target_right_productright. ((((exists fom_beta_height_pfp_execution_common_target_right_productright_entry. fom_beta_height_pfp_execution_common_target_right_productright_entry + S (fom_value_pfp_execution_common_target_right_productright) = S ((S (fom_index_pfp_execution_common_target_right_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_execution_common_target_right_productright_entry. db = fom_beta_quotient_pfp_execution_common_target_right_productright_entry * S ((S (fom_index_pfp_execution_common_target_right_productright)) * dc) + (fom_value_pfp_execution_common_target_right_productright))) /\\ (exists fom_gap_pfp_execution_common_target_right_productright_value_bound. fom_gap_pfp_execution_common_target_right_productright_value_bound + S (fom_value_pfp_execution_common_target_right_productright) = p))) /\\ (((((((pfrd_qlen_execution_common_target_right)=0 \\/ (J)=0) /\\ (((pfrd_plen_execution_common_target_right)=0)))) \\/ (((~((pfrd_qlen_execution_common_target_right)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_execution_common_target_right)+(J)=S (pfrd_plen_execution_common_target_right)))))))) /\\ ((forall pfc_index_execution_common_target_right_productcoefficients. (exists pfa_gap_execution_common_target_right_productcoefficientsbound. pfa_gap_execution_common_target_right_productcoefficientsbound + S (pfc_index_execution_common_target_right_productcoefficients) = (pfrd_plen_execution_common_target_right)) -> exists pfc_value_execution_common_target_right_productcoefficients. ((((exists ff_h_pfp_execution_common_target_right_productcoefficientsentry. ff_h_pfp_execution_common_target_right_productcoefficientsentry + S (pfc_value_execution_common_target_right_productcoefficients) = S ((S (pfc_index_execution_common_target_right_productcoefficients)) * pfrd_pc_execution_common_target_right)) /\\ exists ff_q_pfp_execution_common_target_right_productcoefficientsentry. pfrd_pb_execution_common_target_right = ff_q_pfp_execution_common_target_right_productcoefficientsentry * S ((S (pfc_index_execution_common_target_right_productcoefficients)) * pfrd_pc_execution_common_target_right) + (pfc_value_execution_common_target_right_productcoefficients))) /\\ ((exists pfc_terms_code_execution_common_target_right_productcoefficientscoefficient pfc_terms_scale_execution_common_target_right_productcoefficientscoefficient pfc_natural_sum_execution_common_target_right_productcoefficientscoefficient. ((forall pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_common_target_right_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_common_target_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_common_target_right_productcoefficients))) -> exists pfc_value_execution_common_target_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_common_target_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_common_target_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_common_target_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_target_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_common_target_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_common_target_right_productcoefficientscoefficient = ff_q_pfp_execution_common_target_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_target_right_productcoefficientscoefficient) + (pfc_value_execution_common_target_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_common_target_right_productcoefficientscoefficientdiagonalterm pfc_left_execution_common_target_right_productcoefficientscoefficientdiagonalterm pfc_right_execution_common_target_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal)+pfc_complement_execution_common_target_right_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_common_target_right_productcoefficients)) /\\ ((((((exists pfa_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal) = (pfrd_qlen_execution_common_target_right)) /\\ ((((exists ff_h_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_common_target_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_target_right)) /\\ exists ff_q_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_execution_common_target_right = ff_q_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_target_right) + (pfc_left_execution_common_target_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_execution_common_target_right)=(pfc_index_execution_common_target_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_common_target_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_common_target_right_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_common_target_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_common_target_right_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_common_target_right_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_execution_common_target_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_common_target_right_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_execution_common_target_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_common_target_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_common_target_right_productcoefficientscoefficientdiagonal)=pfc_left_execution_common_target_right_productcoefficientscoefficientdiagonalterm*pfc_right_execution_common_target_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_common_target_right_productcoefficientscoefficientsum fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_common_target_right_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_common_target_right_productcoefficients))) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_common_target_right_productcoefficients))) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_common_target_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_common_target_right_productcoefficients)) -> exists fs_a_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_target_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_common_target_right_productcoefficientscoefficient = fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_target_right_productcoefficientscoefficient) + (fs_a_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum) + (fs_r_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_common_target_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_target_right_productcoefficientscoefficientsum) + (fs_s_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_common_target_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_common_target_right_productcoefficientscoefficientresiduebound. pfa_gap_execution_common_target_right_productcoefficientscoefficientresiduebound + S (pfc_value_execution_common_target_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_common_target_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_common_target_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_common_target_right_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_common_target_right_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_common_target_right_productcoefficients) + (p) * pfa_offset_right_execution_common_target_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_execution_common_target_right_target pfrep_left_execution_common_target_right_target pfrep_right_execution_common_target_right_target. ((exists pfrep_position_execution_common_target_right_targetfirst. ((pfrep_position_execution_common_target_right_targetfirst+S (pfrep_power_execution_common_target_right_target)=(pfrd_plen_execution_common_target_right)) /\\ ((((exists ff_h_pfp_execution_common_target_right_targetfirstentry. ff_h_pfp_execution_common_target_right_targetfirstentry + S (pfrep_left_execution_common_target_right_target) = S ((S (pfrep_position_execution_common_target_right_targetfirst)) * pfrd_pc_execution_common_target_right)) /\\ exists ff_q_pfp_execution_common_target_right_targetfirstentry. pfrd_pb_execution_common_target_right = ff_q_pfp_execution_common_target_right_targetfirstentry * S ((S (pfrep_position_execution_common_target_right_targetfirst)) * pfrd_pc_execution_common_target_right) + (pfrep_left_execution_common_target_right_target)))))) \\/ (((exists pfrep_gap_execution_common_target_right_targetfirstoutside. pfrep_gap_execution_common_target_right_targetfirstoutside+(pfrd_plen_execution_common_target_right)=(pfrep_power_execution_common_target_right_target)) /\\ (((pfrep_left_execution_common_target_right_target)=0))))) -> ((exists pfrep_position_execution_common_target_right_targetsecond. ((pfrep_position_execution_common_target_right_targetsecond+S (pfrep_power_execution_common_target_right_target)=(N)) /\\ ((((exists ff_h_pfp_execution_common_target_right_targetsecondentry. ff_h_pfp_execution_common_target_right_targetsecondentry + S (pfrep_right_execution_common_target_right_target) = S ((S (pfrep_position_execution_common_target_right_targetsecond)) * rc)) /\\ exists ff_q_pfp_execution_common_target_right_targetsecondentry. rb = ff_q_pfp_execution_common_target_right_targetsecondentry * S ((S (pfrep_position_execution_common_target_right_targetsecond)) * rc) + (pfrep_right_execution_common_target_right_target)))))) \\/ (((exists pfrep_gap_execution_common_target_right_targetsecondoutside. pfrep_gap_execution_common_target_right_targetsecondoutside+(N)=(pfrep_power_execution_common_target_right_target)) /\\ (((pfrep_right_execution_common_target_right_target)=0))))) -> pfrep_left_execution_common_target_right_target=pfrep_right_execution_common_target_right_target)))))))))) -> (((((forall fom_index_pfp_execution_common_source_left_canonical. (exists fom_gap_pfp_execution_common_source_left_canonical_index_bound. fom_gap_pfp_execution_common_source_left_canonical_index_bound + S (fom_index_pfp_execution_common_source_left_canonical) = L) -> exists fom_value_pfp_execution_common_source_left_canonical. ((((exists fom_beta_height_pfp_execution_common_source_left_canonical_entry. fom_beta_height_pfp_execution_common_source_left_canonical_entry + S (fom_value_pfp_execution_common_source_left_canonical) = S ((S (fom_index_pfp_execution_common_source_left_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_common_source_left_canonical_entry. ab = fom_beta_quotient_pfp_execution_common_source_left_canonical_entry * S ((S (fom_index_pfp_execution_common_source_left_canonical)) * ac) + (fom_value_pfp_execution_common_source_left_canonical))) /\\ (exists fom_gap_pfp_execution_common_source_left_canonical_value_bound. fom_gap_pfp_execution_common_source_left_canonical_value_bound + S (fom_value_pfp_execution_common_source_left_canonical) = p))) /\\ ((exists pfrd_qb_execution_common_source_left pfrd_qc_execution_common_source_left pfrd_qlen_execution_common_source_left pfrd_pb_execution_common_source_left pfrd_pc_execution_common_source_left pfrd_plen_execution_common_source_left. ((((forall fom_index_pfp_execution_common_source_left_productleft. (exists fom_gap_pfp_execution_common_source_left_productleft_index_bound. fom_gap_pfp_execution_common_source_left_productleft_index_bound + S (fom_index_pfp_execution_common_source_left_productleft) = pfrd_qlen_execution_common_source_left) -> exists fom_value_pfp_execution_common_source_left_productleft. ((((exists fom_beta_height_pfp_execution_common_source_left_productleft_entry. fom_beta_height_pfp_execution_common_source_left_productleft_entry + S (fom_value_pfp_execution_common_source_left_productleft) = S ((S (fom_index_pfp_execution_common_source_left_productleft)) * pfrd_qc_execution_common_source_left)) /\\ exists fom_beta_quotient_pfp_execution_common_source_left_productleft_entry. pfrd_qb_execution_common_source_left = fom_beta_quotient_pfp_execution_common_source_left_productleft_entry * S ((S (fom_index_pfp_execution_common_source_left_productleft)) * pfrd_qc_execution_common_source_left) + (fom_value_pfp_execution_common_source_left_productleft))) /\\ (exists fom_gap_pfp_execution_common_source_left_productleft_value_bound. fom_gap_pfp_execution_common_source_left_productleft_value_bound + S (fom_value_pfp_execution_common_source_left_productleft) = p))) /\\ (((forall fom_index_pfp_execution_common_source_left_productright. (exists fom_gap_pfp_execution_common_source_left_productright_index_bound. fom_gap_pfp_execution_common_source_left_productright_index_bound + S (fom_index_pfp_execution_common_source_left_productright) = J) -> exists fom_value_pfp_execution_common_source_left_productright. ((((exists fom_beta_height_pfp_execution_common_source_left_productright_entry. fom_beta_height_pfp_execution_common_source_left_productright_entry + S (fom_value_pfp_execution_common_source_left_productright) = S ((S (fom_index_pfp_execution_common_source_left_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_execution_common_source_left_productright_entry. db = fom_beta_quotient_pfp_execution_common_source_left_productright_entry * S ((S (fom_index_pfp_execution_common_source_left_productright)) * dc) + (fom_value_pfp_execution_common_source_left_productright))) /\\ (exists fom_gap_pfp_execution_common_source_left_productright_value_bound. fom_gap_pfp_execution_common_source_left_productright_value_bound + S (fom_value_pfp_execution_common_source_left_productright) = p))) /\\ (((((((pfrd_qlen_execution_common_source_left)=0 \\/ (J)=0) /\\ (((pfrd_plen_execution_common_source_left)=0)))) \\/ (((~((pfrd_qlen_execution_common_source_left)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_execution_common_source_left)+(J)=S (pfrd_plen_execution_common_source_left)))))))) /\\ ((forall pfc_index_execution_common_source_left_productcoefficients. (exists pfa_gap_execution_common_source_left_productcoefficientsbound. pfa_gap_execution_common_source_left_productcoefficientsbound + S (pfc_index_execution_common_source_left_productcoefficients) = (pfrd_plen_execution_common_source_left)) -> exists pfc_value_execution_common_source_left_productcoefficients. ((((exists ff_h_pfp_execution_common_source_left_productcoefficientsentry. ff_h_pfp_execution_common_source_left_productcoefficientsentry + S (pfc_value_execution_common_source_left_productcoefficients) = S ((S (pfc_index_execution_common_source_left_productcoefficients)) * pfrd_pc_execution_common_source_left)) /\\ exists ff_q_pfp_execution_common_source_left_productcoefficientsentry. pfrd_pb_execution_common_source_left = ff_q_pfp_execution_common_source_left_productcoefficientsentry * S ((S (pfc_index_execution_common_source_left_productcoefficients)) * pfrd_pc_execution_common_source_left) + (pfc_value_execution_common_source_left_productcoefficients))) /\\ ((exists pfc_terms_code_execution_common_source_left_productcoefficientscoefficient pfc_terms_scale_execution_common_source_left_productcoefficientscoefficient pfc_natural_sum_execution_common_source_left_productcoefficientscoefficient. ((forall pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_common_source_left_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_common_source_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_common_source_left_productcoefficients))) -> exists pfc_value_execution_common_source_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_common_source_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_common_source_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_common_source_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_source_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_common_source_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_common_source_left_productcoefficientscoefficient = ff_q_pfp_execution_common_source_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_source_left_productcoefficientscoefficient) + (pfc_value_execution_common_source_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_common_source_left_productcoefficientscoefficientdiagonalterm pfc_left_execution_common_source_left_productcoefficientscoefficientdiagonalterm pfc_right_execution_common_source_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal)+pfc_complement_execution_common_source_left_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_common_source_left_productcoefficients)) /\\ ((((((exists pfa_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal) = (pfrd_qlen_execution_common_source_left)) /\\ ((((exists ff_h_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_common_source_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_source_left)) /\\ exists ff_q_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_execution_common_source_left = ff_q_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_source_left) + (pfc_left_execution_common_source_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_execution_common_source_left)=(pfc_index_execution_common_source_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_common_source_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_common_source_left_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_common_source_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_common_source_left_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_common_source_left_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_execution_common_source_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_common_source_left_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_execution_common_source_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_common_source_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_common_source_left_productcoefficientscoefficientdiagonal)=pfc_left_execution_common_source_left_productcoefficientscoefficientdiagonalterm*pfc_right_execution_common_source_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_common_source_left_productcoefficientscoefficientsum fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_common_source_left_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_common_source_left_productcoefficients))) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_common_source_left_productcoefficients))) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_common_source_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_common_source_left_productcoefficients)) -> exists fs_a_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_source_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_common_source_left_productcoefficientscoefficient = fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_source_left_productcoefficientscoefficient) + (fs_a_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum) + (fs_r_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_common_source_left_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_left_productcoefficientscoefficientsum) + (fs_s_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_common_source_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_common_source_left_productcoefficientscoefficientresiduebound. pfa_gap_execution_common_source_left_productcoefficientscoefficientresiduebound + S (pfc_value_execution_common_source_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_common_source_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_common_source_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_common_source_left_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_common_source_left_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_common_source_left_productcoefficients) + (p) * pfa_offset_right_execution_common_source_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_execution_common_source_left_target pfrep_left_execution_common_source_left_target pfrep_right_execution_common_source_left_target. ((exists pfrep_position_execution_common_source_left_targetfirst. ((pfrep_position_execution_common_source_left_targetfirst+S (pfrep_power_execution_common_source_left_target)=(pfrd_plen_execution_common_source_left)) /\\ ((((exists ff_h_pfp_execution_common_source_left_targetfirstentry. ff_h_pfp_execution_common_source_left_targetfirstentry + S (pfrep_left_execution_common_source_left_target) = S ((S (pfrep_position_execution_common_source_left_targetfirst)) * pfrd_pc_execution_common_source_left)) /\\ exists ff_q_pfp_execution_common_source_left_targetfirstentry. pfrd_pb_execution_common_source_left = ff_q_pfp_execution_common_source_left_targetfirstentry * S ((S (pfrep_position_execution_common_source_left_targetfirst)) * pfrd_pc_execution_common_source_left) + (pfrep_left_execution_common_source_left_target)))))) \\/ (((exists pfrep_gap_execution_common_source_left_targetfirstoutside. pfrep_gap_execution_common_source_left_targetfirstoutside+(pfrd_plen_execution_common_source_left)=(pfrep_power_execution_common_source_left_target)) /\\ (((pfrep_left_execution_common_source_left_target)=0))))) -> ((exists pfrep_position_execution_common_source_left_targetsecond. ((pfrep_position_execution_common_source_left_targetsecond+S (pfrep_power_execution_common_source_left_target)=(L)) /\\ ((((exists ff_h_pfp_execution_common_source_left_targetsecondentry. ff_h_pfp_execution_common_source_left_targetsecondentry + S (pfrep_right_execution_common_source_left_target) = S ((S (pfrep_position_execution_common_source_left_targetsecond)) * ac)) /\\ exists ff_q_pfp_execution_common_source_left_targetsecondentry. ab = ff_q_pfp_execution_common_source_left_targetsecondentry * S ((S (pfrep_position_execution_common_source_left_targetsecond)) * ac) + (pfrep_right_execution_common_source_left_target)))))) \\/ (((exists pfrep_gap_execution_common_source_left_targetsecondoutside. pfrep_gap_execution_common_source_left_targetsecondoutside+(L)=(pfrep_power_execution_common_source_left_target)) /\\ (((pfrep_right_execution_common_source_left_target)=0))))) -> pfrep_left_execution_common_source_left_target=pfrep_right_execution_common_source_left_target))))))) /\\ ((((forall fom_index_pfp_execution_common_source_right_canonical. (exists fom_gap_pfp_execution_common_source_right_canonical_index_bound. fom_gap_pfp_execution_common_source_right_canonical_index_bound + S (fom_index_pfp_execution_common_source_right_canonical) = S d) -> exists fom_value_pfp_execution_common_source_right_canonical. ((((exists fom_beta_height_pfp_execution_common_source_right_canonical_entry. fom_beta_height_pfp_execution_common_source_right_canonical_entry + S (fom_value_pfp_execution_common_source_right_canonical) = S ((S (fom_index_pfp_execution_common_source_right_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_common_source_right_canonical_entry. bb = fom_beta_quotient_pfp_execution_common_source_right_canonical_entry * S ((S (fom_index_pfp_execution_common_source_right_canonical)) * bc) + (fom_value_pfp_execution_common_source_right_canonical))) /\\ (exists fom_gap_pfp_execution_common_source_right_canonical_value_bound. fom_gap_pfp_execution_common_source_right_canonical_value_bound + S (fom_value_pfp_execution_common_source_right_canonical) = p))) /\\ ((exists pfrd_qb_execution_common_source_right pfrd_qc_execution_common_source_right pfrd_qlen_execution_common_source_right pfrd_pb_execution_common_source_right pfrd_pc_execution_common_source_right pfrd_plen_execution_common_source_right. ((((forall fom_index_pfp_execution_common_source_right_productleft. (exists fom_gap_pfp_execution_common_source_right_productleft_index_bound. fom_gap_pfp_execution_common_source_right_productleft_index_bound + S (fom_index_pfp_execution_common_source_right_productleft) = pfrd_qlen_execution_common_source_right) -> exists fom_value_pfp_execution_common_source_right_productleft. ((((exists fom_beta_height_pfp_execution_common_source_right_productleft_entry. fom_beta_height_pfp_execution_common_source_right_productleft_entry + S (fom_value_pfp_execution_common_source_right_productleft) = S ((S (fom_index_pfp_execution_common_source_right_productleft)) * pfrd_qc_execution_common_source_right)) /\\ exists fom_beta_quotient_pfp_execution_common_source_right_productleft_entry. pfrd_qb_execution_common_source_right = fom_beta_quotient_pfp_execution_common_source_right_productleft_entry * S ((S (fom_index_pfp_execution_common_source_right_productleft)) * pfrd_qc_execution_common_source_right) + (fom_value_pfp_execution_common_source_right_productleft))) /\\ (exists fom_gap_pfp_execution_common_source_right_productleft_value_bound. fom_gap_pfp_execution_common_source_right_productleft_value_bound + S (fom_value_pfp_execution_common_source_right_productleft) = p))) /\\ (((forall fom_index_pfp_execution_common_source_right_productright. (exists fom_gap_pfp_execution_common_source_right_productright_index_bound. fom_gap_pfp_execution_common_source_right_productright_index_bound + S (fom_index_pfp_execution_common_source_right_productright) = J) -> exists fom_value_pfp_execution_common_source_right_productright. ((((exists fom_beta_height_pfp_execution_common_source_right_productright_entry. fom_beta_height_pfp_execution_common_source_right_productright_entry + S (fom_value_pfp_execution_common_source_right_productright) = S ((S (fom_index_pfp_execution_common_source_right_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_execution_common_source_right_productright_entry. db = fom_beta_quotient_pfp_execution_common_source_right_productright_entry * S ((S (fom_index_pfp_execution_common_source_right_productright)) * dc) + (fom_value_pfp_execution_common_source_right_productright))) /\\ (exists fom_gap_pfp_execution_common_source_right_productright_value_bound. fom_gap_pfp_execution_common_source_right_productright_value_bound + S (fom_value_pfp_execution_common_source_right_productright) = p))) /\\ (((((((pfrd_qlen_execution_common_source_right)=0 \\/ (J)=0) /\\ (((pfrd_plen_execution_common_source_right)=0)))) \\/ (((~((pfrd_qlen_execution_common_source_right)=0)) /\\ (((~((J)=0)) /\\ (((pfrd_qlen_execution_common_source_right)+(J)=S (pfrd_plen_execution_common_source_right)))))))) /\\ ((forall pfc_index_execution_common_source_right_productcoefficients. (exists pfa_gap_execution_common_source_right_productcoefficientsbound. pfa_gap_execution_common_source_right_productcoefficientsbound + S (pfc_index_execution_common_source_right_productcoefficients) = (pfrd_plen_execution_common_source_right)) -> exists pfc_value_execution_common_source_right_productcoefficients. ((((exists ff_h_pfp_execution_common_source_right_productcoefficientsentry. ff_h_pfp_execution_common_source_right_productcoefficientsentry + S (pfc_value_execution_common_source_right_productcoefficients) = S ((S (pfc_index_execution_common_source_right_productcoefficients)) * pfrd_pc_execution_common_source_right)) /\\ exists ff_q_pfp_execution_common_source_right_productcoefficientsentry. pfrd_pb_execution_common_source_right = ff_q_pfp_execution_common_source_right_productcoefficientsentry * S ((S (pfc_index_execution_common_source_right_productcoefficients)) * pfrd_pc_execution_common_source_right) + (pfc_value_execution_common_source_right_productcoefficients))) /\\ ((exists pfc_terms_code_execution_common_source_right_productcoefficientscoefficient pfc_terms_scale_execution_common_source_right_productcoefficientscoefficient pfc_natural_sum_execution_common_source_right_productcoefficientscoefficient. ((forall pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_common_source_right_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_common_source_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_common_source_right_productcoefficients))) -> exists pfc_value_execution_common_source_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_common_source_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_common_source_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_common_source_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_source_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_common_source_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_common_source_right_productcoefficientscoefficient = ff_q_pfp_execution_common_source_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_common_source_right_productcoefficientscoefficient) + (pfc_value_execution_common_source_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_common_source_right_productcoefficientscoefficientdiagonalterm pfc_left_execution_common_source_right_productcoefficientscoefficientdiagonalterm pfc_right_execution_common_source_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal)+pfc_complement_execution_common_source_right_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_common_source_right_productcoefficients)) /\\ ((((((exists pfa_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal) = (pfrd_qlen_execution_common_source_right)) /\\ ((((exists ff_h_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_common_source_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_source_right)) /\\ exists ff_q_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_execution_common_source_right = ff_q_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_execution_common_source_right) + (pfc_left_execution_common_source_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_execution_common_source_right)=(pfc_index_execution_common_source_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_common_source_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_common_source_right_productcoefficientscoefficientdiagonalterm) = (J)) /\\ ((((exists ff_h_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_common_source_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_common_source_right_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_common_source_right_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_execution_common_source_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_common_source_right_productcoefficientscoefficientdiagonaltermrightoutside+(J)=(pfc_complement_execution_common_source_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_common_source_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_common_source_right_productcoefficientscoefficientdiagonal)=pfc_left_execution_common_source_right_productcoefficientscoefficientdiagonalterm*pfc_right_execution_common_source_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_common_source_right_productcoefficientscoefficientsum fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_common_source_right_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_common_source_right_productcoefficients))) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_common_source_right_productcoefficients))) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_common_source_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_common_source_right_productcoefficients)) -> exists fs_a_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_source_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_common_source_right_productcoefficientscoefficient = fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_common_source_right_productcoefficientscoefficient) + (fs_a_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum) + (fs_r_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_common_source_right_productcoefficientscoefficientsum = fs_q_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_common_source_right_productcoefficientscoefficientsum) + (fs_s_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_common_source_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_common_source_right_productcoefficientscoefficientresiduebound. pfa_gap_execution_common_source_right_productcoefficientscoefficientresiduebound + S (pfc_value_execution_common_source_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_common_source_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_common_source_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_common_source_right_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_common_source_right_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_common_source_right_productcoefficients) + (p) * pfa_offset_right_execution_common_source_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_execution_common_source_right_target pfrep_left_execution_common_source_right_target pfrep_right_execution_common_source_right_target. ((exists pfrep_position_execution_common_source_right_targetfirst. ((pfrep_position_execution_common_source_right_targetfirst+S (pfrep_power_execution_common_source_right_target)=(pfrd_plen_execution_common_source_right)) /\\ ((((exists ff_h_pfp_execution_common_source_right_targetfirstentry. ff_h_pfp_execution_common_source_right_targetfirstentry + S (pfrep_left_execution_common_source_right_target) = S ((S (pfrep_position_execution_common_source_right_targetfirst)) * pfrd_pc_execution_common_source_right)) /\\ exists ff_q_pfp_execution_common_source_right_targetfirstentry. pfrd_pb_execution_common_source_right = ff_q_pfp_execution_common_source_right_targetfirstentry * S ((S (pfrep_position_execution_common_source_right_targetfirst)) * pfrd_pc_execution_common_source_right) + (pfrep_left_execution_common_source_right_target)))))) \\/ (((exists pfrep_gap_execution_common_source_right_targetfirstoutside. pfrep_gap_execution_common_source_right_targetfirstoutside+(pfrd_plen_execution_common_source_right)=(pfrep_power_execution_common_source_right_target)) /\\ (((pfrep_left_execution_common_source_right_target)=0))))) -> ((exists pfrep_position_execution_common_source_right_targetsecond. ((pfrep_position_execution_common_source_right_targetsecond+S (pfrep_power_execution_common_source_right_target)=(S d)) /\\ ((((exists ff_h_pfp_execution_common_source_right_targetsecondentry. ff_h_pfp_execution_common_source_right_targetsecondentry + S (pfrep_right_execution_common_source_right_target) = S ((S (pfrep_position_execution_common_source_right_targetsecond)) * bc)) /\\ exists ff_q_pfp_execution_common_source_right_targetsecondentry. bb = ff_q_pfp_execution_common_source_right_targetsecondentry * S ((S (pfrep_position_execution_common_source_right_targetsecond)) * bc) + (pfrep_right_execution_common_source_right_target)))))) \\/ (((exists pfrep_gap_execution_common_source_right_targetsecondoutside. pfrep_gap_execution_common_source_right_targetsecondoutside+(S d)=(pfrep_power_execution_common_source_right_target)) /\\ (((pfrep_right_execution_common_source_right_target)=0))))) -> 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"reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "kind": "alpha_v34_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=440]" }, { "document_sha256": "81943a545e7b9f667ad758c24de14bb47280e8cf3fbb3ca714efec55fb096408", "kind": "alpha_v34_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "e5a6d177fa05a3ac54e3f58f9cbccd1f9f4b9a4f9fbfa80573a30e01a6cd4d56", "membership": "alpha_only", "name": "prime_field_polynomial_euclidean_backward_coefficient_identity", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro La", "intro bb", "intro bc", "intro Lb", "intro rb", "intro rc", "intro Lr", "intro qb", "intro qc", "intro Lq", "intro pb", "intro pc", "intro Lp", "intro ub", "intro uc", "intro Lu", "intro vb", "intro vc", "intro Lv", "intro gb", "intro gc", "intro Lg", "intro cb", "intro cc", "intro Lc", "intro db", "intro dc", "intro Ld", "intro wb", "intro wc", "intro Lw", "intro tb", "intro tc", "intro Lt", "intro xb", "intro xc", "intro Lx", "intro yb", "intro yc", "intro Ly", "intro zb", "intro zc", "intro Lz", "intro hb", "intro hc", "intro Lh", "intro hp", "intro hQB", "intro hdivision", "intro hUB", "intro hVR", "intro hold", "intro hVQ", "intro hsub", "intro hVA", "intro hTB", "intro hWB", "intro hVP", "have hXbound : forall fom_index_pfp_hXbound_bounded. (exists fom_gap_pfp_hXbound_bounded_index_bound. fom_gap_pfp_hXbound_bounded_index_bound + S (fom_index_pfp_hXbound_bounded) = Lx) -> exists fom_value_pfp_hXbound_bounded. ((((exists fom_beta_height_pfp_hXbound_bounded_entry. fom_beta_height_pfp_hXbound_bounded_entry + S (fom_value_pfp_hXbound_bounded) = S ((S (fom_index_pfp_hXbound_bounded)) * xc)) /\\ exists fom_beta_quotient_pfp_hXbound_bounded_entry. xb = fom_beta_quotient_pfp_hXbound_bounded_entry * S ((S (fom_index_pfp_hXbound_bounded)) * xc) + (fom_value_pfp_hXbound_bounded))) /\\ (exists fom_gap_pfp_hXbound_bounded_value_bound. fom_gap_pfp_hXbound_bounded_value_bound + S (fom_value_pfp_hXbound_bounded) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (vb)", "specialize prime_field_polynomial_convolution_bounded (vc)", "specialize prime_field_polynomial_convolution_bounded (Lv)", "specialize prime_field_polynomial_convolution_bounded (ab)", "specialize prime_field_polynomial_convolution_bounded (ac)", "specialize prime_field_polynomial_convolution_bounded (La)", "specialize prime_field_polynomial_convolution_bounded (xb)", "specialize prime_field_polynomial_convolution_bounded (xc)", "specialize prime_field_polynomial_convolution_bounded (Lx)", "apply prime_field_polynomial_convolution_bounded", "exact hVA", "have hYbound : forall fom_index_pfp_hYbound_bounded. (exists fom_gap_pfp_hYbound_bounded_index_bound. fom_gap_pfp_hYbound_bounded_index_bound + S (fom_index_pfp_hYbound_bounded) = Ly) -> exists fom_value_pfp_hYbound_bounded. ((((exists fom_beta_height_pfp_hYbound_bounded_entry. fom_beta_height_pfp_hYbound_bounded_entry + S (fom_value_pfp_hYbound_bounded) = S ((S (fom_index_pfp_hYbound_bounded)) * yc)) /\\ exists fom_beta_quotient_pfp_hYbound_bounded_entry. yb = fom_beta_quotient_pfp_hYbound_bounded_entry * S ((S (fom_index_pfp_hYbound_bounded)) * yc) + (fom_value_pfp_hYbound_bounded))) /\\ (exists fom_gap_pfp_hYbound_bounded_value_bound. fom_gap_pfp_hYbound_bounded_value_bound + S (fom_value_pfp_hYbound_bounded) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (tb)", "specialize prime_field_polynomial_convolution_bounded (tc)", "specialize prime_field_polynomial_convolution_bounded (Lt)", "specialize prime_field_polynomial_convolution_bounded (bb)", "specialize prime_field_polynomial_convolution_bounded (bc)", "specialize prime_field_polynomial_convolution_bounded (Lb)", "specialize prime_field_polynomial_convolution_bounded (yb)", "specialize prime_field_polynomial_convolution_bounded (yc)", "specialize prime_field_polynomial_convolution_bounded (Ly)", "apply prime_field_polynomial_convolution_bounded", "exact hTB", "have hZbound : forall fom_index_pfp_hZbound_bounded. (exists fom_gap_pfp_hZbound_bounded_index_bound. fom_gap_pfp_hZbound_bounded_index_bound + S (fom_index_pfp_hZbound_bounded) = Lz) -> exists fom_value_pfp_hZbound_bounded. ((((exists fom_beta_height_pfp_hZbound_bounded_entry. fom_beta_height_pfp_hZbound_bounded_entry + S (fom_value_pfp_hZbound_bounded) = S ((S (fom_index_pfp_hZbound_bounded)) * zc)) /\\ exists fom_beta_quotient_pfp_hZbound_bounded_entry. zb = fom_beta_quotient_pfp_hZbound_bounded_entry * S ((S (fom_index_pfp_hZbound_bounded)) * zc) + (fom_value_pfp_hZbound_bounded))) /\\ (exists fom_gap_pfp_hZbound_bounded_value_bound. fom_gap_pfp_hZbound_bounded_value_bound + S (fom_value_pfp_hZbound_bounded) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (wb)", "specialize prime_field_polynomial_convolution_bounded (wc)", "specialize prime_field_polynomial_convolution_bounded (Lw)", "specialize prime_field_polynomial_convolution_bounded (bb)", "specialize prime_field_polynomial_convolution_bounded (bc)", "specialize prime_field_polynomial_convolution_bounded (Lb)", "specialize prime_field_polynomial_convolution_bounded (zb)", "specialize prime_field_polynomial_convolution_bounded (zc)", "specialize prime_field_polynomial_convolution_bounded (Lz)", "apply prime_field_polynomial_convolution_bounded", "exact hWB", "have hDbound : forall fom_index_pfp_hDbound_bounded. (exists fom_gap_pfp_hDbound_bounded_index_bound. fom_gap_pfp_hDbound_bounded_index_bound + S (fom_index_pfp_hDbound_bounded) = Ld) -> exists fom_value_pfp_hDbound_bounded. ((((exists fom_beta_height_pfp_hDbound_bounded_entry. fom_beta_height_pfp_hDbound_bounded_entry + S (fom_value_pfp_hDbound_bounded) = S ((S (fom_index_pfp_hDbound_bounded)) * dc)) /\\ exists fom_beta_quotient_pfp_hDbound_bounded_entry. db = fom_beta_quotient_pfp_hDbound_bounded_entry * S ((S (fom_index_pfp_hDbound_bounded)) * dc) + (fom_value_pfp_hDbound_bounded))) /\\ (exists fom_gap_pfp_hDbound_bounded_value_bound. fom_gap_pfp_hDbound_bounded_value_bound + S (fom_value_pfp_hDbound_bounded) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (vb)", "specialize prime_field_polynomial_convolution_bounded (vc)", "specialize prime_field_polynomial_convolution_bounded (Lv)", "specialize prime_field_polynomial_convolution_bounded (rb)", "specialize prime_field_polynomial_convolution_bounded (rc)", "specialize prime_field_polynomial_convolution_bounded (Lr)", "specialize prime_field_polynomial_convolution_bounded (db)", "specialize prime_field_polynomial_convolution_bounded (dc)", "specialize prime_field_polynomial_convolution_bounded (Ld)", "apply prime_field_polynomial_convolution_bounded", "exact hVR", "have hGbound : ((forall fom_index_pfp_backward_old_C. (exists fom_gap_pfp_backward_old_C_index_bound. fom_gap_pfp_backward_old_C_index_bound + S (fom_index_pfp_backward_old_C) = Lc) -> exists fom_value_pfp_backward_old_C. ((((exists fom_beta_height_pfp_backward_old_C_entry. fom_beta_height_pfp_backward_old_C_entry + S (fom_value_pfp_backward_old_C) = S ((S (fom_index_pfp_backward_old_C)) * cc)) /\\ exists fom_beta_quotient_pfp_backward_old_C_entry. cb = fom_beta_quotient_pfp_backward_old_C_entry * S ((S (fom_index_pfp_backward_old_C)) * cc) + (fom_value_pfp_backward_old_C))) /\\ (exists fom_gap_pfp_backward_old_C_value_bound. fom_gap_pfp_backward_old_C_value_bound + S (fom_value_pfp_backward_old_C) = p))) /\\ (((forall fom_index_pfp_backward_old_D. (exists fom_gap_pfp_backward_old_D_index_bound. fom_gap_pfp_backward_old_D_index_bound + S (fom_index_pfp_backward_old_D) = Ld) -> exists fom_value_pfp_backward_old_D. ((((exists fom_beta_height_pfp_backward_old_D_entry. fom_beta_height_pfp_backward_old_D_entry + S (fom_value_pfp_backward_old_D) = S ((S (fom_index_pfp_backward_old_D)) * dc)) /\\ exists fom_beta_quotient_pfp_backward_old_D_entry. db = fom_beta_quotient_pfp_backward_old_D_entry * S ((S (fom_index_pfp_backward_old_D)) * dc) + (fom_value_pfp_backward_old_D))) /\\ (exists fom_gap_pfp_backward_old_D_value_bound. fom_gap_pfp_backward_old_D_value_bound + S (fom_value_pfp_backward_old_D) = p))) /\\ ((forall fom_index_pfp_backward_old_G. (exists fom_gap_pfp_backward_old_G_index_bound. fom_gap_pfp_backward_old_G_index_bound + S (fom_index_pfp_backward_old_G) = Lg) -> exists fom_value_pfp_backward_old_G. ((((exists fom_beta_height_pfp_backward_old_G_entry. fom_beta_height_pfp_backward_old_G_entry + S (fom_value_pfp_backward_old_G) = S ((S (fom_index_pfp_backward_old_G)) * gc)) /\\ exists fom_beta_quotient_pfp_backward_old_G_entry. gb = fom_beta_quotient_pfp_backward_old_G_entry * S ((S (fom_index_pfp_backward_old_G)) * gc) + (fom_value_pfp_backward_old_G))) /\\ (exists fom_gap_pfp_backward_old_G_value_bound. fom_gap_pfp_backward_old_G_value_bound + S (fom_value_pfp_backward_old_G) = p)))))))", "specialize prime_field_polynomial_aligned_add_bounded (p)", "specialize prime_field_polynomial_aligned_add_bounded (cb)", "specialize prime_field_polynomial_aligned_add_bounded (cc)", "specialize prime_field_polynomial_aligned_add_bounded (Lc)", "specialize prime_field_polynomial_aligned_add_bounded (db)", "specialize prime_field_polynomial_aligned_add_bounded (dc)", "specialize prime_field_polynomial_aligned_add_bounded (Ld)", "specialize prime_field_polynomial_aligned_add_bounded (gb)", "specialize prime_field_polynomial_aligned_add_bounded (gc)", "specialize prime_field_polynomial_aligned_add_bounded (Lg)", "apply prime_field_polynomial_aligned_add_bounded", "exact hold", "cases hGbound", "cases hGbound_right", "have hright : ((forall fom_index_pfp_backward_right_sum_left_bounded. (exists fom_gap_pfp_backward_right_sum_left_bounded_index_bound. fom_gap_pfp_backward_right_sum_left_bounded_index_bound + S (fom_index_pfp_backward_right_sum_left_bounded) = Ly) -> exists fom_value_pfp_backward_right_sum_left_bounded. ((((exists fom_beta_height_pfp_backward_right_sum_left_bounded_entry. fom_beta_height_pfp_backward_right_sum_left_bounded_entry + S (fom_value_pfp_backward_right_sum_left_bounded) = S ((S (fom_index_pfp_backward_right_sum_left_bounded)) * yc)) /\\ exists fom_beta_quotient_pfp_backward_right_sum_left_bounded_entry. yb = fom_beta_quotient_pfp_backward_right_sum_left_bounded_entry * S ((S (fom_index_pfp_backward_right_sum_left_bounded)) * yc) + (fom_value_pfp_backward_right_sum_left_bounded))) /\\ (exists fom_gap_pfp_backward_right_sum_left_bounded_value_bound. fom_gap_pfp_backward_right_sum_left_bounded_value_bound + S (fom_value_pfp_backward_right_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_backward_right_sum_right_bounded. (exists fom_gap_pfp_backward_right_sum_right_bounded_index_bound. fom_gap_pfp_backward_right_sum_right_bounded_index_bound + S (fom_index_pfp_backward_right_sum_right_bounded) = Lz) -> exists fom_value_pfp_backward_right_sum_right_bounded. ((((exists fom_beta_height_pfp_backward_right_sum_right_bounded_entry. fom_beta_height_pfp_backward_right_sum_right_bounded_entry + S (fom_value_pfp_backward_right_sum_right_bounded) = S ((S (fom_index_pfp_backward_right_sum_right_bounded)) * zc)) /\\ exists fom_beta_quotient_pfp_backward_right_sum_right_bounded_entry. zb = fom_beta_quotient_pfp_backward_right_sum_right_bounded_entry * S ((S (fom_index_pfp_backward_right_sum_right_bounded)) * zc) + (fom_value_pfp_backward_right_sum_right_bounded))) /\\ (exists fom_gap_pfp_backward_right_sum_right_bounded_value_bound. fom_gap_pfp_backward_right_sum_right_bounded_value_bound + S (fom_value_pfp_backward_right_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_backward_right_sum_result_bounded. (exists fom_gap_pfp_backward_right_sum_result_bounded_index_bound. fom_gap_pfp_backward_right_sum_result_bounded_index_bound + S (fom_index_pfp_backward_right_sum_result_bounded) = Lc) -> exists fom_value_pfp_backward_right_sum_result_bounded. ((((exists fom_beta_height_pfp_backward_right_sum_result_bounded_entry. fom_beta_height_pfp_backward_right_sum_result_bounded_entry + S (fom_value_pfp_backward_right_sum_result_bounded) = S ((S (fom_index_pfp_backward_right_sum_result_bounded)) * cc)) /\\ exists fom_beta_quotient_pfp_backward_right_sum_result_bounded_entry. cb = fom_beta_quotient_pfp_backward_right_sum_result_bounded_entry * S ((S (fom_index_pfp_backward_right_sum_result_bounded)) * cc) + (fom_value_pfp_backward_right_sum_result_bounded))) /\\ (exists fom_gap_pfp_backward_right_sum_result_bounded_value_bound. fom_gap_pfp_backward_right_sum_result_bounded_value_bound + S (fom_value_pfp_backward_right_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_backward_right_sum pfaa_left_c_backward_right_sum pfaa_right_b_backward_right_sum pfaa_right_c_backward_right_sum pfaa_sum_b_backward_right_sum pfaa_sum_c_backward_right_sum pfaa_length_backward_right_sum. ((((forall pfrep_power_backward_right_sum_witness_common_left pfrep_left_backward_right_sum_witness_common_left pfrep_right_backward_right_sum_witness_common_left. ((exists pfrep_position_backward_right_sum_witness_common_leftfirst. ((pfrep_position_backward_right_sum_witness_common_leftfirst+S (pfrep_power_backward_right_sum_witness_common_left)=(Ly)) /\\ ((((exists ff_h_pfp_backward_right_sum_witness_common_leftfirstentry. ff_h_pfp_backward_right_sum_witness_common_leftfirstentry + S (pfrep_left_backward_right_sum_witness_common_left) = S ((S (pfrep_position_backward_right_sum_witness_common_leftfirst)) * yc)) /\\ exists ff_q_pfp_backward_right_sum_witness_common_leftfirstentry. yb = ff_q_pfp_backward_right_sum_witness_common_leftfirstentry * S ((S (pfrep_position_backward_right_sum_witness_common_leftfirst)) * yc) + (pfrep_left_backward_right_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_right_sum_witness_common_leftfirstoutside. pfrep_gap_backward_right_sum_witness_common_leftfirstoutside+(Ly)=(pfrep_power_backward_right_sum_witness_common_left)) /\\ (((pfrep_left_backward_right_sum_witness_common_left)=0))))) -> ((exists pfrep_position_backward_right_sum_witness_common_leftsecond. ((pfrep_position_backward_right_sum_witness_common_leftsecond+S (pfrep_power_backward_right_sum_witness_common_left)=(pfaa_length_backward_right_sum)) /\\ ((((exists ff_h_pfp_backward_right_sum_witness_common_leftsecondentry. ff_h_pfp_backward_right_sum_witness_common_leftsecondentry + S (pfrep_right_backward_right_sum_witness_common_left) = S ((S (pfrep_position_backward_right_sum_witness_common_leftsecond)) * pfaa_left_c_backward_right_sum)) /\\ exists ff_q_pfp_backward_right_sum_witness_common_leftsecondentry. pfaa_left_b_backward_right_sum = ff_q_pfp_backward_right_sum_witness_common_leftsecondentry * S ((S (pfrep_position_backward_right_sum_witness_common_leftsecond)) * pfaa_left_c_backward_right_sum) + (pfrep_right_backward_right_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_right_sum_witness_common_leftsecondoutside. pfrep_gap_backward_right_sum_witness_common_leftsecondoutside+(pfaa_length_backward_right_sum)=(pfrep_power_backward_right_sum_witness_common_left)) /\\ (((pfrep_right_backward_right_sum_witness_common_left)=0))))) -> pfrep_left_backward_right_sum_witness_common_left=pfrep_right_backward_right_sum_witness_common_left) /\\ ((forall pfrep_power_backward_right_sum_witness_common_right pfrep_left_backward_right_sum_witness_common_right pfrep_right_backward_right_sum_witness_common_right. ((exists pfrep_position_backward_right_sum_witness_common_rightfirst. ((pfrep_position_backward_right_sum_witness_common_rightfirst+S (pfrep_power_backward_right_sum_witness_common_right)=(Lz)) /\\ ((((exists ff_h_pfp_backward_right_sum_witness_common_rightfirstentry. ff_h_pfp_backward_right_sum_witness_common_rightfirstentry + S (pfrep_left_backward_right_sum_witness_common_right) = S ((S (pfrep_position_backward_right_sum_witness_common_rightfirst)) * zc)) /\\ exists ff_q_pfp_backward_right_sum_witness_common_rightfirstentry. zb = ff_q_pfp_backward_right_sum_witness_common_rightfirstentry * S ((S (pfrep_position_backward_right_sum_witness_common_rightfirst)) * zc) + (pfrep_left_backward_right_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_right_sum_witness_common_rightfirstoutside. pfrep_gap_backward_right_sum_witness_common_rightfirstoutside+(Lz)=(pfrep_power_backward_right_sum_witness_common_right)) /\\ (((pfrep_left_backward_right_sum_witness_common_right)=0))))) -> ((exists pfrep_position_backward_right_sum_witness_common_rightsecond. ((pfrep_position_backward_right_sum_witness_common_rightsecond+S (pfrep_power_backward_right_sum_witness_common_right)=(pfaa_length_backward_right_sum)) /\\ ((((exists ff_h_pfp_backward_right_sum_witness_common_rightsecondentry. ff_h_pfp_backward_right_sum_witness_common_rightsecondentry + S (pfrep_right_backward_right_sum_witness_common_right) = S ((S (pfrep_position_backward_right_sum_witness_common_rightsecond)) * pfaa_right_c_backward_right_sum)) /\\ exists ff_q_pfp_backward_right_sum_witness_common_rightsecondentry. pfaa_right_b_backward_right_sum = ff_q_pfp_backward_right_sum_witness_common_rightsecondentry * S ((S (pfrep_position_backward_right_sum_witness_common_rightsecond)) * pfaa_right_c_backward_right_sum) + (pfrep_right_backward_right_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_right_sum_witness_common_rightsecondoutside. pfrep_gap_backward_right_sum_witness_common_rightsecondoutside+(pfaa_length_backward_right_sum)=(pfrep_power_backward_right_sum_witness_common_right)) /\\ (((pfrep_right_backward_right_sum_witness_common_right)=0))))) -> pfrep_left_backward_right_sum_witness_common_right=pfrep_right_backward_right_sum_witness_common_right)))) /\\ (((forall pfp_index_backward_right_sum_witness_operation. (exists pfa_gap_backward_right_sum_witness_operationindex. pfa_gap_backward_right_sum_witness_operationindex + S (pfp_index_backward_right_sum_witness_operation) = (pfaa_length_backward_right_sum)) -> exists pfp_left_backward_right_sum_witness_operation pfp_right_backward_right_sum_witness_operation pfp_value_backward_right_sum_witness_operation. ((((exists ff_h_pfp_backward_right_sum_witness_operationleft. ff_h_pfp_backward_right_sum_witness_operationleft + S (pfp_left_backward_right_sum_witness_operation) = S ((S (pfp_index_backward_right_sum_witness_operation)) * pfaa_left_c_backward_right_sum)) /\\ exists ff_q_pfp_backward_right_sum_witness_operationleft. pfaa_left_b_backward_right_sum = ff_q_pfp_backward_right_sum_witness_operationleft * S ((S (pfp_index_backward_right_sum_witness_operation)) * pfaa_left_c_backward_right_sum) + (pfp_left_backward_right_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_right_sum_witness_operationright. ff_h_pfp_backward_right_sum_witness_operationright + S (pfp_right_backward_right_sum_witness_operation) = S ((S (pfp_index_backward_right_sum_witness_operation)) * pfaa_right_c_backward_right_sum)) /\\ exists ff_q_pfp_backward_right_sum_witness_operationright. pfaa_right_b_backward_right_sum = ff_q_pfp_backward_right_sum_witness_operationright * S ((S (pfp_index_backward_right_sum_witness_operation)) * pfaa_right_c_backward_right_sum) + (pfp_right_backward_right_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_right_sum_witness_operationtarget. ff_h_pfp_backward_right_sum_witness_operationtarget + S (pfp_value_backward_right_sum_witness_operation) = S ((S (pfp_index_backward_right_sum_witness_operation)) * pfaa_sum_c_backward_right_sum)) /\\ exists ff_q_pfp_backward_right_sum_witness_operationtarget. pfaa_sum_b_backward_right_sum = ff_q_pfp_backward_right_sum_witness_operationtarget * S ((S (pfp_index_backward_right_sum_witness_operation)) * pfaa_sum_c_backward_right_sum) + (pfp_value_backward_right_sum_witness_operation))) /\\ ((((exists pfa_gap_backward_right_sum_witness_operationoperationleft. pfa_gap_backward_right_sum_witness_operationoperationleft + S (pfp_left_backward_right_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_backward_right_sum_witness_operationoperationright. pfa_gap_backward_right_sum_witness_operationoperationright + S (pfp_right_backward_right_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_backward_right_sum_witness_operationoperationresultbound. pfa_gap_backward_right_sum_witness_operationoperationresultbound + S (pfp_value_backward_right_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_backward_right_sum_witness_operationoperationresultcongruence pfa_offset_right_backward_right_sum_witness_operationoperationresultcongruence. ((pfp_left_backward_right_sum_witness_operation) + (pfp_right_backward_right_sum_witness_operation)) + (p) * pfa_offset_left_backward_right_sum_witness_operationoperationresultcongruence = (pfp_value_backward_right_sum_witness_operation) + (p) * pfa_offset_right_backward_right_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_backward_right_sum_witness_output pfrep_left_backward_right_sum_witness_output pfrep_right_backward_right_sum_witness_output. ((exists pfrep_position_backward_right_sum_witness_outputfirst. ((pfrep_position_backward_right_sum_witness_outputfirst+S (pfrep_power_backward_right_sum_witness_output)=(pfaa_length_backward_right_sum)) /\\ ((((exists ff_h_pfp_backward_right_sum_witness_outputfirstentry. ff_h_pfp_backward_right_sum_witness_outputfirstentry + S (pfrep_left_backward_right_sum_witness_output) = S ((S (pfrep_position_backward_right_sum_witness_outputfirst)) * pfaa_sum_c_backward_right_sum)) /\\ exists ff_q_pfp_backward_right_sum_witness_outputfirstentry. pfaa_sum_b_backward_right_sum = ff_q_pfp_backward_right_sum_witness_outputfirstentry * S ((S (pfrep_position_backward_right_sum_witness_outputfirst)) * pfaa_sum_c_backward_right_sum) + (pfrep_left_backward_right_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_right_sum_witness_outputfirstoutside. pfrep_gap_backward_right_sum_witness_outputfirstoutside+(pfaa_length_backward_right_sum)=(pfrep_power_backward_right_sum_witness_output)) /\\ (((pfrep_left_backward_right_sum_witness_output)=0))))) -> ((exists pfrep_position_backward_right_sum_witness_outputsecond. ((pfrep_position_backward_right_sum_witness_outputsecond+S (pfrep_power_backward_right_sum_witness_output)=(Lc)) /\\ ((((exists ff_h_pfp_backward_right_sum_witness_outputsecondentry. ff_h_pfp_backward_right_sum_witness_outputsecondentry + S (pfrep_right_backward_right_sum_witness_output) = S ((S (pfrep_position_backward_right_sum_witness_outputsecond)) * cc)) /\\ exists ff_q_pfp_backward_right_sum_witness_outputsecondentry. cb = ff_q_pfp_backward_right_sum_witness_outputsecondentry * S ((S (pfrep_position_backward_right_sum_witness_outputsecond)) * cc) + (pfrep_right_backward_right_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_right_sum_witness_outputsecondoutside. pfrep_gap_backward_right_sum_witness_outputsecondoutside+(Lc)=(pfrep_power_backward_right_sum_witness_output)) /\\ (((pfrep_right_backward_right_sum_witness_output)=0))))) -> pfrep_left_backward_right_sum_witness_output=pfrep_right_backward_right_sum_witness_output))))))))))))", "specialize prime_field_polynomial_aligned_add_commutative (p)", "specialize prime_field_polynomial_aligned_add_commutative (zb)", "specialize prime_field_polynomial_aligned_add_commutative (zc)", "specialize prime_field_polynomial_aligned_add_commutative (Lz)", "specialize prime_field_polynomial_aligned_add_commutative (yb)", "specialize prime_field_polynomial_aligned_add_commutative (yc)", "specialize prime_field_polynomial_aligned_add_commutative (Ly)", "specialize prime_field_polynomial_aligned_add_commutative (cb)", "specialize prime_field_polynomial_aligned_add_commutative (cc)", "specialize prime_field_polynomial_aligned_add_commutative (Lc)", "apply prime_field_polynomial_aligned_add_commutative", "specialize prime_field_polynomial_aligned_convolution_right_add (p)", "specialize prime_field_polynomial_aligned_convolution_right_add (wb)", "specialize prime_field_polynomial_aligned_convolution_right_add (wc)", "specialize prime_field_polynomial_aligned_convolution_right_add (Lw)", "specialize prime_field_polynomial_aligned_convolution_right_add (tb)", "specialize prime_field_polynomial_aligned_convolution_right_add (tc)", "specialize prime_field_polynomial_aligned_convolution_right_add (Lt)", "specialize prime_field_polynomial_aligned_convolution_right_add (ub)", "specialize prime_field_polynomial_aligned_convolution_right_add (uc)", "specialize prime_field_polynomial_aligned_convolution_right_add (Lu)", "specialize prime_field_polynomial_aligned_convolution_right_add (bb)", "specialize prime_field_polynomial_aligned_convolution_right_add (bc)", "specialize prime_field_polynomial_aligned_convolution_right_add (Lb)", "specialize prime_field_polynomial_aligned_convolution_right_add (zb)", "specialize prime_field_polynomial_aligned_convolution_right_add (zc)", "specialize prime_field_polynomial_aligned_convolution_right_add (Lz)", "specialize prime_field_polynomial_aligned_convolution_right_add (yb)", "specialize prime_field_polynomial_aligned_convolution_right_add (yc)", "specialize prime_field_polynomial_aligned_convolution_right_add (Ly)", "specialize prime_field_polynomial_aligned_convolution_right_add (cb)", "specialize prime_field_polynomial_aligned_convolution_right_add (cc)", "specialize prime_field_polynomial_aligned_convolution_right_add (Lc)", "apply prime_field_polynomial_aligned_convolution_right_add", "exact hp", "exact hsub", "exact hWB", "exact hTB", "exact hUB", "have hleft : ((forall fom_index_pfp_backward_left_sum_left_bounded. (exists fom_gap_pfp_backward_left_sum_left_bounded_index_bound. fom_gap_pfp_backward_left_sum_left_bounded_index_bound + S (fom_index_pfp_backward_left_sum_left_bounded) = Lh) -> exists fom_value_pfp_backward_left_sum_left_bounded. ((((exists fom_beta_height_pfp_backward_left_sum_left_bounded_entry. fom_beta_height_pfp_backward_left_sum_left_bounded_entry + S (fom_value_pfp_backward_left_sum_left_bounded) = S ((S (fom_index_pfp_backward_left_sum_left_bounded)) * hc)) /\\ exists fom_beta_quotient_pfp_backward_left_sum_left_bounded_entry. hb = fom_beta_quotient_pfp_backward_left_sum_left_bounded_entry * S ((S (fom_index_pfp_backward_left_sum_left_bounded)) * hc) + (fom_value_pfp_backward_left_sum_left_bounded))) /\\ (exists fom_gap_pfp_backward_left_sum_left_bounded_value_bound. fom_gap_pfp_backward_left_sum_left_bounded_value_bound + S (fom_value_pfp_backward_left_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_backward_left_sum_right_bounded. (exists fom_gap_pfp_backward_left_sum_right_bounded_index_bound. fom_gap_pfp_backward_left_sum_right_bounded_index_bound + S (fom_index_pfp_backward_left_sum_right_bounded) = Ld) -> exists fom_value_pfp_backward_left_sum_right_bounded. ((((exists fom_beta_height_pfp_backward_left_sum_right_bounded_entry. fom_beta_height_pfp_backward_left_sum_right_bounded_entry + S (fom_value_pfp_backward_left_sum_right_bounded) = S ((S (fom_index_pfp_backward_left_sum_right_bounded)) * dc)) /\\ exists fom_beta_quotient_pfp_backward_left_sum_right_bounded_entry. db = fom_beta_quotient_pfp_backward_left_sum_right_bounded_entry * S ((S (fom_index_pfp_backward_left_sum_right_bounded)) * dc) + (fom_value_pfp_backward_left_sum_right_bounded))) /\\ (exists fom_gap_pfp_backward_left_sum_right_bounded_value_bound. fom_gap_pfp_backward_left_sum_right_bounded_value_bound + S (fom_value_pfp_backward_left_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_backward_left_sum_result_bounded. (exists fom_gap_pfp_backward_left_sum_result_bounded_index_bound. fom_gap_pfp_backward_left_sum_result_bounded_index_bound + S (fom_index_pfp_backward_left_sum_result_bounded) = Lx) -> exists fom_value_pfp_backward_left_sum_result_bounded. ((((exists fom_beta_height_pfp_backward_left_sum_result_bounded_entry. fom_beta_height_pfp_backward_left_sum_result_bounded_entry + S (fom_value_pfp_backward_left_sum_result_bounded) = S ((S (fom_index_pfp_backward_left_sum_result_bounded)) * xc)) /\\ exists fom_beta_quotient_pfp_backward_left_sum_result_bounded_entry. xb = fom_beta_quotient_pfp_backward_left_sum_result_bounded_entry * S ((S (fom_index_pfp_backward_left_sum_result_bounded)) * xc) + (fom_value_pfp_backward_left_sum_result_bounded))) /\\ (exists fom_gap_pfp_backward_left_sum_result_bounded_value_bound. fom_gap_pfp_backward_left_sum_result_bounded_value_bound + S (fom_value_pfp_backward_left_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_backward_left_sum pfaa_left_c_backward_left_sum pfaa_right_b_backward_left_sum pfaa_right_c_backward_left_sum pfaa_sum_b_backward_left_sum pfaa_sum_c_backward_left_sum pfaa_length_backward_left_sum. ((((forall pfrep_power_backward_left_sum_witness_common_left pfrep_left_backward_left_sum_witness_common_left pfrep_right_backward_left_sum_witness_common_left. ((exists pfrep_position_backward_left_sum_witness_common_leftfirst. ((pfrep_position_backward_left_sum_witness_common_leftfirst+S (pfrep_power_backward_left_sum_witness_common_left)=(Lh)) /\\ ((((exists ff_h_pfp_backward_left_sum_witness_common_leftfirstentry. ff_h_pfp_backward_left_sum_witness_common_leftfirstentry + S (pfrep_left_backward_left_sum_witness_common_left) = S ((S (pfrep_position_backward_left_sum_witness_common_leftfirst)) * hc)) /\\ exists ff_q_pfp_backward_left_sum_witness_common_leftfirstentry. hb = ff_q_pfp_backward_left_sum_witness_common_leftfirstentry * S ((S (pfrep_position_backward_left_sum_witness_common_leftfirst)) * hc) + (pfrep_left_backward_left_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_left_sum_witness_common_leftfirstoutside. pfrep_gap_backward_left_sum_witness_common_leftfirstoutside+(Lh)=(pfrep_power_backward_left_sum_witness_common_left)) /\\ (((pfrep_left_backward_left_sum_witness_common_left)=0))))) -> ((exists pfrep_position_backward_left_sum_witness_common_leftsecond. ((pfrep_position_backward_left_sum_witness_common_leftsecond+S (pfrep_power_backward_left_sum_witness_common_left)=(pfaa_length_backward_left_sum)) /\\ ((((exists ff_h_pfp_backward_left_sum_witness_common_leftsecondentry. ff_h_pfp_backward_left_sum_witness_common_leftsecondentry + S (pfrep_right_backward_left_sum_witness_common_left) = S ((S (pfrep_position_backward_left_sum_witness_common_leftsecond)) * pfaa_left_c_backward_left_sum)) /\\ exists ff_q_pfp_backward_left_sum_witness_common_leftsecondentry. pfaa_left_b_backward_left_sum = ff_q_pfp_backward_left_sum_witness_common_leftsecondentry * S ((S (pfrep_position_backward_left_sum_witness_common_leftsecond)) * pfaa_left_c_backward_left_sum) + (pfrep_right_backward_left_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_left_sum_witness_common_leftsecondoutside. pfrep_gap_backward_left_sum_witness_common_leftsecondoutside+(pfaa_length_backward_left_sum)=(pfrep_power_backward_left_sum_witness_common_left)) /\\ (((pfrep_right_backward_left_sum_witness_common_left)=0))))) -> pfrep_left_backward_left_sum_witness_common_left=pfrep_right_backward_left_sum_witness_common_left) /\\ ((forall pfrep_power_backward_left_sum_witness_common_right pfrep_left_backward_left_sum_witness_common_right pfrep_right_backward_left_sum_witness_common_right. ((exists pfrep_position_backward_left_sum_witness_common_rightfirst. ((pfrep_position_backward_left_sum_witness_common_rightfirst+S (pfrep_power_backward_left_sum_witness_common_right)=(Ld)) /\\ ((((exists ff_h_pfp_backward_left_sum_witness_common_rightfirstentry. ff_h_pfp_backward_left_sum_witness_common_rightfirstentry + S (pfrep_left_backward_left_sum_witness_common_right) = S ((S (pfrep_position_backward_left_sum_witness_common_rightfirst)) * dc)) /\\ exists ff_q_pfp_backward_left_sum_witness_common_rightfirstentry. db = ff_q_pfp_backward_left_sum_witness_common_rightfirstentry * S ((S (pfrep_position_backward_left_sum_witness_common_rightfirst)) * dc) + (pfrep_left_backward_left_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_left_sum_witness_common_rightfirstoutside. pfrep_gap_backward_left_sum_witness_common_rightfirstoutside+(Ld)=(pfrep_power_backward_left_sum_witness_common_right)) /\\ (((pfrep_left_backward_left_sum_witness_common_right)=0))))) -> ((exists pfrep_position_backward_left_sum_witness_common_rightsecond. ((pfrep_position_backward_left_sum_witness_common_rightsecond+S (pfrep_power_backward_left_sum_witness_common_right)=(pfaa_length_backward_left_sum)) /\\ ((((exists ff_h_pfp_backward_left_sum_witness_common_rightsecondentry. ff_h_pfp_backward_left_sum_witness_common_rightsecondentry + S (pfrep_right_backward_left_sum_witness_common_right) = S ((S (pfrep_position_backward_left_sum_witness_common_rightsecond)) * pfaa_right_c_backward_left_sum)) /\\ exists ff_q_pfp_backward_left_sum_witness_common_rightsecondentry. pfaa_right_b_backward_left_sum = ff_q_pfp_backward_left_sum_witness_common_rightsecondentry * S ((S (pfrep_position_backward_left_sum_witness_common_rightsecond)) * pfaa_right_c_backward_left_sum) + (pfrep_right_backward_left_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_left_sum_witness_common_rightsecondoutside. pfrep_gap_backward_left_sum_witness_common_rightsecondoutside+(pfaa_length_backward_left_sum)=(pfrep_power_backward_left_sum_witness_common_right)) /\\ (((pfrep_right_backward_left_sum_witness_common_right)=0))))) -> pfrep_left_backward_left_sum_witness_common_right=pfrep_right_backward_left_sum_witness_common_right)))) /\\ (((forall pfp_index_backward_left_sum_witness_operation. (exists pfa_gap_backward_left_sum_witness_operationindex. pfa_gap_backward_left_sum_witness_operationindex + S (pfp_index_backward_left_sum_witness_operation) = (pfaa_length_backward_left_sum)) -> exists pfp_left_backward_left_sum_witness_operation pfp_right_backward_left_sum_witness_operation pfp_value_backward_left_sum_witness_operation. ((((exists ff_h_pfp_backward_left_sum_witness_operationleft. ff_h_pfp_backward_left_sum_witness_operationleft + S (pfp_left_backward_left_sum_witness_operation) = S ((S (pfp_index_backward_left_sum_witness_operation)) * pfaa_left_c_backward_left_sum)) /\\ exists ff_q_pfp_backward_left_sum_witness_operationleft. pfaa_left_b_backward_left_sum = ff_q_pfp_backward_left_sum_witness_operationleft * S ((S (pfp_index_backward_left_sum_witness_operation)) * pfaa_left_c_backward_left_sum) + (pfp_left_backward_left_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_left_sum_witness_operationright. ff_h_pfp_backward_left_sum_witness_operationright + S (pfp_right_backward_left_sum_witness_operation) = S ((S (pfp_index_backward_left_sum_witness_operation)) * pfaa_right_c_backward_left_sum)) /\\ exists ff_q_pfp_backward_left_sum_witness_operationright. pfaa_right_b_backward_left_sum = ff_q_pfp_backward_left_sum_witness_operationright * S ((S (pfp_index_backward_left_sum_witness_operation)) * pfaa_right_c_backward_left_sum) + (pfp_right_backward_left_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_left_sum_witness_operationtarget. ff_h_pfp_backward_left_sum_witness_operationtarget + S (pfp_value_backward_left_sum_witness_operation) = S ((S (pfp_index_backward_left_sum_witness_operation)) * pfaa_sum_c_backward_left_sum)) /\\ exists ff_q_pfp_backward_left_sum_witness_operationtarget. pfaa_sum_b_backward_left_sum = ff_q_pfp_backward_left_sum_witness_operationtarget * S ((S (pfp_index_backward_left_sum_witness_operation)) * pfaa_sum_c_backward_left_sum) + (pfp_value_backward_left_sum_witness_operation))) /\\ ((((exists pfa_gap_backward_left_sum_witness_operationoperationleft. pfa_gap_backward_left_sum_witness_operationoperationleft + S (pfp_left_backward_left_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_backward_left_sum_witness_operationoperationright. pfa_gap_backward_left_sum_witness_operationoperationright + S (pfp_right_backward_left_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_backward_left_sum_witness_operationoperationresultbound. pfa_gap_backward_left_sum_witness_operationoperationresultbound + S (pfp_value_backward_left_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_backward_left_sum_witness_operationoperationresultcongruence pfa_offset_right_backward_left_sum_witness_operationoperationresultcongruence. ((pfp_left_backward_left_sum_witness_operation) + (pfp_right_backward_left_sum_witness_operation)) + (p) * pfa_offset_left_backward_left_sum_witness_operationoperationresultcongruence = (pfp_value_backward_left_sum_witness_operation) + (p) * pfa_offset_right_backward_left_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_backward_left_sum_witness_output pfrep_left_backward_left_sum_witness_output pfrep_right_backward_left_sum_witness_output. ((exists pfrep_position_backward_left_sum_witness_outputfirst. ((pfrep_position_backward_left_sum_witness_outputfirst+S (pfrep_power_backward_left_sum_witness_output)=(pfaa_length_backward_left_sum)) /\\ ((((exists ff_h_pfp_backward_left_sum_witness_outputfirstentry. ff_h_pfp_backward_left_sum_witness_outputfirstentry + S (pfrep_left_backward_left_sum_witness_output) = S ((S (pfrep_position_backward_left_sum_witness_outputfirst)) * pfaa_sum_c_backward_left_sum)) /\\ exists ff_q_pfp_backward_left_sum_witness_outputfirstentry. pfaa_sum_b_backward_left_sum = ff_q_pfp_backward_left_sum_witness_outputfirstentry * S ((S (pfrep_position_backward_left_sum_witness_outputfirst)) * pfaa_sum_c_backward_left_sum) + (pfrep_left_backward_left_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_left_sum_witness_outputfirstoutside. pfrep_gap_backward_left_sum_witness_outputfirstoutside+(pfaa_length_backward_left_sum)=(pfrep_power_backward_left_sum_witness_output)) /\\ (((pfrep_left_backward_left_sum_witness_output)=0))))) -> ((exists pfrep_position_backward_left_sum_witness_outputsecond. ((pfrep_position_backward_left_sum_witness_outputsecond+S (pfrep_power_backward_left_sum_witness_output)=(Lx)) /\\ ((((exists ff_h_pfp_backward_left_sum_witness_outputsecondentry. ff_h_pfp_backward_left_sum_witness_outputsecondentry + S (pfrep_right_backward_left_sum_witness_output) = S ((S (pfrep_position_backward_left_sum_witness_outputsecond)) * xc)) /\\ exists ff_q_pfp_backward_left_sum_witness_outputsecondentry. xb = ff_q_pfp_backward_left_sum_witness_outputsecondentry * S ((S (pfrep_position_backward_left_sum_witness_outputsecond)) * xc) + (pfrep_right_backward_left_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_left_sum_witness_outputsecondoutside. pfrep_gap_backward_left_sum_witness_outputsecondoutside+(Lx)=(pfrep_power_backward_left_sum_witness_output)) /\\ (((pfrep_right_backward_left_sum_witness_output)=0))))) -> pfrep_left_backward_left_sum_witness_output=pfrep_right_backward_left_sum_witness_output))))))))))))", "specialize prime_field_polynomial_aligned_convolution_left_add (p)", "specialize prime_field_polynomial_aligned_convolution_left_add (pb)", "specialize prime_field_polynomial_aligned_convolution_left_add (pc)", "specialize prime_field_polynomial_aligned_convolution_left_add (Lp)", "specialize prime_field_polynomial_aligned_convolution_left_add (rb)", "specialize prime_field_polynomial_aligned_convolution_left_add (rc)", "specialize prime_field_polynomial_aligned_convolution_left_add (Lr)", "specialize prime_field_polynomial_aligned_convolution_left_add (ab)", "specialize prime_field_polynomial_aligned_convolution_left_add (ac)", "specialize prime_field_polynomial_aligned_convolution_left_add (La)", "specialize prime_field_polynomial_aligned_convolution_left_add (vb)", "specialize prime_field_polynomial_aligned_convolution_left_add (vc)", "specialize prime_field_polynomial_aligned_convolution_left_add (Lv)", "specialize prime_field_polynomial_aligned_convolution_left_add (hb)", "specialize prime_field_polynomial_aligned_convolution_left_add (hc)", "specialize prime_field_polynomial_aligned_convolution_left_add (Lh)", "specialize prime_field_polynomial_aligned_convolution_left_add (db)", "specialize prime_field_polynomial_aligned_convolution_left_add (dc)", "specialize prime_field_polynomial_aligned_convolution_left_add (Ld)", "specialize prime_field_polynomial_aligned_convolution_left_add (xb)", "specialize prime_field_polynomial_aligned_convolution_left_add (xc)", "specialize prime_field_polynomial_aligned_convolution_left_add (Lx)", "apply prime_field_polynomial_aligned_convolution_left_add", "exact hp", "exact hdivision", "exact hVP", "exact hVR", "exact hVA", "have heq : forall pfrep_power_backward_associated_equivalent pfrep_left_backward_associated_equivalent pfrep_right_backward_associated_equivalent. ((exists pfrep_position_backward_associated_equivalentfirst. ((pfrep_position_backward_associated_equivalentfirst+S (pfrep_power_backward_associated_equivalent)=(Lz)) /\\ ((((exists ff_h_pfp_backward_associated_equivalentfirstentry. ff_h_pfp_backward_associated_equivalentfirstentry + S (pfrep_left_backward_associated_equivalent) = S ((S (pfrep_position_backward_associated_equivalentfirst)) * zc)) /\\ exists ff_q_pfp_backward_associated_equivalentfirstentry. zb = ff_q_pfp_backward_associated_equivalentfirstentry * S ((S (pfrep_position_backward_associated_equivalentfirst)) * zc) + (pfrep_left_backward_associated_equivalent)))))) \\/ (((exists pfrep_gap_backward_associated_equivalentfirstoutside. pfrep_gap_backward_associated_equivalentfirstoutside+(Lz)=(pfrep_power_backward_associated_equivalent)) /\\ (((pfrep_left_backward_associated_equivalent)=0))))) -> ((exists pfrep_position_backward_associated_equivalentsecond. ((pfrep_position_backward_associated_equivalentsecond+S (pfrep_power_backward_associated_equivalent)=(Lh)) /\\ ((((exists ff_h_pfp_backward_associated_equivalentsecondentry. ff_h_pfp_backward_associated_equivalentsecondentry + S (pfrep_right_backward_associated_equivalent) = S ((S (pfrep_position_backward_associated_equivalentsecond)) * hc)) /\\ exists ff_q_pfp_backward_associated_equivalentsecondentry. hb = ff_q_pfp_backward_associated_equivalentsecondentry * S ((S (pfrep_position_backward_associated_equivalentsecond)) * hc) + (pfrep_right_backward_associated_equivalent)))))) \\/ (((exists pfrep_gap_backward_associated_equivalentsecondoutside. pfrep_gap_backward_associated_equivalentsecondoutside+(Lh)=(pfrep_power_backward_associated_equivalent)) /\\ (((pfrep_right_backward_associated_equivalent)=0))))) -> pfrep_left_backward_associated_equivalent=pfrep_right_backward_associated_equivalent", "specialize prime_field_polynomial_convolution_associative_equivalent (p)", "specialize prime_field_polynomial_convolution_associative_equivalent (vb)", "specialize prime_field_polynomial_convolution_associative_equivalent (vc)", "specialize prime_field_polynomial_convolution_associative_equivalent (Lv)", "specialize prime_field_polynomial_convolution_associative_equivalent (qb)", "specialize prime_field_polynomial_convolution_associative_equivalent (qc)", "specialize prime_field_polynomial_convolution_associative_equivalent (Lq)", "specialize prime_field_polynomial_convolution_associative_equivalent (wb)", "specialize prime_field_polynomial_convolution_associative_equivalent (wc)", "specialize prime_field_polynomial_convolution_associative_equivalent (Lw)", "specialize prime_field_polynomial_convolution_associative_equivalent (bb)", "specialize prime_field_polynomial_convolution_associative_equivalent (bc)", "specialize prime_field_polynomial_convolution_associative_equivalent (Lb)", "specialize prime_field_polynomial_convolution_associative_equivalent (pb)", "specialize prime_field_polynomial_convolution_associative_equivalent (pc)", "specialize prime_field_polynomial_convolution_associative_equivalent (Lp)", "specialize prime_field_polynomial_convolution_associative_equivalent (zb)", "specialize prime_field_polynomial_convolution_associative_equivalent (zc)", "specialize prime_field_polynomial_convolution_associative_equivalent (Lz)", "specialize prime_field_polynomial_convolution_associative_equivalent (hb)", "specialize prime_field_polynomial_convolution_associative_equivalent (hc)", "specialize prime_field_polynomial_convolution_associative_equivalent (Lh)", "apply prime_field_polynomial_convolution_associative_equivalent", "exact hp", "exact hVQ", "exact hQB", "exact hWB", "exact hVP", "have hmiddle : ((forall fom_index_pfp_backward_middle_sum_left_bounded. (exists fom_gap_pfp_backward_middle_sum_left_bounded_index_bound. fom_gap_pfp_backward_middle_sum_left_bounded_index_bound + S (fom_index_pfp_backward_middle_sum_left_bounded) = Lz) -> exists fom_value_pfp_backward_middle_sum_left_bounded. ((((exists fom_beta_height_pfp_backward_middle_sum_left_bounded_entry. fom_beta_height_pfp_backward_middle_sum_left_bounded_entry + S (fom_value_pfp_backward_middle_sum_left_bounded) = S ((S (fom_index_pfp_backward_middle_sum_left_bounded)) * zc)) /\\ exists fom_beta_quotient_pfp_backward_middle_sum_left_bounded_entry. zb = fom_beta_quotient_pfp_backward_middle_sum_left_bounded_entry * S ((S (fom_index_pfp_backward_middle_sum_left_bounded)) * zc) + (fom_value_pfp_backward_middle_sum_left_bounded))) /\\ (exists fom_gap_pfp_backward_middle_sum_left_bounded_value_bound. fom_gap_pfp_backward_middle_sum_left_bounded_value_bound + S (fom_value_pfp_backward_middle_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_backward_middle_sum_right_bounded. (exists fom_gap_pfp_backward_middle_sum_right_bounded_index_bound. fom_gap_pfp_backward_middle_sum_right_bounded_index_bound + S (fom_index_pfp_backward_middle_sum_right_bounded) = Ld) -> exists fom_value_pfp_backward_middle_sum_right_bounded. ((((exists fom_beta_height_pfp_backward_middle_sum_right_bounded_entry. fom_beta_height_pfp_backward_middle_sum_right_bounded_entry + S (fom_value_pfp_backward_middle_sum_right_bounded) = S ((S (fom_index_pfp_backward_middle_sum_right_bounded)) * dc)) /\\ exists fom_beta_quotient_pfp_backward_middle_sum_right_bounded_entry. db = fom_beta_quotient_pfp_backward_middle_sum_right_bounded_entry * S ((S (fom_index_pfp_backward_middle_sum_right_bounded)) * dc) + (fom_value_pfp_backward_middle_sum_right_bounded))) /\\ (exists fom_gap_pfp_backward_middle_sum_right_bounded_value_bound. fom_gap_pfp_backward_middle_sum_right_bounded_value_bound + S (fom_value_pfp_backward_middle_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_backward_middle_sum_result_bounded. (exists fom_gap_pfp_backward_middle_sum_result_bounded_index_bound. fom_gap_pfp_backward_middle_sum_result_bounded_index_bound + S (fom_index_pfp_backward_middle_sum_result_bounded) = Lx) -> exists fom_value_pfp_backward_middle_sum_result_bounded. ((((exists fom_beta_height_pfp_backward_middle_sum_result_bounded_entry. fom_beta_height_pfp_backward_middle_sum_result_bounded_entry + S (fom_value_pfp_backward_middle_sum_result_bounded) = S ((S (fom_index_pfp_backward_middle_sum_result_bounded)) * xc)) /\\ exists fom_beta_quotient_pfp_backward_middle_sum_result_bounded_entry. xb = fom_beta_quotient_pfp_backward_middle_sum_result_bounded_entry * S ((S (fom_index_pfp_backward_middle_sum_result_bounded)) * xc) + (fom_value_pfp_backward_middle_sum_result_bounded))) /\\ (exists fom_gap_pfp_backward_middle_sum_result_bounded_value_bound. fom_gap_pfp_backward_middle_sum_result_bounded_value_bound + S (fom_value_pfp_backward_middle_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_backward_middle_sum pfaa_left_c_backward_middle_sum pfaa_right_b_backward_middle_sum pfaa_right_c_backward_middle_sum pfaa_sum_b_backward_middle_sum pfaa_sum_c_backward_middle_sum pfaa_length_backward_middle_sum. ((((forall pfrep_power_backward_middle_sum_witness_common_left pfrep_left_backward_middle_sum_witness_common_left pfrep_right_backward_middle_sum_witness_common_left. ((exists pfrep_position_backward_middle_sum_witness_common_leftfirst. ((pfrep_position_backward_middle_sum_witness_common_leftfirst+S (pfrep_power_backward_middle_sum_witness_common_left)=(Lz)) /\\ ((((exists ff_h_pfp_backward_middle_sum_witness_common_leftfirstentry. ff_h_pfp_backward_middle_sum_witness_common_leftfirstentry + S (pfrep_left_backward_middle_sum_witness_common_left) = S ((S (pfrep_position_backward_middle_sum_witness_common_leftfirst)) * zc)) /\\ exists ff_q_pfp_backward_middle_sum_witness_common_leftfirstentry. zb = ff_q_pfp_backward_middle_sum_witness_common_leftfirstentry * S ((S (pfrep_position_backward_middle_sum_witness_common_leftfirst)) * zc) + (pfrep_left_backward_middle_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_middle_sum_witness_common_leftfirstoutside. pfrep_gap_backward_middle_sum_witness_common_leftfirstoutside+(Lz)=(pfrep_power_backward_middle_sum_witness_common_left)) /\\ (((pfrep_left_backward_middle_sum_witness_common_left)=0))))) -> ((exists pfrep_position_backward_middle_sum_witness_common_leftsecond. ((pfrep_position_backward_middle_sum_witness_common_leftsecond+S (pfrep_power_backward_middle_sum_witness_common_left)=(pfaa_length_backward_middle_sum)) /\\ ((((exists ff_h_pfp_backward_middle_sum_witness_common_leftsecondentry. ff_h_pfp_backward_middle_sum_witness_common_leftsecondentry + S (pfrep_right_backward_middle_sum_witness_common_left) = S ((S (pfrep_position_backward_middle_sum_witness_common_leftsecond)) * pfaa_left_c_backward_middle_sum)) /\\ exists ff_q_pfp_backward_middle_sum_witness_common_leftsecondentry. pfaa_left_b_backward_middle_sum = ff_q_pfp_backward_middle_sum_witness_common_leftsecondentry * S ((S (pfrep_position_backward_middle_sum_witness_common_leftsecond)) * pfaa_left_c_backward_middle_sum) + (pfrep_right_backward_middle_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_middle_sum_witness_common_leftsecondoutside. pfrep_gap_backward_middle_sum_witness_common_leftsecondoutside+(pfaa_length_backward_middle_sum)=(pfrep_power_backward_middle_sum_witness_common_left)) /\\ (((pfrep_right_backward_middle_sum_witness_common_left)=0))))) -> pfrep_left_backward_middle_sum_witness_common_left=pfrep_right_backward_middle_sum_witness_common_left) /\\ ((forall pfrep_power_backward_middle_sum_witness_common_right pfrep_left_backward_middle_sum_witness_common_right pfrep_right_backward_middle_sum_witness_common_right. ((exists pfrep_position_backward_middle_sum_witness_common_rightfirst. ((pfrep_position_backward_middle_sum_witness_common_rightfirst+S (pfrep_power_backward_middle_sum_witness_common_right)=(Ld)) /\\ ((((exists ff_h_pfp_backward_middle_sum_witness_common_rightfirstentry. ff_h_pfp_backward_middle_sum_witness_common_rightfirstentry + S (pfrep_left_backward_middle_sum_witness_common_right) = S ((S (pfrep_position_backward_middle_sum_witness_common_rightfirst)) * dc)) /\\ exists ff_q_pfp_backward_middle_sum_witness_common_rightfirstentry. db = ff_q_pfp_backward_middle_sum_witness_common_rightfirstentry * S ((S (pfrep_position_backward_middle_sum_witness_common_rightfirst)) * dc) + (pfrep_left_backward_middle_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_middle_sum_witness_common_rightfirstoutside. pfrep_gap_backward_middle_sum_witness_common_rightfirstoutside+(Ld)=(pfrep_power_backward_middle_sum_witness_common_right)) /\\ (((pfrep_left_backward_middle_sum_witness_common_right)=0))))) -> ((exists pfrep_position_backward_middle_sum_witness_common_rightsecond. ((pfrep_position_backward_middle_sum_witness_common_rightsecond+S (pfrep_power_backward_middle_sum_witness_common_right)=(pfaa_length_backward_middle_sum)) /\\ ((((exists ff_h_pfp_backward_middle_sum_witness_common_rightsecondentry. ff_h_pfp_backward_middle_sum_witness_common_rightsecondentry + S (pfrep_right_backward_middle_sum_witness_common_right) = S ((S (pfrep_position_backward_middle_sum_witness_common_rightsecond)) * pfaa_right_c_backward_middle_sum)) /\\ exists ff_q_pfp_backward_middle_sum_witness_common_rightsecondentry. pfaa_right_b_backward_middle_sum = ff_q_pfp_backward_middle_sum_witness_common_rightsecondentry * S ((S (pfrep_position_backward_middle_sum_witness_common_rightsecond)) * pfaa_right_c_backward_middle_sum) + (pfrep_right_backward_middle_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_middle_sum_witness_common_rightsecondoutside. pfrep_gap_backward_middle_sum_witness_common_rightsecondoutside+(pfaa_length_backward_middle_sum)=(pfrep_power_backward_middle_sum_witness_common_right)) /\\ (((pfrep_right_backward_middle_sum_witness_common_right)=0))))) -> pfrep_left_backward_middle_sum_witness_common_right=pfrep_right_backward_middle_sum_witness_common_right)))) /\\ (((forall pfp_index_backward_middle_sum_witness_operation. (exists pfa_gap_backward_middle_sum_witness_operationindex. pfa_gap_backward_middle_sum_witness_operationindex + S (pfp_index_backward_middle_sum_witness_operation) = (pfaa_length_backward_middle_sum)) -> exists pfp_left_backward_middle_sum_witness_operation pfp_right_backward_middle_sum_witness_operation pfp_value_backward_middle_sum_witness_operation. ((((exists ff_h_pfp_backward_middle_sum_witness_operationleft. ff_h_pfp_backward_middle_sum_witness_operationleft + S (pfp_left_backward_middle_sum_witness_operation) = S ((S (pfp_index_backward_middle_sum_witness_operation)) * pfaa_left_c_backward_middle_sum)) /\\ exists ff_q_pfp_backward_middle_sum_witness_operationleft. pfaa_left_b_backward_middle_sum = ff_q_pfp_backward_middle_sum_witness_operationleft * S ((S (pfp_index_backward_middle_sum_witness_operation)) * pfaa_left_c_backward_middle_sum) + (pfp_left_backward_middle_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_middle_sum_witness_operationright. ff_h_pfp_backward_middle_sum_witness_operationright + S (pfp_right_backward_middle_sum_witness_operation) = S ((S (pfp_index_backward_middle_sum_witness_operation)) * pfaa_right_c_backward_middle_sum)) /\\ exists ff_q_pfp_backward_middle_sum_witness_operationright. pfaa_right_b_backward_middle_sum = ff_q_pfp_backward_middle_sum_witness_operationright * S ((S (pfp_index_backward_middle_sum_witness_operation)) * pfaa_right_c_backward_middle_sum) + (pfp_right_backward_middle_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_middle_sum_witness_operationtarget. ff_h_pfp_backward_middle_sum_witness_operationtarget + S (pfp_value_backward_middle_sum_witness_operation) = S ((S (pfp_index_backward_middle_sum_witness_operation)) * pfaa_sum_c_backward_middle_sum)) /\\ exists ff_q_pfp_backward_middle_sum_witness_operationtarget. pfaa_sum_b_backward_middle_sum = ff_q_pfp_backward_middle_sum_witness_operationtarget * S ((S (pfp_index_backward_middle_sum_witness_operation)) * pfaa_sum_c_backward_middle_sum) + (pfp_value_backward_middle_sum_witness_operation))) /\\ ((((exists pfa_gap_backward_middle_sum_witness_operationoperationleft. pfa_gap_backward_middle_sum_witness_operationoperationleft + S (pfp_left_backward_middle_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_backward_middle_sum_witness_operationoperationright. pfa_gap_backward_middle_sum_witness_operationoperationright + S (pfp_right_backward_middle_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_backward_middle_sum_witness_operationoperationresultbound. pfa_gap_backward_middle_sum_witness_operationoperationresultbound + S (pfp_value_backward_middle_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_backward_middle_sum_witness_operationoperationresultcongruence pfa_offset_right_backward_middle_sum_witness_operationoperationresultcongruence. ((pfp_left_backward_middle_sum_witness_operation) + (pfp_right_backward_middle_sum_witness_operation)) + (p) * pfa_offset_left_backward_middle_sum_witness_operationoperationresultcongruence = (pfp_value_backward_middle_sum_witness_operation) + (p) * pfa_offset_right_backward_middle_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_backward_middle_sum_witness_output pfrep_left_backward_middle_sum_witness_output pfrep_right_backward_middle_sum_witness_output. ((exists pfrep_position_backward_middle_sum_witness_outputfirst. ((pfrep_position_backward_middle_sum_witness_outputfirst+S (pfrep_power_backward_middle_sum_witness_output)=(pfaa_length_backward_middle_sum)) /\\ ((((exists ff_h_pfp_backward_middle_sum_witness_outputfirstentry. ff_h_pfp_backward_middle_sum_witness_outputfirstentry + S (pfrep_left_backward_middle_sum_witness_output) = S ((S (pfrep_position_backward_middle_sum_witness_outputfirst)) * pfaa_sum_c_backward_middle_sum)) /\\ exists ff_q_pfp_backward_middle_sum_witness_outputfirstentry. pfaa_sum_b_backward_middle_sum = ff_q_pfp_backward_middle_sum_witness_outputfirstentry * S ((S (pfrep_position_backward_middle_sum_witness_outputfirst)) * pfaa_sum_c_backward_middle_sum) + (pfrep_left_backward_middle_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_middle_sum_witness_outputfirstoutside. pfrep_gap_backward_middle_sum_witness_outputfirstoutside+(pfaa_length_backward_middle_sum)=(pfrep_power_backward_middle_sum_witness_output)) /\\ (((pfrep_left_backward_middle_sum_witness_output)=0))))) -> ((exists pfrep_position_backward_middle_sum_witness_outputsecond. ((pfrep_position_backward_middle_sum_witness_outputsecond+S (pfrep_power_backward_middle_sum_witness_output)=(Lx)) /\\ ((((exists ff_h_pfp_backward_middle_sum_witness_outputsecondentry. ff_h_pfp_backward_middle_sum_witness_outputsecondentry + S (pfrep_right_backward_middle_sum_witness_output) = S ((S (pfrep_position_backward_middle_sum_witness_outputsecond)) * xc)) /\\ exists ff_q_pfp_backward_middle_sum_witness_outputsecondentry. xb = ff_q_pfp_backward_middle_sum_witness_outputsecondentry * S ((S (pfrep_position_backward_middle_sum_witness_outputsecond)) * xc) + (pfrep_right_backward_middle_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_middle_sum_witness_outputsecondoutside. pfrep_gap_backward_middle_sum_witness_outputsecondoutside+(Lx)=(pfrep_power_backward_middle_sum_witness_output)) /\\ (((pfrep_right_backward_middle_sum_witness_output)=0))))) -> pfrep_left_backward_middle_sum_witness_output=pfrep_right_backward_middle_sum_witness_output))))))))))))", "specialize prime_field_polynomial_aligned_add_transport (p)", "specialize prime_field_polynomial_aligned_add_transport (hb)", "specialize prime_field_polynomial_aligned_add_transport (hc)", "specialize prime_field_polynomial_aligned_add_transport (Lh)", "specialize prime_field_polynomial_aligned_add_transport (db)", "specialize prime_field_polynomial_aligned_add_transport (dc)", "specialize prime_field_polynomial_aligned_add_transport (Ld)", "specialize prime_field_polynomial_aligned_add_transport (xb)", "specialize prime_field_polynomial_aligned_add_transport (xc)", "specialize prime_field_polynomial_aligned_add_transport (Lx)", "specialize prime_field_polynomial_aligned_add_transport (zb)", "specialize prime_field_polynomial_aligned_add_transport (zc)", "specialize prime_field_polynomial_aligned_add_transport (Lz)", "specialize prime_field_polynomial_aligned_add_transport (db)", "specialize prime_field_polynomial_aligned_add_transport (dc)", "specialize prime_field_polynomial_aligned_add_transport (Ld)", "specialize prime_field_polynomial_aligned_add_transport (xb)", "specialize prime_field_polynomial_aligned_add_transport (xc)", "specialize prime_field_polynomial_aligned_add_transport (Lx)", "apply prime_field_polynomial_aligned_add_transport", "exact hZbound", "exact hDbound", "exact hXbound", "exact heq", "specialize prime_field_polynomial_power_coefficient_functional (db)", "specialize prime_field_polynomial_power_coefficient_functional (dc)", "specialize prime_field_polynomial_power_coefficient_functional (Ld)", "apply prime_field_polynomial_power_coefficient_functional", "specialize prime_field_polynomial_power_coefficient_functional (xb)", "specialize prime_field_polynomial_power_coefficient_functional (xc)", "specialize prime_field_polynomial_power_coefficient_functional (Lx)", "apply prime_field_polynomial_power_coefficient_functional", "exact hleft", "have hnew : exists ob oc. ((forall fom_index_pfp_backward_new_sum_left_bounded. (exists fom_gap_pfp_backward_new_sum_left_bounded_index_bound. fom_gap_pfp_backward_new_sum_left_bounded_index_bound + S (fom_index_pfp_backward_new_sum_left_bounded) = Ly) -> exists fom_value_pfp_backward_new_sum_left_bounded. ((((exists fom_beta_height_pfp_backward_new_sum_left_bounded_entry. fom_beta_height_pfp_backward_new_sum_left_bounded_entry + S (fom_value_pfp_backward_new_sum_left_bounded) = S ((S (fom_index_pfp_backward_new_sum_left_bounded)) * yc)) /\\ exists fom_beta_quotient_pfp_backward_new_sum_left_bounded_entry. yb = fom_beta_quotient_pfp_backward_new_sum_left_bounded_entry * S ((S (fom_index_pfp_backward_new_sum_left_bounded)) * yc) + (fom_value_pfp_backward_new_sum_left_bounded))) /\\ (exists fom_gap_pfp_backward_new_sum_left_bounded_value_bound. fom_gap_pfp_backward_new_sum_left_bounded_value_bound + S (fom_value_pfp_backward_new_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_backward_new_sum_right_bounded. (exists fom_gap_pfp_backward_new_sum_right_bounded_index_bound. fom_gap_pfp_backward_new_sum_right_bounded_index_bound + S (fom_index_pfp_backward_new_sum_right_bounded) = Lx) -> exists fom_value_pfp_backward_new_sum_right_bounded. ((((exists fom_beta_height_pfp_backward_new_sum_right_bounded_entry. fom_beta_height_pfp_backward_new_sum_right_bounded_entry + S (fom_value_pfp_backward_new_sum_right_bounded) = S ((S (fom_index_pfp_backward_new_sum_right_bounded)) * xc)) /\\ exists fom_beta_quotient_pfp_backward_new_sum_right_bounded_entry. xb = fom_beta_quotient_pfp_backward_new_sum_right_bounded_entry * S ((S (fom_index_pfp_backward_new_sum_right_bounded)) * xc) + (fom_value_pfp_backward_new_sum_right_bounded))) /\\ (exists fom_gap_pfp_backward_new_sum_right_bounded_value_bound. fom_gap_pfp_backward_new_sum_right_bounded_value_bound + S (fom_value_pfp_backward_new_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_backward_new_sum_result_bounded. (exists fom_gap_pfp_backward_new_sum_result_bounded_index_bound. fom_gap_pfp_backward_new_sum_result_bounded_index_bound + S (fom_index_pfp_backward_new_sum_result_bounded) = (Ly)+(Lx)) -> exists fom_value_pfp_backward_new_sum_result_bounded. ((((exists fom_beta_height_pfp_backward_new_sum_result_bounded_entry. fom_beta_height_pfp_backward_new_sum_result_bounded_entry + S (fom_value_pfp_backward_new_sum_result_bounded) = S ((S (fom_index_pfp_backward_new_sum_result_bounded)) * oc)) /\\ exists fom_beta_quotient_pfp_backward_new_sum_result_bounded_entry. ob = fom_beta_quotient_pfp_backward_new_sum_result_bounded_entry * S ((S (fom_index_pfp_backward_new_sum_result_bounded)) * oc) + (fom_value_pfp_backward_new_sum_result_bounded))) /\\ (exists fom_gap_pfp_backward_new_sum_result_bounded_value_bound. fom_gap_pfp_backward_new_sum_result_bounded_value_bound + S (fom_value_pfp_backward_new_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_backward_new_sum pfaa_left_c_backward_new_sum pfaa_right_b_backward_new_sum pfaa_right_c_backward_new_sum pfaa_sum_b_backward_new_sum pfaa_sum_c_backward_new_sum pfaa_length_backward_new_sum. ((((forall pfrep_power_backward_new_sum_witness_common_left pfrep_left_backward_new_sum_witness_common_left pfrep_right_backward_new_sum_witness_common_left. ((exists pfrep_position_backward_new_sum_witness_common_leftfirst. ((pfrep_position_backward_new_sum_witness_common_leftfirst+S (pfrep_power_backward_new_sum_witness_common_left)=(Ly)) /\\ ((((exists ff_h_pfp_backward_new_sum_witness_common_leftfirstentry. ff_h_pfp_backward_new_sum_witness_common_leftfirstentry + S (pfrep_left_backward_new_sum_witness_common_left) = S ((S (pfrep_position_backward_new_sum_witness_common_leftfirst)) * yc)) /\\ exists ff_q_pfp_backward_new_sum_witness_common_leftfirstentry. yb = ff_q_pfp_backward_new_sum_witness_common_leftfirstentry * S ((S (pfrep_position_backward_new_sum_witness_common_leftfirst)) * yc) + (pfrep_left_backward_new_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_new_sum_witness_common_leftfirstoutside. pfrep_gap_backward_new_sum_witness_common_leftfirstoutside+(Ly)=(pfrep_power_backward_new_sum_witness_common_left)) /\\ (((pfrep_left_backward_new_sum_witness_common_left)=0))))) -> ((exists pfrep_position_backward_new_sum_witness_common_leftsecond. ((pfrep_position_backward_new_sum_witness_common_leftsecond+S (pfrep_power_backward_new_sum_witness_common_left)=(pfaa_length_backward_new_sum)) /\\ ((((exists ff_h_pfp_backward_new_sum_witness_common_leftsecondentry. ff_h_pfp_backward_new_sum_witness_common_leftsecondentry + S (pfrep_right_backward_new_sum_witness_common_left) = S ((S (pfrep_position_backward_new_sum_witness_common_leftsecond)) * pfaa_left_c_backward_new_sum)) /\\ exists ff_q_pfp_backward_new_sum_witness_common_leftsecondentry. pfaa_left_b_backward_new_sum = ff_q_pfp_backward_new_sum_witness_common_leftsecondentry * S ((S (pfrep_position_backward_new_sum_witness_common_leftsecond)) * pfaa_left_c_backward_new_sum) + (pfrep_right_backward_new_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_new_sum_witness_common_leftsecondoutside. pfrep_gap_backward_new_sum_witness_common_leftsecondoutside+(pfaa_length_backward_new_sum)=(pfrep_power_backward_new_sum_witness_common_left)) /\\ (((pfrep_right_backward_new_sum_witness_common_left)=0))))) -> pfrep_left_backward_new_sum_witness_common_left=pfrep_right_backward_new_sum_witness_common_left) /\\ ((forall pfrep_power_backward_new_sum_witness_common_right pfrep_left_backward_new_sum_witness_common_right pfrep_right_backward_new_sum_witness_common_right. ((exists pfrep_position_backward_new_sum_witness_common_rightfirst. ((pfrep_position_backward_new_sum_witness_common_rightfirst+S (pfrep_power_backward_new_sum_witness_common_right)=(Lx)) /\\ ((((exists ff_h_pfp_backward_new_sum_witness_common_rightfirstentry. ff_h_pfp_backward_new_sum_witness_common_rightfirstentry + S (pfrep_left_backward_new_sum_witness_common_right) = S ((S (pfrep_position_backward_new_sum_witness_common_rightfirst)) * xc)) /\\ exists ff_q_pfp_backward_new_sum_witness_common_rightfirstentry. xb = ff_q_pfp_backward_new_sum_witness_common_rightfirstentry * S ((S (pfrep_position_backward_new_sum_witness_common_rightfirst)) * xc) + (pfrep_left_backward_new_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_new_sum_witness_common_rightfirstoutside. pfrep_gap_backward_new_sum_witness_common_rightfirstoutside+(Lx)=(pfrep_power_backward_new_sum_witness_common_right)) /\\ (((pfrep_left_backward_new_sum_witness_common_right)=0))))) -> ((exists pfrep_position_backward_new_sum_witness_common_rightsecond. ((pfrep_position_backward_new_sum_witness_common_rightsecond+S (pfrep_power_backward_new_sum_witness_common_right)=(pfaa_length_backward_new_sum)) /\\ ((((exists ff_h_pfp_backward_new_sum_witness_common_rightsecondentry. ff_h_pfp_backward_new_sum_witness_common_rightsecondentry + S (pfrep_right_backward_new_sum_witness_common_right) = S ((S (pfrep_position_backward_new_sum_witness_common_rightsecond)) * pfaa_right_c_backward_new_sum)) /\\ exists ff_q_pfp_backward_new_sum_witness_common_rightsecondentry. pfaa_right_b_backward_new_sum = ff_q_pfp_backward_new_sum_witness_common_rightsecondentry * S ((S (pfrep_position_backward_new_sum_witness_common_rightsecond)) * pfaa_right_c_backward_new_sum) + (pfrep_right_backward_new_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_new_sum_witness_common_rightsecondoutside. pfrep_gap_backward_new_sum_witness_common_rightsecondoutside+(pfaa_length_backward_new_sum)=(pfrep_power_backward_new_sum_witness_common_right)) /\\ (((pfrep_right_backward_new_sum_witness_common_right)=0))))) -> pfrep_left_backward_new_sum_witness_common_right=pfrep_right_backward_new_sum_witness_common_right)))) /\\ (((forall pfp_index_backward_new_sum_witness_operation. (exists pfa_gap_backward_new_sum_witness_operationindex. pfa_gap_backward_new_sum_witness_operationindex + S (pfp_index_backward_new_sum_witness_operation) = (pfaa_length_backward_new_sum)) -> exists pfp_left_backward_new_sum_witness_operation pfp_right_backward_new_sum_witness_operation pfp_value_backward_new_sum_witness_operation. ((((exists ff_h_pfp_backward_new_sum_witness_operationleft. ff_h_pfp_backward_new_sum_witness_operationleft + S (pfp_left_backward_new_sum_witness_operation) = S ((S (pfp_index_backward_new_sum_witness_operation)) * pfaa_left_c_backward_new_sum)) /\\ exists ff_q_pfp_backward_new_sum_witness_operationleft. pfaa_left_b_backward_new_sum = ff_q_pfp_backward_new_sum_witness_operationleft * S ((S (pfp_index_backward_new_sum_witness_operation)) * pfaa_left_c_backward_new_sum) + (pfp_left_backward_new_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_new_sum_witness_operationright. ff_h_pfp_backward_new_sum_witness_operationright + S (pfp_right_backward_new_sum_witness_operation) = S ((S (pfp_index_backward_new_sum_witness_operation)) * pfaa_right_c_backward_new_sum)) /\\ exists ff_q_pfp_backward_new_sum_witness_operationright. pfaa_right_b_backward_new_sum = ff_q_pfp_backward_new_sum_witness_operationright * S ((S (pfp_index_backward_new_sum_witness_operation)) * pfaa_right_c_backward_new_sum) + (pfp_right_backward_new_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_new_sum_witness_operationtarget. ff_h_pfp_backward_new_sum_witness_operationtarget + S (pfp_value_backward_new_sum_witness_operation) = S ((S (pfp_index_backward_new_sum_witness_operation)) * pfaa_sum_c_backward_new_sum)) /\\ exists ff_q_pfp_backward_new_sum_witness_operationtarget. pfaa_sum_b_backward_new_sum = ff_q_pfp_backward_new_sum_witness_operationtarget * S ((S (pfp_index_backward_new_sum_witness_operation)) * pfaa_sum_c_backward_new_sum) + (pfp_value_backward_new_sum_witness_operation))) /\\ ((((exists pfa_gap_backward_new_sum_witness_operationoperationleft. pfa_gap_backward_new_sum_witness_operationoperationleft + S (pfp_left_backward_new_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_backward_new_sum_witness_operationoperationright. pfa_gap_backward_new_sum_witness_operationoperationright + S (pfp_right_backward_new_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_backward_new_sum_witness_operationoperationresultbound. pfa_gap_backward_new_sum_witness_operationoperationresultbound + S (pfp_value_backward_new_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_backward_new_sum_witness_operationoperationresultcongruence pfa_offset_right_backward_new_sum_witness_operationoperationresultcongruence. ((pfp_left_backward_new_sum_witness_operation) + (pfp_right_backward_new_sum_witness_operation)) + (p) * pfa_offset_left_backward_new_sum_witness_operationoperationresultcongruence = (pfp_value_backward_new_sum_witness_operation) + (p) * pfa_offset_right_backward_new_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_backward_new_sum_witness_output pfrep_left_backward_new_sum_witness_output pfrep_right_backward_new_sum_witness_output. ((exists pfrep_position_backward_new_sum_witness_outputfirst. ((pfrep_position_backward_new_sum_witness_outputfirst+S (pfrep_power_backward_new_sum_witness_output)=(pfaa_length_backward_new_sum)) /\\ ((((exists ff_h_pfp_backward_new_sum_witness_outputfirstentry. ff_h_pfp_backward_new_sum_witness_outputfirstentry + S (pfrep_left_backward_new_sum_witness_output) = S ((S (pfrep_position_backward_new_sum_witness_outputfirst)) * pfaa_sum_c_backward_new_sum)) /\\ exists ff_q_pfp_backward_new_sum_witness_outputfirstentry. pfaa_sum_b_backward_new_sum = ff_q_pfp_backward_new_sum_witness_outputfirstentry * S ((S (pfrep_position_backward_new_sum_witness_outputfirst)) * pfaa_sum_c_backward_new_sum) + (pfrep_left_backward_new_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_new_sum_witness_outputfirstoutside. pfrep_gap_backward_new_sum_witness_outputfirstoutside+(pfaa_length_backward_new_sum)=(pfrep_power_backward_new_sum_witness_output)) /\\ (((pfrep_left_backward_new_sum_witness_output)=0))))) -> ((exists pfrep_position_backward_new_sum_witness_outputsecond. ((pfrep_position_backward_new_sum_witness_outputsecond+S (pfrep_power_backward_new_sum_witness_output)=((Ly)+(Lx))) /\\ ((((exists ff_h_pfp_backward_new_sum_witness_outputsecondentry. ff_h_pfp_backward_new_sum_witness_outputsecondentry + S (pfrep_right_backward_new_sum_witness_output) = S ((S (pfrep_position_backward_new_sum_witness_outputsecond)) * oc)) /\\ exists ff_q_pfp_backward_new_sum_witness_outputsecondentry. ob = ff_q_pfp_backward_new_sum_witness_outputsecondentry * S ((S (pfrep_position_backward_new_sum_witness_outputsecond)) * oc) + (pfrep_right_backward_new_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_new_sum_witness_outputsecondoutside. pfrep_gap_backward_new_sum_witness_outputsecondoutside+((Ly)+(Lx))=(pfrep_power_backward_new_sum_witness_output)) /\\ (((pfrep_right_backward_new_sum_witness_output)=0))))) -> pfrep_left_backward_new_sum_witness_output=pfrep_right_backward_new_sum_witness_output))))))))))))", "specialize prime_field_polynomial_aligned_add_exists (p)", "specialize prime_field_polynomial_aligned_add_exists (yb)", "specialize prime_field_polynomial_aligned_add_exists (yc)", "specialize prime_field_polynomial_aligned_add_exists (Ly)", "specialize prime_field_polynomial_aligned_add_exists (xb)", "specialize prime_field_polynomial_aligned_add_exists (xc)", "specialize prime_field_polynomial_aligned_add_exists (Lx)", "apply prime_field_polynomial_aligned_add_exists", "exact hp", "exact hYbound", "exact hXbound", "cases hnew", "cases hnew_witness", "have hresult : forall pfrep_power_backward_equivalent_result pfrep_left_backward_equivalent_result pfrep_right_backward_equivalent_result. ((exists pfrep_position_backward_equivalent_resultfirst. ((pfrep_position_backward_equivalent_resultfirst+S (pfrep_power_backward_equivalent_result)=(Lg)) /\\ ((((exists ff_h_pfp_backward_equivalent_resultfirstentry. ff_h_pfp_backward_equivalent_resultfirstentry + S (pfrep_left_backward_equivalent_result) = S ((S (pfrep_position_backward_equivalent_resultfirst)) * gc)) /\\ exists ff_q_pfp_backward_equivalent_resultfirstentry. gb = ff_q_pfp_backward_equivalent_resultfirstentry * S ((S (pfrep_position_backward_equivalent_resultfirst)) * gc) + (pfrep_left_backward_equivalent_result)))))) \\/ (((exists pfrep_gap_backward_equivalent_resultfirstoutside. pfrep_gap_backward_equivalent_resultfirstoutside+(Lg)=(pfrep_power_backward_equivalent_result)) /\\ (((pfrep_left_backward_equivalent_result)=0))))) -> ((exists pfrep_position_backward_equivalent_resultsecond. ((pfrep_position_backward_equivalent_resultsecond+S (pfrep_power_backward_equivalent_result)=((Ly)+(Lx))) /\\ ((((exists ff_h_pfp_backward_equivalent_resultsecondentry. ff_h_pfp_backward_equivalent_resultsecondentry + S (pfrep_right_backward_equivalent_result) = S ((S (pfrep_position_backward_equivalent_resultsecond)) * x1)) /\\ exists ff_q_pfp_backward_equivalent_resultsecondentry. x = ff_q_pfp_backward_equivalent_resultsecondentry * S ((S (pfrep_position_backward_equivalent_resultsecond)) * x1) + (pfrep_right_backward_equivalent_result)))))) \\/ (((exists pfrep_gap_backward_equivalent_resultsecondoutside. pfrep_gap_backward_equivalent_resultsecondoutside+((Ly)+(Lx))=(pfrep_power_backward_equivalent_result)) /\\ (((pfrep_right_backward_equivalent_result)=0))))) -> pfrep_left_backward_equivalent_result=pfrep_right_backward_equivalent_result", "specialize prime_field_polynomial_aligned_add_associative (p)", "specialize prime_field_polynomial_aligned_add_associative (yb)", "specialize prime_field_polynomial_aligned_add_associative (yc)", "specialize prime_field_polynomial_aligned_add_associative (Ly)", "specialize prime_field_polynomial_aligned_add_associative (zb)", "specialize prime_field_polynomial_aligned_add_associative (zc)", "specialize prime_field_polynomial_aligned_add_associative (Lz)", "specialize prime_field_polynomial_aligned_add_associative (db)", "specialize prime_field_polynomial_aligned_add_associative (dc)", "specialize prime_field_polynomial_aligned_add_associative (Ld)", "specialize prime_field_polynomial_aligned_add_associative (cb)", "specialize prime_field_polynomial_aligned_add_associative (cc)", "specialize prime_field_polynomial_aligned_add_associative (Lc)", "specialize prime_field_polynomial_aligned_add_associative (xb)", "specialize prime_field_polynomial_aligned_add_associative (xc)", "specialize prime_field_polynomial_aligned_add_associative (Lx)", "specialize prime_field_polynomial_aligned_add_associative (gb)", "specialize prime_field_polynomial_aligned_add_associative (gc)", "specialize prime_field_polynomial_aligned_add_associative (Lg)", "specialize prime_field_polynomial_aligned_add_associative (x)", "specialize prime_field_polynomial_aligned_add_associative (x1)", "specialize prime_field_polynomial_aligned_add_associative ((Ly)+(Lx))", "apply prime_field_polynomial_aligned_add_associative", "exact hp", "exact hright", "exact hold", "exact hmiddle", "exact hnew_witness_witness", "specialize prime_field_polynomial_aligned_add_commutative (p)", "specialize prime_field_polynomial_aligned_add_commutative (yb)", "specialize prime_field_polynomial_aligned_add_commutative (yc)", "specialize prime_field_polynomial_aligned_add_commutative (Ly)", "specialize prime_field_polynomial_aligned_add_commutative (xb)", "specialize prime_field_polynomial_aligned_add_commutative (xc)", "specialize prime_field_polynomial_aligned_add_commutative (Lx)", "specialize prime_field_polynomial_aligned_add_commutative (gb)", "specialize prime_field_polynomial_aligned_add_commutative (gc)", "specialize prime_field_polynomial_aligned_add_commutative (Lg)", "apply prime_field_polynomial_aligned_add_commutative", "specialize prime_field_polynomial_aligned_add_transport (p)", "specialize prime_field_polynomial_aligned_add_transport (yb)", "specialize prime_field_polynomial_aligned_add_transport (yc)", "specialize prime_field_polynomial_aligned_add_transport (Ly)", "specialize prime_field_polynomial_aligned_add_transport (xb)", "specialize prime_field_polynomial_aligned_add_transport (xc)", "specialize prime_field_polynomial_aligned_add_transport (Lx)", "specialize prime_field_polynomial_aligned_add_transport (x)", "specialize prime_field_polynomial_aligned_add_transport (x1)", "specialize prime_field_polynomial_aligned_add_transport ((Ly)+(Lx))", "specialize prime_field_polynomial_aligned_add_transport (yb)", "specialize prime_field_polynomial_aligned_add_transport (yc)", "specialize prime_field_polynomial_aligned_add_transport (Ly)", "specialize prime_field_polynomial_aligned_add_transport (xb)", "specialize prime_field_polynomial_aligned_add_transport (xc)", "specialize prime_field_polynomial_aligned_add_transport (Lx)", "specialize prime_field_polynomial_aligned_add_transport (gb)", "specialize prime_field_polynomial_aligned_add_transport (gc)", "specialize prime_field_polynomial_aligned_add_transport (Lg)", "apply prime_field_polynomial_aligned_add_transport", "exact hYbound", "exact hXbound", "exact hGbound_right_right", "specialize prime_field_polynomial_power_coefficient_functional (yb)", "specialize prime_field_polynomial_power_coefficient_functional (yc)", "specialize prime_field_polynomial_power_coefficient_functional (Ly)", "apply prime_field_polynomial_power_coefficient_functional", "specialize prime_field_polynomial_power_coefficient_functional (xb)", "specialize prime_field_polynomial_power_coefficient_functional (xc)", "specialize prime_field_polynomial_power_coefficient_functional (Lx)", "apply prime_field_polynomial_power_coefficient_functional", "specialize prime_field_polynomial_equivalent_symmetric (gb)", "specialize prime_field_polynomial_equivalent_symmetric (gc)", "specialize prime_field_polynomial_equivalent_symmetric (Lg)", "specialize prime_field_polynomial_equivalent_symmetric (x)", "specialize prime_field_polynomial_equivalent_symmetric (x1)", "specialize prime_field_polynomial_equivalent_symmetric ((Ly)+(Lx))", "apply prime_field_polynomial_equivalent_symmetric", "exact hresult", "exact hnew_witness_witness" ], "script_sha256": "0df8ba3af0a6eacb3bfcc2b5eb329d5024bb1ed1306688f6aa2941688eb15f3d", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_bezout_backward_candidate.py", "sha256": "c3903482000c957ac77f84a43a85d135e4caa19e4484328035f91b82cbf3a702" }, "statement": "forall p ab ac La bb bc Lb rb rc Lr qb qc Lq pb pc Lp ub uc Lu vb vc Lv gb gc Lg cb cc Lc db dc Ld wb wc Lw tb tc Lt xb xc Lx yb yc Ly zb zc Lz hb hc Lh. (~((p) = 1) /\\ forall pfa_factor_left_backward_prime pfa_factor_right_backward_prime. (p) = pfa_factor_left_backward_prime * pfa_factor_right_backward_prime -> pfa_factor_left_backward_prime = 1 \\/ pfa_factor_right_backward_prime = 1) -> (((forall fom_index_pfp_backward_division_productleft. (exists fom_gap_pfp_backward_division_productleft_index_bound. fom_gap_pfp_backward_division_productleft_index_bound + S (fom_index_pfp_backward_division_productleft) = Lq) -> exists fom_value_pfp_backward_division_productleft. ((((exists fom_beta_height_pfp_backward_division_productleft_entry. fom_beta_height_pfp_backward_division_productleft_entry + S (fom_value_pfp_backward_division_productleft) = S ((S (fom_index_pfp_backward_division_productleft)) * qc)) /\\ exists fom_beta_quotient_pfp_backward_division_productleft_entry. qb = fom_beta_quotient_pfp_backward_division_productleft_entry * S ((S (fom_index_pfp_backward_division_productleft)) * qc) + (fom_value_pfp_backward_division_productleft))) /\\ (exists fom_gap_pfp_backward_division_productleft_value_bound. fom_gap_pfp_backward_division_productleft_value_bound + S (fom_value_pfp_backward_division_productleft) = p))) /\\ (((forall fom_index_pfp_backward_division_productright. (exists fom_gap_pfp_backward_division_productright_index_bound. fom_gap_pfp_backward_division_productright_index_bound + S (fom_index_pfp_backward_division_productright) = Lb) -> exists fom_value_pfp_backward_division_productright. ((((exists fom_beta_height_pfp_backward_division_productright_entry. fom_beta_height_pfp_backward_division_productright_entry + S (fom_value_pfp_backward_division_productright) = S ((S (fom_index_pfp_backward_division_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_backward_division_productright_entry. bb = fom_beta_quotient_pfp_backward_division_productright_entry * S ((S (fom_index_pfp_backward_division_productright)) * bc) + (fom_value_pfp_backward_division_productright))) /\\ (exists fom_gap_pfp_backward_division_productright_value_bound. fom_gap_pfp_backward_division_productright_value_bound + S (fom_value_pfp_backward_division_productright) = p))) /\\ (((((((Lq)=0 \\/ (Lb)=0) /\\ (((Lp)=0)))) \\/ (((~((Lq)=0)) /\\ (((~((Lb)=0)) /\\ (((Lq)+(Lb)=S (Lp)))))))) /\\ ((forall pfc_index_backward_division_productcoefficients. (exists pfa_gap_backward_division_productcoefficientsbound. pfa_gap_backward_division_productcoefficientsbound + S (pfc_index_backward_division_productcoefficients) = (Lp)) -> exists pfc_value_backward_division_productcoefficients. ((((exists ff_h_pfp_backward_division_productcoefficientsentry. ff_h_pfp_backward_division_productcoefficientsentry + S (pfc_value_backward_division_productcoefficients) = S ((S (pfc_index_backward_division_productcoefficients)) * pc)) /\\ exists ff_q_pfp_backward_division_productcoefficientsentry. pb = ff_q_pfp_backward_division_productcoefficientsentry * S ((S (pfc_index_backward_division_productcoefficients)) * pc) + (pfc_value_backward_division_productcoefficients))) /\\ ((exists pfc_terms_code_backward_division_productcoefficientscoefficient pfc_terms_scale_backward_division_productcoefficientscoefficient pfc_natural_sum_backward_division_productcoefficientscoefficient. ((forall pfc_index_backward_division_productcoefficientscoefficientdiagonal. (exists pfa_gap_backward_division_productcoefficientscoefficientdiagonalbound. pfa_gap_backward_division_productcoefficientscoefficientdiagonalbound + S (pfc_index_backward_division_productcoefficientscoefficientdiagonal) = (S (pfc_index_backward_division_productcoefficients))) -> exists pfc_value_backward_division_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_division_productcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_division_productcoefficientscoefficientdiagonalentry + S (pfc_value_backward_division_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_division_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_division_productcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_division_productcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_division_productcoefficientscoefficient = ff_q_pfp_backward_division_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_division_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_division_productcoefficientscoefficient) + (pfc_value_backward_division_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_division_productcoefficientscoefficientdiagonalterm pfc_left_backward_division_productcoefficientscoefficientdiagonalterm pfc_right_backward_division_productcoefficientscoefficientdiagonalterm. (((pfc_index_backward_division_productcoefficientscoefficientdiagonal)+pfc_complement_backward_division_productcoefficientscoefficientdiagonalterm=(pfc_index_backward_division_productcoefficients)) /\\ ((((((exists pfa_gap_backward_division_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_division_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_division_productcoefficientscoefficientdiagonal) = (Lq)) /\\ ((((exists ff_h_pfp_backward_division_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_division_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_division_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_division_productcoefficientscoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_backward_division_productcoefficientscoefficientdiagonaltermleftentry. qb = ff_q_pfp_backward_division_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_division_productcoefficientscoefficientdiagonal)) * qc) + (pfc_left_backward_division_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_division_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_division_productcoefficientscoefficientdiagonaltermleftoutside+(Lq)=(pfc_index_backward_division_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_division_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_division_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_division_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_division_productcoefficientscoefficientdiagonalterm) = (Lb)) /\\ ((((exists ff_h_pfp_backward_division_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_division_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_division_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_division_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_backward_division_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_backward_division_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_division_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_backward_division_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_division_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_division_productcoefficientscoefficientdiagonaltermrightoutside+(Lb)=(pfc_complement_backward_division_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_division_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_division_productcoefficientscoefficientdiagonal)=pfc_left_backward_division_productcoefficientscoefficientdiagonalterm*pfc_right_backward_division_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_division_productcoefficientscoefficientsum fs_v_pfc_backward_division_productcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_start. fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_division_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_start. fs_u_pfc_backward_division_productcoefficientscoefficientsum = fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_division_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_division_productcoefficientscoefficient) = S ((S (S (pfc_index_backward_division_productcoefficients))) * fs_v_pfc_backward_division_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_division_productcoefficientscoefficientsum = fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_division_productcoefficients))) * fs_v_pfc_backward_division_productcoefficientscoefficientsum) + (pfc_natural_sum_backward_division_productcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_division_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_division_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps = S (pfc_index_backward_division_productcoefficients)) -> exists fs_a_pfc_backward_division_productcoefficientscoefficientsum_body_steps fs_r_pfc_backward_division_productcoefficientscoefficientsum_body_steps fs_s_pfc_backward_division_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_division_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_division_productcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_division_productcoefficientscoefficient = fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_division_productcoefficientscoefficient) + (fs_a_pfc_backward_division_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_division_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_division_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_division_productcoefficientscoefficientsum = fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_division_productcoefficientscoefficientsum) + (fs_r_pfc_backward_division_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_division_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_division_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_division_productcoefficientscoefficientsum = fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_division_productcoefficientscoefficientsum) + (fs_s_pfc_backward_division_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_division_productcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_division_productcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_division_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_division_productcoefficientscoefficientresiduebound. pfa_gap_backward_division_productcoefficientscoefficientresiduebound + S (pfc_value_backward_division_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_division_productcoefficientscoefficientresiduecongruence pfa_offset_right_backward_division_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_division_productcoefficientscoefficient) + (p) * pfa_offset_left_backward_division_productcoefficientscoefficientresiduecongruence = (pfc_value_backward_division_productcoefficients) + (p) * pfa_offset_right_backward_division_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_division_sum_left_bounded. (exists fom_gap_pfp_backward_division_sum_left_bounded_index_bound. fom_gap_pfp_backward_division_sum_left_bounded_index_bound + S (fom_index_pfp_backward_division_sum_left_bounded) = Lp) -> exists fom_value_pfp_backward_division_sum_left_bounded. ((((exists fom_beta_height_pfp_backward_division_sum_left_bounded_entry. fom_beta_height_pfp_backward_division_sum_left_bounded_entry + S (fom_value_pfp_backward_division_sum_left_bounded) = S ((S (fom_index_pfp_backward_division_sum_left_bounded)) * pc)) /\\ exists fom_beta_quotient_pfp_backward_division_sum_left_bounded_entry. pb = fom_beta_quotient_pfp_backward_division_sum_left_bounded_entry * S ((S (fom_index_pfp_backward_division_sum_left_bounded)) * pc) + (fom_value_pfp_backward_division_sum_left_bounded))) /\\ (exists fom_gap_pfp_backward_division_sum_left_bounded_value_bound. fom_gap_pfp_backward_division_sum_left_bounded_value_bound + S (fom_value_pfp_backward_division_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_backward_division_sum_right_bounded. (exists fom_gap_pfp_backward_division_sum_right_bounded_index_bound. fom_gap_pfp_backward_division_sum_right_bounded_index_bound + S (fom_index_pfp_backward_division_sum_right_bounded) = Lr) -> exists fom_value_pfp_backward_division_sum_right_bounded. ((((exists fom_beta_height_pfp_backward_division_sum_right_bounded_entry. fom_beta_height_pfp_backward_division_sum_right_bounded_entry + S (fom_value_pfp_backward_division_sum_right_bounded) = S ((S (fom_index_pfp_backward_division_sum_right_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_backward_division_sum_right_bounded_entry. rb = fom_beta_quotient_pfp_backward_division_sum_right_bounded_entry * S ((S (fom_index_pfp_backward_division_sum_right_bounded)) * rc) + (fom_value_pfp_backward_division_sum_right_bounded))) /\\ (exists fom_gap_pfp_backward_division_sum_right_bounded_value_bound. fom_gap_pfp_backward_division_sum_right_bounded_value_bound + S (fom_value_pfp_backward_division_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_backward_division_sum_result_bounded. (exists fom_gap_pfp_backward_division_sum_result_bounded_index_bound. fom_gap_pfp_backward_division_sum_result_bounded_index_bound + S (fom_index_pfp_backward_division_sum_result_bounded) = La) -> exists fom_value_pfp_backward_division_sum_result_bounded. ((((exists fom_beta_height_pfp_backward_division_sum_result_bounded_entry. fom_beta_height_pfp_backward_division_sum_result_bounded_entry + S (fom_value_pfp_backward_division_sum_result_bounded) = S ((S (fom_index_pfp_backward_division_sum_result_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_backward_division_sum_result_bounded_entry. ab = fom_beta_quotient_pfp_backward_division_sum_result_bounded_entry * S ((S (fom_index_pfp_backward_division_sum_result_bounded)) * ac) + (fom_value_pfp_backward_division_sum_result_bounded))) /\\ (exists fom_gap_pfp_backward_division_sum_result_bounded_value_bound. fom_gap_pfp_backward_division_sum_result_bounded_value_bound + S (fom_value_pfp_backward_division_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_backward_division_sum pfaa_left_c_backward_division_sum pfaa_right_b_backward_division_sum pfaa_right_c_backward_division_sum pfaa_sum_b_backward_division_sum pfaa_sum_c_backward_division_sum pfaa_length_backward_division_sum. ((((forall pfrep_power_backward_division_sum_witness_common_left pfrep_left_backward_division_sum_witness_common_left pfrep_right_backward_division_sum_witness_common_left. ((exists pfrep_position_backward_division_sum_witness_common_leftfirst. ((pfrep_position_backward_division_sum_witness_common_leftfirst+S (pfrep_power_backward_division_sum_witness_common_left)=(Lp)) /\\ ((((exists ff_h_pfp_backward_division_sum_witness_common_leftfirstentry. ff_h_pfp_backward_division_sum_witness_common_leftfirstentry + S (pfrep_left_backward_division_sum_witness_common_left) = S ((S (pfrep_position_backward_division_sum_witness_common_leftfirst)) * pc)) /\\ exists ff_q_pfp_backward_division_sum_witness_common_leftfirstentry. pb = ff_q_pfp_backward_division_sum_witness_common_leftfirstentry * S ((S (pfrep_position_backward_division_sum_witness_common_leftfirst)) * pc) + (pfrep_left_backward_division_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_division_sum_witness_common_leftfirstoutside. pfrep_gap_backward_division_sum_witness_common_leftfirstoutside+(Lp)=(pfrep_power_backward_division_sum_witness_common_left)) /\\ (((pfrep_left_backward_division_sum_witness_common_left)=0))))) -> ((exists pfrep_position_backward_division_sum_witness_common_leftsecond. ((pfrep_position_backward_division_sum_witness_common_leftsecond+S (pfrep_power_backward_division_sum_witness_common_left)=(pfaa_length_backward_division_sum)) /\\ ((((exists ff_h_pfp_backward_division_sum_witness_common_leftsecondentry. ff_h_pfp_backward_division_sum_witness_common_leftsecondentry + S (pfrep_right_backward_division_sum_witness_common_left) = S ((S (pfrep_position_backward_division_sum_witness_common_leftsecond)) * pfaa_left_c_backward_division_sum)) /\\ exists ff_q_pfp_backward_division_sum_witness_common_leftsecondentry. pfaa_left_b_backward_division_sum = ff_q_pfp_backward_division_sum_witness_common_leftsecondentry * S ((S (pfrep_position_backward_division_sum_witness_common_leftsecond)) * pfaa_left_c_backward_division_sum) + (pfrep_right_backward_division_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_division_sum_witness_common_leftsecondoutside. pfrep_gap_backward_division_sum_witness_common_leftsecondoutside+(pfaa_length_backward_division_sum)=(pfrep_power_backward_division_sum_witness_common_left)) /\\ (((pfrep_right_backward_division_sum_witness_common_left)=0))))) -> pfrep_left_backward_division_sum_witness_common_left=pfrep_right_backward_division_sum_witness_common_left) /\\ ((forall pfrep_power_backward_division_sum_witness_common_right pfrep_left_backward_division_sum_witness_common_right pfrep_right_backward_division_sum_witness_common_right. ((exists pfrep_position_backward_division_sum_witness_common_rightfirst. ((pfrep_position_backward_division_sum_witness_common_rightfirst+S (pfrep_power_backward_division_sum_witness_common_right)=(Lr)) /\\ ((((exists ff_h_pfp_backward_division_sum_witness_common_rightfirstentry. ff_h_pfp_backward_division_sum_witness_common_rightfirstentry + S (pfrep_left_backward_division_sum_witness_common_right) = S ((S (pfrep_position_backward_division_sum_witness_common_rightfirst)) * rc)) /\\ exists ff_q_pfp_backward_division_sum_witness_common_rightfirstentry. rb = ff_q_pfp_backward_division_sum_witness_common_rightfirstentry * S ((S (pfrep_position_backward_division_sum_witness_common_rightfirst)) * rc) + (pfrep_left_backward_division_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_division_sum_witness_common_rightfirstoutside. pfrep_gap_backward_division_sum_witness_common_rightfirstoutside+(Lr)=(pfrep_power_backward_division_sum_witness_common_right)) /\\ (((pfrep_left_backward_division_sum_witness_common_right)=0))))) -> ((exists pfrep_position_backward_division_sum_witness_common_rightsecond. ((pfrep_position_backward_division_sum_witness_common_rightsecond+S (pfrep_power_backward_division_sum_witness_common_right)=(pfaa_length_backward_division_sum)) /\\ ((((exists ff_h_pfp_backward_division_sum_witness_common_rightsecondentry. ff_h_pfp_backward_division_sum_witness_common_rightsecondentry + S (pfrep_right_backward_division_sum_witness_common_right) = S ((S (pfrep_position_backward_division_sum_witness_common_rightsecond)) * pfaa_right_c_backward_division_sum)) /\\ exists ff_q_pfp_backward_division_sum_witness_common_rightsecondentry. pfaa_right_b_backward_division_sum = ff_q_pfp_backward_division_sum_witness_common_rightsecondentry * S ((S (pfrep_position_backward_division_sum_witness_common_rightsecond)) * pfaa_right_c_backward_division_sum) + (pfrep_right_backward_division_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_division_sum_witness_common_rightsecondoutside. pfrep_gap_backward_division_sum_witness_common_rightsecondoutside+(pfaa_length_backward_division_sum)=(pfrep_power_backward_division_sum_witness_common_right)) /\\ (((pfrep_right_backward_division_sum_witness_common_right)=0))))) -> pfrep_left_backward_division_sum_witness_common_right=pfrep_right_backward_division_sum_witness_common_right)))) /\\ (((forall pfp_index_backward_division_sum_witness_operation. (exists pfa_gap_backward_division_sum_witness_operationindex. pfa_gap_backward_division_sum_witness_operationindex + S (pfp_index_backward_division_sum_witness_operation) = (pfaa_length_backward_division_sum)) -> exists pfp_left_backward_division_sum_witness_operation pfp_right_backward_division_sum_witness_operation pfp_value_backward_division_sum_witness_operation. ((((exists ff_h_pfp_backward_division_sum_witness_operationleft. ff_h_pfp_backward_division_sum_witness_operationleft + S (pfp_left_backward_division_sum_witness_operation) = S ((S (pfp_index_backward_division_sum_witness_operation)) * pfaa_left_c_backward_division_sum)) /\\ exists ff_q_pfp_backward_division_sum_witness_operationleft. pfaa_left_b_backward_division_sum = ff_q_pfp_backward_division_sum_witness_operationleft * S ((S (pfp_index_backward_division_sum_witness_operation)) * pfaa_left_c_backward_division_sum) + (pfp_left_backward_division_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_division_sum_witness_operationright. ff_h_pfp_backward_division_sum_witness_operationright + S (pfp_right_backward_division_sum_witness_operation) = S ((S (pfp_index_backward_division_sum_witness_operation)) * pfaa_right_c_backward_division_sum)) /\\ exists ff_q_pfp_backward_division_sum_witness_operationright. pfaa_right_b_backward_division_sum = ff_q_pfp_backward_division_sum_witness_operationright * S ((S (pfp_index_backward_division_sum_witness_operation)) * pfaa_right_c_backward_division_sum) + (pfp_right_backward_division_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_division_sum_witness_operationtarget. ff_h_pfp_backward_division_sum_witness_operationtarget + S (pfp_value_backward_division_sum_witness_operation) = S ((S (pfp_index_backward_division_sum_witness_operation)) * pfaa_sum_c_backward_division_sum)) /\\ exists ff_q_pfp_backward_division_sum_witness_operationtarget. pfaa_sum_b_backward_division_sum = ff_q_pfp_backward_division_sum_witness_operationtarget * S ((S (pfp_index_backward_division_sum_witness_operation)) * pfaa_sum_c_backward_division_sum) + (pfp_value_backward_division_sum_witness_operation))) /\\ ((((exists pfa_gap_backward_division_sum_witness_operationoperationleft. pfa_gap_backward_division_sum_witness_operationoperationleft + S (pfp_left_backward_division_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_backward_division_sum_witness_operationoperationright. pfa_gap_backward_division_sum_witness_operationoperationright + S (pfp_right_backward_division_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_backward_division_sum_witness_operationoperationresultbound. pfa_gap_backward_division_sum_witness_operationoperationresultbound + S (pfp_value_backward_division_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_backward_division_sum_witness_operationoperationresultcongruence pfa_offset_right_backward_division_sum_witness_operationoperationresultcongruence. ((pfp_left_backward_division_sum_witness_operation) + (pfp_right_backward_division_sum_witness_operation)) + (p) * pfa_offset_left_backward_division_sum_witness_operationoperationresultcongruence = (pfp_value_backward_division_sum_witness_operation) + (p) * pfa_offset_right_backward_division_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_backward_division_sum_witness_output pfrep_left_backward_division_sum_witness_output pfrep_right_backward_division_sum_witness_output. ((exists pfrep_position_backward_division_sum_witness_outputfirst. ((pfrep_position_backward_division_sum_witness_outputfirst+S (pfrep_power_backward_division_sum_witness_output)=(pfaa_length_backward_division_sum)) /\\ ((((exists ff_h_pfp_backward_division_sum_witness_outputfirstentry. ff_h_pfp_backward_division_sum_witness_outputfirstentry + S (pfrep_left_backward_division_sum_witness_output) = S ((S (pfrep_position_backward_division_sum_witness_outputfirst)) * pfaa_sum_c_backward_division_sum)) /\\ exists ff_q_pfp_backward_division_sum_witness_outputfirstentry. pfaa_sum_b_backward_division_sum = ff_q_pfp_backward_division_sum_witness_outputfirstentry * S ((S (pfrep_position_backward_division_sum_witness_outputfirst)) * pfaa_sum_c_backward_division_sum) + (pfrep_left_backward_division_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_division_sum_witness_outputfirstoutside. pfrep_gap_backward_division_sum_witness_outputfirstoutside+(pfaa_length_backward_division_sum)=(pfrep_power_backward_division_sum_witness_output)) /\\ (((pfrep_left_backward_division_sum_witness_output)=0))))) -> ((exists pfrep_position_backward_division_sum_witness_outputsecond. ((pfrep_position_backward_division_sum_witness_outputsecond+S (pfrep_power_backward_division_sum_witness_output)=(La)) /\\ ((((exists ff_h_pfp_backward_division_sum_witness_outputsecondentry. ff_h_pfp_backward_division_sum_witness_outputsecondentry + S (pfrep_right_backward_division_sum_witness_output) = S ((S (pfrep_position_backward_division_sum_witness_outputsecond)) * ac)) /\\ exists ff_q_pfp_backward_division_sum_witness_outputsecondentry. ab = ff_q_pfp_backward_division_sum_witness_outputsecondentry * S ((S (pfrep_position_backward_division_sum_witness_outputsecond)) * ac) + (pfrep_right_backward_division_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_division_sum_witness_outputsecondoutside. pfrep_gap_backward_division_sum_witness_outputsecondoutside+(La)=(pfrep_power_backward_division_sum_witness_output)) /\\ (((pfrep_right_backward_division_sum_witness_output)=0))))) -> pfrep_left_backward_division_sum_witness_output=pfrep_right_backward_division_sum_witness_output))))))))))))) -> (((forall fom_index_pfp_backward_old_leftleft. (exists fom_gap_pfp_backward_old_leftleft_index_bound. fom_gap_pfp_backward_old_leftleft_index_bound + S (fom_index_pfp_backward_old_leftleft) = Lu) -> exists fom_value_pfp_backward_old_leftleft. ((((exists fom_beta_height_pfp_backward_old_leftleft_entry. fom_beta_height_pfp_backward_old_leftleft_entry + S (fom_value_pfp_backward_old_leftleft) = S ((S (fom_index_pfp_backward_old_leftleft)) * uc)) /\\ exists fom_beta_quotient_pfp_backward_old_leftleft_entry. ub = fom_beta_quotient_pfp_backward_old_leftleft_entry * S ((S (fom_index_pfp_backward_old_leftleft)) * uc) + (fom_value_pfp_backward_old_leftleft))) /\\ (exists fom_gap_pfp_backward_old_leftleft_value_bound. fom_gap_pfp_backward_old_leftleft_value_bound + S (fom_value_pfp_backward_old_leftleft) = p))) /\\ (((forall fom_index_pfp_backward_old_leftright. (exists fom_gap_pfp_backward_old_leftright_index_bound. fom_gap_pfp_backward_old_leftright_index_bound + S (fom_index_pfp_backward_old_leftright) = Lb) -> exists fom_value_pfp_backward_old_leftright. ((((exists fom_beta_height_pfp_backward_old_leftright_entry. fom_beta_height_pfp_backward_old_leftright_entry + S (fom_value_pfp_backward_old_leftright) = S ((S (fom_index_pfp_backward_old_leftright)) * bc)) /\\ exists fom_beta_quotient_pfp_backward_old_leftright_entry. bb = fom_beta_quotient_pfp_backward_old_leftright_entry * S ((S (fom_index_pfp_backward_old_leftright)) * bc) + (fom_value_pfp_backward_old_leftright))) /\\ (exists fom_gap_pfp_backward_old_leftright_value_bound. fom_gap_pfp_backward_old_leftright_value_bound + S (fom_value_pfp_backward_old_leftright) = p))) /\\ (((((((Lu)=0 \\/ (Lb)=0) /\\ (((Lc)=0)))) \\/ (((~((Lu)=0)) /\\ (((~((Lb)=0)) /\\ (((Lu)+(Lb)=S (Lc)))))))) /\\ ((forall pfc_index_backward_old_leftcoefficients. (exists pfa_gap_backward_old_leftcoefficientsbound. pfa_gap_backward_old_leftcoefficientsbound + S (pfc_index_backward_old_leftcoefficients) = (Lc)) -> exists pfc_value_backward_old_leftcoefficients. ((((exists ff_h_pfp_backward_old_leftcoefficientsentry. ff_h_pfp_backward_old_leftcoefficientsentry + S (pfc_value_backward_old_leftcoefficients) = S ((S (pfc_index_backward_old_leftcoefficients)) * cc)) /\\ exists ff_q_pfp_backward_old_leftcoefficientsentry. cb = ff_q_pfp_backward_old_leftcoefficientsentry * S ((S (pfc_index_backward_old_leftcoefficients)) * cc) + (pfc_value_backward_old_leftcoefficients))) /\\ ((exists pfc_terms_code_backward_old_leftcoefficientscoefficient pfc_terms_scale_backward_old_leftcoefficientscoefficient pfc_natural_sum_backward_old_leftcoefficientscoefficient. ((forall pfc_index_backward_old_leftcoefficientscoefficientdiagonal. (exists pfa_gap_backward_old_leftcoefficientscoefficientdiagonalbound. pfa_gap_backward_old_leftcoefficientscoefficientdiagonalbound + S (pfc_index_backward_old_leftcoefficientscoefficientdiagonal) = (S (pfc_index_backward_old_leftcoefficients))) -> exists pfc_value_backward_old_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_old_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_old_leftcoefficientscoefficientdiagonalentry + S (pfc_value_backward_old_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_old_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_old_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_old_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_old_leftcoefficientscoefficient = ff_q_pfp_backward_old_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_old_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_old_leftcoefficientscoefficient) + (pfc_value_backward_old_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_old_leftcoefficientscoefficientdiagonalterm pfc_left_backward_old_leftcoefficientscoefficientdiagonalterm pfc_right_backward_old_leftcoefficientscoefficientdiagonalterm. (((pfc_index_backward_old_leftcoefficientscoefficientdiagonal)+pfc_complement_backward_old_leftcoefficientscoefficientdiagonalterm=(pfc_index_backward_old_leftcoefficients)) /\\ ((((((exists pfa_gap_backward_old_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_old_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_old_leftcoefficientscoefficientdiagonal) = (Lu)) /\\ ((((exists ff_h_pfp_backward_old_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_old_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_old_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_old_leftcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_backward_old_leftcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_backward_old_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_old_leftcoefficientscoefficientdiagonal)) * uc) + (pfc_left_backward_old_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_old_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_old_leftcoefficientscoefficientdiagonaltermleftoutside+(Lu)=(pfc_index_backward_old_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_old_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_old_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_old_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_old_leftcoefficientscoefficientdiagonalterm) = (Lb)) /\\ ((((exists ff_h_pfp_backward_old_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_old_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_old_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_old_leftcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_backward_old_leftcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_backward_old_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_old_leftcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_backward_old_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_old_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_old_leftcoefficientscoefficientdiagonaltermrightoutside+(Lb)=(pfc_complement_backward_old_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_old_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_old_leftcoefficientscoefficientdiagonal)=pfc_left_backward_old_leftcoefficientscoefficientdiagonalterm*pfc_right_backward_old_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_old_leftcoefficientscoefficientsum fs_v_pfc_backward_old_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_start. fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_start. fs_u_pfc_backward_old_leftcoefficientscoefficientsum = fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_old_leftcoefficientscoefficient) = S ((S (S (pfc_index_backward_old_leftcoefficients))) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_old_leftcoefficientscoefficientsum = fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_old_leftcoefficients))) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum) + (pfc_natural_sum_backward_old_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps = S (pfc_index_backward_old_leftcoefficients)) -> exists fs_a_pfc_backward_old_leftcoefficientscoefficientsum_body_steps fs_r_pfc_backward_old_leftcoefficientscoefficientsum_body_steps fs_s_pfc_backward_old_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_old_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_old_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_old_leftcoefficientscoefficient = fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_old_leftcoefficientscoefficient) + (fs_a_pfc_backward_old_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_old_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_old_leftcoefficientscoefficientsum = fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum) + (fs_r_pfc_backward_old_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_old_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_old_leftcoefficientscoefficientsum = fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum) + (fs_s_pfc_backward_old_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_old_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_old_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_old_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_old_leftcoefficientscoefficientresiduebound. pfa_gap_backward_old_leftcoefficientscoefficientresiduebound + S (pfc_value_backward_old_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_old_leftcoefficientscoefficientresiduecongruence pfa_offset_right_backward_old_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_old_leftcoefficientscoefficient) + (p) * pfa_offset_left_backward_old_leftcoefficientscoefficientresiduecongruence = (pfc_value_backward_old_leftcoefficients) + (p) * pfa_offset_right_backward_old_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_old_rightleft. (exists fom_gap_pfp_backward_old_rightleft_index_bound. fom_gap_pfp_backward_old_rightleft_index_bound + S (fom_index_pfp_backward_old_rightleft) = Lv) -> exists fom_value_pfp_backward_old_rightleft. ((((exists fom_beta_height_pfp_backward_old_rightleft_entry. fom_beta_height_pfp_backward_old_rightleft_entry + S (fom_value_pfp_backward_old_rightleft) = S ((S (fom_index_pfp_backward_old_rightleft)) * vc)) /\\ exists fom_beta_quotient_pfp_backward_old_rightleft_entry. vb = fom_beta_quotient_pfp_backward_old_rightleft_entry * S ((S (fom_index_pfp_backward_old_rightleft)) * vc) + (fom_value_pfp_backward_old_rightleft))) /\\ (exists fom_gap_pfp_backward_old_rightleft_value_bound. fom_gap_pfp_backward_old_rightleft_value_bound + S (fom_value_pfp_backward_old_rightleft) = p))) /\\ (((forall fom_index_pfp_backward_old_rightright. (exists fom_gap_pfp_backward_old_rightright_index_bound. fom_gap_pfp_backward_old_rightright_index_bound + S (fom_index_pfp_backward_old_rightright) = Lr) -> exists fom_value_pfp_backward_old_rightright. ((((exists fom_beta_height_pfp_backward_old_rightright_entry. fom_beta_height_pfp_backward_old_rightright_entry + S (fom_value_pfp_backward_old_rightright) = S ((S (fom_index_pfp_backward_old_rightright)) * rc)) /\\ exists fom_beta_quotient_pfp_backward_old_rightright_entry. rb = fom_beta_quotient_pfp_backward_old_rightright_entry * S ((S (fom_index_pfp_backward_old_rightright)) * rc) + (fom_value_pfp_backward_old_rightright))) /\\ (exists fom_gap_pfp_backward_old_rightright_value_bound. fom_gap_pfp_backward_old_rightright_value_bound + S (fom_value_pfp_backward_old_rightright) = p))) /\\ (((((((Lv)=0 \\/ (Lr)=0) /\\ (((Ld)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((Lr)=0)) /\\ (((Lv)+(Lr)=S (Ld)))))))) /\\ ((forall pfc_index_backward_old_rightcoefficients. (exists pfa_gap_backward_old_rightcoefficientsbound. pfa_gap_backward_old_rightcoefficientsbound + S (pfc_index_backward_old_rightcoefficients) = (Ld)) -> exists pfc_value_backward_old_rightcoefficients. ((((exists ff_h_pfp_backward_old_rightcoefficientsentry. ff_h_pfp_backward_old_rightcoefficientsentry + S (pfc_value_backward_old_rightcoefficients) = S ((S (pfc_index_backward_old_rightcoefficients)) * dc)) /\\ exists ff_q_pfp_backward_old_rightcoefficientsentry. db = ff_q_pfp_backward_old_rightcoefficientsentry * S ((S (pfc_index_backward_old_rightcoefficients)) * dc) + (pfc_value_backward_old_rightcoefficients))) /\\ ((exists pfc_terms_code_backward_old_rightcoefficientscoefficient pfc_terms_scale_backward_old_rightcoefficientscoefficient pfc_natural_sum_backward_old_rightcoefficientscoefficient. ((forall pfc_index_backward_old_rightcoefficientscoefficientdiagonal. (exists pfa_gap_backward_old_rightcoefficientscoefficientdiagonalbound. pfa_gap_backward_old_rightcoefficientscoefficientdiagonalbound + S (pfc_index_backward_old_rightcoefficientscoefficientdiagonal) = (S (pfc_index_backward_old_rightcoefficients))) -> exists pfc_value_backward_old_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_old_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_old_rightcoefficientscoefficientdiagonalentry + S (pfc_value_backward_old_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_old_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_old_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_old_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_old_rightcoefficientscoefficient = ff_q_pfp_backward_old_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_old_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_old_rightcoefficientscoefficient) + (pfc_value_backward_old_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_old_rightcoefficientscoefficientdiagonalterm pfc_left_backward_old_rightcoefficientscoefficientdiagonalterm pfc_right_backward_old_rightcoefficientscoefficientdiagonalterm. (((pfc_index_backward_old_rightcoefficientscoefficientdiagonal)+pfc_complement_backward_old_rightcoefficientscoefficientdiagonalterm=(pfc_index_backward_old_rightcoefficients)) /\\ ((((((exists pfa_gap_backward_old_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_old_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_old_rightcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_backward_old_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_old_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_old_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_old_rightcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_backward_old_rightcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_backward_old_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_old_rightcoefficientscoefficientdiagonal)) * vc) + (pfc_left_backward_old_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_old_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_old_rightcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_backward_old_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_old_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_old_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_old_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_old_rightcoefficientscoefficientdiagonalterm) = (Lr)) /\\ ((((exists ff_h_pfp_backward_old_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_old_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_old_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_old_rightcoefficientscoefficientdiagonalterm)) * rc)) /\\ exists ff_q_pfp_backward_old_rightcoefficientscoefficientdiagonaltermrightentry. rb = ff_q_pfp_backward_old_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_old_rightcoefficientscoefficientdiagonalterm)) * rc) + (pfc_right_backward_old_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_old_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_old_rightcoefficientscoefficientdiagonaltermrightoutside+(Lr)=(pfc_complement_backward_old_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_old_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_old_rightcoefficientscoefficientdiagonal)=pfc_left_backward_old_rightcoefficientscoefficientdiagonalterm*pfc_right_backward_old_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_old_rightcoefficientscoefficientsum fs_v_pfc_backward_old_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_start. fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_start. fs_u_pfc_backward_old_rightcoefficientscoefficientsum = fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_old_rightcoefficientscoefficient) = S ((S (S (pfc_index_backward_old_rightcoefficients))) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_old_rightcoefficientscoefficientsum = fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_old_rightcoefficients))) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum) + (pfc_natural_sum_backward_old_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps = S (pfc_index_backward_old_rightcoefficients)) -> exists fs_a_pfc_backward_old_rightcoefficientscoefficientsum_body_steps fs_r_pfc_backward_old_rightcoefficientscoefficientsum_body_steps fs_s_pfc_backward_old_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_old_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_old_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_old_rightcoefficientscoefficient = fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_old_rightcoefficientscoefficient) + (fs_a_pfc_backward_old_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_old_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_old_rightcoefficientscoefficientsum = fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum) + (fs_r_pfc_backward_old_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_old_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_old_rightcoefficientscoefficientsum = fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum) + (fs_s_pfc_backward_old_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_old_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_old_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_old_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_old_rightcoefficientscoefficientresiduebound. pfa_gap_backward_old_rightcoefficientscoefficientresiduebound + S (pfc_value_backward_old_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_old_rightcoefficientscoefficientresiduecongruence pfa_offset_right_backward_old_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_old_rightcoefficientscoefficient) + (p) * pfa_offset_left_backward_old_rightcoefficientscoefficientresiduecongruence = (pfc_value_backward_old_rightcoefficients) + (p) * pfa_offset_right_backward_old_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_old_sum_left_bounded. (exists fom_gap_pfp_backward_old_sum_left_bounded_index_bound. fom_gap_pfp_backward_old_sum_left_bounded_index_bound + S (fom_index_pfp_backward_old_sum_left_bounded) = Lc) -> exists fom_value_pfp_backward_old_sum_left_bounded. ((((exists fom_beta_height_pfp_backward_old_sum_left_bounded_entry. fom_beta_height_pfp_backward_old_sum_left_bounded_entry + S (fom_value_pfp_backward_old_sum_left_bounded) = S ((S (fom_index_pfp_backward_old_sum_left_bounded)) * cc)) /\\ exists fom_beta_quotient_pfp_backward_old_sum_left_bounded_entry. cb = fom_beta_quotient_pfp_backward_old_sum_left_bounded_entry * S ((S (fom_index_pfp_backward_old_sum_left_bounded)) * cc) + (fom_value_pfp_backward_old_sum_left_bounded))) /\\ (exists fom_gap_pfp_backward_old_sum_left_bounded_value_bound. fom_gap_pfp_backward_old_sum_left_bounded_value_bound + S (fom_value_pfp_backward_old_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_backward_old_sum_right_bounded. (exists fom_gap_pfp_backward_old_sum_right_bounded_index_bound. fom_gap_pfp_backward_old_sum_right_bounded_index_bound + S (fom_index_pfp_backward_old_sum_right_bounded) = Ld) -> exists fom_value_pfp_backward_old_sum_right_bounded. ((((exists fom_beta_height_pfp_backward_old_sum_right_bounded_entry. fom_beta_height_pfp_backward_old_sum_right_bounded_entry + S (fom_value_pfp_backward_old_sum_right_bounded) = S ((S (fom_index_pfp_backward_old_sum_right_bounded)) * dc)) /\\ exists fom_beta_quotient_pfp_backward_old_sum_right_bounded_entry. db = fom_beta_quotient_pfp_backward_old_sum_right_bounded_entry * S ((S (fom_index_pfp_backward_old_sum_right_bounded)) * dc) + (fom_value_pfp_backward_old_sum_right_bounded))) /\\ (exists fom_gap_pfp_backward_old_sum_right_bounded_value_bound. fom_gap_pfp_backward_old_sum_right_bounded_value_bound + S (fom_value_pfp_backward_old_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_backward_old_sum_result_bounded. (exists fom_gap_pfp_backward_old_sum_result_bounded_index_bound. fom_gap_pfp_backward_old_sum_result_bounded_index_bound + S (fom_index_pfp_backward_old_sum_result_bounded) = Lg) -> exists fom_value_pfp_backward_old_sum_result_bounded. ((((exists fom_beta_height_pfp_backward_old_sum_result_bounded_entry. fom_beta_height_pfp_backward_old_sum_result_bounded_entry + S (fom_value_pfp_backward_old_sum_result_bounded) = S ((S (fom_index_pfp_backward_old_sum_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_backward_old_sum_result_bounded_entry. gb = fom_beta_quotient_pfp_backward_old_sum_result_bounded_entry * S ((S (fom_index_pfp_backward_old_sum_result_bounded)) * gc) + (fom_value_pfp_backward_old_sum_result_bounded))) /\\ (exists fom_gap_pfp_backward_old_sum_result_bounded_value_bound. fom_gap_pfp_backward_old_sum_result_bounded_value_bound + S (fom_value_pfp_backward_old_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_backward_old_sum pfaa_left_c_backward_old_sum pfaa_right_b_backward_old_sum pfaa_right_c_backward_old_sum pfaa_sum_b_backward_old_sum pfaa_sum_c_backward_old_sum pfaa_length_backward_old_sum. ((((forall pfrep_power_backward_old_sum_witness_common_left pfrep_left_backward_old_sum_witness_common_left pfrep_right_backward_old_sum_witness_common_left. ((exists pfrep_position_backward_old_sum_witness_common_leftfirst. ((pfrep_position_backward_old_sum_witness_common_leftfirst+S (pfrep_power_backward_old_sum_witness_common_left)=(Lc)) /\\ ((((exists ff_h_pfp_backward_old_sum_witness_common_leftfirstentry. ff_h_pfp_backward_old_sum_witness_common_leftfirstentry + S (pfrep_left_backward_old_sum_witness_common_left) = S ((S (pfrep_position_backward_old_sum_witness_common_leftfirst)) * cc)) /\\ exists ff_q_pfp_backward_old_sum_witness_common_leftfirstentry. cb = ff_q_pfp_backward_old_sum_witness_common_leftfirstentry * S ((S (pfrep_position_backward_old_sum_witness_common_leftfirst)) * cc) + (pfrep_left_backward_old_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_old_sum_witness_common_leftfirstoutside. pfrep_gap_backward_old_sum_witness_common_leftfirstoutside+(Lc)=(pfrep_power_backward_old_sum_witness_common_left)) /\\ (((pfrep_left_backward_old_sum_witness_common_left)=0))))) -> ((exists pfrep_position_backward_old_sum_witness_common_leftsecond. ((pfrep_position_backward_old_sum_witness_common_leftsecond+S (pfrep_power_backward_old_sum_witness_common_left)=(pfaa_length_backward_old_sum)) /\\ ((((exists ff_h_pfp_backward_old_sum_witness_common_leftsecondentry. ff_h_pfp_backward_old_sum_witness_common_leftsecondentry + S (pfrep_right_backward_old_sum_witness_common_left) = S ((S (pfrep_position_backward_old_sum_witness_common_leftsecond)) * pfaa_left_c_backward_old_sum)) /\\ exists ff_q_pfp_backward_old_sum_witness_common_leftsecondentry. pfaa_left_b_backward_old_sum = ff_q_pfp_backward_old_sum_witness_common_leftsecondentry * S ((S (pfrep_position_backward_old_sum_witness_common_leftsecond)) * pfaa_left_c_backward_old_sum) + (pfrep_right_backward_old_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_old_sum_witness_common_leftsecondoutside. pfrep_gap_backward_old_sum_witness_common_leftsecondoutside+(pfaa_length_backward_old_sum)=(pfrep_power_backward_old_sum_witness_common_left)) /\\ (((pfrep_right_backward_old_sum_witness_common_left)=0))))) -> pfrep_left_backward_old_sum_witness_common_left=pfrep_right_backward_old_sum_witness_common_left) /\\ ((forall pfrep_power_backward_old_sum_witness_common_right pfrep_left_backward_old_sum_witness_common_right pfrep_right_backward_old_sum_witness_common_right. ((exists pfrep_position_backward_old_sum_witness_common_rightfirst. ((pfrep_position_backward_old_sum_witness_common_rightfirst+S (pfrep_power_backward_old_sum_witness_common_right)=(Ld)) /\\ ((((exists ff_h_pfp_backward_old_sum_witness_common_rightfirstentry. ff_h_pfp_backward_old_sum_witness_common_rightfirstentry + S (pfrep_left_backward_old_sum_witness_common_right) = S ((S (pfrep_position_backward_old_sum_witness_common_rightfirst)) * dc)) /\\ exists ff_q_pfp_backward_old_sum_witness_common_rightfirstentry. db = ff_q_pfp_backward_old_sum_witness_common_rightfirstentry * S ((S (pfrep_position_backward_old_sum_witness_common_rightfirst)) * dc) + (pfrep_left_backward_old_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_old_sum_witness_common_rightfirstoutside. pfrep_gap_backward_old_sum_witness_common_rightfirstoutside+(Ld)=(pfrep_power_backward_old_sum_witness_common_right)) /\\ (((pfrep_left_backward_old_sum_witness_common_right)=0))))) -> ((exists pfrep_position_backward_old_sum_witness_common_rightsecond. ((pfrep_position_backward_old_sum_witness_common_rightsecond+S (pfrep_power_backward_old_sum_witness_common_right)=(pfaa_length_backward_old_sum)) /\\ ((((exists ff_h_pfp_backward_old_sum_witness_common_rightsecondentry. ff_h_pfp_backward_old_sum_witness_common_rightsecondentry + S (pfrep_right_backward_old_sum_witness_common_right) = S ((S (pfrep_position_backward_old_sum_witness_common_rightsecond)) * pfaa_right_c_backward_old_sum)) /\\ exists ff_q_pfp_backward_old_sum_witness_common_rightsecondentry. pfaa_right_b_backward_old_sum = ff_q_pfp_backward_old_sum_witness_common_rightsecondentry * S ((S (pfrep_position_backward_old_sum_witness_common_rightsecond)) * pfaa_right_c_backward_old_sum) + (pfrep_right_backward_old_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_old_sum_witness_common_rightsecondoutside. pfrep_gap_backward_old_sum_witness_common_rightsecondoutside+(pfaa_length_backward_old_sum)=(pfrep_power_backward_old_sum_witness_common_right)) /\\ (((pfrep_right_backward_old_sum_witness_common_right)=0))))) -> pfrep_left_backward_old_sum_witness_common_right=pfrep_right_backward_old_sum_witness_common_right)))) /\\ (((forall pfp_index_backward_old_sum_witness_operation. (exists pfa_gap_backward_old_sum_witness_operationindex. pfa_gap_backward_old_sum_witness_operationindex + S (pfp_index_backward_old_sum_witness_operation) = (pfaa_length_backward_old_sum)) -> exists pfp_left_backward_old_sum_witness_operation pfp_right_backward_old_sum_witness_operation pfp_value_backward_old_sum_witness_operation. ((((exists ff_h_pfp_backward_old_sum_witness_operationleft. ff_h_pfp_backward_old_sum_witness_operationleft + S (pfp_left_backward_old_sum_witness_operation) = S ((S (pfp_index_backward_old_sum_witness_operation)) * pfaa_left_c_backward_old_sum)) /\\ exists ff_q_pfp_backward_old_sum_witness_operationleft. pfaa_left_b_backward_old_sum = ff_q_pfp_backward_old_sum_witness_operationleft * S ((S (pfp_index_backward_old_sum_witness_operation)) * pfaa_left_c_backward_old_sum) + (pfp_left_backward_old_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_old_sum_witness_operationright. ff_h_pfp_backward_old_sum_witness_operationright + S (pfp_right_backward_old_sum_witness_operation) = S ((S (pfp_index_backward_old_sum_witness_operation)) * pfaa_right_c_backward_old_sum)) /\\ exists ff_q_pfp_backward_old_sum_witness_operationright. pfaa_right_b_backward_old_sum = ff_q_pfp_backward_old_sum_witness_operationright * S ((S (pfp_index_backward_old_sum_witness_operation)) * pfaa_right_c_backward_old_sum) + (pfp_right_backward_old_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_old_sum_witness_operationtarget. ff_h_pfp_backward_old_sum_witness_operationtarget + S (pfp_value_backward_old_sum_witness_operation) = S ((S (pfp_index_backward_old_sum_witness_operation)) * pfaa_sum_c_backward_old_sum)) /\\ exists ff_q_pfp_backward_old_sum_witness_operationtarget. pfaa_sum_b_backward_old_sum = ff_q_pfp_backward_old_sum_witness_operationtarget * S ((S (pfp_index_backward_old_sum_witness_operation)) * pfaa_sum_c_backward_old_sum) + (pfp_value_backward_old_sum_witness_operation))) /\\ ((((exists pfa_gap_backward_old_sum_witness_operationoperationleft. pfa_gap_backward_old_sum_witness_operationoperationleft + S (pfp_left_backward_old_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_backward_old_sum_witness_operationoperationright. pfa_gap_backward_old_sum_witness_operationoperationright + S (pfp_right_backward_old_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_backward_old_sum_witness_operationoperationresultbound. pfa_gap_backward_old_sum_witness_operationoperationresultbound + S (pfp_value_backward_old_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_backward_old_sum_witness_operationoperationresultcongruence pfa_offset_right_backward_old_sum_witness_operationoperationresultcongruence. ((pfp_left_backward_old_sum_witness_operation) + (pfp_right_backward_old_sum_witness_operation)) + (p) * pfa_offset_left_backward_old_sum_witness_operationoperationresultcongruence = (pfp_value_backward_old_sum_witness_operation) + (p) * pfa_offset_right_backward_old_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_backward_old_sum_witness_output pfrep_left_backward_old_sum_witness_output pfrep_right_backward_old_sum_witness_output. ((exists pfrep_position_backward_old_sum_witness_outputfirst. ((pfrep_position_backward_old_sum_witness_outputfirst+S (pfrep_power_backward_old_sum_witness_output)=(pfaa_length_backward_old_sum)) /\\ ((((exists ff_h_pfp_backward_old_sum_witness_outputfirstentry. ff_h_pfp_backward_old_sum_witness_outputfirstentry + S (pfrep_left_backward_old_sum_witness_output) = S ((S (pfrep_position_backward_old_sum_witness_outputfirst)) * pfaa_sum_c_backward_old_sum)) /\\ exists ff_q_pfp_backward_old_sum_witness_outputfirstentry. pfaa_sum_b_backward_old_sum = ff_q_pfp_backward_old_sum_witness_outputfirstentry * S ((S (pfrep_position_backward_old_sum_witness_outputfirst)) * pfaa_sum_c_backward_old_sum) + (pfrep_left_backward_old_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_old_sum_witness_outputfirstoutside. pfrep_gap_backward_old_sum_witness_outputfirstoutside+(pfaa_length_backward_old_sum)=(pfrep_power_backward_old_sum_witness_output)) /\\ (((pfrep_left_backward_old_sum_witness_output)=0))))) -> ((exists pfrep_position_backward_old_sum_witness_outputsecond. ((pfrep_position_backward_old_sum_witness_outputsecond+S (pfrep_power_backward_old_sum_witness_output)=(Lg)) /\\ ((((exists ff_h_pfp_backward_old_sum_witness_outputsecondentry. ff_h_pfp_backward_old_sum_witness_outputsecondentry + S (pfrep_right_backward_old_sum_witness_output) = S ((S (pfrep_position_backward_old_sum_witness_outputsecond)) * gc)) /\\ exists ff_q_pfp_backward_old_sum_witness_outputsecondentry. gb = ff_q_pfp_backward_old_sum_witness_outputsecondentry * S ((S (pfrep_position_backward_old_sum_witness_outputsecond)) * gc) + (pfrep_right_backward_old_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_old_sum_witness_outputsecondoutside. pfrep_gap_backward_old_sum_witness_outputsecondoutside+(Lg)=(pfrep_power_backward_old_sum_witness_output)) /\\ (((pfrep_right_backward_old_sum_witness_output)=0))))) -> pfrep_left_backward_old_sum_witness_output=pfrep_right_backward_old_sum_witness_output))))))))))))) -> (((forall fom_index_pfp_backward_coefficient_productleft. (exists fom_gap_pfp_backward_coefficient_productleft_index_bound. fom_gap_pfp_backward_coefficient_productleft_index_bound + S (fom_index_pfp_backward_coefficient_productleft) = Lv) -> exists fom_value_pfp_backward_coefficient_productleft. ((((exists fom_beta_height_pfp_backward_coefficient_productleft_entry. fom_beta_height_pfp_backward_coefficient_productleft_entry + S (fom_value_pfp_backward_coefficient_productleft) = S ((S (fom_index_pfp_backward_coefficient_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_backward_coefficient_productleft_entry. vb = fom_beta_quotient_pfp_backward_coefficient_productleft_entry * S ((S (fom_index_pfp_backward_coefficient_productleft)) * vc) + (fom_value_pfp_backward_coefficient_productleft))) /\\ (exists fom_gap_pfp_backward_coefficient_productleft_value_bound. fom_gap_pfp_backward_coefficient_productleft_value_bound + S (fom_value_pfp_backward_coefficient_productleft) = p))) /\\ (((forall fom_index_pfp_backward_coefficient_productright. (exists fom_gap_pfp_backward_coefficient_productright_index_bound. fom_gap_pfp_backward_coefficient_productright_index_bound + S (fom_index_pfp_backward_coefficient_productright) = Lq) -> exists fom_value_pfp_backward_coefficient_productright. ((((exists fom_beta_height_pfp_backward_coefficient_productright_entry. fom_beta_height_pfp_backward_coefficient_productright_entry + S (fom_value_pfp_backward_coefficient_productright) = S ((S (fom_index_pfp_backward_coefficient_productright)) * qc)) /\\ exists fom_beta_quotient_pfp_backward_coefficient_productright_entry. qb = fom_beta_quotient_pfp_backward_coefficient_productright_entry * S ((S (fom_index_pfp_backward_coefficient_productright)) * qc) + (fom_value_pfp_backward_coefficient_productright))) /\\ (exists fom_gap_pfp_backward_coefficient_productright_value_bound. fom_gap_pfp_backward_coefficient_productright_value_bound + S (fom_value_pfp_backward_coefficient_productright) = p))) /\\ (((((((Lv)=0 \\/ (Lq)=0) /\\ (((Lw)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((Lq)=0)) /\\ (((Lv)+(Lq)=S (Lw)))))))) /\\ ((forall pfc_index_backward_coefficient_productcoefficients. (exists pfa_gap_backward_coefficient_productcoefficientsbound. pfa_gap_backward_coefficient_productcoefficientsbound + S (pfc_index_backward_coefficient_productcoefficients) = (Lw)) -> exists pfc_value_backward_coefficient_productcoefficients. ((((exists ff_h_pfp_backward_coefficient_productcoefficientsentry. ff_h_pfp_backward_coefficient_productcoefficientsentry + S (pfc_value_backward_coefficient_productcoefficients) = S ((S (pfc_index_backward_coefficient_productcoefficients)) * wc)) /\\ exists ff_q_pfp_backward_coefficient_productcoefficientsentry. wb = ff_q_pfp_backward_coefficient_productcoefficientsentry * S ((S (pfc_index_backward_coefficient_productcoefficients)) * wc) + (pfc_value_backward_coefficient_productcoefficients))) /\\ ((exists pfc_terms_code_backward_coefficient_productcoefficientscoefficient pfc_terms_scale_backward_coefficient_productcoefficientscoefficient pfc_natural_sum_backward_coefficient_productcoefficientscoefficient. ((forall pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal. (exists pfa_gap_backward_coefficient_productcoefficientscoefficientdiagonalbound. pfa_gap_backward_coefficient_productcoefficientscoefficientdiagonalbound + S (pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal) = (S (pfc_index_backward_coefficient_productcoefficients))) -> exists pfc_value_backward_coefficient_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_coefficient_productcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_coefficient_productcoefficientscoefficientdiagonalentry + S (pfc_value_backward_coefficient_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_coefficient_productcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_coefficient_productcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_coefficient_productcoefficientscoefficient = ff_q_pfp_backward_coefficient_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_coefficient_productcoefficientscoefficient) + (pfc_value_backward_coefficient_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_coefficient_productcoefficientscoefficientdiagonalterm pfc_left_backward_coefficient_productcoefficientscoefficientdiagonalterm pfc_right_backward_coefficient_productcoefficientscoefficientdiagonalterm. (((pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal)+pfc_complement_backward_coefficient_productcoefficientscoefficientdiagonalterm=(pfc_index_backward_coefficient_productcoefficients)) /\\ ((((((exists pfa_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_coefficient_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_backward_coefficient_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_coefficient_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_coefficient_productcoefficientscoefficientdiagonalterm) = (Lq)) /\\ ((((exists ff_h_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_coefficient_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_coefficient_productcoefficientscoefficientdiagonalterm)) * qc)) /\\ exists ff_q_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermrightentry. qb = ff_q_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_coefficient_productcoefficientscoefficientdiagonalterm)) * qc) + (pfc_right_backward_coefficient_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermrightoutside+(Lq)=(pfc_complement_backward_coefficient_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_coefficient_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_coefficient_productcoefficientscoefficientdiagonal)=pfc_left_backward_coefficient_productcoefficientscoefficientdiagonalterm*pfc_right_backward_coefficient_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_coefficient_productcoefficientscoefficientsum fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_start. fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_start. fs_u_pfc_backward_coefficient_productcoefficientscoefficientsum = fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_coefficient_productcoefficientscoefficient) = S ((S (S (pfc_index_backward_coefficient_productcoefficients))) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_coefficient_productcoefficientscoefficientsum = fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_coefficient_productcoefficients))) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum) + (pfc_natural_sum_backward_coefficient_productcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps = S (pfc_index_backward_coefficient_productcoefficients)) -> exists fs_a_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps fs_r_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps fs_s_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_coefficient_productcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_coefficient_productcoefficientscoefficient = fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_coefficient_productcoefficientscoefficient) + (fs_a_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_coefficient_productcoefficientscoefficientsum = fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum) + (fs_r_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_coefficient_productcoefficientscoefficientsum = fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum) + (fs_s_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_coefficient_productcoefficientscoefficientresiduebound. pfa_gap_backward_coefficient_productcoefficientscoefficientresiduebound + S (pfc_value_backward_coefficient_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_coefficient_productcoefficientscoefficientresiduecongruence pfa_offset_right_backward_coefficient_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_coefficient_productcoefficientscoefficient) + (p) * pfa_offset_left_backward_coefficient_productcoefficientscoefficientresiduecongruence = (pfc_value_backward_coefficient_productcoefficients) + (p) * pfa_offset_right_backward_coefficient_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_coefficient_difference_left_bounded. (exists fom_gap_pfp_backward_coefficient_difference_left_bounded_index_bound. fom_gap_pfp_backward_coefficient_difference_left_bounded_index_bound + S (fom_index_pfp_backward_coefficient_difference_left_bounded) = Lw) -> exists fom_value_pfp_backward_coefficient_difference_left_bounded. ((((exists fom_beta_height_pfp_backward_coefficient_difference_left_bounded_entry. fom_beta_height_pfp_backward_coefficient_difference_left_bounded_entry + S (fom_value_pfp_backward_coefficient_difference_left_bounded) = S ((S (fom_index_pfp_backward_coefficient_difference_left_bounded)) * wc)) /\\ exists fom_beta_quotient_pfp_backward_coefficient_difference_left_bounded_entry. wb = fom_beta_quotient_pfp_backward_coefficient_difference_left_bounded_entry * S ((S (fom_index_pfp_backward_coefficient_difference_left_bounded)) * wc) + (fom_value_pfp_backward_coefficient_difference_left_bounded))) /\\ (exists fom_gap_pfp_backward_coefficient_difference_left_bounded_value_bound. fom_gap_pfp_backward_coefficient_difference_left_bounded_value_bound + S (fom_value_pfp_backward_coefficient_difference_left_bounded) = p))) /\\ (((forall fom_index_pfp_backward_coefficient_difference_right_bounded. (exists fom_gap_pfp_backward_coefficient_difference_right_bounded_index_bound. fom_gap_pfp_backward_coefficient_difference_right_bounded_index_bound + S (fom_index_pfp_backward_coefficient_difference_right_bounded) = Lt) -> exists fom_value_pfp_backward_coefficient_difference_right_bounded. ((((exists fom_beta_height_pfp_backward_coefficient_difference_right_bounded_entry. fom_beta_height_pfp_backward_coefficient_difference_right_bounded_entry + S (fom_value_pfp_backward_coefficient_difference_right_bounded) = S ((S (fom_index_pfp_backward_coefficient_difference_right_bounded)) * tc)) /\\ exists fom_beta_quotient_pfp_backward_coefficient_difference_right_bounded_entry. tb = fom_beta_quotient_pfp_backward_coefficient_difference_right_bounded_entry * S ((S (fom_index_pfp_backward_coefficient_difference_right_bounded)) * tc) + (fom_value_pfp_backward_coefficient_difference_right_bounded))) /\\ (exists fom_gap_pfp_backward_coefficient_difference_right_bounded_value_bound. fom_gap_pfp_backward_coefficient_difference_right_bounded_value_bound + S (fom_value_pfp_backward_coefficient_difference_right_bounded) = p))) /\\ (((forall fom_index_pfp_backward_coefficient_difference_result_bounded. (exists fom_gap_pfp_backward_coefficient_difference_result_bounded_index_bound. fom_gap_pfp_backward_coefficient_difference_result_bounded_index_bound + S (fom_index_pfp_backward_coefficient_difference_result_bounded) = Lu) -> exists fom_value_pfp_backward_coefficient_difference_result_bounded. ((((exists fom_beta_height_pfp_backward_coefficient_difference_result_bounded_entry. fom_beta_height_pfp_backward_coefficient_difference_result_bounded_entry + S (fom_value_pfp_backward_coefficient_difference_result_bounded) = S ((S (fom_index_pfp_backward_coefficient_difference_result_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_backward_coefficient_difference_result_bounded_entry. ub = fom_beta_quotient_pfp_backward_coefficient_difference_result_bounded_entry * S ((S (fom_index_pfp_backward_coefficient_difference_result_bounded)) * uc) + (fom_value_pfp_backward_coefficient_difference_result_bounded))) /\\ (exists fom_gap_pfp_backward_coefficient_difference_result_bounded_value_bound. fom_gap_pfp_backward_coefficient_difference_result_bounded_value_bound + S (fom_value_pfp_backward_coefficient_difference_result_bounded) = p))) /\\ ((exists pfaa_left_b_backward_coefficient_difference pfaa_left_c_backward_coefficient_difference pfaa_right_b_backward_coefficient_difference pfaa_right_c_backward_coefficient_difference pfaa_sum_b_backward_coefficient_difference pfaa_sum_c_backward_coefficient_difference pfaa_length_backward_coefficient_difference. ((((forall pfrep_power_backward_coefficient_difference_witness_common_left pfrep_left_backward_coefficient_difference_witness_common_left pfrep_right_backward_coefficient_difference_witness_common_left. ((exists pfrep_position_backward_coefficient_difference_witness_common_leftfirst. ((pfrep_position_backward_coefficient_difference_witness_common_leftfirst+S (pfrep_power_backward_coefficient_difference_witness_common_left)=(Lw)) /\\ ((((exists ff_h_pfp_backward_coefficient_difference_witness_common_leftfirstentry. ff_h_pfp_backward_coefficient_difference_witness_common_leftfirstentry + S (pfrep_left_backward_coefficient_difference_witness_common_left) = S ((S (pfrep_position_backward_coefficient_difference_witness_common_leftfirst)) * wc)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_common_leftfirstentry. wb = ff_q_pfp_backward_coefficient_difference_witness_common_leftfirstentry * S ((S (pfrep_position_backward_coefficient_difference_witness_common_leftfirst)) * wc) + (pfrep_left_backward_coefficient_difference_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_coefficient_difference_witness_common_leftfirstoutside. pfrep_gap_backward_coefficient_difference_witness_common_leftfirstoutside+(Lw)=(pfrep_power_backward_coefficient_difference_witness_common_left)) /\\ (((pfrep_left_backward_coefficient_difference_witness_common_left)=0))))) -> ((exists pfrep_position_backward_coefficient_difference_witness_common_leftsecond. ((pfrep_position_backward_coefficient_difference_witness_common_leftsecond+S (pfrep_power_backward_coefficient_difference_witness_common_left)=(pfaa_length_backward_coefficient_difference)) /\\ ((((exists ff_h_pfp_backward_coefficient_difference_witness_common_leftsecondentry. ff_h_pfp_backward_coefficient_difference_witness_common_leftsecondentry + S (pfrep_right_backward_coefficient_difference_witness_common_left) = S ((S (pfrep_position_backward_coefficient_difference_witness_common_leftsecond)) * pfaa_left_c_backward_coefficient_difference)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_common_leftsecondentry. pfaa_left_b_backward_coefficient_difference = ff_q_pfp_backward_coefficient_difference_witness_common_leftsecondentry * S ((S (pfrep_position_backward_coefficient_difference_witness_common_leftsecond)) * pfaa_left_c_backward_coefficient_difference) + (pfrep_right_backward_coefficient_difference_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_coefficient_difference_witness_common_leftsecondoutside. pfrep_gap_backward_coefficient_difference_witness_common_leftsecondoutside+(pfaa_length_backward_coefficient_difference)=(pfrep_power_backward_coefficient_difference_witness_common_left)) /\\ (((pfrep_right_backward_coefficient_difference_witness_common_left)=0))))) -> pfrep_left_backward_coefficient_difference_witness_common_left=pfrep_right_backward_coefficient_difference_witness_common_left) /\\ ((forall pfrep_power_backward_coefficient_difference_witness_common_right pfrep_left_backward_coefficient_difference_witness_common_right pfrep_right_backward_coefficient_difference_witness_common_right. ((exists pfrep_position_backward_coefficient_difference_witness_common_rightfirst. ((pfrep_position_backward_coefficient_difference_witness_common_rightfirst+S (pfrep_power_backward_coefficient_difference_witness_common_right)=(Lt)) /\\ ((((exists ff_h_pfp_backward_coefficient_difference_witness_common_rightfirstentry. ff_h_pfp_backward_coefficient_difference_witness_common_rightfirstentry + S (pfrep_left_backward_coefficient_difference_witness_common_right) = S ((S (pfrep_position_backward_coefficient_difference_witness_common_rightfirst)) * tc)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_common_rightfirstentry. tb = ff_q_pfp_backward_coefficient_difference_witness_common_rightfirstentry * S ((S (pfrep_position_backward_coefficient_difference_witness_common_rightfirst)) * tc) + (pfrep_left_backward_coefficient_difference_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_coefficient_difference_witness_common_rightfirstoutside. pfrep_gap_backward_coefficient_difference_witness_common_rightfirstoutside+(Lt)=(pfrep_power_backward_coefficient_difference_witness_common_right)) /\\ (((pfrep_left_backward_coefficient_difference_witness_common_right)=0))))) -> ((exists pfrep_position_backward_coefficient_difference_witness_common_rightsecond. ((pfrep_position_backward_coefficient_difference_witness_common_rightsecond+S (pfrep_power_backward_coefficient_difference_witness_common_right)=(pfaa_length_backward_coefficient_difference)) /\\ ((((exists ff_h_pfp_backward_coefficient_difference_witness_common_rightsecondentry. ff_h_pfp_backward_coefficient_difference_witness_common_rightsecondentry + S (pfrep_right_backward_coefficient_difference_witness_common_right) = S ((S (pfrep_position_backward_coefficient_difference_witness_common_rightsecond)) * pfaa_right_c_backward_coefficient_difference)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_common_rightsecondentry. pfaa_right_b_backward_coefficient_difference = ff_q_pfp_backward_coefficient_difference_witness_common_rightsecondentry * S ((S (pfrep_position_backward_coefficient_difference_witness_common_rightsecond)) * pfaa_right_c_backward_coefficient_difference) + (pfrep_right_backward_coefficient_difference_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_coefficient_difference_witness_common_rightsecondoutside. pfrep_gap_backward_coefficient_difference_witness_common_rightsecondoutside+(pfaa_length_backward_coefficient_difference)=(pfrep_power_backward_coefficient_difference_witness_common_right)) /\\ (((pfrep_right_backward_coefficient_difference_witness_common_right)=0))))) -> pfrep_left_backward_coefficient_difference_witness_common_right=pfrep_right_backward_coefficient_difference_witness_common_right)))) /\\ (((forall pfp_index_backward_coefficient_difference_witness_operation. (exists pfa_gap_backward_coefficient_difference_witness_operationindex. pfa_gap_backward_coefficient_difference_witness_operationindex + S (pfp_index_backward_coefficient_difference_witness_operation) = (pfaa_length_backward_coefficient_difference)) -> exists pfp_left_backward_coefficient_difference_witness_operation pfp_right_backward_coefficient_difference_witness_operation pfp_value_backward_coefficient_difference_witness_operation. ((((exists ff_h_pfp_backward_coefficient_difference_witness_operationleft. ff_h_pfp_backward_coefficient_difference_witness_operationleft + S (pfp_left_backward_coefficient_difference_witness_operation) = S ((S (pfp_index_backward_coefficient_difference_witness_operation)) * pfaa_left_c_backward_coefficient_difference)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_operationleft. pfaa_left_b_backward_coefficient_difference = ff_q_pfp_backward_coefficient_difference_witness_operationleft * S ((S (pfp_index_backward_coefficient_difference_witness_operation)) * pfaa_left_c_backward_coefficient_difference) + (pfp_left_backward_coefficient_difference_witness_operation))) /\\ (((((exists ff_h_pfp_backward_coefficient_difference_witness_operationright. ff_h_pfp_backward_coefficient_difference_witness_operationright + S (pfp_right_backward_coefficient_difference_witness_operation) = S ((S (pfp_index_backward_coefficient_difference_witness_operation)) * pfaa_right_c_backward_coefficient_difference)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_operationright. pfaa_right_b_backward_coefficient_difference = ff_q_pfp_backward_coefficient_difference_witness_operationright * S ((S (pfp_index_backward_coefficient_difference_witness_operation)) * pfaa_right_c_backward_coefficient_difference) + (pfp_right_backward_coefficient_difference_witness_operation))) /\\ (((((exists ff_h_pfp_backward_coefficient_difference_witness_operationtarget. ff_h_pfp_backward_coefficient_difference_witness_operationtarget + S (pfp_value_backward_coefficient_difference_witness_operation) = S ((S (pfp_index_backward_coefficient_difference_witness_operation)) * pfaa_sum_c_backward_coefficient_difference)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_operationtarget. pfaa_sum_b_backward_coefficient_difference = ff_q_pfp_backward_coefficient_difference_witness_operationtarget * S ((S (pfp_index_backward_coefficient_difference_witness_operation)) * pfaa_sum_c_backward_coefficient_difference) + (pfp_value_backward_coefficient_difference_witness_operation))) /\\ ((((exists pfa_gap_backward_coefficient_difference_witness_operationoperationleft. pfa_gap_backward_coefficient_difference_witness_operationoperationleft + S (pfp_left_backward_coefficient_difference_witness_operation) = (p)) /\\ (((exists pfa_gap_backward_coefficient_difference_witness_operationoperationright. pfa_gap_backward_coefficient_difference_witness_operationoperationright + S (pfp_right_backward_coefficient_difference_witness_operation) = (p)) /\\ ((((exists pfa_gap_backward_coefficient_difference_witness_operationoperationresultbound. pfa_gap_backward_coefficient_difference_witness_operationoperationresultbound + S (pfp_value_backward_coefficient_difference_witness_operation) = (p)) /\\ ((exists pfa_offset_left_backward_coefficient_difference_witness_operationoperationresultcongruence pfa_offset_right_backward_coefficient_difference_witness_operationoperationresultcongruence. ((pfp_left_backward_coefficient_difference_witness_operation) + (pfp_right_backward_coefficient_difference_witness_operation)) + (p) * pfa_offset_left_backward_coefficient_difference_witness_operationoperationresultcongruence = (pfp_value_backward_coefficient_difference_witness_operation) + (p) * pfa_offset_right_backward_coefficient_difference_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_backward_coefficient_difference_witness_output pfrep_left_backward_coefficient_difference_witness_output pfrep_right_backward_coefficient_difference_witness_output. ((exists pfrep_position_backward_coefficient_difference_witness_outputfirst. ((pfrep_position_backward_coefficient_difference_witness_outputfirst+S (pfrep_power_backward_coefficient_difference_witness_output)=(pfaa_length_backward_coefficient_difference)) /\\ ((((exists ff_h_pfp_backward_coefficient_difference_witness_outputfirstentry. ff_h_pfp_backward_coefficient_difference_witness_outputfirstentry + S (pfrep_left_backward_coefficient_difference_witness_output) = S ((S (pfrep_position_backward_coefficient_difference_witness_outputfirst)) * pfaa_sum_c_backward_coefficient_difference)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_outputfirstentry. pfaa_sum_b_backward_coefficient_difference = ff_q_pfp_backward_coefficient_difference_witness_outputfirstentry * S ((S (pfrep_position_backward_coefficient_difference_witness_outputfirst)) * pfaa_sum_c_backward_coefficient_difference) + (pfrep_left_backward_coefficient_difference_witness_output)))))) \\/ (((exists pfrep_gap_backward_coefficient_difference_witness_outputfirstoutside. pfrep_gap_backward_coefficient_difference_witness_outputfirstoutside+(pfaa_length_backward_coefficient_difference)=(pfrep_power_backward_coefficient_difference_witness_output)) /\\ (((pfrep_left_backward_coefficient_difference_witness_output)=0))))) -> ((exists pfrep_position_backward_coefficient_difference_witness_outputsecond. ((pfrep_position_backward_coefficient_difference_witness_outputsecond+S (pfrep_power_backward_coefficient_difference_witness_output)=(Lu)) /\\ ((((exists ff_h_pfp_backward_coefficient_difference_witness_outputsecondentry. ff_h_pfp_backward_coefficient_difference_witness_outputsecondentry + S (pfrep_right_backward_coefficient_difference_witness_output) = S ((S (pfrep_position_backward_coefficient_difference_witness_outputsecond)) * uc)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_outputsecondentry. ub = ff_q_pfp_backward_coefficient_difference_witness_outputsecondentry * S ((S (pfrep_position_backward_coefficient_difference_witness_outputsecond)) * uc) + (pfrep_right_backward_coefficient_difference_witness_output)))))) \\/ (((exists pfrep_gap_backward_coefficient_difference_witness_outputsecondoutside. pfrep_gap_backward_coefficient_difference_witness_outputsecondoutside+(Lu)=(pfrep_power_backward_coefficient_difference_witness_output)) /\\ (((pfrep_right_backward_coefficient_difference_witness_output)=0))))) -> pfrep_left_backward_coefficient_difference_witness_output=pfrep_right_backward_coefficient_difference_witness_output))))))))))))) -> (((forall fom_index_pfp_backward_new_leftleft. (exists fom_gap_pfp_backward_new_leftleft_index_bound. fom_gap_pfp_backward_new_leftleft_index_bound + S (fom_index_pfp_backward_new_leftleft) = Lv) -> exists fom_value_pfp_backward_new_leftleft. ((((exists fom_beta_height_pfp_backward_new_leftleft_entry. fom_beta_height_pfp_backward_new_leftleft_entry + S (fom_value_pfp_backward_new_leftleft) = S ((S (fom_index_pfp_backward_new_leftleft)) * vc)) /\\ exists fom_beta_quotient_pfp_backward_new_leftleft_entry. vb = fom_beta_quotient_pfp_backward_new_leftleft_entry * S ((S (fom_index_pfp_backward_new_leftleft)) * vc) + (fom_value_pfp_backward_new_leftleft))) /\\ (exists fom_gap_pfp_backward_new_leftleft_value_bound. fom_gap_pfp_backward_new_leftleft_value_bound + S (fom_value_pfp_backward_new_leftleft) = p))) /\\ (((forall fom_index_pfp_backward_new_leftright. (exists fom_gap_pfp_backward_new_leftright_index_bound. fom_gap_pfp_backward_new_leftright_index_bound + S (fom_index_pfp_backward_new_leftright) = La) -> exists fom_value_pfp_backward_new_leftright. ((((exists fom_beta_height_pfp_backward_new_leftright_entry. fom_beta_height_pfp_backward_new_leftright_entry + S (fom_value_pfp_backward_new_leftright) = S ((S (fom_index_pfp_backward_new_leftright)) * ac)) /\\ exists fom_beta_quotient_pfp_backward_new_leftright_entry. ab = fom_beta_quotient_pfp_backward_new_leftright_entry * S ((S (fom_index_pfp_backward_new_leftright)) * ac) + (fom_value_pfp_backward_new_leftright))) /\\ (exists fom_gap_pfp_backward_new_leftright_value_bound. fom_gap_pfp_backward_new_leftright_value_bound + S (fom_value_pfp_backward_new_leftright) = p))) /\\ (((((((Lv)=0 \\/ (La)=0) /\\ (((Lx)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((La)=0)) /\\ (((Lv)+(La)=S (Lx)))))))) /\\ ((forall pfc_index_backward_new_leftcoefficients. (exists pfa_gap_backward_new_leftcoefficientsbound. pfa_gap_backward_new_leftcoefficientsbound + S (pfc_index_backward_new_leftcoefficients) = (Lx)) -> exists pfc_value_backward_new_leftcoefficients. ((((exists ff_h_pfp_backward_new_leftcoefficientsentry. ff_h_pfp_backward_new_leftcoefficientsentry + S (pfc_value_backward_new_leftcoefficients) = S ((S (pfc_index_backward_new_leftcoefficients)) * xc)) /\\ exists ff_q_pfp_backward_new_leftcoefficientsentry. xb = ff_q_pfp_backward_new_leftcoefficientsentry * S ((S (pfc_index_backward_new_leftcoefficients)) * xc) + (pfc_value_backward_new_leftcoefficients))) /\\ ((exists pfc_terms_code_backward_new_leftcoefficientscoefficient pfc_terms_scale_backward_new_leftcoefficientscoefficient pfc_natural_sum_backward_new_leftcoefficientscoefficient. ((forall pfc_index_backward_new_leftcoefficientscoefficientdiagonal. (exists pfa_gap_backward_new_leftcoefficientscoefficientdiagonalbound. pfa_gap_backward_new_leftcoefficientscoefficientdiagonalbound + S (pfc_index_backward_new_leftcoefficientscoefficientdiagonal) = (S (pfc_index_backward_new_leftcoefficients))) -> exists pfc_value_backward_new_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_new_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_new_leftcoefficientscoefficientdiagonalentry + S (pfc_value_backward_new_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_new_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_new_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_new_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_new_leftcoefficientscoefficient = ff_q_pfp_backward_new_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_new_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_new_leftcoefficientscoefficient) + (pfc_value_backward_new_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_new_leftcoefficientscoefficientdiagonalterm pfc_left_backward_new_leftcoefficientscoefficientdiagonalterm pfc_right_backward_new_leftcoefficientscoefficientdiagonalterm. (((pfc_index_backward_new_leftcoefficientscoefficientdiagonal)+pfc_complement_backward_new_leftcoefficientscoefficientdiagonalterm=(pfc_index_backward_new_leftcoefficients)) /\\ ((((((exists pfa_gap_backward_new_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_new_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_new_leftcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_backward_new_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_new_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_new_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_new_leftcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_backward_new_leftcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_backward_new_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_new_leftcoefficientscoefficientdiagonal)) * vc) + (pfc_left_backward_new_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_new_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_new_leftcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_backward_new_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_new_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_new_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_new_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_new_leftcoefficientscoefficientdiagonalterm) = (La)) /\\ ((((exists ff_h_pfp_backward_new_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_new_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_new_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_new_leftcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_backward_new_leftcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_backward_new_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_new_leftcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_backward_new_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_new_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_new_leftcoefficientscoefficientdiagonaltermrightoutside+(La)=(pfc_complement_backward_new_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_new_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_new_leftcoefficientscoefficientdiagonal)=pfc_left_backward_new_leftcoefficientscoefficientdiagonalterm*pfc_right_backward_new_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_new_leftcoefficientscoefficientsum fs_v_pfc_backward_new_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_start. fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_start. fs_u_pfc_backward_new_leftcoefficientscoefficientsum = fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_new_leftcoefficientscoefficient) = S ((S (S (pfc_index_backward_new_leftcoefficients))) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_new_leftcoefficientscoefficientsum = fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_new_leftcoefficients))) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum) + (pfc_natural_sum_backward_new_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps = S (pfc_index_backward_new_leftcoefficients)) -> exists fs_a_pfc_backward_new_leftcoefficientscoefficientsum_body_steps fs_r_pfc_backward_new_leftcoefficientscoefficientsum_body_steps fs_s_pfc_backward_new_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_new_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_new_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_new_leftcoefficientscoefficient = fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_new_leftcoefficientscoefficient) + (fs_a_pfc_backward_new_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_new_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_new_leftcoefficientscoefficientsum = fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum) + (fs_r_pfc_backward_new_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_new_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_new_leftcoefficientscoefficientsum = fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum) + (fs_s_pfc_backward_new_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_new_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_new_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_new_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_new_leftcoefficientscoefficientresiduebound. pfa_gap_backward_new_leftcoefficientscoefficientresiduebound + S (pfc_value_backward_new_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_new_leftcoefficientscoefficientresiduecongruence pfa_offset_right_backward_new_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_new_leftcoefficientscoefficient) + (p) * pfa_offset_left_backward_new_leftcoefficientscoefficientresiduecongruence = (pfc_value_backward_new_leftcoefficients) + (p) * pfa_offset_right_backward_new_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_new_rightleft. (exists fom_gap_pfp_backward_new_rightleft_index_bound. fom_gap_pfp_backward_new_rightleft_index_bound + S (fom_index_pfp_backward_new_rightleft) = Lt) -> exists fom_value_pfp_backward_new_rightleft. ((((exists fom_beta_height_pfp_backward_new_rightleft_entry. fom_beta_height_pfp_backward_new_rightleft_entry + S (fom_value_pfp_backward_new_rightleft) = S ((S (fom_index_pfp_backward_new_rightleft)) * tc)) /\\ exists fom_beta_quotient_pfp_backward_new_rightleft_entry. tb = fom_beta_quotient_pfp_backward_new_rightleft_entry * S ((S (fom_index_pfp_backward_new_rightleft)) * tc) + (fom_value_pfp_backward_new_rightleft))) /\\ (exists fom_gap_pfp_backward_new_rightleft_value_bound. fom_gap_pfp_backward_new_rightleft_value_bound + S (fom_value_pfp_backward_new_rightleft) = p))) /\\ (((forall fom_index_pfp_backward_new_rightright. (exists fom_gap_pfp_backward_new_rightright_index_bound. fom_gap_pfp_backward_new_rightright_index_bound + S (fom_index_pfp_backward_new_rightright) = Lb) -> exists fom_value_pfp_backward_new_rightright. ((((exists fom_beta_height_pfp_backward_new_rightright_entry. fom_beta_height_pfp_backward_new_rightright_entry + S (fom_value_pfp_backward_new_rightright) = S ((S (fom_index_pfp_backward_new_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_backward_new_rightright_entry. bb = fom_beta_quotient_pfp_backward_new_rightright_entry * S ((S (fom_index_pfp_backward_new_rightright)) * bc) + (fom_value_pfp_backward_new_rightright))) /\\ (exists fom_gap_pfp_backward_new_rightright_value_bound. fom_gap_pfp_backward_new_rightright_value_bound + S (fom_value_pfp_backward_new_rightright) = p))) /\\ (((((((Lt)=0 \\/ (Lb)=0) /\\ (((Ly)=0)))) \\/ (((~((Lt)=0)) /\\ (((~((Lb)=0)) /\\ (((Lt)+(Lb)=S (Ly)))))))) /\\ ((forall pfc_index_backward_new_rightcoefficients. (exists pfa_gap_backward_new_rightcoefficientsbound. pfa_gap_backward_new_rightcoefficientsbound + S (pfc_index_backward_new_rightcoefficients) = (Ly)) -> exists pfc_value_backward_new_rightcoefficients. ((((exists ff_h_pfp_backward_new_rightcoefficientsentry. ff_h_pfp_backward_new_rightcoefficientsentry + S (pfc_value_backward_new_rightcoefficients) = S ((S (pfc_index_backward_new_rightcoefficients)) * yc)) /\\ exists ff_q_pfp_backward_new_rightcoefficientsentry. yb = ff_q_pfp_backward_new_rightcoefficientsentry * S ((S (pfc_index_backward_new_rightcoefficients)) * yc) + (pfc_value_backward_new_rightcoefficients))) /\\ ((exists pfc_terms_code_backward_new_rightcoefficientscoefficient pfc_terms_scale_backward_new_rightcoefficientscoefficient pfc_natural_sum_backward_new_rightcoefficientscoefficient. ((forall pfc_index_backward_new_rightcoefficientscoefficientdiagonal. (exists pfa_gap_backward_new_rightcoefficientscoefficientdiagonalbound. pfa_gap_backward_new_rightcoefficientscoefficientdiagonalbound + S (pfc_index_backward_new_rightcoefficientscoefficientdiagonal) = (S (pfc_index_backward_new_rightcoefficients))) -> exists pfc_value_backward_new_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_new_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_new_rightcoefficientscoefficientdiagonalentry + S (pfc_value_backward_new_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_new_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_new_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_new_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_new_rightcoefficientscoefficient = ff_q_pfp_backward_new_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_new_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_new_rightcoefficientscoefficient) + (pfc_value_backward_new_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_new_rightcoefficientscoefficientdiagonalterm pfc_left_backward_new_rightcoefficientscoefficientdiagonalterm pfc_right_backward_new_rightcoefficientscoefficientdiagonalterm. (((pfc_index_backward_new_rightcoefficientscoefficientdiagonal)+pfc_complement_backward_new_rightcoefficientscoefficientdiagonalterm=(pfc_index_backward_new_rightcoefficients)) /\\ ((((((exists pfa_gap_backward_new_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_new_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_new_rightcoefficientscoefficientdiagonal) = (Lt)) /\\ ((((exists ff_h_pfp_backward_new_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_new_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_new_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_new_rightcoefficientscoefficientdiagonal)) * tc)) /\\ exists ff_q_pfp_backward_new_rightcoefficientscoefficientdiagonaltermleftentry. tb = ff_q_pfp_backward_new_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_new_rightcoefficientscoefficientdiagonal)) * tc) + (pfc_left_backward_new_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_new_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_new_rightcoefficientscoefficientdiagonaltermleftoutside+(Lt)=(pfc_index_backward_new_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_new_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_new_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_new_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_new_rightcoefficientscoefficientdiagonalterm) = (Lb)) /\\ ((((exists ff_h_pfp_backward_new_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_new_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_new_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_new_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_backward_new_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_backward_new_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_new_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_backward_new_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_new_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_new_rightcoefficientscoefficientdiagonaltermrightoutside+(Lb)=(pfc_complement_backward_new_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_new_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_new_rightcoefficientscoefficientdiagonal)=pfc_left_backward_new_rightcoefficientscoefficientdiagonalterm*pfc_right_backward_new_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_new_rightcoefficientscoefficientsum fs_v_pfc_backward_new_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_start. fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_start. fs_u_pfc_backward_new_rightcoefficientscoefficientsum = fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_new_rightcoefficientscoefficient) = S ((S (S (pfc_index_backward_new_rightcoefficients))) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_new_rightcoefficientscoefficientsum = fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_new_rightcoefficients))) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum) + (pfc_natural_sum_backward_new_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps = S (pfc_index_backward_new_rightcoefficients)) -> exists fs_a_pfc_backward_new_rightcoefficientscoefficientsum_body_steps fs_r_pfc_backward_new_rightcoefficientscoefficientsum_body_steps fs_s_pfc_backward_new_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_new_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_new_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_new_rightcoefficientscoefficient = fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_new_rightcoefficientscoefficient) + (fs_a_pfc_backward_new_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_new_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_new_rightcoefficientscoefficientsum = fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum) + (fs_r_pfc_backward_new_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_new_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_new_rightcoefficientscoefficientsum = fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum) + (fs_s_pfc_backward_new_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_new_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_new_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_new_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_new_rightcoefficientscoefficientresiduebound. pfa_gap_backward_new_rightcoefficientscoefficientresiduebound + S (pfc_value_backward_new_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_new_rightcoefficientscoefficientresiduecongruence pfa_offset_right_backward_new_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_new_rightcoefficientscoefficient) + (p) * pfa_offset_left_backward_new_rightcoefficientscoefficientresiduecongruence = (pfc_value_backward_new_rightcoefficients) + (p) * pfa_offset_right_backward_new_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_associated_productleft. (exists fom_gap_pfp_backward_associated_productleft_index_bound. fom_gap_pfp_backward_associated_productleft_index_bound + S (fom_index_pfp_backward_associated_productleft) = Lw) -> exists fom_value_pfp_backward_associated_productleft. ((((exists fom_beta_height_pfp_backward_associated_productleft_entry. fom_beta_height_pfp_backward_associated_productleft_entry + S (fom_value_pfp_backward_associated_productleft) = S ((S (fom_index_pfp_backward_associated_productleft)) * wc)) /\\ exists fom_beta_quotient_pfp_backward_associated_productleft_entry. wb = fom_beta_quotient_pfp_backward_associated_productleft_entry * S ((S (fom_index_pfp_backward_associated_productleft)) * wc) + (fom_value_pfp_backward_associated_productleft))) /\\ (exists fom_gap_pfp_backward_associated_productleft_value_bound. fom_gap_pfp_backward_associated_productleft_value_bound + S (fom_value_pfp_backward_associated_productleft) = p))) /\\ (((forall fom_index_pfp_backward_associated_productright. (exists fom_gap_pfp_backward_associated_productright_index_bound. fom_gap_pfp_backward_associated_productright_index_bound + S (fom_index_pfp_backward_associated_productright) = Lb) -> exists fom_value_pfp_backward_associated_productright. ((((exists fom_beta_height_pfp_backward_associated_productright_entry. fom_beta_height_pfp_backward_associated_productright_entry + S (fom_value_pfp_backward_associated_productright) = S ((S (fom_index_pfp_backward_associated_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_backward_associated_productright_entry. bb = fom_beta_quotient_pfp_backward_associated_productright_entry * S ((S (fom_index_pfp_backward_associated_productright)) * bc) + (fom_value_pfp_backward_associated_productright))) /\\ (exists fom_gap_pfp_backward_associated_productright_value_bound. fom_gap_pfp_backward_associated_productright_value_bound + S (fom_value_pfp_backward_associated_productright) = p))) /\\ (((((((Lw)=0 \\/ (Lb)=0) /\\ (((Lz)=0)))) \\/ (((~((Lw)=0)) /\\ (((~((Lb)=0)) /\\ (((Lw)+(Lb)=S (Lz)))))))) /\\ ((forall pfc_index_backward_associated_productcoefficients. (exists pfa_gap_backward_associated_productcoefficientsbound. pfa_gap_backward_associated_productcoefficientsbound + S (pfc_index_backward_associated_productcoefficients) = (Lz)) -> exists pfc_value_backward_associated_productcoefficients. ((((exists ff_h_pfp_backward_associated_productcoefficientsentry. ff_h_pfp_backward_associated_productcoefficientsentry + S (pfc_value_backward_associated_productcoefficients) = S ((S (pfc_index_backward_associated_productcoefficients)) * zc)) /\\ exists ff_q_pfp_backward_associated_productcoefficientsentry. zb = ff_q_pfp_backward_associated_productcoefficientsentry * S ((S (pfc_index_backward_associated_productcoefficients)) * zc) + (pfc_value_backward_associated_productcoefficients))) /\\ ((exists pfc_terms_code_backward_associated_productcoefficientscoefficient pfc_terms_scale_backward_associated_productcoefficientscoefficient pfc_natural_sum_backward_associated_productcoefficientscoefficient. ((forall pfc_index_backward_associated_productcoefficientscoefficientdiagonal. (exists pfa_gap_backward_associated_productcoefficientscoefficientdiagonalbound. pfa_gap_backward_associated_productcoefficientscoefficientdiagonalbound + S (pfc_index_backward_associated_productcoefficientscoefficientdiagonal) = (S (pfc_index_backward_associated_productcoefficients))) -> exists pfc_value_backward_associated_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_associated_productcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_associated_productcoefficientscoefficientdiagonalentry + S (pfc_value_backward_associated_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_associated_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_associated_productcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_associated_productcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_associated_productcoefficientscoefficient = ff_q_pfp_backward_associated_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_associated_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_associated_productcoefficientscoefficient) + (pfc_value_backward_associated_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_associated_productcoefficientscoefficientdiagonalterm pfc_left_backward_associated_productcoefficientscoefficientdiagonalterm pfc_right_backward_associated_productcoefficientscoefficientdiagonalterm. (((pfc_index_backward_associated_productcoefficientscoefficientdiagonal)+pfc_complement_backward_associated_productcoefficientscoefficientdiagonalterm=(pfc_index_backward_associated_productcoefficients)) /\\ ((((((exists pfa_gap_backward_associated_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_associated_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_associated_productcoefficientscoefficientdiagonal) = (Lw)) /\\ ((((exists ff_h_pfp_backward_associated_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_associated_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_associated_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_associated_productcoefficientscoefficientdiagonal)) * wc)) /\\ exists ff_q_pfp_backward_associated_productcoefficientscoefficientdiagonaltermleftentry. wb = ff_q_pfp_backward_associated_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_associated_productcoefficientscoefficientdiagonal)) * wc) + (pfc_left_backward_associated_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_associated_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_associated_productcoefficientscoefficientdiagonaltermleftoutside+(Lw)=(pfc_index_backward_associated_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_associated_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_associated_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_associated_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_associated_productcoefficientscoefficientdiagonalterm) = (Lb)) /\\ ((((exists ff_h_pfp_backward_associated_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_associated_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_associated_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_associated_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_backward_associated_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_backward_associated_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_associated_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_backward_associated_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_associated_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_associated_productcoefficientscoefficientdiagonaltermrightoutside+(Lb)=(pfc_complement_backward_associated_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_associated_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_associated_productcoefficientscoefficientdiagonal)=pfc_left_backward_associated_productcoefficientscoefficientdiagonalterm*pfc_right_backward_associated_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_associated_productcoefficientscoefficientsum fs_v_pfc_backward_associated_productcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_start. fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_start. fs_u_pfc_backward_associated_productcoefficientscoefficientsum = fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_associated_productcoefficientscoefficient) = S ((S (S (pfc_index_backward_associated_productcoefficients))) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_associated_productcoefficientscoefficientsum = fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_associated_productcoefficients))) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum) + (pfc_natural_sum_backward_associated_productcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps = S (pfc_index_backward_associated_productcoefficients)) -> exists fs_a_pfc_backward_associated_productcoefficientscoefficientsum_body_steps fs_r_pfc_backward_associated_productcoefficientscoefficientsum_body_steps fs_s_pfc_backward_associated_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_associated_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_associated_productcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_associated_productcoefficientscoefficient = fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_associated_productcoefficientscoefficient) + (fs_a_pfc_backward_associated_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_associated_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_associated_productcoefficientscoefficientsum = fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum) + (fs_r_pfc_backward_associated_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_associated_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_associated_productcoefficientscoefficientsum = fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum) + (fs_s_pfc_backward_associated_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_associated_productcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_associated_productcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_associated_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_associated_productcoefficientscoefficientresiduebound. pfa_gap_backward_associated_productcoefficientscoefficientresiduebound + S (pfc_value_backward_associated_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_associated_productcoefficientscoefficientresiduecongruence pfa_offset_right_backward_associated_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_associated_productcoefficientscoefficient) + (p) * pfa_offset_left_backward_associated_productcoefficientscoefficientresiduecongruence = (pfc_value_backward_associated_productcoefficients) + (p) * pfa_offset_right_backward_associated_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_distributed_productleft. (exists fom_gap_pfp_backward_distributed_productleft_index_bound. fom_gap_pfp_backward_distributed_productleft_index_bound + S (fom_index_pfp_backward_distributed_productleft) = Lv) -> exists fom_value_pfp_backward_distributed_productleft. ((((exists fom_beta_height_pfp_backward_distributed_productleft_entry. fom_beta_height_pfp_backward_distributed_productleft_entry + S (fom_value_pfp_backward_distributed_productleft) = S ((S (fom_index_pfp_backward_distributed_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_backward_distributed_productleft_entry. vb = fom_beta_quotient_pfp_backward_distributed_productleft_entry * S ((S (fom_index_pfp_backward_distributed_productleft)) * vc) + (fom_value_pfp_backward_distributed_productleft))) /\\ (exists fom_gap_pfp_backward_distributed_productleft_value_bound. fom_gap_pfp_backward_distributed_productleft_value_bound + S (fom_value_pfp_backward_distributed_productleft) = p))) /\\ (((forall fom_index_pfp_backward_distributed_productright. (exists fom_gap_pfp_backward_distributed_productright_index_bound. fom_gap_pfp_backward_distributed_productright_index_bound + S (fom_index_pfp_backward_distributed_productright) = Lp) -> exists fom_value_pfp_backward_distributed_productright. ((((exists fom_beta_height_pfp_backward_distributed_productright_entry. fom_beta_height_pfp_backward_distributed_productright_entry + S (fom_value_pfp_backward_distributed_productright) = S ((S (fom_index_pfp_backward_distributed_productright)) * pc)) /\\ exists fom_beta_quotient_pfp_backward_distributed_productright_entry. pb = fom_beta_quotient_pfp_backward_distributed_productright_entry * S ((S (fom_index_pfp_backward_distributed_productright)) * pc) + (fom_value_pfp_backward_distributed_productright))) /\\ (exists fom_gap_pfp_backward_distributed_productright_value_bound. fom_gap_pfp_backward_distributed_productright_value_bound + S (fom_value_pfp_backward_distributed_productright) = p))) /\\ (((((((Lv)=0 \\/ (Lp)=0) /\\ (((Lh)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((Lp)=0)) /\\ (((Lv)+(Lp)=S (Lh)))))))) /\\ ((forall pfc_index_backward_distributed_productcoefficients. (exists pfa_gap_backward_distributed_productcoefficientsbound. pfa_gap_backward_distributed_productcoefficientsbound + S (pfc_index_backward_distributed_productcoefficients) = (Lh)) -> exists pfc_value_backward_distributed_productcoefficients. ((((exists ff_h_pfp_backward_distributed_productcoefficientsentry. ff_h_pfp_backward_distributed_productcoefficientsentry + S (pfc_value_backward_distributed_productcoefficients) = S ((S (pfc_index_backward_distributed_productcoefficients)) * hc)) /\\ exists ff_q_pfp_backward_distributed_productcoefficientsentry. hb = ff_q_pfp_backward_distributed_productcoefficientsentry * S ((S (pfc_index_backward_distributed_productcoefficients)) * hc) + (pfc_value_backward_distributed_productcoefficients))) /\\ ((exists pfc_terms_code_backward_distributed_productcoefficientscoefficient pfc_terms_scale_backward_distributed_productcoefficientscoefficient pfc_natural_sum_backward_distributed_productcoefficientscoefficient. ((forall pfc_index_backward_distributed_productcoefficientscoefficientdiagonal. (exists pfa_gap_backward_distributed_productcoefficientscoefficientdiagonalbound. pfa_gap_backward_distributed_productcoefficientscoefficientdiagonalbound + S (pfc_index_backward_distributed_productcoefficientscoefficientdiagonal) = (S (pfc_index_backward_distributed_productcoefficients))) -> exists pfc_value_backward_distributed_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_distributed_productcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_distributed_productcoefficientscoefficientdiagonalentry + S (pfc_value_backward_distributed_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_distributed_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_distributed_productcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_distributed_productcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_distributed_productcoefficientscoefficient = ff_q_pfp_backward_distributed_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_distributed_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_distributed_productcoefficientscoefficient) + (pfc_value_backward_distributed_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_distributed_productcoefficientscoefficientdiagonalterm pfc_left_backward_distributed_productcoefficientscoefficientdiagonalterm pfc_right_backward_distributed_productcoefficientscoefficientdiagonalterm. (((pfc_index_backward_distributed_productcoefficientscoefficientdiagonal)+pfc_complement_backward_distributed_productcoefficientscoefficientdiagonalterm=(pfc_index_backward_distributed_productcoefficients)) /\\ ((((((exists pfa_gap_backward_distributed_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_distributed_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_distributed_productcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_distributed_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_distributed_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_distributed_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_backward_distributed_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_distributed_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_distributed_productcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_backward_distributed_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_distributed_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_distributed_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_distributed_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_distributed_productcoefficientscoefficientdiagonalterm) = (Lp)) /\\ ((((exists ff_h_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_distributed_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_distributed_productcoefficientscoefficientdiagonalterm)) * pc)) /\\ exists ff_q_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermrightentry. pb = ff_q_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_distributed_productcoefficientscoefficientdiagonalterm)) * pc) + (pfc_right_backward_distributed_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_distributed_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_distributed_productcoefficientscoefficientdiagonaltermrightoutside+(Lp)=(pfc_complement_backward_distributed_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_distributed_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_distributed_productcoefficientscoefficientdiagonal)=pfc_left_backward_distributed_productcoefficientscoefficientdiagonalterm*pfc_right_backward_distributed_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_distributed_productcoefficientscoefficientsum fs_v_pfc_backward_distributed_productcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_start. fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_start. fs_u_pfc_backward_distributed_productcoefficientscoefficientsum = fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_distributed_productcoefficientscoefficient) = S ((S (S (pfc_index_backward_distributed_productcoefficients))) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_distributed_productcoefficientscoefficientsum = fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_distributed_productcoefficients))) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum) + (pfc_natural_sum_backward_distributed_productcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps = S (pfc_index_backward_distributed_productcoefficients)) -> exists fs_a_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps fs_r_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps fs_s_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_distributed_productcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_distributed_productcoefficientscoefficient = fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_distributed_productcoefficientscoefficient) + (fs_a_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_distributed_productcoefficientscoefficientsum = fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum) + (fs_r_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_distributed_productcoefficientscoefficientsum = fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum) + (fs_s_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_distributed_productcoefficientscoefficientresiduebound. pfa_gap_backward_distributed_productcoefficientscoefficientresiduebound + S (pfc_value_backward_distributed_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_distributed_productcoefficientscoefficientresiduecongruence pfa_offset_right_backward_distributed_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_distributed_productcoefficientscoefficient) + (p) * pfa_offset_left_backward_distributed_productcoefficientscoefficientresiduecongruence = (pfc_value_backward_distributed_productcoefficients) + (p) * pfa_offset_right_backward_distributed_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_result_left_bounded. (exists fom_gap_pfp_backward_result_left_bounded_index_bound. fom_gap_pfp_backward_result_left_bounded_index_bound + S (fom_index_pfp_backward_result_left_bounded) = Lx) -> exists fom_value_pfp_backward_result_left_bounded. ((((exists fom_beta_height_pfp_backward_result_left_bounded_entry. fom_beta_height_pfp_backward_result_left_bounded_entry + S (fom_value_pfp_backward_result_left_bounded) = S ((S (fom_index_pfp_backward_result_left_bounded)) * xc)) /\\ exists fom_beta_quotient_pfp_backward_result_left_bounded_entry. xb = fom_beta_quotient_pfp_backward_result_left_bounded_entry * S ((S (fom_index_pfp_backward_result_left_bounded)) * xc) + (fom_value_pfp_backward_result_left_bounded))) /\\ (exists fom_gap_pfp_backward_result_left_bounded_value_bound. fom_gap_pfp_backward_result_left_bounded_value_bound + S (fom_value_pfp_backward_result_left_bounded) = p))) /\\ (((forall fom_index_pfp_backward_result_right_bounded. (exists fom_gap_pfp_backward_result_right_bounded_index_bound. fom_gap_pfp_backward_result_right_bounded_index_bound + S (fom_index_pfp_backward_result_right_bounded) = Ly) -> exists fom_value_pfp_backward_result_right_bounded. ((((exists fom_beta_height_pfp_backward_result_right_bounded_entry. fom_beta_height_pfp_backward_result_right_bounded_entry + S (fom_value_pfp_backward_result_right_bounded) = S ((S (fom_index_pfp_backward_result_right_bounded)) * yc)) /\\ exists fom_beta_quotient_pfp_backward_result_right_bounded_entry. yb = fom_beta_quotient_pfp_backward_result_right_bounded_entry * S ((S (fom_index_pfp_backward_result_right_bounded)) * yc) + (fom_value_pfp_backward_result_right_bounded))) /\\ (exists fom_gap_pfp_backward_result_right_bounded_value_bound. fom_gap_pfp_backward_result_right_bounded_value_bound + S (fom_value_pfp_backward_result_right_bounded) = p))) /\\ (((forall fom_index_pfp_backward_result_result_bounded. (exists fom_gap_pfp_backward_result_result_bounded_index_bound. fom_gap_pfp_backward_result_result_bounded_index_bound + S (fom_index_pfp_backward_result_result_bounded) = Lg) -> exists fom_value_pfp_backward_result_result_bounded. ((((exists fom_beta_height_pfp_backward_result_result_bounded_entry. fom_beta_height_pfp_backward_result_result_bounded_entry + S (fom_value_pfp_backward_result_result_bounded) = S ((S (fom_index_pfp_backward_result_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_backward_result_result_bounded_entry. gb = fom_beta_quotient_pfp_backward_result_result_bounded_entry * S ((S (fom_index_pfp_backward_result_result_bounded)) * gc) + (fom_value_pfp_backward_result_result_bounded))) /\\ (exists fom_gap_pfp_backward_result_result_bounded_value_bound. fom_gap_pfp_backward_result_result_bounded_value_bound + S (fom_value_pfp_backward_result_result_bounded) = p))) /\\ ((exists pfaa_left_b_backward_result pfaa_left_c_backward_result pfaa_right_b_backward_result pfaa_right_c_backward_result pfaa_sum_b_backward_result pfaa_sum_c_backward_result pfaa_length_backward_result. ((((forall pfrep_power_backward_result_witness_common_left pfrep_left_backward_result_witness_common_left pfrep_right_backward_result_witness_common_left. ((exists pfrep_position_backward_result_witness_common_leftfirst. ((pfrep_position_backward_result_witness_common_leftfirst+S (pfrep_power_backward_result_witness_common_left)=(Lx)) /\\ ((((exists ff_h_pfp_backward_result_witness_common_leftfirstentry. ff_h_pfp_backward_result_witness_common_leftfirstentry + S (pfrep_left_backward_result_witness_common_left) = S ((S (pfrep_position_backward_result_witness_common_leftfirst)) * xc)) /\\ exists ff_q_pfp_backward_result_witness_common_leftfirstentry. xb = ff_q_pfp_backward_result_witness_common_leftfirstentry * S ((S (pfrep_position_backward_result_witness_common_leftfirst)) * xc) + (pfrep_left_backward_result_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_result_witness_common_leftfirstoutside. pfrep_gap_backward_result_witness_common_leftfirstoutside+(Lx)=(pfrep_power_backward_result_witness_common_left)) /\\ (((pfrep_left_backward_result_witness_common_left)=0))))) -> ((exists pfrep_position_backward_result_witness_common_leftsecond. ((pfrep_position_backward_result_witness_common_leftsecond+S (pfrep_power_backward_result_witness_common_left)=(pfaa_length_backward_result)) /\\ ((((exists ff_h_pfp_backward_result_witness_common_leftsecondentry. ff_h_pfp_backward_result_witness_common_leftsecondentry + S (pfrep_right_backward_result_witness_common_left) = S ((S (pfrep_position_backward_result_witness_common_leftsecond)) * pfaa_left_c_backward_result)) /\\ exists ff_q_pfp_backward_result_witness_common_leftsecondentry. pfaa_left_b_backward_result = ff_q_pfp_backward_result_witness_common_leftsecondentry * S ((S (pfrep_position_backward_result_witness_common_leftsecond)) * pfaa_left_c_backward_result) + (pfrep_right_backward_result_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_result_witness_common_leftsecondoutside. pfrep_gap_backward_result_witness_common_leftsecondoutside+(pfaa_length_backward_result)=(pfrep_power_backward_result_witness_common_left)) /\\ (((pfrep_right_backward_result_witness_common_left)=0))))) -> pfrep_left_backward_result_witness_common_left=pfrep_right_backward_result_witness_common_left) /\\ ((forall pfrep_power_backward_result_witness_common_right pfrep_left_backward_result_witness_common_right pfrep_right_backward_result_witness_common_right. ((exists pfrep_position_backward_result_witness_common_rightfirst. ((pfrep_position_backward_result_witness_common_rightfirst+S (pfrep_power_backward_result_witness_common_right)=(Ly)) /\\ ((((exists ff_h_pfp_backward_result_witness_common_rightfirstentry. ff_h_pfp_backward_result_witness_common_rightfirstentry + S (pfrep_left_backward_result_witness_common_right) = S ((S (pfrep_position_backward_result_witness_common_rightfirst)) * yc)) /\\ exists ff_q_pfp_backward_result_witness_common_rightfirstentry. yb = ff_q_pfp_backward_result_witness_common_rightfirstentry * S ((S (pfrep_position_backward_result_witness_common_rightfirst)) * yc) + (pfrep_left_backward_result_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_result_witness_common_rightfirstoutside. pfrep_gap_backward_result_witness_common_rightfirstoutside+(Ly)=(pfrep_power_backward_result_witness_common_right)) /\\ (((pfrep_left_backward_result_witness_common_right)=0))))) -> ((exists pfrep_position_backward_result_witness_common_rightsecond. ((pfrep_position_backward_result_witness_common_rightsecond+S (pfrep_power_backward_result_witness_common_right)=(pfaa_length_backward_result)) /\\ ((((exists ff_h_pfp_backward_result_witness_common_rightsecondentry. ff_h_pfp_backward_result_witness_common_rightsecondentry + S (pfrep_right_backward_result_witness_common_right) = S ((S (pfrep_position_backward_result_witness_common_rightsecond)) * pfaa_right_c_backward_result)) /\\ exists ff_q_pfp_backward_result_witness_common_rightsecondentry. pfaa_right_b_backward_result = ff_q_pfp_backward_result_witness_common_rightsecondentry * S ((S (pfrep_position_backward_result_witness_common_rightsecond)) * pfaa_right_c_backward_result) + (pfrep_right_backward_result_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_result_witness_common_rightsecondoutside. pfrep_gap_backward_result_witness_common_rightsecondoutside+(pfaa_length_backward_result)=(pfrep_power_backward_result_witness_common_right)) /\\ (((pfrep_right_backward_result_witness_common_right)=0))))) -> pfrep_left_backward_result_witness_common_right=pfrep_right_backward_result_witness_common_right)))) /\\ (((forall pfp_index_backward_result_witness_operation. (exists pfa_gap_backward_result_witness_operationindex. pfa_gap_backward_result_witness_operationindex + S (pfp_index_backward_result_witness_operation) = (pfaa_length_backward_result)) -> exists pfp_left_backward_result_witness_operation pfp_right_backward_result_witness_operation pfp_value_backward_result_witness_operation. ((((exists ff_h_pfp_backward_result_witness_operationleft. ff_h_pfp_backward_result_witness_operationleft + S (pfp_left_backward_result_witness_operation) = S ((S (pfp_index_backward_result_witness_operation)) * pfaa_left_c_backward_result)) /\\ exists ff_q_pfp_backward_result_witness_operationleft. pfaa_left_b_backward_result = ff_q_pfp_backward_result_witness_operationleft * S ((S (pfp_index_backward_result_witness_operation)) * pfaa_left_c_backward_result) + (pfp_left_backward_result_witness_operation))) /\\ (((((exists ff_h_pfp_backward_result_witness_operationright. ff_h_pfp_backward_result_witness_operationright + S (pfp_right_backward_result_witness_operation) = S ((S (pfp_index_backward_result_witness_operation)) * pfaa_right_c_backward_result)) /\\ exists ff_q_pfp_backward_result_witness_operationright. pfaa_right_b_backward_result = ff_q_pfp_backward_result_witness_operationright * S ((S (pfp_index_backward_result_witness_operation)) * pfaa_right_c_backward_result) + (pfp_right_backward_result_witness_operation))) /\\ (((((exists ff_h_pfp_backward_result_witness_operationtarget. ff_h_pfp_backward_result_witness_operationtarget + S (pfp_value_backward_result_witness_operation) = S ((S (pfp_index_backward_result_witness_operation)) * pfaa_sum_c_backward_result)) /\\ exists ff_q_pfp_backward_result_witness_operationtarget. pfaa_sum_b_backward_result = ff_q_pfp_backward_result_witness_operationtarget * S ((S (pfp_index_backward_result_witness_operation)) * pfaa_sum_c_backward_result) + (pfp_value_backward_result_witness_operation))) /\\ ((((exists pfa_gap_backward_result_witness_operationoperationleft. pfa_gap_backward_result_witness_operationoperationleft + S (pfp_left_backward_result_witness_operation) = (p)) /\\ (((exists pfa_gap_backward_result_witness_operationoperationright. pfa_gap_backward_result_witness_operationoperationright + S (pfp_right_backward_result_witness_operation) = (p)) /\\ ((((exists pfa_gap_backward_result_witness_operationoperationresultbound. pfa_gap_backward_result_witness_operationoperationresultbound + S (pfp_value_backward_result_witness_operation) = (p)) /\\ ((exists pfa_offset_left_backward_result_witness_operationoperationresultcongruence pfa_offset_right_backward_result_witness_operationoperationresultcongruence. ((pfp_left_backward_result_witness_operation) + (pfp_right_backward_result_witness_operation)) + (p) * pfa_offset_left_backward_result_witness_operationoperationresultcongruence = (pfp_value_backward_result_witness_operation) + (p) * pfa_offset_right_backward_result_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_backward_result_witness_output pfrep_left_backward_result_witness_output pfrep_right_backward_result_witness_output. ((exists pfrep_position_backward_result_witness_outputfirst. ((pfrep_position_backward_result_witness_outputfirst+S (pfrep_power_backward_result_witness_output)=(pfaa_length_backward_result)) /\\ ((((exists ff_h_pfp_backward_result_witness_outputfirstentry. ff_h_pfp_backward_result_witness_outputfirstentry + S (pfrep_left_backward_result_witness_output) = S ((S (pfrep_position_backward_result_witness_outputfirst)) * pfaa_sum_c_backward_result)) /\\ exists ff_q_pfp_backward_result_witness_outputfirstentry. pfaa_sum_b_backward_result = ff_q_pfp_backward_result_witness_outputfirstentry * S ((S (pfrep_position_backward_result_witness_outputfirst)) * pfaa_sum_c_backward_result) + (pfrep_left_backward_result_witness_output)))))) \\/ (((exists pfrep_gap_backward_result_witness_outputfirstoutside. pfrep_gap_backward_result_witness_outputfirstoutside+(pfaa_length_backward_result)=(pfrep_power_backward_result_witness_output)) /\\ (((pfrep_left_backward_result_witness_output)=0))))) -> ((exists pfrep_position_backward_result_witness_outputsecond. ((pfrep_position_backward_result_witness_outputsecond+S (pfrep_power_backward_result_witness_output)=(Lg)) /\\ ((((exists ff_h_pfp_backward_result_witness_outputsecondentry. ff_h_pfp_backward_result_witness_outputsecondentry + S (pfrep_right_backward_result_witness_output) = S ((S (pfrep_position_backward_result_witness_outputsecond)) * gc)) /\\ exists ff_q_pfp_backward_result_witness_outputsecondentry. gb = ff_q_pfp_backward_result_witness_outputsecondentry * S ((S (pfrep_position_backward_result_witness_outputsecond)) * gc) + (pfrep_right_backward_result_witness_output)))))) \\/ (((exists pfrep_gap_backward_result_witness_outputsecondoutside. pfrep_gap_backward_result_witness_outputsecondoutside+(Lg)=(pfrep_power_backward_result_witness_output)) /\\ (((pfrep_right_backward_result_witness_output)=0))))) -> pfrep_left_backward_result_witness_output=pfrep_right_backward_result_witness_output)))))))))))))", "statement_sha256": "402455ee9077bd171093cbd17381141a628a864ecc05cdba7b9fc2369f80cf2a", "summary": "From genuine products and the actual difference T=U-V*Q, prove G=V*A+T*B by ordered distributivity, actual convolution associativity and aligned addition reassociation; 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(~((p) = 1) /\\ forall pfa_factor_left_backward_prime pfa_factor_right_backward_prime. (p) = pfa_factor_left_backward_prime * pfa_factor_right_backward_prime -> pfa_factor_left_backward_prime = 1 \\/ pfa_factor_right_backward_prime = 1) -> (((forall fom_index_pfp_backward_division_productleft. (exists fom_gap_pfp_backward_division_productleft_index_bound. fom_gap_pfp_backward_division_productleft_index_bound + S (fom_index_pfp_backward_division_productleft) = Lq) -> exists fom_value_pfp_backward_division_productleft. ((((exists fom_beta_height_pfp_backward_division_productleft_entry. fom_beta_height_pfp_backward_division_productleft_entry + S (fom_value_pfp_backward_division_productleft) = S ((S (fom_index_pfp_backward_division_productleft)) * qc)) /\\ exists fom_beta_quotient_pfp_backward_division_productleft_entry. qb = fom_beta_quotient_pfp_backward_division_productleft_entry * S ((S (fom_index_pfp_backward_division_productleft)) * qc) + (fom_value_pfp_backward_division_productleft))) /\\ (exists fom_gap_pfp_backward_division_productleft_value_bound. fom_gap_pfp_backward_division_productleft_value_bound + S (fom_value_pfp_backward_division_productleft) = p))) /\\ (((forall fom_index_pfp_backward_division_productright. (exists fom_gap_pfp_backward_division_productright_index_bound. fom_gap_pfp_backward_division_productright_index_bound + S (fom_index_pfp_backward_division_productright) = Lb) -> exists fom_value_pfp_backward_division_productright. ((((exists fom_beta_height_pfp_backward_division_productright_entry. fom_beta_height_pfp_backward_division_productright_entry + S (fom_value_pfp_backward_division_productright) = S ((S (fom_index_pfp_backward_division_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_backward_division_productright_entry. bb = fom_beta_quotient_pfp_backward_division_productright_entry * S ((S (fom_index_pfp_backward_division_productright)) * bc) + (fom_value_pfp_backward_division_productright))) /\\ (exists fom_gap_pfp_backward_division_productright_value_bound. fom_gap_pfp_backward_division_productright_value_bound + S (fom_value_pfp_backward_division_productright) = p))) /\\ (((((((Lq)=0 \\/ (Lb)=0) /\\ (((Lp)=0)))) \\/ (((~((Lq)=0)) /\\ (((~((Lb)=0)) /\\ (((Lq)+(Lb)=S (Lp)))))))) /\\ ((forall pfc_index_backward_division_productcoefficients. (exists pfa_gap_backward_division_productcoefficientsbound. pfa_gap_backward_division_productcoefficientsbound + S (pfc_index_backward_division_productcoefficients) = (Lp)) -> exists pfc_value_backward_division_productcoefficients. ((((exists ff_h_pfp_backward_division_productcoefficientsentry. ff_h_pfp_backward_division_productcoefficientsentry + S (pfc_value_backward_division_productcoefficients) = S ((S (pfc_index_backward_division_productcoefficients)) * pc)) /\\ exists ff_q_pfp_backward_division_productcoefficientsentry. pb = ff_q_pfp_backward_division_productcoefficientsentry * S ((S (pfc_index_backward_division_productcoefficients)) * pc) + (pfc_value_backward_division_productcoefficients))) /\\ ((exists pfc_terms_code_backward_division_productcoefficientscoefficient pfc_terms_scale_backward_division_productcoefficientscoefficient pfc_natural_sum_backward_division_productcoefficientscoefficient. ((forall pfc_index_backward_division_productcoefficientscoefficientdiagonal. (exists pfa_gap_backward_division_productcoefficientscoefficientdiagonalbound. pfa_gap_backward_division_productcoefficientscoefficientdiagonalbound + S (pfc_index_backward_division_productcoefficientscoefficientdiagonal) = (S (pfc_index_backward_division_productcoefficients))) -> exists pfc_value_backward_division_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_division_productcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_division_productcoefficientscoefficientdiagonalentry + S (pfc_value_backward_division_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_division_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_division_productcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_division_productcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_division_productcoefficientscoefficient = ff_q_pfp_backward_division_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_division_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_division_productcoefficientscoefficient) + (pfc_value_backward_division_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_division_productcoefficientscoefficientdiagonalterm pfc_left_backward_division_productcoefficientscoefficientdiagonalterm pfc_right_backward_division_productcoefficientscoefficientdiagonalterm. (((pfc_index_backward_division_productcoefficientscoefficientdiagonal)+pfc_complement_backward_division_productcoefficientscoefficientdiagonalterm=(pfc_index_backward_division_productcoefficients)) /\\ ((((((exists pfa_gap_backward_division_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_division_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_division_productcoefficientscoefficientdiagonal) = (Lq)) /\\ ((((exists ff_h_pfp_backward_division_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_division_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_division_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_division_productcoefficientscoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_backward_division_productcoefficientscoefficientdiagonaltermleftentry. qb = ff_q_pfp_backward_division_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_division_productcoefficientscoefficientdiagonal)) * qc) + (pfc_left_backward_division_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_division_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_division_productcoefficientscoefficientdiagonaltermleftoutside+(Lq)=(pfc_index_backward_division_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_division_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_division_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_division_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_division_productcoefficientscoefficientdiagonalterm) = (Lb)) /\\ ((((exists ff_h_pfp_backward_division_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_division_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_division_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_division_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_backward_division_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_backward_division_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_division_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_backward_division_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_division_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_division_productcoefficientscoefficientdiagonaltermrightoutside+(Lb)=(pfc_complement_backward_division_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_division_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_division_productcoefficientscoefficientdiagonal)=pfc_left_backward_division_productcoefficientscoefficientdiagonalterm*pfc_right_backward_division_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_division_productcoefficientscoefficientsum fs_v_pfc_backward_division_productcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_start. fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_division_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_start. fs_u_pfc_backward_division_productcoefficientscoefficientsum = fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_division_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_division_productcoefficientscoefficient) = S ((S (S (pfc_index_backward_division_productcoefficients))) * fs_v_pfc_backward_division_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_division_productcoefficientscoefficientsum = fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_division_productcoefficients))) * fs_v_pfc_backward_division_productcoefficientscoefficientsum) + (pfc_natural_sum_backward_division_productcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_division_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_division_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps = S (pfc_index_backward_division_productcoefficients)) -> exists fs_a_pfc_backward_division_productcoefficientscoefficientsum_body_steps fs_r_pfc_backward_division_productcoefficientscoefficientsum_body_steps fs_s_pfc_backward_division_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_division_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_division_productcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_division_productcoefficientscoefficient = fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_division_productcoefficientscoefficient) + (fs_a_pfc_backward_division_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_division_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_division_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_division_productcoefficientscoefficientsum = fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_division_productcoefficientscoefficientsum) + (fs_r_pfc_backward_division_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_division_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_division_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_division_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_division_productcoefficientscoefficientsum = fs_q_pfc_backward_division_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_division_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_division_productcoefficientscoefficientsum) + (fs_s_pfc_backward_division_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_division_productcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_division_productcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_division_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_division_productcoefficientscoefficientresiduebound. pfa_gap_backward_division_productcoefficientscoefficientresiduebound + S (pfc_value_backward_division_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_division_productcoefficientscoefficientresiduecongruence pfa_offset_right_backward_division_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_division_productcoefficientscoefficient) + (p) * pfa_offset_left_backward_division_productcoefficientscoefficientresiduecongruence = (pfc_value_backward_division_productcoefficients) + (p) * pfa_offset_right_backward_division_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_division_sum_left_bounded. (exists fom_gap_pfp_backward_division_sum_left_bounded_index_bound. fom_gap_pfp_backward_division_sum_left_bounded_index_bound + S (fom_index_pfp_backward_division_sum_left_bounded) = Lp) -> exists fom_value_pfp_backward_division_sum_left_bounded. ((((exists fom_beta_height_pfp_backward_division_sum_left_bounded_entry. fom_beta_height_pfp_backward_division_sum_left_bounded_entry + S (fom_value_pfp_backward_division_sum_left_bounded) = S ((S (fom_index_pfp_backward_division_sum_left_bounded)) * pc)) /\\ exists fom_beta_quotient_pfp_backward_division_sum_left_bounded_entry. pb = fom_beta_quotient_pfp_backward_division_sum_left_bounded_entry * S ((S (fom_index_pfp_backward_division_sum_left_bounded)) * pc) + (fom_value_pfp_backward_division_sum_left_bounded))) /\\ (exists fom_gap_pfp_backward_division_sum_left_bounded_value_bound. fom_gap_pfp_backward_division_sum_left_bounded_value_bound + S (fom_value_pfp_backward_division_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_backward_division_sum_right_bounded. (exists fom_gap_pfp_backward_division_sum_right_bounded_index_bound. fom_gap_pfp_backward_division_sum_right_bounded_index_bound + S (fom_index_pfp_backward_division_sum_right_bounded) = Lr) -> exists fom_value_pfp_backward_division_sum_right_bounded. ((((exists fom_beta_height_pfp_backward_division_sum_right_bounded_entry. fom_beta_height_pfp_backward_division_sum_right_bounded_entry + S (fom_value_pfp_backward_division_sum_right_bounded) = S ((S (fom_index_pfp_backward_division_sum_right_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_backward_division_sum_right_bounded_entry. rb = fom_beta_quotient_pfp_backward_division_sum_right_bounded_entry * S ((S (fom_index_pfp_backward_division_sum_right_bounded)) * rc) + (fom_value_pfp_backward_division_sum_right_bounded))) /\\ (exists fom_gap_pfp_backward_division_sum_right_bounded_value_bound. fom_gap_pfp_backward_division_sum_right_bounded_value_bound + S (fom_value_pfp_backward_division_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_backward_division_sum_result_bounded. (exists fom_gap_pfp_backward_division_sum_result_bounded_index_bound. fom_gap_pfp_backward_division_sum_result_bounded_index_bound + S (fom_index_pfp_backward_division_sum_result_bounded) = La) -> exists fom_value_pfp_backward_division_sum_result_bounded. ((((exists fom_beta_height_pfp_backward_division_sum_result_bounded_entry. fom_beta_height_pfp_backward_division_sum_result_bounded_entry + S (fom_value_pfp_backward_division_sum_result_bounded) = S ((S (fom_index_pfp_backward_division_sum_result_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_backward_division_sum_result_bounded_entry. ab = fom_beta_quotient_pfp_backward_division_sum_result_bounded_entry * S ((S (fom_index_pfp_backward_division_sum_result_bounded)) * ac) + (fom_value_pfp_backward_division_sum_result_bounded))) /\\ (exists fom_gap_pfp_backward_division_sum_result_bounded_value_bound. fom_gap_pfp_backward_division_sum_result_bounded_value_bound + S (fom_value_pfp_backward_division_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_backward_division_sum pfaa_left_c_backward_division_sum pfaa_right_b_backward_division_sum pfaa_right_c_backward_division_sum pfaa_sum_b_backward_division_sum pfaa_sum_c_backward_division_sum pfaa_length_backward_division_sum. ((((forall pfrep_power_backward_division_sum_witness_common_left pfrep_left_backward_division_sum_witness_common_left pfrep_right_backward_division_sum_witness_common_left. ((exists pfrep_position_backward_division_sum_witness_common_leftfirst. ((pfrep_position_backward_division_sum_witness_common_leftfirst+S (pfrep_power_backward_division_sum_witness_common_left)=(Lp)) /\\ ((((exists ff_h_pfp_backward_division_sum_witness_common_leftfirstentry. ff_h_pfp_backward_division_sum_witness_common_leftfirstentry + S (pfrep_left_backward_division_sum_witness_common_left) = S ((S (pfrep_position_backward_division_sum_witness_common_leftfirst)) * pc)) /\\ exists ff_q_pfp_backward_division_sum_witness_common_leftfirstentry. pb = ff_q_pfp_backward_division_sum_witness_common_leftfirstentry * S ((S (pfrep_position_backward_division_sum_witness_common_leftfirst)) * pc) + (pfrep_left_backward_division_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_division_sum_witness_common_leftfirstoutside. pfrep_gap_backward_division_sum_witness_common_leftfirstoutside+(Lp)=(pfrep_power_backward_division_sum_witness_common_left)) /\\ (((pfrep_left_backward_division_sum_witness_common_left)=0))))) -> ((exists pfrep_position_backward_division_sum_witness_common_leftsecond. ((pfrep_position_backward_division_sum_witness_common_leftsecond+S (pfrep_power_backward_division_sum_witness_common_left)=(pfaa_length_backward_division_sum)) /\\ ((((exists ff_h_pfp_backward_division_sum_witness_common_leftsecondentry. ff_h_pfp_backward_division_sum_witness_common_leftsecondentry + S (pfrep_right_backward_division_sum_witness_common_left) = S ((S (pfrep_position_backward_division_sum_witness_common_leftsecond)) * pfaa_left_c_backward_division_sum)) /\\ exists ff_q_pfp_backward_division_sum_witness_common_leftsecondentry. pfaa_left_b_backward_division_sum = ff_q_pfp_backward_division_sum_witness_common_leftsecondentry * S ((S (pfrep_position_backward_division_sum_witness_common_leftsecond)) * pfaa_left_c_backward_division_sum) + (pfrep_right_backward_division_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_division_sum_witness_common_leftsecondoutside. pfrep_gap_backward_division_sum_witness_common_leftsecondoutside+(pfaa_length_backward_division_sum)=(pfrep_power_backward_division_sum_witness_common_left)) /\\ (((pfrep_right_backward_division_sum_witness_common_left)=0))))) -> pfrep_left_backward_division_sum_witness_common_left=pfrep_right_backward_division_sum_witness_common_left) /\\ ((forall pfrep_power_backward_division_sum_witness_common_right pfrep_left_backward_division_sum_witness_common_right pfrep_right_backward_division_sum_witness_common_right. ((exists pfrep_position_backward_division_sum_witness_common_rightfirst. ((pfrep_position_backward_division_sum_witness_common_rightfirst+S (pfrep_power_backward_division_sum_witness_common_right)=(Lr)) /\\ ((((exists ff_h_pfp_backward_division_sum_witness_common_rightfirstentry. ff_h_pfp_backward_division_sum_witness_common_rightfirstentry + S (pfrep_left_backward_division_sum_witness_common_right) = S ((S (pfrep_position_backward_division_sum_witness_common_rightfirst)) * rc)) /\\ exists ff_q_pfp_backward_division_sum_witness_common_rightfirstentry. rb = ff_q_pfp_backward_division_sum_witness_common_rightfirstentry * S ((S (pfrep_position_backward_division_sum_witness_common_rightfirst)) * rc) + (pfrep_left_backward_division_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_division_sum_witness_common_rightfirstoutside. pfrep_gap_backward_division_sum_witness_common_rightfirstoutside+(Lr)=(pfrep_power_backward_division_sum_witness_common_right)) /\\ (((pfrep_left_backward_division_sum_witness_common_right)=0))))) -> ((exists pfrep_position_backward_division_sum_witness_common_rightsecond. ((pfrep_position_backward_division_sum_witness_common_rightsecond+S (pfrep_power_backward_division_sum_witness_common_right)=(pfaa_length_backward_division_sum)) /\\ ((((exists ff_h_pfp_backward_division_sum_witness_common_rightsecondentry. ff_h_pfp_backward_division_sum_witness_common_rightsecondentry + S (pfrep_right_backward_division_sum_witness_common_right) = S ((S (pfrep_position_backward_division_sum_witness_common_rightsecond)) * pfaa_right_c_backward_division_sum)) /\\ exists ff_q_pfp_backward_division_sum_witness_common_rightsecondentry. pfaa_right_b_backward_division_sum = ff_q_pfp_backward_division_sum_witness_common_rightsecondentry * S ((S (pfrep_position_backward_division_sum_witness_common_rightsecond)) * pfaa_right_c_backward_division_sum) + (pfrep_right_backward_division_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_division_sum_witness_common_rightsecondoutside. pfrep_gap_backward_division_sum_witness_common_rightsecondoutside+(pfaa_length_backward_division_sum)=(pfrep_power_backward_division_sum_witness_common_right)) /\\ (((pfrep_right_backward_division_sum_witness_common_right)=0))))) -> pfrep_left_backward_division_sum_witness_common_right=pfrep_right_backward_division_sum_witness_common_right)))) /\\ (((forall pfp_index_backward_division_sum_witness_operation. (exists pfa_gap_backward_division_sum_witness_operationindex. pfa_gap_backward_division_sum_witness_operationindex + S (pfp_index_backward_division_sum_witness_operation) = (pfaa_length_backward_division_sum)) -> exists pfp_left_backward_division_sum_witness_operation pfp_right_backward_division_sum_witness_operation pfp_value_backward_division_sum_witness_operation. ((((exists ff_h_pfp_backward_division_sum_witness_operationleft. ff_h_pfp_backward_division_sum_witness_operationleft + S (pfp_left_backward_division_sum_witness_operation) = S ((S (pfp_index_backward_division_sum_witness_operation)) * pfaa_left_c_backward_division_sum)) /\\ exists ff_q_pfp_backward_division_sum_witness_operationleft. pfaa_left_b_backward_division_sum = ff_q_pfp_backward_division_sum_witness_operationleft * S ((S (pfp_index_backward_division_sum_witness_operation)) * pfaa_left_c_backward_division_sum) + (pfp_left_backward_division_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_division_sum_witness_operationright. ff_h_pfp_backward_division_sum_witness_operationright + S (pfp_right_backward_division_sum_witness_operation) = S ((S (pfp_index_backward_division_sum_witness_operation)) * pfaa_right_c_backward_division_sum)) /\\ exists ff_q_pfp_backward_division_sum_witness_operationright. pfaa_right_b_backward_division_sum = ff_q_pfp_backward_division_sum_witness_operationright * S ((S (pfp_index_backward_division_sum_witness_operation)) * pfaa_right_c_backward_division_sum) + (pfp_right_backward_division_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_division_sum_witness_operationtarget. ff_h_pfp_backward_division_sum_witness_operationtarget + S (pfp_value_backward_division_sum_witness_operation) = S ((S (pfp_index_backward_division_sum_witness_operation)) * pfaa_sum_c_backward_division_sum)) /\\ exists ff_q_pfp_backward_division_sum_witness_operationtarget. pfaa_sum_b_backward_division_sum = ff_q_pfp_backward_division_sum_witness_operationtarget * S ((S (pfp_index_backward_division_sum_witness_operation)) * pfaa_sum_c_backward_division_sum) + (pfp_value_backward_division_sum_witness_operation))) /\\ ((((exists pfa_gap_backward_division_sum_witness_operationoperationleft. pfa_gap_backward_division_sum_witness_operationoperationleft + S (pfp_left_backward_division_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_backward_division_sum_witness_operationoperationright. pfa_gap_backward_division_sum_witness_operationoperationright + S (pfp_right_backward_division_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_backward_division_sum_witness_operationoperationresultbound. pfa_gap_backward_division_sum_witness_operationoperationresultbound + S (pfp_value_backward_division_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_backward_division_sum_witness_operationoperationresultcongruence pfa_offset_right_backward_division_sum_witness_operationoperationresultcongruence. ((pfp_left_backward_division_sum_witness_operation) + (pfp_right_backward_division_sum_witness_operation)) + (p) * pfa_offset_left_backward_division_sum_witness_operationoperationresultcongruence = (pfp_value_backward_division_sum_witness_operation) + (p) * pfa_offset_right_backward_division_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_backward_division_sum_witness_output pfrep_left_backward_division_sum_witness_output pfrep_right_backward_division_sum_witness_output. ((exists pfrep_position_backward_division_sum_witness_outputfirst. ((pfrep_position_backward_division_sum_witness_outputfirst+S (pfrep_power_backward_division_sum_witness_output)=(pfaa_length_backward_division_sum)) /\\ ((((exists ff_h_pfp_backward_division_sum_witness_outputfirstentry. ff_h_pfp_backward_division_sum_witness_outputfirstentry + S (pfrep_left_backward_division_sum_witness_output) = S ((S (pfrep_position_backward_division_sum_witness_outputfirst)) * pfaa_sum_c_backward_division_sum)) /\\ exists ff_q_pfp_backward_division_sum_witness_outputfirstentry. pfaa_sum_b_backward_division_sum = ff_q_pfp_backward_division_sum_witness_outputfirstentry * S ((S (pfrep_position_backward_division_sum_witness_outputfirst)) * pfaa_sum_c_backward_division_sum) + (pfrep_left_backward_division_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_division_sum_witness_outputfirstoutside. pfrep_gap_backward_division_sum_witness_outputfirstoutside+(pfaa_length_backward_division_sum)=(pfrep_power_backward_division_sum_witness_output)) /\\ (((pfrep_left_backward_division_sum_witness_output)=0))))) -> ((exists pfrep_position_backward_division_sum_witness_outputsecond. ((pfrep_position_backward_division_sum_witness_outputsecond+S (pfrep_power_backward_division_sum_witness_output)=(La)) /\\ ((((exists ff_h_pfp_backward_division_sum_witness_outputsecondentry. ff_h_pfp_backward_division_sum_witness_outputsecondentry + S (pfrep_right_backward_division_sum_witness_output) = S ((S (pfrep_position_backward_division_sum_witness_outputsecond)) * ac)) /\\ exists ff_q_pfp_backward_division_sum_witness_outputsecondentry. ab = ff_q_pfp_backward_division_sum_witness_outputsecondentry * S ((S (pfrep_position_backward_division_sum_witness_outputsecond)) * ac) + (pfrep_right_backward_division_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_division_sum_witness_outputsecondoutside. pfrep_gap_backward_division_sum_witness_outputsecondoutside+(La)=(pfrep_power_backward_division_sum_witness_output)) /\\ (((pfrep_right_backward_division_sum_witness_output)=0))))) -> pfrep_left_backward_division_sum_witness_output=pfrep_right_backward_division_sum_witness_output))))))))))))) -> (((forall fom_index_pfp_backward_old_leftleft. (exists fom_gap_pfp_backward_old_leftleft_index_bound. fom_gap_pfp_backward_old_leftleft_index_bound + S (fom_index_pfp_backward_old_leftleft) = Lu) -> exists fom_value_pfp_backward_old_leftleft. ((((exists fom_beta_height_pfp_backward_old_leftleft_entry. fom_beta_height_pfp_backward_old_leftleft_entry + S (fom_value_pfp_backward_old_leftleft) = S ((S (fom_index_pfp_backward_old_leftleft)) * uc)) /\\ exists fom_beta_quotient_pfp_backward_old_leftleft_entry. ub = fom_beta_quotient_pfp_backward_old_leftleft_entry * S ((S (fom_index_pfp_backward_old_leftleft)) * uc) + (fom_value_pfp_backward_old_leftleft))) /\\ (exists fom_gap_pfp_backward_old_leftleft_value_bound. fom_gap_pfp_backward_old_leftleft_value_bound + S (fom_value_pfp_backward_old_leftleft) = p))) /\\ (((forall fom_index_pfp_backward_old_leftright. (exists fom_gap_pfp_backward_old_leftright_index_bound. fom_gap_pfp_backward_old_leftright_index_bound + S (fom_index_pfp_backward_old_leftright) = Lb) -> exists fom_value_pfp_backward_old_leftright. ((((exists fom_beta_height_pfp_backward_old_leftright_entry. fom_beta_height_pfp_backward_old_leftright_entry + S (fom_value_pfp_backward_old_leftright) = S ((S (fom_index_pfp_backward_old_leftright)) * bc)) /\\ exists fom_beta_quotient_pfp_backward_old_leftright_entry. bb = fom_beta_quotient_pfp_backward_old_leftright_entry * S ((S (fom_index_pfp_backward_old_leftright)) * bc) + (fom_value_pfp_backward_old_leftright))) /\\ (exists fom_gap_pfp_backward_old_leftright_value_bound. fom_gap_pfp_backward_old_leftright_value_bound + S (fom_value_pfp_backward_old_leftright) = p))) /\\ (((((((Lu)=0 \\/ (Lb)=0) /\\ (((Lc)=0)))) \\/ (((~((Lu)=0)) /\\ (((~((Lb)=0)) /\\ (((Lu)+(Lb)=S (Lc)))))))) /\\ ((forall pfc_index_backward_old_leftcoefficients. (exists pfa_gap_backward_old_leftcoefficientsbound. pfa_gap_backward_old_leftcoefficientsbound + S (pfc_index_backward_old_leftcoefficients) = (Lc)) -> exists pfc_value_backward_old_leftcoefficients. ((((exists ff_h_pfp_backward_old_leftcoefficientsentry. ff_h_pfp_backward_old_leftcoefficientsentry + S (pfc_value_backward_old_leftcoefficients) = S ((S (pfc_index_backward_old_leftcoefficients)) * cc)) /\\ exists ff_q_pfp_backward_old_leftcoefficientsentry. cb = ff_q_pfp_backward_old_leftcoefficientsentry * S ((S (pfc_index_backward_old_leftcoefficients)) * cc) + (pfc_value_backward_old_leftcoefficients))) /\\ ((exists pfc_terms_code_backward_old_leftcoefficientscoefficient pfc_terms_scale_backward_old_leftcoefficientscoefficient pfc_natural_sum_backward_old_leftcoefficientscoefficient. ((forall pfc_index_backward_old_leftcoefficientscoefficientdiagonal. (exists pfa_gap_backward_old_leftcoefficientscoefficientdiagonalbound. pfa_gap_backward_old_leftcoefficientscoefficientdiagonalbound + S (pfc_index_backward_old_leftcoefficientscoefficientdiagonal) = (S (pfc_index_backward_old_leftcoefficients))) -> exists pfc_value_backward_old_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_old_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_old_leftcoefficientscoefficientdiagonalentry + S (pfc_value_backward_old_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_old_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_old_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_old_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_old_leftcoefficientscoefficient = ff_q_pfp_backward_old_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_old_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_old_leftcoefficientscoefficient) + (pfc_value_backward_old_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_old_leftcoefficientscoefficientdiagonalterm pfc_left_backward_old_leftcoefficientscoefficientdiagonalterm pfc_right_backward_old_leftcoefficientscoefficientdiagonalterm. (((pfc_index_backward_old_leftcoefficientscoefficientdiagonal)+pfc_complement_backward_old_leftcoefficientscoefficientdiagonalterm=(pfc_index_backward_old_leftcoefficients)) /\\ ((((((exists pfa_gap_backward_old_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_old_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_old_leftcoefficientscoefficientdiagonal) = (Lu)) /\\ ((((exists ff_h_pfp_backward_old_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_old_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_old_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_old_leftcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_backward_old_leftcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_backward_old_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_old_leftcoefficientscoefficientdiagonal)) * uc) + (pfc_left_backward_old_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_old_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_old_leftcoefficientscoefficientdiagonaltermleftoutside+(Lu)=(pfc_index_backward_old_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_old_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_old_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_old_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_old_leftcoefficientscoefficientdiagonalterm) = (Lb)) /\\ ((((exists ff_h_pfp_backward_old_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_old_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_old_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_old_leftcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_backward_old_leftcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_backward_old_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_old_leftcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_backward_old_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_old_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_old_leftcoefficientscoefficientdiagonaltermrightoutside+(Lb)=(pfc_complement_backward_old_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_old_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_old_leftcoefficientscoefficientdiagonal)=pfc_left_backward_old_leftcoefficientscoefficientdiagonalterm*pfc_right_backward_old_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_old_leftcoefficientscoefficientsum fs_v_pfc_backward_old_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_start. fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_start. fs_u_pfc_backward_old_leftcoefficientscoefficientsum = fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_old_leftcoefficientscoefficient) = S ((S (S (pfc_index_backward_old_leftcoefficients))) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_old_leftcoefficientscoefficientsum = fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_old_leftcoefficients))) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum) + (pfc_natural_sum_backward_old_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps = S (pfc_index_backward_old_leftcoefficients)) -> exists fs_a_pfc_backward_old_leftcoefficientscoefficientsum_body_steps fs_r_pfc_backward_old_leftcoefficientscoefficientsum_body_steps fs_s_pfc_backward_old_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_old_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_old_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_old_leftcoefficientscoefficient = fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_old_leftcoefficientscoefficient) + (fs_a_pfc_backward_old_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_old_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_old_leftcoefficientscoefficientsum = fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum) + (fs_r_pfc_backward_old_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_old_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_old_leftcoefficientscoefficientsum = fs_q_pfc_backward_old_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_leftcoefficientscoefficientsum) + (fs_s_pfc_backward_old_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_old_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_old_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_old_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_old_leftcoefficientscoefficientresiduebound. pfa_gap_backward_old_leftcoefficientscoefficientresiduebound + S (pfc_value_backward_old_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_old_leftcoefficientscoefficientresiduecongruence pfa_offset_right_backward_old_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_old_leftcoefficientscoefficient) + (p) * pfa_offset_left_backward_old_leftcoefficientscoefficientresiduecongruence = (pfc_value_backward_old_leftcoefficients) + (p) * pfa_offset_right_backward_old_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_old_rightleft. (exists fom_gap_pfp_backward_old_rightleft_index_bound. fom_gap_pfp_backward_old_rightleft_index_bound + S (fom_index_pfp_backward_old_rightleft) = Lv) -> exists fom_value_pfp_backward_old_rightleft. ((((exists fom_beta_height_pfp_backward_old_rightleft_entry. fom_beta_height_pfp_backward_old_rightleft_entry + S (fom_value_pfp_backward_old_rightleft) = S ((S (fom_index_pfp_backward_old_rightleft)) * vc)) /\\ exists fom_beta_quotient_pfp_backward_old_rightleft_entry. vb = fom_beta_quotient_pfp_backward_old_rightleft_entry * S ((S (fom_index_pfp_backward_old_rightleft)) * vc) + (fom_value_pfp_backward_old_rightleft))) /\\ (exists fom_gap_pfp_backward_old_rightleft_value_bound. fom_gap_pfp_backward_old_rightleft_value_bound + S (fom_value_pfp_backward_old_rightleft) = p))) /\\ (((forall fom_index_pfp_backward_old_rightright. (exists fom_gap_pfp_backward_old_rightright_index_bound. fom_gap_pfp_backward_old_rightright_index_bound + S (fom_index_pfp_backward_old_rightright) = Lr) -> exists fom_value_pfp_backward_old_rightright. ((((exists fom_beta_height_pfp_backward_old_rightright_entry. fom_beta_height_pfp_backward_old_rightright_entry + S (fom_value_pfp_backward_old_rightright) = S ((S (fom_index_pfp_backward_old_rightright)) * rc)) /\\ exists fom_beta_quotient_pfp_backward_old_rightright_entry. rb = fom_beta_quotient_pfp_backward_old_rightright_entry * S ((S (fom_index_pfp_backward_old_rightright)) * rc) + (fom_value_pfp_backward_old_rightright))) /\\ (exists fom_gap_pfp_backward_old_rightright_value_bound. fom_gap_pfp_backward_old_rightright_value_bound + S (fom_value_pfp_backward_old_rightright) = p))) /\\ (((((((Lv)=0 \\/ (Lr)=0) /\\ (((Ld)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((Lr)=0)) /\\ (((Lv)+(Lr)=S (Ld)))))))) /\\ ((forall pfc_index_backward_old_rightcoefficients. (exists pfa_gap_backward_old_rightcoefficientsbound. pfa_gap_backward_old_rightcoefficientsbound + S (pfc_index_backward_old_rightcoefficients) = (Ld)) -> exists pfc_value_backward_old_rightcoefficients. ((((exists ff_h_pfp_backward_old_rightcoefficientsentry. ff_h_pfp_backward_old_rightcoefficientsentry + S (pfc_value_backward_old_rightcoefficients) = S ((S (pfc_index_backward_old_rightcoefficients)) * dc)) /\\ exists ff_q_pfp_backward_old_rightcoefficientsentry. db = ff_q_pfp_backward_old_rightcoefficientsentry * S ((S (pfc_index_backward_old_rightcoefficients)) * dc) + (pfc_value_backward_old_rightcoefficients))) /\\ ((exists pfc_terms_code_backward_old_rightcoefficientscoefficient pfc_terms_scale_backward_old_rightcoefficientscoefficient pfc_natural_sum_backward_old_rightcoefficientscoefficient. ((forall pfc_index_backward_old_rightcoefficientscoefficientdiagonal. (exists pfa_gap_backward_old_rightcoefficientscoefficientdiagonalbound. pfa_gap_backward_old_rightcoefficientscoefficientdiagonalbound + S (pfc_index_backward_old_rightcoefficientscoefficientdiagonal) = (S (pfc_index_backward_old_rightcoefficients))) -> exists pfc_value_backward_old_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_old_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_old_rightcoefficientscoefficientdiagonalentry + S (pfc_value_backward_old_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_old_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_old_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_old_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_old_rightcoefficientscoefficient = ff_q_pfp_backward_old_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_old_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_old_rightcoefficientscoefficient) + (pfc_value_backward_old_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_old_rightcoefficientscoefficientdiagonalterm pfc_left_backward_old_rightcoefficientscoefficientdiagonalterm pfc_right_backward_old_rightcoefficientscoefficientdiagonalterm. (((pfc_index_backward_old_rightcoefficientscoefficientdiagonal)+pfc_complement_backward_old_rightcoefficientscoefficientdiagonalterm=(pfc_index_backward_old_rightcoefficients)) /\\ ((((((exists pfa_gap_backward_old_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_old_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_old_rightcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_backward_old_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_old_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_old_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_old_rightcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_backward_old_rightcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_backward_old_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_old_rightcoefficientscoefficientdiagonal)) * vc) + (pfc_left_backward_old_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_old_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_old_rightcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_backward_old_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_old_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_old_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_old_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_old_rightcoefficientscoefficientdiagonalterm) = (Lr)) /\\ ((((exists ff_h_pfp_backward_old_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_old_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_old_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_old_rightcoefficientscoefficientdiagonalterm)) * rc)) /\\ exists ff_q_pfp_backward_old_rightcoefficientscoefficientdiagonaltermrightentry. rb = ff_q_pfp_backward_old_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_old_rightcoefficientscoefficientdiagonalterm)) * rc) + (pfc_right_backward_old_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_old_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_old_rightcoefficientscoefficientdiagonaltermrightoutside+(Lr)=(pfc_complement_backward_old_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_old_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_old_rightcoefficientscoefficientdiagonal)=pfc_left_backward_old_rightcoefficientscoefficientdiagonalterm*pfc_right_backward_old_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_old_rightcoefficientscoefficientsum fs_v_pfc_backward_old_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_start. fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_start. fs_u_pfc_backward_old_rightcoefficientscoefficientsum = fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_old_rightcoefficientscoefficient) = S ((S (S (pfc_index_backward_old_rightcoefficients))) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_old_rightcoefficientscoefficientsum = fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_old_rightcoefficients))) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum) + (pfc_natural_sum_backward_old_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps = S (pfc_index_backward_old_rightcoefficients)) -> exists fs_a_pfc_backward_old_rightcoefficientscoefficientsum_body_steps fs_r_pfc_backward_old_rightcoefficientscoefficientsum_body_steps fs_s_pfc_backward_old_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_old_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_old_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_old_rightcoefficientscoefficient = fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_old_rightcoefficientscoefficient) + (fs_a_pfc_backward_old_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_old_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_old_rightcoefficientscoefficientsum = fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum) + (fs_r_pfc_backward_old_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_old_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_old_rightcoefficientscoefficientsum = fs_q_pfc_backward_old_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_old_rightcoefficientscoefficientsum) + (fs_s_pfc_backward_old_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_old_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_old_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_old_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_old_rightcoefficientscoefficientresiduebound. pfa_gap_backward_old_rightcoefficientscoefficientresiduebound + S (pfc_value_backward_old_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_old_rightcoefficientscoefficientresiduecongruence pfa_offset_right_backward_old_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_old_rightcoefficientscoefficient) + (p) * pfa_offset_left_backward_old_rightcoefficientscoefficientresiduecongruence = (pfc_value_backward_old_rightcoefficients) + (p) * pfa_offset_right_backward_old_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_old_sum_left_bounded. (exists fom_gap_pfp_backward_old_sum_left_bounded_index_bound. fom_gap_pfp_backward_old_sum_left_bounded_index_bound + S (fom_index_pfp_backward_old_sum_left_bounded) = Lc) -> exists fom_value_pfp_backward_old_sum_left_bounded. ((((exists fom_beta_height_pfp_backward_old_sum_left_bounded_entry. fom_beta_height_pfp_backward_old_sum_left_bounded_entry + S (fom_value_pfp_backward_old_sum_left_bounded) = S ((S (fom_index_pfp_backward_old_sum_left_bounded)) * cc)) /\\ exists fom_beta_quotient_pfp_backward_old_sum_left_bounded_entry. cb = fom_beta_quotient_pfp_backward_old_sum_left_bounded_entry * S ((S (fom_index_pfp_backward_old_sum_left_bounded)) * cc) + (fom_value_pfp_backward_old_sum_left_bounded))) /\\ (exists fom_gap_pfp_backward_old_sum_left_bounded_value_bound. fom_gap_pfp_backward_old_sum_left_bounded_value_bound + S (fom_value_pfp_backward_old_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_backward_old_sum_right_bounded. (exists fom_gap_pfp_backward_old_sum_right_bounded_index_bound. fom_gap_pfp_backward_old_sum_right_bounded_index_bound + S (fom_index_pfp_backward_old_sum_right_bounded) = Ld) -> exists fom_value_pfp_backward_old_sum_right_bounded. ((((exists fom_beta_height_pfp_backward_old_sum_right_bounded_entry. fom_beta_height_pfp_backward_old_sum_right_bounded_entry + S (fom_value_pfp_backward_old_sum_right_bounded) = S ((S (fom_index_pfp_backward_old_sum_right_bounded)) * dc)) /\\ exists fom_beta_quotient_pfp_backward_old_sum_right_bounded_entry. db = fom_beta_quotient_pfp_backward_old_sum_right_bounded_entry * S ((S (fom_index_pfp_backward_old_sum_right_bounded)) * dc) + (fom_value_pfp_backward_old_sum_right_bounded))) /\\ (exists fom_gap_pfp_backward_old_sum_right_bounded_value_bound. fom_gap_pfp_backward_old_sum_right_bounded_value_bound + S (fom_value_pfp_backward_old_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_backward_old_sum_result_bounded. (exists fom_gap_pfp_backward_old_sum_result_bounded_index_bound. fom_gap_pfp_backward_old_sum_result_bounded_index_bound + S (fom_index_pfp_backward_old_sum_result_bounded) = Lg) -> exists fom_value_pfp_backward_old_sum_result_bounded. ((((exists fom_beta_height_pfp_backward_old_sum_result_bounded_entry. fom_beta_height_pfp_backward_old_sum_result_bounded_entry + S (fom_value_pfp_backward_old_sum_result_bounded) = S ((S (fom_index_pfp_backward_old_sum_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_backward_old_sum_result_bounded_entry. gb = fom_beta_quotient_pfp_backward_old_sum_result_bounded_entry * S ((S (fom_index_pfp_backward_old_sum_result_bounded)) * gc) + (fom_value_pfp_backward_old_sum_result_bounded))) /\\ (exists fom_gap_pfp_backward_old_sum_result_bounded_value_bound. fom_gap_pfp_backward_old_sum_result_bounded_value_bound + S (fom_value_pfp_backward_old_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_backward_old_sum pfaa_left_c_backward_old_sum pfaa_right_b_backward_old_sum pfaa_right_c_backward_old_sum pfaa_sum_b_backward_old_sum pfaa_sum_c_backward_old_sum pfaa_length_backward_old_sum. ((((forall pfrep_power_backward_old_sum_witness_common_left pfrep_left_backward_old_sum_witness_common_left pfrep_right_backward_old_sum_witness_common_left. ((exists pfrep_position_backward_old_sum_witness_common_leftfirst. ((pfrep_position_backward_old_sum_witness_common_leftfirst+S (pfrep_power_backward_old_sum_witness_common_left)=(Lc)) /\\ ((((exists ff_h_pfp_backward_old_sum_witness_common_leftfirstentry. ff_h_pfp_backward_old_sum_witness_common_leftfirstentry + S (pfrep_left_backward_old_sum_witness_common_left) = S ((S (pfrep_position_backward_old_sum_witness_common_leftfirst)) * cc)) /\\ exists ff_q_pfp_backward_old_sum_witness_common_leftfirstentry. cb = ff_q_pfp_backward_old_sum_witness_common_leftfirstentry * S ((S (pfrep_position_backward_old_sum_witness_common_leftfirst)) * cc) + (pfrep_left_backward_old_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_old_sum_witness_common_leftfirstoutside. pfrep_gap_backward_old_sum_witness_common_leftfirstoutside+(Lc)=(pfrep_power_backward_old_sum_witness_common_left)) /\\ (((pfrep_left_backward_old_sum_witness_common_left)=0))))) -> ((exists pfrep_position_backward_old_sum_witness_common_leftsecond. ((pfrep_position_backward_old_sum_witness_common_leftsecond+S (pfrep_power_backward_old_sum_witness_common_left)=(pfaa_length_backward_old_sum)) /\\ ((((exists ff_h_pfp_backward_old_sum_witness_common_leftsecondentry. ff_h_pfp_backward_old_sum_witness_common_leftsecondentry + S (pfrep_right_backward_old_sum_witness_common_left) = S ((S (pfrep_position_backward_old_sum_witness_common_leftsecond)) * pfaa_left_c_backward_old_sum)) /\\ exists ff_q_pfp_backward_old_sum_witness_common_leftsecondentry. pfaa_left_b_backward_old_sum = ff_q_pfp_backward_old_sum_witness_common_leftsecondentry * S ((S (pfrep_position_backward_old_sum_witness_common_leftsecond)) * pfaa_left_c_backward_old_sum) + (pfrep_right_backward_old_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_old_sum_witness_common_leftsecondoutside. pfrep_gap_backward_old_sum_witness_common_leftsecondoutside+(pfaa_length_backward_old_sum)=(pfrep_power_backward_old_sum_witness_common_left)) /\\ (((pfrep_right_backward_old_sum_witness_common_left)=0))))) -> pfrep_left_backward_old_sum_witness_common_left=pfrep_right_backward_old_sum_witness_common_left) /\\ ((forall pfrep_power_backward_old_sum_witness_common_right pfrep_left_backward_old_sum_witness_common_right pfrep_right_backward_old_sum_witness_common_right. ((exists pfrep_position_backward_old_sum_witness_common_rightfirst. ((pfrep_position_backward_old_sum_witness_common_rightfirst+S (pfrep_power_backward_old_sum_witness_common_right)=(Ld)) /\\ ((((exists ff_h_pfp_backward_old_sum_witness_common_rightfirstentry. ff_h_pfp_backward_old_sum_witness_common_rightfirstentry + S (pfrep_left_backward_old_sum_witness_common_right) = S ((S (pfrep_position_backward_old_sum_witness_common_rightfirst)) * dc)) /\\ exists ff_q_pfp_backward_old_sum_witness_common_rightfirstentry. db = ff_q_pfp_backward_old_sum_witness_common_rightfirstentry * S ((S (pfrep_position_backward_old_sum_witness_common_rightfirst)) * dc) + (pfrep_left_backward_old_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_old_sum_witness_common_rightfirstoutside. pfrep_gap_backward_old_sum_witness_common_rightfirstoutside+(Ld)=(pfrep_power_backward_old_sum_witness_common_right)) /\\ (((pfrep_left_backward_old_sum_witness_common_right)=0))))) -> ((exists pfrep_position_backward_old_sum_witness_common_rightsecond. ((pfrep_position_backward_old_sum_witness_common_rightsecond+S (pfrep_power_backward_old_sum_witness_common_right)=(pfaa_length_backward_old_sum)) /\\ ((((exists ff_h_pfp_backward_old_sum_witness_common_rightsecondentry. ff_h_pfp_backward_old_sum_witness_common_rightsecondentry + S (pfrep_right_backward_old_sum_witness_common_right) = S ((S (pfrep_position_backward_old_sum_witness_common_rightsecond)) * pfaa_right_c_backward_old_sum)) /\\ exists ff_q_pfp_backward_old_sum_witness_common_rightsecondentry. pfaa_right_b_backward_old_sum = ff_q_pfp_backward_old_sum_witness_common_rightsecondentry * S ((S (pfrep_position_backward_old_sum_witness_common_rightsecond)) * pfaa_right_c_backward_old_sum) + (pfrep_right_backward_old_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_old_sum_witness_common_rightsecondoutside. pfrep_gap_backward_old_sum_witness_common_rightsecondoutside+(pfaa_length_backward_old_sum)=(pfrep_power_backward_old_sum_witness_common_right)) /\\ (((pfrep_right_backward_old_sum_witness_common_right)=0))))) -> pfrep_left_backward_old_sum_witness_common_right=pfrep_right_backward_old_sum_witness_common_right)))) /\\ (((forall pfp_index_backward_old_sum_witness_operation. (exists pfa_gap_backward_old_sum_witness_operationindex. pfa_gap_backward_old_sum_witness_operationindex + S (pfp_index_backward_old_sum_witness_operation) = (pfaa_length_backward_old_sum)) -> exists pfp_left_backward_old_sum_witness_operation pfp_right_backward_old_sum_witness_operation pfp_value_backward_old_sum_witness_operation. ((((exists ff_h_pfp_backward_old_sum_witness_operationleft. ff_h_pfp_backward_old_sum_witness_operationleft + S (pfp_left_backward_old_sum_witness_operation) = S ((S (pfp_index_backward_old_sum_witness_operation)) * pfaa_left_c_backward_old_sum)) /\\ exists ff_q_pfp_backward_old_sum_witness_operationleft. pfaa_left_b_backward_old_sum = ff_q_pfp_backward_old_sum_witness_operationleft * S ((S (pfp_index_backward_old_sum_witness_operation)) * pfaa_left_c_backward_old_sum) + (pfp_left_backward_old_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_old_sum_witness_operationright. ff_h_pfp_backward_old_sum_witness_operationright + S (pfp_right_backward_old_sum_witness_operation) = S ((S (pfp_index_backward_old_sum_witness_operation)) * pfaa_right_c_backward_old_sum)) /\\ exists ff_q_pfp_backward_old_sum_witness_operationright. pfaa_right_b_backward_old_sum = ff_q_pfp_backward_old_sum_witness_operationright * S ((S (pfp_index_backward_old_sum_witness_operation)) * pfaa_right_c_backward_old_sum) + (pfp_right_backward_old_sum_witness_operation))) /\\ (((((exists ff_h_pfp_backward_old_sum_witness_operationtarget. ff_h_pfp_backward_old_sum_witness_operationtarget + S (pfp_value_backward_old_sum_witness_operation) = S ((S (pfp_index_backward_old_sum_witness_operation)) * pfaa_sum_c_backward_old_sum)) /\\ exists ff_q_pfp_backward_old_sum_witness_operationtarget. pfaa_sum_b_backward_old_sum = ff_q_pfp_backward_old_sum_witness_operationtarget * S ((S (pfp_index_backward_old_sum_witness_operation)) * pfaa_sum_c_backward_old_sum) + (pfp_value_backward_old_sum_witness_operation))) /\\ ((((exists pfa_gap_backward_old_sum_witness_operationoperationleft. pfa_gap_backward_old_sum_witness_operationoperationleft + S (pfp_left_backward_old_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_backward_old_sum_witness_operationoperationright. pfa_gap_backward_old_sum_witness_operationoperationright + S (pfp_right_backward_old_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_backward_old_sum_witness_operationoperationresultbound. pfa_gap_backward_old_sum_witness_operationoperationresultbound + S (pfp_value_backward_old_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_backward_old_sum_witness_operationoperationresultcongruence pfa_offset_right_backward_old_sum_witness_operationoperationresultcongruence. ((pfp_left_backward_old_sum_witness_operation) + (pfp_right_backward_old_sum_witness_operation)) + (p) * pfa_offset_left_backward_old_sum_witness_operationoperationresultcongruence = (pfp_value_backward_old_sum_witness_operation) + (p) * pfa_offset_right_backward_old_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_backward_old_sum_witness_output pfrep_left_backward_old_sum_witness_output pfrep_right_backward_old_sum_witness_output. ((exists pfrep_position_backward_old_sum_witness_outputfirst. ((pfrep_position_backward_old_sum_witness_outputfirst+S (pfrep_power_backward_old_sum_witness_output)=(pfaa_length_backward_old_sum)) /\\ ((((exists ff_h_pfp_backward_old_sum_witness_outputfirstentry. ff_h_pfp_backward_old_sum_witness_outputfirstentry + S (pfrep_left_backward_old_sum_witness_output) = S ((S (pfrep_position_backward_old_sum_witness_outputfirst)) * pfaa_sum_c_backward_old_sum)) /\\ exists ff_q_pfp_backward_old_sum_witness_outputfirstentry. pfaa_sum_b_backward_old_sum = ff_q_pfp_backward_old_sum_witness_outputfirstentry * S ((S (pfrep_position_backward_old_sum_witness_outputfirst)) * pfaa_sum_c_backward_old_sum) + (pfrep_left_backward_old_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_old_sum_witness_outputfirstoutside. pfrep_gap_backward_old_sum_witness_outputfirstoutside+(pfaa_length_backward_old_sum)=(pfrep_power_backward_old_sum_witness_output)) /\\ (((pfrep_left_backward_old_sum_witness_output)=0))))) -> ((exists pfrep_position_backward_old_sum_witness_outputsecond. ((pfrep_position_backward_old_sum_witness_outputsecond+S (pfrep_power_backward_old_sum_witness_output)=(Lg)) /\\ ((((exists ff_h_pfp_backward_old_sum_witness_outputsecondentry. ff_h_pfp_backward_old_sum_witness_outputsecondentry + S (pfrep_right_backward_old_sum_witness_output) = S ((S (pfrep_position_backward_old_sum_witness_outputsecond)) * gc)) /\\ exists ff_q_pfp_backward_old_sum_witness_outputsecondentry. gb = ff_q_pfp_backward_old_sum_witness_outputsecondentry * S ((S (pfrep_position_backward_old_sum_witness_outputsecond)) * gc) + (pfrep_right_backward_old_sum_witness_output)))))) \\/ (((exists pfrep_gap_backward_old_sum_witness_outputsecondoutside. pfrep_gap_backward_old_sum_witness_outputsecondoutside+(Lg)=(pfrep_power_backward_old_sum_witness_output)) /\\ (((pfrep_right_backward_old_sum_witness_output)=0))))) -> pfrep_left_backward_old_sum_witness_output=pfrep_right_backward_old_sum_witness_output))))))))))))) -> (((forall fom_index_pfp_backward_coefficient_productleft. (exists fom_gap_pfp_backward_coefficient_productleft_index_bound. fom_gap_pfp_backward_coefficient_productleft_index_bound + S (fom_index_pfp_backward_coefficient_productleft) = Lv) -> exists fom_value_pfp_backward_coefficient_productleft. ((((exists fom_beta_height_pfp_backward_coefficient_productleft_entry. fom_beta_height_pfp_backward_coefficient_productleft_entry + S (fom_value_pfp_backward_coefficient_productleft) = S ((S (fom_index_pfp_backward_coefficient_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_backward_coefficient_productleft_entry. vb = fom_beta_quotient_pfp_backward_coefficient_productleft_entry * S ((S (fom_index_pfp_backward_coefficient_productleft)) * vc) + (fom_value_pfp_backward_coefficient_productleft))) /\\ (exists fom_gap_pfp_backward_coefficient_productleft_value_bound. fom_gap_pfp_backward_coefficient_productleft_value_bound + S (fom_value_pfp_backward_coefficient_productleft) = p))) /\\ (((forall fom_index_pfp_backward_coefficient_productright. (exists fom_gap_pfp_backward_coefficient_productright_index_bound. fom_gap_pfp_backward_coefficient_productright_index_bound + S (fom_index_pfp_backward_coefficient_productright) = Lq) -> exists fom_value_pfp_backward_coefficient_productright. ((((exists fom_beta_height_pfp_backward_coefficient_productright_entry. fom_beta_height_pfp_backward_coefficient_productright_entry + S (fom_value_pfp_backward_coefficient_productright) = S ((S (fom_index_pfp_backward_coefficient_productright)) * qc)) /\\ exists fom_beta_quotient_pfp_backward_coefficient_productright_entry. qb = fom_beta_quotient_pfp_backward_coefficient_productright_entry * S ((S (fom_index_pfp_backward_coefficient_productright)) * qc) + (fom_value_pfp_backward_coefficient_productright))) /\\ (exists fom_gap_pfp_backward_coefficient_productright_value_bound. fom_gap_pfp_backward_coefficient_productright_value_bound + S (fom_value_pfp_backward_coefficient_productright) = p))) /\\ (((((((Lv)=0 \\/ (Lq)=0) /\\ (((Lw)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((Lq)=0)) /\\ (((Lv)+(Lq)=S (Lw)))))))) /\\ ((forall pfc_index_backward_coefficient_productcoefficients. (exists pfa_gap_backward_coefficient_productcoefficientsbound. pfa_gap_backward_coefficient_productcoefficientsbound + S (pfc_index_backward_coefficient_productcoefficients) = (Lw)) -> exists pfc_value_backward_coefficient_productcoefficients. ((((exists ff_h_pfp_backward_coefficient_productcoefficientsentry. ff_h_pfp_backward_coefficient_productcoefficientsentry + S (pfc_value_backward_coefficient_productcoefficients) = S ((S (pfc_index_backward_coefficient_productcoefficients)) * wc)) /\\ exists ff_q_pfp_backward_coefficient_productcoefficientsentry. wb = ff_q_pfp_backward_coefficient_productcoefficientsentry * S ((S (pfc_index_backward_coefficient_productcoefficients)) * wc) + (pfc_value_backward_coefficient_productcoefficients))) /\\ ((exists pfc_terms_code_backward_coefficient_productcoefficientscoefficient pfc_terms_scale_backward_coefficient_productcoefficientscoefficient pfc_natural_sum_backward_coefficient_productcoefficientscoefficient. ((forall pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal. (exists pfa_gap_backward_coefficient_productcoefficientscoefficientdiagonalbound. pfa_gap_backward_coefficient_productcoefficientscoefficientdiagonalbound + S (pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal) = (S (pfc_index_backward_coefficient_productcoefficients))) -> exists pfc_value_backward_coefficient_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_coefficient_productcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_coefficient_productcoefficientscoefficientdiagonalentry + S (pfc_value_backward_coefficient_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_coefficient_productcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_coefficient_productcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_coefficient_productcoefficientscoefficient = ff_q_pfp_backward_coefficient_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_coefficient_productcoefficientscoefficient) + (pfc_value_backward_coefficient_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_coefficient_productcoefficientscoefficientdiagonalterm pfc_left_backward_coefficient_productcoefficientscoefficientdiagonalterm pfc_right_backward_coefficient_productcoefficientscoefficientdiagonalterm. (((pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal)+pfc_complement_backward_coefficient_productcoefficientscoefficientdiagonalterm=(pfc_index_backward_coefficient_productcoefficients)) /\\ ((((((exists pfa_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_coefficient_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_backward_coefficient_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_backward_coefficient_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_coefficient_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_coefficient_productcoefficientscoefficientdiagonalterm) = (Lq)) /\\ ((((exists ff_h_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_coefficient_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_coefficient_productcoefficientscoefficientdiagonalterm)) * qc)) /\\ exists ff_q_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermrightentry. qb = ff_q_pfp_backward_coefficient_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_coefficient_productcoefficientscoefficientdiagonalterm)) * qc) + (pfc_right_backward_coefficient_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_coefficient_productcoefficientscoefficientdiagonaltermrightoutside+(Lq)=(pfc_complement_backward_coefficient_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_coefficient_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_coefficient_productcoefficientscoefficientdiagonal)=pfc_left_backward_coefficient_productcoefficientscoefficientdiagonalterm*pfc_right_backward_coefficient_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_coefficient_productcoefficientscoefficientsum fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_start. fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_start. fs_u_pfc_backward_coefficient_productcoefficientscoefficientsum = fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_coefficient_productcoefficientscoefficient) = S ((S (S (pfc_index_backward_coefficient_productcoefficients))) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_coefficient_productcoefficientscoefficientsum = fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_coefficient_productcoefficients))) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum) + (pfc_natural_sum_backward_coefficient_productcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps = S (pfc_index_backward_coefficient_productcoefficients)) -> exists fs_a_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps fs_r_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps fs_s_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_coefficient_productcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_coefficient_productcoefficientscoefficient = fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_coefficient_productcoefficientscoefficient) + (fs_a_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_coefficient_productcoefficientscoefficientsum = fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum) + (fs_r_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_coefficient_productcoefficientscoefficientsum = fs_q_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_coefficient_productcoefficientscoefficientsum) + (fs_s_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_coefficient_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_coefficient_productcoefficientscoefficientresiduebound. pfa_gap_backward_coefficient_productcoefficientscoefficientresiduebound + S (pfc_value_backward_coefficient_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_coefficient_productcoefficientscoefficientresiduecongruence pfa_offset_right_backward_coefficient_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_coefficient_productcoefficientscoefficient) + (p) * pfa_offset_left_backward_coefficient_productcoefficientscoefficientresiduecongruence = (pfc_value_backward_coefficient_productcoefficients) + (p) * pfa_offset_right_backward_coefficient_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_coefficient_difference_left_bounded. (exists fom_gap_pfp_backward_coefficient_difference_left_bounded_index_bound. fom_gap_pfp_backward_coefficient_difference_left_bounded_index_bound + S (fom_index_pfp_backward_coefficient_difference_left_bounded) = Lw) -> exists fom_value_pfp_backward_coefficient_difference_left_bounded. ((((exists fom_beta_height_pfp_backward_coefficient_difference_left_bounded_entry. fom_beta_height_pfp_backward_coefficient_difference_left_bounded_entry + S (fom_value_pfp_backward_coefficient_difference_left_bounded) = S ((S (fom_index_pfp_backward_coefficient_difference_left_bounded)) * wc)) /\\ exists fom_beta_quotient_pfp_backward_coefficient_difference_left_bounded_entry. wb = fom_beta_quotient_pfp_backward_coefficient_difference_left_bounded_entry * S ((S (fom_index_pfp_backward_coefficient_difference_left_bounded)) * wc) + (fom_value_pfp_backward_coefficient_difference_left_bounded))) /\\ (exists fom_gap_pfp_backward_coefficient_difference_left_bounded_value_bound. fom_gap_pfp_backward_coefficient_difference_left_bounded_value_bound + S (fom_value_pfp_backward_coefficient_difference_left_bounded) = p))) /\\ (((forall fom_index_pfp_backward_coefficient_difference_right_bounded. (exists fom_gap_pfp_backward_coefficient_difference_right_bounded_index_bound. fom_gap_pfp_backward_coefficient_difference_right_bounded_index_bound + S (fom_index_pfp_backward_coefficient_difference_right_bounded) = Lt) -> exists fom_value_pfp_backward_coefficient_difference_right_bounded. ((((exists fom_beta_height_pfp_backward_coefficient_difference_right_bounded_entry. fom_beta_height_pfp_backward_coefficient_difference_right_bounded_entry + S (fom_value_pfp_backward_coefficient_difference_right_bounded) = S ((S (fom_index_pfp_backward_coefficient_difference_right_bounded)) * tc)) /\\ exists fom_beta_quotient_pfp_backward_coefficient_difference_right_bounded_entry. tb = fom_beta_quotient_pfp_backward_coefficient_difference_right_bounded_entry * S ((S (fom_index_pfp_backward_coefficient_difference_right_bounded)) * tc) + (fom_value_pfp_backward_coefficient_difference_right_bounded))) /\\ (exists fom_gap_pfp_backward_coefficient_difference_right_bounded_value_bound. fom_gap_pfp_backward_coefficient_difference_right_bounded_value_bound + S (fom_value_pfp_backward_coefficient_difference_right_bounded) = p))) /\\ (((forall fom_index_pfp_backward_coefficient_difference_result_bounded. (exists fom_gap_pfp_backward_coefficient_difference_result_bounded_index_bound. fom_gap_pfp_backward_coefficient_difference_result_bounded_index_bound + S (fom_index_pfp_backward_coefficient_difference_result_bounded) = Lu) -> exists fom_value_pfp_backward_coefficient_difference_result_bounded. ((((exists fom_beta_height_pfp_backward_coefficient_difference_result_bounded_entry. fom_beta_height_pfp_backward_coefficient_difference_result_bounded_entry + S (fom_value_pfp_backward_coefficient_difference_result_bounded) = S ((S (fom_index_pfp_backward_coefficient_difference_result_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_backward_coefficient_difference_result_bounded_entry. ub = fom_beta_quotient_pfp_backward_coefficient_difference_result_bounded_entry * S ((S (fom_index_pfp_backward_coefficient_difference_result_bounded)) * uc) + (fom_value_pfp_backward_coefficient_difference_result_bounded))) /\\ (exists fom_gap_pfp_backward_coefficient_difference_result_bounded_value_bound. fom_gap_pfp_backward_coefficient_difference_result_bounded_value_bound + S (fom_value_pfp_backward_coefficient_difference_result_bounded) = p))) /\\ ((exists pfaa_left_b_backward_coefficient_difference pfaa_left_c_backward_coefficient_difference pfaa_right_b_backward_coefficient_difference pfaa_right_c_backward_coefficient_difference pfaa_sum_b_backward_coefficient_difference pfaa_sum_c_backward_coefficient_difference pfaa_length_backward_coefficient_difference. ((((forall pfrep_power_backward_coefficient_difference_witness_common_left pfrep_left_backward_coefficient_difference_witness_common_left pfrep_right_backward_coefficient_difference_witness_common_left. ((exists pfrep_position_backward_coefficient_difference_witness_common_leftfirst. ((pfrep_position_backward_coefficient_difference_witness_common_leftfirst+S (pfrep_power_backward_coefficient_difference_witness_common_left)=(Lw)) /\\ ((((exists ff_h_pfp_backward_coefficient_difference_witness_common_leftfirstentry. ff_h_pfp_backward_coefficient_difference_witness_common_leftfirstentry + S (pfrep_left_backward_coefficient_difference_witness_common_left) = S ((S (pfrep_position_backward_coefficient_difference_witness_common_leftfirst)) * wc)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_common_leftfirstentry. wb = ff_q_pfp_backward_coefficient_difference_witness_common_leftfirstentry * S ((S (pfrep_position_backward_coefficient_difference_witness_common_leftfirst)) * wc) + (pfrep_left_backward_coefficient_difference_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_coefficient_difference_witness_common_leftfirstoutside. pfrep_gap_backward_coefficient_difference_witness_common_leftfirstoutside+(Lw)=(pfrep_power_backward_coefficient_difference_witness_common_left)) /\\ (((pfrep_left_backward_coefficient_difference_witness_common_left)=0))))) -> ((exists pfrep_position_backward_coefficient_difference_witness_common_leftsecond. ((pfrep_position_backward_coefficient_difference_witness_common_leftsecond+S (pfrep_power_backward_coefficient_difference_witness_common_left)=(pfaa_length_backward_coefficient_difference)) /\\ ((((exists ff_h_pfp_backward_coefficient_difference_witness_common_leftsecondentry. ff_h_pfp_backward_coefficient_difference_witness_common_leftsecondentry + S (pfrep_right_backward_coefficient_difference_witness_common_left) = S ((S (pfrep_position_backward_coefficient_difference_witness_common_leftsecond)) * pfaa_left_c_backward_coefficient_difference)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_common_leftsecondentry. pfaa_left_b_backward_coefficient_difference = ff_q_pfp_backward_coefficient_difference_witness_common_leftsecondentry * S ((S (pfrep_position_backward_coefficient_difference_witness_common_leftsecond)) * pfaa_left_c_backward_coefficient_difference) + (pfrep_right_backward_coefficient_difference_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_coefficient_difference_witness_common_leftsecondoutside. pfrep_gap_backward_coefficient_difference_witness_common_leftsecondoutside+(pfaa_length_backward_coefficient_difference)=(pfrep_power_backward_coefficient_difference_witness_common_left)) /\\ (((pfrep_right_backward_coefficient_difference_witness_common_left)=0))))) -> pfrep_left_backward_coefficient_difference_witness_common_left=pfrep_right_backward_coefficient_difference_witness_common_left) /\\ ((forall pfrep_power_backward_coefficient_difference_witness_common_right pfrep_left_backward_coefficient_difference_witness_common_right pfrep_right_backward_coefficient_difference_witness_common_right. ((exists pfrep_position_backward_coefficient_difference_witness_common_rightfirst. ((pfrep_position_backward_coefficient_difference_witness_common_rightfirst+S (pfrep_power_backward_coefficient_difference_witness_common_right)=(Lt)) /\\ ((((exists ff_h_pfp_backward_coefficient_difference_witness_common_rightfirstentry. ff_h_pfp_backward_coefficient_difference_witness_common_rightfirstentry + S (pfrep_left_backward_coefficient_difference_witness_common_right) = S ((S (pfrep_position_backward_coefficient_difference_witness_common_rightfirst)) * tc)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_common_rightfirstentry. tb = ff_q_pfp_backward_coefficient_difference_witness_common_rightfirstentry * S ((S (pfrep_position_backward_coefficient_difference_witness_common_rightfirst)) * tc) + (pfrep_left_backward_coefficient_difference_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_coefficient_difference_witness_common_rightfirstoutside. pfrep_gap_backward_coefficient_difference_witness_common_rightfirstoutside+(Lt)=(pfrep_power_backward_coefficient_difference_witness_common_right)) /\\ (((pfrep_left_backward_coefficient_difference_witness_common_right)=0))))) -> ((exists pfrep_position_backward_coefficient_difference_witness_common_rightsecond. ((pfrep_position_backward_coefficient_difference_witness_common_rightsecond+S (pfrep_power_backward_coefficient_difference_witness_common_right)=(pfaa_length_backward_coefficient_difference)) /\\ ((((exists ff_h_pfp_backward_coefficient_difference_witness_common_rightsecondentry. ff_h_pfp_backward_coefficient_difference_witness_common_rightsecondentry + S (pfrep_right_backward_coefficient_difference_witness_common_right) = S ((S (pfrep_position_backward_coefficient_difference_witness_common_rightsecond)) * pfaa_right_c_backward_coefficient_difference)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_common_rightsecondentry. pfaa_right_b_backward_coefficient_difference = ff_q_pfp_backward_coefficient_difference_witness_common_rightsecondentry * S ((S (pfrep_position_backward_coefficient_difference_witness_common_rightsecond)) * pfaa_right_c_backward_coefficient_difference) + (pfrep_right_backward_coefficient_difference_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_coefficient_difference_witness_common_rightsecondoutside. pfrep_gap_backward_coefficient_difference_witness_common_rightsecondoutside+(pfaa_length_backward_coefficient_difference)=(pfrep_power_backward_coefficient_difference_witness_common_right)) /\\ (((pfrep_right_backward_coefficient_difference_witness_common_right)=0))))) -> pfrep_left_backward_coefficient_difference_witness_common_right=pfrep_right_backward_coefficient_difference_witness_common_right)))) /\\ (((forall pfp_index_backward_coefficient_difference_witness_operation. (exists pfa_gap_backward_coefficient_difference_witness_operationindex. pfa_gap_backward_coefficient_difference_witness_operationindex + S (pfp_index_backward_coefficient_difference_witness_operation) = (pfaa_length_backward_coefficient_difference)) -> exists pfp_left_backward_coefficient_difference_witness_operation pfp_right_backward_coefficient_difference_witness_operation pfp_value_backward_coefficient_difference_witness_operation. ((((exists ff_h_pfp_backward_coefficient_difference_witness_operationleft. ff_h_pfp_backward_coefficient_difference_witness_operationleft + S (pfp_left_backward_coefficient_difference_witness_operation) = S ((S (pfp_index_backward_coefficient_difference_witness_operation)) * pfaa_left_c_backward_coefficient_difference)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_operationleft. pfaa_left_b_backward_coefficient_difference = ff_q_pfp_backward_coefficient_difference_witness_operationleft * S ((S (pfp_index_backward_coefficient_difference_witness_operation)) * pfaa_left_c_backward_coefficient_difference) + (pfp_left_backward_coefficient_difference_witness_operation))) /\\ (((((exists ff_h_pfp_backward_coefficient_difference_witness_operationright. ff_h_pfp_backward_coefficient_difference_witness_operationright + S (pfp_right_backward_coefficient_difference_witness_operation) = S ((S (pfp_index_backward_coefficient_difference_witness_operation)) * pfaa_right_c_backward_coefficient_difference)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_operationright. pfaa_right_b_backward_coefficient_difference = ff_q_pfp_backward_coefficient_difference_witness_operationright * S ((S (pfp_index_backward_coefficient_difference_witness_operation)) * pfaa_right_c_backward_coefficient_difference) + (pfp_right_backward_coefficient_difference_witness_operation))) /\\ (((((exists ff_h_pfp_backward_coefficient_difference_witness_operationtarget. ff_h_pfp_backward_coefficient_difference_witness_operationtarget + S (pfp_value_backward_coefficient_difference_witness_operation) = S ((S (pfp_index_backward_coefficient_difference_witness_operation)) * pfaa_sum_c_backward_coefficient_difference)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_operationtarget. pfaa_sum_b_backward_coefficient_difference = ff_q_pfp_backward_coefficient_difference_witness_operationtarget * S ((S (pfp_index_backward_coefficient_difference_witness_operation)) * pfaa_sum_c_backward_coefficient_difference) + (pfp_value_backward_coefficient_difference_witness_operation))) /\\ ((((exists pfa_gap_backward_coefficient_difference_witness_operationoperationleft. pfa_gap_backward_coefficient_difference_witness_operationoperationleft + S (pfp_left_backward_coefficient_difference_witness_operation) = (p)) /\\ (((exists pfa_gap_backward_coefficient_difference_witness_operationoperationright. pfa_gap_backward_coefficient_difference_witness_operationoperationright + S (pfp_right_backward_coefficient_difference_witness_operation) = (p)) /\\ ((((exists pfa_gap_backward_coefficient_difference_witness_operationoperationresultbound. pfa_gap_backward_coefficient_difference_witness_operationoperationresultbound + S (pfp_value_backward_coefficient_difference_witness_operation) = (p)) /\\ ((exists pfa_offset_left_backward_coefficient_difference_witness_operationoperationresultcongruence pfa_offset_right_backward_coefficient_difference_witness_operationoperationresultcongruence. ((pfp_left_backward_coefficient_difference_witness_operation) + (pfp_right_backward_coefficient_difference_witness_operation)) + (p) * pfa_offset_left_backward_coefficient_difference_witness_operationoperationresultcongruence = (pfp_value_backward_coefficient_difference_witness_operation) + (p) * pfa_offset_right_backward_coefficient_difference_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_backward_coefficient_difference_witness_output pfrep_left_backward_coefficient_difference_witness_output pfrep_right_backward_coefficient_difference_witness_output. ((exists pfrep_position_backward_coefficient_difference_witness_outputfirst. ((pfrep_position_backward_coefficient_difference_witness_outputfirst+S (pfrep_power_backward_coefficient_difference_witness_output)=(pfaa_length_backward_coefficient_difference)) /\\ ((((exists ff_h_pfp_backward_coefficient_difference_witness_outputfirstentry. ff_h_pfp_backward_coefficient_difference_witness_outputfirstentry + S (pfrep_left_backward_coefficient_difference_witness_output) = S ((S (pfrep_position_backward_coefficient_difference_witness_outputfirst)) * pfaa_sum_c_backward_coefficient_difference)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_outputfirstentry. pfaa_sum_b_backward_coefficient_difference = ff_q_pfp_backward_coefficient_difference_witness_outputfirstentry * S ((S (pfrep_position_backward_coefficient_difference_witness_outputfirst)) * pfaa_sum_c_backward_coefficient_difference) + (pfrep_left_backward_coefficient_difference_witness_output)))))) \\/ (((exists pfrep_gap_backward_coefficient_difference_witness_outputfirstoutside. pfrep_gap_backward_coefficient_difference_witness_outputfirstoutside+(pfaa_length_backward_coefficient_difference)=(pfrep_power_backward_coefficient_difference_witness_output)) /\\ (((pfrep_left_backward_coefficient_difference_witness_output)=0))))) -> ((exists pfrep_position_backward_coefficient_difference_witness_outputsecond. ((pfrep_position_backward_coefficient_difference_witness_outputsecond+S (pfrep_power_backward_coefficient_difference_witness_output)=(Lu)) /\\ ((((exists ff_h_pfp_backward_coefficient_difference_witness_outputsecondentry. ff_h_pfp_backward_coefficient_difference_witness_outputsecondentry + S (pfrep_right_backward_coefficient_difference_witness_output) = S ((S (pfrep_position_backward_coefficient_difference_witness_outputsecond)) * uc)) /\\ exists ff_q_pfp_backward_coefficient_difference_witness_outputsecondentry. ub = ff_q_pfp_backward_coefficient_difference_witness_outputsecondentry * S ((S (pfrep_position_backward_coefficient_difference_witness_outputsecond)) * uc) + (pfrep_right_backward_coefficient_difference_witness_output)))))) \\/ (((exists pfrep_gap_backward_coefficient_difference_witness_outputsecondoutside. pfrep_gap_backward_coefficient_difference_witness_outputsecondoutside+(Lu)=(pfrep_power_backward_coefficient_difference_witness_output)) /\\ (((pfrep_right_backward_coefficient_difference_witness_output)=0))))) -> pfrep_left_backward_coefficient_difference_witness_output=pfrep_right_backward_coefficient_difference_witness_output))))))))))))) -> (((forall fom_index_pfp_backward_new_leftleft. (exists fom_gap_pfp_backward_new_leftleft_index_bound. fom_gap_pfp_backward_new_leftleft_index_bound + S (fom_index_pfp_backward_new_leftleft) = Lv) -> exists fom_value_pfp_backward_new_leftleft. ((((exists fom_beta_height_pfp_backward_new_leftleft_entry. fom_beta_height_pfp_backward_new_leftleft_entry + S (fom_value_pfp_backward_new_leftleft) = S ((S (fom_index_pfp_backward_new_leftleft)) * vc)) /\\ exists fom_beta_quotient_pfp_backward_new_leftleft_entry. vb = fom_beta_quotient_pfp_backward_new_leftleft_entry * S ((S (fom_index_pfp_backward_new_leftleft)) * vc) + (fom_value_pfp_backward_new_leftleft))) /\\ (exists fom_gap_pfp_backward_new_leftleft_value_bound. fom_gap_pfp_backward_new_leftleft_value_bound + S (fom_value_pfp_backward_new_leftleft) = p))) /\\ (((forall fom_index_pfp_backward_new_leftright. (exists fom_gap_pfp_backward_new_leftright_index_bound. fom_gap_pfp_backward_new_leftright_index_bound + S (fom_index_pfp_backward_new_leftright) = La) -> exists fom_value_pfp_backward_new_leftright. ((((exists fom_beta_height_pfp_backward_new_leftright_entry. fom_beta_height_pfp_backward_new_leftright_entry + S (fom_value_pfp_backward_new_leftright) = S ((S (fom_index_pfp_backward_new_leftright)) * ac)) /\\ exists fom_beta_quotient_pfp_backward_new_leftright_entry. ab = fom_beta_quotient_pfp_backward_new_leftright_entry * S ((S (fom_index_pfp_backward_new_leftright)) * ac) + (fom_value_pfp_backward_new_leftright))) /\\ (exists fom_gap_pfp_backward_new_leftright_value_bound. fom_gap_pfp_backward_new_leftright_value_bound + S (fom_value_pfp_backward_new_leftright) = p))) /\\ (((((((Lv)=0 \\/ (La)=0) /\\ (((Lx)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((La)=0)) /\\ (((Lv)+(La)=S (Lx)))))))) /\\ ((forall pfc_index_backward_new_leftcoefficients. (exists pfa_gap_backward_new_leftcoefficientsbound. pfa_gap_backward_new_leftcoefficientsbound + S (pfc_index_backward_new_leftcoefficients) = (Lx)) -> exists pfc_value_backward_new_leftcoefficients. ((((exists ff_h_pfp_backward_new_leftcoefficientsentry. ff_h_pfp_backward_new_leftcoefficientsentry + S (pfc_value_backward_new_leftcoefficients) = S ((S (pfc_index_backward_new_leftcoefficients)) * xc)) /\\ exists ff_q_pfp_backward_new_leftcoefficientsentry. xb = ff_q_pfp_backward_new_leftcoefficientsentry * S ((S (pfc_index_backward_new_leftcoefficients)) * xc) + (pfc_value_backward_new_leftcoefficients))) /\\ ((exists pfc_terms_code_backward_new_leftcoefficientscoefficient pfc_terms_scale_backward_new_leftcoefficientscoefficient pfc_natural_sum_backward_new_leftcoefficientscoefficient. ((forall pfc_index_backward_new_leftcoefficientscoefficientdiagonal. (exists pfa_gap_backward_new_leftcoefficientscoefficientdiagonalbound. pfa_gap_backward_new_leftcoefficientscoefficientdiagonalbound + S (pfc_index_backward_new_leftcoefficientscoefficientdiagonal) = (S (pfc_index_backward_new_leftcoefficients))) -> exists pfc_value_backward_new_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_new_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_new_leftcoefficientscoefficientdiagonalentry + S (pfc_value_backward_new_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_new_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_new_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_new_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_new_leftcoefficientscoefficient = ff_q_pfp_backward_new_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_new_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_new_leftcoefficientscoefficient) + (pfc_value_backward_new_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_new_leftcoefficientscoefficientdiagonalterm pfc_left_backward_new_leftcoefficientscoefficientdiagonalterm pfc_right_backward_new_leftcoefficientscoefficientdiagonalterm. (((pfc_index_backward_new_leftcoefficientscoefficientdiagonal)+pfc_complement_backward_new_leftcoefficientscoefficientdiagonalterm=(pfc_index_backward_new_leftcoefficients)) /\\ ((((((exists pfa_gap_backward_new_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_new_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_new_leftcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_backward_new_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_new_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_new_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_new_leftcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_backward_new_leftcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_backward_new_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_new_leftcoefficientscoefficientdiagonal)) * vc) + (pfc_left_backward_new_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_new_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_new_leftcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_backward_new_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_new_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_new_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_new_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_new_leftcoefficientscoefficientdiagonalterm) = (La)) /\\ ((((exists ff_h_pfp_backward_new_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_new_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_new_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_new_leftcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_backward_new_leftcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_backward_new_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_new_leftcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_backward_new_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_new_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_new_leftcoefficientscoefficientdiagonaltermrightoutside+(La)=(pfc_complement_backward_new_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_new_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_new_leftcoefficientscoefficientdiagonal)=pfc_left_backward_new_leftcoefficientscoefficientdiagonalterm*pfc_right_backward_new_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_new_leftcoefficientscoefficientsum fs_v_pfc_backward_new_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_start. fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_start. fs_u_pfc_backward_new_leftcoefficientscoefficientsum = fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_new_leftcoefficientscoefficient) = S ((S (S (pfc_index_backward_new_leftcoefficients))) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_new_leftcoefficientscoefficientsum = fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_new_leftcoefficients))) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum) + (pfc_natural_sum_backward_new_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps = S (pfc_index_backward_new_leftcoefficients)) -> exists fs_a_pfc_backward_new_leftcoefficientscoefficientsum_body_steps fs_r_pfc_backward_new_leftcoefficientscoefficientsum_body_steps fs_s_pfc_backward_new_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_new_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_new_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_new_leftcoefficientscoefficient = fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_new_leftcoefficientscoefficient) + (fs_a_pfc_backward_new_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_new_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_new_leftcoefficientscoefficientsum = fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum) + (fs_r_pfc_backward_new_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_new_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_new_leftcoefficientscoefficientsum = fs_q_pfc_backward_new_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_new_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_leftcoefficientscoefficientsum) + (fs_s_pfc_backward_new_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_new_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_new_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_new_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_new_leftcoefficientscoefficientresiduebound. pfa_gap_backward_new_leftcoefficientscoefficientresiduebound + S (pfc_value_backward_new_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_new_leftcoefficientscoefficientresiduecongruence pfa_offset_right_backward_new_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_new_leftcoefficientscoefficient) + (p) * pfa_offset_left_backward_new_leftcoefficientscoefficientresiduecongruence = (pfc_value_backward_new_leftcoefficients) + (p) * pfa_offset_right_backward_new_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_new_rightleft. (exists fom_gap_pfp_backward_new_rightleft_index_bound. fom_gap_pfp_backward_new_rightleft_index_bound + S (fom_index_pfp_backward_new_rightleft) = Lt) -> exists fom_value_pfp_backward_new_rightleft. ((((exists fom_beta_height_pfp_backward_new_rightleft_entry. fom_beta_height_pfp_backward_new_rightleft_entry + S (fom_value_pfp_backward_new_rightleft) = S ((S (fom_index_pfp_backward_new_rightleft)) * tc)) /\\ exists fom_beta_quotient_pfp_backward_new_rightleft_entry. tb = fom_beta_quotient_pfp_backward_new_rightleft_entry * S ((S (fom_index_pfp_backward_new_rightleft)) * tc) + (fom_value_pfp_backward_new_rightleft))) /\\ (exists fom_gap_pfp_backward_new_rightleft_value_bound. fom_gap_pfp_backward_new_rightleft_value_bound + S (fom_value_pfp_backward_new_rightleft) = p))) /\\ (((forall fom_index_pfp_backward_new_rightright. (exists fom_gap_pfp_backward_new_rightright_index_bound. fom_gap_pfp_backward_new_rightright_index_bound + S (fom_index_pfp_backward_new_rightright) = Lb) -> exists fom_value_pfp_backward_new_rightright. ((((exists fom_beta_height_pfp_backward_new_rightright_entry. fom_beta_height_pfp_backward_new_rightright_entry + S (fom_value_pfp_backward_new_rightright) = S ((S (fom_index_pfp_backward_new_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_backward_new_rightright_entry. bb = fom_beta_quotient_pfp_backward_new_rightright_entry * S ((S (fom_index_pfp_backward_new_rightright)) * bc) + (fom_value_pfp_backward_new_rightright))) /\\ (exists fom_gap_pfp_backward_new_rightright_value_bound. fom_gap_pfp_backward_new_rightright_value_bound + S (fom_value_pfp_backward_new_rightright) = p))) /\\ (((((((Lt)=0 \\/ (Lb)=0) /\\ (((Ly)=0)))) \\/ (((~((Lt)=0)) /\\ (((~((Lb)=0)) /\\ (((Lt)+(Lb)=S (Ly)))))))) /\\ ((forall pfc_index_backward_new_rightcoefficients. (exists pfa_gap_backward_new_rightcoefficientsbound. pfa_gap_backward_new_rightcoefficientsbound + S (pfc_index_backward_new_rightcoefficients) = (Ly)) -> exists pfc_value_backward_new_rightcoefficients. ((((exists ff_h_pfp_backward_new_rightcoefficientsentry. ff_h_pfp_backward_new_rightcoefficientsentry + S (pfc_value_backward_new_rightcoefficients) = S ((S (pfc_index_backward_new_rightcoefficients)) * yc)) /\\ exists ff_q_pfp_backward_new_rightcoefficientsentry. yb = ff_q_pfp_backward_new_rightcoefficientsentry * S ((S (pfc_index_backward_new_rightcoefficients)) * yc) + (pfc_value_backward_new_rightcoefficients))) /\\ ((exists pfc_terms_code_backward_new_rightcoefficientscoefficient pfc_terms_scale_backward_new_rightcoefficientscoefficient pfc_natural_sum_backward_new_rightcoefficientscoefficient. ((forall pfc_index_backward_new_rightcoefficientscoefficientdiagonal. (exists pfa_gap_backward_new_rightcoefficientscoefficientdiagonalbound. pfa_gap_backward_new_rightcoefficientscoefficientdiagonalbound + S (pfc_index_backward_new_rightcoefficientscoefficientdiagonal) = (S (pfc_index_backward_new_rightcoefficients))) -> exists pfc_value_backward_new_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_new_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_new_rightcoefficientscoefficientdiagonalentry + S (pfc_value_backward_new_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_new_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_new_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_new_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_new_rightcoefficientscoefficient = ff_q_pfp_backward_new_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_new_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_new_rightcoefficientscoefficient) + (pfc_value_backward_new_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_new_rightcoefficientscoefficientdiagonalterm pfc_left_backward_new_rightcoefficientscoefficientdiagonalterm pfc_right_backward_new_rightcoefficientscoefficientdiagonalterm. (((pfc_index_backward_new_rightcoefficientscoefficientdiagonal)+pfc_complement_backward_new_rightcoefficientscoefficientdiagonalterm=(pfc_index_backward_new_rightcoefficients)) /\\ ((((((exists pfa_gap_backward_new_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_new_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_new_rightcoefficientscoefficientdiagonal) = (Lt)) /\\ ((((exists ff_h_pfp_backward_new_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_new_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_new_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_new_rightcoefficientscoefficientdiagonal)) * tc)) /\\ exists ff_q_pfp_backward_new_rightcoefficientscoefficientdiagonaltermleftentry. tb = ff_q_pfp_backward_new_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_new_rightcoefficientscoefficientdiagonal)) * tc) + (pfc_left_backward_new_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_new_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_new_rightcoefficientscoefficientdiagonaltermleftoutside+(Lt)=(pfc_index_backward_new_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_new_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_new_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_new_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_new_rightcoefficientscoefficientdiagonalterm) = (Lb)) /\\ ((((exists ff_h_pfp_backward_new_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_new_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_new_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_new_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_backward_new_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_backward_new_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_new_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_backward_new_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_new_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_new_rightcoefficientscoefficientdiagonaltermrightoutside+(Lb)=(pfc_complement_backward_new_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_new_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_new_rightcoefficientscoefficientdiagonal)=pfc_left_backward_new_rightcoefficientscoefficientdiagonalterm*pfc_right_backward_new_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_new_rightcoefficientscoefficientsum fs_v_pfc_backward_new_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_start. fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_start. fs_u_pfc_backward_new_rightcoefficientscoefficientsum = fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_new_rightcoefficientscoefficient) = S ((S (S (pfc_index_backward_new_rightcoefficients))) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_new_rightcoefficientscoefficientsum = fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_new_rightcoefficients))) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum) + (pfc_natural_sum_backward_new_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps = S (pfc_index_backward_new_rightcoefficients)) -> exists fs_a_pfc_backward_new_rightcoefficientscoefficientsum_body_steps fs_r_pfc_backward_new_rightcoefficientscoefficientsum_body_steps fs_s_pfc_backward_new_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_new_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_new_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_new_rightcoefficientscoefficient = fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_new_rightcoefficientscoefficient) + (fs_a_pfc_backward_new_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_new_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_new_rightcoefficientscoefficientsum = fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum) + (fs_r_pfc_backward_new_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_new_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_new_rightcoefficientscoefficientsum = fs_q_pfc_backward_new_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_new_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_new_rightcoefficientscoefficientsum) + (fs_s_pfc_backward_new_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_new_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_new_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_new_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_new_rightcoefficientscoefficientresiduebound. pfa_gap_backward_new_rightcoefficientscoefficientresiduebound + S (pfc_value_backward_new_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_new_rightcoefficientscoefficientresiduecongruence pfa_offset_right_backward_new_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_new_rightcoefficientscoefficient) + (p) * pfa_offset_left_backward_new_rightcoefficientscoefficientresiduecongruence = (pfc_value_backward_new_rightcoefficients) + (p) * pfa_offset_right_backward_new_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_associated_productleft. (exists fom_gap_pfp_backward_associated_productleft_index_bound. fom_gap_pfp_backward_associated_productleft_index_bound + S (fom_index_pfp_backward_associated_productleft) = Lw) -> exists fom_value_pfp_backward_associated_productleft. ((((exists fom_beta_height_pfp_backward_associated_productleft_entry. fom_beta_height_pfp_backward_associated_productleft_entry + S (fom_value_pfp_backward_associated_productleft) = S ((S (fom_index_pfp_backward_associated_productleft)) * wc)) /\\ exists fom_beta_quotient_pfp_backward_associated_productleft_entry. wb = fom_beta_quotient_pfp_backward_associated_productleft_entry * S ((S (fom_index_pfp_backward_associated_productleft)) * wc) + (fom_value_pfp_backward_associated_productleft))) /\\ (exists fom_gap_pfp_backward_associated_productleft_value_bound. fom_gap_pfp_backward_associated_productleft_value_bound + S (fom_value_pfp_backward_associated_productleft) = p))) /\\ (((forall fom_index_pfp_backward_associated_productright. (exists fom_gap_pfp_backward_associated_productright_index_bound. fom_gap_pfp_backward_associated_productright_index_bound + S (fom_index_pfp_backward_associated_productright) = Lb) -> exists fom_value_pfp_backward_associated_productright. ((((exists fom_beta_height_pfp_backward_associated_productright_entry. fom_beta_height_pfp_backward_associated_productright_entry + S (fom_value_pfp_backward_associated_productright) = S ((S (fom_index_pfp_backward_associated_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_backward_associated_productright_entry. bb = fom_beta_quotient_pfp_backward_associated_productright_entry * S ((S (fom_index_pfp_backward_associated_productright)) * bc) + (fom_value_pfp_backward_associated_productright))) /\\ (exists fom_gap_pfp_backward_associated_productright_value_bound. fom_gap_pfp_backward_associated_productright_value_bound + S (fom_value_pfp_backward_associated_productright) = p))) /\\ (((((((Lw)=0 \\/ (Lb)=0) /\\ (((Lz)=0)))) \\/ (((~((Lw)=0)) /\\ (((~((Lb)=0)) /\\ (((Lw)+(Lb)=S (Lz)))))))) /\\ ((forall pfc_index_backward_associated_productcoefficients. (exists pfa_gap_backward_associated_productcoefficientsbound. pfa_gap_backward_associated_productcoefficientsbound + S (pfc_index_backward_associated_productcoefficients) = (Lz)) -> exists pfc_value_backward_associated_productcoefficients. ((((exists ff_h_pfp_backward_associated_productcoefficientsentry. ff_h_pfp_backward_associated_productcoefficientsentry + S (pfc_value_backward_associated_productcoefficients) = S ((S (pfc_index_backward_associated_productcoefficients)) * zc)) /\\ exists ff_q_pfp_backward_associated_productcoefficientsentry. zb = ff_q_pfp_backward_associated_productcoefficientsentry * S ((S (pfc_index_backward_associated_productcoefficients)) * zc) + (pfc_value_backward_associated_productcoefficients))) /\\ ((exists pfc_terms_code_backward_associated_productcoefficientscoefficient pfc_terms_scale_backward_associated_productcoefficientscoefficient pfc_natural_sum_backward_associated_productcoefficientscoefficient. ((forall pfc_index_backward_associated_productcoefficientscoefficientdiagonal. (exists pfa_gap_backward_associated_productcoefficientscoefficientdiagonalbound. pfa_gap_backward_associated_productcoefficientscoefficientdiagonalbound + S (pfc_index_backward_associated_productcoefficientscoefficientdiagonal) = (S (pfc_index_backward_associated_productcoefficients))) -> exists pfc_value_backward_associated_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_associated_productcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_associated_productcoefficientscoefficientdiagonalentry + S (pfc_value_backward_associated_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_associated_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_associated_productcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_associated_productcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_associated_productcoefficientscoefficient = ff_q_pfp_backward_associated_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_associated_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_associated_productcoefficientscoefficient) + (pfc_value_backward_associated_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_associated_productcoefficientscoefficientdiagonalterm pfc_left_backward_associated_productcoefficientscoefficientdiagonalterm pfc_right_backward_associated_productcoefficientscoefficientdiagonalterm. (((pfc_index_backward_associated_productcoefficientscoefficientdiagonal)+pfc_complement_backward_associated_productcoefficientscoefficientdiagonalterm=(pfc_index_backward_associated_productcoefficients)) /\\ ((((((exists pfa_gap_backward_associated_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_associated_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_associated_productcoefficientscoefficientdiagonal) = (Lw)) /\\ ((((exists ff_h_pfp_backward_associated_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_associated_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_associated_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_associated_productcoefficientscoefficientdiagonal)) * wc)) /\\ exists ff_q_pfp_backward_associated_productcoefficientscoefficientdiagonaltermleftentry. wb = ff_q_pfp_backward_associated_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_associated_productcoefficientscoefficientdiagonal)) * wc) + (pfc_left_backward_associated_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_associated_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_associated_productcoefficientscoefficientdiagonaltermleftoutside+(Lw)=(pfc_index_backward_associated_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_associated_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_associated_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_associated_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_associated_productcoefficientscoefficientdiagonalterm) = (Lb)) /\\ ((((exists ff_h_pfp_backward_associated_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_associated_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_associated_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_associated_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_backward_associated_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_backward_associated_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_associated_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_backward_associated_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_associated_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_associated_productcoefficientscoefficientdiagonaltermrightoutside+(Lb)=(pfc_complement_backward_associated_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_associated_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_associated_productcoefficientscoefficientdiagonal)=pfc_left_backward_associated_productcoefficientscoefficientdiagonalterm*pfc_right_backward_associated_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_associated_productcoefficientscoefficientsum fs_v_pfc_backward_associated_productcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_start. fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_start. fs_u_pfc_backward_associated_productcoefficientscoefficientsum = fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_associated_productcoefficientscoefficient) = S ((S (S (pfc_index_backward_associated_productcoefficients))) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_associated_productcoefficientscoefficientsum = fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_associated_productcoefficients))) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum) + (pfc_natural_sum_backward_associated_productcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps = S (pfc_index_backward_associated_productcoefficients)) -> exists fs_a_pfc_backward_associated_productcoefficientscoefficientsum_body_steps fs_r_pfc_backward_associated_productcoefficientscoefficientsum_body_steps fs_s_pfc_backward_associated_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_associated_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_associated_productcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_associated_productcoefficientscoefficient = fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_associated_productcoefficientscoefficient) + (fs_a_pfc_backward_associated_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_associated_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_associated_productcoefficientscoefficientsum = fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum) + (fs_r_pfc_backward_associated_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_associated_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_associated_productcoefficientscoefficientsum = fs_q_pfc_backward_associated_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_associated_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_associated_productcoefficientscoefficientsum) + (fs_s_pfc_backward_associated_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_associated_productcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_associated_productcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_associated_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_associated_productcoefficientscoefficientresiduebound. pfa_gap_backward_associated_productcoefficientscoefficientresiduebound + S (pfc_value_backward_associated_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_associated_productcoefficientscoefficientresiduecongruence pfa_offset_right_backward_associated_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_associated_productcoefficientscoefficient) + (p) * pfa_offset_left_backward_associated_productcoefficientscoefficientresiduecongruence = (pfc_value_backward_associated_productcoefficients) + (p) * pfa_offset_right_backward_associated_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_distributed_productleft. (exists fom_gap_pfp_backward_distributed_productleft_index_bound. fom_gap_pfp_backward_distributed_productleft_index_bound + S (fom_index_pfp_backward_distributed_productleft) = Lv) -> exists fom_value_pfp_backward_distributed_productleft. ((((exists fom_beta_height_pfp_backward_distributed_productleft_entry. fom_beta_height_pfp_backward_distributed_productleft_entry + S (fom_value_pfp_backward_distributed_productleft) = S ((S (fom_index_pfp_backward_distributed_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_backward_distributed_productleft_entry. vb = fom_beta_quotient_pfp_backward_distributed_productleft_entry * S ((S (fom_index_pfp_backward_distributed_productleft)) * vc) + (fom_value_pfp_backward_distributed_productleft))) /\\ (exists fom_gap_pfp_backward_distributed_productleft_value_bound. fom_gap_pfp_backward_distributed_productleft_value_bound + S (fom_value_pfp_backward_distributed_productleft) = p))) /\\ (((forall fom_index_pfp_backward_distributed_productright. (exists fom_gap_pfp_backward_distributed_productright_index_bound. fom_gap_pfp_backward_distributed_productright_index_bound + S (fom_index_pfp_backward_distributed_productright) = Lp) -> exists fom_value_pfp_backward_distributed_productright. ((((exists fom_beta_height_pfp_backward_distributed_productright_entry. fom_beta_height_pfp_backward_distributed_productright_entry + S (fom_value_pfp_backward_distributed_productright) = S ((S (fom_index_pfp_backward_distributed_productright)) * pc)) /\\ exists fom_beta_quotient_pfp_backward_distributed_productright_entry. pb = fom_beta_quotient_pfp_backward_distributed_productright_entry * S ((S (fom_index_pfp_backward_distributed_productright)) * pc) + (fom_value_pfp_backward_distributed_productright))) /\\ (exists fom_gap_pfp_backward_distributed_productright_value_bound. fom_gap_pfp_backward_distributed_productright_value_bound + S (fom_value_pfp_backward_distributed_productright) = p))) /\\ (((((((Lv)=0 \\/ (Lp)=0) /\\ (((Lh)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((Lp)=0)) /\\ (((Lv)+(Lp)=S (Lh)))))))) /\\ ((forall pfc_index_backward_distributed_productcoefficients. (exists pfa_gap_backward_distributed_productcoefficientsbound. pfa_gap_backward_distributed_productcoefficientsbound + S (pfc_index_backward_distributed_productcoefficients) = (Lh)) -> exists pfc_value_backward_distributed_productcoefficients. ((((exists ff_h_pfp_backward_distributed_productcoefficientsentry. ff_h_pfp_backward_distributed_productcoefficientsentry + S (pfc_value_backward_distributed_productcoefficients) = S ((S (pfc_index_backward_distributed_productcoefficients)) * hc)) /\\ exists ff_q_pfp_backward_distributed_productcoefficientsentry. hb = ff_q_pfp_backward_distributed_productcoefficientsentry * S ((S (pfc_index_backward_distributed_productcoefficients)) * hc) + (pfc_value_backward_distributed_productcoefficients))) /\\ ((exists pfc_terms_code_backward_distributed_productcoefficientscoefficient pfc_terms_scale_backward_distributed_productcoefficientscoefficient pfc_natural_sum_backward_distributed_productcoefficientscoefficient. ((forall pfc_index_backward_distributed_productcoefficientscoefficientdiagonal. (exists pfa_gap_backward_distributed_productcoefficientscoefficientdiagonalbound. pfa_gap_backward_distributed_productcoefficientscoefficientdiagonalbound + S (pfc_index_backward_distributed_productcoefficientscoefficientdiagonal) = (S (pfc_index_backward_distributed_productcoefficients))) -> exists pfc_value_backward_distributed_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_backward_distributed_productcoefficientscoefficientdiagonalentry. ff_h_pfp_backward_distributed_productcoefficientscoefficientdiagonalentry + S (pfc_value_backward_distributed_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_backward_distributed_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_distributed_productcoefficientscoefficient)) /\\ exists ff_q_pfp_backward_distributed_productcoefficientscoefficientdiagonalentry. pfc_terms_code_backward_distributed_productcoefficientscoefficient = ff_q_pfp_backward_distributed_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_backward_distributed_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_backward_distributed_productcoefficientscoefficient) + (pfc_value_backward_distributed_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_backward_distributed_productcoefficientscoefficientdiagonalterm pfc_left_backward_distributed_productcoefficientscoefficientdiagonalterm pfc_right_backward_distributed_productcoefficientscoefficientdiagonalterm. (((pfc_index_backward_distributed_productcoefficientscoefficientdiagonal)+pfc_complement_backward_distributed_productcoefficientscoefficientdiagonalterm=(pfc_index_backward_distributed_productcoefficients)) /\\ ((((((exists pfa_gap_backward_distributed_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_backward_distributed_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_backward_distributed_productcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_backward_distributed_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_backward_distributed_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_backward_distributed_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_backward_distributed_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_distributed_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_backward_distributed_productcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_backward_distributed_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_backward_distributed_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_backward_distributed_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_backward_distributed_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_backward_distributed_productcoefficientscoefficientdiagonalterm) = (Lp)) /\\ ((((exists ff_h_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_backward_distributed_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_backward_distributed_productcoefficientscoefficientdiagonalterm)) * pc)) /\\ exists ff_q_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermrightentry. pb = ff_q_pfp_backward_distributed_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_backward_distributed_productcoefficientscoefficientdiagonalterm)) * pc) + (pfc_right_backward_distributed_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_backward_distributed_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_backward_distributed_productcoefficientscoefficientdiagonaltermrightoutside+(Lp)=(pfc_complement_backward_distributed_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_backward_distributed_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_backward_distributed_productcoefficientscoefficientdiagonal)=pfc_left_backward_distributed_productcoefficientscoefficientdiagonalterm*pfc_right_backward_distributed_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_backward_distributed_productcoefficientscoefficientsum fs_v_pfc_backward_distributed_productcoefficientscoefficientsum. ((((exists fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_start. fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_start. fs_u_pfc_backward_distributed_productcoefficientscoefficientsum = fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_backward_distributed_productcoefficientscoefficient) = S ((S (S (pfc_index_backward_distributed_productcoefficients))) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_backward_distributed_productcoefficientscoefficientsum = fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_backward_distributed_productcoefficients))) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum) + (pfc_natural_sum_backward_distributed_productcoefficientscoefficient))) /\\ forall fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps = S (pfc_index_backward_distributed_productcoefficients)) -> exists fs_a_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps fs_r_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps fs_s_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_distributed_productcoefficientscoefficient)) /\\ exists fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_backward_distributed_productcoefficientscoefficient = fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_backward_distributed_productcoefficientscoefficient) + (fs_a_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_backward_distributed_productcoefficientscoefficientsum = fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum) + (fs_r_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_backward_distributed_productcoefficientscoefficientsum = fs_q_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_backward_distributed_productcoefficientscoefficientsum) + (fs_s_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps = fs_r_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps + fs_a_pfc_backward_distributed_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_backward_distributed_productcoefficientscoefficientresiduebound. pfa_gap_backward_distributed_productcoefficientscoefficientresiduebound + S (pfc_value_backward_distributed_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_backward_distributed_productcoefficientscoefficientresiduecongruence pfa_offset_right_backward_distributed_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_backward_distributed_productcoefficientscoefficient) + (p) * pfa_offset_left_backward_distributed_productcoefficientscoefficientresiduecongruence = (pfc_value_backward_distributed_productcoefficients) + (p) * pfa_offset_right_backward_distributed_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_backward_result_left_bounded. (exists fom_gap_pfp_backward_result_left_bounded_index_bound. fom_gap_pfp_backward_result_left_bounded_index_bound + S (fom_index_pfp_backward_result_left_bounded) = Lx) -> exists fom_value_pfp_backward_result_left_bounded. ((((exists fom_beta_height_pfp_backward_result_left_bounded_entry. fom_beta_height_pfp_backward_result_left_bounded_entry + S (fom_value_pfp_backward_result_left_bounded) = S ((S (fom_index_pfp_backward_result_left_bounded)) * xc)) /\\ exists fom_beta_quotient_pfp_backward_result_left_bounded_entry. xb = fom_beta_quotient_pfp_backward_result_left_bounded_entry * S ((S (fom_index_pfp_backward_result_left_bounded)) * xc) + (fom_value_pfp_backward_result_left_bounded))) /\\ (exists fom_gap_pfp_backward_result_left_bounded_value_bound. fom_gap_pfp_backward_result_left_bounded_value_bound + S (fom_value_pfp_backward_result_left_bounded) = p))) /\\ (((forall fom_index_pfp_backward_result_right_bounded. (exists fom_gap_pfp_backward_result_right_bounded_index_bound. fom_gap_pfp_backward_result_right_bounded_index_bound + S (fom_index_pfp_backward_result_right_bounded) = Ly) -> exists fom_value_pfp_backward_result_right_bounded. ((((exists fom_beta_height_pfp_backward_result_right_bounded_entry. fom_beta_height_pfp_backward_result_right_bounded_entry + S (fom_value_pfp_backward_result_right_bounded) = S ((S (fom_index_pfp_backward_result_right_bounded)) * yc)) /\\ exists fom_beta_quotient_pfp_backward_result_right_bounded_entry. yb = fom_beta_quotient_pfp_backward_result_right_bounded_entry * S ((S (fom_index_pfp_backward_result_right_bounded)) * yc) + (fom_value_pfp_backward_result_right_bounded))) /\\ (exists fom_gap_pfp_backward_result_right_bounded_value_bound. fom_gap_pfp_backward_result_right_bounded_value_bound + S (fom_value_pfp_backward_result_right_bounded) = p))) /\\ (((forall fom_index_pfp_backward_result_result_bounded. (exists fom_gap_pfp_backward_result_result_bounded_index_bound. fom_gap_pfp_backward_result_result_bounded_index_bound + S (fom_index_pfp_backward_result_result_bounded) = Lg) -> exists fom_value_pfp_backward_result_result_bounded. ((((exists fom_beta_height_pfp_backward_result_result_bounded_entry. fom_beta_height_pfp_backward_result_result_bounded_entry + S (fom_value_pfp_backward_result_result_bounded) = S ((S (fom_index_pfp_backward_result_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_backward_result_result_bounded_entry. gb = fom_beta_quotient_pfp_backward_result_result_bounded_entry * S ((S (fom_index_pfp_backward_result_result_bounded)) * gc) + (fom_value_pfp_backward_result_result_bounded))) /\\ (exists fom_gap_pfp_backward_result_result_bounded_value_bound. fom_gap_pfp_backward_result_result_bounded_value_bound + S (fom_value_pfp_backward_result_result_bounded) = p))) /\\ ((exists pfaa_left_b_backward_result pfaa_left_c_backward_result pfaa_right_b_backward_result pfaa_right_c_backward_result pfaa_sum_b_backward_result pfaa_sum_c_backward_result pfaa_length_backward_result. ((((forall pfrep_power_backward_result_witness_common_left pfrep_left_backward_result_witness_common_left pfrep_right_backward_result_witness_common_left. ((exists pfrep_position_backward_result_witness_common_leftfirst. ((pfrep_position_backward_result_witness_common_leftfirst+S (pfrep_power_backward_result_witness_common_left)=(Lx)) /\\ ((((exists ff_h_pfp_backward_result_witness_common_leftfirstentry. ff_h_pfp_backward_result_witness_common_leftfirstentry + S (pfrep_left_backward_result_witness_common_left) = S ((S (pfrep_position_backward_result_witness_common_leftfirst)) * xc)) /\\ exists ff_q_pfp_backward_result_witness_common_leftfirstentry. xb = ff_q_pfp_backward_result_witness_common_leftfirstentry * S ((S (pfrep_position_backward_result_witness_common_leftfirst)) * xc) + (pfrep_left_backward_result_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_result_witness_common_leftfirstoutside. pfrep_gap_backward_result_witness_common_leftfirstoutside+(Lx)=(pfrep_power_backward_result_witness_common_left)) /\\ (((pfrep_left_backward_result_witness_common_left)=0))))) -> ((exists pfrep_position_backward_result_witness_common_leftsecond. ((pfrep_position_backward_result_witness_common_leftsecond+S (pfrep_power_backward_result_witness_common_left)=(pfaa_length_backward_result)) /\\ ((((exists ff_h_pfp_backward_result_witness_common_leftsecondentry. ff_h_pfp_backward_result_witness_common_leftsecondentry + S (pfrep_right_backward_result_witness_common_left) = S ((S (pfrep_position_backward_result_witness_common_leftsecond)) * pfaa_left_c_backward_result)) /\\ exists ff_q_pfp_backward_result_witness_common_leftsecondentry. pfaa_left_b_backward_result = ff_q_pfp_backward_result_witness_common_leftsecondentry * S ((S (pfrep_position_backward_result_witness_common_leftsecond)) * pfaa_left_c_backward_result) + (pfrep_right_backward_result_witness_common_left)))))) \\/ (((exists pfrep_gap_backward_result_witness_common_leftsecondoutside. pfrep_gap_backward_result_witness_common_leftsecondoutside+(pfaa_length_backward_result)=(pfrep_power_backward_result_witness_common_left)) /\\ (((pfrep_right_backward_result_witness_common_left)=0))))) -> pfrep_left_backward_result_witness_common_left=pfrep_right_backward_result_witness_common_left) /\\ ((forall pfrep_power_backward_result_witness_common_right pfrep_left_backward_result_witness_common_right pfrep_right_backward_result_witness_common_right. ((exists pfrep_position_backward_result_witness_common_rightfirst. ((pfrep_position_backward_result_witness_common_rightfirst+S (pfrep_power_backward_result_witness_common_right)=(Ly)) /\\ ((((exists ff_h_pfp_backward_result_witness_common_rightfirstentry. ff_h_pfp_backward_result_witness_common_rightfirstentry + S (pfrep_left_backward_result_witness_common_right) = S ((S (pfrep_position_backward_result_witness_common_rightfirst)) * yc)) /\\ exists ff_q_pfp_backward_result_witness_common_rightfirstentry. yb = ff_q_pfp_backward_result_witness_common_rightfirstentry * S ((S (pfrep_position_backward_result_witness_common_rightfirst)) * yc) + (pfrep_left_backward_result_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_result_witness_common_rightfirstoutside. pfrep_gap_backward_result_witness_common_rightfirstoutside+(Ly)=(pfrep_power_backward_result_witness_common_right)) /\\ (((pfrep_left_backward_result_witness_common_right)=0))))) -> ((exists pfrep_position_backward_result_witness_common_rightsecond. ((pfrep_position_backward_result_witness_common_rightsecond+S (pfrep_power_backward_result_witness_common_right)=(pfaa_length_backward_result)) /\\ ((((exists ff_h_pfp_backward_result_witness_common_rightsecondentry. ff_h_pfp_backward_result_witness_common_rightsecondentry + S (pfrep_right_backward_result_witness_common_right) = S ((S (pfrep_position_backward_result_witness_common_rightsecond)) * pfaa_right_c_backward_result)) /\\ exists ff_q_pfp_backward_result_witness_common_rightsecondentry. pfaa_right_b_backward_result = ff_q_pfp_backward_result_witness_common_rightsecondentry * S ((S (pfrep_position_backward_result_witness_common_rightsecond)) * pfaa_right_c_backward_result) + (pfrep_right_backward_result_witness_common_right)))))) \\/ (((exists pfrep_gap_backward_result_witness_common_rightsecondoutside. pfrep_gap_backward_result_witness_common_rightsecondoutside+(pfaa_length_backward_result)=(pfrep_power_backward_result_witness_common_right)) /\\ (((pfrep_right_backward_result_witness_common_right)=0))))) -> pfrep_left_backward_result_witness_common_right=pfrep_right_backward_result_witness_common_right)))) /\\ (((forall pfp_index_backward_result_witness_operation. (exists pfa_gap_backward_result_witness_operationindex. pfa_gap_backward_result_witness_operationindex + S (pfp_index_backward_result_witness_operation) = (pfaa_length_backward_result)) -> exists pfp_left_backward_result_witness_operation pfp_right_backward_result_witness_operation pfp_value_backward_result_witness_operation. ((((exists ff_h_pfp_backward_result_witness_operationleft. ff_h_pfp_backward_result_witness_operationleft + S (pfp_left_backward_result_witness_operation) = S ((S (pfp_index_backward_result_witness_operation)) * pfaa_left_c_backward_result)) /\\ exists ff_q_pfp_backward_result_witness_operationleft. pfaa_left_b_backward_result = ff_q_pfp_backward_result_witness_operationleft * S ((S (pfp_index_backward_result_witness_operation)) * pfaa_left_c_backward_result) + (pfp_left_backward_result_witness_operation))) /\\ (((((exists ff_h_pfp_backward_result_witness_operationright. ff_h_pfp_backward_result_witness_operationright + S (pfp_right_backward_result_witness_operation) = S ((S (pfp_index_backward_result_witness_operation)) * pfaa_right_c_backward_result)) /\\ exists ff_q_pfp_backward_result_witness_operationright. pfaa_right_b_backward_result = ff_q_pfp_backward_result_witness_operationright * S ((S (pfp_index_backward_result_witness_operation)) * pfaa_right_c_backward_result) + (pfp_right_backward_result_witness_operation))) /\\ (((((exists ff_h_pfp_backward_result_witness_operationtarget. ff_h_pfp_backward_result_witness_operationtarget + S (pfp_value_backward_result_witness_operation) = S ((S (pfp_index_backward_result_witness_operation)) * pfaa_sum_c_backward_result)) /\\ exists ff_q_pfp_backward_result_witness_operationtarget. pfaa_sum_b_backward_result = ff_q_pfp_backward_result_witness_operationtarget * S ((S (pfp_index_backward_result_witness_operation)) * pfaa_sum_c_backward_result) + (pfp_value_backward_result_witness_operation))) /\\ ((((exists pfa_gap_backward_result_witness_operationoperationleft. pfa_gap_backward_result_witness_operationoperationleft + S (pfp_left_backward_result_witness_operation) = (p)) /\\ (((exists pfa_gap_backward_result_witness_operationoperationright. pfa_gap_backward_result_witness_operationoperationright + S (pfp_right_backward_result_witness_operation) = (p)) /\\ ((((exists pfa_gap_backward_result_witness_operationoperationresultbound. pfa_gap_backward_result_witness_operationoperationresultbound + S (pfp_value_backward_result_witness_operation) = (p)) /\\ ((exists pfa_offset_left_backward_result_witness_operationoperationresultcongruence pfa_offset_right_backward_result_witness_operationoperationresultcongruence. ((pfp_left_backward_result_witness_operation) + (pfp_right_backward_result_witness_operation)) + (p) * pfa_offset_left_backward_result_witness_operationoperationresultcongruence = (pfp_value_backward_result_witness_operation) + (p) * pfa_offset_right_backward_result_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_backward_result_witness_output pfrep_left_backward_result_witness_output pfrep_right_backward_result_witness_output. ((exists pfrep_position_backward_result_witness_outputfirst. ((pfrep_position_backward_result_witness_outputfirst+S (pfrep_power_backward_result_witness_output)=(pfaa_length_backward_result)) /\\ ((((exists ff_h_pfp_backward_result_witness_outputfirstentry. ff_h_pfp_backward_result_witness_outputfirstentry + S (pfrep_left_backward_result_witness_output) = S ((S (pfrep_position_backward_result_witness_outputfirst)) * pfaa_sum_c_backward_result)) /\\ exists ff_q_pfp_backward_result_witness_outputfirstentry. pfaa_sum_b_backward_result = ff_q_pfp_backward_result_witness_outputfirstentry * S ((S (pfrep_position_backward_result_witness_outputfirst)) * pfaa_sum_c_backward_result) + (pfrep_left_backward_result_witness_output)))))) \\/ (((exists pfrep_gap_backward_result_witness_outputfirstoutside. pfrep_gap_backward_result_witness_outputfirstoutside+(pfaa_length_backward_result)=(pfrep_power_backward_result_witness_output)) /\\ (((pfrep_left_backward_result_witness_output)=0))))) -> ((exists pfrep_position_backward_result_witness_outputsecond. ((pfrep_position_backward_result_witness_outputsecond+S (pfrep_power_backward_result_witness_output)=(Lg)) /\\ ((((exists ff_h_pfp_backward_result_witness_outputsecondentry. ff_h_pfp_backward_result_witness_outputsecondentry + S (pfrep_right_backward_result_witness_output) = S ((S (pfrep_position_backward_result_witness_outputsecond)) * gc)) /\\ exists ff_q_pfp_backward_result_witness_outputsecondentry. gb = ff_q_pfp_backward_result_witness_outputsecondentry * S ((S (pfrep_position_backward_result_witness_outputsecond)) * gc) + (pfrep_right_backward_result_witness_output)))))) \\/ (((exists pfrep_gap_backward_result_witness_outputsecondoutside. pfrep_gap_backward_result_witness_outputsecondoutside+(Lg)=(pfrep_power_backward_result_witness_output)) /\\ (((pfrep_right_backward_result_witness_output)=0))))) -> pfrep_left_backward_result_witness_output=pfrep_right_backward_result_witness_output)))))))))))))", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (p)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (ab)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (ac)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (La)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (bb)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (bc)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (Lb)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (rb)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (rc)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (Lr)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (qb)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (qc)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (Lq)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (pb)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (pc)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (Lp)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (ub)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (uc)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (Lu)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (vb)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (vc)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (Lv)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (gb)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (gc)", "specialize prime_field_polynomial_euclidean_backward_coefficient_identity (Lg)", "exact prime_field_polynomial_euclidean_backward_coefficient_identity", "intro hp", "intro hQB", "intro hdivision", "intro hbezout", "have hp0 : ~(p=0)", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hz", "cases hbezout", "cases hbezout_witness", "cases hbezout_witness_witness", "cases hbezout_witness_witness_witness", "cases hbezout_witness_witness_witness_witness", "cases hbezout_witness_witness_witness_witness_witness", "cases hbezout_witness_witness_witness_witness_witness_witness", "cases hbezout_witness_witness_witness_witness_witness_witness_right", "have hUbound : forall fom_index_pfp_hUbound_local. (exists fom_gap_pfp_hUbound_local_index_bound. fom_gap_pfp_hUbound_local_index_bound + S (fom_index_pfp_hUbound_local) = Lu) -> exists fom_value_pfp_hUbound_local. ((((exists fom_beta_height_pfp_hUbound_local_entry. fom_beta_height_pfp_hUbound_local_entry + S (fom_value_pfp_hUbound_local) = S ((S (fom_index_pfp_hUbound_local)) * uc)) /\\ exists fom_beta_quotient_pfp_hUbound_local_entry. ub = fom_beta_quotient_pfp_hUbound_local_entry * S ((S (fom_index_pfp_hUbound_local)) * uc) + (fom_value_pfp_hUbound_local))) /\\ (exists fom_gap_pfp_hUbound_local_value_bound. fom_gap_pfp_hUbound_local_value_bound + S (fom_value_pfp_hUbound_local) = p))", "cases hbezout_witness_witness_witness_witness_witness_witness_left", "exact hbezout_witness_witness_witness_witness_witness_witness_left_left", "have hBbound : forall fom_index_pfp_hBbound_local. (exists fom_gap_pfp_hBbound_local_index_bound. fom_gap_pfp_hBbound_local_index_bound + S (fom_index_pfp_hBbound_local) = Lb) -> exists fom_value_pfp_hBbound_local. ((((exists fom_beta_height_pfp_hBbound_local_entry. fom_beta_height_pfp_hBbound_local_entry + S (fom_value_pfp_hBbound_local) = S ((S (fom_index_pfp_hBbound_local)) * bc)) /\\ exists fom_beta_quotient_pfp_hBbound_local_entry. bb = fom_beta_quotient_pfp_hBbound_local_entry * S ((S (fom_index_pfp_hBbound_local)) * bc) + (fom_value_pfp_hBbound_local))) /\\ (exists fom_gap_pfp_hBbound_local_value_bound. fom_gap_pfp_hBbound_local_value_bound + S (fom_value_pfp_hBbound_local) = p))", "cases hbezout_witness_witness_witness_witness_witness_witness_left", "cases hbezout_witness_witness_witness_witness_witness_witness_left_right", "exact hbezout_witness_witness_witness_witness_witness_witness_left_right_left", "have hVbound : forall fom_index_pfp_hVbound_local. (exists fom_gap_pfp_hVbound_local_index_bound. fom_gap_pfp_hVbound_local_index_bound + S (fom_index_pfp_hVbound_local) = Lv) -> exists fom_value_pfp_hVbound_local. ((((exists fom_beta_height_pfp_hVbound_local_entry. fom_beta_height_pfp_hVbound_local_entry + S (fom_value_pfp_hVbound_local) = S ((S (fom_index_pfp_hVbound_local)) * vc)) /\\ exists fom_beta_quotient_pfp_hVbound_local_entry. vb = fom_beta_quotient_pfp_hVbound_local_entry * S ((S (fom_index_pfp_hVbound_local)) * vc) + (fom_value_pfp_hVbound_local))) /\\ (exists fom_gap_pfp_hVbound_local_value_bound. fom_gap_pfp_hVbound_local_value_bound + S (fom_value_pfp_hVbound_local) = p))", "cases hbezout_witness_witness_witness_witness_witness_witness_right_left", "exact hbezout_witness_witness_witness_witness_witness_witness_right_left_left", "have hQbound : forall fom_index_pfp_hQbound_local. (exists fom_gap_pfp_hQbound_local_index_bound. fom_gap_pfp_hQbound_local_index_bound + S (fom_index_pfp_hQbound_local) = Lq) -> exists fom_value_pfp_hQbound_local. ((((exists fom_beta_height_pfp_hQbound_local_entry. fom_beta_height_pfp_hQbound_local_entry + S (fom_value_pfp_hQbound_local) = S ((S (fom_index_pfp_hQbound_local)) * qc)) /\\ exists fom_beta_quotient_pfp_hQbound_local_entry. qb = fom_beta_quotient_pfp_hQbound_local_entry * S ((S (fom_index_pfp_hQbound_local)) * qc) + (fom_value_pfp_hQbound_local))) /\\ (exists fom_gap_pfp_hQbound_local_value_bound. fom_gap_pfp_hQbound_local_value_bound + S (fom_value_pfp_hQbound_local) = p))", "cases hQB", "exact hQB_left", "have hPbound : forall fom_index_pfp_hPbound_local. (exists fom_gap_pfp_hPbound_local_index_bound. fom_gap_pfp_hPbound_local_index_bound + S (fom_index_pfp_hPbound_local) = Lp) -> exists fom_value_pfp_hPbound_local. ((((exists fom_beta_height_pfp_hPbound_local_entry. fom_beta_height_pfp_hPbound_local_entry + S (fom_value_pfp_hPbound_local) = S ((S (fom_index_pfp_hPbound_local)) * pc)) /\\ exists fom_beta_quotient_pfp_hPbound_local_entry. pb = fom_beta_quotient_pfp_hPbound_local_entry * S ((S (fom_index_pfp_hPbound_local)) * pc) + (fom_value_pfp_hPbound_local))) /\\ (exists fom_gap_pfp_hPbound_local_value_bound. fom_gap_pfp_hPbound_local_value_bound + S (fom_value_pfp_hPbound_local) = p))", "cases hdivision", "exact hdivision_left", "have hAbound : forall fom_index_pfp_hAbound_local. (exists fom_gap_pfp_hAbound_local_index_bound. fom_gap_pfp_hAbound_local_index_bound + S (fom_index_pfp_hAbound_local) = La) -> exists fom_value_pfp_hAbound_local. ((((exists fom_beta_height_pfp_hAbound_local_entry. fom_beta_height_pfp_hAbound_local_entry + S (fom_value_pfp_hAbound_local) = S ((S (fom_index_pfp_hAbound_local)) * ac)) /\\ exists fom_beta_quotient_pfp_hAbound_local_entry. ab = fom_beta_quotient_pfp_hAbound_local_entry * S ((S (fom_index_pfp_hAbound_local)) * ac) + (fom_value_pfp_hAbound_local))) /\\ (exists fom_gap_pfp_hAbound_local_value_bound. fom_gap_pfp_hAbound_local_value_bound + S (fom_value_pfp_hAbound_local) = p))", "cases hdivision", "cases hdivision_right", "cases hdivision_right_right", "exact hdivision_right_right_left", "have hcoefficient_length : exists n. ((((Lv)=0 \\/ (Lq)=0) /\\ (((n)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((Lq)=0)) /\\ (((Lv)+(Lq)=S (n)))))))", "specialize polynomial_product_length_exists (Lv)", "specialize polynomial_product_length_exists (Lq)", "apply polynomial_product_length_exists", "cases hcoefficient_length", "have hcoefficient_product : exists b c. ((forall fom_index_pfp_hcoefficient_productleft. (exists fom_gap_pfp_hcoefficient_productleft_index_bound. fom_gap_pfp_hcoefficient_productleft_index_bound + S (fom_index_pfp_hcoefficient_productleft) = Lv) -> exists fom_value_pfp_hcoefficient_productleft. ((((exists fom_beta_height_pfp_hcoefficient_productleft_entry. fom_beta_height_pfp_hcoefficient_productleft_entry + S (fom_value_pfp_hcoefficient_productleft) = S ((S (fom_index_pfp_hcoefficient_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_hcoefficient_productleft_entry. vb = fom_beta_quotient_pfp_hcoefficient_productleft_entry * S ((S (fom_index_pfp_hcoefficient_productleft)) * vc) + (fom_value_pfp_hcoefficient_productleft))) /\\ (exists fom_gap_pfp_hcoefficient_productleft_value_bound. fom_gap_pfp_hcoefficient_productleft_value_bound + S (fom_value_pfp_hcoefficient_productleft) = p))) /\\ (((forall fom_index_pfp_hcoefficient_productright. (exists fom_gap_pfp_hcoefficient_productright_index_bound. fom_gap_pfp_hcoefficient_productright_index_bound + S (fom_index_pfp_hcoefficient_productright) = Lq) -> exists fom_value_pfp_hcoefficient_productright. ((((exists fom_beta_height_pfp_hcoefficient_productright_entry. fom_beta_height_pfp_hcoefficient_productright_entry + S (fom_value_pfp_hcoefficient_productright) = S ((S (fom_index_pfp_hcoefficient_productright)) * qc)) /\\ exists fom_beta_quotient_pfp_hcoefficient_productright_entry. qb = fom_beta_quotient_pfp_hcoefficient_productright_entry * S ((S (fom_index_pfp_hcoefficient_productright)) * qc) + (fom_value_pfp_hcoefficient_productright))) /\\ (exists fom_gap_pfp_hcoefficient_productright_value_bound. fom_gap_pfp_hcoefficient_productright_value_bound + S (fom_value_pfp_hcoefficient_productright) = p))) /\\ (((((((Lv)=0 \\/ (Lq)=0) /\\ (((x6)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((Lq)=0)) /\\ (((Lv)+(Lq)=S (x6)))))))) /\\ ((forall pfc_index_hcoefficient_productcoefficients. (exists pfa_gap_hcoefficient_productcoefficientsbound. pfa_gap_hcoefficient_productcoefficientsbound + S (pfc_index_hcoefficient_productcoefficients) = (x6)) -> exists pfc_value_hcoefficient_productcoefficients. ((((exists ff_h_pfp_hcoefficient_productcoefficientsentry. ff_h_pfp_hcoefficient_productcoefficientsentry + S (pfc_value_hcoefficient_productcoefficients) = S ((S (pfc_index_hcoefficient_productcoefficients)) * c)) /\\ exists ff_q_pfp_hcoefficient_productcoefficientsentry. b = ff_q_pfp_hcoefficient_productcoefficientsentry * S ((S (pfc_index_hcoefficient_productcoefficients)) * c) + (pfc_value_hcoefficient_productcoefficients))) /\\ ((exists pfc_terms_code_hcoefficient_productcoefficientscoefficient pfc_terms_scale_hcoefficient_productcoefficientscoefficient pfc_natural_sum_hcoefficient_productcoefficientscoefficient. ((forall pfc_index_hcoefficient_productcoefficientscoefficientdiagonal. (exists pfa_gap_hcoefficient_productcoefficientscoefficientdiagonalbound. pfa_gap_hcoefficient_productcoefficientscoefficientdiagonalbound + S (pfc_index_hcoefficient_productcoefficientscoefficientdiagonal) = (S (pfc_index_hcoefficient_productcoefficients))) -> exists pfc_value_hcoefficient_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_hcoefficient_productcoefficientscoefficientdiagonalentry. ff_h_pfp_hcoefficient_productcoefficientscoefficientdiagonalentry + S (pfc_value_hcoefficient_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_hcoefficient_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hcoefficient_productcoefficientscoefficient)) /\\ exists ff_q_pfp_hcoefficient_productcoefficientscoefficientdiagonalentry. pfc_terms_code_hcoefficient_productcoefficientscoefficient = ff_q_pfp_hcoefficient_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_hcoefficient_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hcoefficient_productcoefficientscoefficient) + (pfc_value_hcoefficient_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_hcoefficient_productcoefficientscoefficientdiagonalterm pfc_left_hcoefficient_productcoefficientscoefficientdiagonalterm pfc_right_hcoefficient_productcoefficientscoefficientdiagonalterm. (((pfc_index_hcoefficient_productcoefficientscoefficientdiagonal)+pfc_complement_hcoefficient_productcoefficientscoefficientdiagonalterm=(pfc_index_hcoefficient_productcoefficients)) /\\ ((((((exists pfa_gap_hcoefficient_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_hcoefficient_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_hcoefficient_productcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_hcoefficient_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_hcoefficient_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_hcoefficient_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_hcoefficient_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_hcoefficient_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_hcoefficient_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_hcoefficient_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_hcoefficient_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hcoefficient_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_hcoefficient_productcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_hcoefficient_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_hcoefficient_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_hcoefficient_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_hcoefficient_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_hcoefficient_productcoefficientscoefficientdiagonalterm) = (Lq)) /\\ ((((exists ff_h_pfp_hcoefficient_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_hcoefficient_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_hcoefficient_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_hcoefficient_productcoefficientscoefficientdiagonalterm)) * qc)) /\\ exists ff_q_pfp_hcoefficient_productcoefficientscoefficientdiagonaltermrightentry. qb = ff_q_pfp_hcoefficient_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_hcoefficient_productcoefficientscoefficientdiagonalterm)) * qc) + (pfc_right_hcoefficient_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hcoefficient_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_hcoefficient_productcoefficientscoefficientdiagonaltermrightoutside+(Lq)=(pfc_complement_hcoefficient_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_hcoefficient_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_hcoefficient_productcoefficientscoefficientdiagonal)=pfc_left_hcoefficient_productcoefficientscoefficientdiagonalterm*pfc_right_hcoefficient_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_hcoefficient_productcoefficientscoefficientsum fs_v_pfc_hcoefficient_productcoefficientscoefficientsum. ((((exists fs_h_pfc_hcoefficient_productcoefficientscoefficientsum_body_start. fs_h_pfc_hcoefficient_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_hcoefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hcoefficient_productcoefficientscoefficientsum_body_start. fs_u_pfc_hcoefficient_productcoefficientscoefficientsum = fs_q_pfc_hcoefficient_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_hcoefficient_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_hcoefficient_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_hcoefficient_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_hcoefficient_productcoefficientscoefficient) = S ((S (S (pfc_index_hcoefficient_productcoefficients))) * fs_v_pfc_hcoefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hcoefficient_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_hcoefficient_productcoefficientscoefficientsum = fs_q_pfc_hcoefficient_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_hcoefficient_productcoefficients))) * fs_v_pfc_hcoefficient_productcoefficientscoefficientsum) + (pfc_natural_sum_hcoefficient_productcoefficientscoefficient))) /\\ forall fs_i_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps = S (pfc_index_hcoefficient_productcoefficients)) -> exists fs_a_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps fs_r_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps fs_s_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hcoefficient_productcoefficientscoefficient)) /\\ exists fs_q_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_hcoefficient_productcoefficientscoefficient = fs_q_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hcoefficient_productcoefficientscoefficient) + (fs_a_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hcoefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_hcoefficient_productcoefficientscoefficientsum = fs_q_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hcoefficient_productcoefficientscoefficientsum) + (fs_r_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hcoefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_hcoefficient_productcoefficientscoefficientsum = fs_q_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hcoefficient_productcoefficientscoefficientsum) + (fs_s_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps = fs_r_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps + fs_a_pfc_hcoefficient_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_hcoefficient_productcoefficientscoefficientresiduebound. pfa_gap_hcoefficient_productcoefficientscoefficientresiduebound + S (pfc_value_hcoefficient_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_hcoefficient_productcoefficientscoefficientresiduecongruence pfa_offset_right_hcoefficient_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_hcoefficient_productcoefficientscoefficient) + (p) * pfa_offset_left_hcoefficient_productcoefficientscoefficientresiduecongruence = (pfc_value_hcoefficient_productcoefficients) + (p) * pfa_offset_right_hcoefficient_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (vb)", "specialize prime_field_polynomial_convolution_at_length_exists (vc)", "specialize prime_field_polynomial_convolution_at_length_exists (Lv)", "specialize prime_field_polynomial_convolution_at_length_exists (qb)", "specialize prime_field_polynomial_convolution_at_length_exists (qc)", "specialize prime_field_polynomial_convolution_at_length_exists (Lq)", "specialize prime_field_polynomial_convolution_at_length_exists (x6)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hVbound", "exact hQbound", "exact hcoefficient_length_witness", "cases hcoefficient_product", "cases hcoefficient_product_witness", "have hWbound : forall fom_index_pfp_hWbound_bounded. (exists fom_gap_pfp_hWbound_bounded_index_bound. fom_gap_pfp_hWbound_bounded_index_bound + S (fom_index_pfp_hWbound_bounded) = x6) -> exists fom_value_pfp_hWbound_bounded. ((((exists fom_beta_height_pfp_hWbound_bounded_entry. fom_beta_height_pfp_hWbound_bounded_entry + S (fom_value_pfp_hWbound_bounded) = S ((S (fom_index_pfp_hWbound_bounded)) * x8)) /\\ exists fom_beta_quotient_pfp_hWbound_bounded_entry. x7 = fom_beta_quotient_pfp_hWbound_bounded_entry * S ((S (fom_index_pfp_hWbound_bounded)) * x8) + (fom_value_pfp_hWbound_bounded))) /\\ (exists fom_gap_pfp_hWbound_bounded_value_bound. fom_gap_pfp_hWbound_bounded_value_bound + S (fom_value_pfp_hWbound_bounded) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (vb)", "specialize prime_field_polynomial_convolution_bounded (vc)", "specialize prime_field_polynomial_convolution_bounded (Lv)", "specialize prime_field_polynomial_convolution_bounded (qb)", "specialize prime_field_polynomial_convolution_bounded (qc)", "specialize prime_field_polynomial_convolution_bounded (Lq)", "specialize prime_field_polynomial_convolution_bounded (x7)", "specialize prime_field_polynomial_convolution_bounded (x8)", "specialize prime_field_polynomial_convolution_bounded (x6)", "apply prime_field_polynomial_convolution_bounded", "exact hcoefficient_product_witness_witness", "have hdifference : exists tb tc. ((forall fom_index_pfp_exists_coefficient_difference_left_bounded. (exists fom_gap_pfp_exists_coefficient_difference_left_bounded_index_bound. fom_gap_pfp_exists_coefficient_difference_left_bounded_index_bound + S (fom_index_pfp_exists_coefficient_difference_left_bounded) = x6) -> exists fom_value_pfp_exists_coefficient_difference_left_bounded. ((((exists fom_beta_height_pfp_exists_coefficient_difference_left_bounded_entry. fom_beta_height_pfp_exists_coefficient_difference_left_bounded_entry + S (fom_value_pfp_exists_coefficient_difference_left_bounded) = S ((S (fom_index_pfp_exists_coefficient_difference_left_bounded)) * x8)) /\\ exists fom_beta_quotient_pfp_exists_coefficient_difference_left_bounded_entry. x7 = fom_beta_quotient_pfp_exists_coefficient_difference_left_bounded_entry * S ((S (fom_index_pfp_exists_coefficient_difference_left_bounded)) * x8) + (fom_value_pfp_exists_coefficient_difference_left_bounded))) /\\ (exists fom_gap_pfp_exists_coefficient_difference_left_bounded_value_bound. fom_gap_pfp_exists_coefficient_difference_left_bounded_value_bound + S (fom_value_pfp_exists_coefficient_difference_left_bounded) = p))) /\\ (((forall fom_index_pfp_exists_coefficient_difference_right_bounded. (exists fom_gap_pfp_exists_coefficient_difference_right_bounded_index_bound. fom_gap_pfp_exists_coefficient_difference_right_bounded_index_bound + S (fom_index_pfp_exists_coefficient_difference_right_bounded) = (Lu)+(x6)) -> exists fom_value_pfp_exists_coefficient_difference_right_bounded. ((((exists fom_beta_height_pfp_exists_coefficient_difference_right_bounded_entry. fom_beta_height_pfp_exists_coefficient_difference_right_bounded_entry + S (fom_value_pfp_exists_coefficient_difference_right_bounded) = S ((S (fom_index_pfp_exists_coefficient_difference_right_bounded)) * tc)) /\\ exists fom_beta_quotient_pfp_exists_coefficient_difference_right_bounded_entry. tb = fom_beta_quotient_pfp_exists_coefficient_difference_right_bounded_entry * S ((S (fom_index_pfp_exists_coefficient_difference_right_bounded)) * tc) + (fom_value_pfp_exists_coefficient_difference_right_bounded))) /\\ (exists fom_gap_pfp_exists_coefficient_difference_right_bounded_value_bound. fom_gap_pfp_exists_coefficient_difference_right_bounded_value_bound + S (fom_value_pfp_exists_coefficient_difference_right_bounded) = p))) /\\ (((forall fom_index_pfp_exists_coefficient_difference_result_bounded. (exists fom_gap_pfp_exists_coefficient_difference_result_bounded_index_bound. fom_gap_pfp_exists_coefficient_difference_result_bounded_index_bound + S (fom_index_pfp_exists_coefficient_difference_result_bounded) = Lu) -> exists fom_value_pfp_exists_coefficient_difference_result_bounded. ((((exists fom_beta_height_pfp_exists_coefficient_difference_result_bounded_entry. fom_beta_height_pfp_exists_coefficient_difference_result_bounded_entry + S (fom_value_pfp_exists_coefficient_difference_result_bounded) = S ((S (fom_index_pfp_exists_coefficient_difference_result_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_exists_coefficient_difference_result_bounded_entry. ub = fom_beta_quotient_pfp_exists_coefficient_difference_result_bounded_entry * S ((S (fom_index_pfp_exists_coefficient_difference_result_bounded)) * uc) + (fom_value_pfp_exists_coefficient_difference_result_bounded))) /\\ (exists fom_gap_pfp_exists_coefficient_difference_result_bounded_value_bound. fom_gap_pfp_exists_coefficient_difference_result_bounded_value_bound + S (fom_value_pfp_exists_coefficient_difference_result_bounded) = p))) /\\ ((exists pfaa_left_b_exists_coefficient_difference pfaa_left_c_exists_coefficient_difference pfaa_right_b_exists_coefficient_difference pfaa_right_c_exists_coefficient_difference pfaa_sum_b_exists_coefficient_difference pfaa_sum_c_exists_coefficient_difference pfaa_length_exists_coefficient_difference. ((((forall pfrep_power_exists_coefficient_difference_witness_common_left pfrep_left_exists_coefficient_difference_witness_common_left pfrep_right_exists_coefficient_difference_witness_common_left. ((exists pfrep_position_exists_coefficient_difference_witness_common_leftfirst. ((pfrep_position_exists_coefficient_difference_witness_common_leftfirst+S (pfrep_power_exists_coefficient_difference_witness_common_left)=(x6)) /\\ ((((exists ff_h_pfp_exists_coefficient_difference_witness_common_leftfirstentry. ff_h_pfp_exists_coefficient_difference_witness_common_leftfirstentry + S (pfrep_left_exists_coefficient_difference_witness_common_left) = S ((S (pfrep_position_exists_coefficient_difference_witness_common_leftfirst)) * x8)) /\\ exists ff_q_pfp_exists_coefficient_difference_witness_common_leftfirstentry. x7 = ff_q_pfp_exists_coefficient_difference_witness_common_leftfirstentry * S ((S (pfrep_position_exists_coefficient_difference_witness_common_leftfirst)) * x8) + (pfrep_left_exists_coefficient_difference_witness_common_left)))))) \\/ (((exists pfrep_gap_exists_coefficient_difference_witness_common_leftfirstoutside. pfrep_gap_exists_coefficient_difference_witness_common_leftfirstoutside+(x6)=(pfrep_power_exists_coefficient_difference_witness_common_left)) /\\ (((pfrep_left_exists_coefficient_difference_witness_common_left)=0))))) -> ((exists pfrep_position_exists_coefficient_difference_witness_common_leftsecond. ((pfrep_position_exists_coefficient_difference_witness_common_leftsecond+S (pfrep_power_exists_coefficient_difference_witness_common_left)=(pfaa_length_exists_coefficient_difference)) /\\ ((((exists ff_h_pfp_exists_coefficient_difference_witness_common_leftsecondentry. ff_h_pfp_exists_coefficient_difference_witness_common_leftsecondentry + S (pfrep_right_exists_coefficient_difference_witness_common_left) = S ((S (pfrep_position_exists_coefficient_difference_witness_common_leftsecond)) * pfaa_left_c_exists_coefficient_difference)) /\\ exists ff_q_pfp_exists_coefficient_difference_witness_common_leftsecondentry. pfaa_left_b_exists_coefficient_difference = ff_q_pfp_exists_coefficient_difference_witness_common_leftsecondentry * S ((S (pfrep_position_exists_coefficient_difference_witness_common_leftsecond)) * pfaa_left_c_exists_coefficient_difference) + (pfrep_right_exists_coefficient_difference_witness_common_left)))))) \\/ (((exists pfrep_gap_exists_coefficient_difference_witness_common_leftsecondoutside. pfrep_gap_exists_coefficient_difference_witness_common_leftsecondoutside+(pfaa_length_exists_coefficient_difference)=(pfrep_power_exists_coefficient_difference_witness_common_left)) /\\ (((pfrep_right_exists_coefficient_difference_witness_common_left)=0))))) -> pfrep_left_exists_coefficient_difference_witness_common_left=pfrep_right_exists_coefficient_difference_witness_common_left) /\\ ((forall pfrep_power_exists_coefficient_difference_witness_common_right pfrep_left_exists_coefficient_difference_witness_common_right pfrep_right_exists_coefficient_difference_witness_common_right. ((exists pfrep_position_exists_coefficient_difference_witness_common_rightfirst. ((pfrep_position_exists_coefficient_difference_witness_common_rightfirst+S (pfrep_power_exists_coefficient_difference_witness_common_right)=((Lu)+(x6))) /\\ ((((exists ff_h_pfp_exists_coefficient_difference_witness_common_rightfirstentry. ff_h_pfp_exists_coefficient_difference_witness_common_rightfirstentry + S (pfrep_left_exists_coefficient_difference_witness_common_right) = S ((S (pfrep_position_exists_coefficient_difference_witness_common_rightfirst)) * tc)) /\\ exists ff_q_pfp_exists_coefficient_difference_witness_common_rightfirstentry. tb = ff_q_pfp_exists_coefficient_difference_witness_common_rightfirstentry * S ((S (pfrep_position_exists_coefficient_difference_witness_common_rightfirst)) * tc) + (pfrep_left_exists_coefficient_difference_witness_common_right)))))) \\/ (((exists pfrep_gap_exists_coefficient_difference_witness_common_rightfirstoutside. pfrep_gap_exists_coefficient_difference_witness_common_rightfirstoutside+((Lu)+(x6))=(pfrep_power_exists_coefficient_difference_witness_common_right)) /\\ (((pfrep_left_exists_coefficient_difference_witness_common_right)=0))))) -> ((exists pfrep_position_exists_coefficient_difference_witness_common_rightsecond. ((pfrep_position_exists_coefficient_difference_witness_common_rightsecond+S (pfrep_power_exists_coefficient_difference_witness_common_right)=(pfaa_length_exists_coefficient_difference)) /\\ ((((exists ff_h_pfp_exists_coefficient_difference_witness_common_rightsecondentry. ff_h_pfp_exists_coefficient_difference_witness_common_rightsecondentry + S (pfrep_right_exists_coefficient_difference_witness_common_right) = S ((S (pfrep_position_exists_coefficient_difference_witness_common_rightsecond)) * pfaa_right_c_exists_coefficient_difference)) /\\ exists ff_q_pfp_exists_coefficient_difference_witness_common_rightsecondentry. pfaa_right_b_exists_coefficient_difference = ff_q_pfp_exists_coefficient_difference_witness_common_rightsecondentry * S ((S (pfrep_position_exists_coefficient_difference_witness_common_rightsecond)) * pfaa_right_c_exists_coefficient_difference) + (pfrep_right_exists_coefficient_difference_witness_common_right)))))) \\/ (((exists pfrep_gap_exists_coefficient_difference_witness_common_rightsecondoutside. pfrep_gap_exists_coefficient_difference_witness_common_rightsecondoutside+(pfaa_length_exists_coefficient_difference)=(pfrep_power_exists_coefficient_difference_witness_common_right)) /\\ (((pfrep_right_exists_coefficient_difference_witness_common_right)=0))))) -> pfrep_left_exists_coefficient_difference_witness_common_right=pfrep_right_exists_coefficient_difference_witness_common_right)))) /\\ (((forall pfp_index_exists_coefficient_difference_witness_operation. (exists pfa_gap_exists_coefficient_difference_witness_operationindex. pfa_gap_exists_coefficient_difference_witness_operationindex + S (pfp_index_exists_coefficient_difference_witness_operation) = (pfaa_length_exists_coefficient_difference)) -> exists pfp_left_exists_coefficient_difference_witness_operation pfp_right_exists_coefficient_difference_witness_operation pfp_value_exists_coefficient_difference_witness_operation. ((((exists ff_h_pfp_exists_coefficient_difference_witness_operationleft. ff_h_pfp_exists_coefficient_difference_witness_operationleft + S (pfp_left_exists_coefficient_difference_witness_operation) = S ((S (pfp_index_exists_coefficient_difference_witness_operation)) * pfaa_left_c_exists_coefficient_difference)) /\\ exists ff_q_pfp_exists_coefficient_difference_witness_operationleft. pfaa_left_b_exists_coefficient_difference = ff_q_pfp_exists_coefficient_difference_witness_operationleft * S ((S (pfp_index_exists_coefficient_difference_witness_operation)) * pfaa_left_c_exists_coefficient_difference) + (pfp_left_exists_coefficient_difference_witness_operation))) /\\ (((((exists ff_h_pfp_exists_coefficient_difference_witness_operationright. ff_h_pfp_exists_coefficient_difference_witness_operationright + S (pfp_right_exists_coefficient_difference_witness_operation) = S ((S (pfp_index_exists_coefficient_difference_witness_operation)) * pfaa_right_c_exists_coefficient_difference)) /\\ exists ff_q_pfp_exists_coefficient_difference_witness_operationright. pfaa_right_b_exists_coefficient_difference = ff_q_pfp_exists_coefficient_difference_witness_operationright * S ((S (pfp_index_exists_coefficient_difference_witness_operation)) * pfaa_right_c_exists_coefficient_difference) + (pfp_right_exists_coefficient_difference_witness_operation))) /\\ (((((exists ff_h_pfp_exists_coefficient_difference_witness_operationtarget. ff_h_pfp_exists_coefficient_difference_witness_operationtarget + S (pfp_value_exists_coefficient_difference_witness_operation) = S ((S (pfp_index_exists_coefficient_difference_witness_operation)) * pfaa_sum_c_exists_coefficient_difference)) /\\ exists ff_q_pfp_exists_coefficient_difference_witness_operationtarget. pfaa_sum_b_exists_coefficient_difference = ff_q_pfp_exists_coefficient_difference_witness_operationtarget * S ((S (pfp_index_exists_coefficient_difference_witness_operation)) * pfaa_sum_c_exists_coefficient_difference) + (pfp_value_exists_coefficient_difference_witness_operation))) /\\ ((((exists pfa_gap_exists_coefficient_difference_witness_operationoperationleft. pfa_gap_exists_coefficient_difference_witness_operationoperationleft + S (pfp_left_exists_coefficient_difference_witness_operation) = (p)) /\\ (((exists pfa_gap_exists_coefficient_difference_witness_operationoperationright. pfa_gap_exists_coefficient_difference_witness_operationoperationright + S (pfp_right_exists_coefficient_difference_witness_operation) = (p)) /\\ ((((exists pfa_gap_exists_coefficient_difference_witness_operationoperationresultbound. pfa_gap_exists_coefficient_difference_witness_operationoperationresultbound + S (pfp_value_exists_coefficient_difference_witness_operation) = (p)) /\\ ((exists pfa_offset_left_exists_coefficient_difference_witness_operationoperationresultcongruence pfa_offset_right_exists_coefficient_difference_witness_operationoperationresultcongruence. ((pfp_left_exists_coefficient_difference_witness_operation) + (pfp_right_exists_coefficient_difference_witness_operation)) + (p) * pfa_offset_left_exists_coefficient_difference_witness_operationoperationresultcongruence = (pfp_value_exists_coefficient_difference_witness_operation) + (p) * pfa_offset_right_exists_coefficient_difference_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_exists_coefficient_difference_witness_output pfrep_left_exists_coefficient_difference_witness_output pfrep_right_exists_coefficient_difference_witness_output. ((exists pfrep_position_exists_coefficient_difference_witness_outputfirst. ((pfrep_position_exists_coefficient_difference_witness_outputfirst+S (pfrep_power_exists_coefficient_difference_witness_output)=(pfaa_length_exists_coefficient_difference)) /\\ ((((exists ff_h_pfp_exists_coefficient_difference_witness_outputfirstentry. ff_h_pfp_exists_coefficient_difference_witness_outputfirstentry + S (pfrep_left_exists_coefficient_difference_witness_output) = S ((S (pfrep_position_exists_coefficient_difference_witness_outputfirst)) * pfaa_sum_c_exists_coefficient_difference)) /\\ exists ff_q_pfp_exists_coefficient_difference_witness_outputfirstentry. pfaa_sum_b_exists_coefficient_difference = ff_q_pfp_exists_coefficient_difference_witness_outputfirstentry * S ((S (pfrep_position_exists_coefficient_difference_witness_outputfirst)) * pfaa_sum_c_exists_coefficient_difference) + (pfrep_left_exists_coefficient_difference_witness_output)))))) \\/ (((exists pfrep_gap_exists_coefficient_difference_witness_outputfirstoutside. pfrep_gap_exists_coefficient_difference_witness_outputfirstoutside+(pfaa_length_exists_coefficient_difference)=(pfrep_power_exists_coefficient_difference_witness_output)) /\\ (((pfrep_left_exists_coefficient_difference_witness_output)=0))))) -> ((exists pfrep_position_exists_coefficient_difference_witness_outputsecond. ((pfrep_position_exists_coefficient_difference_witness_outputsecond+S (pfrep_power_exists_coefficient_difference_witness_output)=(Lu)) /\\ ((((exists ff_h_pfp_exists_coefficient_difference_witness_outputsecondentry. ff_h_pfp_exists_coefficient_difference_witness_outputsecondentry + S (pfrep_right_exists_coefficient_difference_witness_output) = S ((S (pfrep_position_exists_coefficient_difference_witness_outputsecond)) * uc)) /\\ exists ff_q_pfp_exists_coefficient_difference_witness_outputsecondentry. ub = ff_q_pfp_exists_coefficient_difference_witness_outputsecondentry * S ((S (pfrep_position_exists_coefficient_difference_witness_outputsecond)) * uc) + (pfrep_right_exists_coefficient_difference_witness_output)))))) \\/ (((exists pfrep_gap_exists_coefficient_difference_witness_outputsecondoutside. pfrep_gap_exists_coefficient_difference_witness_outputsecondoutside+(Lu)=(pfrep_power_exists_coefficient_difference_witness_output)) /\\ (((pfrep_right_exists_coefficient_difference_witness_output)=0))))) -> pfrep_left_exists_coefficient_difference_witness_output=pfrep_right_exists_coefficient_difference_witness_output))))))))))))", "specialize prime_field_polynomial_aligned_subtract_exists (p)", "specialize prime_field_polynomial_aligned_subtract_exists (ub)", "specialize prime_field_polynomial_aligned_subtract_exists (uc)", "specialize prime_field_polynomial_aligned_subtract_exists (Lu)", "specialize prime_field_polynomial_aligned_subtract_exists (x7)", "specialize prime_field_polynomial_aligned_subtract_exists (x8)", "specialize prime_field_polynomial_aligned_subtract_exists (x6)", "apply prime_field_polynomial_aligned_subtract_exists", "exact hp", "exact hUbound", "exact hWbound", "cases hdifference", "cases hdifference_witness", "have htbound : forall fom_index_pfp_exists_difference_T. (exists fom_gap_pfp_exists_difference_T_index_bound. fom_gap_pfp_exists_difference_T_index_bound + S (fom_index_pfp_exists_difference_T) = (Lu)+(x6)) -> exists fom_value_pfp_exists_difference_T. ((((exists fom_beta_height_pfp_exists_difference_T_entry. fom_beta_height_pfp_exists_difference_T_entry + S (fom_value_pfp_exists_difference_T) = S ((S (fom_index_pfp_exists_difference_T)) * x10)) /\\ exists fom_beta_quotient_pfp_exists_difference_T_entry. x9 = fom_beta_quotient_pfp_exists_difference_T_entry * S ((S (fom_index_pfp_exists_difference_T)) * x10) + (fom_value_pfp_exists_difference_T))) /\\ (exists fom_gap_pfp_exists_difference_T_value_bound. fom_gap_pfp_exists_difference_T_value_bound + S (fom_value_pfp_exists_difference_T) = p))", "have hsmall : ((forall fom_index_pfp_exists_difference_W. (exists fom_gap_pfp_exists_difference_W_index_bound. fom_gap_pfp_exists_difference_W_index_bound + S (fom_index_pfp_exists_difference_W) = x6) -> exists fom_value_pfp_exists_difference_W. ((((exists fom_beta_height_pfp_exists_difference_W_entry. fom_beta_height_pfp_exists_difference_W_entry + S (fom_value_pfp_exists_difference_W) = S ((S (fom_index_pfp_exists_difference_W)) * x8)) /\\ exists fom_beta_quotient_pfp_exists_difference_W_entry. x7 = fom_beta_quotient_pfp_exists_difference_W_entry * S ((S (fom_index_pfp_exists_difference_W)) * x8) + (fom_value_pfp_exists_difference_W))) /\\ (exists fom_gap_pfp_exists_difference_W_value_bound. fom_gap_pfp_exists_difference_W_value_bound + S (fom_value_pfp_exists_difference_W) = p))) /\\ (((forall fom_index_pfp_exists_difference_T. (exists fom_gap_pfp_exists_difference_T_index_bound. fom_gap_pfp_exists_difference_T_index_bound + S (fom_index_pfp_exists_difference_T) = (Lu)+(x6)) -> exists fom_value_pfp_exists_difference_T. ((((exists fom_beta_height_pfp_exists_difference_T_entry. fom_beta_height_pfp_exists_difference_T_entry + S (fom_value_pfp_exists_difference_T) = S ((S (fom_index_pfp_exists_difference_T)) * x10)) /\\ exists fom_beta_quotient_pfp_exists_difference_T_entry. x9 = fom_beta_quotient_pfp_exists_difference_T_entry * S ((S (fom_index_pfp_exists_difference_T)) * x10) + (fom_value_pfp_exists_difference_T))) /\\ (exists fom_gap_pfp_exists_difference_T_value_bound. fom_gap_pfp_exists_difference_T_value_bound + S (fom_value_pfp_exists_difference_T) = p))) /\\ ((forall fom_index_pfp_exists_difference_U. (exists fom_gap_pfp_exists_difference_U_index_bound. fom_gap_pfp_exists_difference_U_index_bound + S (fom_index_pfp_exists_difference_U) = Lu) -> exists fom_value_pfp_exists_difference_U. ((((exists fom_beta_height_pfp_exists_difference_U_entry. fom_beta_height_pfp_exists_difference_U_entry + S (fom_value_pfp_exists_difference_U) = S ((S (fom_index_pfp_exists_difference_U)) * uc)) /\\ exists fom_beta_quotient_pfp_exists_difference_U_entry. ub = fom_beta_quotient_pfp_exists_difference_U_entry * S ((S (fom_index_pfp_exists_difference_U)) * uc) + (fom_value_pfp_exists_difference_U))) /\\ (exists fom_gap_pfp_exists_difference_U_value_bound. fom_gap_pfp_exists_difference_U_value_bound + S (fom_value_pfp_exists_difference_U) = p)))))))", "specialize prime_field_polynomial_aligned_add_bounded (p)", "specialize prime_field_polynomial_aligned_add_bounded (x7)", "specialize prime_field_polynomial_aligned_add_bounded (x8)", "specialize prime_field_polynomial_aligned_add_bounded (x6)", "specialize prime_field_polynomial_aligned_add_bounded (x9)", "specialize prime_field_polynomial_aligned_add_bounded (x10)", "specialize prime_field_polynomial_aligned_add_bounded ((Lu)+(x6))", "specialize prime_field_polynomial_aligned_add_bounded (ub)", "specialize prime_field_polynomial_aligned_add_bounded (uc)", "specialize prime_field_polynomial_aligned_add_bounded (Lu)", "apply prime_field_polynomial_aligned_add_bounded", "exact hdifference_witness_witness", "cases hsmall", "cases hsmall_right", "exact hsmall_right_left", "have hnewleft_length : exists n. ((((Lv)=0 \\/ (La)=0) /\\ (((n)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((La)=0)) /\\ (((Lv)+(La)=S (n)))))))", "specialize polynomial_product_length_exists (Lv)", "specialize polynomial_product_length_exists (La)", "apply polynomial_product_length_exists", "cases hnewleft_length", "have hnewleft_product : exists b c. ((forall fom_index_pfp_hnewleft_productleft. (exists fom_gap_pfp_hnewleft_productleft_index_bound. fom_gap_pfp_hnewleft_productleft_index_bound + S (fom_index_pfp_hnewleft_productleft) = Lv) -> exists fom_value_pfp_hnewleft_productleft. ((((exists fom_beta_height_pfp_hnewleft_productleft_entry. fom_beta_height_pfp_hnewleft_productleft_entry + S (fom_value_pfp_hnewleft_productleft) = S ((S (fom_index_pfp_hnewleft_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_hnewleft_productleft_entry. vb = fom_beta_quotient_pfp_hnewleft_productleft_entry * S ((S (fom_index_pfp_hnewleft_productleft)) * vc) + (fom_value_pfp_hnewleft_productleft))) /\\ (exists fom_gap_pfp_hnewleft_productleft_value_bound. fom_gap_pfp_hnewleft_productleft_value_bound + S (fom_value_pfp_hnewleft_productleft) = p))) /\\ (((forall fom_index_pfp_hnewleft_productright. (exists fom_gap_pfp_hnewleft_productright_index_bound. fom_gap_pfp_hnewleft_productright_index_bound + S (fom_index_pfp_hnewleft_productright) = La) -> exists fom_value_pfp_hnewleft_productright. ((((exists fom_beta_height_pfp_hnewleft_productright_entry. fom_beta_height_pfp_hnewleft_productright_entry + S (fom_value_pfp_hnewleft_productright) = S ((S (fom_index_pfp_hnewleft_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_hnewleft_productright_entry. ab = fom_beta_quotient_pfp_hnewleft_productright_entry * S ((S (fom_index_pfp_hnewleft_productright)) * ac) + (fom_value_pfp_hnewleft_productright))) /\\ (exists fom_gap_pfp_hnewleft_productright_value_bound. fom_gap_pfp_hnewleft_productright_value_bound + S (fom_value_pfp_hnewleft_productright) = p))) /\\ (((((((Lv)=0 \\/ (La)=0) /\\ (((x11)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((La)=0)) /\\ (((Lv)+(La)=S (x11)))))))) /\\ ((forall pfc_index_hnewleft_productcoefficients. (exists pfa_gap_hnewleft_productcoefficientsbound. pfa_gap_hnewleft_productcoefficientsbound + S (pfc_index_hnewleft_productcoefficients) = (x11)) -> exists pfc_value_hnewleft_productcoefficients. ((((exists ff_h_pfp_hnewleft_productcoefficientsentry. ff_h_pfp_hnewleft_productcoefficientsentry + S (pfc_value_hnewleft_productcoefficients) = S ((S (pfc_index_hnewleft_productcoefficients)) * c)) /\\ exists ff_q_pfp_hnewleft_productcoefficientsentry. b = ff_q_pfp_hnewleft_productcoefficientsentry * S ((S (pfc_index_hnewleft_productcoefficients)) * c) + (pfc_value_hnewleft_productcoefficients))) /\\ ((exists pfc_terms_code_hnewleft_productcoefficientscoefficient pfc_terms_scale_hnewleft_productcoefficientscoefficient pfc_natural_sum_hnewleft_productcoefficientscoefficient. ((forall pfc_index_hnewleft_productcoefficientscoefficientdiagonal. (exists pfa_gap_hnewleft_productcoefficientscoefficientdiagonalbound. pfa_gap_hnewleft_productcoefficientscoefficientdiagonalbound + S (pfc_index_hnewleft_productcoefficientscoefficientdiagonal) = (S (pfc_index_hnewleft_productcoefficients))) -> exists pfc_value_hnewleft_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_hnewleft_productcoefficientscoefficientdiagonalentry. ff_h_pfp_hnewleft_productcoefficientscoefficientdiagonalentry + S (pfc_value_hnewleft_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_hnewleft_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hnewleft_productcoefficientscoefficient)) /\\ exists ff_q_pfp_hnewleft_productcoefficientscoefficientdiagonalentry. pfc_terms_code_hnewleft_productcoefficientscoefficient = ff_q_pfp_hnewleft_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_hnewleft_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hnewleft_productcoefficientscoefficient) + (pfc_value_hnewleft_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_hnewleft_productcoefficientscoefficientdiagonalterm pfc_left_hnewleft_productcoefficientscoefficientdiagonalterm pfc_right_hnewleft_productcoefficientscoefficientdiagonalterm. (((pfc_index_hnewleft_productcoefficientscoefficientdiagonal)+pfc_complement_hnewleft_productcoefficientscoefficientdiagonalterm=(pfc_index_hnewleft_productcoefficients)) /\\ ((((((exists pfa_gap_hnewleft_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_hnewleft_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_hnewleft_productcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_hnewleft_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_hnewleft_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_hnewleft_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_hnewleft_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_hnewleft_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_hnewleft_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_hnewleft_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_hnewleft_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hnewleft_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_hnewleft_productcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_hnewleft_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_hnewleft_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_hnewleft_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_hnewleft_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_hnewleft_productcoefficientscoefficientdiagonalterm) = (La)) /\\ ((((exists ff_h_pfp_hnewleft_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_hnewleft_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_hnewleft_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_hnewleft_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_hnewleft_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_hnewleft_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_hnewleft_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_hnewleft_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hnewleft_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_hnewleft_productcoefficientscoefficientdiagonaltermrightoutside+(La)=(pfc_complement_hnewleft_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_hnewleft_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_hnewleft_productcoefficientscoefficientdiagonal)=pfc_left_hnewleft_productcoefficientscoefficientdiagonalterm*pfc_right_hnewleft_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_hnewleft_productcoefficientscoefficientsum fs_v_pfc_hnewleft_productcoefficientscoefficientsum. ((((exists fs_h_pfc_hnewleft_productcoefficientscoefficientsum_body_start. fs_h_pfc_hnewleft_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_hnewleft_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnewleft_productcoefficientscoefficientsum_body_start. fs_u_pfc_hnewleft_productcoefficientscoefficientsum = fs_q_pfc_hnewleft_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_hnewleft_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_hnewleft_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_hnewleft_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_hnewleft_productcoefficientscoefficient) = S ((S (S (pfc_index_hnewleft_productcoefficients))) * fs_v_pfc_hnewleft_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnewleft_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_hnewleft_productcoefficientscoefficientsum = fs_q_pfc_hnewleft_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_hnewleft_productcoefficients))) * fs_v_pfc_hnewleft_productcoefficientscoefficientsum) + (pfc_natural_sum_hnewleft_productcoefficientscoefficient))) /\\ forall fs_i_pfc_hnewleft_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_hnewleft_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_hnewleft_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_hnewleft_productcoefficientscoefficientsum_body_steps = S (pfc_index_hnewleft_productcoefficients)) -> exists fs_a_pfc_hnewleft_productcoefficientscoefficientsum_body_steps fs_r_pfc_hnewleft_productcoefficientscoefficientsum_body_steps fs_s_pfc_hnewleft_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_hnewleft_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_hnewleft_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_hnewleft_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hnewleft_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hnewleft_productcoefficientscoefficient)) /\\ exists fs_q_pfc_hnewleft_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_hnewleft_productcoefficientscoefficient = fs_q_pfc_hnewleft_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_hnewleft_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hnewleft_productcoefficientscoefficient) + (fs_a_pfc_hnewleft_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hnewleft_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_hnewleft_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_hnewleft_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hnewleft_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnewleft_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnewleft_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_hnewleft_productcoefficientscoefficientsum = fs_q_pfc_hnewleft_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_hnewleft_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnewleft_productcoefficientscoefficientsum) + (fs_r_pfc_hnewleft_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hnewleft_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_hnewleft_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_hnewleft_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_hnewleft_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnewleft_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnewleft_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_hnewleft_productcoefficientscoefficientsum = fs_q_pfc_hnewleft_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_hnewleft_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnewleft_productcoefficientscoefficientsum) + (fs_s_pfc_hnewleft_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_hnewleft_productcoefficientscoefficientsum_body_steps = fs_r_pfc_hnewleft_productcoefficientscoefficientsum_body_steps + fs_a_pfc_hnewleft_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_hnewleft_productcoefficientscoefficientresiduebound. pfa_gap_hnewleft_productcoefficientscoefficientresiduebound + S (pfc_value_hnewleft_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_hnewleft_productcoefficientscoefficientresiduecongruence pfa_offset_right_hnewleft_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_hnewleft_productcoefficientscoefficient) + (p) * pfa_offset_left_hnewleft_productcoefficientscoefficientresiduecongruence = (pfc_value_hnewleft_productcoefficients) + (p) * pfa_offset_right_hnewleft_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (vb)", "specialize prime_field_polynomial_convolution_at_length_exists (vc)", "specialize prime_field_polynomial_convolution_at_length_exists (Lv)", "specialize prime_field_polynomial_convolution_at_length_exists (ab)", "specialize prime_field_polynomial_convolution_at_length_exists (ac)", "specialize prime_field_polynomial_convolution_at_length_exists (La)", "specialize prime_field_polynomial_convolution_at_length_exists (x11)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hVbound", "exact hAbound", "exact hnewleft_length_witness", "cases hnewleft_product", "cases hnewleft_product_witness", "have hnewright_length : exists n. (((((Lu)+(x6))=0 \\/ (Lb)=0) /\\ (((n)=0)))) \\/ (((~(((Lu)+(x6))=0)) /\\ (((~((Lb)=0)) /\\ ((((Lu)+(x6))+(Lb)=S (n)))))))", "specialize polynomial_product_length_exists ((Lu)+(x6))", "specialize polynomial_product_length_exists (Lb)", "apply polynomial_product_length_exists", "cases hnewright_length", "have hnewright_product : exists b c. ((forall fom_index_pfp_hnewright_productleft. (exists fom_gap_pfp_hnewright_productleft_index_bound. fom_gap_pfp_hnewright_productleft_index_bound + S (fom_index_pfp_hnewright_productleft) = (Lu)+(x6)) -> exists fom_value_pfp_hnewright_productleft. ((((exists fom_beta_height_pfp_hnewright_productleft_entry. fom_beta_height_pfp_hnewright_productleft_entry + S (fom_value_pfp_hnewright_productleft) = S ((S (fom_index_pfp_hnewright_productleft)) * x10)) /\\ exists fom_beta_quotient_pfp_hnewright_productleft_entry. x9 = fom_beta_quotient_pfp_hnewright_productleft_entry * S ((S (fom_index_pfp_hnewright_productleft)) * x10) + (fom_value_pfp_hnewright_productleft))) /\\ (exists fom_gap_pfp_hnewright_productleft_value_bound. fom_gap_pfp_hnewright_productleft_value_bound + S (fom_value_pfp_hnewright_productleft) = p))) /\\ (((forall fom_index_pfp_hnewright_productright. (exists fom_gap_pfp_hnewright_productright_index_bound. fom_gap_pfp_hnewright_productright_index_bound + S (fom_index_pfp_hnewright_productright) = Lb) -> exists fom_value_pfp_hnewright_productright. ((((exists fom_beta_height_pfp_hnewright_productright_entry. fom_beta_height_pfp_hnewright_productright_entry + S (fom_value_pfp_hnewright_productright) = S ((S (fom_index_pfp_hnewright_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_hnewright_productright_entry. bb = fom_beta_quotient_pfp_hnewright_productright_entry * S ((S (fom_index_pfp_hnewright_productright)) * bc) + (fom_value_pfp_hnewright_productright))) /\\ (exists fom_gap_pfp_hnewright_productright_value_bound. fom_gap_pfp_hnewright_productright_value_bound + S (fom_value_pfp_hnewright_productright) = p))) /\\ ((((((((Lu)+(x6))=0 \\/ (Lb)=0) /\\ (((x14)=0)))) \\/ (((~(((Lu)+(x6))=0)) /\\ (((~((Lb)=0)) /\\ ((((Lu)+(x6))+(Lb)=S (x14)))))))) /\\ ((forall pfc_index_hnewright_productcoefficients. (exists pfa_gap_hnewright_productcoefficientsbound. pfa_gap_hnewright_productcoefficientsbound + S (pfc_index_hnewright_productcoefficients) = (x14)) -> exists pfc_value_hnewright_productcoefficients. ((((exists ff_h_pfp_hnewright_productcoefficientsentry. ff_h_pfp_hnewright_productcoefficientsentry + S (pfc_value_hnewright_productcoefficients) = S ((S (pfc_index_hnewright_productcoefficients)) * c)) /\\ exists ff_q_pfp_hnewright_productcoefficientsentry. b = ff_q_pfp_hnewright_productcoefficientsentry * S ((S (pfc_index_hnewright_productcoefficients)) * c) + (pfc_value_hnewright_productcoefficients))) /\\ ((exists pfc_terms_code_hnewright_productcoefficientscoefficient pfc_terms_scale_hnewright_productcoefficientscoefficient pfc_natural_sum_hnewright_productcoefficientscoefficient. ((forall pfc_index_hnewright_productcoefficientscoefficientdiagonal. (exists pfa_gap_hnewright_productcoefficientscoefficientdiagonalbound. pfa_gap_hnewright_productcoefficientscoefficientdiagonalbound + S (pfc_index_hnewright_productcoefficientscoefficientdiagonal) = (S (pfc_index_hnewright_productcoefficients))) -> exists pfc_value_hnewright_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_hnewright_productcoefficientscoefficientdiagonalentry. ff_h_pfp_hnewright_productcoefficientscoefficientdiagonalentry + S (pfc_value_hnewright_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_hnewright_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hnewright_productcoefficientscoefficient)) /\\ exists ff_q_pfp_hnewright_productcoefficientscoefficientdiagonalentry. pfc_terms_code_hnewright_productcoefficientscoefficient = ff_q_pfp_hnewright_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_hnewright_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hnewright_productcoefficientscoefficient) + (pfc_value_hnewright_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_hnewright_productcoefficientscoefficientdiagonalterm pfc_left_hnewright_productcoefficientscoefficientdiagonalterm pfc_right_hnewright_productcoefficientscoefficientdiagonalterm. (((pfc_index_hnewright_productcoefficientscoefficientdiagonal)+pfc_complement_hnewright_productcoefficientscoefficientdiagonalterm=(pfc_index_hnewright_productcoefficients)) /\\ ((((((exists pfa_gap_hnewright_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_hnewright_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_hnewright_productcoefficientscoefficientdiagonal) = ((Lu)+(x6))) /\\ ((((exists ff_h_pfp_hnewright_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_hnewright_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_hnewright_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_hnewright_productcoefficientscoefficientdiagonal)) * x10)) /\\ exists ff_q_pfp_hnewright_productcoefficientscoefficientdiagonaltermleftentry. x9 = ff_q_pfp_hnewright_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_hnewright_productcoefficientscoefficientdiagonal)) * x10) + (pfc_left_hnewright_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hnewright_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_hnewright_productcoefficientscoefficientdiagonaltermleftoutside+((Lu)+(x6))=(pfc_index_hnewright_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_hnewright_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_hnewright_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_hnewright_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_hnewright_productcoefficientscoefficientdiagonalterm) = (Lb)) /\\ ((((exists ff_h_pfp_hnewright_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_hnewright_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_hnewright_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_hnewright_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_hnewright_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_hnewright_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_hnewright_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_hnewright_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hnewright_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_hnewright_productcoefficientscoefficientdiagonaltermrightoutside+(Lb)=(pfc_complement_hnewright_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_hnewright_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_hnewright_productcoefficientscoefficientdiagonal)=pfc_left_hnewright_productcoefficientscoefficientdiagonalterm*pfc_right_hnewright_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_hnewright_productcoefficientscoefficientsum fs_v_pfc_hnewright_productcoefficientscoefficientsum. ((((exists fs_h_pfc_hnewright_productcoefficientscoefficientsum_body_start. fs_h_pfc_hnewright_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_hnewright_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnewright_productcoefficientscoefficientsum_body_start. fs_u_pfc_hnewright_productcoefficientscoefficientsum = fs_q_pfc_hnewright_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_hnewright_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_hnewright_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_hnewright_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_hnewright_productcoefficientscoefficient) = S ((S (S (pfc_index_hnewright_productcoefficients))) * fs_v_pfc_hnewright_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnewright_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_hnewright_productcoefficientscoefficientsum = fs_q_pfc_hnewright_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_hnewright_productcoefficients))) * fs_v_pfc_hnewright_productcoefficientscoefficientsum) + (pfc_natural_sum_hnewright_productcoefficientscoefficient))) /\\ forall fs_i_pfc_hnewright_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_hnewright_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_hnewright_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_hnewright_productcoefficientscoefficientsum_body_steps = S (pfc_index_hnewright_productcoefficients)) -> exists fs_a_pfc_hnewright_productcoefficientscoefficientsum_body_steps fs_r_pfc_hnewright_productcoefficientscoefficientsum_body_steps fs_s_pfc_hnewright_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_hnewright_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_hnewright_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_hnewright_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hnewright_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hnewright_productcoefficientscoefficient)) /\\ exists fs_q_pfc_hnewright_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_hnewright_productcoefficientscoefficient = fs_q_pfc_hnewright_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_hnewright_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hnewright_productcoefficientscoefficient) + (fs_a_pfc_hnewright_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hnewright_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_hnewright_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_hnewright_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hnewright_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnewright_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnewright_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_hnewright_productcoefficientscoefficientsum = fs_q_pfc_hnewright_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_hnewright_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnewright_productcoefficientscoefficientsum) + (fs_r_pfc_hnewright_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hnewright_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_hnewright_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_hnewright_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_hnewright_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnewright_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnewright_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_hnewright_productcoefficientscoefficientsum = fs_q_pfc_hnewright_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_hnewright_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnewright_productcoefficientscoefficientsum) + (fs_s_pfc_hnewright_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_hnewright_productcoefficientscoefficientsum_body_steps = fs_r_pfc_hnewright_productcoefficientscoefficientsum_body_steps + fs_a_pfc_hnewright_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_hnewright_productcoefficientscoefficientresiduebound. pfa_gap_hnewright_productcoefficientscoefficientresiduebound + S (pfc_value_hnewright_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_hnewright_productcoefficientscoefficientresiduecongruence pfa_offset_right_hnewright_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_hnewright_productcoefficientscoefficient) + (p) * pfa_offset_left_hnewright_productcoefficientscoefficientresiduecongruence = (pfc_value_hnewright_productcoefficients) + (p) * pfa_offset_right_hnewright_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (x9)", "specialize prime_field_polynomial_convolution_at_length_exists (x10)", "specialize prime_field_polynomial_convolution_at_length_exists ((Lu)+(x6))", "specialize prime_field_polynomial_convolution_at_length_exists (bb)", "specialize prime_field_polynomial_convolution_at_length_exists (bc)", "specialize prime_field_polynomial_convolution_at_length_exists (Lb)", "specialize prime_field_polynomial_convolution_at_length_exists (x14)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact htbound", "exact hBbound", "exact hnewright_length_witness", "cases hnewright_product", "cases hnewright_product_witness", "have hresult : ((forall fom_index_pfp_exists_backward_identity_left_bounded. (exists fom_gap_pfp_exists_backward_identity_left_bounded_index_bound. fom_gap_pfp_exists_backward_identity_left_bounded_index_bound + S (fom_index_pfp_exists_backward_identity_left_bounded) = x11) -> exists fom_value_pfp_exists_backward_identity_left_bounded. ((((exists fom_beta_height_pfp_exists_backward_identity_left_bounded_entry. fom_beta_height_pfp_exists_backward_identity_left_bounded_entry + S (fom_value_pfp_exists_backward_identity_left_bounded) = S ((S (fom_index_pfp_exists_backward_identity_left_bounded)) * x13)) /\\ exists fom_beta_quotient_pfp_exists_backward_identity_left_bounded_entry. x12 = fom_beta_quotient_pfp_exists_backward_identity_left_bounded_entry * S ((S (fom_index_pfp_exists_backward_identity_left_bounded)) * x13) + (fom_value_pfp_exists_backward_identity_left_bounded))) /\\ (exists fom_gap_pfp_exists_backward_identity_left_bounded_value_bound. fom_gap_pfp_exists_backward_identity_left_bounded_value_bound + S (fom_value_pfp_exists_backward_identity_left_bounded) = p))) /\\ (((forall fom_index_pfp_exists_backward_identity_right_bounded. (exists fom_gap_pfp_exists_backward_identity_right_bounded_index_bound. fom_gap_pfp_exists_backward_identity_right_bounded_index_bound + S (fom_index_pfp_exists_backward_identity_right_bounded) = x14) -> exists fom_value_pfp_exists_backward_identity_right_bounded. ((((exists fom_beta_height_pfp_exists_backward_identity_right_bounded_entry. fom_beta_height_pfp_exists_backward_identity_right_bounded_entry + S (fom_value_pfp_exists_backward_identity_right_bounded) = S ((S (fom_index_pfp_exists_backward_identity_right_bounded)) * x16)) /\\ exists fom_beta_quotient_pfp_exists_backward_identity_right_bounded_entry. x15 = fom_beta_quotient_pfp_exists_backward_identity_right_bounded_entry * S ((S (fom_index_pfp_exists_backward_identity_right_bounded)) * x16) + (fom_value_pfp_exists_backward_identity_right_bounded))) /\\ (exists fom_gap_pfp_exists_backward_identity_right_bounded_value_bound. fom_gap_pfp_exists_backward_identity_right_bounded_value_bound + S (fom_value_pfp_exists_backward_identity_right_bounded) = p))) /\\ (((forall fom_index_pfp_exists_backward_identity_result_bounded. (exists fom_gap_pfp_exists_backward_identity_result_bounded_index_bound. fom_gap_pfp_exists_backward_identity_result_bounded_index_bound + S (fom_index_pfp_exists_backward_identity_result_bounded) = Lg) -> exists fom_value_pfp_exists_backward_identity_result_bounded. ((((exists fom_beta_height_pfp_exists_backward_identity_result_bounded_entry. fom_beta_height_pfp_exists_backward_identity_result_bounded_entry + S (fom_value_pfp_exists_backward_identity_result_bounded) = S ((S (fom_index_pfp_exists_backward_identity_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_exists_backward_identity_result_bounded_entry. gb = fom_beta_quotient_pfp_exists_backward_identity_result_bounded_entry * S ((S (fom_index_pfp_exists_backward_identity_result_bounded)) * gc) + (fom_value_pfp_exists_backward_identity_result_bounded))) /\\ (exists fom_gap_pfp_exists_backward_identity_result_bounded_value_bound. fom_gap_pfp_exists_backward_identity_result_bounded_value_bound + S (fom_value_pfp_exists_backward_identity_result_bounded) = p))) /\\ ((exists pfaa_left_b_exists_backward_identity pfaa_left_c_exists_backward_identity pfaa_right_b_exists_backward_identity pfaa_right_c_exists_backward_identity pfaa_sum_b_exists_backward_identity pfaa_sum_c_exists_backward_identity pfaa_length_exists_backward_identity. ((((forall pfrep_power_exists_backward_identity_witness_common_left pfrep_left_exists_backward_identity_witness_common_left pfrep_right_exists_backward_identity_witness_common_left. ((exists pfrep_position_exists_backward_identity_witness_common_leftfirst. ((pfrep_position_exists_backward_identity_witness_common_leftfirst+S (pfrep_power_exists_backward_identity_witness_common_left)=(x11)) /\\ ((((exists ff_h_pfp_exists_backward_identity_witness_common_leftfirstentry. ff_h_pfp_exists_backward_identity_witness_common_leftfirstentry + S (pfrep_left_exists_backward_identity_witness_common_left) = S ((S (pfrep_position_exists_backward_identity_witness_common_leftfirst)) * x13)) /\\ exists ff_q_pfp_exists_backward_identity_witness_common_leftfirstentry. x12 = ff_q_pfp_exists_backward_identity_witness_common_leftfirstentry * S ((S (pfrep_position_exists_backward_identity_witness_common_leftfirst)) * x13) + (pfrep_left_exists_backward_identity_witness_common_left)))))) \\/ (((exists pfrep_gap_exists_backward_identity_witness_common_leftfirstoutside. pfrep_gap_exists_backward_identity_witness_common_leftfirstoutside+(x11)=(pfrep_power_exists_backward_identity_witness_common_left)) /\\ (((pfrep_left_exists_backward_identity_witness_common_left)=0))))) -> ((exists pfrep_position_exists_backward_identity_witness_common_leftsecond. ((pfrep_position_exists_backward_identity_witness_common_leftsecond+S (pfrep_power_exists_backward_identity_witness_common_left)=(pfaa_length_exists_backward_identity)) /\\ ((((exists ff_h_pfp_exists_backward_identity_witness_common_leftsecondentry. ff_h_pfp_exists_backward_identity_witness_common_leftsecondentry + S (pfrep_right_exists_backward_identity_witness_common_left) = S ((S (pfrep_position_exists_backward_identity_witness_common_leftsecond)) * pfaa_left_c_exists_backward_identity)) /\\ exists ff_q_pfp_exists_backward_identity_witness_common_leftsecondentry. pfaa_left_b_exists_backward_identity = ff_q_pfp_exists_backward_identity_witness_common_leftsecondentry * S ((S (pfrep_position_exists_backward_identity_witness_common_leftsecond)) * pfaa_left_c_exists_backward_identity) + (pfrep_right_exists_backward_identity_witness_common_left)))))) \\/ (((exists pfrep_gap_exists_backward_identity_witness_common_leftsecondoutside. pfrep_gap_exists_backward_identity_witness_common_leftsecondoutside+(pfaa_length_exists_backward_identity)=(pfrep_power_exists_backward_identity_witness_common_left)) /\\ (((pfrep_right_exists_backward_identity_witness_common_left)=0))))) -> pfrep_left_exists_backward_identity_witness_common_left=pfrep_right_exists_backward_identity_witness_common_left) /\\ ((forall pfrep_power_exists_backward_identity_witness_common_right pfrep_left_exists_backward_identity_witness_common_right pfrep_right_exists_backward_identity_witness_common_right. ((exists pfrep_position_exists_backward_identity_witness_common_rightfirst. ((pfrep_position_exists_backward_identity_witness_common_rightfirst+S (pfrep_power_exists_backward_identity_witness_common_right)=(x14)) /\\ ((((exists ff_h_pfp_exists_backward_identity_witness_common_rightfirstentry. ff_h_pfp_exists_backward_identity_witness_common_rightfirstentry + S (pfrep_left_exists_backward_identity_witness_common_right) = S ((S (pfrep_position_exists_backward_identity_witness_common_rightfirst)) * x16)) /\\ exists ff_q_pfp_exists_backward_identity_witness_common_rightfirstentry. x15 = ff_q_pfp_exists_backward_identity_witness_common_rightfirstentry * S ((S (pfrep_position_exists_backward_identity_witness_common_rightfirst)) * x16) + (pfrep_left_exists_backward_identity_witness_common_right)))))) \\/ (((exists pfrep_gap_exists_backward_identity_witness_common_rightfirstoutside. pfrep_gap_exists_backward_identity_witness_common_rightfirstoutside+(x14)=(pfrep_power_exists_backward_identity_witness_common_right)) /\\ (((pfrep_left_exists_backward_identity_witness_common_right)=0))))) -> ((exists pfrep_position_exists_backward_identity_witness_common_rightsecond. ((pfrep_position_exists_backward_identity_witness_common_rightsecond+S (pfrep_power_exists_backward_identity_witness_common_right)=(pfaa_length_exists_backward_identity)) /\\ ((((exists ff_h_pfp_exists_backward_identity_witness_common_rightsecondentry. ff_h_pfp_exists_backward_identity_witness_common_rightsecondentry + S (pfrep_right_exists_backward_identity_witness_common_right) = S ((S (pfrep_position_exists_backward_identity_witness_common_rightsecond)) * pfaa_right_c_exists_backward_identity)) /\\ exists ff_q_pfp_exists_backward_identity_witness_common_rightsecondentry. pfaa_right_b_exists_backward_identity = ff_q_pfp_exists_backward_identity_witness_common_rightsecondentry * S ((S (pfrep_position_exists_backward_identity_witness_common_rightsecond)) * pfaa_right_c_exists_backward_identity) + (pfrep_right_exists_backward_identity_witness_common_right)))))) \\/ (((exists pfrep_gap_exists_backward_identity_witness_common_rightsecondoutside. pfrep_gap_exists_backward_identity_witness_common_rightsecondoutside+(pfaa_length_exists_backward_identity)=(pfrep_power_exists_backward_identity_witness_common_right)) /\\ (((pfrep_right_exists_backward_identity_witness_common_right)=0))))) -> pfrep_left_exists_backward_identity_witness_common_right=pfrep_right_exists_backward_identity_witness_common_right)))) /\\ (((forall pfp_index_exists_backward_identity_witness_operation. (exists pfa_gap_exists_backward_identity_witness_operationindex. pfa_gap_exists_backward_identity_witness_operationindex + S (pfp_index_exists_backward_identity_witness_operation) = (pfaa_length_exists_backward_identity)) -> exists pfp_left_exists_backward_identity_witness_operation pfp_right_exists_backward_identity_witness_operation pfp_value_exists_backward_identity_witness_operation. ((((exists ff_h_pfp_exists_backward_identity_witness_operationleft. ff_h_pfp_exists_backward_identity_witness_operationleft + S (pfp_left_exists_backward_identity_witness_operation) = S ((S (pfp_index_exists_backward_identity_witness_operation)) * pfaa_left_c_exists_backward_identity)) /\\ exists ff_q_pfp_exists_backward_identity_witness_operationleft. pfaa_left_b_exists_backward_identity = ff_q_pfp_exists_backward_identity_witness_operationleft * S ((S (pfp_index_exists_backward_identity_witness_operation)) * pfaa_left_c_exists_backward_identity) + (pfp_left_exists_backward_identity_witness_operation))) /\\ (((((exists ff_h_pfp_exists_backward_identity_witness_operationright. ff_h_pfp_exists_backward_identity_witness_operationright + S (pfp_right_exists_backward_identity_witness_operation) = S ((S (pfp_index_exists_backward_identity_witness_operation)) * pfaa_right_c_exists_backward_identity)) /\\ exists ff_q_pfp_exists_backward_identity_witness_operationright. pfaa_right_b_exists_backward_identity = ff_q_pfp_exists_backward_identity_witness_operationright * S ((S (pfp_index_exists_backward_identity_witness_operation)) * pfaa_right_c_exists_backward_identity) + (pfp_right_exists_backward_identity_witness_operation))) /\\ (((((exists ff_h_pfp_exists_backward_identity_witness_operationtarget. ff_h_pfp_exists_backward_identity_witness_operationtarget + S (pfp_value_exists_backward_identity_witness_operation) = S ((S (pfp_index_exists_backward_identity_witness_operation)) * pfaa_sum_c_exists_backward_identity)) /\\ exists ff_q_pfp_exists_backward_identity_witness_operationtarget. pfaa_sum_b_exists_backward_identity = ff_q_pfp_exists_backward_identity_witness_operationtarget * S ((S (pfp_index_exists_backward_identity_witness_operation)) * pfaa_sum_c_exists_backward_identity) + (pfp_value_exists_backward_identity_witness_operation))) /\\ ((((exists pfa_gap_exists_backward_identity_witness_operationoperationleft. pfa_gap_exists_backward_identity_witness_operationoperationleft + S (pfp_left_exists_backward_identity_witness_operation) = (p)) /\\ (((exists pfa_gap_exists_backward_identity_witness_operationoperationright. pfa_gap_exists_backward_identity_witness_operationoperationright + S (pfp_right_exists_backward_identity_witness_operation) = (p)) /\\ ((((exists pfa_gap_exists_backward_identity_witness_operationoperationresultbound. pfa_gap_exists_backward_identity_witness_operationoperationresultbound + S (pfp_value_exists_backward_identity_witness_operation) = (p)) /\\ ((exists pfa_offset_left_exists_backward_identity_witness_operationoperationresultcongruence pfa_offset_right_exists_backward_identity_witness_operationoperationresultcongruence. ((pfp_left_exists_backward_identity_witness_operation) + (pfp_right_exists_backward_identity_witness_operation)) + (p) * pfa_offset_left_exists_backward_identity_witness_operationoperationresultcongruence = (pfp_value_exists_backward_identity_witness_operation) + (p) * pfa_offset_right_exists_backward_identity_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_exists_backward_identity_witness_output pfrep_left_exists_backward_identity_witness_output pfrep_right_exists_backward_identity_witness_output. ((exists pfrep_position_exists_backward_identity_witness_outputfirst. ((pfrep_position_exists_backward_identity_witness_outputfirst+S (pfrep_power_exists_backward_identity_witness_output)=(pfaa_length_exists_backward_identity)) /\\ ((((exists ff_h_pfp_exists_backward_identity_witness_outputfirstentry. ff_h_pfp_exists_backward_identity_witness_outputfirstentry + S (pfrep_left_exists_backward_identity_witness_output) = S ((S (pfrep_position_exists_backward_identity_witness_outputfirst)) * pfaa_sum_c_exists_backward_identity)) /\\ exists ff_q_pfp_exists_backward_identity_witness_outputfirstentry. pfaa_sum_b_exists_backward_identity = ff_q_pfp_exists_backward_identity_witness_outputfirstentry * S ((S (pfrep_position_exists_backward_identity_witness_outputfirst)) * pfaa_sum_c_exists_backward_identity) + (pfrep_left_exists_backward_identity_witness_output)))))) \\/ (((exists pfrep_gap_exists_backward_identity_witness_outputfirstoutside. pfrep_gap_exists_backward_identity_witness_outputfirstoutside+(pfaa_length_exists_backward_identity)=(pfrep_power_exists_backward_identity_witness_output)) /\\ (((pfrep_left_exists_backward_identity_witness_output)=0))))) -> ((exists pfrep_position_exists_backward_identity_witness_outputsecond. ((pfrep_position_exists_backward_identity_witness_outputsecond+S (pfrep_power_exists_backward_identity_witness_output)=(Lg)) /\\ ((((exists ff_h_pfp_exists_backward_identity_witness_outputsecondentry. ff_h_pfp_exists_backward_identity_witness_outputsecondentry + S (pfrep_right_exists_backward_identity_witness_output) = S ((S (pfrep_position_exists_backward_identity_witness_outputsecond)) * gc)) /\\ exists ff_q_pfp_exists_backward_identity_witness_outputsecondentry. gb = ff_q_pfp_exists_backward_identity_witness_outputsecondentry * S ((S (pfrep_position_exists_backward_identity_witness_outputsecond)) * gc) + (pfrep_right_exists_backward_identity_witness_output)))))) \\/ (((exists pfrep_gap_exists_backward_identity_witness_outputsecondoutside. pfrep_gap_exists_backward_identity_witness_outputsecondoutside+(Lg)=(pfrep_power_exists_backward_identity_witness_output)) /\\ (((pfrep_right_exists_backward_identity_witness_output)=0))))) -> pfrep_left_exists_backward_identity_witness_output=pfrep_right_exists_backward_identity_witness_output))))))))))))", "have hassociate_length : exists n. ((((x6)=0 \\/ (Lb)=0) /\\ (((n)=0)))) \\/ (((~((x6)=0)) /\\ (((~((Lb)=0)) /\\ (((x6)+(Lb)=S (n)))))))", "specialize polynomial_product_length_exists (x6)", "specialize polynomial_product_length_exists (Lb)", "apply polynomial_product_length_exists", "cases hassociate_length", "have hassociate_product : exists b c. ((forall fom_index_pfp_hassociate_productleft. (exists fom_gap_pfp_hassociate_productleft_index_bound. fom_gap_pfp_hassociate_productleft_index_bound + S (fom_index_pfp_hassociate_productleft) = x6) -> exists fom_value_pfp_hassociate_productleft. ((((exists fom_beta_height_pfp_hassociate_productleft_entry. fom_beta_height_pfp_hassociate_productleft_entry + S (fom_value_pfp_hassociate_productleft) = S ((S (fom_index_pfp_hassociate_productleft)) * x8)) /\\ exists fom_beta_quotient_pfp_hassociate_productleft_entry. x7 = fom_beta_quotient_pfp_hassociate_productleft_entry * S ((S (fom_index_pfp_hassociate_productleft)) * x8) + (fom_value_pfp_hassociate_productleft))) /\\ (exists fom_gap_pfp_hassociate_productleft_value_bound. fom_gap_pfp_hassociate_productleft_value_bound + S (fom_value_pfp_hassociate_productleft) = p))) /\\ (((forall fom_index_pfp_hassociate_productright. (exists fom_gap_pfp_hassociate_productright_index_bound. fom_gap_pfp_hassociate_productright_index_bound + S (fom_index_pfp_hassociate_productright) = Lb) -> exists fom_value_pfp_hassociate_productright. ((((exists fom_beta_height_pfp_hassociate_productright_entry. fom_beta_height_pfp_hassociate_productright_entry + S (fom_value_pfp_hassociate_productright) = S ((S (fom_index_pfp_hassociate_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_hassociate_productright_entry. bb = fom_beta_quotient_pfp_hassociate_productright_entry * S ((S (fom_index_pfp_hassociate_productright)) * bc) + (fom_value_pfp_hassociate_productright))) /\\ (exists fom_gap_pfp_hassociate_productright_value_bound. fom_gap_pfp_hassociate_productright_value_bound + S (fom_value_pfp_hassociate_productright) = p))) /\\ (((((((x6)=0 \\/ (Lb)=0) /\\ (((x17)=0)))) \\/ (((~((x6)=0)) /\\ (((~((Lb)=0)) /\\ (((x6)+(Lb)=S (x17)))))))) /\\ ((forall pfc_index_hassociate_productcoefficients. (exists pfa_gap_hassociate_productcoefficientsbound. pfa_gap_hassociate_productcoefficientsbound + S (pfc_index_hassociate_productcoefficients) = (x17)) -> exists pfc_value_hassociate_productcoefficients. ((((exists ff_h_pfp_hassociate_productcoefficientsentry. ff_h_pfp_hassociate_productcoefficientsentry + S (pfc_value_hassociate_productcoefficients) = S ((S (pfc_index_hassociate_productcoefficients)) * c)) /\\ exists ff_q_pfp_hassociate_productcoefficientsentry. b = ff_q_pfp_hassociate_productcoefficientsentry * S ((S (pfc_index_hassociate_productcoefficients)) * c) + (pfc_value_hassociate_productcoefficients))) /\\ ((exists pfc_terms_code_hassociate_productcoefficientscoefficient pfc_terms_scale_hassociate_productcoefficientscoefficient pfc_natural_sum_hassociate_productcoefficientscoefficient. ((forall pfc_index_hassociate_productcoefficientscoefficientdiagonal. (exists pfa_gap_hassociate_productcoefficientscoefficientdiagonalbound. pfa_gap_hassociate_productcoefficientscoefficientdiagonalbound + S (pfc_index_hassociate_productcoefficientscoefficientdiagonal) = (S (pfc_index_hassociate_productcoefficients))) -> exists pfc_value_hassociate_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_hassociate_productcoefficientscoefficientdiagonalentry. ff_h_pfp_hassociate_productcoefficientscoefficientdiagonalentry + S (pfc_value_hassociate_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_hassociate_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hassociate_productcoefficientscoefficient)) /\\ exists ff_q_pfp_hassociate_productcoefficientscoefficientdiagonalentry. pfc_terms_code_hassociate_productcoefficientscoefficient = ff_q_pfp_hassociate_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_hassociate_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hassociate_productcoefficientscoefficient) + (pfc_value_hassociate_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_hassociate_productcoefficientscoefficientdiagonalterm pfc_left_hassociate_productcoefficientscoefficientdiagonalterm pfc_right_hassociate_productcoefficientscoefficientdiagonalterm. (((pfc_index_hassociate_productcoefficientscoefficientdiagonal)+pfc_complement_hassociate_productcoefficientscoefficientdiagonalterm=(pfc_index_hassociate_productcoefficients)) /\\ ((((((exists pfa_gap_hassociate_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_hassociate_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_hassociate_productcoefficientscoefficientdiagonal) = (x6)) /\\ ((((exists ff_h_pfp_hassociate_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_hassociate_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_hassociate_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_hassociate_productcoefficientscoefficientdiagonal)) * x8)) /\\ exists ff_q_pfp_hassociate_productcoefficientscoefficientdiagonaltermleftentry. x7 = ff_q_pfp_hassociate_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_hassociate_productcoefficientscoefficientdiagonal)) * x8) + (pfc_left_hassociate_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hassociate_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_hassociate_productcoefficientscoefficientdiagonaltermleftoutside+(x6)=(pfc_index_hassociate_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_hassociate_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_hassociate_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_hassociate_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_hassociate_productcoefficientscoefficientdiagonalterm) = (Lb)) /\\ ((((exists ff_h_pfp_hassociate_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_hassociate_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_hassociate_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_hassociate_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_hassociate_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_hassociate_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_hassociate_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_hassociate_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hassociate_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_hassociate_productcoefficientscoefficientdiagonaltermrightoutside+(Lb)=(pfc_complement_hassociate_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_hassociate_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_hassociate_productcoefficientscoefficientdiagonal)=pfc_left_hassociate_productcoefficientscoefficientdiagonalterm*pfc_right_hassociate_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_hassociate_productcoefficientscoefficientsum fs_v_pfc_hassociate_productcoefficientscoefficientsum. ((((exists fs_h_pfc_hassociate_productcoefficientscoefficientsum_body_start. fs_h_pfc_hassociate_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_hassociate_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hassociate_productcoefficientscoefficientsum_body_start. fs_u_pfc_hassociate_productcoefficientscoefficientsum = fs_q_pfc_hassociate_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_hassociate_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_hassociate_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_hassociate_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_hassociate_productcoefficientscoefficient) = S ((S (S (pfc_index_hassociate_productcoefficients))) * fs_v_pfc_hassociate_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hassociate_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_hassociate_productcoefficientscoefficientsum = fs_q_pfc_hassociate_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_hassociate_productcoefficients))) * fs_v_pfc_hassociate_productcoefficientscoefficientsum) + (pfc_natural_sum_hassociate_productcoefficientscoefficient))) /\\ forall fs_i_pfc_hassociate_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_hassociate_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_hassociate_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_hassociate_productcoefficientscoefficientsum_body_steps = S (pfc_index_hassociate_productcoefficients)) -> exists fs_a_pfc_hassociate_productcoefficientscoefficientsum_body_steps fs_r_pfc_hassociate_productcoefficientscoefficientsum_body_steps fs_s_pfc_hassociate_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_hassociate_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_hassociate_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_hassociate_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hassociate_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hassociate_productcoefficientscoefficient)) /\\ exists fs_q_pfc_hassociate_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_hassociate_productcoefficientscoefficient = fs_q_pfc_hassociate_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_hassociate_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hassociate_productcoefficientscoefficient) + (fs_a_pfc_hassociate_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hassociate_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_hassociate_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_hassociate_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hassociate_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hassociate_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hassociate_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_hassociate_productcoefficientscoefficientsum = fs_q_pfc_hassociate_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_hassociate_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hassociate_productcoefficientscoefficientsum) + (fs_r_pfc_hassociate_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hassociate_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_hassociate_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_hassociate_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_hassociate_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hassociate_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hassociate_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_hassociate_productcoefficientscoefficientsum = fs_q_pfc_hassociate_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_hassociate_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hassociate_productcoefficientscoefficientsum) + (fs_s_pfc_hassociate_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_hassociate_productcoefficientscoefficientsum_body_steps = fs_r_pfc_hassociate_productcoefficientscoefficientsum_body_steps + fs_a_pfc_hassociate_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_hassociate_productcoefficientscoefficientresiduebound. pfa_gap_hassociate_productcoefficientscoefficientresiduebound + S (pfc_value_hassociate_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_hassociate_productcoefficientscoefficientresiduecongruence pfa_offset_right_hassociate_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_hassociate_productcoefficientscoefficient) + (p) * pfa_offset_left_hassociate_productcoefficientscoefficientresiduecongruence = (pfc_value_hassociate_productcoefficients) + (p) * pfa_offset_right_hassociate_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (x7)", "specialize prime_field_polynomial_convolution_at_length_exists (x8)", "specialize prime_field_polynomial_convolution_at_length_exists (x6)", "specialize prime_field_polynomial_convolution_at_length_exists (bb)", "specialize prime_field_polynomial_convolution_at_length_exists (bc)", "specialize prime_field_polynomial_convolution_at_length_exists (Lb)", "specialize prime_field_polynomial_convolution_at_length_exists (x17)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hWbound", "exact hBbound", "exact hassociate_length_witness", "cases hassociate_product", "cases hassociate_product_witness", "have hdistribute_length : exists n. ((((Lv)=0 \\/ (Lp)=0) /\\ (((n)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((Lp)=0)) /\\ (((Lv)+(Lp)=S (n)))))))", "specialize polynomial_product_length_exists (Lv)", "specialize polynomial_product_length_exists (Lp)", "apply polynomial_product_length_exists", "cases hdistribute_length", "have hdistribute_product : exists b c. ((forall fom_index_pfp_hdistribute_productleft. (exists fom_gap_pfp_hdistribute_productleft_index_bound. fom_gap_pfp_hdistribute_productleft_index_bound + S (fom_index_pfp_hdistribute_productleft) = Lv) -> exists fom_value_pfp_hdistribute_productleft. ((((exists fom_beta_height_pfp_hdistribute_productleft_entry. fom_beta_height_pfp_hdistribute_productleft_entry + S (fom_value_pfp_hdistribute_productleft) = S ((S (fom_index_pfp_hdistribute_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_hdistribute_productleft_entry. vb = fom_beta_quotient_pfp_hdistribute_productleft_entry * S ((S (fom_index_pfp_hdistribute_productleft)) * vc) + (fom_value_pfp_hdistribute_productleft))) /\\ (exists fom_gap_pfp_hdistribute_productleft_value_bound. fom_gap_pfp_hdistribute_productleft_value_bound + S (fom_value_pfp_hdistribute_productleft) = p))) /\\ (((forall fom_index_pfp_hdistribute_productright. (exists fom_gap_pfp_hdistribute_productright_index_bound. fom_gap_pfp_hdistribute_productright_index_bound + S (fom_index_pfp_hdistribute_productright) = Lp) -> exists fom_value_pfp_hdistribute_productright. ((((exists fom_beta_height_pfp_hdistribute_productright_entry. fom_beta_height_pfp_hdistribute_productright_entry + S (fom_value_pfp_hdistribute_productright) = S ((S (fom_index_pfp_hdistribute_productright)) * pc)) /\\ exists fom_beta_quotient_pfp_hdistribute_productright_entry. pb = fom_beta_quotient_pfp_hdistribute_productright_entry * S ((S (fom_index_pfp_hdistribute_productright)) * pc) + (fom_value_pfp_hdistribute_productright))) /\\ (exists fom_gap_pfp_hdistribute_productright_value_bound. fom_gap_pfp_hdistribute_productright_value_bound + S (fom_value_pfp_hdistribute_productright) = p))) /\\ (((((((Lv)=0 \\/ (Lp)=0) /\\ (((x20)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((Lp)=0)) /\\ (((Lv)+(Lp)=S (x20)))))))) /\\ ((forall pfc_index_hdistribute_productcoefficients. (exists pfa_gap_hdistribute_productcoefficientsbound. pfa_gap_hdistribute_productcoefficientsbound + S (pfc_index_hdistribute_productcoefficients) = (x20)) -> exists pfc_value_hdistribute_productcoefficients. ((((exists ff_h_pfp_hdistribute_productcoefficientsentry. ff_h_pfp_hdistribute_productcoefficientsentry + S (pfc_value_hdistribute_productcoefficients) = S ((S (pfc_index_hdistribute_productcoefficients)) * c)) /\\ exists ff_q_pfp_hdistribute_productcoefficientsentry. b = ff_q_pfp_hdistribute_productcoefficientsentry * S ((S (pfc_index_hdistribute_productcoefficients)) * c) + (pfc_value_hdistribute_productcoefficients))) /\\ ((exists pfc_terms_code_hdistribute_productcoefficientscoefficient pfc_terms_scale_hdistribute_productcoefficientscoefficient pfc_natural_sum_hdistribute_productcoefficientscoefficient. ((forall pfc_index_hdistribute_productcoefficientscoefficientdiagonal. (exists pfa_gap_hdistribute_productcoefficientscoefficientdiagonalbound. pfa_gap_hdistribute_productcoefficientscoefficientdiagonalbound + S (pfc_index_hdistribute_productcoefficientscoefficientdiagonal) = (S (pfc_index_hdistribute_productcoefficients))) -> exists pfc_value_hdistribute_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_hdistribute_productcoefficientscoefficientdiagonalentry. ff_h_pfp_hdistribute_productcoefficientscoefficientdiagonalentry + S (pfc_value_hdistribute_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_hdistribute_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hdistribute_productcoefficientscoefficient)) /\\ exists ff_q_pfp_hdistribute_productcoefficientscoefficientdiagonalentry. pfc_terms_code_hdistribute_productcoefficientscoefficient = ff_q_pfp_hdistribute_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_hdistribute_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hdistribute_productcoefficientscoefficient) + (pfc_value_hdistribute_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_hdistribute_productcoefficientscoefficientdiagonalterm pfc_left_hdistribute_productcoefficientscoefficientdiagonalterm pfc_right_hdistribute_productcoefficientscoefficientdiagonalterm. (((pfc_index_hdistribute_productcoefficientscoefficientdiagonal)+pfc_complement_hdistribute_productcoefficientscoefficientdiagonalterm=(pfc_index_hdistribute_productcoefficients)) /\\ ((((((exists pfa_gap_hdistribute_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_hdistribute_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_hdistribute_productcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_hdistribute_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_hdistribute_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_hdistribute_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_hdistribute_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_hdistribute_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_hdistribute_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_hdistribute_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_hdistribute_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hdistribute_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_hdistribute_productcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_hdistribute_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_hdistribute_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_hdistribute_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_hdistribute_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_hdistribute_productcoefficientscoefficientdiagonalterm) = (Lp)) /\\ ((((exists ff_h_pfp_hdistribute_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_hdistribute_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_hdistribute_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_hdistribute_productcoefficientscoefficientdiagonalterm)) * pc)) /\\ exists ff_q_pfp_hdistribute_productcoefficientscoefficientdiagonaltermrightentry. pb = ff_q_pfp_hdistribute_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_hdistribute_productcoefficientscoefficientdiagonalterm)) * pc) + (pfc_right_hdistribute_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hdistribute_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_hdistribute_productcoefficientscoefficientdiagonaltermrightoutside+(Lp)=(pfc_complement_hdistribute_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_hdistribute_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_hdistribute_productcoefficientscoefficientdiagonal)=pfc_left_hdistribute_productcoefficientscoefficientdiagonalterm*pfc_right_hdistribute_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_hdistribute_productcoefficientscoefficientsum fs_v_pfc_hdistribute_productcoefficientscoefficientsum. ((((exists fs_h_pfc_hdistribute_productcoefficientscoefficientsum_body_start. fs_h_pfc_hdistribute_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_hdistribute_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hdistribute_productcoefficientscoefficientsum_body_start. fs_u_pfc_hdistribute_productcoefficientscoefficientsum = fs_q_pfc_hdistribute_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_hdistribute_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_hdistribute_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_hdistribute_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_hdistribute_productcoefficientscoefficient) = S ((S (S (pfc_index_hdistribute_productcoefficients))) * fs_v_pfc_hdistribute_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hdistribute_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_hdistribute_productcoefficientscoefficientsum = fs_q_pfc_hdistribute_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_hdistribute_productcoefficients))) * fs_v_pfc_hdistribute_productcoefficientscoefficientsum) + (pfc_natural_sum_hdistribute_productcoefficientscoefficient))) /\\ forall fs_i_pfc_hdistribute_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_hdistribute_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_hdistribute_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_hdistribute_productcoefficientscoefficientsum_body_steps = S (pfc_index_hdistribute_productcoefficients)) -> exists fs_a_pfc_hdistribute_productcoefficientscoefficientsum_body_steps fs_r_pfc_hdistribute_productcoefficientscoefficientsum_body_steps fs_s_pfc_hdistribute_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_hdistribute_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_hdistribute_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_hdistribute_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hdistribute_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hdistribute_productcoefficientscoefficient)) /\\ exists fs_q_pfc_hdistribute_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_hdistribute_productcoefficientscoefficient = fs_q_pfc_hdistribute_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_hdistribute_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hdistribute_productcoefficientscoefficient) + (fs_a_pfc_hdistribute_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hdistribute_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_hdistribute_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_hdistribute_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hdistribute_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hdistribute_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hdistribute_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_hdistribute_productcoefficientscoefficientsum = fs_q_pfc_hdistribute_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_hdistribute_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hdistribute_productcoefficientscoefficientsum) + (fs_r_pfc_hdistribute_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hdistribute_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_hdistribute_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_hdistribute_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_hdistribute_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hdistribute_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hdistribute_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_hdistribute_productcoefficientscoefficientsum = fs_q_pfc_hdistribute_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_hdistribute_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hdistribute_productcoefficientscoefficientsum) + (fs_s_pfc_hdistribute_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_hdistribute_productcoefficientscoefficientsum_body_steps = fs_r_pfc_hdistribute_productcoefficientscoefficientsum_body_steps + fs_a_pfc_hdistribute_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_hdistribute_productcoefficientscoefficientresiduebound. pfa_gap_hdistribute_productcoefficientscoefficientresiduebound + S (pfc_value_hdistribute_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_hdistribute_productcoefficientscoefficientresiduecongruence pfa_offset_right_hdistribute_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_hdistribute_productcoefficientscoefficient) + (p) * pfa_offset_left_hdistribute_productcoefficientscoefficientresiduecongruence = (pfc_value_hdistribute_productcoefficients) + (p) * pfa_offset_right_hdistribute_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (vb)", "specialize prime_field_polynomial_convolution_at_length_exists (vc)", "specialize prime_field_polynomial_convolution_at_length_exists (Lv)", "specialize prime_field_polynomial_convolution_at_length_exists (pb)", "specialize prime_field_polynomial_convolution_at_length_exists (pc)", "specialize prime_field_polynomial_convolution_at_length_exists (Lp)", "specialize prime_field_polynomial_convolution_at_length_exists (x20)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hVbound", "exact hPbound", "exact hdistribute_length_witness", "cases hdistribute_product", "cases hdistribute_product_witness", "specialize hstep (x)", "specialize hstep (x1)", "specialize hstep (x2)", "specialize hstep (x3)", "specialize hstep (x4)", "specialize hstep (x5)", "specialize hstep (x7)", "specialize hstep (x8)", "specialize hstep (x6)", "specialize hstep (x9)", "specialize hstep (x10)", "specialize hstep ((Lu)+(x6))", "specialize hstep (x12)", "specialize hstep (x13)", "specialize hstep (x11)", "specialize hstep (x15)", "specialize hstep (x16)", "specialize hstep (x14)", "specialize hstep (x18)", "specialize hstep (x19)", "specialize hstep (x17)", "specialize hstep (x21)", "specialize hstep (x22)", "specialize hstep (x20)", "apply hstep", "exact hp", "exact hQB", "exact hdivision", "exact hbezout_witness_witness_witness_witness_witness_witness_left", "exact hbezout_witness_witness_witness_witness_witness_witness_right_left", "exact hbezout_witness_witness_witness_witness_witness_witness_right_right", "exact hcoefficient_product_witness_witness", "exact hdifference_witness_witness", "exact hnewleft_product_witness_witness", "exact hnewright_product_witness_witness", "exact hassociate_product_witness_witness", "exact hdistribute_product_witness_witness", "exists x7", "exists x8", "exists x6", "exists x9", "exists x10", "exists (Lu)+(x6)", "split", "exact hcoefficient_product_witness_witness", "split", "exact hdifference_witness_witness", "exists x12", "exists x13", "exists x11", "exists x15", "exists x16", "exists x14", "split", "exact hnewleft_product_witness_witness", "split", "exact hnewright_product_witness_witness", "exact hresult" ], "script_sha256": "6861d715cb13590eab9bf723d99ebcdbc17bea85dddce8e7b45ce3ccb33e6ab8", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_bezout_backward_candidate.py", "sha256": "c3903482000c957ac77f84a43a85d135e4caa19e4484328035f91b82cbf3a702" }, "statement": "forall p ab ac La bb bc Lb rb rc Lr qb qc Lq pb pc Lp gb gc Lg ub uc Lu vb vc Lv. (~((p) = 1) /\\ forall pfa_factor_left_bezout_backward_prime pfa_factor_right_bezout_backward_prime. (p) = pfa_factor_left_bezout_backward_prime * pfa_factor_right_bezout_backward_prime -> pfa_factor_left_bezout_backward_prime = 1 \\/ pfa_factor_right_bezout_backward_prime = 1) -> (((forall fom_index_pfp_bezout_backward_productleft. (exists fom_gap_pfp_bezout_backward_productleft_index_bound. fom_gap_pfp_bezout_backward_productleft_index_bound + S (fom_index_pfp_bezout_backward_productleft) = Lq) -> exists fom_value_pfp_bezout_backward_productleft. ((((exists fom_beta_height_pfp_bezout_backward_productleft_entry. fom_beta_height_pfp_bezout_backward_productleft_entry + S (fom_value_pfp_bezout_backward_productleft) = S ((S (fom_index_pfp_bezout_backward_productleft)) * qc)) /\\ exists fom_beta_quotient_pfp_bezout_backward_productleft_entry. qb = fom_beta_quotient_pfp_bezout_backward_productleft_entry * S ((S (fom_index_pfp_bezout_backward_productleft)) * qc) + (fom_value_pfp_bezout_backward_productleft))) /\\ (exists fom_gap_pfp_bezout_backward_productleft_value_bound. fom_gap_pfp_bezout_backward_productleft_value_bound + S (fom_value_pfp_bezout_backward_productleft) = p))) /\\ (((forall fom_index_pfp_bezout_backward_productright. (exists fom_gap_pfp_bezout_backward_productright_index_bound. fom_gap_pfp_bezout_backward_productright_index_bound + S (fom_index_pfp_bezout_backward_productright) = Lb) -> exists fom_value_pfp_bezout_backward_productright. ((((exists fom_beta_height_pfp_bezout_backward_productright_entry. fom_beta_height_pfp_bezout_backward_productright_entry + S (fom_value_pfp_bezout_backward_productright) = S ((S (fom_index_pfp_bezout_backward_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_bezout_backward_productright_entry. bb = fom_beta_quotient_pfp_bezout_backward_productright_entry * S ((S (fom_index_pfp_bezout_backward_productright)) * bc) + (fom_value_pfp_bezout_backward_productright))) /\\ (exists fom_gap_pfp_bezout_backward_productright_value_bound. fom_gap_pfp_bezout_backward_productright_value_bound + S (fom_value_pfp_bezout_backward_productright) = p))) /\\ (((((((Lq)=0 \\/ (Lb)=0) /\\ (((Lp)=0)))) \\/ (((~((Lq)=0)) /\\ (((~((Lb)=0)) /\\ (((Lq)+(Lb)=S (Lp)))))))) /\\ ((forall pfc_index_bezout_backward_productcoefficients. (exists pfa_gap_bezout_backward_productcoefficientsbound. pfa_gap_bezout_backward_productcoefficientsbound + S (pfc_index_bezout_backward_productcoefficients) = (Lp)) -> exists pfc_value_bezout_backward_productcoefficients. ((((exists ff_h_pfp_bezout_backward_productcoefficientsentry. ff_h_pfp_bezout_backward_productcoefficientsentry + S (pfc_value_bezout_backward_productcoefficients) = S ((S (pfc_index_bezout_backward_productcoefficients)) * pc)) /\\ exists ff_q_pfp_bezout_backward_productcoefficientsentry. pb = ff_q_pfp_bezout_backward_productcoefficientsentry * S ((S (pfc_index_bezout_backward_productcoefficients)) * pc) + (pfc_value_bezout_backward_productcoefficients))) /\\ ((exists pfc_terms_code_bezout_backward_productcoefficientscoefficient pfc_terms_scale_bezout_backward_productcoefficientscoefficient pfc_natural_sum_bezout_backward_productcoefficientscoefficient. ((forall pfc_index_bezout_backward_productcoefficientscoefficientdiagonal. (exists pfa_gap_bezout_backward_productcoefficientscoefficientdiagonalbound. pfa_gap_bezout_backward_productcoefficientscoefficientdiagonalbound + S (pfc_index_bezout_backward_productcoefficientscoefficientdiagonal) = (S (pfc_index_bezout_backward_productcoefficients))) -> exists pfc_value_bezout_backward_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_bezout_backward_productcoefficientscoefficientdiagonalentry. ff_h_pfp_bezout_backward_productcoefficientscoefficientdiagonalentry + S (pfc_value_bezout_backward_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_bezout_backward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bezout_backward_productcoefficientscoefficient)) /\\ exists ff_q_pfp_bezout_backward_productcoefficientscoefficientdiagonalentry. pfc_terms_code_bezout_backward_productcoefficientscoefficient = ff_q_pfp_bezout_backward_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_bezout_backward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bezout_backward_productcoefficientscoefficient) + (pfc_value_bezout_backward_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_bezout_backward_productcoefficientscoefficientdiagonalterm pfc_left_bezout_backward_productcoefficientscoefficientdiagonalterm pfc_right_bezout_backward_productcoefficientscoefficientdiagonalterm. (((pfc_index_bezout_backward_productcoefficientscoefficientdiagonal)+pfc_complement_bezout_backward_productcoefficientscoefficientdiagonalterm=(pfc_index_bezout_backward_productcoefficients)) /\\ ((((((exists pfa_gap_bezout_backward_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_bezout_backward_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_bezout_backward_productcoefficientscoefficientdiagonal) = (Lq)) /\\ ((((exists ff_h_pfp_bezout_backward_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_bezout_backward_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_bezout_backward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_bezout_backward_productcoefficientscoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_bezout_backward_productcoefficientscoefficientdiagonaltermleftentry. qb = ff_q_pfp_bezout_backward_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_bezout_backward_productcoefficientscoefficientdiagonal)) * qc) + (pfc_left_bezout_backward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bezout_backward_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_bezout_backward_productcoefficientscoefficientdiagonaltermleftoutside+(Lq)=(pfc_index_bezout_backward_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_bezout_backward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_bezout_backward_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_bezout_backward_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_bezout_backward_productcoefficientscoefficientdiagonalterm) = (Lb)) /\\ ((((exists ff_h_pfp_bezout_backward_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_bezout_backward_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_bezout_backward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_bezout_backward_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_bezout_backward_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_bezout_backward_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_bezout_backward_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_bezout_backward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bezout_backward_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_bezout_backward_productcoefficientscoefficientdiagonaltermrightoutside+(Lb)=(pfc_complement_bezout_backward_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_bezout_backward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_bezout_backward_productcoefficientscoefficientdiagonal)=pfc_left_bezout_backward_productcoefficientscoefficientdiagonalterm*pfc_right_bezout_backward_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_bezout_backward_productcoefficientscoefficientsum fs_v_pfc_bezout_backward_productcoefficientscoefficientsum. ((((exists fs_h_pfc_bezout_backward_productcoefficientscoefficientsum_body_start. fs_h_pfc_bezout_backward_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_bezout_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_productcoefficientscoefficientsum_body_start. fs_u_pfc_bezout_backward_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_bezout_backward_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_bezout_backward_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_bezout_backward_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_bezout_backward_productcoefficientscoefficient) = S ((S (S (pfc_index_bezout_backward_productcoefficients))) * fs_v_pfc_bezout_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_bezout_backward_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_bezout_backward_productcoefficients))) * fs_v_pfc_bezout_backward_productcoefficientscoefficientsum) + (pfc_natural_sum_bezout_backward_productcoefficientscoefficient))) /\\ forall fs_i_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps = S (pfc_index_bezout_backward_productcoefficients)) -> exists fs_a_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps fs_r_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps fs_s_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bezout_backward_productcoefficientscoefficient)) /\\ exists fs_q_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_bezout_backward_productcoefficientscoefficient = fs_q_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bezout_backward_productcoefficientscoefficient) + (fs_a_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_bezout_backward_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_productcoefficientscoefficientsum) + (fs_r_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_bezout_backward_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_productcoefficientscoefficientsum) + (fs_s_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps = fs_r_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps + fs_a_pfc_bezout_backward_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_bezout_backward_productcoefficientscoefficientresiduebound. pfa_gap_bezout_backward_productcoefficientscoefficientresiduebound + S (pfc_value_bezout_backward_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_bezout_backward_productcoefficientscoefficientresiduecongruence pfa_offset_right_bezout_backward_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_bezout_backward_productcoefficientscoefficient) + (p) * pfa_offset_left_bezout_backward_productcoefficientscoefficientresiduecongruence = (pfc_value_bezout_backward_productcoefficients) + (p) * pfa_offset_right_bezout_backward_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (((forall fom_index_pfp_bezout_backward_identity_left_bounded. (exists fom_gap_pfp_bezout_backward_identity_left_bounded_index_bound. fom_gap_pfp_bezout_backward_identity_left_bounded_index_bound + S (fom_index_pfp_bezout_backward_identity_left_bounded) = Lp) -> exists fom_value_pfp_bezout_backward_identity_left_bounded. ((((exists fom_beta_height_pfp_bezout_backward_identity_left_bounded_entry. fom_beta_height_pfp_bezout_backward_identity_left_bounded_entry + S (fom_value_pfp_bezout_backward_identity_left_bounded) = S ((S (fom_index_pfp_bezout_backward_identity_left_bounded)) * pc)) /\\ exists fom_beta_quotient_pfp_bezout_backward_identity_left_bounded_entry. pb = fom_beta_quotient_pfp_bezout_backward_identity_left_bounded_entry * S ((S (fom_index_pfp_bezout_backward_identity_left_bounded)) * pc) + (fom_value_pfp_bezout_backward_identity_left_bounded))) /\\ (exists fom_gap_pfp_bezout_backward_identity_left_bounded_value_bound. fom_gap_pfp_bezout_backward_identity_left_bounded_value_bound + S (fom_value_pfp_bezout_backward_identity_left_bounded) = p))) /\\ (((forall fom_index_pfp_bezout_backward_identity_right_bounded. (exists fom_gap_pfp_bezout_backward_identity_right_bounded_index_bound. fom_gap_pfp_bezout_backward_identity_right_bounded_index_bound + S (fom_index_pfp_bezout_backward_identity_right_bounded) = Lr) -> exists fom_value_pfp_bezout_backward_identity_right_bounded. ((((exists fom_beta_height_pfp_bezout_backward_identity_right_bounded_entry. fom_beta_height_pfp_bezout_backward_identity_right_bounded_entry + S (fom_value_pfp_bezout_backward_identity_right_bounded) = S ((S (fom_index_pfp_bezout_backward_identity_right_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_bezout_backward_identity_right_bounded_entry. rb = fom_beta_quotient_pfp_bezout_backward_identity_right_bounded_entry * S ((S (fom_index_pfp_bezout_backward_identity_right_bounded)) * rc) + (fom_value_pfp_bezout_backward_identity_right_bounded))) /\\ (exists fom_gap_pfp_bezout_backward_identity_right_bounded_value_bound. fom_gap_pfp_bezout_backward_identity_right_bounded_value_bound + S (fom_value_pfp_bezout_backward_identity_right_bounded) = p))) /\\ (((forall fom_index_pfp_bezout_backward_identity_result_bounded. (exists fom_gap_pfp_bezout_backward_identity_result_bounded_index_bound. fom_gap_pfp_bezout_backward_identity_result_bounded_index_bound + S (fom_index_pfp_bezout_backward_identity_result_bounded) = La) -> exists fom_value_pfp_bezout_backward_identity_result_bounded. ((((exists fom_beta_height_pfp_bezout_backward_identity_result_bounded_entry. fom_beta_height_pfp_bezout_backward_identity_result_bounded_entry + S (fom_value_pfp_bezout_backward_identity_result_bounded) = S ((S (fom_index_pfp_bezout_backward_identity_result_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_bezout_backward_identity_result_bounded_entry. ab = fom_beta_quotient_pfp_bezout_backward_identity_result_bounded_entry * S ((S (fom_index_pfp_bezout_backward_identity_result_bounded)) * ac) + (fom_value_pfp_bezout_backward_identity_result_bounded))) /\\ (exists fom_gap_pfp_bezout_backward_identity_result_bounded_value_bound. fom_gap_pfp_bezout_backward_identity_result_bounded_value_bound + S (fom_value_pfp_bezout_backward_identity_result_bounded) = p))) /\\ ((exists pfaa_left_b_bezout_backward_identity pfaa_left_c_bezout_backward_identity pfaa_right_b_bezout_backward_identity pfaa_right_c_bezout_backward_identity pfaa_sum_b_bezout_backward_identity pfaa_sum_c_bezout_backward_identity pfaa_length_bezout_backward_identity. ((((forall pfrep_power_bezout_backward_identity_witness_common_left pfrep_left_bezout_backward_identity_witness_common_left pfrep_right_bezout_backward_identity_witness_common_left. ((exists pfrep_position_bezout_backward_identity_witness_common_leftfirst. ((pfrep_position_bezout_backward_identity_witness_common_leftfirst+S (pfrep_power_bezout_backward_identity_witness_common_left)=(Lp)) /\\ ((((exists ff_h_pfp_bezout_backward_identity_witness_common_leftfirstentry. ff_h_pfp_bezout_backward_identity_witness_common_leftfirstentry + S (pfrep_left_bezout_backward_identity_witness_common_left) = S ((S (pfrep_position_bezout_backward_identity_witness_common_leftfirst)) * pc)) /\\ exists ff_q_pfp_bezout_backward_identity_witness_common_leftfirstentry. pb = ff_q_pfp_bezout_backward_identity_witness_common_leftfirstentry * S ((S (pfrep_position_bezout_backward_identity_witness_common_leftfirst)) * pc) + (pfrep_left_bezout_backward_identity_witness_common_left)))))) \\/ (((exists pfrep_gap_bezout_backward_identity_witness_common_leftfirstoutside. pfrep_gap_bezout_backward_identity_witness_common_leftfirstoutside+(Lp)=(pfrep_power_bezout_backward_identity_witness_common_left)) /\\ (((pfrep_left_bezout_backward_identity_witness_common_left)=0))))) -> ((exists pfrep_position_bezout_backward_identity_witness_common_leftsecond. ((pfrep_position_bezout_backward_identity_witness_common_leftsecond+S (pfrep_power_bezout_backward_identity_witness_common_left)=(pfaa_length_bezout_backward_identity)) /\\ ((((exists ff_h_pfp_bezout_backward_identity_witness_common_leftsecondentry. ff_h_pfp_bezout_backward_identity_witness_common_leftsecondentry + S (pfrep_right_bezout_backward_identity_witness_common_left) = S ((S (pfrep_position_bezout_backward_identity_witness_common_leftsecond)) * pfaa_left_c_bezout_backward_identity)) /\\ exists ff_q_pfp_bezout_backward_identity_witness_common_leftsecondentry. pfaa_left_b_bezout_backward_identity = ff_q_pfp_bezout_backward_identity_witness_common_leftsecondentry * S ((S (pfrep_position_bezout_backward_identity_witness_common_leftsecond)) * pfaa_left_c_bezout_backward_identity) + (pfrep_right_bezout_backward_identity_witness_common_left)))))) \\/ (((exists pfrep_gap_bezout_backward_identity_witness_common_leftsecondoutside. pfrep_gap_bezout_backward_identity_witness_common_leftsecondoutside+(pfaa_length_bezout_backward_identity)=(pfrep_power_bezout_backward_identity_witness_common_left)) /\\ (((pfrep_right_bezout_backward_identity_witness_common_left)=0))))) -> pfrep_left_bezout_backward_identity_witness_common_left=pfrep_right_bezout_backward_identity_witness_common_left) /\\ ((forall pfrep_power_bezout_backward_identity_witness_common_right pfrep_left_bezout_backward_identity_witness_common_right pfrep_right_bezout_backward_identity_witness_common_right. ((exists pfrep_position_bezout_backward_identity_witness_common_rightfirst. ((pfrep_position_bezout_backward_identity_witness_common_rightfirst+S (pfrep_power_bezout_backward_identity_witness_common_right)=(Lr)) /\\ ((((exists ff_h_pfp_bezout_backward_identity_witness_common_rightfirstentry. ff_h_pfp_bezout_backward_identity_witness_common_rightfirstentry + S (pfrep_left_bezout_backward_identity_witness_common_right) = S ((S (pfrep_position_bezout_backward_identity_witness_common_rightfirst)) * rc)) /\\ exists ff_q_pfp_bezout_backward_identity_witness_common_rightfirstentry. rb = ff_q_pfp_bezout_backward_identity_witness_common_rightfirstentry * S ((S (pfrep_position_bezout_backward_identity_witness_common_rightfirst)) * rc) + (pfrep_left_bezout_backward_identity_witness_common_right)))))) \\/ (((exists pfrep_gap_bezout_backward_identity_witness_common_rightfirstoutside. pfrep_gap_bezout_backward_identity_witness_common_rightfirstoutside+(Lr)=(pfrep_power_bezout_backward_identity_witness_common_right)) /\\ (((pfrep_left_bezout_backward_identity_witness_common_right)=0))))) -> ((exists pfrep_position_bezout_backward_identity_witness_common_rightsecond. ((pfrep_position_bezout_backward_identity_witness_common_rightsecond+S (pfrep_power_bezout_backward_identity_witness_common_right)=(pfaa_length_bezout_backward_identity)) /\\ ((((exists ff_h_pfp_bezout_backward_identity_witness_common_rightsecondentry. ff_h_pfp_bezout_backward_identity_witness_common_rightsecondentry + S (pfrep_right_bezout_backward_identity_witness_common_right) = S ((S (pfrep_position_bezout_backward_identity_witness_common_rightsecond)) * pfaa_right_c_bezout_backward_identity)) /\\ exists ff_q_pfp_bezout_backward_identity_witness_common_rightsecondentry. pfaa_right_b_bezout_backward_identity = ff_q_pfp_bezout_backward_identity_witness_common_rightsecondentry * S ((S (pfrep_position_bezout_backward_identity_witness_common_rightsecond)) * pfaa_right_c_bezout_backward_identity) + (pfrep_right_bezout_backward_identity_witness_common_right)))))) \\/ (((exists pfrep_gap_bezout_backward_identity_witness_common_rightsecondoutside. pfrep_gap_bezout_backward_identity_witness_common_rightsecondoutside+(pfaa_length_bezout_backward_identity)=(pfrep_power_bezout_backward_identity_witness_common_right)) /\\ (((pfrep_right_bezout_backward_identity_witness_common_right)=0))))) -> pfrep_left_bezout_backward_identity_witness_common_right=pfrep_right_bezout_backward_identity_witness_common_right)))) /\\ (((forall pfp_index_bezout_backward_identity_witness_operation. (exists pfa_gap_bezout_backward_identity_witness_operationindex. pfa_gap_bezout_backward_identity_witness_operationindex + S (pfp_index_bezout_backward_identity_witness_operation) = (pfaa_length_bezout_backward_identity)) -> exists pfp_left_bezout_backward_identity_witness_operation pfp_right_bezout_backward_identity_witness_operation pfp_value_bezout_backward_identity_witness_operation. ((((exists ff_h_pfp_bezout_backward_identity_witness_operationleft. ff_h_pfp_bezout_backward_identity_witness_operationleft + S (pfp_left_bezout_backward_identity_witness_operation) = S ((S (pfp_index_bezout_backward_identity_witness_operation)) * pfaa_left_c_bezout_backward_identity)) /\\ exists ff_q_pfp_bezout_backward_identity_witness_operationleft. pfaa_left_b_bezout_backward_identity = ff_q_pfp_bezout_backward_identity_witness_operationleft * S ((S (pfp_index_bezout_backward_identity_witness_operation)) * pfaa_left_c_bezout_backward_identity) + (pfp_left_bezout_backward_identity_witness_operation))) /\\ (((((exists ff_h_pfp_bezout_backward_identity_witness_operationright. ff_h_pfp_bezout_backward_identity_witness_operationright + S (pfp_right_bezout_backward_identity_witness_operation) = S ((S (pfp_index_bezout_backward_identity_witness_operation)) * pfaa_right_c_bezout_backward_identity)) /\\ exists ff_q_pfp_bezout_backward_identity_witness_operationright. pfaa_right_b_bezout_backward_identity = ff_q_pfp_bezout_backward_identity_witness_operationright * S ((S (pfp_index_bezout_backward_identity_witness_operation)) * pfaa_right_c_bezout_backward_identity) + (pfp_right_bezout_backward_identity_witness_operation))) /\\ (((((exists ff_h_pfp_bezout_backward_identity_witness_operationtarget. ff_h_pfp_bezout_backward_identity_witness_operationtarget + S (pfp_value_bezout_backward_identity_witness_operation) = S ((S (pfp_index_bezout_backward_identity_witness_operation)) * pfaa_sum_c_bezout_backward_identity)) /\\ exists ff_q_pfp_bezout_backward_identity_witness_operationtarget. pfaa_sum_b_bezout_backward_identity = ff_q_pfp_bezout_backward_identity_witness_operationtarget * S ((S (pfp_index_bezout_backward_identity_witness_operation)) * pfaa_sum_c_bezout_backward_identity) + (pfp_value_bezout_backward_identity_witness_operation))) /\\ ((((exists pfa_gap_bezout_backward_identity_witness_operationoperationleft. pfa_gap_bezout_backward_identity_witness_operationoperationleft + S (pfp_left_bezout_backward_identity_witness_operation) = (p)) /\\ (((exists pfa_gap_bezout_backward_identity_witness_operationoperationright. pfa_gap_bezout_backward_identity_witness_operationoperationright + S (pfp_right_bezout_backward_identity_witness_operation) = (p)) /\\ ((((exists pfa_gap_bezout_backward_identity_witness_operationoperationresultbound. pfa_gap_bezout_backward_identity_witness_operationoperationresultbound + S (pfp_value_bezout_backward_identity_witness_operation) = (p)) /\\ ((exists pfa_offset_left_bezout_backward_identity_witness_operationoperationresultcongruence pfa_offset_right_bezout_backward_identity_witness_operationoperationresultcongruence. ((pfp_left_bezout_backward_identity_witness_operation) + (pfp_right_bezout_backward_identity_witness_operation)) + (p) * pfa_offset_left_bezout_backward_identity_witness_operationoperationresultcongruence = (pfp_value_bezout_backward_identity_witness_operation) + (p) * pfa_offset_right_bezout_backward_identity_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_bezout_backward_identity_witness_output pfrep_left_bezout_backward_identity_witness_output pfrep_right_bezout_backward_identity_witness_output. ((exists pfrep_position_bezout_backward_identity_witness_outputfirst. ((pfrep_position_bezout_backward_identity_witness_outputfirst+S (pfrep_power_bezout_backward_identity_witness_output)=(pfaa_length_bezout_backward_identity)) /\\ ((((exists ff_h_pfp_bezout_backward_identity_witness_outputfirstentry. ff_h_pfp_bezout_backward_identity_witness_outputfirstentry + S (pfrep_left_bezout_backward_identity_witness_output) = S ((S (pfrep_position_bezout_backward_identity_witness_outputfirst)) * pfaa_sum_c_bezout_backward_identity)) /\\ exists ff_q_pfp_bezout_backward_identity_witness_outputfirstentry. pfaa_sum_b_bezout_backward_identity = ff_q_pfp_bezout_backward_identity_witness_outputfirstentry * S ((S (pfrep_position_bezout_backward_identity_witness_outputfirst)) * pfaa_sum_c_bezout_backward_identity) + (pfrep_left_bezout_backward_identity_witness_output)))))) \\/ (((exists pfrep_gap_bezout_backward_identity_witness_outputfirstoutside. pfrep_gap_bezout_backward_identity_witness_outputfirstoutside+(pfaa_length_bezout_backward_identity)=(pfrep_power_bezout_backward_identity_witness_output)) /\\ (((pfrep_left_bezout_backward_identity_witness_output)=0))))) -> ((exists pfrep_position_bezout_backward_identity_witness_outputsecond. ((pfrep_position_bezout_backward_identity_witness_outputsecond+S (pfrep_power_bezout_backward_identity_witness_output)=(La)) /\\ ((((exists ff_h_pfp_bezout_backward_identity_witness_outputsecondentry. ff_h_pfp_bezout_backward_identity_witness_outputsecondentry + S (pfrep_right_bezout_backward_identity_witness_output) = S ((S (pfrep_position_bezout_backward_identity_witness_outputsecond)) * ac)) /\\ exists ff_q_pfp_bezout_backward_identity_witness_outputsecondentry. ab = ff_q_pfp_bezout_backward_identity_witness_outputsecondentry * S ((S (pfrep_position_bezout_backward_identity_witness_outputsecond)) * ac) + (pfrep_right_bezout_backward_identity_witness_output)))))) \\/ (((exists pfrep_gap_bezout_backward_identity_witness_outputsecondoutside. pfrep_gap_bezout_backward_identity_witness_outputsecondoutside+(La)=(pfrep_power_bezout_backward_identity_witness_output)) /\\ (((pfrep_right_bezout_backward_identity_witness_output)=0))))) -> pfrep_left_bezout_backward_identity_witness_output=pfrep_right_bezout_backward_identity_witness_output))))))))))))) -> (exists pfbz_left_code_bezout_backward_original pfbz_left_scale_bezout_backward_original pfbz_left_length_bezout_backward_original pfbz_right_code_bezout_backward_original pfbz_right_scale_bezout_backward_original pfbz_right_length_bezout_backward_original. ((((forall fom_index_pfp_bezout_backward_original_left_productleft. (exists fom_gap_pfp_bezout_backward_original_left_productleft_index_bound. fom_gap_pfp_bezout_backward_original_left_productleft_index_bound + S (fom_index_pfp_bezout_backward_original_left_productleft) = Lu) -> exists fom_value_pfp_bezout_backward_original_left_productleft. ((((exists fom_beta_height_pfp_bezout_backward_original_left_productleft_entry. fom_beta_height_pfp_bezout_backward_original_left_productleft_entry + S (fom_value_pfp_bezout_backward_original_left_productleft) = S ((S (fom_index_pfp_bezout_backward_original_left_productleft)) * uc)) /\\ exists fom_beta_quotient_pfp_bezout_backward_original_left_productleft_entry. ub = fom_beta_quotient_pfp_bezout_backward_original_left_productleft_entry * S ((S (fom_index_pfp_bezout_backward_original_left_productleft)) * uc) + (fom_value_pfp_bezout_backward_original_left_productleft))) /\\ (exists fom_gap_pfp_bezout_backward_original_left_productleft_value_bound. fom_gap_pfp_bezout_backward_original_left_productleft_value_bound + S (fom_value_pfp_bezout_backward_original_left_productleft) = p))) /\\ (((forall fom_index_pfp_bezout_backward_original_left_productright. (exists fom_gap_pfp_bezout_backward_original_left_productright_index_bound. fom_gap_pfp_bezout_backward_original_left_productright_index_bound + S (fom_index_pfp_bezout_backward_original_left_productright) = Lb) -> exists fom_value_pfp_bezout_backward_original_left_productright. ((((exists fom_beta_height_pfp_bezout_backward_original_left_productright_entry. fom_beta_height_pfp_bezout_backward_original_left_productright_entry + S (fom_value_pfp_bezout_backward_original_left_productright) = S ((S (fom_index_pfp_bezout_backward_original_left_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_bezout_backward_original_left_productright_entry. bb = fom_beta_quotient_pfp_bezout_backward_original_left_productright_entry * S ((S (fom_index_pfp_bezout_backward_original_left_productright)) * bc) + (fom_value_pfp_bezout_backward_original_left_productright))) /\\ (exists fom_gap_pfp_bezout_backward_original_left_productright_value_bound. fom_gap_pfp_bezout_backward_original_left_productright_value_bound + S (fom_value_pfp_bezout_backward_original_left_productright) = p))) /\\ (((((((Lu)=0 \\/ (Lb)=0) /\\ (((pfbz_left_length_bezout_backward_original)=0)))) \\/ (((~((Lu)=0)) /\\ (((~((Lb)=0)) /\\ (((Lu)+(Lb)=S (pfbz_left_length_bezout_backward_original)))))))) /\\ ((forall pfc_index_bezout_backward_original_left_productcoefficients. (exists pfa_gap_bezout_backward_original_left_productcoefficientsbound. pfa_gap_bezout_backward_original_left_productcoefficientsbound + S (pfc_index_bezout_backward_original_left_productcoefficients) = (pfbz_left_length_bezout_backward_original)) -> exists pfc_value_bezout_backward_original_left_productcoefficients. ((((exists ff_h_pfp_bezout_backward_original_left_productcoefficientsentry. ff_h_pfp_bezout_backward_original_left_productcoefficientsentry + S (pfc_value_bezout_backward_original_left_productcoefficients) = S ((S (pfc_index_bezout_backward_original_left_productcoefficients)) * pfbz_left_scale_bezout_backward_original)) /\\ exists ff_q_pfp_bezout_backward_original_left_productcoefficientsentry. pfbz_left_code_bezout_backward_original = ff_q_pfp_bezout_backward_original_left_productcoefficientsentry * S ((S (pfc_index_bezout_backward_original_left_productcoefficients)) * pfbz_left_scale_bezout_backward_original) + (pfc_value_bezout_backward_original_left_productcoefficients))) /\\ ((exists pfc_terms_code_bezout_backward_original_left_productcoefficientscoefficient pfc_terms_scale_bezout_backward_original_left_productcoefficientscoefficient pfc_natural_sum_bezout_backward_original_left_productcoefficientscoefficient. ((forall pfc_index_bezout_backward_original_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_bezout_backward_original_left_productcoefficientscoefficientdiagonalbound. pfa_gap_bezout_backward_original_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_bezout_backward_original_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_bezout_backward_original_left_productcoefficients))) -> exists pfc_value_bezout_backward_original_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_bezout_backward_original_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_bezout_backward_original_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_bezout_backward_original_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_bezout_backward_original_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bezout_backward_original_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_bezout_backward_original_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_bezout_backward_original_left_productcoefficientscoefficient = ff_q_pfp_bezout_backward_original_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_bezout_backward_original_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bezout_backward_original_left_productcoefficientscoefficient) + (pfc_value_bezout_backward_original_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm pfc_left_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm pfc_right_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_bezout_backward_original_left_productcoefficientscoefficientdiagonal)+pfc_complement_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm=(pfc_index_bezout_backward_original_left_productcoefficients)) /\\ ((((((exists pfa_gap_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_bezout_backward_original_left_productcoefficientscoefficientdiagonal) = (Lu)) /\\ ((((exists ff_h_pfp_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_bezout_backward_original_left_productcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_bezout_backward_original_left_productcoefficientscoefficientdiagonal)) * uc) + (pfc_left_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermleftoutside+(Lu)=(pfc_index_bezout_backward_original_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm) = (Lb)) /\\ ((((exists ff_h_pfp_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_bezout_backward_original_left_productcoefficientscoefficientdiagonaltermrightoutside+(Lb)=(pfc_complement_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_bezout_backward_original_left_productcoefficientscoefficientdiagonal)=pfc_left_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm*pfc_right_bezout_backward_original_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_bezout_backward_original_left_productcoefficientscoefficientsum fs_v_pfc_bezout_backward_original_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_bezout_backward_original_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_bezout_backward_original_left_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_bezout_backward_original_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_bezout_backward_original_left_productcoefficientscoefficient) = S ((S (S (pfc_index_bezout_backward_original_left_productcoefficients))) * fs_v_pfc_bezout_backward_original_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_bezout_backward_original_left_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_bezout_backward_original_left_productcoefficients))) * fs_v_pfc_bezout_backward_original_left_productcoefficientscoefficientsum) + (pfc_natural_sum_bezout_backward_original_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_bezout_backward_original_left_productcoefficients)) -> exists fs_a_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bezout_backward_original_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_bezout_backward_original_left_productcoefficientscoefficient = fs_q_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bezout_backward_original_left_productcoefficientscoefficient) + (fs_a_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_original_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_bezout_backward_original_left_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_original_left_productcoefficientscoefficientsum) + (fs_r_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_original_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_bezout_backward_original_left_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_original_left_productcoefficientscoefficientsum) + (fs_s_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_bezout_backward_original_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_bezout_backward_original_left_productcoefficientscoefficientresiduebound. pfa_gap_bezout_backward_original_left_productcoefficientscoefficientresiduebound + S (pfc_value_bezout_backward_original_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_bezout_backward_original_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_bezout_backward_original_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_bezout_backward_original_left_productcoefficientscoefficient) + (p) * pfa_offset_left_bezout_backward_original_left_productcoefficientscoefficientresiduecongruence = (pfc_value_bezout_backward_original_left_productcoefficients) + (p) * pfa_offset_right_bezout_backward_original_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_bezout_backward_original_right_productleft. (exists fom_gap_pfp_bezout_backward_original_right_productleft_index_bound. fom_gap_pfp_bezout_backward_original_right_productleft_index_bound + S (fom_index_pfp_bezout_backward_original_right_productleft) = Lv) -> exists fom_value_pfp_bezout_backward_original_right_productleft. ((((exists fom_beta_height_pfp_bezout_backward_original_right_productleft_entry. fom_beta_height_pfp_bezout_backward_original_right_productleft_entry + S (fom_value_pfp_bezout_backward_original_right_productleft) = S ((S (fom_index_pfp_bezout_backward_original_right_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_bezout_backward_original_right_productleft_entry. vb = fom_beta_quotient_pfp_bezout_backward_original_right_productleft_entry * S ((S (fom_index_pfp_bezout_backward_original_right_productleft)) * vc) + (fom_value_pfp_bezout_backward_original_right_productleft))) /\\ (exists fom_gap_pfp_bezout_backward_original_right_productleft_value_bound. fom_gap_pfp_bezout_backward_original_right_productleft_value_bound + S (fom_value_pfp_bezout_backward_original_right_productleft) = p))) /\\ (((forall fom_index_pfp_bezout_backward_original_right_productright. (exists fom_gap_pfp_bezout_backward_original_right_productright_index_bound. fom_gap_pfp_bezout_backward_original_right_productright_index_bound + S (fom_index_pfp_bezout_backward_original_right_productright) = Lr) -> exists fom_value_pfp_bezout_backward_original_right_productright. ((((exists fom_beta_height_pfp_bezout_backward_original_right_productright_entry. fom_beta_height_pfp_bezout_backward_original_right_productright_entry + S (fom_value_pfp_bezout_backward_original_right_productright) = S ((S (fom_index_pfp_bezout_backward_original_right_productright)) * rc)) /\\ exists fom_beta_quotient_pfp_bezout_backward_original_right_productright_entry. rb = fom_beta_quotient_pfp_bezout_backward_original_right_productright_entry * S ((S (fom_index_pfp_bezout_backward_original_right_productright)) * rc) + (fom_value_pfp_bezout_backward_original_right_productright))) /\\ (exists fom_gap_pfp_bezout_backward_original_right_productright_value_bound. fom_gap_pfp_bezout_backward_original_right_productright_value_bound + S (fom_value_pfp_bezout_backward_original_right_productright) = p))) /\\ (((((((Lv)=0 \\/ (Lr)=0) /\\ (((pfbz_right_length_bezout_backward_original)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((Lr)=0)) /\\ (((Lv)+(Lr)=S (pfbz_right_length_bezout_backward_original)))))))) /\\ ((forall pfc_index_bezout_backward_original_right_productcoefficients. (exists pfa_gap_bezout_backward_original_right_productcoefficientsbound. pfa_gap_bezout_backward_original_right_productcoefficientsbound + S (pfc_index_bezout_backward_original_right_productcoefficients) = (pfbz_right_length_bezout_backward_original)) -> exists pfc_value_bezout_backward_original_right_productcoefficients. ((((exists ff_h_pfp_bezout_backward_original_right_productcoefficientsentry. ff_h_pfp_bezout_backward_original_right_productcoefficientsentry + S (pfc_value_bezout_backward_original_right_productcoefficients) = S ((S (pfc_index_bezout_backward_original_right_productcoefficients)) * pfbz_right_scale_bezout_backward_original)) /\\ exists ff_q_pfp_bezout_backward_original_right_productcoefficientsentry. pfbz_right_code_bezout_backward_original = ff_q_pfp_bezout_backward_original_right_productcoefficientsentry * S ((S (pfc_index_bezout_backward_original_right_productcoefficients)) * pfbz_right_scale_bezout_backward_original) + (pfc_value_bezout_backward_original_right_productcoefficients))) /\\ ((exists pfc_terms_code_bezout_backward_original_right_productcoefficientscoefficient pfc_terms_scale_bezout_backward_original_right_productcoefficientscoefficient pfc_natural_sum_bezout_backward_original_right_productcoefficientscoefficient. ((forall pfc_index_bezout_backward_original_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_bezout_backward_original_right_productcoefficientscoefficientdiagonalbound. pfa_gap_bezout_backward_original_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_bezout_backward_original_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_bezout_backward_original_right_productcoefficients))) -> exists pfc_value_bezout_backward_original_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_bezout_backward_original_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_bezout_backward_original_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_bezout_backward_original_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_bezout_backward_original_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bezout_backward_original_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_bezout_backward_original_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_bezout_backward_original_right_productcoefficientscoefficient = ff_q_pfp_bezout_backward_original_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_bezout_backward_original_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bezout_backward_original_right_productcoefficientscoefficient) + (pfc_value_bezout_backward_original_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm pfc_left_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm pfc_right_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_bezout_backward_original_right_productcoefficientscoefficientdiagonal)+pfc_complement_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm=(pfc_index_bezout_backward_original_right_productcoefficients)) /\\ ((((((exists pfa_gap_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_bezout_backward_original_right_productcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_bezout_backward_original_right_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_bezout_backward_original_right_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_bezout_backward_original_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm) = (Lr)) /\\ ((((exists ff_h_pfp_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm)) * rc)) /\\ exists ff_q_pfp_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermrightentry. rb = ff_q_pfp_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm)) * rc) + (pfc_right_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_bezout_backward_original_right_productcoefficientscoefficientdiagonaltermrightoutside+(Lr)=(pfc_complement_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_bezout_backward_original_right_productcoefficientscoefficientdiagonal)=pfc_left_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm*pfc_right_bezout_backward_original_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_bezout_backward_original_right_productcoefficientscoefficientsum fs_v_pfc_bezout_backward_original_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_bezout_backward_original_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_bezout_backward_original_right_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_bezout_backward_original_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_bezout_backward_original_right_productcoefficientscoefficient) = S ((S (S (pfc_index_bezout_backward_original_right_productcoefficients))) * fs_v_pfc_bezout_backward_original_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_bezout_backward_original_right_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_bezout_backward_original_right_productcoefficients))) * fs_v_pfc_bezout_backward_original_right_productcoefficientscoefficientsum) + (pfc_natural_sum_bezout_backward_original_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_bezout_backward_original_right_productcoefficients)) -> exists fs_a_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bezout_backward_original_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_bezout_backward_original_right_productcoefficientscoefficient = fs_q_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bezout_backward_original_right_productcoefficientscoefficient) + (fs_a_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_original_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_bezout_backward_original_right_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_original_right_productcoefficientscoefficientsum) + (fs_r_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_original_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_bezout_backward_original_right_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_original_right_productcoefficientscoefficientsum) + (fs_s_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_bezout_backward_original_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_bezout_backward_original_right_productcoefficientscoefficientresiduebound. pfa_gap_bezout_backward_original_right_productcoefficientscoefficientresiduebound + S (pfc_value_bezout_backward_original_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_bezout_backward_original_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_bezout_backward_original_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_bezout_backward_original_right_productcoefficientscoefficient) + (p) * pfa_offset_left_bezout_backward_original_right_productcoefficientscoefficientresiduecongruence = (pfc_value_bezout_backward_original_right_productcoefficients) + (p) * pfa_offset_right_bezout_backward_original_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_bezout_backward_original_sum_left_bounded. (exists fom_gap_pfp_bezout_backward_original_sum_left_bounded_index_bound. fom_gap_pfp_bezout_backward_original_sum_left_bounded_index_bound + S (fom_index_pfp_bezout_backward_original_sum_left_bounded) = pfbz_left_length_bezout_backward_original) -> exists fom_value_pfp_bezout_backward_original_sum_left_bounded. ((((exists fom_beta_height_pfp_bezout_backward_original_sum_left_bounded_entry. fom_beta_height_pfp_bezout_backward_original_sum_left_bounded_entry + S (fom_value_pfp_bezout_backward_original_sum_left_bounded) = S ((S (fom_index_pfp_bezout_backward_original_sum_left_bounded)) * pfbz_left_scale_bezout_backward_original)) /\\ exists fom_beta_quotient_pfp_bezout_backward_original_sum_left_bounded_entry. pfbz_left_code_bezout_backward_original = fom_beta_quotient_pfp_bezout_backward_original_sum_left_bounded_entry * S ((S (fom_index_pfp_bezout_backward_original_sum_left_bounded)) * pfbz_left_scale_bezout_backward_original) + (fom_value_pfp_bezout_backward_original_sum_left_bounded))) /\\ (exists fom_gap_pfp_bezout_backward_original_sum_left_bounded_value_bound. fom_gap_pfp_bezout_backward_original_sum_left_bounded_value_bound + S (fom_value_pfp_bezout_backward_original_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_bezout_backward_original_sum_right_bounded. (exists fom_gap_pfp_bezout_backward_original_sum_right_bounded_index_bound. fom_gap_pfp_bezout_backward_original_sum_right_bounded_index_bound + S (fom_index_pfp_bezout_backward_original_sum_right_bounded) = pfbz_right_length_bezout_backward_original) -> exists fom_value_pfp_bezout_backward_original_sum_right_bounded. ((((exists fom_beta_height_pfp_bezout_backward_original_sum_right_bounded_entry. fom_beta_height_pfp_bezout_backward_original_sum_right_bounded_entry + S (fom_value_pfp_bezout_backward_original_sum_right_bounded) = S ((S (fom_index_pfp_bezout_backward_original_sum_right_bounded)) * pfbz_right_scale_bezout_backward_original)) /\\ exists fom_beta_quotient_pfp_bezout_backward_original_sum_right_bounded_entry. pfbz_right_code_bezout_backward_original = fom_beta_quotient_pfp_bezout_backward_original_sum_right_bounded_entry * S ((S (fom_index_pfp_bezout_backward_original_sum_right_bounded)) * pfbz_right_scale_bezout_backward_original) + (fom_value_pfp_bezout_backward_original_sum_right_bounded))) /\\ (exists fom_gap_pfp_bezout_backward_original_sum_right_bounded_value_bound. fom_gap_pfp_bezout_backward_original_sum_right_bounded_value_bound + S (fom_value_pfp_bezout_backward_original_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_bezout_backward_original_sum_result_bounded. (exists fom_gap_pfp_bezout_backward_original_sum_result_bounded_index_bound. fom_gap_pfp_bezout_backward_original_sum_result_bounded_index_bound + S (fom_index_pfp_bezout_backward_original_sum_result_bounded) = Lg) -> exists fom_value_pfp_bezout_backward_original_sum_result_bounded. ((((exists fom_beta_height_pfp_bezout_backward_original_sum_result_bounded_entry. fom_beta_height_pfp_bezout_backward_original_sum_result_bounded_entry + S (fom_value_pfp_bezout_backward_original_sum_result_bounded) = S ((S (fom_index_pfp_bezout_backward_original_sum_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_bezout_backward_original_sum_result_bounded_entry. gb = fom_beta_quotient_pfp_bezout_backward_original_sum_result_bounded_entry * S ((S (fom_index_pfp_bezout_backward_original_sum_result_bounded)) * gc) + (fom_value_pfp_bezout_backward_original_sum_result_bounded))) /\\ (exists fom_gap_pfp_bezout_backward_original_sum_result_bounded_value_bound. fom_gap_pfp_bezout_backward_original_sum_result_bounded_value_bound + S (fom_value_pfp_bezout_backward_original_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_bezout_backward_original_sum pfaa_left_c_bezout_backward_original_sum pfaa_right_b_bezout_backward_original_sum pfaa_right_c_bezout_backward_original_sum pfaa_sum_b_bezout_backward_original_sum pfaa_sum_c_bezout_backward_original_sum pfaa_length_bezout_backward_original_sum. ((((forall pfrep_power_bezout_backward_original_sum_witness_common_left pfrep_left_bezout_backward_original_sum_witness_common_left pfrep_right_bezout_backward_original_sum_witness_common_left. ((exists pfrep_position_bezout_backward_original_sum_witness_common_leftfirst. ((pfrep_position_bezout_backward_original_sum_witness_common_leftfirst+S (pfrep_power_bezout_backward_original_sum_witness_common_left)=(pfbz_left_length_bezout_backward_original)) /\\ ((((exists ff_h_pfp_bezout_backward_original_sum_witness_common_leftfirstentry. ff_h_pfp_bezout_backward_original_sum_witness_common_leftfirstentry + S (pfrep_left_bezout_backward_original_sum_witness_common_left) = S ((S (pfrep_position_bezout_backward_original_sum_witness_common_leftfirst)) * pfbz_left_scale_bezout_backward_original)) /\\ exists ff_q_pfp_bezout_backward_original_sum_witness_common_leftfirstentry. pfbz_left_code_bezout_backward_original = ff_q_pfp_bezout_backward_original_sum_witness_common_leftfirstentry * S ((S (pfrep_position_bezout_backward_original_sum_witness_common_leftfirst)) * pfbz_left_scale_bezout_backward_original) + (pfrep_left_bezout_backward_original_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_bezout_backward_original_sum_witness_common_leftfirstoutside. pfrep_gap_bezout_backward_original_sum_witness_common_leftfirstoutside+(pfbz_left_length_bezout_backward_original)=(pfrep_power_bezout_backward_original_sum_witness_common_left)) /\\ (((pfrep_left_bezout_backward_original_sum_witness_common_left)=0))))) -> ((exists pfrep_position_bezout_backward_original_sum_witness_common_leftsecond. ((pfrep_position_bezout_backward_original_sum_witness_common_leftsecond+S (pfrep_power_bezout_backward_original_sum_witness_common_left)=(pfaa_length_bezout_backward_original_sum)) /\\ ((((exists ff_h_pfp_bezout_backward_original_sum_witness_common_leftsecondentry. ff_h_pfp_bezout_backward_original_sum_witness_common_leftsecondentry + S (pfrep_right_bezout_backward_original_sum_witness_common_left) = S ((S (pfrep_position_bezout_backward_original_sum_witness_common_leftsecond)) * pfaa_left_c_bezout_backward_original_sum)) /\\ exists ff_q_pfp_bezout_backward_original_sum_witness_common_leftsecondentry. pfaa_left_b_bezout_backward_original_sum = ff_q_pfp_bezout_backward_original_sum_witness_common_leftsecondentry * S ((S (pfrep_position_bezout_backward_original_sum_witness_common_leftsecond)) * pfaa_left_c_bezout_backward_original_sum) + (pfrep_right_bezout_backward_original_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_bezout_backward_original_sum_witness_common_leftsecondoutside. pfrep_gap_bezout_backward_original_sum_witness_common_leftsecondoutside+(pfaa_length_bezout_backward_original_sum)=(pfrep_power_bezout_backward_original_sum_witness_common_left)) /\\ (((pfrep_right_bezout_backward_original_sum_witness_common_left)=0))))) -> pfrep_left_bezout_backward_original_sum_witness_common_left=pfrep_right_bezout_backward_original_sum_witness_common_left) /\\ ((forall pfrep_power_bezout_backward_original_sum_witness_common_right pfrep_left_bezout_backward_original_sum_witness_common_right pfrep_right_bezout_backward_original_sum_witness_common_right. ((exists pfrep_position_bezout_backward_original_sum_witness_common_rightfirst. ((pfrep_position_bezout_backward_original_sum_witness_common_rightfirst+S (pfrep_power_bezout_backward_original_sum_witness_common_right)=(pfbz_right_length_bezout_backward_original)) /\\ ((((exists ff_h_pfp_bezout_backward_original_sum_witness_common_rightfirstentry. ff_h_pfp_bezout_backward_original_sum_witness_common_rightfirstentry + S (pfrep_left_bezout_backward_original_sum_witness_common_right) = S ((S (pfrep_position_bezout_backward_original_sum_witness_common_rightfirst)) * pfbz_right_scale_bezout_backward_original)) /\\ exists ff_q_pfp_bezout_backward_original_sum_witness_common_rightfirstentry. pfbz_right_code_bezout_backward_original = ff_q_pfp_bezout_backward_original_sum_witness_common_rightfirstentry * S ((S (pfrep_position_bezout_backward_original_sum_witness_common_rightfirst)) * pfbz_right_scale_bezout_backward_original) + (pfrep_left_bezout_backward_original_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_bezout_backward_original_sum_witness_common_rightfirstoutside. pfrep_gap_bezout_backward_original_sum_witness_common_rightfirstoutside+(pfbz_right_length_bezout_backward_original)=(pfrep_power_bezout_backward_original_sum_witness_common_right)) /\\ (((pfrep_left_bezout_backward_original_sum_witness_common_right)=0))))) -> ((exists pfrep_position_bezout_backward_original_sum_witness_common_rightsecond. ((pfrep_position_bezout_backward_original_sum_witness_common_rightsecond+S (pfrep_power_bezout_backward_original_sum_witness_common_right)=(pfaa_length_bezout_backward_original_sum)) /\\ ((((exists ff_h_pfp_bezout_backward_original_sum_witness_common_rightsecondentry. ff_h_pfp_bezout_backward_original_sum_witness_common_rightsecondentry + S (pfrep_right_bezout_backward_original_sum_witness_common_right) = S ((S (pfrep_position_bezout_backward_original_sum_witness_common_rightsecond)) * pfaa_right_c_bezout_backward_original_sum)) /\\ exists ff_q_pfp_bezout_backward_original_sum_witness_common_rightsecondentry. pfaa_right_b_bezout_backward_original_sum = ff_q_pfp_bezout_backward_original_sum_witness_common_rightsecondentry * S ((S (pfrep_position_bezout_backward_original_sum_witness_common_rightsecond)) * pfaa_right_c_bezout_backward_original_sum) + (pfrep_right_bezout_backward_original_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_bezout_backward_original_sum_witness_common_rightsecondoutside. pfrep_gap_bezout_backward_original_sum_witness_common_rightsecondoutside+(pfaa_length_bezout_backward_original_sum)=(pfrep_power_bezout_backward_original_sum_witness_common_right)) /\\ (((pfrep_right_bezout_backward_original_sum_witness_common_right)=0))))) -> pfrep_left_bezout_backward_original_sum_witness_common_right=pfrep_right_bezout_backward_original_sum_witness_common_right)))) /\\ (((forall pfp_index_bezout_backward_original_sum_witness_operation. (exists pfa_gap_bezout_backward_original_sum_witness_operationindex. pfa_gap_bezout_backward_original_sum_witness_operationindex + S (pfp_index_bezout_backward_original_sum_witness_operation) = (pfaa_length_bezout_backward_original_sum)) -> exists pfp_left_bezout_backward_original_sum_witness_operation pfp_right_bezout_backward_original_sum_witness_operation pfp_value_bezout_backward_original_sum_witness_operation. ((((exists ff_h_pfp_bezout_backward_original_sum_witness_operationleft. ff_h_pfp_bezout_backward_original_sum_witness_operationleft + S (pfp_left_bezout_backward_original_sum_witness_operation) = S ((S (pfp_index_bezout_backward_original_sum_witness_operation)) * pfaa_left_c_bezout_backward_original_sum)) /\\ exists ff_q_pfp_bezout_backward_original_sum_witness_operationleft. pfaa_left_b_bezout_backward_original_sum = ff_q_pfp_bezout_backward_original_sum_witness_operationleft * S ((S (pfp_index_bezout_backward_original_sum_witness_operation)) * pfaa_left_c_bezout_backward_original_sum) + (pfp_left_bezout_backward_original_sum_witness_operation))) /\\ (((((exists ff_h_pfp_bezout_backward_original_sum_witness_operationright. ff_h_pfp_bezout_backward_original_sum_witness_operationright + S (pfp_right_bezout_backward_original_sum_witness_operation) = S ((S (pfp_index_bezout_backward_original_sum_witness_operation)) * pfaa_right_c_bezout_backward_original_sum)) /\\ exists ff_q_pfp_bezout_backward_original_sum_witness_operationright. pfaa_right_b_bezout_backward_original_sum = ff_q_pfp_bezout_backward_original_sum_witness_operationright * S ((S (pfp_index_bezout_backward_original_sum_witness_operation)) * pfaa_right_c_bezout_backward_original_sum) + (pfp_right_bezout_backward_original_sum_witness_operation))) /\\ (((((exists ff_h_pfp_bezout_backward_original_sum_witness_operationtarget. ff_h_pfp_bezout_backward_original_sum_witness_operationtarget + S (pfp_value_bezout_backward_original_sum_witness_operation) = S ((S (pfp_index_bezout_backward_original_sum_witness_operation)) * pfaa_sum_c_bezout_backward_original_sum)) /\\ exists ff_q_pfp_bezout_backward_original_sum_witness_operationtarget. pfaa_sum_b_bezout_backward_original_sum = ff_q_pfp_bezout_backward_original_sum_witness_operationtarget * S ((S (pfp_index_bezout_backward_original_sum_witness_operation)) * pfaa_sum_c_bezout_backward_original_sum) + (pfp_value_bezout_backward_original_sum_witness_operation))) /\\ ((((exists pfa_gap_bezout_backward_original_sum_witness_operationoperationleft. pfa_gap_bezout_backward_original_sum_witness_operationoperationleft + S (pfp_left_bezout_backward_original_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_bezout_backward_original_sum_witness_operationoperationright. pfa_gap_bezout_backward_original_sum_witness_operationoperationright + S (pfp_right_bezout_backward_original_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_bezout_backward_original_sum_witness_operationoperationresultbound. pfa_gap_bezout_backward_original_sum_witness_operationoperationresultbound + S (pfp_value_bezout_backward_original_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_bezout_backward_original_sum_witness_operationoperationresultcongruence pfa_offset_right_bezout_backward_original_sum_witness_operationoperationresultcongruence. ((pfp_left_bezout_backward_original_sum_witness_operation) + (pfp_right_bezout_backward_original_sum_witness_operation)) + (p) * pfa_offset_left_bezout_backward_original_sum_witness_operationoperationresultcongruence = (pfp_value_bezout_backward_original_sum_witness_operation) + (p) * pfa_offset_right_bezout_backward_original_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_bezout_backward_original_sum_witness_output pfrep_left_bezout_backward_original_sum_witness_output pfrep_right_bezout_backward_original_sum_witness_output. ((exists pfrep_position_bezout_backward_original_sum_witness_outputfirst. ((pfrep_position_bezout_backward_original_sum_witness_outputfirst+S (pfrep_power_bezout_backward_original_sum_witness_output)=(pfaa_length_bezout_backward_original_sum)) /\\ ((((exists ff_h_pfp_bezout_backward_original_sum_witness_outputfirstentry. ff_h_pfp_bezout_backward_original_sum_witness_outputfirstentry + S (pfrep_left_bezout_backward_original_sum_witness_output) = S ((S (pfrep_position_bezout_backward_original_sum_witness_outputfirst)) * pfaa_sum_c_bezout_backward_original_sum)) /\\ exists ff_q_pfp_bezout_backward_original_sum_witness_outputfirstentry. pfaa_sum_b_bezout_backward_original_sum = ff_q_pfp_bezout_backward_original_sum_witness_outputfirstentry * S ((S (pfrep_position_bezout_backward_original_sum_witness_outputfirst)) * pfaa_sum_c_bezout_backward_original_sum) + (pfrep_left_bezout_backward_original_sum_witness_output)))))) \\/ (((exists pfrep_gap_bezout_backward_original_sum_witness_outputfirstoutside. pfrep_gap_bezout_backward_original_sum_witness_outputfirstoutside+(pfaa_length_bezout_backward_original_sum)=(pfrep_power_bezout_backward_original_sum_witness_output)) /\\ (((pfrep_left_bezout_backward_original_sum_witness_output)=0))))) -> ((exists pfrep_position_bezout_backward_original_sum_witness_outputsecond. ((pfrep_position_bezout_backward_original_sum_witness_outputsecond+S (pfrep_power_bezout_backward_original_sum_witness_output)=(Lg)) /\\ ((((exists ff_h_pfp_bezout_backward_original_sum_witness_outputsecondentry. ff_h_pfp_bezout_backward_original_sum_witness_outputsecondentry + S (pfrep_right_bezout_backward_original_sum_witness_output) = S ((S (pfrep_position_bezout_backward_original_sum_witness_outputsecond)) * gc)) /\\ exists ff_q_pfp_bezout_backward_original_sum_witness_outputsecondentry. gb = ff_q_pfp_bezout_backward_original_sum_witness_outputsecondentry * S ((S (pfrep_position_bezout_backward_original_sum_witness_outputsecond)) * gc) + (pfrep_right_bezout_backward_original_sum_witness_output)))))) \\/ (((exists pfrep_gap_bezout_backward_original_sum_witness_outputsecondoutside. pfrep_gap_bezout_backward_original_sum_witness_outputsecondoutside+(Lg)=(pfrep_power_bezout_backward_original_sum_witness_output)) /\\ (((pfrep_right_bezout_backward_original_sum_witness_output)=0))))) -> pfrep_left_bezout_backward_original_sum_witness_output=pfrep_right_bezout_backward_original_sum_witness_output)))))))))))))))))) -> (exists pfbz_update_wb_bezout_backward_result pfbz_update_wc_bezout_backward_result pfbz_update_W_bezout_backward_result pfbz_update_tb_bezout_backward_result pfbz_update_tc_bezout_backward_result pfbz_update_T_bezout_backward_result. ((((forall fom_index_pfp_bezout_backward_result_coefficient_productleft. (exists fom_gap_pfp_bezout_backward_result_coefficient_productleft_index_bound. fom_gap_pfp_bezout_backward_result_coefficient_productleft_index_bound + S (fom_index_pfp_bezout_backward_result_coefficient_productleft) = Lv) -> exists fom_value_pfp_bezout_backward_result_coefficient_productleft. ((((exists fom_beta_height_pfp_bezout_backward_result_coefficient_productleft_entry. fom_beta_height_pfp_bezout_backward_result_coefficient_productleft_entry + S (fom_value_pfp_bezout_backward_result_coefficient_productleft) = S ((S (fom_index_pfp_bezout_backward_result_coefficient_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_bezout_backward_result_coefficient_productleft_entry. vb = fom_beta_quotient_pfp_bezout_backward_result_coefficient_productleft_entry * S ((S (fom_index_pfp_bezout_backward_result_coefficient_productleft)) * vc) + (fom_value_pfp_bezout_backward_result_coefficient_productleft))) /\\ (exists fom_gap_pfp_bezout_backward_result_coefficient_productleft_value_bound. fom_gap_pfp_bezout_backward_result_coefficient_productleft_value_bound + S (fom_value_pfp_bezout_backward_result_coefficient_productleft) = p))) /\\ (((forall fom_index_pfp_bezout_backward_result_coefficient_productright. (exists fom_gap_pfp_bezout_backward_result_coefficient_productright_index_bound. fom_gap_pfp_bezout_backward_result_coefficient_productright_index_bound + S (fom_index_pfp_bezout_backward_result_coefficient_productright) = Lq) -> exists fom_value_pfp_bezout_backward_result_coefficient_productright. ((((exists fom_beta_height_pfp_bezout_backward_result_coefficient_productright_entry. fom_beta_height_pfp_bezout_backward_result_coefficient_productright_entry + S (fom_value_pfp_bezout_backward_result_coefficient_productright) = S ((S (fom_index_pfp_bezout_backward_result_coefficient_productright)) * qc)) /\\ exists fom_beta_quotient_pfp_bezout_backward_result_coefficient_productright_entry. qb = fom_beta_quotient_pfp_bezout_backward_result_coefficient_productright_entry * S ((S (fom_index_pfp_bezout_backward_result_coefficient_productright)) * qc) + (fom_value_pfp_bezout_backward_result_coefficient_productright))) /\\ (exists fom_gap_pfp_bezout_backward_result_coefficient_productright_value_bound. fom_gap_pfp_bezout_backward_result_coefficient_productright_value_bound + S (fom_value_pfp_bezout_backward_result_coefficient_productright) = p))) /\\ (((((((Lv)=0 \\/ (Lq)=0) /\\ (((pfbz_update_W_bezout_backward_result)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((Lq)=0)) /\\ (((Lv)+(Lq)=S (pfbz_update_W_bezout_backward_result)))))))) /\\ ((forall pfc_index_bezout_backward_result_coefficient_productcoefficients. (exists pfa_gap_bezout_backward_result_coefficient_productcoefficientsbound. pfa_gap_bezout_backward_result_coefficient_productcoefficientsbound + S (pfc_index_bezout_backward_result_coefficient_productcoefficients) = (pfbz_update_W_bezout_backward_result)) -> exists pfc_value_bezout_backward_result_coefficient_productcoefficients. ((((exists ff_h_pfp_bezout_backward_result_coefficient_productcoefficientsentry. ff_h_pfp_bezout_backward_result_coefficient_productcoefficientsentry + S (pfc_value_bezout_backward_result_coefficient_productcoefficients) = S ((S (pfc_index_bezout_backward_result_coefficient_productcoefficients)) * pfbz_update_wc_bezout_backward_result)) /\\ exists ff_q_pfp_bezout_backward_result_coefficient_productcoefficientsentry. pfbz_update_wb_bezout_backward_result = ff_q_pfp_bezout_backward_result_coefficient_productcoefficientsentry * S ((S (pfc_index_bezout_backward_result_coefficient_productcoefficients)) * pfbz_update_wc_bezout_backward_result) + (pfc_value_bezout_backward_result_coefficient_productcoefficients))) /\\ ((exists pfc_terms_code_bezout_backward_result_coefficient_productcoefficientscoefficient pfc_terms_scale_bezout_backward_result_coefficient_productcoefficientscoefficient pfc_natural_sum_bezout_backward_result_coefficient_productcoefficientscoefficient. ((forall pfc_index_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonal. (exists pfa_gap_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalbound. pfa_gap_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalbound + S (pfc_index_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonal) = (S (pfc_index_bezout_backward_result_coefficient_productcoefficients))) -> exists pfc_value_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalentry. ff_h_pfp_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalentry + S (pfc_value_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bezout_backward_result_coefficient_productcoefficientscoefficient)) /\\ exists ff_q_pfp_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalentry. pfc_terms_code_bezout_backward_result_coefficient_productcoefficientscoefficient = ff_q_pfp_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bezout_backward_result_coefficient_productcoefficientscoefficient) + (pfc_value_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm pfc_left_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm pfc_right_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm. (((pfc_index_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonal)+pfc_complement_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm=(pfc_index_bezout_backward_result_coefficient_productcoefficients)) /\\ ((((((exists pfa_gap_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm) = (Lq)) /\\ ((((exists ff_h_pfp_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm)) * qc)) /\\ exists ff_q_pfp_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightentry. qb = ff_q_pfp_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm)) * qc) + (pfc_right_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightoutside+(Lq)=(pfc_complement_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonal)=pfc_left_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm*pfc_right_bezout_backward_result_coefficient_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum fs_v_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum. ((((exists fs_h_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_start. fs_h_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_start. fs_u_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_bezout_backward_result_coefficient_productcoefficientscoefficient) = S ((S (S (pfc_index_bezout_backward_result_coefficient_productcoefficients))) * fs_v_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_bezout_backward_result_coefficient_productcoefficients))) * fs_v_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum) + (pfc_natural_sum_bezout_backward_result_coefficient_productcoefficientscoefficient))) /\\ forall fs_i_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps = S (pfc_index_bezout_backward_result_coefficient_productcoefficients)) -> exists fs_a_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps fs_r_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps fs_s_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bezout_backward_result_coefficient_productcoefficientscoefficient)) /\\ exists fs_q_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_bezout_backward_result_coefficient_productcoefficientscoefficient = fs_q_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bezout_backward_result_coefficient_productcoefficientscoefficient) + (fs_a_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum) + (fs_r_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum) + (fs_s_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps = fs_r_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps + fs_a_pfc_bezout_backward_result_coefficient_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_bezout_backward_result_coefficient_productcoefficientscoefficientresiduebound. pfa_gap_bezout_backward_result_coefficient_productcoefficientscoefficientresiduebound + S (pfc_value_bezout_backward_result_coefficient_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_bezout_backward_result_coefficient_productcoefficientscoefficientresiduecongruence pfa_offset_right_bezout_backward_result_coefficient_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_bezout_backward_result_coefficient_productcoefficientscoefficient) + (p) * pfa_offset_left_bezout_backward_result_coefficient_productcoefficientscoefficientresiduecongruence = (pfc_value_bezout_backward_result_coefficient_productcoefficients) + (p) * pfa_offset_right_bezout_backward_result_coefficient_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_bezout_backward_result_coefficient_difference_left_bounded. (exists fom_gap_pfp_bezout_backward_result_coefficient_difference_left_bounded_index_bound. fom_gap_pfp_bezout_backward_result_coefficient_difference_left_bounded_index_bound + S (fom_index_pfp_bezout_backward_result_coefficient_difference_left_bounded) = pfbz_update_W_bezout_backward_result) -> exists fom_value_pfp_bezout_backward_result_coefficient_difference_left_bounded. ((((exists fom_beta_height_pfp_bezout_backward_result_coefficient_difference_left_bounded_entry. fom_beta_height_pfp_bezout_backward_result_coefficient_difference_left_bounded_entry + S (fom_value_pfp_bezout_backward_result_coefficient_difference_left_bounded) = S ((S (fom_index_pfp_bezout_backward_result_coefficient_difference_left_bounded)) * pfbz_update_wc_bezout_backward_result)) /\\ exists fom_beta_quotient_pfp_bezout_backward_result_coefficient_difference_left_bounded_entry. pfbz_update_wb_bezout_backward_result = fom_beta_quotient_pfp_bezout_backward_result_coefficient_difference_left_bounded_entry * S ((S (fom_index_pfp_bezout_backward_result_coefficient_difference_left_bounded)) * pfbz_update_wc_bezout_backward_result) + (fom_value_pfp_bezout_backward_result_coefficient_difference_left_bounded))) /\\ (exists fom_gap_pfp_bezout_backward_result_coefficient_difference_left_bounded_value_bound. fom_gap_pfp_bezout_backward_result_coefficient_difference_left_bounded_value_bound + S (fom_value_pfp_bezout_backward_result_coefficient_difference_left_bounded) = p))) /\\ (((forall fom_index_pfp_bezout_backward_result_coefficient_difference_right_bounded. (exists fom_gap_pfp_bezout_backward_result_coefficient_difference_right_bounded_index_bound. fom_gap_pfp_bezout_backward_result_coefficient_difference_right_bounded_index_bound + S (fom_index_pfp_bezout_backward_result_coefficient_difference_right_bounded) = pfbz_update_T_bezout_backward_result) -> exists fom_value_pfp_bezout_backward_result_coefficient_difference_right_bounded. ((((exists fom_beta_height_pfp_bezout_backward_result_coefficient_difference_right_bounded_entry. fom_beta_height_pfp_bezout_backward_result_coefficient_difference_right_bounded_entry + S (fom_value_pfp_bezout_backward_result_coefficient_difference_right_bounded) = S ((S (fom_index_pfp_bezout_backward_result_coefficient_difference_right_bounded)) * pfbz_update_tc_bezout_backward_result)) /\\ exists fom_beta_quotient_pfp_bezout_backward_result_coefficient_difference_right_bounded_entry. pfbz_update_tb_bezout_backward_result = fom_beta_quotient_pfp_bezout_backward_result_coefficient_difference_right_bounded_entry * S ((S (fom_index_pfp_bezout_backward_result_coefficient_difference_right_bounded)) * pfbz_update_tc_bezout_backward_result) + (fom_value_pfp_bezout_backward_result_coefficient_difference_right_bounded))) /\\ (exists fom_gap_pfp_bezout_backward_result_coefficient_difference_right_bounded_value_bound. fom_gap_pfp_bezout_backward_result_coefficient_difference_right_bounded_value_bound + S (fom_value_pfp_bezout_backward_result_coefficient_difference_right_bounded) = p))) /\\ (((forall fom_index_pfp_bezout_backward_result_coefficient_difference_result_bounded. (exists fom_gap_pfp_bezout_backward_result_coefficient_difference_result_bounded_index_bound. fom_gap_pfp_bezout_backward_result_coefficient_difference_result_bounded_index_bound + S (fom_index_pfp_bezout_backward_result_coefficient_difference_result_bounded) = Lu) -> exists fom_value_pfp_bezout_backward_result_coefficient_difference_result_bounded. ((((exists fom_beta_height_pfp_bezout_backward_result_coefficient_difference_result_bounded_entry. fom_beta_height_pfp_bezout_backward_result_coefficient_difference_result_bounded_entry + S (fom_value_pfp_bezout_backward_result_coefficient_difference_result_bounded) = S ((S (fom_index_pfp_bezout_backward_result_coefficient_difference_result_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_bezout_backward_result_coefficient_difference_result_bounded_entry. ub = fom_beta_quotient_pfp_bezout_backward_result_coefficient_difference_result_bounded_entry * S ((S (fom_index_pfp_bezout_backward_result_coefficient_difference_result_bounded)) * uc) + (fom_value_pfp_bezout_backward_result_coefficient_difference_result_bounded))) /\\ (exists fom_gap_pfp_bezout_backward_result_coefficient_difference_result_bounded_value_bound. fom_gap_pfp_bezout_backward_result_coefficient_difference_result_bounded_value_bound + S (fom_value_pfp_bezout_backward_result_coefficient_difference_result_bounded) = p))) /\\ ((exists pfaa_left_b_bezout_backward_result_coefficient_difference pfaa_left_c_bezout_backward_result_coefficient_difference pfaa_right_b_bezout_backward_result_coefficient_difference pfaa_right_c_bezout_backward_result_coefficient_difference pfaa_sum_b_bezout_backward_result_coefficient_difference pfaa_sum_c_bezout_backward_result_coefficient_difference pfaa_length_bezout_backward_result_coefficient_difference. ((((forall pfrep_power_bezout_backward_result_coefficient_difference_witness_common_left pfrep_left_bezout_backward_result_coefficient_difference_witness_common_left pfrep_right_bezout_backward_result_coefficient_difference_witness_common_left. ((exists pfrep_position_bezout_backward_result_coefficient_difference_witness_common_leftfirst. ((pfrep_position_bezout_backward_result_coefficient_difference_witness_common_leftfirst+S (pfrep_power_bezout_backward_result_coefficient_difference_witness_common_left)=(pfbz_update_W_bezout_backward_result)) /\\ ((((exists ff_h_pfp_bezout_backward_result_coefficient_difference_witness_common_leftfirstentry. ff_h_pfp_bezout_backward_result_coefficient_difference_witness_common_leftfirstentry + S (pfrep_left_bezout_backward_result_coefficient_difference_witness_common_left) = S ((S (pfrep_position_bezout_backward_result_coefficient_difference_witness_common_leftfirst)) * pfbz_update_wc_bezout_backward_result)) /\\ exists ff_q_pfp_bezout_backward_result_coefficient_difference_witness_common_leftfirstentry. pfbz_update_wb_bezout_backward_result = ff_q_pfp_bezout_backward_result_coefficient_difference_witness_common_leftfirstentry * S ((S (pfrep_position_bezout_backward_result_coefficient_difference_witness_common_leftfirst)) * pfbz_update_wc_bezout_backward_result) + (pfrep_left_bezout_backward_result_coefficient_difference_witness_common_left)))))) \\/ (((exists pfrep_gap_bezout_backward_result_coefficient_difference_witness_common_leftfirstoutside. pfrep_gap_bezout_backward_result_coefficient_difference_witness_common_leftfirstoutside+(pfbz_update_W_bezout_backward_result)=(pfrep_power_bezout_backward_result_coefficient_difference_witness_common_left)) /\\ (((pfrep_left_bezout_backward_result_coefficient_difference_witness_common_left)=0))))) -> ((exists pfrep_position_bezout_backward_result_coefficient_difference_witness_common_leftsecond. ((pfrep_position_bezout_backward_result_coefficient_difference_witness_common_leftsecond+S (pfrep_power_bezout_backward_result_coefficient_difference_witness_common_left)=(pfaa_length_bezout_backward_result_coefficient_difference)) /\\ ((((exists ff_h_pfp_bezout_backward_result_coefficient_difference_witness_common_leftsecondentry. ff_h_pfp_bezout_backward_result_coefficient_difference_witness_common_leftsecondentry + S (pfrep_right_bezout_backward_result_coefficient_difference_witness_common_left) = S ((S (pfrep_position_bezout_backward_result_coefficient_difference_witness_common_leftsecond)) * pfaa_left_c_bezout_backward_result_coefficient_difference)) /\\ exists ff_q_pfp_bezout_backward_result_coefficient_difference_witness_common_leftsecondentry. pfaa_left_b_bezout_backward_result_coefficient_difference = ff_q_pfp_bezout_backward_result_coefficient_difference_witness_common_leftsecondentry * S ((S (pfrep_position_bezout_backward_result_coefficient_difference_witness_common_leftsecond)) * pfaa_left_c_bezout_backward_result_coefficient_difference) + (pfrep_right_bezout_backward_result_coefficient_difference_witness_common_left)))))) \\/ (((exists pfrep_gap_bezout_backward_result_coefficient_difference_witness_common_leftsecondoutside. pfrep_gap_bezout_backward_result_coefficient_difference_witness_common_leftsecondoutside+(pfaa_length_bezout_backward_result_coefficient_difference)=(pfrep_power_bezout_backward_result_coefficient_difference_witness_common_left)) /\\ (((pfrep_right_bezout_backward_result_coefficient_difference_witness_common_left)=0))))) -> pfrep_left_bezout_backward_result_coefficient_difference_witness_common_left=pfrep_right_bezout_backward_result_coefficient_difference_witness_common_left) /\\ ((forall pfrep_power_bezout_backward_result_coefficient_difference_witness_common_right pfrep_left_bezout_backward_result_coefficient_difference_witness_common_right pfrep_right_bezout_backward_result_coefficient_difference_witness_common_right. ((exists pfrep_position_bezout_backward_result_coefficient_difference_witness_common_rightfirst. ((pfrep_position_bezout_backward_result_coefficient_difference_witness_common_rightfirst+S (pfrep_power_bezout_backward_result_coefficient_difference_witness_common_right)=(pfbz_update_T_bezout_backward_result)) /\\ ((((exists ff_h_pfp_bezout_backward_result_coefficient_difference_witness_common_rightfirstentry. ff_h_pfp_bezout_backward_result_coefficient_difference_witness_common_rightfirstentry + S (pfrep_left_bezout_backward_result_coefficient_difference_witness_common_right) = S ((S (pfrep_position_bezout_backward_result_coefficient_difference_witness_common_rightfirst)) * pfbz_update_tc_bezout_backward_result)) /\\ exists ff_q_pfp_bezout_backward_result_coefficient_difference_witness_common_rightfirstentry. pfbz_update_tb_bezout_backward_result = ff_q_pfp_bezout_backward_result_coefficient_difference_witness_common_rightfirstentry * S ((S (pfrep_position_bezout_backward_result_coefficient_difference_witness_common_rightfirst)) * pfbz_update_tc_bezout_backward_result) + (pfrep_left_bezout_backward_result_coefficient_difference_witness_common_right)))))) \\/ (((exists pfrep_gap_bezout_backward_result_coefficient_difference_witness_common_rightfirstoutside. pfrep_gap_bezout_backward_result_coefficient_difference_witness_common_rightfirstoutside+(pfbz_update_T_bezout_backward_result)=(pfrep_power_bezout_backward_result_coefficient_difference_witness_common_right)) /\\ (((pfrep_left_bezout_backward_result_coefficient_difference_witness_common_right)=0))))) -> ((exists pfrep_position_bezout_backward_result_coefficient_difference_witness_common_rightsecond. ((pfrep_position_bezout_backward_result_coefficient_difference_witness_common_rightsecond+S (pfrep_power_bezout_backward_result_coefficient_difference_witness_common_right)=(pfaa_length_bezout_backward_result_coefficient_difference)) /\\ ((((exists ff_h_pfp_bezout_backward_result_coefficient_difference_witness_common_rightsecondentry. ff_h_pfp_bezout_backward_result_coefficient_difference_witness_common_rightsecondentry + S (pfrep_right_bezout_backward_result_coefficient_difference_witness_common_right) = S ((S (pfrep_position_bezout_backward_result_coefficient_difference_witness_common_rightsecond)) * pfaa_right_c_bezout_backward_result_coefficient_difference)) /\\ exists ff_q_pfp_bezout_backward_result_coefficient_difference_witness_common_rightsecondentry. pfaa_right_b_bezout_backward_result_coefficient_difference = ff_q_pfp_bezout_backward_result_coefficient_difference_witness_common_rightsecondentry * S ((S (pfrep_position_bezout_backward_result_coefficient_difference_witness_common_rightsecond)) * pfaa_right_c_bezout_backward_result_coefficient_difference) + (pfrep_right_bezout_backward_result_coefficient_difference_witness_common_right)))))) \\/ (((exists pfrep_gap_bezout_backward_result_coefficient_difference_witness_common_rightsecondoutside. pfrep_gap_bezout_backward_result_coefficient_difference_witness_common_rightsecondoutside+(pfaa_length_bezout_backward_result_coefficient_difference)=(pfrep_power_bezout_backward_result_coefficient_difference_witness_common_right)) /\\ (((pfrep_right_bezout_backward_result_coefficient_difference_witness_common_right)=0))))) -> pfrep_left_bezout_backward_result_coefficient_difference_witness_common_right=pfrep_right_bezout_backward_result_coefficient_difference_witness_common_right)))) /\\ (((forall pfp_index_bezout_backward_result_coefficient_difference_witness_operation. (exists pfa_gap_bezout_backward_result_coefficient_difference_witness_operationindex. pfa_gap_bezout_backward_result_coefficient_difference_witness_operationindex + S (pfp_index_bezout_backward_result_coefficient_difference_witness_operation) = (pfaa_length_bezout_backward_result_coefficient_difference)) -> exists pfp_left_bezout_backward_result_coefficient_difference_witness_operation pfp_right_bezout_backward_result_coefficient_difference_witness_operation pfp_value_bezout_backward_result_coefficient_difference_witness_operation. ((((exists ff_h_pfp_bezout_backward_result_coefficient_difference_witness_operationleft. ff_h_pfp_bezout_backward_result_coefficient_difference_witness_operationleft + S (pfp_left_bezout_backward_result_coefficient_difference_witness_operation) = S ((S (pfp_index_bezout_backward_result_coefficient_difference_witness_operation)) * pfaa_left_c_bezout_backward_result_coefficient_difference)) /\\ exists ff_q_pfp_bezout_backward_result_coefficient_difference_witness_operationleft. pfaa_left_b_bezout_backward_result_coefficient_difference = ff_q_pfp_bezout_backward_result_coefficient_difference_witness_operationleft * S ((S (pfp_index_bezout_backward_result_coefficient_difference_witness_operation)) * pfaa_left_c_bezout_backward_result_coefficient_difference) + (pfp_left_bezout_backward_result_coefficient_difference_witness_operation))) /\\ (((((exists ff_h_pfp_bezout_backward_result_coefficient_difference_witness_operationright. ff_h_pfp_bezout_backward_result_coefficient_difference_witness_operationright + S (pfp_right_bezout_backward_result_coefficient_difference_witness_operation) = S ((S (pfp_index_bezout_backward_result_coefficient_difference_witness_operation)) * pfaa_right_c_bezout_backward_result_coefficient_difference)) /\\ exists ff_q_pfp_bezout_backward_result_coefficient_difference_witness_operationright. pfaa_right_b_bezout_backward_result_coefficient_difference = ff_q_pfp_bezout_backward_result_coefficient_difference_witness_operationright * S ((S (pfp_index_bezout_backward_result_coefficient_difference_witness_operation)) * pfaa_right_c_bezout_backward_result_coefficient_difference) + (pfp_right_bezout_backward_result_coefficient_difference_witness_operation))) /\\ (((((exists ff_h_pfp_bezout_backward_result_coefficient_difference_witness_operationtarget. ff_h_pfp_bezout_backward_result_coefficient_difference_witness_operationtarget + S (pfp_value_bezout_backward_result_coefficient_difference_witness_operation) = S ((S (pfp_index_bezout_backward_result_coefficient_difference_witness_operation)) * pfaa_sum_c_bezout_backward_result_coefficient_difference)) /\\ exists ff_q_pfp_bezout_backward_result_coefficient_difference_witness_operationtarget. pfaa_sum_b_bezout_backward_result_coefficient_difference = ff_q_pfp_bezout_backward_result_coefficient_difference_witness_operationtarget * S ((S (pfp_index_bezout_backward_result_coefficient_difference_witness_operation)) * pfaa_sum_c_bezout_backward_result_coefficient_difference) + (pfp_value_bezout_backward_result_coefficient_difference_witness_operation))) /\\ ((((exists pfa_gap_bezout_backward_result_coefficient_difference_witness_operationoperationleft. pfa_gap_bezout_backward_result_coefficient_difference_witness_operationoperationleft + S (pfp_left_bezout_backward_result_coefficient_difference_witness_operation) = (p)) /\\ (((exists pfa_gap_bezout_backward_result_coefficient_difference_witness_operationoperationright. pfa_gap_bezout_backward_result_coefficient_difference_witness_operationoperationright + S (pfp_right_bezout_backward_result_coefficient_difference_witness_operation) = (p)) /\\ ((((exists pfa_gap_bezout_backward_result_coefficient_difference_witness_operationoperationresultbound. pfa_gap_bezout_backward_result_coefficient_difference_witness_operationoperationresultbound + S (pfp_value_bezout_backward_result_coefficient_difference_witness_operation) = (p)) /\\ ((exists pfa_offset_left_bezout_backward_result_coefficient_difference_witness_operationoperationresultcongruence pfa_offset_right_bezout_backward_result_coefficient_difference_witness_operationoperationresultcongruence. ((pfp_left_bezout_backward_result_coefficient_difference_witness_operation) + (pfp_right_bezout_backward_result_coefficient_difference_witness_operation)) + (p) * pfa_offset_left_bezout_backward_result_coefficient_difference_witness_operationoperationresultcongruence = (pfp_value_bezout_backward_result_coefficient_difference_witness_operation) + (p) * pfa_offset_right_bezout_backward_result_coefficient_difference_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_bezout_backward_result_coefficient_difference_witness_output pfrep_left_bezout_backward_result_coefficient_difference_witness_output pfrep_right_bezout_backward_result_coefficient_difference_witness_output. ((exists pfrep_position_bezout_backward_result_coefficient_difference_witness_outputfirst. ((pfrep_position_bezout_backward_result_coefficient_difference_witness_outputfirst+S (pfrep_power_bezout_backward_result_coefficient_difference_witness_output)=(pfaa_length_bezout_backward_result_coefficient_difference)) /\\ ((((exists ff_h_pfp_bezout_backward_result_coefficient_difference_witness_outputfirstentry. ff_h_pfp_bezout_backward_result_coefficient_difference_witness_outputfirstentry + S (pfrep_left_bezout_backward_result_coefficient_difference_witness_output) = S ((S (pfrep_position_bezout_backward_result_coefficient_difference_witness_outputfirst)) * pfaa_sum_c_bezout_backward_result_coefficient_difference)) /\\ exists ff_q_pfp_bezout_backward_result_coefficient_difference_witness_outputfirstentry. pfaa_sum_b_bezout_backward_result_coefficient_difference = ff_q_pfp_bezout_backward_result_coefficient_difference_witness_outputfirstentry * S ((S (pfrep_position_bezout_backward_result_coefficient_difference_witness_outputfirst)) * pfaa_sum_c_bezout_backward_result_coefficient_difference) + (pfrep_left_bezout_backward_result_coefficient_difference_witness_output)))))) \\/ (((exists pfrep_gap_bezout_backward_result_coefficient_difference_witness_outputfirstoutside. pfrep_gap_bezout_backward_result_coefficient_difference_witness_outputfirstoutside+(pfaa_length_bezout_backward_result_coefficient_difference)=(pfrep_power_bezout_backward_result_coefficient_difference_witness_output)) /\\ (((pfrep_left_bezout_backward_result_coefficient_difference_witness_output)=0))))) -> ((exists pfrep_position_bezout_backward_result_coefficient_difference_witness_outputsecond. ((pfrep_position_bezout_backward_result_coefficient_difference_witness_outputsecond+S (pfrep_power_bezout_backward_result_coefficient_difference_witness_output)=(Lu)) /\\ ((((exists ff_h_pfp_bezout_backward_result_coefficient_difference_witness_outputsecondentry. ff_h_pfp_bezout_backward_result_coefficient_difference_witness_outputsecondentry + S (pfrep_right_bezout_backward_result_coefficient_difference_witness_output) = S ((S (pfrep_position_bezout_backward_result_coefficient_difference_witness_outputsecond)) * uc)) /\\ exists ff_q_pfp_bezout_backward_result_coefficient_difference_witness_outputsecondentry. ub = ff_q_pfp_bezout_backward_result_coefficient_difference_witness_outputsecondentry * S ((S (pfrep_position_bezout_backward_result_coefficient_difference_witness_outputsecond)) * uc) + (pfrep_right_bezout_backward_result_coefficient_difference_witness_output)))))) \\/ (((exists pfrep_gap_bezout_backward_result_coefficient_difference_witness_outputsecondoutside. pfrep_gap_bezout_backward_result_coefficient_difference_witness_outputsecondoutside+(Lu)=(pfrep_power_bezout_backward_result_coefficient_difference_witness_output)) /\\ (((pfrep_right_bezout_backward_result_coefficient_difference_witness_output)=0))))) -> pfrep_left_bezout_backward_result_coefficient_difference_witness_output=pfrep_right_bezout_backward_result_coefficient_difference_witness_output))))))))))))) /\\ ((exists pfbz_left_code_bezout_backward_result_bezout pfbz_left_scale_bezout_backward_result_bezout pfbz_left_length_bezout_backward_result_bezout pfbz_right_code_bezout_backward_result_bezout pfbz_right_scale_bezout_backward_result_bezout pfbz_right_length_bezout_backward_result_bezout. ((((forall fom_index_pfp_bezout_backward_result_bezout_left_productleft. (exists fom_gap_pfp_bezout_backward_result_bezout_left_productleft_index_bound. fom_gap_pfp_bezout_backward_result_bezout_left_productleft_index_bound + S (fom_index_pfp_bezout_backward_result_bezout_left_productleft) = Lv) -> exists fom_value_pfp_bezout_backward_result_bezout_left_productleft. ((((exists fom_beta_height_pfp_bezout_backward_result_bezout_left_productleft_entry. fom_beta_height_pfp_bezout_backward_result_bezout_left_productleft_entry + S (fom_value_pfp_bezout_backward_result_bezout_left_productleft) = S ((S (fom_index_pfp_bezout_backward_result_bezout_left_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_bezout_backward_result_bezout_left_productleft_entry. vb = fom_beta_quotient_pfp_bezout_backward_result_bezout_left_productleft_entry * S ((S (fom_index_pfp_bezout_backward_result_bezout_left_productleft)) * vc) + (fom_value_pfp_bezout_backward_result_bezout_left_productleft))) /\\ (exists fom_gap_pfp_bezout_backward_result_bezout_left_productleft_value_bound. fom_gap_pfp_bezout_backward_result_bezout_left_productleft_value_bound + S (fom_value_pfp_bezout_backward_result_bezout_left_productleft) = p))) /\\ (((forall fom_index_pfp_bezout_backward_result_bezout_left_productright. (exists fom_gap_pfp_bezout_backward_result_bezout_left_productright_index_bound. fom_gap_pfp_bezout_backward_result_bezout_left_productright_index_bound + S (fom_index_pfp_bezout_backward_result_bezout_left_productright) = La) -> exists fom_value_pfp_bezout_backward_result_bezout_left_productright. ((((exists fom_beta_height_pfp_bezout_backward_result_bezout_left_productright_entry. fom_beta_height_pfp_bezout_backward_result_bezout_left_productright_entry + S (fom_value_pfp_bezout_backward_result_bezout_left_productright) = S ((S (fom_index_pfp_bezout_backward_result_bezout_left_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_bezout_backward_result_bezout_left_productright_entry. ab = fom_beta_quotient_pfp_bezout_backward_result_bezout_left_productright_entry * S ((S (fom_index_pfp_bezout_backward_result_bezout_left_productright)) * ac) + (fom_value_pfp_bezout_backward_result_bezout_left_productright))) /\\ (exists fom_gap_pfp_bezout_backward_result_bezout_left_productright_value_bound. fom_gap_pfp_bezout_backward_result_bezout_left_productright_value_bound + S (fom_value_pfp_bezout_backward_result_bezout_left_productright) = p))) /\\ (((((((Lv)=0 \\/ (La)=0) /\\ (((pfbz_left_length_bezout_backward_result_bezout)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((La)=0)) /\\ (((Lv)+(La)=S (pfbz_left_length_bezout_backward_result_bezout)))))))) /\\ ((forall pfc_index_bezout_backward_result_bezout_left_productcoefficients. (exists pfa_gap_bezout_backward_result_bezout_left_productcoefficientsbound. pfa_gap_bezout_backward_result_bezout_left_productcoefficientsbound + S (pfc_index_bezout_backward_result_bezout_left_productcoefficients) = (pfbz_left_length_bezout_backward_result_bezout)) -> exists pfc_value_bezout_backward_result_bezout_left_productcoefficients. ((((exists ff_h_pfp_bezout_backward_result_bezout_left_productcoefficientsentry. ff_h_pfp_bezout_backward_result_bezout_left_productcoefficientsentry + S (pfc_value_bezout_backward_result_bezout_left_productcoefficients) = S ((S (pfc_index_bezout_backward_result_bezout_left_productcoefficients)) * pfbz_left_scale_bezout_backward_result_bezout)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_left_productcoefficientsentry. pfbz_left_code_bezout_backward_result_bezout = ff_q_pfp_bezout_backward_result_bezout_left_productcoefficientsentry * S ((S (pfc_index_bezout_backward_result_bezout_left_productcoefficients)) * pfbz_left_scale_bezout_backward_result_bezout) + (pfc_value_bezout_backward_result_bezout_left_productcoefficients))) /\\ ((exists pfc_terms_code_bezout_backward_result_bezout_left_productcoefficientscoefficient pfc_terms_scale_bezout_backward_result_bezout_left_productcoefficientscoefficient pfc_natural_sum_bezout_backward_result_bezout_left_productcoefficientscoefficient. ((forall pfc_index_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalbound. pfa_gap_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_bezout_backward_result_bezout_left_productcoefficients))) -> exists pfc_value_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bezout_backward_result_bezout_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_bezout_backward_result_bezout_left_productcoefficientscoefficient = ff_q_pfp_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bezout_backward_result_bezout_left_productcoefficientscoefficient) + (pfc_value_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm pfc_left_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm pfc_right_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonal)+pfc_complement_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm=(pfc_index_bezout_backward_result_bezout_left_productcoefficients)) /\\ ((((((exists pfa_gap_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm) = (La)) /\\ ((((exists ff_h_pfp_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightoutside+(La)=(pfc_complement_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonal)=pfc_left_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm*pfc_right_bezout_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum fs_v_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_bezout_backward_result_bezout_left_productcoefficientscoefficient) = S ((S (S (pfc_index_bezout_backward_result_bezout_left_productcoefficients))) * fs_v_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_bezout_backward_result_bezout_left_productcoefficients))) * fs_v_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum) + (pfc_natural_sum_bezout_backward_result_bezout_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_bezout_backward_result_bezout_left_productcoefficients)) -> exists fs_a_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bezout_backward_result_bezout_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_bezout_backward_result_bezout_left_productcoefficientscoefficient = fs_q_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bezout_backward_result_bezout_left_productcoefficientscoefficient) + (fs_a_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum) + (fs_r_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum) + (fs_s_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_bezout_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_bezout_backward_result_bezout_left_productcoefficientscoefficientresiduebound. pfa_gap_bezout_backward_result_bezout_left_productcoefficientscoefficientresiduebound + S (pfc_value_bezout_backward_result_bezout_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_bezout_backward_result_bezout_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_bezout_backward_result_bezout_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_bezout_backward_result_bezout_left_productcoefficientscoefficient) + (p) * pfa_offset_left_bezout_backward_result_bezout_left_productcoefficientscoefficientresiduecongruence = (pfc_value_bezout_backward_result_bezout_left_productcoefficients) + (p) * pfa_offset_right_bezout_backward_result_bezout_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_bezout_backward_result_bezout_right_productleft. (exists fom_gap_pfp_bezout_backward_result_bezout_right_productleft_index_bound. fom_gap_pfp_bezout_backward_result_bezout_right_productleft_index_bound + S (fom_index_pfp_bezout_backward_result_bezout_right_productleft) = pfbz_update_T_bezout_backward_result) -> exists fom_value_pfp_bezout_backward_result_bezout_right_productleft. ((((exists fom_beta_height_pfp_bezout_backward_result_bezout_right_productleft_entry. fom_beta_height_pfp_bezout_backward_result_bezout_right_productleft_entry + S (fom_value_pfp_bezout_backward_result_bezout_right_productleft) = S ((S (fom_index_pfp_bezout_backward_result_bezout_right_productleft)) * pfbz_update_tc_bezout_backward_result)) /\\ exists fom_beta_quotient_pfp_bezout_backward_result_bezout_right_productleft_entry. pfbz_update_tb_bezout_backward_result = fom_beta_quotient_pfp_bezout_backward_result_bezout_right_productleft_entry * S ((S (fom_index_pfp_bezout_backward_result_bezout_right_productleft)) * pfbz_update_tc_bezout_backward_result) + (fom_value_pfp_bezout_backward_result_bezout_right_productleft))) /\\ (exists fom_gap_pfp_bezout_backward_result_bezout_right_productleft_value_bound. fom_gap_pfp_bezout_backward_result_bezout_right_productleft_value_bound + S (fom_value_pfp_bezout_backward_result_bezout_right_productleft) = p))) /\\ (((forall fom_index_pfp_bezout_backward_result_bezout_right_productright. (exists fom_gap_pfp_bezout_backward_result_bezout_right_productright_index_bound. fom_gap_pfp_bezout_backward_result_bezout_right_productright_index_bound + S (fom_index_pfp_bezout_backward_result_bezout_right_productright) = Lb) -> exists fom_value_pfp_bezout_backward_result_bezout_right_productright. ((((exists fom_beta_height_pfp_bezout_backward_result_bezout_right_productright_entry. fom_beta_height_pfp_bezout_backward_result_bezout_right_productright_entry + S (fom_value_pfp_bezout_backward_result_bezout_right_productright) = S ((S (fom_index_pfp_bezout_backward_result_bezout_right_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_bezout_backward_result_bezout_right_productright_entry. bb = fom_beta_quotient_pfp_bezout_backward_result_bezout_right_productright_entry * S ((S (fom_index_pfp_bezout_backward_result_bezout_right_productright)) * bc) + (fom_value_pfp_bezout_backward_result_bezout_right_productright))) /\\ (exists fom_gap_pfp_bezout_backward_result_bezout_right_productright_value_bound. fom_gap_pfp_bezout_backward_result_bezout_right_productright_value_bound + S (fom_value_pfp_bezout_backward_result_bezout_right_productright) = p))) /\\ (((((((pfbz_update_T_bezout_backward_result)=0 \\/ (Lb)=0) /\\ (((pfbz_right_length_bezout_backward_result_bezout)=0)))) \\/ (((~((pfbz_update_T_bezout_backward_result)=0)) /\\ (((~((Lb)=0)) /\\ (((pfbz_update_T_bezout_backward_result)+(Lb)=S (pfbz_right_length_bezout_backward_result_bezout)))))))) /\\ ((forall pfc_index_bezout_backward_result_bezout_right_productcoefficients. (exists pfa_gap_bezout_backward_result_bezout_right_productcoefficientsbound. pfa_gap_bezout_backward_result_bezout_right_productcoefficientsbound + S (pfc_index_bezout_backward_result_bezout_right_productcoefficients) = (pfbz_right_length_bezout_backward_result_bezout)) -> exists pfc_value_bezout_backward_result_bezout_right_productcoefficients. ((((exists ff_h_pfp_bezout_backward_result_bezout_right_productcoefficientsentry. ff_h_pfp_bezout_backward_result_bezout_right_productcoefficientsentry + S (pfc_value_bezout_backward_result_bezout_right_productcoefficients) = S ((S (pfc_index_bezout_backward_result_bezout_right_productcoefficients)) * pfbz_right_scale_bezout_backward_result_bezout)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_right_productcoefficientsentry. pfbz_right_code_bezout_backward_result_bezout = ff_q_pfp_bezout_backward_result_bezout_right_productcoefficientsentry * S ((S (pfc_index_bezout_backward_result_bezout_right_productcoefficients)) * pfbz_right_scale_bezout_backward_result_bezout) + (pfc_value_bezout_backward_result_bezout_right_productcoefficients))) /\\ ((exists pfc_terms_code_bezout_backward_result_bezout_right_productcoefficientscoefficient pfc_terms_scale_bezout_backward_result_bezout_right_productcoefficientscoefficient pfc_natural_sum_bezout_backward_result_bezout_right_productcoefficientscoefficient. ((forall pfc_index_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalbound. pfa_gap_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_bezout_backward_result_bezout_right_productcoefficients))) -> exists pfc_value_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bezout_backward_result_bezout_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_bezout_backward_result_bezout_right_productcoefficientscoefficient = ff_q_pfp_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bezout_backward_result_bezout_right_productcoefficientscoefficient) + (pfc_value_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm pfc_left_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm pfc_right_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonal)+pfc_complement_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm=(pfc_index_bezout_backward_result_bezout_right_productcoefficients)) /\\ ((((((exists pfa_gap_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonal) = (pfbz_update_T_bezout_backward_result)) /\\ ((((exists ff_h_pfp_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonal)) * pfbz_update_tc_bezout_backward_result)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftentry. pfbz_update_tb_bezout_backward_result = ff_q_pfp_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonal)) * pfbz_update_tc_bezout_backward_result) + (pfc_left_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfbz_update_T_bezout_backward_result)=(pfc_index_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm) = (Lb)) /\\ ((((exists ff_h_pfp_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightoutside+(Lb)=(pfc_complement_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonal)=pfc_left_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm*pfc_right_bezout_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum fs_v_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_bezout_backward_result_bezout_right_productcoefficientscoefficient) = S ((S (S (pfc_index_bezout_backward_result_bezout_right_productcoefficients))) * fs_v_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_bezout_backward_result_bezout_right_productcoefficients))) * fs_v_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum) + (pfc_natural_sum_bezout_backward_result_bezout_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_bezout_backward_result_bezout_right_productcoefficients)) -> exists fs_a_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bezout_backward_result_bezout_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_bezout_backward_result_bezout_right_productcoefficientscoefficient = fs_q_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bezout_backward_result_bezout_right_productcoefficientscoefficient) + (fs_a_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum) + (fs_r_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum = fs_q_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum) + (fs_s_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_bezout_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_bezout_backward_result_bezout_right_productcoefficientscoefficientresiduebound. pfa_gap_bezout_backward_result_bezout_right_productcoefficientscoefficientresiduebound + S (pfc_value_bezout_backward_result_bezout_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_bezout_backward_result_bezout_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_bezout_backward_result_bezout_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_bezout_backward_result_bezout_right_productcoefficientscoefficient) + (p) * pfa_offset_left_bezout_backward_result_bezout_right_productcoefficientscoefficientresiduecongruence = (pfc_value_bezout_backward_result_bezout_right_productcoefficients) + (p) * pfa_offset_right_bezout_backward_result_bezout_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_bezout_backward_result_bezout_sum_left_bounded. (exists fom_gap_pfp_bezout_backward_result_bezout_sum_left_bounded_index_bound. fom_gap_pfp_bezout_backward_result_bezout_sum_left_bounded_index_bound + S (fom_index_pfp_bezout_backward_result_bezout_sum_left_bounded) = pfbz_left_length_bezout_backward_result_bezout) -> exists fom_value_pfp_bezout_backward_result_bezout_sum_left_bounded. ((((exists fom_beta_height_pfp_bezout_backward_result_bezout_sum_left_bounded_entry. fom_beta_height_pfp_bezout_backward_result_bezout_sum_left_bounded_entry + S (fom_value_pfp_bezout_backward_result_bezout_sum_left_bounded) = S ((S (fom_index_pfp_bezout_backward_result_bezout_sum_left_bounded)) * pfbz_left_scale_bezout_backward_result_bezout)) /\\ exists fom_beta_quotient_pfp_bezout_backward_result_bezout_sum_left_bounded_entry. pfbz_left_code_bezout_backward_result_bezout = fom_beta_quotient_pfp_bezout_backward_result_bezout_sum_left_bounded_entry * S ((S (fom_index_pfp_bezout_backward_result_bezout_sum_left_bounded)) * pfbz_left_scale_bezout_backward_result_bezout) + (fom_value_pfp_bezout_backward_result_bezout_sum_left_bounded))) /\\ (exists fom_gap_pfp_bezout_backward_result_bezout_sum_left_bounded_value_bound. fom_gap_pfp_bezout_backward_result_bezout_sum_left_bounded_value_bound + S (fom_value_pfp_bezout_backward_result_bezout_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_bezout_backward_result_bezout_sum_right_bounded. (exists fom_gap_pfp_bezout_backward_result_bezout_sum_right_bounded_index_bound. fom_gap_pfp_bezout_backward_result_bezout_sum_right_bounded_index_bound + S (fom_index_pfp_bezout_backward_result_bezout_sum_right_bounded) = pfbz_right_length_bezout_backward_result_bezout) -> exists fom_value_pfp_bezout_backward_result_bezout_sum_right_bounded. ((((exists fom_beta_height_pfp_bezout_backward_result_bezout_sum_right_bounded_entry. fom_beta_height_pfp_bezout_backward_result_bezout_sum_right_bounded_entry + S (fom_value_pfp_bezout_backward_result_bezout_sum_right_bounded) = S ((S (fom_index_pfp_bezout_backward_result_bezout_sum_right_bounded)) * pfbz_right_scale_bezout_backward_result_bezout)) /\\ exists fom_beta_quotient_pfp_bezout_backward_result_bezout_sum_right_bounded_entry. pfbz_right_code_bezout_backward_result_bezout = fom_beta_quotient_pfp_bezout_backward_result_bezout_sum_right_bounded_entry * S ((S (fom_index_pfp_bezout_backward_result_bezout_sum_right_bounded)) * pfbz_right_scale_bezout_backward_result_bezout) + (fom_value_pfp_bezout_backward_result_bezout_sum_right_bounded))) /\\ (exists fom_gap_pfp_bezout_backward_result_bezout_sum_right_bounded_value_bound. fom_gap_pfp_bezout_backward_result_bezout_sum_right_bounded_value_bound + S (fom_value_pfp_bezout_backward_result_bezout_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_bezout_backward_result_bezout_sum_result_bounded. (exists fom_gap_pfp_bezout_backward_result_bezout_sum_result_bounded_index_bound. fom_gap_pfp_bezout_backward_result_bezout_sum_result_bounded_index_bound + S (fom_index_pfp_bezout_backward_result_bezout_sum_result_bounded) = Lg) -> exists fom_value_pfp_bezout_backward_result_bezout_sum_result_bounded. ((((exists fom_beta_height_pfp_bezout_backward_result_bezout_sum_result_bounded_entry. fom_beta_height_pfp_bezout_backward_result_bezout_sum_result_bounded_entry + S (fom_value_pfp_bezout_backward_result_bezout_sum_result_bounded) = S ((S (fom_index_pfp_bezout_backward_result_bezout_sum_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_bezout_backward_result_bezout_sum_result_bounded_entry. gb = fom_beta_quotient_pfp_bezout_backward_result_bezout_sum_result_bounded_entry * S ((S (fom_index_pfp_bezout_backward_result_bezout_sum_result_bounded)) * gc) + (fom_value_pfp_bezout_backward_result_bezout_sum_result_bounded))) /\\ (exists fom_gap_pfp_bezout_backward_result_bezout_sum_result_bounded_value_bound. fom_gap_pfp_bezout_backward_result_bezout_sum_result_bounded_value_bound + S (fom_value_pfp_bezout_backward_result_bezout_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_bezout_backward_result_bezout_sum pfaa_left_c_bezout_backward_result_bezout_sum pfaa_right_b_bezout_backward_result_bezout_sum pfaa_right_c_bezout_backward_result_bezout_sum pfaa_sum_b_bezout_backward_result_bezout_sum pfaa_sum_c_bezout_backward_result_bezout_sum pfaa_length_bezout_backward_result_bezout_sum. ((((forall pfrep_power_bezout_backward_result_bezout_sum_witness_common_left pfrep_left_bezout_backward_result_bezout_sum_witness_common_left pfrep_right_bezout_backward_result_bezout_sum_witness_common_left. ((exists pfrep_position_bezout_backward_result_bezout_sum_witness_common_leftfirst. ((pfrep_position_bezout_backward_result_bezout_sum_witness_common_leftfirst+S (pfrep_power_bezout_backward_result_bezout_sum_witness_common_left)=(pfbz_left_length_bezout_backward_result_bezout)) /\\ ((((exists ff_h_pfp_bezout_backward_result_bezout_sum_witness_common_leftfirstentry. ff_h_pfp_bezout_backward_result_bezout_sum_witness_common_leftfirstentry + S (pfrep_left_bezout_backward_result_bezout_sum_witness_common_left) = S ((S (pfrep_position_bezout_backward_result_bezout_sum_witness_common_leftfirst)) * pfbz_left_scale_bezout_backward_result_bezout)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_sum_witness_common_leftfirstentry. pfbz_left_code_bezout_backward_result_bezout = ff_q_pfp_bezout_backward_result_bezout_sum_witness_common_leftfirstentry * S ((S (pfrep_position_bezout_backward_result_bezout_sum_witness_common_leftfirst)) * pfbz_left_scale_bezout_backward_result_bezout) + (pfrep_left_bezout_backward_result_bezout_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_bezout_backward_result_bezout_sum_witness_common_leftfirstoutside. pfrep_gap_bezout_backward_result_bezout_sum_witness_common_leftfirstoutside+(pfbz_left_length_bezout_backward_result_bezout)=(pfrep_power_bezout_backward_result_bezout_sum_witness_common_left)) /\\ (((pfrep_left_bezout_backward_result_bezout_sum_witness_common_left)=0))))) -> ((exists pfrep_position_bezout_backward_result_bezout_sum_witness_common_leftsecond. ((pfrep_position_bezout_backward_result_bezout_sum_witness_common_leftsecond+S (pfrep_power_bezout_backward_result_bezout_sum_witness_common_left)=(pfaa_length_bezout_backward_result_bezout_sum)) /\\ ((((exists ff_h_pfp_bezout_backward_result_bezout_sum_witness_common_leftsecondentry. ff_h_pfp_bezout_backward_result_bezout_sum_witness_common_leftsecondentry + S (pfrep_right_bezout_backward_result_bezout_sum_witness_common_left) = S ((S (pfrep_position_bezout_backward_result_bezout_sum_witness_common_leftsecond)) * pfaa_left_c_bezout_backward_result_bezout_sum)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_sum_witness_common_leftsecondentry. pfaa_left_b_bezout_backward_result_bezout_sum = ff_q_pfp_bezout_backward_result_bezout_sum_witness_common_leftsecondentry * S ((S (pfrep_position_bezout_backward_result_bezout_sum_witness_common_leftsecond)) * pfaa_left_c_bezout_backward_result_bezout_sum) + (pfrep_right_bezout_backward_result_bezout_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_bezout_backward_result_bezout_sum_witness_common_leftsecondoutside. pfrep_gap_bezout_backward_result_bezout_sum_witness_common_leftsecondoutside+(pfaa_length_bezout_backward_result_bezout_sum)=(pfrep_power_bezout_backward_result_bezout_sum_witness_common_left)) /\\ (((pfrep_right_bezout_backward_result_bezout_sum_witness_common_left)=0))))) -> pfrep_left_bezout_backward_result_bezout_sum_witness_common_left=pfrep_right_bezout_backward_result_bezout_sum_witness_common_left) /\\ ((forall pfrep_power_bezout_backward_result_bezout_sum_witness_common_right pfrep_left_bezout_backward_result_bezout_sum_witness_common_right pfrep_right_bezout_backward_result_bezout_sum_witness_common_right. ((exists pfrep_position_bezout_backward_result_bezout_sum_witness_common_rightfirst. ((pfrep_position_bezout_backward_result_bezout_sum_witness_common_rightfirst+S (pfrep_power_bezout_backward_result_bezout_sum_witness_common_right)=(pfbz_right_length_bezout_backward_result_bezout)) /\\ ((((exists ff_h_pfp_bezout_backward_result_bezout_sum_witness_common_rightfirstentry. ff_h_pfp_bezout_backward_result_bezout_sum_witness_common_rightfirstentry + S (pfrep_left_bezout_backward_result_bezout_sum_witness_common_right) = S ((S (pfrep_position_bezout_backward_result_bezout_sum_witness_common_rightfirst)) * pfbz_right_scale_bezout_backward_result_bezout)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_sum_witness_common_rightfirstentry. pfbz_right_code_bezout_backward_result_bezout = ff_q_pfp_bezout_backward_result_bezout_sum_witness_common_rightfirstentry * S ((S (pfrep_position_bezout_backward_result_bezout_sum_witness_common_rightfirst)) * pfbz_right_scale_bezout_backward_result_bezout) + (pfrep_left_bezout_backward_result_bezout_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_bezout_backward_result_bezout_sum_witness_common_rightfirstoutside. pfrep_gap_bezout_backward_result_bezout_sum_witness_common_rightfirstoutside+(pfbz_right_length_bezout_backward_result_bezout)=(pfrep_power_bezout_backward_result_bezout_sum_witness_common_right)) /\\ (((pfrep_left_bezout_backward_result_bezout_sum_witness_common_right)=0))))) -> ((exists pfrep_position_bezout_backward_result_bezout_sum_witness_common_rightsecond. ((pfrep_position_bezout_backward_result_bezout_sum_witness_common_rightsecond+S (pfrep_power_bezout_backward_result_bezout_sum_witness_common_right)=(pfaa_length_bezout_backward_result_bezout_sum)) /\\ ((((exists ff_h_pfp_bezout_backward_result_bezout_sum_witness_common_rightsecondentry. ff_h_pfp_bezout_backward_result_bezout_sum_witness_common_rightsecondentry + S (pfrep_right_bezout_backward_result_bezout_sum_witness_common_right) = S ((S (pfrep_position_bezout_backward_result_bezout_sum_witness_common_rightsecond)) * pfaa_right_c_bezout_backward_result_bezout_sum)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_sum_witness_common_rightsecondentry. pfaa_right_b_bezout_backward_result_bezout_sum = ff_q_pfp_bezout_backward_result_bezout_sum_witness_common_rightsecondentry * S ((S (pfrep_position_bezout_backward_result_bezout_sum_witness_common_rightsecond)) * pfaa_right_c_bezout_backward_result_bezout_sum) + (pfrep_right_bezout_backward_result_bezout_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_bezout_backward_result_bezout_sum_witness_common_rightsecondoutside. pfrep_gap_bezout_backward_result_bezout_sum_witness_common_rightsecondoutside+(pfaa_length_bezout_backward_result_bezout_sum)=(pfrep_power_bezout_backward_result_bezout_sum_witness_common_right)) /\\ (((pfrep_right_bezout_backward_result_bezout_sum_witness_common_right)=0))))) -> pfrep_left_bezout_backward_result_bezout_sum_witness_common_right=pfrep_right_bezout_backward_result_bezout_sum_witness_common_right)))) /\\ (((forall pfp_index_bezout_backward_result_bezout_sum_witness_operation. (exists pfa_gap_bezout_backward_result_bezout_sum_witness_operationindex. pfa_gap_bezout_backward_result_bezout_sum_witness_operationindex + S (pfp_index_bezout_backward_result_bezout_sum_witness_operation) = (pfaa_length_bezout_backward_result_bezout_sum)) -> exists pfp_left_bezout_backward_result_bezout_sum_witness_operation pfp_right_bezout_backward_result_bezout_sum_witness_operation pfp_value_bezout_backward_result_bezout_sum_witness_operation. ((((exists ff_h_pfp_bezout_backward_result_bezout_sum_witness_operationleft. ff_h_pfp_bezout_backward_result_bezout_sum_witness_operationleft + S (pfp_left_bezout_backward_result_bezout_sum_witness_operation) = S ((S (pfp_index_bezout_backward_result_bezout_sum_witness_operation)) * pfaa_left_c_bezout_backward_result_bezout_sum)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_sum_witness_operationleft. pfaa_left_b_bezout_backward_result_bezout_sum = ff_q_pfp_bezout_backward_result_bezout_sum_witness_operationleft * S ((S (pfp_index_bezout_backward_result_bezout_sum_witness_operation)) * pfaa_left_c_bezout_backward_result_bezout_sum) + (pfp_left_bezout_backward_result_bezout_sum_witness_operation))) /\\ (((((exists ff_h_pfp_bezout_backward_result_bezout_sum_witness_operationright. ff_h_pfp_bezout_backward_result_bezout_sum_witness_operationright + S (pfp_right_bezout_backward_result_bezout_sum_witness_operation) = S ((S (pfp_index_bezout_backward_result_bezout_sum_witness_operation)) * pfaa_right_c_bezout_backward_result_bezout_sum)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_sum_witness_operationright. pfaa_right_b_bezout_backward_result_bezout_sum = ff_q_pfp_bezout_backward_result_bezout_sum_witness_operationright * S ((S (pfp_index_bezout_backward_result_bezout_sum_witness_operation)) * pfaa_right_c_bezout_backward_result_bezout_sum) + (pfp_right_bezout_backward_result_bezout_sum_witness_operation))) /\\ (((((exists ff_h_pfp_bezout_backward_result_bezout_sum_witness_operationtarget. ff_h_pfp_bezout_backward_result_bezout_sum_witness_operationtarget + S (pfp_value_bezout_backward_result_bezout_sum_witness_operation) = S ((S (pfp_index_bezout_backward_result_bezout_sum_witness_operation)) * pfaa_sum_c_bezout_backward_result_bezout_sum)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_sum_witness_operationtarget. pfaa_sum_b_bezout_backward_result_bezout_sum = ff_q_pfp_bezout_backward_result_bezout_sum_witness_operationtarget * S ((S (pfp_index_bezout_backward_result_bezout_sum_witness_operation)) * pfaa_sum_c_bezout_backward_result_bezout_sum) + (pfp_value_bezout_backward_result_bezout_sum_witness_operation))) /\\ ((((exists pfa_gap_bezout_backward_result_bezout_sum_witness_operationoperationleft. pfa_gap_bezout_backward_result_bezout_sum_witness_operationoperationleft + S (pfp_left_bezout_backward_result_bezout_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_bezout_backward_result_bezout_sum_witness_operationoperationright. pfa_gap_bezout_backward_result_bezout_sum_witness_operationoperationright + S (pfp_right_bezout_backward_result_bezout_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_bezout_backward_result_bezout_sum_witness_operationoperationresultbound. pfa_gap_bezout_backward_result_bezout_sum_witness_operationoperationresultbound + S (pfp_value_bezout_backward_result_bezout_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_bezout_backward_result_bezout_sum_witness_operationoperationresultcongruence pfa_offset_right_bezout_backward_result_bezout_sum_witness_operationoperationresultcongruence. ((pfp_left_bezout_backward_result_bezout_sum_witness_operation) + (pfp_right_bezout_backward_result_bezout_sum_witness_operation)) + (p) * pfa_offset_left_bezout_backward_result_bezout_sum_witness_operationoperationresultcongruence = (pfp_value_bezout_backward_result_bezout_sum_witness_operation) + (p) * pfa_offset_right_bezout_backward_result_bezout_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_bezout_backward_result_bezout_sum_witness_output pfrep_left_bezout_backward_result_bezout_sum_witness_output pfrep_right_bezout_backward_result_bezout_sum_witness_output. ((exists pfrep_position_bezout_backward_result_bezout_sum_witness_outputfirst. ((pfrep_position_bezout_backward_result_bezout_sum_witness_outputfirst+S (pfrep_power_bezout_backward_result_bezout_sum_witness_output)=(pfaa_length_bezout_backward_result_bezout_sum)) /\\ ((((exists ff_h_pfp_bezout_backward_result_bezout_sum_witness_outputfirstentry. ff_h_pfp_bezout_backward_result_bezout_sum_witness_outputfirstentry + S (pfrep_left_bezout_backward_result_bezout_sum_witness_output) = S ((S (pfrep_position_bezout_backward_result_bezout_sum_witness_outputfirst)) * pfaa_sum_c_bezout_backward_result_bezout_sum)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_sum_witness_outputfirstentry. pfaa_sum_b_bezout_backward_result_bezout_sum = ff_q_pfp_bezout_backward_result_bezout_sum_witness_outputfirstentry * S ((S (pfrep_position_bezout_backward_result_bezout_sum_witness_outputfirst)) * pfaa_sum_c_bezout_backward_result_bezout_sum) + (pfrep_left_bezout_backward_result_bezout_sum_witness_output)))))) \\/ (((exists pfrep_gap_bezout_backward_result_bezout_sum_witness_outputfirstoutside. pfrep_gap_bezout_backward_result_bezout_sum_witness_outputfirstoutside+(pfaa_length_bezout_backward_result_bezout_sum)=(pfrep_power_bezout_backward_result_bezout_sum_witness_output)) /\\ (((pfrep_left_bezout_backward_result_bezout_sum_witness_output)=0))))) -> ((exists pfrep_position_bezout_backward_result_bezout_sum_witness_outputsecond. ((pfrep_position_bezout_backward_result_bezout_sum_witness_outputsecond+S (pfrep_power_bezout_backward_result_bezout_sum_witness_output)=(Lg)) /\\ ((((exists ff_h_pfp_bezout_backward_result_bezout_sum_witness_outputsecondentry. ff_h_pfp_bezout_backward_result_bezout_sum_witness_outputsecondentry + S (pfrep_right_bezout_backward_result_bezout_sum_witness_output) = S ((S (pfrep_position_bezout_backward_result_bezout_sum_witness_outputsecond)) * gc)) /\\ exists ff_q_pfp_bezout_backward_result_bezout_sum_witness_outputsecondentry. gb = ff_q_pfp_bezout_backward_result_bezout_sum_witness_outputsecondentry * S ((S (pfrep_position_bezout_backward_result_bezout_sum_witness_outputsecond)) * gc) + (pfrep_right_bezout_backward_result_bezout_sum_witness_output)))))) \\/ (((exists pfrep_gap_bezout_backward_result_bezout_sum_witness_outputsecondoutside. pfrep_gap_bezout_backward_result_bezout_sum_witness_outputsecondoutside+(Lg)=(pfrep_power_bezout_backward_result_bezout_sum_witness_output)) /\\ (((pfrep_right_bezout_backward_result_bezout_sum_witness_output)=0))))) -> pfrep_left_bezout_backward_result_bezout_sum_witness_output=pfrep_right_bezout_backward_result_bezout_sum_witness_output)))))))))))))))))))))))", "statement_sha256": "97de166d816f36b6f5ddee9433d7c55b0f4649dd0d4782d620d26fafeef4652b", "summary": "Construct W=V*Q, T=U-W, and genuine new products to turn an actual Bezout representation for (B,R) into one for (A,B), with both coefficient-update graphs returned as witnesses.", "summary_sha256": "f814a0cc3ef2dca8d410d0e15ebbe8da6eee49802abd3bd8b9b4fd9e68302862" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "bcd80181a8680b3d30280520e1dffc3684f4a330f8769a7255cc03aaf36084cf", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 442, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "c3903482000c957ac77f84a43a85d135e4caa19e4484328035f91b82cbf3a702", "test_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85" }, "body_checked": true, "body_receipt": { "command_count": 76, "dependency_count": 2, "dne_command_count": 0, "name": "prime_field_polynomial_division_execution_bezout_backward", "proof_depth": 60, "proof_edges": 85, "proof_nodes": 86, "proof_objects": 86, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "prime_field_polynomial_division_execution_aligned_identity", "prime_field_polynomial_bezout_euclidean_backward" ], "dependencies_sha256": "cf2b45e32c22586fe997a50642d98ac2605d96fa72d76f0bb2269c3d5a9527bc", "empty_context_closure": { "body_proof_depth": 60, "body_proof_nodes": 86, "bundle_campaign": "polynomial-gcd-bezout", "bundle_dependency_edge_count": 1578, "bundle_node_count": 493, "bundle_node_id": 442, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "bundle_root_id": 492, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "0308b3b63fea32baa1eb473222ffa0dd93c119908c6c324191e08969d0d7b76c", "status": "checked" }, "enrollment_index": 4186, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "c3903482000c957ac77f84a43a85d135e4caa19e4484328035f91b82cbf3a702", "kind": "alpha_v34_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_bezout_backward_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85", "kind": "alpha_v34_frontier_executable_audit", "path": "peano-lab/py/tests/test_campaign_research_v34_closure.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "kind": "alpha_v34_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-field-polynomial-gcd-bezout-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "kind": "alpha_v34_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=442]" }, { "document_sha256": "81943a545e7b9f667ad758c24de14bb47280e8cf3fbb3ca714efec55fb096408", "kind": "alpha_v34_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "537897abd3c66a55172a725c42c9bae9f2c742c9a0c75490df40f4846fb3b6a9", "membership": "alpha_only", "name": "prime_field_polynomial_division_execution_bezout_backward", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro La", "intro bb", "intro bc", "intro d", "intro qb", "intro qc", "intro Lq", "intro rb", "intro rc", "intro Lr", "intro gb", "intro gc", "intro Lg", "intro ub", "intro uc", "intro Lu", "intro vb", "intro vc", "intro Lv", "intro hp", "intro he", "intro hb", "have hi : exists pb pc N. ((((forall fom_index_pfp_execution_backward_productleft. (exists fom_gap_pfp_execution_backward_productleft_index_bound. fom_gap_pfp_execution_backward_productleft_index_bound + S (fom_index_pfp_execution_backward_productleft) = Lq) -> exists fom_value_pfp_execution_backward_productleft. ((((exists fom_beta_height_pfp_execution_backward_productleft_entry. fom_beta_height_pfp_execution_backward_productleft_entry + S (fom_value_pfp_execution_backward_productleft) = S ((S (fom_index_pfp_execution_backward_productleft)) * qc)) /\\ exists fom_beta_quotient_pfp_execution_backward_productleft_entry. qb = fom_beta_quotient_pfp_execution_backward_productleft_entry * S ((S (fom_index_pfp_execution_backward_productleft)) * qc) + (fom_value_pfp_execution_backward_productleft))) /\\ (exists fom_gap_pfp_execution_backward_productleft_value_bound. fom_gap_pfp_execution_backward_productleft_value_bound + S (fom_value_pfp_execution_backward_productleft) = p))) /\\ (((forall fom_index_pfp_execution_backward_productright. (exists fom_gap_pfp_execution_backward_productright_index_bound. fom_gap_pfp_execution_backward_productright_index_bound + S (fom_index_pfp_execution_backward_productright) = S d) -> exists fom_value_pfp_execution_backward_productright. ((((exists fom_beta_height_pfp_execution_backward_productright_entry. fom_beta_height_pfp_execution_backward_productright_entry + S (fom_value_pfp_execution_backward_productright) = S ((S (fom_index_pfp_execution_backward_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_backward_productright_entry. bb = fom_beta_quotient_pfp_execution_backward_productright_entry * S ((S (fom_index_pfp_execution_backward_productright)) * bc) + (fom_value_pfp_execution_backward_productright))) /\\ (exists fom_gap_pfp_execution_backward_productright_value_bound. fom_gap_pfp_execution_backward_productright_value_bound + S (fom_value_pfp_execution_backward_productright) = p))) /\\ (((((((Lq)=0 \\/ (S d)=0) /\\ (((N)=0)))) \\/ (((~((Lq)=0)) /\\ (((~((S d)=0)) /\\ (((Lq)+(S d)=S (N)))))))) /\\ ((forall pfc_index_execution_backward_productcoefficients. (exists pfa_gap_execution_backward_productcoefficientsbound. pfa_gap_execution_backward_productcoefficientsbound + S (pfc_index_execution_backward_productcoefficients) = (N)) -> exists pfc_value_execution_backward_productcoefficients. ((((exists ff_h_pfp_execution_backward_productcoefficientsentry. ff_h_pfp_execution_backward_productcoefficientsentry + S (pfc_value_execution_backward_productcoefficients) = S ((S (pfc_index_execution_backward_productcoefficients)) * pc)) /\\ exists ff_q_pfp_execution_backward_productcoefficientsentry. pb = ff_q_pfp_execution_backward_productcoefficientsentry * S ((S (pfc_index_execution_backward_productcoefficients)) * pc) + (pfc_value_execution_backward_productcoefficients))) /\\ ((exists pfc_terms_code_execution_backward_productcoefficientscoefficient pfc_terms_scale_execution_backward_productcoefficientscoefficient pfc_natural_sum_execution_backward_productcoefficientscoefficient. ((forall pfc_index_execution_backward_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_backward_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_backward_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_backward_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_backward_productcoefficients))) -> exists pfc_value_execution_backward_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_backward_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_backward_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_backward_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_backward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_backward_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_backward_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_backward_productcoefficientscoefficient = ff_q_pfp_execution_backward_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_backward_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_backward_productcoefficientscoefficient) + (pfc_value_execution_backward_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_backward_productcoefficientscoefficientdiagonalterm pfc_left_execution_backward_productcoefficientscoefficientdiagonalterm pfc_right_execution_backward_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_backward_productcoefficientscoefficientdiagonal)+pfc_complement_execution_backward_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_backward_productcoefficients)) /\\ ((((((exists pfa_gap_execution_backward_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_backward_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_backward_productcoefficientscoefficientdiagonal) = (Lq)) /\\ ((((exists ff_h_pfp_execution_backward_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_backward_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_backward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_backward_productcoefficientscoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_backward_productcoefficientscoefficientdiagonaltermleftentry. qb = ff_q_pfp_execution_backward_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_backward_productcoefficientscoefficientdiagonal)) * qc) + (pfc_left_execution_backward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_backward_productcoefficientscoefficientdiagonaltermleftoutside+(Lq)=(pfc_index_execution_backward_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_backward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_backward_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_backward_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_backward_productcoefficientscoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_execution_backward_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_backward_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_backward_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_backward_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_backward_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_backward_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_backward_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_execution_backward_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_backward_productcoefficientscoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_execution_backward_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_backward_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_backward_productcoefficientscoefficientdiagonal)=pfc_left_execution_backward_productcoefficientscoefficientdiagonalterm*pfc_right_execution_backward_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_backward_productcoefficientscoefficientsum fs_v_pfc_execution_backward_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_backward_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_backward_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_backward_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_backward_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_backward_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_backward_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_backward_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_backward_productcoefficients))) * fs_v_pfc_execution_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_backward_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_backward_productcoefficients))) * fs_v_pfc_execution_backward_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_backward_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_backward_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_backward_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_backward_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_backward_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_backward_productcoefficients)) -> exists fs_a_pfc_execution_backward_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_backward_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_backward_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_backward_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_backward_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_backward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_backward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_backward_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_backward_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_backward_productcoefficientscoefficient = fs_q_pfc_execution_backward_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_backward_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_backward_productcoefficientscoefficient) + (fs_a_pfc_execution_backward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_backward_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_backward_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_backward_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_productcoefficientscoefficientsum) + (fs_r_pfc_execution_backward_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_backward_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_backward_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_backward_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_backward_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_productcoefficientscoefficientsum) + (fs_s_pfc_execution_backward_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_backward_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_backward_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_backward_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_backward_productcoefficientscoefficientresiduebound. pfa_gap_execution_backward_productcoefficientscoefficientresiduebound + S (pfc_value_execution_backward_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_backward_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_backward_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_backward_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_backward_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_backward_productcoefficients) + (p) * pfa_offset_right_execution_backward_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_execution_backward_identity_left_bounded. (exists fom_gap_pfp_execution_backward_identity_left_bounded_index_bound. fom_gap_pfp_execution_backward_identity_left_bounded_index_bound + S (fom_index_pfp_execution_backward_identity_left_bounded) = N) -> exists fom_value_pfp_execution_backward_identity_left_bounded. ((((exists fom_beta_height_pfp_execution_backward_identity_left_bounded_entry. fom_beta_height_pfp_execution_backward_identity_left_bounded_entry + S (fom_value_pfp_execution_backward_identity_left_bounded) = S ((S (fom_index_pfp_execution_backward_identity_left_bounded)) * pc)) /\\ exists fom_beta_quotient_pfp_execution_backward_identity_left_bounded_entry. pb = fom_beta_quotient_pfp_execution_backward_identity_left_bounded_entry * S ((S (fom_index_pfp_execution_backward_identity_left_bounded)) * pc) + (fom_value_pfp_execution_backward_identity_left_bounded))) /\\ (exists fom_gap_pfp_execution_backward_identity_left_bounded_value_bound. fom_gap_pfp_execution_backward_identity_left_bounded_value_bound + S (fom_value_pfp_execution_backward_identity_left_bounded) = p))) /\\ (((forall fom_index_pfp_execution_backward_identity_right_bounded. (exists fom_gap_pfp_execution_backward_identity_right_bounded_index_bound. fom_gap_pfp_execution_backward_identity_right_bounded_index_bound + S (fom_index_pfp_execution_backward_identity_right_bounded) = Lr) -> exists fom_value_pfp_execution_backward_identity_right_bounded. ((((exists fom_beta_height_pfp_execution_backward_identity_right_bounded_entry. fom_beta_height_pfp_execution_backward_identity_right_bounded_entry + S (fom_value_pfp_execution_backward_identity_right_bounded) = S ((S (fom_index_pfp_execution_backward_identity_right_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_execution_backward_identity_right_bounded_entry. rb = fom_beta_quotient_pfp_execution_backward_identity_right_bounded_entry * S ((S (fom_index_pfp_execution_backward_identity_right_bounded)) * rc) + (fom_value_pfp_execution_backward_identity_right_bounded))) /\\ (exists fom_gap_pfp_execution_backward_identity_right_bounded_value_bound. fom_gap_pfp_execution_backward_identity_right_bounded_value_bound + S (fom_value_pfp_execution_backward_identity_right_bounded) = p))) /\\ (((forall fom_index_pfp_execution_backward_identity_result_bounded. (exists fom_gap_pfp_execution_backward_identity_result_bounded_index_bound. fom_gap_pfp_execution_backward_identity_result_bounded_index_bound + S (fom_index_pfp_execution_backward_identity_result_bounded) = La) -> exists fom_value_pfp_execution_backward_identity_result_bounded. ((((exists fom_beta_height_pfp_execution_backward_identity_result_bounded_entry. fom_beta_height_pfp_execution_backward_identity_result_bounded_entry + S (fom_value_pfp_execution_backward_identity_result_bounded) = S ((S (fom_index_pfp_execution_backward_identity_result_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_backward_identity_result_bounded_entry. ab = fom_beta_quotient_pfp_execution_backward_identity_result_bounded_entry * S ((S (fom_index_pfp_execution_backward_identity_result_bounded)) * ac) + (fom_value_pfp_execution_backward_identity_result_bounded))) /\\ (exists fom_gap_pfp_execution_backward_identity_result_bounded_value_bound. fom_gap_pfp_execution_backward_identity_result_bounded_value_bound + S (fom_value_pfp_execution_backward_identity_result_bounded) = p))) /\\ ((exists pfaa_left_b_execution_backward_identity pfaa_left_c_execution_backward_identity pfaa_right_b_execution_backward_identity pfaa_right_c_execution_backward_identity pfaa_sum_b_execution_backward_identity pfaa_sum_c_execution_backward_identity pfaa_length_execution_backward_identity. ((((forall pfrep_power_execution_backward_identity_witness_common_left pfrep_left_execution_backward_identity_witness_common_left pfrep_right_execution_backward_identity_witness_common_left. ((exists pfrep_position_execution_backward_identity_witness_common_leftfirst. ((pfrep_position_execution_backward_identity_witness_common_leftfirst+S (pfrep_power_execution_backward_identity_witness_common_left)=(N)) /\\ ((((exists ff_h_pfp_execution_backward_identity_witness_common_leftfirstentry. ff_h_pfp_execution_backward_identity_witness_common_leftfirstentry + S (pfrep_left_execution_backward_identity_witness_common_left) = S ((S (pfrep_position_execution_backward_identity_witness_common_leftfirst)) * pc)) /\\ exists ff_q_pfp_execution_backward_identity_witness_common_leftfirstentry. pb = ff_q_pfp_execution_backward_identity_witness_common_leftfirstentry * S ((S (pfrep_position_execution_backward_identity_witness_common_leftfirst)) * pc) + (pfrep_left_execution_backward_identity_witness_common_left)))))) \\/ (((exists pfrep_gap_execution_backward_identity_witness_common_leftfirstoutside. pfrep_gap_execution_backward_identity_witness_common_leftfirstoutside+(N)=(pfrep_power_execution_backward_identity_witness_common_left)) /\\ (((pfrep_left_execution_backward_identity_witness_common_left)=0))))) -> ((exists pfrep_position_execution_backward_identity_witness_common_leftsecond. ((pfrep_position_execution_backward_identity_witness_common_leftsecond+S (pfrep_power_execution_backward_identity_witness_common_left)=(pfaa_length_execution_backward_identity)) /\\ ((((exists ff_h_pfp_execution_backward_identity_witness_common_leftsecondentry. ff_h_pfp_execution_backward_identity_witness_common_leftsecondentry + S (pfrep_right_execution_backward_identity_witness_common_left) = S ((S (pfrep_position_execution_backward_identity_witness_common_leftsecond)) * pfaa_left_c_execution_backward_identity)) /\\ exists ff_q_pfp_execution_backward_identity_witness_common_leftsecondentry. pfaa_left_b_execution_backward_identity = ff_q_pfp_execution_backward_identity_witness_common_leftsecondentry * S ((S (pfrep_position_execution_backward_identity_witness_common_leftsecond)) * pfaa_left_c_execution_backward_identity) + (pfrep_right_execution_backward_identity_witness_common_left)))))) \\/ (((exists pfrep_gap_execution_backward_identity_witness_common_leftsecondoutside. pfrep_gap_execution_backward_identity_witness_common_leftsecondoutside+(pfaa_length_execution_backward_identity)=(pfrep_power_execution_backward_identity_witness_common_left)) /\\ (((pfrep_right_execution_backward_identity_witness_common_left)=0))))) -> pfrep_left_execution_backward_identity_witness_common_left=pfrep_right_execution_backward_identity_witness_common_left) /\\ ((forall pfrep_power_execution_backward_identity_witness_common_right pfrep_left_execution_backward_identity_witness_common_right pfrep_right_execution_backward_identity_witness_common_right. ((exists pfrep_position_execution_backward_identity_witness_common_rightfirst. ((pfrep_position_execution_backward_identity_witness_common_rightfirst+S (pfrep_power_execution_backward_identity_witness_common_right)=(Lr)) /\\ ((((exists ff_h_pfp_execution_backward_identity_witness_common_rightfirstentry. ff_h_pfp_execution_backward_identity_witness_common_rightfirstentry + S (pfrep_left_execution_backward_identity_witness_common_right) = S ((S (pfrep_position_execution_backward_identity_witness_common_rightfirst)) * rc)) /\\ exists ff_q_pfp_execution_backward_identity_witness_common_rightfirstentry. rb = ff_q_pfp_execution_backward_identity_witness_common_rightfirstentry * S ((S (pfrep_position_execution_backward_identity_witness_common_rightfirst)) * rc) + (pfrep_left_execution_backward_identity_witness_common_right)))))) \\/ (((exists pfrep_gap_execution_backward_identity_witness_common_rightfirstoutside. pfrep_gap_execution_backward_identity_witness_common_rightfirstoutside+(Lr)=(pfrep_power_execution_backward_identity_witness_common_right)) /\\ (((pfrep_left_execution_backward_identity_witness_common_right)=0))))) -> ((exists pfrep_position_execution_backward_identity_witness_common_rightsecond. ((pfrep_position_execution_backward_identity_witness_common_rightsecond+S (pfrep_power_execution_backward_identity_witness_common_right)=(pfaa_length_execution_backward_identity)) /\\ ((((exists ff_h_pfp_execution_backward_identity_witness_common_rightsecondentry. ff_h_pfp_execution_backward_identity_witness_common_rightsecondentry + S (pfrep_right_execution_backward_identity_witness_common_right) = S ((S (pfrep_position_execution_backward_identity_witness_common_rightsecond)) * pfaa_right_c_execution_backward_identity)) /\\ exists ff_q_pfp_execution_backward_identity_witness_common_rightsecondentry. pfaa_right_b_execution_backward_identity = ff_q_pfp_execution_backward_identity_witness_common_rightsecondentry * S ((S (pfrep_position_execution_backward_identity_witness_common_rightsecond)) * pfaa_right_c_execution_backward_identity) + (pfrep_right_execution_backward_identity_witness_common_right)))))) \\/ (((exists pfrep_gap_execution_backward_identity_witness_common_rightsecondoutside. pfrep_gap_execution_backward_identity_witness_common_rightsecondoutside+(pfaa_length_execution_backward_identity)=(pfrep_power_execution_backward_identity_witness_common_right)) /\\ (((pfrep_right_execution_backward_identity_witness_common_right)=0))))) -> pfrep_left_execution_backward_identity_witness_common_right=pfrep_right_execution_backward_identity_witness_common_right)))) /\\ (((forall pfp_index_execution_backward_identity_witness_operation. (exists pfa_gap_execution_backward_identity_witness_operationindex. pfa_gap_execution_backward_identity_witness_operationindex + S (pfp_index_execution_backward_identity_witness_operation) = (pfaa_length_execution_backward_identity)) -> exists pfp_left_execution_backward_identity_witness_operation pfp_right_execution_backward_identity_witness_operation pfp_value_execution_backward_identity_witness_operation. ((((exists ff_h_pfp_execution_backward_identity_witness_operationleft. ff_h_pfp_execution_backward_identity_witness_operationleft + S (pfp_left_execution_backward_identity_witness_operation) = S ((S (pfp_index_execution_backward_identity_witness_operation)) * pfaa_left_c_execution_backward_identity)) /\\ exists ff_q_pfp_execution_backward_identity_witness_operationleft. pfaa_left_b_execution_backward_identity = ff_q_pfp_execution_backward_identity_witness_operationleft * S ((S (pfp_index_execution_backward_identity_witness_operation)) * pfaa_left_c_execution_backward_identity) + (pfp_left_execution_backward_identity_witness_operation))) /\\ (((((exists ff_h_pfp_execution_backward_identity_witness_operationright. ff_h_pfp_execution_backward_identity_witness_operationright + S (pfp_right_execution_backward_identity_witness_operation) = S ((S (pfp_index_execution_backward_identity_witness_operation)) * pfaa_right_c_execution_backward_identity)) /\\ exists ff_q_pfp_execution_backward_identity_witness_operationright. pfaa_right_b_execution_backward_identity = ff_q_pfp_execution_backward_identity_witness_operationright * S ((S (pfp_index_execution_backward_identity_witness_operation)) * pfaa_right_c_execution_backward_identity) + (pfp_right_execution_backward_identity_witness_operation))) /\\ (((((exists ff_h_pfp_execution_backward_identity_witness_operationtarget. ff_h_pfp_execution_backward_identity_witness_operationtarget + S (pfp_value_execution_backward_identity_witness_operation) = S ((S (pfp_index_execution_backward_identity_witness_operation)) * pfaa_sum_c_execution_backward_identity)) /\\ exists ff_q_pfp_execution_backward_identity_witness_operationtarget. pfaa_sum_b_execution_backward_identity = ff_q_pfp_execution_backward_identity_witness_operationtarget * S ((S (pfp_index_execution_backward_identity_witness_operation)) * pfaa_sum_c_execution_backward_identity) + (pfp_value_execution_backward_identity_witness_operation))) /\\ ((((exists pfa_gap_execution_backward_identity_witness_operationoperationleft. pfa_gap_execution_backward_identity_witness_operationoperationleft + S (pfp_left_execution_backward_identity_witness_operation) = (p)) /\\ (((exists pfa_gap_execution_backward_identity_witness_operationoperationright. pfa_gap_execution_backward_identity_witness_operationoperationright + S (pfp_right_execution_backward_identity_witness_operation) = (p)) /\\ ((((exists pfa_gap_execution_backward_identity_witness_operationoperationresultbound. pfa_gap_execution_backward_identity_witness_operationoperationresultbound + S (pfp_value_execution_backward_identity_witness_operation) = (p)) /\\ ((exists pfa_offset_left_execution_backward_identity_witness_operationoperationresultcongruence pfa_offset_right_execution_backward_identity_witness_operationoperationresultcongruence. ((pfp_left_execution_backward_identity_witness_operation) + (pfp_right_execution_backward_identity_witness_operation)) + (p) * pfa_offset_left_execution_backward_identity_witness_operationoperationresultcongruence = (pfp_value_execution_backward_identity_witness_operation) + (p) * pfa_offset_right_execution_backward_identity_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_execution_backward_identity_witness_output pfrep_left_execution_backward_identity_witness_output pfrep_right_execution_backward_identity_witness_output. ((exists pfrep_position_execution_backward_identity_witness_outputfirst. ((pfrep_position_execution_backward_identity_witness_outputfirst+S (pfrep_power_execution_backward_identity_witness_output)=(pfaa_length_execution_backward_identity)) /\\ ((((exists ff_h_pfp_execution_backward_identity_witness_outputfirstentry. ff_h_pfp_execution_backward_identity_witness_outputfirstentry + S (pfrep_left_execution_backward_identity_witness_output) = S ((S (pfrep_position_execution_backward_identity_witness_outputfirst)) * pfaa_sum_c_execution_backward_identity)) /\\ exists ff_q_pfp_execution_backward_identity_witness_outputfirstentry. pfaa_sum_b_execution_backward_identity = ff_q_pfp_execution_backward_identity_witness_outputfirstentry * S ((S (pfrep_position_execution_backward_identity_witness_outputfirst)) * pfaa_sum_c_execution_backward_identity) + (pfrep_left_execution_backward_identity_witness_output)))))) \\/ (((exists pfrep_gap_execution_backward_identity_witness_outputfirstoutside. pfrep_gap_execution_backward_identity_witness_outputfirstoutside+(pfaa_length_execution_backward_identity)=(pfrep_power_execution_backward_identity_witness_output)) /\\ (((pfrep_left_execution_backward_identity_witness_output)=0))))) -> ((exists pfrep_position_execution_backward_identity_witness_outputsecond. ((pfrep_position_execution_backward_identity_witness_outputsecond+S (pfrep_power_execution_backward_identity_witness_output)=(La)) /\\ ((((exists ff_h_pfp_execution_backward_identity_witness_outputsecondentry. ff_h_pfp_execution_backward_identity_witness_outputsecondentry + S (pfrep_right_execution_backward_identity_witness_output) = S ((S (pfrep_position_execution_backward_identity_witness_outputsecond)) * ac)) /\\ exists ff_q_pfp_execution_backward_identity_witness_outputsecondentry. ab = ff_q_pfp_execution_backward_identity_witness_outputsecondentry * S ((S (pfrep_position_execution_backward_identity_witness_outputsecond)) * ac) + (pfrep_right_execution_backward_identity_witness_output)))))) \\/ (((exists pfrep_gap_execution_backward_identity_witness_outputsecondoutside. pfrep_gap_execution_backward_identity_witness_outputsecondoutside+(La)=(pfrep_power_execution_backward_identity_witness_output)) /\\ (((pfrep_right_execution_backward_identity_witness_output)=0))))) -> pfrep_left_execution_backward_identity_witness_output=pfrep_right_execution_backward_identity_witness_output)))))))))))))))", "specialize prime_field_polynomial_division_execution_aligned_identity (p)", "specialize prime_field_polynomial_division_execution_aligned_identity (ab)", "specialize prime_field_polynomial_division_execution_aligned_identity (ac)", "specialize prime_field_polynomial_division_execution_aligned_identity (La)", "specialize prime_field_polynomial_division_execution_aligned_identity (bb)", "specialize prime_field_polynomial_division_execution_aligned_identity (bc)", "specialize prime_field_polynomial_division_execution_aligned_identity (d)", "specialize prime_field_polynomial_division_execution_aligned_identity (qb)", "specialize prime_field_polynomial_division_execution_aligned_identity (qc)", "specialize prime_field_polynomial_division_execution_aligned_identity (Lq)", "specialize prime_field_polynomial_division_execution_aligned_identity (rb)", "specialize prime_field_polynomial_division_execution_aligned_identity (rc)", "specialize prime_field_polynomial_division_execution_aligned_identity (Lr)", "apply prime_field_polynomial_division_execution_aligned_identity", "exact hp", "exact he", "cases hi", "cases hi_witness", "cases hi_witness_witness", "cases hi_witness_witness_witness", "specialize prime_field_polynomial_bezout_euclidean_backward (p)", "specialize prime_field_polynomial_bezout_euclidean_backward (ab)", "specialize prime_field_polynomial_bezout_euclidean_backward (ac)", "specialize prime_field_polynomial_bezout_euclidean_backward (La)", "specialize prime_field_polynomial_bezout_euclidean_backward (bb)", "specialize prime_field_polynomial_bezout_euclidean_backward (bc)", "specialize prime_field_polynomial_bezout_euclidean_backward (S d)", "specialize prime_field_polynomial_bezout_euclidean_backward (rb)", "specialize prime_field_polynomial_bezout_euclidean_backward (rc)", "specialize prime_field_polynomial_bezout_euclidean_backward (Lr)", "specialize prime_field_polynomial_bezout_euclidean_backward (qb)", "specialize prime_field_polynomial_bezout_euclidean_backward (qc)", "specialize prime_field_polynomial_bezout_euclidean_backward (Lq)", "specialize prime_field_polynomial_bezout_euclidean_backward (x)", "specialize prime_field_polynomial_bezout_euclidean_backward (x1)", "specialize prime_field_polynomial_bezout_euclidean_backward (x2)", "specialize prime_field_polynomial_bezout_euclidean_backward (gb)", "specialize prime_field_polynomial_bezout_euclidean_backward (gc)", "specialize prime_field_polynomial_bezout_euclidean_backward (Lg)", "specialize prime_field_polynomial_bezout_euclidean_backward (ub)", "specialize prime_field_polynomial_bezout_euclidean_backward (uc)", "specialize prime_field_polynomial_bezout_euclidean_backward (Lu)", "specialize prime_field_polynomial_bezout_euclidean_backward (vb)", "specialize prime_field_polynomial_bezout_euclidean_backward (vc)", "specialize prime_field_polynomial_bezout_euclidean_backward (Lv)", "apply prime_field_polynomial_bezout_euclidean_backward", "exact hp", "exact hi_witness_witness_witness_left", "exact hi_witness_witness_witness_right", "exact hb" ], "script_sha256": "380eb88f0098a5fc363ab525713c6ddac1b3348bc1ebb4f1039be4caabe3ed69", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_bezout_backward_candidate.py", "sha256": "c3903482000c957ac77f84a43a85d135e4caa19e4484328035f91b82cbf3a702" }, "statement": "forall p ab ac La bb bc d qb qc Lq rb rc Lr gb gc Lg ub uc Lu vb vc Lv. (~((p) = 1) /\\ forall pfa_factor_left_execution_backward_prime pfa_factor_right_execution_backward_prime. (p) = pfa_factor_left_execution_backward_prime * pfa_factor_right_execution_backward_prime -> pfa_factor_left_execution_backward_prime = 1 \\/ pfa_factor_right_execution_backward_prime = 1) -> (((forall fom_index_pfp_execution_backward_actualinput. (exists fom_gap_pfp_execution_backward_actualinput_index_bound. fom_gap_pfp_execution_backward_actualinput_index_bound + S (fom_index_pfp_execution_backward_actualinput) = La) -> exists fom_value_pfp_execution_backward_actualinput. ((((exists fom_beta_height_pfp_execution_backward_actualinput_entry. fom_beta_height_pfp_execution_backward_actualinput_entry + S (fom_value_pfp_execution_backward_actualinput) = S ((S (fom_index_pfp_execution_backward_actualinput)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_backward_actualinput_entry. ab = fom_beta_quotient_pfp_execution_backward_actualinput_entry * S ((S (fom_index_pfp_execution_backward_actualinput)) * ac) + (fom_value_pfp_execution_backward_actualinput))) /\\ (exists fom_gap_pfp_execution_backward_actualinput_value_bound. fom_gap_pfp_execution_backward_actualinput_value_bound + S (fom_value_pfp_execution_backward_actualinput) = p))) /\\ (((forall fom_index_pfp_execution_backward_actualdivisor. (exists fom_gap_pfp_execution_backward_actualdivisor_index_bound. fom_gap_pfp_execution_backward_actualdivisor_index_bound + S (fom_index_pfp_execution_backward_actualdivisor) = S (d)) -> exists fom_value_pfp_execution_backward_actualdivisor. ((((exists fom_beta_height_pfp_execution_backward_actualdivisor_entry. fom_beta_height_pfp_execution_backward_actualdivisor_entry + S (fom_value_pfp_execution_backward_actualdivisor) = S ((S (fom_index_pfp_execution_backward_actualdivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_backward_actualdivisor_entry. bb = fom_beta_quotient_pfp_execution_backward_actualdivisor_entry * S ((S (fom_index_pfp_execution_backward_actualdivisor)) * bc) + (fom_value_pfp_execution_backward_actualdivisor))) /\\ (exists fom_gap_pfp_execution_backward_actualdivisor_value_bound. fom_gap_pfp_execution_backward_actualdivisor_value_bound + S (fom_value_pfp_execution_backward_actualdivisor) = p))) /\\ (((((((Lq)=0) /\\ ((exists pfc_gap_execution_backward_actuallengthshort. pfc_gap_execution_backward_actuallengthshort+(La)=(d))))) \\/ (((~((Lq)=0)) /\\ (((Lq)+(d)=(La)))))) /\\ ((exists pfd_head_execution_backward_actual pfd_inverse_execution_backward_actual pfd_product_code_execution_backward_actual pfd_product_scale_execution_backward_actual pfd_residual_code_execution_backward_actual pfd_residual_scale_execution_backward_actual pfd_cut_execution_backward_actual. ((((exists ff_h_pfp_execution_backward_actualhead. ff_h_pfp_execution_backward_actualhead + S (pfd_head_execution_backward_actual) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_execution_backward_actualhead. bb = ff_q_pfp_execution_backward_actualhead * S ((S (0)) * bc) + (pfd_head_execution_backward_actual))) /\\ (((((~((pfd_head_execution_backward_actual) = 0)) /\\ ((((exists pfa_gap_execution_backward_actualinversemultiplicationleft. pfa_gap_execution_backward_actualinversemultiplicationleft + S (pfd_head_execution_backward_actual) = (p)) /\\ (((exists pfa_gap_execution_backward_actualinversemultiplicationright. pfa_gap_execution_backward_actualinversemultiplicationright + S (pfd_inverse_execution_backward_actual) = (p)) /\\ ((((exists pfa_gap_execution_backward_actualinversemultiplicationresultbound. pfa_gap_execution_backward_actualinversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_execution_backward_actualinversemultiplicationresultcongruence pfa_offset_right_execution_backward_actualinversemultiplicationresultcongruence. ((pfd_head_execution_backward_actual) * (pfd_inverse_execution_backward_actual)) + (p) * pfa_offset_left_execution_backward_actualinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_execution_backward_actualinversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_execution_backward_actualquotient. (exists pfa_gap_execution_backward_actualquotientbound. pfa_gap_execution_backward_actualquotientbound + S (pfd_index_execution_backward_actualquotient) = (Lq)) -> exists pfd_value_execution_backward_actualquotient. ((((exists ff_h_pfp_execution_backward_actualquotiententry. ff_h_pfp_execution_backward_actualquotiententry + S (pfd_value_execution_backward_actualquotient) = S ((S (pfd_index_execution_backward_actualquotient)) * qc)) /\\ exists ff_q_pfp_execution_backward_actualquotiententry. qb = ff_q_pfp_execution_backward_actualquotiententry * S ((S (pfd_index_execution_backward_actualquotient)) * qc) + (pfd_value_execution_backward_actualquotient))) /\\ ((exists pfd_input_execution_backward_actualquotientstep pfd_previous_execution_backward_actualquotientstep pfd_difference_execution_backward_actualquotientstep. ((((exists ff_h_pfp_execution_backward_actualquotientstepinput. ff_h_pfp_execution_backward_actualquotientstepinput + S (pfd_input_execution_backward_actualquotientstep) = S ((S (pfd_index_execution_backward_actualquotient)) * ac)) /\\ exists ff_q_pfp_execution_backward_actualquotientstepinput. ab = ff_q_pfp_execution_backward_actualquotientstepinput * S ((S (pfd_index_execution_backward_actualquotient)) * ac) + (pfd_input_execution_backward_actualquotientstep))) /\\ (((exists pfc_terms_code_execution_backward_actualquotientstepprevious pfc_terms_scale_execution_backward_actualquotientstepprevious pfc_natural_sum_execution_backward_actualquotientstepprevious. ((forall pfc_index_execution_backward_actualquotientsteppreviousdiagonal. (exists pfa_gap_execution_backward_actualquotientsteppreviousdiagonalbound. pfa_gap_execution_backward_actualquotientsteppreviousdiagonalbound + S (pfc_index_execution_backward_actualquotientsteppreviousdiagonal) = (S (pfd_index_execution_backward_actualquotient))) -> exists pfc_value_execution_backward_actualquotientsteppreviousdiagonal. ((((exists ff_h_pfp_execution_backward_actualquotientsteppreviousdiagonalentry. ff_h_pfp_execution_backward_actualquotientsteppreviousdiagonalentry + S (pfc_value_execution_backward_actualquotientsteppreviousdiagonal) = S ((S (pfc_index_execution_backward_actualquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_backward_actualquotientstepprevious)) /\\ exists ff_q_pfp_execution_backward_actualquotientsteppreviousdiagonalentry. pfc_terms_code_execution_backward_actualquotientstepprevious = ff_q_pfp_execution_backward_actualquotientsteppreviousdiagonalentry * S ((S (pfc_index_execution_backward_actualquotientsteppreviousdiagonal)) * pfc_terms_scale_execution_backward_actualquotientstepprevious) + (pfc_value_execution_backward_actualquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_execution_backward_actualquotientsteppreviousdiagonalterm pfc_left_execution_backward_actualquotientsteppreviousdiagonalterm pfc_right_execution_backward_actualquotientsteppreviousdiagonalterm. (((pfc_index_execution_backward_actualquotientsteppreviousdiagonal)+pfc_complement_execution_backward_actualquotientsteppreviousdiagonalterm=(pfd_index_execution_backward_actualquotient)) /\\ ((((((exists pfa_gap_execution_backward_actualquotientsteppreviousdiagonaltermleftinside. pfa_gap_execution_backward_actualquotientsteppreviousdiagonaltermleftinside + S (pfc_index_execution_backward_actualquotientsteppreviousdiagonal) = (pfd_index_execution_backward_actualquotient)) /\\ ((((exists ff_h_pfp_execution_backward_actualquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_execution_backward_actualquotientsteppreviousdiagonaltermleftentry + S (pfc_left_execution_backward_actualquotientsteppreviousdiagonalterm) = S ((S (pfc_index_execution_backward_actualquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_backward_actualquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_execution_backward_actualquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_execution_backward_actualquotientsteppreviousdiagonal)) * qc) + (pfc_left_execution_backward_actualquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_actualquotientsteppreviousdiagonaltermleftoutside. pfc_gap_execution_backward_actualquotientsteppreviousdiagonaltermleftoutside+(pfd_index_execution_backward_actualquotient)=(pfc_index_execution_backward_actualquotientsteppreviousdiagonal)) /\\ (((pfc_left_execution_backward_actualquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_backward_actualquotientsteppreviousdiagonaltermrightinside. pfa_gap_execution_backward_actualquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_execution_backward_actualquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_backward_actualquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_execution_backward_actualquotientsteppreviousdiagonaltermrightentry + S (pfc_right_execution_backward_actualquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_execution_backward_actualquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_backward_actualquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_execution_backward_actualquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_execution_backward_actualquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_execution_backward_actualquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_actualquotientsteppreviousdiagonaltermrightoutside. pfc_gap_execution_backward_actualquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_backward_actualquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_execution_backward_actualquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_execution_backward_actualquotientsteppreviousdiagonal)=pfc_left_execution_backward_actualquotientsteppreviousdiagonalterm*pfc_right_execution_backward_actualquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_backward_actualquotientstepprevioussum fs_v_pfc_execution_backward_actualquotientstepprevioussum. ((((exists fs_h_pfc_execution_backward_actualquotientstepprevioussum_body_start. fs_h_pfc_execution_backward_actualquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_backward_actualquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_backward_actualquotientstepprevioussum_body_start. fs_u_pfc_execution_backward_actualquotientstepprevioussum = fs_q_pfc_execution_backward_actualquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_execution_backward_actualquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_execution_backward_actualquotientstepprevioussum_body_terminal. fs_h_pfc_execution_backward_actualquotientstepprevioussum_body_terminal + S (pfc_natural_sum_execution_backward_actualquotientstepprevious) = S ((S (S (pfd_index_execution_backward_actualquotient))) * fs_v_pfc_execution_backward_actualquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_backward_actualquotientstepprevioussum_body_terminal. fs_u_pfc_execution_backward_actualquotientstepprevioussum = fs_q_pfc_execution_backward_actualquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_execution_backward_actualquotient))) * fs_v_pfc_execution_backward_actualquotientstepprevioussum) + (pfc_natural_sum_execution_backward_actualquotientstepprevious))) /\\ forall fs_i_pfc_execution_backward_actualquotientstepprevioussum_body_steps. (exists fs_lt_pfc_execution_backward_actualquotientstepprevioussum_body_steps_bound. fs_lt_pfc_execution_backward_actualquotientstepprevioussum_body_steps_bound + S fs_i_pfc_execution_backward_actualquotientstepprevioussum_body_steps = S (pfd_index_execution_backward_actualquotient)) -> exists fs_a_pfc_execution_backward_actualquotientstepprevioussum_body_steps fs_r_pfc_execution_backward_actualquotientstepprevioussum_body_steps fs_s_pfc_execution_backward_actualquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_execution_backward_actualquotientstepprevioussum_body_steps_summand. fs_h_pfc_execution_backward_actualquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_execution_backward_actualquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_backward_actualquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_backward_actualquotientstepprevious)) /\\ exists fs_q_pfc_execution_backward_actualquotientstepprevioussum_body_steps_summand. pfc_terms_code_execution_backward_actualquotientstepprevious = fs_q_pfc_execution_backward_actualquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_execution_backward_actualquotientstepprevioussum_body_steps)) * pfc_terms_scale_execution_backward_actualquotientstepprevious) + (fs_a_pfc_execution_backward_actualquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_actualquotientstepprevioussum_body_steps_partial. fs_h_pfc_execution_backward_actualquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_execution_backward_actualquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_execution_backward_actualquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_backward_actualquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_backward_actualquotientstepprevioussum_body_steps_partial. fs_u_pfc_execution_backward_actualquotientstepprevioussum = fs_q_pfc_execution_backward_actualquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_execution_backward_actualquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_backward_actualquotientstepprevioussum) + (fs_r_pfc_execution_backward_actualquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_actualquotientstepprevioussum_body_steps_successor. fs_h_pfc_execution_backward_actualquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_execution_backward_actualquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_execution_backward_actualquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_backward_actualquotientstepprevioussum)) /\\ exists fs_q_pfc_execution_backward_actualquotientstepprevioussum_body_steps_successor. fs_u_pfc_execution_backward_actualquotientstepprevioussum = fs_q_pfc_execution_backward_actualquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_execution_backward_actualquotientstepprevioussum_body_steps)) * fs_v_pfc_execution_backward_actualquotientstepprevioussum) + (fs_s_pfc_execution_backward_actualquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_execution_backward_actualquotientstepprevioussum_body_steps = fs_r_pfc_execution_backward_actualquotientstepprevioussum_body_steps + fs_a_pfc_execution_backward_actualquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_execution_backward_actualquotientsteppreviousresiduebound. pfa_gap_execution_backward_actualquotientsteppreviousresiduebound + S (pfd_previous_execution_backward_actualquotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_backward_actualquotientsteppreviousresiduecongruence pfa_offset_right_execution_backward_actualquotientsteppreviousresiduecongruence. (pfc_natural_sum_execution_backward_actualquotientstepprevious) + (p) * pfa_offset_left_execution_backward_actualquotientsteppreviousresiduecongruence = (pfd_previous_execution_backward_actualquotientstep) + (p) * pfa_offset_right_execution_backward_actualquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_execution_backward_actualquotientstepsubtractleft. pfa_gap_execution_backward_actualquotientstepsubtractleft + S (pfd_previous_execution_backward_actualquotientstep) = (p)) /\\ (((exists pfa_gap_execution_backward_actualquotientstepsubtractright. pfa_gap_execution_backward_actualquotientstepsubtractright + S (pfd_difference_execution_backward_actualquotientstep) = (p)) /\\ ((((exists pfa_gap_execution_backward_actualquotientstepsubtractresultbound. pfa_gap_execution_backward_actualquotientstepsubtractresultbound + S (pfd_input_execution_backward_actualquotientstep) = (p)) /\\ ((exists pfa_offset_left_execution_backward_actualquotientstepsubtractresultcongruence pfa_offset_right_execution_backward_actualquotientstepsubtractresultcongruence. ((pfd_previous_execution_backward_actualquotientstep) + (pfd_difference_execution_backward_actualquotientstep)) + (p) * pfa_offset_left_execution_backward_actualquotientstepsubtractresultcongruence = (pfd_input_execution_backward_actualquotientstep) + (p) * pfa_offset_right_execution_backward_actualquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_execution_backward_actualquotientstepmultiplyleft. pfa_gap_execution_backward_actualquotientstepmultiplyleft + S (pfd_inverse_execution_backward_actual) = (p)) /\\ (((exists pfa_gap_execution_backward_actualquotientstepmultiplyright. pfa_gap_execution_backward_actualquotientstepmultiplyright + S (pfd_difference_execution_backward_actualquotientstep) = (p)) /\\ ((((exists pfa_gap_execution_backward_actualquotientstepmultiplyresultbound. pfa_gap_execution_backward_actualquotientstepmultiplyresultbound + S (pfd_value_execution_backward_actualquotient) = (p)) /\\ ((exists pfa_offset_left_execution_backward_actualquotientstepmultiplyresultcongruence pfa_offset_right_execution_backward_actualquotientstepmultiplyresultcongruence. ((pfd_inverse_execution_backward_actual) * (pfd_difference_execution_backward_actualquotientstep)) + (p) * pfa_offset_left_execution_backward_actualquotientstepmultiplyresultcongruence = (pfd_value_execution_backward_actualquotient) + (p) * pfa_offset_right_execution_backward_actualquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_execution_backward_actualproduct. (exists pfa_gap_execution_backward_actualproductbound. pfa_gap_execution_backward_actualproductbound + S (pfc_index_execution_backward_actualproduct) = (La)) -> exists pfc_value_execution_backward_actualproduct. ((((exists ff_h_pfp_execution_backward_actualproductentry. ff_h_pfp_execution_backward_actualproductentry + S (pfc_value_execution_backward_actualproduct) = S ((S (pfc_index_execution_backward_actualproduct)) * pfd_product_scale_execution_backward_actual)) /\\ exists ff_q_pfp_execution_backward_actualproductentry. pfd_product_code_execution_backward_actual = ff_q_pfp_execution_backward_actualproductentry * S ((S (pfc_index_execution_backward_actualproduct)) * pfd_product_scale_execution_backward_actual) + (pfc_value_execution_backward_actualproduct))) /\\ ((exists pfc_terms_code_execution_backward_actualproductcoefficient pfc_terms_scale_execution_backward_actualproductcoefficient pfc_natural_sum_execution_backward_actualproductcoefficient. ((forall pfc_index_execution_backward_actualproductcoefficientdiagonal. (exists pfa_gap_execution_backward_actualproductcoefficientdiagonalbound. pfa_gap_execution_backward_actualproductcoefficientdiagonalbound + S (pfc_index_execution_backward_actualproductcoefficientdiagonal) = (S (pfc_index_execution_backward_actualproduct))) -> exists pfc_value_execution_backward_actualproductcoefficientdiagonal. ((((exists ff_h_pfp_execution_backward_actualproductcoefficientdiagonalentry. ff_h_pfp_execution_backward_actualproductcoefficientdiagonalentry + S (pfc_value_execution_backward_actualproductcoefficientdiagonal) = S ((S (pfc_index_execution_backward_actualproductcoefficientdiagonal)) * pfc_terms_scale_execution_backward_actualproductcoefficient)) /\\ exists ff_q_pfp_execution_backward_actualproductcoefficientdiagonalentry. pfc_terms_code_execution_backward_actualproductcoefficient = ff_q_pfp_execution_backward_actualproductcoefficientdiagonalentry * S ((S (pfc_index_execution_backward_actualproductcoefficientdiagonal)) * pfc_terms_scale_execution_backward_actualproductcoefficient) + (pfc_value_execution_backward_actualproductcoefficientdiagonal))) /\\ ((exists pfc_complement_execution_backward_actualproductcoefficientdiagonalterm pfc_left_execution_backward_actualproductcoefficientdiagonalterm pfc_right_execution_backward_actualproductcoefficientdiagonalterm. (((pfc_index_execution_backward_actualproductcoefficientdiagonal)+pfc_complement_execution_backward_actualproductcoefficientdiagonalterm=(pfc_index_execution_backward_actualproduct)) /\\ ((((((exists pfa_gap_execution_backward_actualproductcoefficientdiagonaltermleftinside. pfa_gap_execution_backward_actualproductcoefficientdiagonaltermleftinside + S (pfc_index_execution_backward_actualproductcoefficientdiagonal) = (Lq)) /\\ ((((exists ff_h_pfp_execution_backward_actualproductcoefficientdiagonaltermleftentry. ff_h_pfp_execution_backward_actualproductcoefficientdiagonaltermleftentry + S (pfc_left_execution_backward_actualproductcoefficientdiagonalterm) = S ((S (pfc_index_execution_backward_actualproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_execution_backward_actualproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_execution_backward_actualproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_backward_actualproductcoefficientdiagonal)) * qc) + (pfc_left_execution_backward_actualproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_actualproductcoefficientdiagonaltermleftoutside. pfc_gap_execution_backward_actualproductcoefficientdiagonaltermleftoutside+(Lq)=(pfc_index_execution_backward_actualproductcoefficientdiagonal)) /\\ (((pfc_left_execution_backward_actualproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_backward_actualproductcoefficientdiagonaltermrightinside. pfa_gap_execution_backward_actualproductcoefficientdiagonaltermrightinside + S (pfc_complement_execution_backward_actualproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_execution_backward_actualproductcoefficientdiagonaltermrightentry. ff_h_pfp_execution_backward_actualproductcoefficientdiagonaltermrightentry + S (pfc_right_execution_backward_actualproductcoefficientdiagonalterm) = S ((S (pfc_complement_execution_backward_actualproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_backward_actualproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_backward_actualproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_backward_actualproductcoefficientdiagonalterm)) * bc) + (pfc_right_execution_backward_actualproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_actualproductcoefficientdiagonaltermrightoutside. pfc_gap_execution_backward_actualproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_execution_backward_actualproductcoefficientdiagonalterm)) /\\ (((pfc_right_execution_backward_actualproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_backward_actualproductcoefficientdiagonal)=pfc_left_execution_backward_actualproductcoefficientdiagonalterm*pfc_right_execution_backward_actualproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_backward_actualproductcoefficientsum fs_v_pfc_execution_backward_actualproductcoefficientsum. ((((exists fs_h_pfc_execution_backward_actualproductcoefficientsum_body_start. fs_h_pfc_execution_backward_actualproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_backward_actualproductcoefficientsum)) /\\ exists fs_q_pfc_execution_backward_actualproductcoefficientsum_body_start. fs_u_pfc_execution_backward_actualproductcoefficientsum = fs_q_pfc_execution_backward_actualproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_backward_actualproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_backward_actualproductcoefficientsum_body_terminal. fs_h_pfc_execution_backward_actualproductcoefficientsum_body_terminal + S (pfc_natural_sum_execution_backward_actualproductcoefficient) = S ((S (S (pfc_index_execution_backward_actualproduct))) * fs_v_pfc_execution_backward_actualproductcoefficientsum)) /\\ exists fs_q_pfc_execution_backward_actualproductcoefficientsum_body_terminal. fs_u_pfc_execution_backward_actualproductcoefficientsum = fs_q_pfc_execution_backward_actualproductcoefficientsum_body_terminal * S ((S (S (pfc_index_execution_backward_actualproduct))) * fs_v_pfc_execution_backward_actualproductcoefficientsum) + (pfc_natural_sum_execution_backward_actualproductcoefficient))) /\\ forall fs_i_pfc_execution_backward_actualproductcoefficientsum_body_steps. (exists fs_lt_pfc_execution_backward_actualproductcoefficientsum_body_steps_bound. fs_lt_pfc_execution_backward_actualproductcoefficientsum_body_steps_bound + S fs_i_pfc_execution_backward_actualproductcoefficientsum_body_steps = S (pfc_index_execution_backward_actualproduct)) -> exists fs_a_pfc_execution_backward_actualproductcoefficientsum_body_steps fs_r_pfc_execution_backward_actualproductcoefficientsum_body_steps fs_s_pfc_execution_backward_actualproductcoefficientsum_body_steps. ((((exists fs_h_pfc_execution_backward_actualproductcoefficientsum_body_steps_summand. fs_h_pfc_execution_backward_actualproductcoefficientsum_body_steps_summand + S (fs_a_pfc_execution_backward_actualproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_backward_actualproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_backward_actualproductcoefficient)) /\\ exists fs_q_pfc_execution_backward_actualproductcoefficientsum_body_steps_summand. pfc_terms_code_execution_backward_actualproductcoefficient = fs_q_pfc_execution_backward_actualproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_backward_actualproductcoefficientsum_body_steps)) * pfc_terms_scale_execution_backward_actualproductcoefficient) + (fs_a_pfc_execution_backward_actualproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_actualproductcoefficientsum_body_steps_partial. fs_h_pfc_execution_backward_actualproductcoefficientsum_body_steps_partial + S (fs_r_pfc_execution_backward_actualproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_backward_actualproductcoefficientsum_body_steps)) * fs_v_pfc_execution_backward_actualproductcoefficientsum)) /\\ exists fs_q_pfc_execution_backward_actualproductcoefficientsum_body_steps_partial. fs_u_pfc_execution_backward_actualproductcoefficientsum = fs_q_pfc_execution_backward_actualproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_backward_actualproductcoefficientsum_body_steps)) * fs_v_pfc_execution_backward_actualproductcoefficientsum) + (fs_r_pfc_execution_backward_actualproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_actualproductcoefficientsum_body_steps_successor. fs_h_pfc_execution_backward_actualproductcoefficientsum_body_steps_successor + S (fs_s_pfc_execution_backward_actualproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_backward_actualproductcoefficientsum_body_steps)) * fs_v_pfc_execution_backward_actualproductcoefficientsum)) /\\ exists fs_q_pfc_execution_backward_actualproductcoefficientsum_body_steps_successor. fs_u_pfc_execution_backward_actualproductcoefficientsum = fs_q_pfc_execution_backward_actualproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_backward_actualproductcoefficientsum_body_steps)) * fs_v_pfc_execution_backward_actualproductcoefficientsum) + (fs_s_pfc_execution_backward_actualproductcoefficientsum_body_steps))) /\\ fs_s_pfc_execution_backward_actualproductcoefficientsum_body_steps = fs_r_pfc_execution_backward_actualproductcoefficientsum_body_steps + fs_a_pfc_execution_backward_actualproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_backward_actualproductcoefficientresiduebound. pfa_gap_execution_backward_actualproductcoefficientresiduebound + S (pfc_value_execution_backward_actualproduct) = (p)) /\\ ((exists pfa_offset_left_execution_backward_actualproductcoefficientresiduecongruence pfa_offset_right_execution_backward_actualproductcoefficientresiduecongruence. (pfc_natural_sum_execution_backward_actualproductcoefficient) + (p) * pfa_offset_left_execution_backward_actualproductcoefficientresiduecongruence = (pfc_value_execution_backward_actualproduct) + (p) * pfa_offset_right_execution_backward_actualproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_execution_backward_actualdifference. (exists pfa_gap_execution_backward_actualdifferenceindex. pfa_gap_execution_backward_actualdifferenceindex + S (pfs_index_execution_backward_actualdifference) = (La)) -> exists pfs_left_execution_backward_actualdifference pfs_right_execution_backward_actualdifference pfs_result_execution_backward_actualdifference. ((((exists ff_h_pfp_execution_backward_actualdifferenceleft. ff_h_pfp_execution_backward_actualdifferenceleft + S (pfs_left_execution_backward_actualdifference) = S ((S (pfs_index_execution_backward_actualdifference)) * ac)) /\\ exists ff_q_pfp_execution_backward_actualdifferenceleft. ab = ff_q_pfp_execution_backward_actualdifferenceleft * S ((S (pfs_index_execution_backward_actualdifference)) * ac) + (pfs_left_execution_backward_actualdifference))) /\\ (((((exists ff_h_pfp_execution_backward_actualdifferenceright. ff_h_pfp_execution_backward_actualdifferenceright + S (pfs_right_execution_backward_actualdifference) = S ((S (pfs_index_execution_backward_actualdifference)) * pfd_product_scale_execution_backward_actual)) /\\ exists ff_q_pfp_execution_backward_actualdifferenceright. pfd_product_code_execution_backward_actual = ff_q_pfp_execution_backward_actualdifferenceright * S ((S (pfs_index_execution_backward_actualdifference)) * pfd_product_scale_execution_backward_actual) + (pfs_right_execution_backward_actualdifference))) /\\ (((((exists ff_h_pfp_execution_backward_actualdifferenceresult. ff_h_pfp_execution_backward_actualdifferenceresult + S (pfs_result_execution_backward_actualdifference) = S ((S (pfs_index_execution_backward_actualdifference)) * pfd_residual_scale_execution_backward_actual)) /\\ exists ff_q_pfp_execution_backward_actualdifferenceresult. pfd_residual_code_execution_backward_actual = ff_q_pfp_execution_backward_actualdifferenceresult * S ((S (pfs_index_execution_backward_actualdifference)) * pfd_residual_scale_execution_backward_actual) + (pfs_result_execution_backward_actualdifference))) /\\ ((((exists pfa_gap_execution_backward_actualdifferenceoperationleft. pfa_gap_execution_backward_actualdifferenceoperationleft + S (pfs_right_execution_backward_actualdifference) = (p)) /\\ (((exists pfa_gap_execution_backward_actualdifferenceoperationright. pfa_gap_execution_backward_actualdifferenceoperationright + S (pfs_result_execution_backward_actualdifference) = (p)) /\\ ((((exists pfa_gap_execution_backward_actualdifferenceoperationresultbound. pfa_gap_execution_backward_actualdifferenceoperationresultbound + S (pfs_left_execution_backward_actualdifference) = (p)) /\\ ((exists pfa_offset_left_execution_backward_actualdifferenceoperationresultcongruence pfa_offset_right_execution_backward_actualdifferenceoperationresultcongruence. ((pfs_right_execution_backward_actualdifference) + (pfs_result_execution_backward_actualdifference)) + (p) * pfa_offset_left_execution_backward_actualdifferenceoperationresultcongruence = (pfs_left_execution_backward_actualdifference) + (p) * pfa_offset_right_execution_backward_actualdifferenceoperationresultcongruence)))))))))))))))) /\\ (((((La)=(pfd_cut_execution_backward_actual)+(Lr)) /\\ (((forall fom_index_pfp_execution_backward_actualtriminput. (exists fom_gap_pfp_execution_backward_actualtriminput_index_bound. fom_gap_pfp_execution_backward_actualtriminput_index_bound + S (fom_index_pfp_execution_backward_actualtriminput) = La) -> exists fom_value_pfp_execution_backward_actualtriminput. ((((exists fom_beta_height_pfp_execution_backward_actualtriminput_entry. fom_beta_height_pfp_execution_backward_actualtriminput_entry + S (fom_value_pfp_execution_backward_actualtriminput) = S ((S (fom_index_pfp_execution_backward_actualtriminput)) * pfd_residual_scale_execution_backward_actual)) /\\ exists fom_beta_quotient_pfp_execution_backward_actualtriminput_entry. pfd_residual_code_execution_backward_actual = fom_beta_quotient_pfp_execution_backward_actualtriminput_entry * S ((S (fom_index_pfp_execution_backward_actualtriminput)) * pfd_residual_scale_execution_backward_actual) + (fom_value_pfp_execution_backward_actualtriminput))) /\\ (exists fom_gap_pfp_execution_backward_actualtriminput_value_bound. fom_gap_pfp_execution_backward_actualtriminput_value_bound + S (fom_value_pfp_execution_backward_actualtriminput) = p))) /\\ (((forall pfp_repeat_index_execution_backward_actualtrimremoved. (exists pfa_gap_execution_backward_actualtrimremovedindex. pfa_gap_execution_backward_actualtrimremovedindex + S (pfp_repeat_index_execution_backward_actualtrimremoved) = (pfd_cut_execution_backward_actual)) -> (((exists ff_h_pfp_execution_backward_actualtrimremovedentry. ff_h_pfp_execution_backward_actualtrimremovedentry + S (0) = S ((S (pfp_repeat_index_execution_backward_actualtrimremoved)) * pfd_residual_scale_execution_backward_actual)) /\\ exists ff_q_pfp_execution_backward_actualtrimremovedentry. pfd_residual_code_execution_backward_actual = ff_q_pfp_execution_backward_actualtrimremovedentry * S ((S (pfp_repeat_index_execution_backward_actualtrimremoved)) * pfd_residual_scale_execution_backward_actual) + (0)))) /\\ (((forall pftrim_index_execution_backward_actualtrimsuffix pftrim_value_execution_backward_actualtrimsuffix. (exists pfa_gap_execution_backward_actualtrimsuffixbound. pfa_gap_execution_backward_actualtrimsuffixbound + S (pftrim_index_execution_backward_actualtrimsuffix) = (Lr)) -> (((exists ff_h_pfp_execution_backward_actualtrimsuffixsource. ff_h_pfp_execution_backward_actualtrimsuffixsource + S (pftrim_value_execution_backward_actualtrimsuffix) = S ((S ((pfd_cut_execution_backward_actual)+pftrim_index_execution_backward_actualtrimsuffix)) * pfd_residual_scale_execution_backward_actual)) /\\ exists ff_q_pfp_execution_backward_actualtrimsuffixsource. pfd_residual_code_execution_backward_actual = ff_q_pfp_execution_backward_actualtrimsuffixsource * S ((S ((pfd_cut_execution_backward_actual)+pftrim_index_execution_backward_actualtrimsuffix)) * pfd_residual_scale_execution_backward_actual) + (pftrim_value_execution_backward_actualtrimsuffix))) -> (((exists ff_h_pfp_execution_backward_actualtrimsuffixoutput. ff_h_pfp_execution_backward_actualtrimsuffixoutput + S (pftrim_value_execution_backward_actualtrimsuffix) = S ((S (pftrim_index_execution_backward_actualtrimsuffix)) * rc)) /\\ exists ff_q_pfp_execution_backward_actualtrimsuffixoutput. rb = ff_q_pfp_execution_backward_actualtrimsuffixoutput * S ((S (pftrim_index_execution_backward_actualtrimsuffix)) * rc) + (pftrim_value_execution_backward_actualtrimsuffix)))) /\\ (((Lr)=0 \\/ (exists pftrim_leading_execution_backward_actualtrimnormal. ((((exists ff_h_pfp_execution_backward_actualtrimnormalentry. ff_h_pfp_execution_backward_actualtrimnormalentry + S (pftrim_leading_execution_backward_actualtrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_execution_backward_actualtrimnormalentry. rb = ff_q_pfp_execution_backward_actualtrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_execution_backward_actualtrimnormal))) /\\ ((~(pftrim_leading_execution_backward_actualtrimnormal=0))))))))))))))))))))))))))))))))) -> (exists pfbz_left_code_execution_backward_original pfbz_left_scale_execution_backward_original pfbz_left_length_execution_backward_original pfbz_right_code_execution_backward_original pfbz_right_scale_execution_backward_original pfbz_right_length_execution_backward_original. ((((forall fom_index_pfp_execution_backward_original_left_productleft. (exists fom_gap_pfp_execution_backward_original_left_productleft_index_bound. fom_gap_pfp_execution_backward_original_left_productleft_index_bound + S (fom_index_pfp_execution_backward_original_left_productleft) = Lu) -> exists fom_value_pfp_execution_backward_original_left_productleft. ((((exists fom_beta_height_pfp_execution_backward_original_left_productleft_entry. fom_beta_height_pfp_execution_backward_original_left_productleft_entry + S (fom_value_pfp_execution_backward_original_left_productleft) = S ((S (fom_index_pfp_execution_backward_original_left_productleft)) * uc)) /\\ exists fom_beta_quotient_pfp_execution_backward_original_left_productleft_entry. ub = fom_beta_quotient_pfp_execution_backward_original_left_productleft_entry * S ((S (fom_index_pfp_execution_backward_original_left_productleft)) * uc) + (fom_value_pfp_execution_backward_original_left_productleft))) /\\ (exists fom_gap_pfp_execution_backward_original_left_productleft_value_bound. fom_gap_pfp_execution_backward_original_left_productleft_value_bound + S (fom_value_pfp_execution_backward_original_left_productleft) = p))) /\\ (((forall fom_index_pfp_execution_backward_original_left_productright. (exists fom_gap_pfp_execution_backward_original_left_productright_index_bound. fom_gap_pfp_execution_backward_original_left_productright_index_bound + S (fom_index_pfp_execution_backward_original_left_productright) = S d) -> exists fom_value_pfp_execution_backward_original_left_productright. ((((exists fom_beta_height_pfp_execution_backward_original_left_productright_entry. fom_beta_height_pfp_execution_backward_original_left_productright_entry + S (fom_value_pfp_execution_backward_original_left_productright) = S ((S (fom_index_pfp_execution_backward_original_left_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_backward_original_left_productright_entry. bb = fom_beta_quotient_pfp_execution_backward_original_left_productright_entry * S ((S (fom_index_pfp_execution_backward_original_left_productright)) * bc) + (fom_value_pfp_execution_backward_original_left_productright))) /\\ (exists fom_gap_pfp_execution_backward_original_left_productright_value_bound. fom_gap_pfp_execution_backward_original_left_productright_value_bound + S (fom_value_pfp_execution_backward_original_left_productright) = p))) /\\ (((((((Lu)=0 \\/ (S d)=0) /\\ (((pfbz_left_length_execution_backward_original)=0)))) \\/ (((~((Lu)=0)) /\\ (((~((S d)=0)) /\\ (((Lu)+(S d)=S (pfbz_left_length_execution_backward_original)))))))) /\\ ((forall pfc_index_execution_backward_original_left_productcoefficients. (exists pfa_gap_execution_backward_original_left_productcoefficientsbound. pfa_gap_execution_backward_original_left_productcoefficientsbound + S (pfc_index_execution_backward_original_left_productcoefficients) = (pfbz_left_length_execution_backward_original)) -> exists pfc_value_execution_backward_original_left_productcoefficients. ((((exists ff_h_pfp_execution_backward_original_left_productcoefficientsentry. ff_h_pfp_execution_backward_original_left_productcoefficientsentry + S (pfc_value_execution_backward_original_left_productcoefficients) = S ((S (pfc_index_execution_backward_original_left_productcoefficients)) * pfbz_left_scale_execution_backward_original)) /\\ exists ff_q_pfp_execution_backward_original_left_productcoefficientsentry. pfbz_left_code_execution_backward_original = ff_q_pfp_execution_backward_original_left_productcoefficientsentry * S ((S (pfc_index_execution_backward_original_left_productcoefficients)) * pfbz_left_scale_execution_backward_original) + (pfc_value_execution_backward_original_left_productcoefficients))) /\\ ((exists pfc_terms_code_execution_backward_original_left_productcoefficientscoefficient pfc_terms_scale_execution_backward_original_left_productcoefficientscoefficient pfc_natural_sum_execution_backward_original_left_productcoefficientscoefficient. ((forall pfc_index_execution_backward_original_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_backward_original_left_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_backward_original_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_backward_original_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_backward_original_left_productcoefficients))) -> exists pfc_value_execution_backward_original_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_backward_original_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_backward_original_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_backward_original_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_backward_original_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_backward_original_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_backward_original_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_backward_original_left_productcoefficientscoefficient = ff_q_pfp_execution_backward_original_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_backward_original_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_backward_original_left_productcoefficientscoefficient) + (pfc_value_execution_backward_original_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_backward_original_left_productcoefficientscoefficientdiagonalterm pfc_left_execution_backward_original_left_productcoefficientscoefficientdiagonalterm pfc_right_execution_backward_original_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_backward_original_left_productcoefficientscoefficientdiagonal)+pfc_complement_execution_backward_original_left_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_backward_original_left_productcoefficients)) /\\ ((((((exists pfa_gap_execution_backward_original_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_backward_original_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_backward_original_left_productcoefficientscoefficientdiagonal) = (Lu)) /\\ ((((exists ff_h_pfp_execution_backward_original_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_backward_original_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_backward_original_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_backward_original_left_productcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_execution_backward_original_left_productcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_execution_backward_original_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_backward_original_left_productcoefficientscoefficientdiagonal)) * uc) + (pfc_left_execution_backward_original_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_original_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_backward_original_left_productcoefficientscoefficientdiagonaltermleftoutside+(Lu)=(pfc_index_execution_backward_original_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_backward_original_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_backward_original_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_backward_original_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_backward_original_left_productcoefficientscoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_execution_backward_original_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_backward_original_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_backward_original_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_backward_original_left_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_backward_original_left_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_backward_original_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_backward_original_left_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_execution_backward_original_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_original_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_backward_original_left_productcoefficientscoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_execution_backward_original_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_backward_original_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_backward_original_left_productcoefficientscoefficientdiagonal)=pfc_left_execution_backward_original_left_productcoefficientscoefficientdiagonalterm*pfc_right_execution_backward_original_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_backward_original_left_productcoefficientscoefficientsum fs_v_pfc_execution_backward_original_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_backward_original_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_backward_original_left_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_backward_original_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_backward_original_left_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_backward_original_left_productcoefficients))) * fs_v_pfc_execution_backward_original_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_backward_original_left_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_backward_original_left_productcoefficients))) * fs_v_pfc_execution_backward_original_left_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_backward_original_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_backward_original_left_productcoefficients)) -> exists fs_a_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_backward_original_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_backward_original_left_productcoefficientscoefficient = fs_q_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_backward_original_left_productcoefficientscoefficient) + (fs_a_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_original_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_backward_original_left_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_original_left_productcoefficientscoefficientsum) + (fs_r_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_original_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_backward_original_left_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_original_left_productcoefficientscoefficientsum) + (fs_s_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_backward_original_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_backward_original_left_productcoefficientscoefficientresiduebound. pfa_gap_execution_backward_original_left_productcoefficientscoefficientresiduebound + S (pfc_value_execution_backward_original_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_backward_original_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_backward_original_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_backward_original_left_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_backward_original_left_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_backward_original_left_productcoefficients) + (p) * pfa_offset_right_execution_backward_original_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_execution_backward_original_right_productleft. (exists fom_gap_pfp_execution_backward_original_right_productleft_index_bound. fom_gap_pfp_execution_backward_original_right_productleft_index_bound + S (fom_index_pfp_execution_backward_original_right_productleft) = Lv) -> exists fom_value_pfp_execution_backward_original_right_productleft. ((((exists fom_beta_height_pfp_execution_backward_original_right_productleft_entry. fom_beta_height_pfp_execution_backward_original_right_productleft_entry + S (fom_value_pfp_execution_backward_original_right_productleft) = S ((S (fom_index_pfp_execution_backward_original_right_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_execution_backward_original_right_productleft_entry. vb = fom_beta_quotient_pfp_execution_backward_original_right_productleft_entry * S ((S (fom_index_pfp_execution_backward_original_right_productleft)) * vc) + (fom_value_pfp_execution_backward_original_right_productleft))) /\\ (exists fom_gap_pfp_execution_backward_original_right_productleft_value_bound. fom_gap_pfp_execution_backward_original_right_productleft_value_bound + S (fom_value_pfp_execution_backward_original_right_productleft) = p))) /\\ (((forall fom_index_pfp_execution_backward_original_right_productright. (exists fom_gap_pfp_execution_backward_original_right_productright_index_bound. fom_gap_pfp_execution_backward_original_right_productright_index_bound + S (fom_index_pfp_execution_backward_original_right_productright) = Lr) -> exists fom_value_pfp_execution_backward_original_right_productright. ((((exists fom_beta_height_pfp_execution_backward_original_right_productright_entry. fom_beta_height_pfp_execution_backward_original_right_productright_entry + S (fom_value_pfp_execution_backward_original_right_productright) = S ((S (fom_index_pfp_execution_backward_original_right_productright)) * rc)) /\\ exists fom_beta_quotient_pfp_execution_backward_original_right_productright_entry. rb = fom_beta_quotient_pfp_execution_backward_original_right_productright_entry * S ((S (fom_index_pfp_execution_backward_original_right_productright)) * rc) + (fom_value_pfp_execution_backward_original_right_productright))) /\\ (exists fom_gap_pfp_execution_backward_original_right_productright_value_bound. fom_gap_pfp_execution_backward_original_right_productright_value_bound + S (fom_value_pfp_execution_backward_original_right_productright) = p))) /\\ (((((((Lv)=0 \\/ (Lr)=0) /\\ (((pfbz_right_length_execution_backward_original)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((Lr)=0)) /\\ (((Lv)+(Lr)=S (pfbz_right_length_execution_backward_original)))))))) /\\ ((forall pfc_index_execution_backward_original_right_productcoefficients. (exists pfa_gap_execution_backward_original_right_productcoefficientsbound. pfa_gap_execution_backward_original_right_productcoefficientsbound + S (pfc_index_execution_backward_original_right_productcoefficients) = (pfbz_right_length_execution_backward_original)) -> exists pfc_value_execution_backward_original_right_productcoefficients. ((((exists ff_h_pfp_execution_backward_original_right_productcoefficientsentry. ff_h_pfp_execution_backward_original_right_productcoefficientsentry + S (pfc_value_execution_backward_original_right_productcoefficients) = S ((S (pfc_index_execution_backward_original_right_productcoefficients)) * pfbz_right_scale_execution_backward_original)) /\\ exists ff_q_pfp_execution_backward_original_right_productcoefficientsentry. pfbz_right_code_execution_backward_original = ff_q_pfp_execution_backward_original_right_productcoefficientsentry * S ((S (pfc_index_execution_backward_original_right_productcoefficients)) * pfbz_right_scale_execution_backward_original) + (pfc_value_execution_backward_original_right_productcoefficients))) /\\ ((exists pfc_terms_code_execution_backward_original_right_productcoefficientscoefficient pfc_terms_scale_execution_backward_original_right_productcoefficientscoefficient pfc_natural_sum_execution_backward_original_right_productcoefficientscoefficient. ((forall pfc_index_execution_backward_original_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_backward_original_right_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_backward_original_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_backward_original_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_backward_original_right_productcoefficients))) -> exists pfc_value_execution_backward_original_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_backward_original_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_backward_original_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_backward_original_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_backward_original_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_backward_original_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_backward_original_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_backward_original_right_productcoefficientscoefficient = ff_q_pfp_execution_backward_original_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_backward_original_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_backward_original_right_productcoefficientscoefficient) + (pfc_value_execution_backward_original_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_backward_original_right_productcoefficientscoefficientdiagonalterm pfc_left_execution_backward_original_right_productcoefficientscoefficientdiagonalterm pfc_right_execution_backward_original_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_backward_original_right_productcoefficientscoefficientdiagonal)+pfc_complement_execution_backward_original_right_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_backward_original_right_productcoefficients)) /\\ ((((((exists pfa_gap_execution_backward_original_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_backward_original_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_backward_original_right_productcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_execution_backward_original_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_backward_original_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_backward_original_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_backward_original_right_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_execution_backward_original_right_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_execution_backward_original_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_backward_original_right_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_execution_backward_original_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_original_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_backward_original_right_productcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_execution_backward_original_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_backward_original_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_backward_original_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_backward_original_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_backward_original_right_productcoefficientscoefficientdiagonalterm) = (Lr)) /\\ ((((exists ff_h_pfp_execution_backward_original_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_backward_original_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_backward_original_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_backward_original_right_productcoefficientscoefficientdiagonalterm)) * rc)) /\\ exists ff_q_pfp_execution_backward_original_right_productcoefficientscoefficientdiagonaltermrightentry. rb = ff_q_pfp_execution_backward_original_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_backward_original_right_productcoefficientscoefficientdiagonalterm)) * rc) + (pfc_right_execution_backward_original_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_original_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_backward_original_right_productcoefficientscoefficientdiagonaltermrightoutside+(Lr)=(pfc_complement_execution_backward_original_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_backward_original_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_backward_original_right_productcoefficientscoefficientdiagonal)=pfc_left_execution_backward_original_right_productcoefficientscoefficientdiagonalterm*pfc_right_execution_backward_original_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_backward_original_right_productcoefficientscoefficientsum fs_v_pfc_execution_backward_original_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_backward_original_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_backward_original_right_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_backward_original_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_backward_original_right_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_backward_original_right_productcoefficients))) * fs_v_pfc_execution_backward_original_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_backward_original_right_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_backward_original_right_productcoefficients))) * fs_v_pfc_execution_backward_original_right_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_backward_original_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_backward_original_right_productcoefficients)) -> exists fs_a_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_backward_original_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_backward_original_right_productcoefficientscoefficient = fs_q_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_backward_original_right_productcoefficientscoefficient) + (fs_a_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_original_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_backward_original_right_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_original_right_productcoefficientscoefficientsum) + (fs_r_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_original_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_backward_original_right_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_original_right_productcoefficientscoefficientsum) + (fs_s_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_backward_original_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_backward_original_right_productcoefficientscoefficientresiduebound. pfa_gap_execution_backward_original_right_productcoefficientscoefficientresiduebound + S (pfc_value_execution_backward_original_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_backward_original_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_backward_original_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_backward_original_right_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_backward_original_right_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_backward_original_right_productcoefficients) + (p) * pfa_offset_right_execution_backward_original_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_execution_backward_original_sum_left_bounded. (exists fom_gap_pfp_execution_backward_original_sum_left_bounded_index_bound. fom_gap_pfp_execution_backward_original_sum_left_bounded_index_bound + S (fom_index_pfp_execution_backward_original_sum_left_bounded) = pfbz_left_length_execution_backward_original) -> exists fom_value_pfp_execution_backward_original_sum_left_bounded. ((((exists fom_beta_height_pfp_execution_backward_original_sum_left_bounded_entry. fom_beta_height_pfp_execution_backward_original_sum_left_bounded_entry + S (fom_value_pfp_execution_backward_original_sum_left_bounded) = S ((S (fom_index_pfp_execution_backward_original_sum_left_bounded)) * pfbz_left_scale_execution_backward_original)) /\\ exists fom_beta_quotient_pfp_execution_backward_original_sum_left_bounded_entry. pfbz_left_code_execution_backward_original = fom_beta_quotient_pfp_execution_backward_original_sum_left_bounded_entry * S ((S (fom_index_pfp_execution_backward_original_sum_left_bounded)) * pfbz_left_scale_execution_backward_original) + (fom_value_pfp_execution_backward_original_sum_left_bounded))) /\\ (exists fom_gap_pfp_execution_backward_original_sum_left_bounded_value_bound. fom_gap_pfp_execution_backward_original_sum_left_bounded_value_bound + S (fom_value_pfp_execution_backward_original_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_execution_backward_original_sum_right_bounded. (exists fom_gap_pfp_execution_backward_original_sum_right_bounded_index_bound. fom_gap_pfp_execution_backward_original_sum_right_bounded_index_bound + S (fom_index_pfp_execution_backward_original_sum_right_bounded) = pfbz_right_length_execution_backward_original) -> exists fom_value_pfp_execution_backward_original_sum_right_bounded. ((((exists fom_beta_height_pfp_execution_backward_original_sum_right_bounded_entry. fom_beta_height_pfp_execution_backward_original_sum_right_bounded_entry + S (fom_value_pfp_execution_backward_original_sum_right_bounded) = S ((S (fom_index_pfp_execution_backward_original_sum_right_bounded)) * pfbz_right_scale_execution_backward_original)) /\\ exists fom_beta_quotient_pfp_execution_backward_original_sum_right_bounded_entry. pfbz_right_code_execution_backward_original = fom_beta_quotient_pfp_execution_backward_original_sum_right_bounded_entry * S ((S (fom_index_pfp_execution_backward_original_sum_right_bounded)) * pfbz_right_scale_execution_backward_original) + (fom_value_pfp_execution_backward_original_sum_right_bounded))) /\\ (exists fom_gap_pfp_execution_backward_original_sum_right_bounded_value_bound. fom_gap_pfp_execution_backward_original_sum_right_bounded_value_bound + S (fom_value_pfp_execution_backward_original_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_execution_backward_original_sum_result_bounded. (exists fom_gap_pfp_execution_backward_original_sum_result_bounded_index_bound. fom_gap_pfp_execution_backward_original_sum_result_bounded_index_bound + S (fom_index_pfp_execution_backward_original_sum_result_bounded) = Lg) -> exists fom_value_pfp_execution_backward_original_sum_result_bounded. ((((exists fom_beta_height_pfp_execution_backward_original_sum_result_bounded_entry. fom_beta_height_pfp_execution_backward_original_sum_result_bounded_entry + S (fom_value_pfp_execution_backward_original_sum_result_bounded) = S ((S (fom_index_pfp_execution_backward_original_sum_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_execution_backward_original_sum_result_bounded_entry. gb = fom_beta_quotient_pfp_execution_backward_original_sum_result_bounded_entry * S ((S (fom_index_pfp_execution_backward_original_sum_result_bounded)) * gc) + (fom_value_pfp_execution_backward_original_sum_result_bounded))) /\\ (exists fom_gap_pfp_execution_backward_original_sum_result_bounded_value_bound. fom_gap_pfp_execution_backward_original_sum_result_bounded_value_bound + S (fom_value_pfp_execution_backward_original_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_execution_backward_original_sum pfaa_left_c_execution_backward_original_sum pfaa_right_b_execution_backward_original_sum pfaa_right_c_execution_backward_original_sum pfaa_sum_b_execution_backward_original_sum pfaa_sum_c_execution_backward_original_sum pfaa_length_execution_backward_original_sum. ((((forall pfrep_power_execution_backward_original_sum_witness_common_left pfrep_left_execution_backward_original_sum_witness_common_left pfrep_right_execution_backward_original_sum_witness_common_left. ((exists pfrep_position_execution_backward_original_sum_witness_common_leftfirst. ((pfrep_position_execution_backward_original_sum_witness_common_leftfirst+S (pfrep_power_execution_backward_original_sum_witness_common_left)=(pfbz_left_length_execution_backward_original)) /\\ ((((exists ff_h_pfp_execution_backward_original_sum_witness_common_leftfirstentry. ff_h_pfp_execution_backward_original_sum_witness_common_leftfirstentry + S (pfrep_left_execution_backward_original_sum_witness_common_left) = S ((S (pfrep_position_execution_backward_original_sum_witness_common_leftfirst)) * pfbz_left_scale_execution_backward_original)) /\\ exists ff_q_pfp_execution_backward_original_sum_witness_common_leftfirstentry. pfbz_left_code_execution_backward_original = ff_q_pfp_execution_backward_original_sum_witness_common_leftfirstentry * S ((S (pfrep_position_execution_backward_original_sum_witness_common_leftfirst)) * pfbz_left_scale_execution_backward_original) + (pfrep_left_execution_backward_original_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_execution_backward_original_sum_witness_common_leftfirstoutside. pfrep_gap_execution_backward_original_sum_witness_common_leftfirstoutside+(pfbz_left_length_execution_backward_original)=(pfrep_power_execution_backward_original_sum_witness_common_left)) /\\ (((pfrep_left_execution_backward_original_sum_witness_common_left)=0))))) -> ((exists pfrep_position_execution_backward_original_sum_witness_common_leftsecond. ((pfrep_position_execution_backward_original_sum_witness_common_leftsecond+S (pfrep_power_execution_backward_original_sum_witness_common_left)=(pfaa_length_execution_backward_original_sum)) /\\ ((((exists ff_h_pfp_execution_backward_original_sum_witness_common_leftsecondentry. ff_h_pfp_execution_backward_original_sum_witness_common_leftsecondentry + S (pfrep_right_execution_backward_original_sum_witness_common_left) = S ((S (pfrep_position_execution_backward_original_sum_witness_common_leftsecond)) * pfaa_left_c_execution_backward_original_sum)) /\\ exists ff_q_pfp_execution_backward_original_sum_witness_common_leftsecondentry. pfaa_left_b_execution_backward_original_sum = ff_q_pfp_execution_backward_original_sum_witness_common_leftsecondentry * S ((S (pfrep_position_execution_backward_original_sum_witness_common_leftsecond)) * pfaa_left_c_execution_backward_original_sum) + (pfrep_right_execution_backward_original_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_execution_backward_original_sum_witness_common_leftsecondoutside. pfrep_gap_execution_backward_original_sum_witness_common_leftsecondoutside+(pfaa_length_execution_backward_original_sum)=(pfrep_power_execution_backward_original_sum_witness_common_left)) /\\ (((pfrep_right_execution_backward_original_sum_witness_common_left)=0))))) -> pfrep_left_execution_backward_original_sum_witness_common_left=pfrep_right_execution_backward_original_sum_witness_common_left) /\\ ((forall pfrep_power_execution_backward_original_sum_witness_common_right pfrep_left_execution_backward_original_sum_witness_common_right pfrep_right_execution_backward_original_sum_witness_common_right. ((exists pfrep_position_execution_backward_original_sum_witness_common_rightfirst. ((pfrep_position_execution_backward_original_sum_witness_common_rightfirst+S (pfrep_power_execution_backward_original_sum_witness_common_right)=(pfbz_right_length_execution_backward_original)) /\\ ((((exists ff_h_pfp_execution_backward_original_sum_witness_common_rightfirstentry. ff_h_pfp_execution_backward_original_sum_witness_common_rightfirstentry + S (pfrep_left_execution_backward_original_sum_witness_common_right) = S ((S (pfrep_position_execution_backward_original_sum_witness_common_rightfirst)) * pfbz_right_scale_execution_backward_original)) /\\ exists ff_q_pfp_execution_backward_original_sum_witness_common_rightfirstentry. pfbz_right_code_execution_backward_original = ff_q_pfp_execution_backward_original_sum_witness_common_rightfirstentry * S ((S (pfrep_position_execution_backward_original_sum_witness_common_rightfirst)) * pfbz_right_scale_execution_backward_original) + (pfrep_left_execution_backward_original_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_execution_backward_original_sum_witness_common_rightfirstoutside. pfrep_gap_execution_backward_original_sum_witness_common_rightfirstoutside+(pfbz_right_length_execution_backward_original)=(pfrep_power_execution_backward_original_sum_witness_common_right)) /\\ (((pfrep_left_execution_backward_original_sum_witness_common_right)=0))))) -> ((exists pfrep_position_execution_backward_original_sum_witness_common_rightsecond. ((pfrep_position_execution_backward_original_sum_witness_common_rightsecond+S (pfrep_power_execution_backward_original_sum_witness_common_right)=(pfaa_length_execution_backward_original_sum)) /\\ ((((exists ff_h_pfp_execution_backward_original_sum_witness_common_rightsecondentry. ff_h_pfp_execution_backward_original_sum_witness_common_rightsecondentry + S (pfrep_right_execution_backward_original_sum_witness_common_right) = S ((S (pfrep_position_execution_backward_original_sum_witness_common_rightsecond)) * pfaa_right_c_execution_backward_original_sum)) /\\ exists ff_q_pfp_execution_backward_original_sum_witness_common_rightsecondentry. pfaa_right_b_execution_backward_original_sum = ff_q_pfp_execution_backward_original_sum_witness_common_rightsecondentry * S ((S (pfrep_position_execution_backward_original_sum_witness_common_rightsecond)) * pfaa_right_c_execution_backward_original_sum) + (pfrep_right_execution_backward_original_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_execution_backward_original_sum_witness_common_rightsecondoutside. pfrep_gap_execution_backward_original_sum_witness_common_rightsecondoutside+(pfaa_length_execution_backward_original_sum)=(pfrep_power_execution_backward_original_sum_witness_common_right)) /\\ (((pfrep_right_execution_backward_original_sum_witness_common_right)=0))))) -> pfrep_left_execution_backward_original_sum_witness_common_right=pfrep_right_execution_backward_original_sum_witness_common_right)))) /\\ (((forall pfp_index_execution_backward_original_sum_witness_operation. (exists pfa_gap_execution_backward_original_sum_witness_operationindex. pfa_gap_execution_backward_original_sum_witness_operationindex + S (pfp_index_execution_backward_original_sum_witness_operation) = (pfaa_length_execution_backward_original_sum)) -> exists pfp_left_execution_backward_original_sum_witness_operation pfp_right_execution_backward_original_sum_witness_operation pfp_value_execution_backward_original_sum_witness_operation. ((((exists ff_h_pfp_execution_backward_original_sum_witness_operationleft. ff_h_pfp_execution_backward_original_sum_witness_operationleft + S (pfp_left_execution_backward_original_sum_witness_operation) = S ((S (pfp_index_execution_backward_original_sum_witness_operation)) * pfaa_left_c_execution_backward_original_sum)) /\\ exists ff_q_pfp_execution_backward_original_sum_witness_operationleft. pfaa_left_b_execution_backward_original_sum = ff_q_pfp_execution_backward_original_sum_witness_operationleft * S ((S (pfp_index_execution_backward_original_sum_witness_operation)) * pfaa_left_c_execution_backward_original_sum) + (pfp_left_execution_backward_original_sum_witness_operation))) /\\ (((((exists ff_h_pfp_execution_backward_original_sum_witness_operationright. ff_h_pfp_execution_backward_original_sum_witness_operationright + S (pfp_right_execution_backward_original_sum_witness_operation) = S ((S (pfp_index_execution_backward_original_sum_witness_operation)) * pfaa_right_c_execution_backward_original_sum)) /\\ exists ff_q_pfp_execution_backward_original_sum_witness_operationright. pfaa_right_b_execution_backward_original_sum = ff_q_pfp_execution_backward_original_sum_witness_operationright * S ((S (pfp_index_execution_backward_original_sum_witness_operation)) * pfaa_right_c_execution_backward_original_sum) + (pfp_right_execution_backward_original_sum_witness_operation))) /\\ (((((exists ff_h_pfp_execution_backward_original_sum_witness_operationtarget. ff_h_pfp_execution_backward_original_sum_witness_operationtarget + S (pfp_value_execution_backward_original_sum_witness_operation) = S ((S (pfp_index_execution_backward_original_sum_witness_operation)) * pfaa_sum_c_execution_backward_original_sum)) /\\ exists ff_q_pfp_execution_backward_original_sum_witness_operationtarget. pfaa_sum_b_execution_backward_original_sum = ff_q_pfp_execution_backward_original_sum_witness_operationtarget * S ((S (pfp_index_execution_backward_original_sum_witness_operation)) * pfaa_sum_c_execution_backward_original_sum) + (pfp_value_execution_backward_original_sum_witness_operation))) /\\ ((((exists pfa_gap_execution_backward_original_sum_witness_operationoperationleft. pfa_gap_execution_backward_original_sum_witness_operationoperationleft + S (pfp_left_execution_backward_original_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_execution_backward_original_sum_witness_operationoperationright. pfa_gap_execution_backward_original_sum_witness_operationoperationright + S (pfp_right_execution_backward_original_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_execution_backward_original_sum_witness_operationoperationresultbound. pfa_gap_execution_backward_original_sum_witness_operationoperationresultbound + S (pfp_value_execution_backward_original_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_execution_backward_original_sum_witness_operationoperationresultcongruence pfa_offset_right_execution_backward_original_sum_witness_operationoperationresultcongruence. ((pfp_left_execution_backward_original_sum_witness_operation) + (pfp_right_execution_backward_original_sum_witness_operation)) + (p) * pfa_offset_left_execution_backward_original_sum_witness_operationoperationresultcongruence = (pfp_value_execution_backward_original_sum_witness_operation) + (p) * pfa_offset_right_execution_backward_original_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_execution_backward_original_sum_witness_output pfrep_left_execution_backward_original_sum_witness_output pfrep_right_execution_backward_original_sum_witness_output. ((exists pfrep_position_execution_backward_original_sum_witness_outputfirst. ((pfrep_position_execution_backward_original_sum_witness_outputfirst+S (pfrep_power_execution_backward_original_sum_witness_output)=(pfaa_length_execution_backward_original_sum)) /\\ ((((exists ff_h_pfp_execution_backward_original_sum_witness_outputfirstentry. ff_h_pfp_execution_backward_original_sum_witness_outputfirstentry + S (pfrep_left_execution_backward_original_sum_witness_output) = S ((S (pfrep_position_execution_backward_original_sum_witness_outputfirst)) * pfaa_sum_c_execution_backward_original_sum)) /\\ exists ff_q_pfp_execution_backward_original_sum_witness_outputfirstentry. pfaa_sum_b_execution_backward_original_sum = ff_q_pfp_execution_backward_original_sum_witness_outputfirstentry * S ((S (pfrep_position_execution_backward_original_sum_witness_outputfirst)) * pfaa_sum_c_execution_backward_original_sum) + (pfrep_left_execution_backward_original_sum_witness_output)))))) \\/ (((exists pfrep_gap_execution_backward_original_sum_witness_outputfirstoutside. pfrep_gap_execution_backward_original_sum_witness_outputfirstoutside+(pfaa_length_execution_backward_original_sum)=(pfrep_power_execution_backward_original_sum_witness_output)) /\\ (((pfrep_left_execution_backward_original_sum_witness_output)=0))))) -> ((exists pfrep_position_execution_backward_original_sum_witness_outputsecond. ((pfrep_position_execution_backward_original_sum_witness_outputsecond+S (pfrep_power_execution_backward_original_sum_witness_output)=(Lg)) /\\ ((((exists ff_h_pfp_execution_backward_original_sum_witness_outputsecondentry. ff_h_pfp_execution_backward_original_sum_witness_outputsecondentry + S (pfrep_right_execution_backward_original_sum_witness_output) = S ((S (pfrep_position_execution_backward_original_sum_witness_outputsecond)) * gc)) /\\ exists ff_q_pfp_execution_backward_original_sum_witness_outputsecondentry. gb = ff_q_pfp_execution_backward_original_sum_witness_outputsecondentry * S ((S (pfrep_position_execution_backward_original_sum_witness_outputsecond)) * gc) + (pfrep_right_execution_backward_original_sum_witness_output)))))) \\/ (((exists pfrep_gap_execution_backward_original_sum_witness_outputsecondoutside. pfrep_gap_execution_backward_original_sum_witness_outputsecondoutside+(Lg)=(pfrep_power_execution_backward_original_sum_witness_output)) /\\ (((pfrep_right_execution_backward_original_sum_witness_output)=0))))) -> pfrep_left_execution_backward_original_sum_witness_output=pfrep_right_execution_backward_original_sum_witness_output)))))))))))))))))) -> (exists pfbz_update_wb_execution_backward_result pfbz_update_wc_execution_backward_result pfbz_update_W_execution_backward_result pfbz_update_tb_execution_backward_result pfbz_update_tc_execution_backward_result pfbz_update_T_execution_backward_result. ((((forall fom_index_pfp_execution_backward_result_coefficient_productleft. (exists fom_gap_pfp_execution_backward_result_coefficient_productleft_index_bound. fom_gap_pfp_execution_backward_result_coefficient_productleft_index_bound + S (fom_index_pfp_execution_backward_result_coefficient_productleft) = Lv) -> exists fom_value_pfp_execution_backward_result_coefficient_productleft. ((((exists fom_beta_height_pfp_execution_backward_result_coefficient_productleft_entry. fom_beta_height_pfp_execution_backward_result_coefficient_productleft_entry + S (fom_value_pfp_execution_backward_result_coefficient_productleft) = S ((S (fom_index_pfp_execution_backward_result_coefficient_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_execution_backward_result_coefficient_productleft_entry. vb = fom_beta_quotient_pfp_execution_backward_result_coefficient_productleft_entry * S ((S (fom_index_pfp_execution_backward_result_coefficient_productleft)) * vc) + (fom_value_pfp_execution_backward_result_coefficient_productleft))) /\\ (exists fom_gap_pfp_execution_backward_result_coefficient_productleft_value_bound. fom_gap_pfp_execution_backward_result_coefficient_productleft_value_bound + S (fom_value_pfp_execution_backward_result_coefficient_productleft) = p))) /\\ (((forall fom_index_pfp_execution_backward_result_coefficient_productright. (exists fom_gap_pfp_execution_backward_result_coefficient_productright_index_bound. fom_gap_pfp_execution_backward_result_coefficient_productright_index_bound + S (fom_index_pfp_execution_backward_result_coefficient_productright) = Lq) -> exists fom_value_pfp_execution_backward_result_coefficient_productright. ((((exists fom_beta_height_pfp_execution_backward_result_coefficient_productright_entry. fom_beta_height_pfp_execution_backward_result_coefficient_productright_entry + S (fom_value_pfp_execution_backward_result_coefficient_productright) = S ((S (fom_index_pfp_execution_backward_result_coefficient_productright)) * qc)) /\\ exists fom_beta_quotient_pfp_execution_backward_result_coefficient_productright_entry. qb = fom_beta_quotient_pfp_execution_backward_result_coefficient_productright_entry * S ((S (fom_index_pfp_execution_backward_result_coefficient_productright)) * qc) + (fom_value_pfp_execution_backward_result_coefficient_productright))) /\\ (exists fom_gap_pfp_execution_backward_result_coefficient_productright_value_bound. fom_gap_pfp_execution_backward_result_coefficient_productright_value_bound + S (fom_value_pfp_execution_backward_result_coefficient_productright) = p))) /\\ (((((((Lv)=0 \\/ (Lq)=0) /\\ (((pfbz_update_W_execution_backward_result)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((Lq)=0)) /\\ (((Lv)+(Lq)=S (pfbz_update_W_execution_backward_result)))))))) /\\ ((forall pfc_index_execution_backward_result_coefficient_productcoefficients. (exists pfa_gap_execution_backward_result_coefficient_productcoefficientsbound. pfa_gap_execution_backward_result_coefficient_productcoefficientsbound + S (pfc_index_execution_backward_result_coefficient_productcoefficients) = (pfbz_update_W_execution_backward_result)) -> exists pfc_value_execution_backward_result_coefficient_productcoefficients. ((((exists ff_h_pfp_execution_backward_result_coefficient_productcoefficientsentry. ff_h_pfp_execution_backward_result_coefficient_productcoefficientsentry + S (pfc_value_execution_backward_result_coefficient_productcoefficients) = S ((S (pfc_index_execution_backward_result_coefficient_productcoefficients)) * pfbz_update_wc_execution_backward_result)) /\\ exists ff_q_pfp_execution_backward_result_coefficient_productcoefficientsentry. pfbz_update_wb_execution_backward_result = ff_q_pfp_execution_backward_result_coefficient_productcoefficientsentry * S ((S (pfc_index_execution_backward_result_coefficient_productcoefficients)) * pfbz_update_wc_execution_backward_result) + (pfc_value_execution_backward_result_coefficient_productcoefficients))) /\\ ((exists pfc_terms_code_execution_backward_result_coefficient_productcoefficientscoefficient pfc_terms_scale_execution_backward_result_coefficient_productcoefficientscoefficient pfc_natural_sum_execution_backward_result_coefficient_productcoefficientscoefficient. ((forall pfc_index_execution_backward_result_coefficient_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_backward_result_coefficient_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_backward_result_coefficient_productcoefficients))) -> exists pfc_value_execution_backward_result_coefficient_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_backward_result_coefficient_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_backward_result_coefficient_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_backward_result_coefficient_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_backward_result_coefficient_productcoefficientscoefficient = ff_q_pfp_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_backward_result_coefficient_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_backward_result_coefficient_productcoefficientscoefficient) + (pfc_value_execution_backward_result_coefficient_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm pfc_left_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm pfc_right_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_backward_result_coefficient_productcoefficientscoefficientdiagonal)+pfc_complement_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_backward_result_coefficient_productcoefficients)) /\\ ((((((exists pfa_gap_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_backward_result_coefficient_productcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_backward_result_coefficient_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_backward_result_coefficient_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_execution_backward_result_coefficient_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm) = (Lq)) /\\ ((((exists ff_h_pfp_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm)) * qc)) /\\ exists ff_q_pfp_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightentry. qb = ff_q_pfp_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm)) * qc) + (pfc_right_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_backward_result_coefficient_productcoefficientscoefficientdiagonaltermrightoutside+(Lq)=(pfc_complement_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_backward_result_coefficient_productcoefficientscoefficientdiagonal)=pfc_left_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm*pfc_right_execution_backward_result_coefficient_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum fs_v_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_backward_result_coefficient_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_backward_result_coefficient_productcoefficients))) * fs_v_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_backward_result_coefficient_productcoefficients))) * fs_v_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_backward_result_coefficient_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_backward_result_coefficient_productcoefficients)) -> exists fs_a_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_backward_result_coefficient_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_backward_result_coefficient_productcoefficientscoefficient = fs_q_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_backward_result_coefficient_productcoefficientscoefficient) + (fs_a_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum) + (fs_r_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum) + (fs_s_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_backward_result_coefficient_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_backward_result_coefficient_productcoefficientscoefficientresiduebound. pfa_gap_execution_backward_result_coefficient_productcoefficientscoefficientresiduebound + S (pfc_value_execution_backward_result_coefficient_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_backward_result_coefficient_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_backward_result_coefficient_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_backward_result_coefficient_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_backward_result_coefficient_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_backward_result_coefficient_productcoefficients) + (p) * pfa_offset_right_execution_backward_result_coefficient_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_execution_backward_result_coefficient_difference_left_bounded. (exists fom_gap_pfp_execution_backward_result_coefficient_difference_left_bounded_index_bound. fom_gap_pfp_execution_backward_result_coefficient_difference_left_bounded_index_bound + S (fom_index_pfp_execution_backward_result_coefficient_difference_left_bounded) = pfbz_update_W_execution_backward_result) -> exists fom_value_pfp_execution_backward_result_coefficient_difference_left_bounded. ((((exists fom_beta_height_pfp_execution_backward_result_coefficient_difference_left_bounded_entry. fom_beta_height_pfp_execution_backward_result_coefficient_difference_left_bounded_entry + S (fom_value_pfp_execution_backward_result_coefficient_difference_left_bounded) = S ((S (fom_index_pfp_execution_backward_result_coefficient_difference_left_bounded)) * pfbz_update_wc_execution_backward_result)) /\\ exists fom_beta_quotient_pfp_execution_backward_result_coefficient_difference_left_bounded_entry. pfbz_update_wb_execution_backward_result = fom_beta_quotient_pfp_execution_backward_result_coefficient_difference_left_bounded_entry * S ((S (fom_index_pfp_execution_backward_result_coefficient_difference_left_bounded)) * pfbz_update_wc_execution_backward_result) + (fom_value_pfp_execution_backward_result_coefficient_difference_left_bounded))) /\\ (exists fom_gap_pfp_execution_backward_result_coefficient_difference_left_bounded_value_bound. fom_gap_pfp_execution_backward_result_coefficient_difference_left_bounded_value_bound + S (fom_value_pfp_execution_backward_result_coefficient_difference_left_bounded) = p))) /\\ (((forall fom_index_pfp_execution_backward_result_coefficient_difference_right_bounded. (exists fom_gap_pfp_execution_backward_result_coefficient_difference_right_bounded_index_bound. fom_gap_pfp_execution_backward_result_coefficient_difference_right_bounded_index_bound + S (fom_index_pfp_execution_backward_result_coefficient_difference_right_bounded) = pfbz_update_T_execution_backward_result) -> exists fom_value_pfp_execution_backward_result_coefficient_difference_right_bounded. ((((exists fom_beta_height_pfp_execution_backward_result_coefficient_difference_right_bounded_entry. fom_beta_height_pfp_execution_backward_result_coefficient_difference_right_bounded_entry + S (fom_value_pfp_execution_backward_result_coefficient_difference_right_bounded) = S ((S (fom_index_pfp_execution_backward_result_coefficient_difference_right_bounded)) * pfbz_update_tc_execution_backward_result)) /\\ exists fom_beta_quotient_pfp_execution_backward_result_coefficient_difference_right_bounded_entry. pfbz_update_tb_execution_backward_result = fom_beta_quotient_pfp_execution_backward_result_coefficient_difference_right_bounded_entry * S ((S (fom_index_pfp_execution_backward_result_coefficient_difference_right_bounded)) * pfbz_update_tc_execution_backward_result) + (fom_value_pfp_execution_backward_result_coefficient_difference_right_bounded))) /\\ (exists fom_gap_pfp_execution_backward_result_coefficient_difference_right_bounded_value_bound. fom_gap_pfp_execution_backward_result_coefficient_difference_right_bounded_value_bound + S (fom_value_pfp_execution_backward_result_coefficient_difference_right_bounded) = p))) /\\ (((forall fom_index_pfp_execution_backward_result_coefficient_difference_result_bounded. (exists fom_gap_pfp_execution_backward_result_coefficient_difference_result_bounded_index_bound. fom_gap_pfp_execution_backward_result_coefficient_difference_result_bounded_index_bound + S (fom_index_pfp_execution_backward_result_coefficient_difference_result_bounded) = Lu) -> exists fom_value_pfp_execution_backward_result_coefficient_difference_result_bounded. ((((exists fom_beta_height_pfp_execution_backward_result_coefficient_difference_result_bounded_entry. fom_beta_height_pfp_execution_backward_result_coefficient_difference_result_bounded_entry + S (fom_value_pfp_execution_backward_result_coefficient_difference_result_bounded) = S ((S (fom_index_pfp_execution_backward_result_coefficient_difference_result_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_execution_backward_result_coefficient_difference_result_bounded_entry. ub = fom_beta_quotient_pfp_execution_backward_result_coefficient_difference_result_bounded_entry * S ((S (fom_index_pfp_execution_backward_result_coefficient_difference_result_bounded)) * uc) + (fom_value_pfp_execution_backward_result_coefficient_difference_result_bounded))) /\\ (exists fom_gap_pfp_execution_backward_result_coefficient_difference_result_bounded_value_bound. fom_gap_pfp_execution_backward_result_coefficient_difference_result_bounded_value_bound + S (fom_value_pfp_execution_backward_result_coefficient_difference_result_bounded) = p))) /\\ ((exists pfaa_left_b_execution_backward_result_coefficient_difference pfaa_left_c_execution_backward_result_coefficient_difference pfaa_right_b_execution_backward_result_coefficient_difference pfaa_right_c_execution_backward_result_coefficient_difference pfaa_sum_b_execution_backward_result_coefficient_difference pfaa_sum_c_execution_backward_result_coefficient_difference pfaa_length_execution_backward_result_coefficient_difference. ((((forall pfrep_power_execution_backward_result_coefficient_difference_witness_common_left pfrep_left_execution_backward_result_coefficient_difference_witness_common_left pfrep_right_execution_backward_result_coefficient_difference_witness_common_left. ((exists pfrep_position_execution_backward_result_coefficient_difference_witness_common_leftfirst. ((pfrep_position_execution_backward_result_coefficient_difference_witness_common_leftfirst+S (pfrep_power_execution_backward_result_coefficient_difference_witness_common_left)=(pfbz_update_W_execution_backward_result)) /\\ ((((exists ff_h_pfp_execution_backward_result_coefficient_difference_witness_common_leftfirstentry. ff_h_pfp_execution_backward_result_coefficient_difference_witness_common_leftfirstentry + S (pfrep_left_execution_backward_result_coefficient_difference_witness_common_left) = S ((S (pfrep_position_execution_backward_result_coefficient_difference_witness_common_leftfirst)) * pfbz_update_wc_execution_backward_result)) /\\ exists ff_q_pfp_execution_backward_result_coefficient_difference_witness_common_leftfirstentry. pfbz_update_wb_execution_backward_result = ff_q_pfp_execution_backward_result_coefficient_difference_witness_common_leftfirstentry * S ((S (pfrep_position_execution_backward_result_coefficient_difference_witness_common_leftfirst)) * pfbz_update_wc_execution_backward_result) + (pfrep_left_execution_backward_result_coefficient_difference_witness_common_left)))))) \\/ (((exists pfrep_gap_execution_backward_result_coefficient_difference_witness_common_leftfirstoutside. pfrep_gap_execution_backward_result_coefficient_difference_witness_common_leftfirstoutside+(pfbz_update_W_execution_backward_result)=(pfrep_power_execution_backward_result_coefficient_difference_witness_common_left)) /\\ (((pfrep_left_execution_backward_result_coefficient_difference_witness_common_left)=0))))) -> ((exists pfrep_position_execution_backward_result_coefficient_difference_witness_common_leftsecond. ((pfrep_position_execution_backward_result_coefficient_difference_witness_common_leftsecond+S (pfrep_power_execution_backward_result_coefficient_difference_witness_common_left)=(pfaa_length_execution_backward_result_coefficient_difference)) /\\ ((((exists ff_h_pfp_execution_backward_result_coefficient_difference_witness_common_leftsecondentry. ff_h_pfp_execution_backward_result_coefficient_difference_witness_common_leftsecondentry + S (pfrep_right_execution_backward_result_coefficient_difference_witness_common_left) = S ((S (pfrep_position_execution_backward_result_coefficient_difference_witness_common_leftsecond)) * pfaa_left_c_execution_backward_result_coefficient_difference)) /\\ exists ff_q_pfp_execution_backward_result_coefficient_difference_witness_common_leftsecondentry. pfaa_left_b_execution_backward_result_coefficient_difference = ff_q_pfp_execution_backward_result_coefficient_difference_witness_common_leftsecondentry * S ((S (pfrep_position_execution_backward_result_coefficient_difference_witness_common_leftsecond)) * pfaa_left_c_execution_backward_result_coefficient_difference) + (pfrep_right_execution_backward_result_coefficient_difference_witness_common_left)))))) \\/ (((exists pfrep_gap_execution_backward_result_coefficient_difference_witness_common_leftsecondoutside. pfrep_gap_execution_backward_result_coefficient_difference_witness_common_leftsecondoutside+(pfaa_length_execution_backward_result_coefficient_difference)=(pfrep_power_execution_backward_result_coefficient_difference_witness_common_left)) /\\ (((pfrep_right_execution_backward_result_coefficient_difference_witness_common_left)=0))))) -> pfrep_left_execution_backward_result_coefficient_difference_witness_common_left=pfrep_right_execution_backward_result_coefficient_difference_witness_common_left) /\\ ((forall pfrep_power_execution_backward_result_coefficient_difference_witness_common_right pfrep_left_execution_backward_result_coefficient_difference_witness_common_right pfrep_right_execution_backward_result_coefficient_difference_witness_common_right. ((exists pfrep_position_execution_backward_result_coefficient_difference_witness_common_rightfirst. ((pfrep_position_execution_backward_result_coefficient_difference_witness_common_rightfirst+S (pfrep_power_execution_backward_result_coefficient_difference_witness_common_right)=(pfbz_update_T_execution_backward_result)) /\\ ((((exists ff_h_pfp_execution_backward_result_coefficient_difference_witness_common_rightfirstentry. ff_h_pfp_execution_backward_result_coefficient_difference_witness_common_rightfirstentry + S (pfrep_left_execution_backward_result_coefficient_difference_witness_common_right) = S ((S (pfrep_position_execution_backward_result_coefficient_difference_witness_common_rightfirst)) * pfbz_update_tc_execution_backward_result)) /\\ exists ff_q_pfp_execution_backward_result_coefficient_difference_witness_common_rightfirstentry. pfbz_update_tb_execution_backward_result = ff_q_pfp_execution_backward_result_coefficient_difference_witness_common_rightfirstentry * S ((S (pfrep_position_execution_backward_result_coefficient_difference_witness_common_rightfirst)) * pfbz_update_tc_execution_backward_result) + (pfrep_left_execution_backward_result_coefficient_difference_witness_common_right)))))) \\/ (((exists pfrep_gap_execution_backward_result_coefficient_difference_witness_common_rightfirstoutside. pfrep_gap_execution_backward_result_coefficient_difference_witness_common_rightfirstoutside+(pfbz_update_T_execution_backward_result)=(pfrep_power_execution_backward_result_coefficient_difference_witness_common_right)) /\\ (((pfrep_left_execution_backward_result_coefficient_difference_witness_common_right)=0))))) -> ((exists pfrep_position_execution_backward_result_coefficient_difference_witness_common_rightsecond. ((pfrep_position_execution_backward_result_coefficient_difference_witness_common_rightsecond+S (pfrep_power_execution_backward_result_coefficient_difference_witness_common_right)=(pfaa_length_execution_backward_result_coefficient_difference)) /\\ ((((exists ff_h_pfp_execution_backward_result_coefficient_difference_witness_common_rightsecondentry. ff_h_pfp_execution_backward_result_coefficient_difference_witness_common_rightsecondentry + S (pfrep_right_execution_backward_result_coefficient_difference_witness_common_right) = S ((S (pfrep_position_execution_backward_result_coefficient_difference_witness_common_rightsecond)) * pfaa_right_c_execution_backward_result_coefficient_difference)) /\\ exists ff_q_pfp_execution_backward_result_coefficient_difference_witness_common_rightsecondentry. pfaa_right_b_execution_backward_result_coefficient_difference = ff_q_pfp_execution_backward_result_coefficient_difference_witness_common_rightsecondentry * S ((S (pfrep_position_execution_backward_result_coefficient_difference_witness_common_rightsecond)) * pfaa_right_c_execution_backward_result_coefficient_difference) + (pfrep_right_execution_backward_result_coefficient_difference_witness_common_right)))))) \\/ (((exists pfrep_gap_execution_backward_result_coefficient_difference_witness_common_rightsecondoutside. pfrep_gap_execution_backward_result_coefficient_difference_witness_common_rightsecondoutside+(pfaa_length_execution_backward_result_coefficient_difference)=(pfrep_power_execution_backward_result_coefficient_difference_witness_common_right)) /\\ (((pfrep_right_execution_backward_result_coefficient_difference_witness_common_right)=0))))) -> pfrep_left_execution_backward_result_coefficient_difference_witness_common_right=pfrep_right_execution_backward_result_coefficient_difference_witness_common_right)))) /\\ (((forall pfp_index_execution_backward_result_coefficient_difference_witness_operation. (exists pfa_gap_execution_backward_result_coefficient_difference_witness_operationindex. pfa_gap_execution_backward_result_coefficient_difference_witness_operationindex + S (pfp_index_execution_backward_result_coefficient_difference_witness_operation) = (pfaa_length_execution_backward_result_coefficient_difference)) -> exists pfp_left_execution_backward_result_coefficient_difference_witness_operation pfp_right_execution_backward_result_coefficient_difference_witness_operation pfp_value_execution_backward_result_coefficient_difference_witness_operation. ((((exists ff_h_pfp_execution_backward_result_coefficient_difference_witness_operationleft. ff_h_pfp_execution_backward_result_coefficient_difference_witness_operationleft + S (pfp_left_execution_backward_result_coefficient_difference_witness_operation) = S ((S (pfp_index_execution_backward_result_coefficient_difference_witness_operation)) * pfaa_left_c_execution_backward_result_coefficient_difference)) /\\ exists ff_q_pfp_execution_backward_result_coefficient_difference_witness_operationleft. pfaa_left_b_execution_backward_result_coefficient_difference = ff_q_pfp_execution_backward_result_coefficient_difference_witness_operationleft * S ((S (pfp_index_execution_backward_result_coefficient_difference_witness_operation)) * pfaa_left_c_execution_backward_result_coefficient_difference) + (pfp_left_execution_backward_result_coefficient_difference_witness_operation))) /\\ (((((exists ff_h_pfp_execution_backward_result_coefficient_difference_witness_operationright. ff_h_pfp_execution_backward_result_coefficient_difference_witness_operationright + S (pfp_right_execution_backward_result_coefficient_difference_witness_operation) = S ((S (pfp_index_execution_backward_result_coefficient_difference_witness_operation)) * pfaa_right_c_execution_backward_result_coefficient_difference)) /\\ exists ff_q_pfp_execution_backward_result_coefficient_difference_witness_operationright. pfaa_right_b_execution_backward_result_coefficient_difference = ff_q_pfp_execution_backward_result_coefficient_difference_witness_operationright * S ((S (pfp_index_execution_backward_result_coefficient_difference_witness_operation)) * pfaa_right_c_execution_backward_result_coefficient_difference) + (pfp_right_execution_backward_result_coefficient_difference_witness_operation))) /\\ (((((exists ff_h_pfp_execution_backward_result_coefficient_difference_witness_operationtarget. ff_h_pfp_execution_backward_result_coefficient_difference_witness_operationtarget + S (pfp_value_execution_backward_result_coefficient_difference_witness_operation) = S ((S (pfp_index_execution_backward_result_coefficient_difference_witness_operation)) * pfaa_sum_c_execution_backward_result_coefficient_difference)) /\\ exists ff_q_pfp_execution_backward_result_coefficient_difference_witness_operationtarget. pfaa_sum_b_execution_backward_result_coefficient_difference = ff_q_pfp_execution_backward_result_coefficient_difference_witness_operationtarget * S ((S (pfp_index_execution_backward_result_coefficient_difference_witness_operation)) * pfaa_sum_c_execution_backward_result_coefficient_difference) + (pfp_value_execution_backward_result_coefficient_difference_witness_operation))) /\\ ((((exists pfa_gap_execution_backward_result_coefficient_difference_witness_operationoperationleft. pfa_gap_execution_backward_result_coefficient_difference_witness_operationoperationleft + S (pfp_left_execution_backward_result_coefficient_difference_witness_operation) = (p)) /\\ (((exists pfa_gap_execution_backward_result_coefficient_difference_witness_operationoperationright. pfa_gap_execution_backward_result_coefficient_difference_witness_operationoperationright + S (pfp_right_execution_backward_result_coefficient_difference_witness_operation) = (p)) /\\ ((((exists pfa_gap_execution_backward_result_coefficient_difference_witness_operationoperationresultbound. pfa_gap_execution_backward_result_coefficient_difference_witness_operationoperationresultbound + S (pfp_value_execution_backward_result_coefficient_difference_witness_operation) = (p)) /\\ ((exists pfa_offset_left_execution_backward_result_coefficient_difference_witness_operationoperationresultcongruence pfa_offset_right_execution_backward_result_coefficient_difference_witness_operationoperationresultcongruence. ((pfp_left_execution_backward_result_coefficient_difference_witness_operation) + (pfp_right_execution_backward_result_coefficient_difference_witness_operation)) + (p) * pfa_offset_left_execution_backward_result_coefficient_difference_witness_operationoperationresultcongruence = (pfp_value_execution_backward_result_coefficient_difference_witness_operation) + (p) * pfa_offset_right_execution_backward_result_coefficient_difference_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_execution_backward_result_coefficient_difference_witness_output pfrep_left_execution_backward_result_coefficient_difference_witness_output pfrep_right_execution_backward_result_coefficient_difference_witness_output. ((exists pfrep_position_execution_backward_result_coefficient_difference_witness_outputfirst. ((pfrep_position_execution_backward_result_coefficient_difference_witness_outputfirst+S (pfrep_power_execution_backward_result_coefficient_difference_witness_output)=(pfaa_length_execution_backward_result_coefficient_difference)) /\\ ((((exists ff_h_pfp_execution_backward_result_coefficient_difference_witness_outputfirstentry. ff_h_pfp_execution_backward_result_coefficient_difference_witness_outputfirstentry + S (pfrep_left_execution_backward_result_coefficient_difference_witness_output) = S ((S (pfrep_position_execution_backward_result_coefficient_difference_witness_outputfirst)) * pfaa_sum_c_execution_backward_result_coefficient_difference)) /\\ exists ff_q_pfp_execution_backward_result_coefficient_difference_witness_outputfirstentry. pfaa_sum_b_execution_backward_result_coefficient_difference = ff_q_pfp_execution_backward_result_coefficient_difference_witness_outputfirstentry * S ((S (pfrep_position_execution_backward_result_coefficient_difference_witness_outputfirst)) * pfaa_sum_c_execution_backward_result_coefficient_difference) + (pfrep_left_execution_backward_result_coefficient_difference_witness_output)))))) \\/ (((exists pfrep_gap_execution_backward_result_coefficient_difference_witness_outputfirstoutside. pfrep_gap_execution_backward_result_coefficient_difference_witness_outputfirstoutside+(pfaa_length_execution_backward_result_coefficient_difference)=(pfrep_power_execution_backward_result_coefficient_difference_witness_output)) /\\ (((pfrep_left_execution_backward_result_coefficient_difference_witness_output)=0))))) -> ((exists pfrep_position_execution_backward_result_coefficient_difference_witness_outputsecond. ((pfrep_position_execution_backward_result_coefficient_difference_witness_outputsecond+S (pfrep_power_execution_backward_result_coefficient_difference_witness_output)=(Lu)) /\\ ((((exists ff_h_pfp_execution_backward_result_coefficient_difference_witness_outputsecondentry. ff_h_pfp_execution_backward_result_coefficient_difference_witness_outputsecondentry + S (pfrep_right_execution_backward_result_coefficient_difference_witness_output) = S ((S (pfrep_position_execution_backward_result_coefficient_difference_witness_outputsecond)) * uc)) /\\ exists ff_q_pfp_execution_backward_result_coefficient_difference_witness_outputsecondentry. ub = ff_q_pfp_execution_backward_result_coefficient_difference_witness_outputsecondentry * S ((S (pfrep_position_execution_backward_result_coefficient_difference_witness_outputsecond)) * uc) + (pfrep_right_execution_backward_result_coefficient_difference_witness_output)))))) \\/ (((exists pfrep_gap_execution_backward_result_coefficient_difference_witness_outputsecondoutside. pfrep_gap_execution_backward_result_coefficient_difference_witness_outputsecondoutside+(Lu)=(pfrep_power_execution_backward_result_coefficient_difference_witness_output)) /\\ (((pfrep_right_execution_backward_result_coefficient_difference_witness_output)=0))))) -> pfrep_left_execution_backward_result_coefficient_difference_witness_output=pfrep_right_execution_backward_result_coefficient_difference_witness_output))))))))))))) /\\ ((exists pfbz_left_code_execution_backward_result_bezout pfbz_left_scale_execution_backward_result_bezout pfbz_left_length_execution_backward_result_bezout pfbz_right_code_execution_backward_result_bezout pfbz_right_scale_execution_backward_result_bezout pfbz_right_length_execution_backward_result_bezout. ((((forall fom_index_pfp_execution_backward_result_bezout_left_productleft. (exists fom_gap_pfp_execution_backward_result_bezout_left_productleft_index_bound. fom_gap_pfp_execution_backward_result_bezout_left_productleft_index_bound + S (fom_index_pfp_execution_backward_result_bezout_left_productleft) = Lv) -> exists fom_value_pfp_execution_backward_result_bezout_left_productleft. ((((exists fom_beta_height_pfp_execution_backward_result_bezout_left_productleft_entry. fom_beta_height_pfp_execution_backward_result_bezout_left_productleft_entry + S (fom_value_pfp_execution_backward_result_bezout_left_productleft) = S ((S (fom_index_pfp_execution_backward_result_bezout_left_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_execution_backward_result_bezout_left_productleft_entry. vb = fom_beta_quotient_pfp_execution_backward_result_bezout_left_productleft_entry * S ((S (fom_index_pfp_execution_backward_result_bezout_left_productleft)) * vc) + (fom_value_pfp_execution_backward_result_bezout_left_productleft))) /\\ (exists fom_gap_pfp_execution_backward_result_bezout_left_productleft_value_bound. fom_gap_pfp_execution_backward_result_bezout_left_productleft_value_bound + S (fom_value_pfp_execution_backward_result_bezout_left_productleft) = p))) /\\ (((forall fom_index_pfp_execution_backward_result_bezout_left_productright. (exists fom_gap_pfp_execution_backward_result_bezout_left_productright_index_bound. fom_gap_pfp_execution_backward_result_bezout_left_productright_index_bound + S (fom_index_pfp_execution_backward_result_bezout_left_productright) = La) -> exists fom_value_pfp_execution_backward_result_bezout_left_productright. ((((exists fom_beta_height_pfp_execution_backward_result_bezout_left_productright_entry. fom_beta_height_pfp_execution_backward_result_bezout_left_productright_entry + S (fom_value_pfp_execution_backward_result_bezout_left_productright) = S ((S (fom_index_pfp_execution_backward_result_bezout_left_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_execution_backward_result_bezout_left_productright_entry. ab = fom_beta_quotient_pfp_execution_backward_result_bezout_left_productright_entry * S ((S (fom_index_pfp_execution_backward_result_bezout_left_productright)) * ac) + (fom_value_pfp_execution_backward_result_bezout_left_productright))) /\\ (exists fom_gap_pfp_execution_backward_result_bezout_left_productright_value_bound. fom_gap_pfp_execution_backward_result_bezout_left_productright_value_bound + S (fom_value_pfp_execution_backward_result_bezout_left_productright) = p))) /\\ (((((((Lv)=0 \\/ (La)=0) /\\ (((pfbz_left_length_execution_backward_result_bezout)=0)))) \\/ (((~((Lv)=0)) /\\ (((~((La)=0)) /\\ (((Lv)+(La)=S (pfbz_left_length_execution_backward_result_bezout)))))))) /\\ ((forall pfc_index_execution_backward_result_bezout_left_productcoefficients. (exists pfa_gap_execution_backward_result_bezout_left_productcoefficientsbound. pfa_gap_execution_backward_result_bezout_left_productcoefficientsbound + S (pfc_index_execution_backward_result_bezout_left_productcoefficients) = (pfbz_left_length_execution_backward_result_bezout)) -> exists pfc_value_execution_backward_result_bezout_left_productcoefficients. ((((exists ff_h_pfp_execution_backward_result_bezout_left_productcoefficientsentry. ff_h_pfp_execution_backward_result_bezout_left_productcoefficientsentry + S (pfc_value_execution_backward_result_bezout_left_productcoefficients) = S ((S (pfc_index_execution_backward_result_bezout_left_productcoefficients)) * pfbz_left_scale_execution_backward_result_bezout)) /\\ exists ff_q_pfp_execution_backward_result_bezout_left_productcoefficientsentry. pfbz_left_code_execution_backward_result_bezout = ff_q_pfp_execution_backward_result_bezout_left_productcoefficientsentry * S ((S (pfc_index_execution_backward_result_bezout_left_productcoefficients)) * pfbz_left_scale_execution_backward_result_bezout) + (pfc_value_execution_backward_result_bezout_left_productcoefficients))) /\\ ((exists pfc_terms_code_execution_backward_result_bezout_left_productcoefficientscoefficient pfc_terms_scale_execution_backward_result_bezout_left_productcoefficientscoefficient pfc_natural_sum_execution_backward_result_bezout_left_productcoefficientscoefficient. ((forall pfc_index_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_backward_result_bezout_left_productcoefficients))) -> exists pfc_value_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_backward_result_bezout_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_backward_result_bezout_left_productcoefficientscoefficient = ff_q_pfp_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_backward_result_bezout_left_productcoefficientscoefficient) + (pfc_value_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm pfc_left_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm pfc_right_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonal)+pfc_complement_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_backward_result_bezout_left_productcoefficients)) /\\ ((((((exists pfa_gap_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonal) = (Lv)) /\\ ((((exists ff_h_pfp_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermleftoutside+(Lv)=(pfc_index_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm) = (La)) /\\ ((((exists ff_h_pfp_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonaltermrightoutside+(La)=(pfc_complement_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonal)=pfc_left_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm*pfc_right_execution_backward_result_bezout_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum fs_v_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_backward_result_bezout_left_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_backward_result_bezout_left_productcoefficients))) * fs_v_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_backward_result_bezout_left_productcoefficients))) * fs_v_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_backward_result_bezout_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_backward_result_bezout_left_productcoefficients)) -> exists fs_a_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_backward_result_bezout_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_backward_result_bezout_left_productcoefficientscoefficient = fs_q_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_backward_result_bezout_left_productcoefficientscoefficient) + (fs_a_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum) + (fs_r_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum) + (fs_s_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_backward_result_bezout_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_backward_result_bezout_left_productcoefficientscoefficientresiduebound. pfa_gap_execution_backward_result_bezout_left_productcoefficientscoefficientresiduebound + S (pfc_value_execution_backward_result_bezout_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_backward_result_bezout_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_backward_result_bezout_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_backward_result_bezout_left_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_backward_result_bezout_left_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_backward_result_bezout_left_productcoefficients) + (p) * pfa_offset_right_execution_backward_result_bezout_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_execution_backward_result_bezout_right_productleft. (exists fom_gap_pfp_execution_backward_result_bezout_right_productleft_index_bound. fom_gap_pfp_execution_backward_result_bezout_right_productleft_index_bound + S (fom_index_pfp_execution_backward_result_bezout_right_productleft) = pfbz_update_T_execution_backward_result) -> exists fom_value_pfp_execution_backward_result_bezout_right_productleft. ((((exists fom_beta_height_pfp_execution_backward_result_bezout_right_productleft_entry. fom_beta_height_pfp_execution_backward_result_bezout_right_productleft_entry + S (fom_value_pfp_execution_backward_result_bezout_right_productleft) = S ((S (fom_index_pfp_execution_backward_result_bezout_right_productleft)) * pfbz_update_tc_execution_backward_result)) /\\ exists fom_beta_quotient_pfp_execution_backward_result_bezout_right_productleft_entry. pfbz_update_tb_execution_backward_result = fom_beta_quotient_pfp_execution_backward_result_bezout_right_productleft_entry * S ((S (fom_index_pfp_execution_backward_result_bezout_right_productleft)) * pfbz_update_tc_execution_backward_result) + (fom_value_pfp_execution_backward_result_bezout_right_productleft))) /\\ (exists fom_gap_pfp_execution_backward_result_bezout_right_productleft_value_bound. fom_gap_pfp_execution_backward_result_bezout_right_productleft_value_bound + S (fom_value_pfp_execution_backward_result_bezout_right_productleft) = p))) /\\ (((forall fom_index_pfp_execution_backward_result_bezout_right_productright. (exists fom_gap_pfp_execution_backward_result_bezout_right_productright_index_bound. fom_gap_pfp_execution_backward_result_bezout_right_productright_index_bound + S (fom_index_pfp_execution_backward_result_bezout_right_productright) = S d) -> exists fom_value_pfp_execution_backward_result_bezout_right_productright. ((((exists fom_beta_height_pfp_execution_backward_result_bezout_right_productright_entry. fom_beta_height_pfp_execution_backward_result_bezout_right_productright_entry + S (fom_value_pfp_execution_backward_result_bezout_right_productright) = S ((S (fom_index_pfp_execution_backward_result_bezout_right_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_execution_backward_result_bezout_right_productright_entry. bb = fom_beta_quotient_pfp_execution_backward_result_bezout_right_productright_entry * S ((S (fom_index_pfp_execution_backward_result_bezout_right_productright)) * bc) + (fom_value_pfp_execution_backward_result_bezout_right_productright))) /\\ (exists fom_gap_pfp_execution_backward_result_bezout_right_productright_value_bound. fom_gap_pfp_execution_backward_result_bezout_right_productright_value_bound + S (fom_value_pfp_execution_backward_result_bezout_right_productright) = p))) /\\ (((((((pfbz_update_T_execution_backward_result)=0 \\/ (S d)=0) /\\ (((pfbz_right_length_execution_backward_result_bezout)=0)))) \\/ (((~((pfbz_update_T_execution_backward_result)=0)) /\\ (((~((S d)=0)) /\\ (((pfbz_update_T_execution_backward_result)+(S d)=S (pfbz_right_length_execution_backward_result_bezout)))))))) /\\ ((forall pfc_index_execution_backward_result_bezout_right_productcoefficients. (exists pfa_gap_execution_backward_result_bezout_right_productcoefficientsbound. pfa_gap_execution_backward_result_bezout_right_productcoefficientsbound + S (pfc_index_execution_backward_result_bezout_right_productcoefficients) = (pfbz_right_length_execution_backward_result_bezout)) -> exists pfc_value_execution_backward_result_bezout_right_productcoefficients. ((((exists ff_h_pfp_execution_backward_result_bezout_right_productcoefficientsentry. ff_h_pfp_execution_backward_result_bezout_right_productcoefficientsentry + S (pfc_value_execution_backward_result_bezout_right_productcoefficients) = S ((S (pfc_index_execution_backward_result_bezout_right_productcoefficients)) * pfbz_right_scale_execution_backward_result_bezout)) /\\ exists ff_q_pfp_execution_backward_result_bezout_right_productcoefficientsentry. pfbz_right_code_execution_backward_result_bezout = ff_q_pfp_execution_backward_result_bezout_right_productcoefficientsentry * S ((S (pfc_index_execution_backward_result_bezout_right_productcoefficients)) * pfbz_right_scale_execution_backward_result_bezout) + (pfc_value_execution_backward_result_bezout_right_productcoefficients))) /\\ ((exists pfc_terms_code_execution_backward_result_bezout_right_productcoefficientscoefficient pfc_terms_scale_execution_backward_result_bezout_right_productcoefficientscoefficient pfc_natural_sum_execution_backward_result_bezout_right_productcoefficientscoefficient. ((forall pfc_index_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalbound. pfa_gap_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_execution_backward_result_bezout_right_productcoefficients))) -> exists pfc_value_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_backward_result_bezout_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_execution_backward_result_bezout_right_productcoefficientscoefficient = ff_q_pfp_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_execution_backward_result_bezout_right_productcoefficientscoefficient) + (pfc_value_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm pfc_left_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm pfc_right_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonal)+pfc_complement_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm=(pfc_index_execution_backward_result_bezout_right_productcoefficients)) /\\ ((((((exists pfa_gap_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonal) = (pfbz_update_T_execution_backward_result)) /\\ ((((exists ff_h_pfp_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonal)) * pfbz_update_tc_execution_backward_result)) /\\ exists ff_q_pfp_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftentry. pfbz_update_tb_execution_backward_result = ff_q_pfp_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonal)) * pfbz_update_tc_execution_backward_result) + (pfc_left_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfbz_update_T_execution_backward_result)=(pfc_index_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonal)=pfc_left_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm*pfc_right_execution_backward_result_bezout_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum fs_v_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_execution_backward_result_bezout_right_productcoefficientscoefficient) = S ((S (S (pfc_index_execution_backward_result_bezout_right_productcoefficients))) * fs_v_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_execution_backward_result_bezout_right_productcoefficients))) * fs_v_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum) + (pfc_natural_sum_execution_backward_result_bezout_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_execution_backward_result_bezout_right_productcoefficients)) -> exists fs_a_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_backward_result_bezout_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_execution_backward_result_bezout_right_productcoefficientscoefficient = fs_q_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_execution_backward_result_bezout_right_productcoefficientscoefficient) + (fs_a_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum) + (fs_r_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum = fs_q_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum) + (fs_s_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_execution_backward_result_bezout_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_execution_backward_result_bezout_right_productcoefficientscoefficientresiduebound. pfa_gap_execution_backward_result_bezout_right_productcoefficientscoefficientresiduebound + S (pfc_value_execution_backward_result_bezout_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_execution_backward_result_bezout_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_execution_backward_result_bezout_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_execution_backward_result_bezout_right_productcoefficientscoefficient) + (p) * pfa_offset_left_execution_backward_result_bezout_right_productcoefficientscoefficientresiduecongruence = (pfc_value_execution_backward_result_bezout_right_productcoefficients) + (p) * pfa_offset_right_execution_backward_result_bezout_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_execution_backward_result_bezout_sum_left_bounded. (exists fom_gap_pfp_execution_backward_result_bezout_sum_left_bounded_index_bound. fom_gap_pfp_execution_backward_result_bezout_sum_left_bounded_index_bound + S (fom_index_pfp_execution_backward_result_bezout_sum_left_bounded) = pfbz_left_length_execution_backward_result_bezout) -> exists fom_value_pfp_execution_backward_result_bezout_sum_left_bounded. ((((exists fom_beta_height_pfp_execution_backward_result_bezout_sum_left_bounded_entry. fom_beta_height_pfp_execution_backward_result_bezout_sum_left_bounded_entry + S (fom_value_pfp_execution_backward_result_bezout_sum_left_bounded) = S ((S (fom_index_pfp_execution_backward_result_bezout_sum_left_bounded)) * pfbz_left_scale_execution_backward_result_bezout)) /\\ exists fom_beta_quotient_pfp_execution_backward_result_bezout_sum_left_bounded_entry. pfbz_left_code_execution_backward_result_bezout = fom_beta_quotient_pfp_execution_backward_result_bezout_sum_left_bounded_entry * S ((S (fom_index_pfp_execution_backward_result_bezout_sum_left_bounded)) * pfbz_left_scale_execution_backward_result_bezout) + (fom_value_pfp_execution_backward_result_bezout_sum_left_bounded))) /\\ (exists fom_gap_pfp_execution_backward_result_bezout_sum_left_bounded_value_bound. fom_gap_pfp_execution_backward_result_bezout_sum_left_bounded_value_bound + S (fom_value_pfp_execution_backward_result_bezout_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_execution_backward_result_bezout_sum_right_bounded. (exists fom_gap_pfp_execution_backward_result_bezout_sum_right_bounded_index_bound. fom_gap_pfp_execution_backward_result_bezout_sum_right_bounded_index_bound + S (fom_index_pfp_execution_backward_result_bezout_sum_right_bounded) = pfbz_right_length_execution_backward_result_bezout) -> exists fom_value_pfp_execution_backward_result_bezout_sum_right_bounded. ((((exists fom_beta_height_pfp_execution_backward_result_bezout_sum_right_bounded_entry. fom_beta_height_pfp_execution_backward_result_bezout_sum_right_bounded_entry + S (fom_value_pfp_execution_backward_result_bezout_sum_right_bounded) = S ((S (fom_index_pfp_execution_backward_result_bezout_sum_right_bounded)) * pfbz_right_scale_execution_backward_result_bezout)) /\\ exists fom_beta_quotient_pfp_execution_backward_result_bezout_sum_right_bounded_entry. pfbz_right_code_execution_backward_result_bezout = fom_beta_quotient_pfp_execution_backward_result_bezout_sum_right_bounded_entry * S ((S (fom_index_pfp_execution_backward_result_bezout_sum_right_bounded)) * pfbz_right_scale_execution_backward_result_bezout) + (fom_value_pfp_execution_backward_result_bezout_sum_right_bounded))) /\\ (exists fom_gap_pfp_execution_backward_result_bezout_sum_right_bounded_value_bound. fom_gap_pfp_execution_backward_result_bezout_sum_right_bounded_value_bound + S (fom_value_pfp_execution_backward_result_bezout_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_execution_backward_result_bezout_sum_result_bounded. (exists fom_gap_pfp_execution_backward_result_bezout_sum_result_bounded_index_bound. fom_gap_pfp_execution_backward_result_bezout_sum_result_bounded_index_bound + S (fom_index_pfp_execution_backward_result_bezout_sum_result_bounded) = Lg) -> exists fom_value_pfp_execution_backward_result_bezout_sum_result_bounded. ((((exists fom_beta_height_pfp_execution_backward_result_bezout_sum_result_bounded_entry. fom_beta_height_pfp_execution_backward_result_bezout_sum_result_bounded_entry + S (fom_value_pfp_execution_backward_result_bezout_sum_result_bounded) = S ((S (fom_index_pfp_execution_backward_result_bezout_sum_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_execution_backward_result_bezout_sum_result_bounded_entry. gb = fom_beta_quotient_pfp_execution_backward_result_bezout_sum_result_bounded_entry * S ((S (fom_index_pfp_execution_backward_result_bezout_sum_result_bounded)) * gc) + (fom_value_pfp_execution_backward_result_bezout_sum_result_bounded))) /\\ (exists fom_gap_pfp_execution_backward_result_bezout_sum_result_bounded_value_bound. fom_gap_pfp_execution_backward_result_bezout_sum_result_bounded_value_bound + S (fom_value_pfp_execution_backward_result_bezout_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_execution_backward_result_bezout_sum pfaa_left_c_execution_backward_result_bezout_sum pfaa_right_b_execution_backward_result_bezout_sum pfaa_right_c_execution_backward_result_bezout_sum pfaa_sum_b_execution_backward_result_bezout_sum pfaa_sum_c_execution_backward_result_bezout_sum pfaa_length_execution_backward_result_bezout_sum. ((((forall pfrep_power_execution_backward_result_bezout_sum_witness_common_left pfrep_left_execution_backward_result_bezout_sum_witness_common_left pfrep_right_execution_backward_result_bezout_sum_witness_common_left. ((exists pfrep_position_execution_backward_result_bezout_sum_witness_common_leftfirst. ((pfrep_position_execution_backward_result_bezout_sum_witness_common_leftfirst+S (pfrep_power_execution_backward_result_bezout_sum_witness_common_left)=(pfbz_left_length_execution_backward_result_bezout)) /\\ ((((exists ff_h_pfp_execution_backward_result_bezout_sum_witness_common_leftfirstentry. ff_h_pfp_execution_backward_result_bezout_sum_witness_common_leftfirstentry + S (pfrep_left_execution_backward_result_bezout_sum_witness_common_left) = S ((S (pfrep_position_execution_backward_result_bezout_sum_witness_common_leftfirst)) * pfbz_left_scale_execution_backward_result_bezout)) /\\ exists ff_q_pfp_execution_backward_result_bezout_sum_witness_common_leftfirstentry. pfbz_left_code_execution_backward_result_bezout = ff_q_pfp_execution_backward_result_bezout_sum_witness_common_leftfirstentry * S ((S (pfrep_position_execution_backward_result_bezout_sum_witness_common_leftfirst)) * pfbz_left_scale_execution_backward_result_bezout) + (pfrep_left_execution_backward_result_bezout_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_execution_backward_result_bezout_sum_witness_common_leftfirstoutside. pfrep_gap_execution_backward_result_bezout_sum_witness_common_leftfirstoutside+(pfbz_left_length_execution_backward_result_bezout)=(pfrep_power_execution_backward_result_bezout_sum_witness_common_left)) /\\ (((pfrep_left_execution_backward_result_bezout_sum_witness_common_left)=0))))) -> ((exists pfrep_position_execution_backward_result_bezout_sum_witness_common_leftsecond. ((pfrep_position_execution_backward_result_bezout_sum_witness_common_leftsecond+S (pfrep_power_execution_backward_result_bezout_sum_witness_common_left)=(pfaa_length_execution_backward_result_bezout_sum)) /\\ ((((exists ff_h_pfp_execution_backward_result_bezout_sum_witness_common_leftsecondentry. ff_h_pfp_execution_backward_result_bezout_sum_witness_common_leftsecondentry + S (pfrep_right_execution_backward_result_bezout_sum_witness_common_left) = S ((S (pfrep_position_execution_backward_result_bezout_sum_witness_common_leftsecond)) * pfaa_left_c_execution_backward_result_bezout_sum)) /\\ exists ff_q_pfp_execution_backward_result_bezout_sum_witness_common_leftsecondentry. pfaa_left_b_execution_backward_result_bezout_sum = ff_q_pfp_execution_backward_result_bezout_sum_witness_common_leftsecondentry * S ((S (pfrep_position_execution_backward_result_bezout_sum_witness_common_leftsecond)) * pfaa_left_c_execution_backward_result_bezout_sum) + (pfrep_right_execution_backward_result_bezout_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_execution_backward_result_bezout_sum_witness_common_leftsecondoutside. pfrep_gap_execution_backward_result_bezout_sum_witness_common_leftsecondoutside+(pfaa_length_execution_backward_result_bezout_sum)=(pfrep_power_execution_backward_result_bezout_sum_witness_common_left)) /\\ (((pfrep_right_execution_backward_result_bezout_sum_witness_common_left)=0))))) -> pfrep_left_execution_backward_result_bezout_sum_witness_common_left=pfrep_right_execution_backward_result_bezout_sum_witness_common_left) /\\ ((forall pfrep_power_execution_backward_result_bezout_sum_witness_common_right pfrep_left_execution_backward_result_bezout_sum_witness_common_right pfrep_right_execution_backward_result_bezout_sum_witness_common_right. ((exists pfrep_position_execution_backward_result_bezout_sum_witness_common_rightfirst. ((pfrep_position_execution_backward_result_bezout_sum_witness_common_rightfirst+S (pfrep_power_execution_backward_result_bezout_sum_witness_common_right)=(pfbz_right_length_execution_backward_result_bezout)) /\\ ((((exists ff_h_pfp_execution_backward_result_bezout_sum_witness_common_rightfirstentry. ff_h_pfp_execution_backward_result_bezout_sum_witness_common_rightfirstentry + S (pfrep_left_execution_backward_result_bezout_sum_witness_common_right) = S ((S (pfrep_position_execution_backward_result_bezout_sum_witness_common_rightfirst)) * pfbz_right_scale_execution_backward_result_bezout)) /\\ exists ff_q_pfp_execution_backward_result_bezout_sum_witness_common_rightfirstentry. pfbz_right_code_execution_backward_result_bezout = ff_q_pfp_execution_backward_result_bezout_sum_witness_common_rightfirstentry * S ((S (pfrep_position_execution_backward_result_bezout_sum_witness_common_rightfirst)) * pfbz_right_scale_execution_backward_result_bezout) + (pfrep_left_execution_backward_result_bezout_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_execution_backward_result_bezout_sum_witness_common_rightfirstoutside. pfrep_gap_execution_backward_result_bezout_sum_witness_common_rightfirstoutside+(pfbz_right_length_execution_backward_result_bezout)=(pfrep_power_execution_backward_result_bezout_sum_witness_common_right)) /\\ (((pfrep_left_execution_backward_result_bezout_sum_witness_common_right)=0))))) -> ((exists pfrep_position_execution_backward_result_bezout_sum_witness_common_rightsecond. ((pfrep_position_execution_backward_result_bezout_sum_witness_common_rightsecond+S (pfrep_power_execution_backward_result_bezout_sum_witness_common_right)=(pfaa_length_execution_backward_result_bezout_sum)) /\\ ((((exists ff_h_pfp_execution_backward_result_bezout_sum_witness_common_rightsecondentry. ff_h_pfp_execution_backward_result_bezout_sum_witness_common_rightsecondentry + S (pfrep_right_execution_backward_result_bezout_sum_witness_common_right) = S ((S (pfrep_position_execution_backward_result_bezout_sum_witness_common_rightsecond)) * pfaa_right_c_execution_backward_result_bezout_sum)) /\\ exists ff_q_pfp_execution_backward_result_bezout_sum_witness_common_rightsecondentry. pfaa_right_b_execution_backward_result_bezout_sum = ff_q_pfp_execution_backward_result_bezout_sum_witness_common_rightsecondentry * S ((S (pfrep_position_execution_backward_result_bezout_sum_witness_common_rightsecond)) * pfaa_right_c_execution_backward_result_bezout_sum) + (pfrep_right_execution_backward_result_bezout_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_execution_backward_result_bezout_sum_witness_common_rightsecondoutside. pfrep_gap_execution_backward_result_bezout_sum_witness_common_rightsecondoutside+(pfaa_length_execution_backward_result_bezout_sum)=(pfrep_power_execution_backward_result_bezout_sum_witness_common_right)) /\\ (((pfrep_right_execution_backward_result_bezout_sum_witness_common_right)=0))))) -> pfrep_left_execution_backward_result_bezout_sum_witness_common_right=pfrep_right_execution_backward_result_bezout_sum_witness_common_right)))) /\\ (((forall pfp_index_execution_backward_result_bezout_sum_witness_operation. (exists pfa_gap_execution_backward_result_bezout_sum_witness_operationindex. pfa_gap_execution_backward_result_bezout_sum_witness_operationindex + S (pfp_index_execution_backward_result_bezout_sum_witness_operation) = (pfaa_length_execution_backward_result_bezout_sum)) -> exists pfp_left_execution_backward_result_bezout_sum_witness_operation pfp_right_execution_backward_result_bezout_sum_witness_operation pfp_value_execution_backward_result_bezout_sum_witness_operation. ((((exists ff_h_pfp_execution_backward_result_bezout_sum_witness_operationleft. ff_h_pfp_execution_backward_result_bezout_sum_witness_operationleft + S (pfp_left_execution_backward_result_bezout_sum_witness_operation) = S ((S (pfp_index_execution_backward_result_bezout_sum_witness_operation)) * pfaa_left_c_execution_backward_result_bezout_sum)) /\\ exists ff_q_pfp_execution_backward_result_bezout_sum_witness_operationleft. pfaa_left_b_execution_backward_result_bezout_sum = ff_q_pfp_execution_backward_result_bezout_sum_witness_operationleft * S ((S (pfp_index_execution_backward_result_bezout_sum_witness_operation)) * pfaa_left_c_execution_backward_result_bezout_sum) + (pfp_left_execution_backward_result_bezout_sum_witness_operation))) /\\ (((((exists ff_h_pfp_execution_backward_result_bezout_sum_witness_operationright. ff_h_pfp_execution_backward_result_bezout_sum_witness_operationright + S (pfp_right_execution_backward_result_bezout_sum_witness_operation) = S ((S (pfp_index_execution_backward_result_bezout_sum_witness_operation)) * pfaa_right_c_execution_backward_result_bezout_sum)) /\\ exists ff_q_pfp_execution_backward_result_bezout_sum_witness_operationright. pfaa_right_b_execution_backward_result_bezout_sum = ff_q_pfp_execution_backward_result_bezout_sum_witness_operationright * S ((S (pfp_index_execution_backward_result_bezout_sum_witness_operation)) * pfaa_right_c_execution_backward_result_bezout_sum) + (pfp_right_execution_backward_result_bezout_sum_witness_operation))) /\\ (((((exists ff_h_pfp_execution_backward_result_bezout_sum_witness_operationtarget. ff_h_pfp_execution_backward_result_bezout_sum_witness_operationtarget + S (pfp_value_execution_backward_result_bezout_sum_witness_operation) = S ((S (pfp_index_execution_backward_result_bezout_sum_witness_operation)) * pfaa_sum_c_execution_backward_result_bezout_sum)) /\\ exists ff_q_pfp_execution_backward_result_bezout_sum_witness_operationtarget. pfaa_sum_b_execution_backward_result_bezout_sum = ff_q_pfp_execution_backward_result_bezout_sum_witness_operationtarget * S ((S (pfp_index_execution_backward_result_bezout_sum_witness_operation)) * pfaa_sum_c_execution_backward_result_bezout_sum) + (pfp_value_execution_backward_result_bezout_sum_witness_operation))) /\\ ((((exists pfa_gap_execution_backward_result_bezout_sum_witness_operationoperationleft. pfa_gap_execution_backward_result_bezout_sum_witness_operationoperationleft + S (pfp_left_execution_backward_result_bezout_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_execution_backward_result_bezout_sum_witness_operationoperationright. pfa_gap_execution_backward_result_bezout_sum_witness_operationoperationright + S (pfp_right_execution_backward_result_bezout_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_execution_backward_result_bezout_sum_witness_operationoperationresultbound. pfa_gap_execution_backward_result_bezout_sum_witness_operationoperationresultbound + S (pfp_value_execution_backward_result_bezout_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_execution_backward_result_bezout_sum_witness_operationoperationresultcongruence pfa_offset_right_execution_backward_result_bezout_sum_witness_operationoperationresultcongruence. ((pfp_left_execution_backward_result_bezout_sum_witness_operation) + (pfp_right_execution_backward_result_bezout_sum_witness_operation)) + (p) * pfa_offset_left_execution_backward_result_bezout_sum_witness_operationoperationresultcongruence = (pfp_value_execution_backward_result_bezout_sum_witness_operation) + (p) * pfa_offset_right_execution_backward_result_bezout_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_execution_backward_result_bezout_sum_witness_output pfrep_left_execution_backward_result_bezout_sum_witness_output pfrep_right_execution_backward_result_bezout_sum_witness_output. ((exists pfrep_position_execution_backward_result_bezout_sum_witness_outputfirst. ((pfrep_position_execution_backward_result_bezout_sum_witness_outputfirst+S (pfrep_power_execution_backward_result_bezout_sum_witness_output)=(pfaa_length_execution_backward_result_bezout_sum)) /\\ ((((exists ff_h_pfp_execution_backward_result_bezout_sum_witness_outputfirstentry. ff_h_pfp_execution_backward_result_bezout_sum_witness_outputfirstentry + S (pfrep_left_execution_backward_result_bezout_sum_witness_output) = S ((S (pfrep_position_execution_backward_result_bezout_sum_witness_outputfirst)) * pfaa_sum_c_execution_backward_result_bezout_sum)) /\\ exists ff_q_pfp_execution_backward_result_bezout_sum_witness_outputfirstentry. pfaa_sum_b_execution_backward_result_bezout_sum = ff_q_pfp_execution_backward_result_bezout_sum_witness_outputfirstentry * S ((S (pfrep_position_execution_backward_result_bezout_sum_witness_outputfirst)) * pfaa_sum_c_execution_backward_result_bezout_sum) + (pfrep_left_execution_backward_result_bezout_sum_witness_output)))))) \\/ (((exists pfrep_gap_execution_backward_result_bezout_sum_witness_outputfirstoutside. pfrep_gap_execution_backward_result_bezout_sum_witness_outputfirstoutside+(pfaa_length_execution_backward_result_bezout_sum)=(pfrep_power_execution_backward_result_bezout_sum_witness_output)) /\\ (((pfrep_left_execution_backward_result_bezout_sum_witness_output)=0))))) -> ((exists pfrep_position_execution_backward_result_bezout_sum_witness_outputsecond. ((pfrep_position_execution_backward_result_bezout_sum_witness_outputsecond+S (pfrep_power_execution_backward_result_bezout_sum_witness_output)=(Lg)) /\\ ((((exists ff_h_pfp_execution_backward_result_bezout_sum_witness_outputsecondentry. ff_h_pfp_execution_backward_result_bezout_sum_witness_outputsecondentry + S (pfrep_right_execution_backward_result_bezout_sum_witness_output) = S ((S (pfrep_position_execution_backward_result_bezout_sum_witness_outputsecond)) * gc)) /\\ exists ff_q_pfp_execution_backward_result_bezout_sum_witness_outputsecondentry. gb = ff_q_pfp_execution_backward_result_bezout_sum_witness_outputsecondentry * S ((S (pfrep_position_execution_backward_result_bezout_sum_witness_outputsecond)) * gc) + (pfrep_right_execution_backward_result_bezout_sum_witness_output)))))) \\/ (((exists pfrep_gap_execution_backward_result_bezout_sum_witness_outputsecondoutside. pfrep_gap_execution_backward_result_bezout_sum_witness_outputsecondoutside+(Lg)=(pfrep_power_execution_backward_result_bezout_sum_witness_output)) /\\ (((pfrep_right_execution_backward_result_bezout_sum_witness_output)=0))))) -> pfrep_left_execution_backward_result_bezout_sum_witness_output=pfrep_right_execution_backward_result_bezout_sum_witness_output)))))))))))))))))))))))", "statement_sha256": "0308b3b63fea32baa1eb473222ffa0dd93c119908c6c324191e08969d0d7b76c", "summary": "A real division execution automatically supplies the proper Euclidean identity and constructs the exact backward Bezout coefficient update, including an empty quotient.", "summary_sha256": "b80f52d485a6d3bcff739f48dab04dc14d00901c5d6de7a7bd82fd0d2ed4b04d" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "1172229bfdef6c6f215e3467ee57b5d3568fe0e98926def07e5f337f839eb4af", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 444, "bundle_sha256": 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"prime_field_polynomial_power_coefficient_functional", "prime_field_polynomial_equivalent_symmetric", "prime_field_polynomial_zero_prefix_equivalent_empty", "prime_field_polynomial_add_zero_right" ], "dependencies_sha256": "ffe834e08d8e529ddcd84933103e8115f8343af93b6d583df985458f0c64705c", "empty_context_closure": { "body_proof_depth": 39, "body_proof_nodes": 85, "bundle_campaign": "polynomial-gcd-bezout", "bundle_dependency_edge_count": 1578, "bundle_node_count": 493, "bundle_node_id": 444, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "bundle_root_id": 492, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "b2abf4a7f78ffa4b231d6a6745a6f3a47493ff1099d5857279065ee6d4a5fc75", "status": "checked" }, "enrollment_index": 4187, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "76b90226e5e29fdde3d9bb49accccf8d9b4c0cc17a4de406af253e999102533c", "kind": "alpha_v34_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_bezout_laws_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85", "kind": "alpha_v34_frontier_executable_audit", "path": "peano-lab/py/tests/test_campaign_research_v34_closure.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "kind": "alpha_v34_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-field-polynomial-gcd-bezout-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "kind": "alpha_v34_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=444]" }, { "document_sha256": "81943a545e7b9f667ad758c24de14bb47280e8cf3fbb3ca714efec55fb096408", "kind": "alpha_v34_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "e209b047bd740daca85f6f1e8cf0a74e426715a7b0ab69929a96a41331b4e88e", "membership": "alpha_only", "name": "prime_field_polynomial_aligned_add_empty_right", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro A", "intro hp", "intro ha", "have hz : exists zb zc. forall pfp_repeat_index_empty_add_zeros. (exists pfa_gap_empty_add_zerosindex. pfa_gap_empty_add_zerosindex + S (pfp_repeat_index_empty_add_zeros) = (A)) -> (((exists ff_h_pfp_empty_add_zerosentry. ff_h_pfp_empty_add_zerosentry + S (0) = S ((S (pfp_repeat_index_empty_add_zeros)) * zc)) /\\ exists ff_q_pfp_empty_add_zerosentry. zb = ff_q_pfp_empty_add_zerosentry * S ((S (pfp_repeat_index_empty_add_zeros)) * zc) + (0)))", "specialize beta_repeat_exists (0)", "specialize beta_repeat_exists (A)", "apply beta_repeat_exists", "cases hz", "cases hz_witness", "specialize prime_field_polynomial_aligned_add_from_common (p)", "specialize prime_field_polynomial_aligned_add_from_common (ab)", "specialize prime_field_polynomial_aligned_add_from_common (ac)", "specialize prime_field_polynomial_aligned_add_from_common (A)", "specialize prime_field_polynomial_aligned_add_from_common (0)", "specialize prime_field_polynomial_aligned_add_from_common (0)", "specialize prime_field_polynomial_aligned_add_from_common (0)", "specialize prime_field_polynomial_aligned_add_from_common (ab)", "specialize prime_field_polynomial_aligned_add_from_common (ac)", "specialize prime_field_polynomial_aligned_add_from_common (A)", "specialize prime_field_polynomial_aligned_add_from_common (ab)", "specialize prime_field_polynomial_aligned_add_from_common (ac)", "specialize prime_field_polynomial_aligned_add_from_common (x)", "specialize prime_field_polynomial_aligned_add_from_common (x1)", "specialize prime_field_polynomial_aligned_add_from_common (ab)", "specialize prime_field_polynomial_aligned_add_from_common (ac)", "specialize prime_field_polynomial_aligned_add_from_common (A)", "apply prime_field_polynomial_aligned_add_from_common", "exact ha", "specialize matrix_rank_bounded_prefix_empty (0)", "specialize matrix_rank_bounded_prefix_empty (0)", "specialize matrix_rank_bounded_prefix_empty (p)", "apply matrix_rank_bounded_prefix_empty", "exact ha", "split", "specialize prime_field_polynomial_power_coefficient_functional (ab)", "specialize prime_field_polynomial_power_coefficient_functional (ac)", "specialize prime_field_polynomial_power_coefficient_functional (A)", "apply prime_field_polynomial_power_coefficient_functional", "specialize prime_field_polynomial_equivalent_symmetric (x)", "specialize prime_field_polynomial_equivalent_symmetric (x1)", "specialize prime_field_polynomial_equivalent_symmetric (A)", "specialize prime_field_polynomial_equivalent_symmetric (0)", "specialize prime_field_polynomial_equivalent_symmetric (0)", "specialize prime_field_polynomial_equivalent_symmetric (0)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (x)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (x1)", "specialize prime_field_polynomial_zero_prefix_equivalent_empty (A)", "apply prime_field_polynomial_zero_prefix_equivalent_empty", "exact hz_witness_witness", "specialize prime_field_polynomial_add_zero_right (p)", "specialize prime_field_polynomial_add_zero_right (ab)", "specialize prime_field_polynomial_add_zero_right (ac)", "specialize prime_field_polynomial_add_zero_right (x)", "specialize prime_field_polynomial_add_zero_right (x1)", "specialize prime_field_polynomial_add_zero_right (A)", "apply prime_field_polynomial_add_zero_right", "exact hp", "exact ha", "exact hz_witness_witness", "specialize prime_field_polynomial_power_coefficient_functional (ab)", "specialize prime_field_polynomial_power_coefficient_functional (ac)", "specialize prime_field_polynomial_power_coefficient_functional (A)", "apply prime_field_polynomial_power_coefficient_functional" ], "script_sha256": "8a4686a3aa0dd92436efdfbedb6e35da356cfd5e78ca7f4fb3b456ac86c35478", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_bezout_laws_candidate.py", "sha256": "76b90226e5e29fdde3d9bb49accccf8d9b4c0cc17a4de406af253e999102533c" }, "statement": "forall p ab ac A. (~((p) = 1) /\\ forall pfa_factor_left_empty_add_prime pfa_factor_right_empty_add_prime. (p) = pfa_factor_left_empty_add_prime * pfa_factor_right_empty_add_prime -> pfa_factor_left_empty_add_prime = 1 \\/ pfa_factor_right_empty_add_prime = 1) -> (forall fom_index_pfp_empty_add_bound. (exists fom_gap_pfp_empty_add_bound_index_bound. fom_gap_pfp_empty_add_bound_index_bound + S (fom_index_pfp_empty_add_bound) = A) -> exists fom_value_pfp_empty_add_bound. ((((exists fom_beta_height_pfp_empty_add_bound_entry. fom_beta_height_pfp_empty_add_bound_entry + S (fom_value_pfp_empty_add_bound) = S ((S (fom_index_pfp_empty_add_bound)) * ac)) /\\ exists fom_beta_quotient_pfp_empty_add_bound_entry. ab = fom_beta_quotient_pfp_empty_add_bound_entry * S ((S (fom_index_pfp_empty_add_bound)) * ac) + (fom_value_pfp_empty_add_bound))) /\\ (exists fom_gap_pfp_empty_add_bound_value_bound. fom_gap_pfp_empty_add_bound_value_bound + S (fom_value_pfp_empty_add_bound) = p))) -> (((forall fom_index_pfp_empty_add_result_left_bounded. (exists fom_gap_pfp_empty_add_result_left_bounded_index_bound. fom_gap_pfp_empty_add_result_left_bounded_index_bound + S (fom_index_pfp_empty_add_result_left_bounded) = A) -> exists fom_value_pfp_empty_add_result_left_bounded. ((((exists fom_beta_height_pfp_empty_add_result_left_bounded_entry. fom_beta_height_pfp_empty_add_result_left_bounded_entry + S (fom_value_pfp_empty_add_result_left_bounded) = S ((S (fom_index_pfp_empty_add_result_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_empty_add_result_left_bounded_entry. ab = fom_beta_quotient_pfp_empty_add_result_left_bounded_entry * S ((S (fom_index_pfp_empty_add_result_left_bounded)) * ac) + (fom_value_pfp_empty_add_result_left_bounded))) /\\ (exists fom_gap_pfp_empty_add_result_left_bounded_value_bound. fom_gap_pfp_empty_add_result_left_bounded_value_bound + S (fom_value_pfp_empty_add_result_left_bounded) = p))) /\\ (((forall fom_index_pfp_empty_add_result_right_bounded. (exists fom_gap_pfp_empty_add_result_right_bounded_index_bound. fom_gap_pfp_empty_add_result_right_bounded_index_bound + S (fom_index_pfp_empty_add_result_right_bounded) = 0) -> exists fom_value_pfp_empty_add_result_right_bounded. ((((exists fom_beta_height_pfp_empty_add_result_right_bounded_entry. fom_beta_height_pfp_empty_add_result_right_bounded_entry + S (fom_value_pfp_empty_add_result_right_bounded) = S ((S (fom_index_pfp_empty_add_result_right_bounded)) * 0)) /\\ exists fom_beta_quotient_pfp_empty_add_result_right_bounded_entry. 0 = fom_beta_quotient_pfp_empty_add_result_right_bounded_entry * S ((S (fom_index_pfp_empty_add_result_right_bounded)) * 0) + (fom_value_pfp_empty_add_result_right_bounded))) /\\ (exists fom_gap_pfp_empty_add_result_right_bounded_value_bound. fom_gap_pfp_empty_add_result_right_bounded_value_bound + S (fom_value_pfp_empty_add_result_right_bounded) = p))) /\\ (((forall fom_index_pfp_empty_add_result_result_bounded. (exists fom_gap_pfp_empty_add_result_result_bounded_index_bound. fom_gap_pfp_empty_add_result_result_bounded_index_bound + S (fom_index_pfp_empty_add_result_result_bounded) = A) -> exists fom_value_pfp_empty_add_result_result_bounded. ((((exists fom_beta_height_pfp_empty_add_result_result_bounded_entry. fom_beta_height_pfp_empty_add_result_result_bounded_entry + S (fom_value_pfp_empty_add_result_result_bounded) = S ((S (fom_index_pfp_empty_add_result_result_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_empty_add_result_result_bounded_entry. ab = fom_beta_quotient_pfp_empty_add_result_result_bounded_entry * S ((S (fom_index_pfp_empty_add_result_result_bounded)) * ac) + (fom_value_pfp_empty_add_result_result_bounded))) /\\ (exists fom_gap_pfp_empty_add_result_result_bounded_value_bound. fom_gap_pfp_empty_add_result_result_bounded_value_bound + S (fom_value_pfp_empty_add_result_result_bounded) = p))) /\\ ((exists pfaa_left_b_empty_add_result pfaa_left_c_empty_add_result pfaa_right_b_empty_add_result pfaa_right_c_empty_add_result pfaa_sum_b_empty_add_result pfaa_sum_c_empty_add_result pfaa_length_empty_add_result. ((((forall pfrep_power_empty_add_result_witness_common_left pfrep_left_empty_add_result_witness_common_left pfrep_right_empty_add_result_witness_common_left. ((exists pfrep_position_empty_add_result_witness_common_leftfirst. ((pfrep_position_empty_add_result_witness_common_leftfirst+S (pfrep_power_empty_add_result_witness_common_left)=(A)) /\\ ((((exists ff_h_pfp_empty_add_result_witness_common_leftfirstentry. ff_h_pfp_empty_add_result_witness_common_leftfirstentry + S (pfrep_left_empty_add_result_witness_common_left) = S ((S (pfrep_position_empty_add_result_witness_common_leftfirst)) * ac)) /\\ exists ff_q_pfp_empty_add_result_witness_common_leftfirstentry. ab = ff_q_pfp_empty_add_result_witness_common_leftfirstentry * S ((S (pfrep_position_empty_add_result_witness_common_leftfirst)) * ac) + (pfrep_left_empty_add_result_witness_common_left)))))) \\/ (((exists pfrep_gap_empty_add_result_witness_common_leftfirstoutside. pfrep_gap_empty_add_result_witness_common_leftfirstoutside+(A)=(pfrep_power_empty_add_result_witness_common_left)) /\\ (((pfrep_left_empty_add_result_witness_common_left)=0))))) -> ((exists pfrep_position_empty_add_result_witness_common_leftsecond. ((pfrep_position_empty_add_result_witness_common_leftsecond+S (pfrep_power_empty_add_result_witness_common_left)=(pfaa_length_empty_add_result)) /\\ ((((exists ff_h_pfp_empty_add_result_witness_common_leftsecondentry. ff_h_pfp_empty_add_result_witness_common_leftsecondentry + S (pfrep_right_empty_add_result_witness_common_left) = S ((S (pfrep_position_empty_add_result_witness_common_leftsecond)) * pfaa_left_c_empty_add_result)) /\\ exists ff_q_pfp_empty_add_result_witness_common_leftsecondentry. pfaa_left_b_empty_add_result = ff_q_pfp_empty_add_result_witness_common_leftsecondentry * S ((S (pfrep_position_empty_add_result_witness_common_leftsecond)) * pfaa_left_c_empty_add_result) + (pfrep_right_empty_add_result_witness_common_left)))))) \\/ (((exists pfrep_gap_empty_add_result_witness_common_leftsecondoutside. pfrep_gap_empty_add_result_witness_common_leftsecondoutside+(pfaa_length_empty_add_result)=(pfrep_power_empty_add_result_witness_common_left)) /\\ (((pfrep_right_empty_add_result_witness_common_left)=0))))) -> pfrep_left_empty_add_result_witness_common_left=pfrep_right_empty_add_result_witness_common_left) /\\ ((forall pfrep_power_empty_add_result_witness_common_right pfrep_left_empty_add_result_witness_common_right pfrep_right_empty_add_result_witness_common_right. ((exists pfrep_position_empty_add_result_witness_common_rightfirst. ((pfrep_position_empty_add_result_witness_common_rightfirst+S (pfrep_power_empty_add_result_witness_common_right)=(0)) /\\ ((((exists ff_h_pfp_empty_add_result_witness_common_rightfirstentry. ff_h_pfp_empty_add_result_witness_common_rightfirstentry + S (pfrep_left_empty_add_result_witness_common_right) = S ((S (pfrep_position_empty_add_result_witness_common_rightfirst)) * 0)) /\\ exists ff_q_pfp_empty_add_result_witness_common_rightfirstentry. 0 = ff_q_pfp_empty_add_result_witness_common_rightfirstentry * S ((S (pfrep_position_empty_add_result_witness_common_rightfirst)) * 0) + (pfrep_left_empty_add_result_witness_common_right)))))) \\/ (((exists pfrep_gap_empty_add_result_witness_common_rightfirstoutside. pfrep_gap_empty_add_result_witness_common_rightfirstoutside+(0)=(pfrep_power_empty_add_result_witness_common_right)) /\\ (((pfrep_left_empty_add_result_witness_common_right)=0))))) -> ((exists pfrep_position_empty_add_result_witness_common_rightsecond. ((pfrep_position_empty_add_result_witness_common_rightsecond+S (pfrep_power_empty_add_result_witness_common_right)=(pfaa_length_empty_add_result)) /\\ ((((exists ff_h_pfp_empty_add_result_witness_common_rightsecondentry. ff_h_pfp_empty_add_result_witness_common_rightsecondentry + S (pfrep_right_empty_add_result_witness_common_right) = S ((S (pfrep_position_empty_add_result_witness_common_rightsecond)) * pfaa_right_c_empty_add_result)) /\\ exists ff_q_pfp_empty_add_result_witness_common_rightsecondentry. pfaa_right_b_empty_add_result = ff_q_pfp_empty_add_result_witness_common_rightsecondentry * S ((S (pfrep_position_empty_add_result_witness_common_rightsecond)) * pfaa_right_c_empty_add_result) + (pfrep_right_empty_add_result_witness_common_right)))))) \\/ (((exists pfrep_gap_empty_add_result_witness_common_rightsecondoutside. pfrep_gap_empty_add_result_witness_common_rightsecondoutside+(pfaa_length_empty_add_result)=(pfrep_power_empty_add_result_witness_common_right)) /\\ (((pfrep_right_empty_add_result_witness_common_right)=0))))) -> pfrep_left_empty_add_result_witness_common_right=pfrep_right_empty_add_result_witness_common_right)))) /\\ (((forall pfp_index_empty_add_result_witness_operation. (exists pfa_gap_empty_add_result_witness_operationindex. pfa_gap_empty_add_result_witness_operationindex + S (pfp_index_empty_add_result_witness_operation) = (pfaa_length_empty_add_result)) -> exists pfp_left_empty_add_result_witness_operation pfp_right_empty_add_result_witness_operation pfp_value_empty_add_result_witness_operation. ((((exists ff_h_pfp_empty_add_result_witness_operationleft. ff_h_pfp_empty_add_result_witness_operationleft + S (pfp_left_empty_add_result_witness_operation) = S ((S (pfp_index_empty_add_result_witness_operation)) * pfaa_left_c_empty_add_result)) /\\ exists ff_q_pfp_empty_add_result_witness_operationleft. pfaa_left_b_empty_add_result = ff_q_pfp_empty_add_result_witness_operationleft * S ((S (pfp_index_empty_add_result_witness_operation)) * pfaa_left_c_empty_add_result) + (pfp_left_empty_add_result_witness_operation))) /\\ (((((exists ff_h_pfp_empty_add_result_witness_operationright. ff_h_pfp_empty_add_result_witness_operationright + S (pfp_right_empty_add_result_witness_operation) = S ((S (pfp_index_empty_add_result_witness_operation)) * pfaa_right_c_empty_add_result)) /\\ exists ff_q_pfp_empty_add_result_witness_operationright. pfaa_right_b_empty_add_result = ff_q_pfp_empty_add_result_witness_operationright * S ((S (pfp_index_empty_add_result_witness_operation)) * pfaa_right_c_empty_add_result) + (pfp_right_empty_add_result_witness_operation))) /\\ (((((exists ff_h_pfp_empty_add_result_witness_operationtarget. ff_h_pfp_empty_add_result_witness_operationtarget + S (pfp_value_empty_add_result_witness_operation) = S ((S (pfp_index_empty_add_result_witness_operation)) * pfaa_sum_c_empty_add_result)) /\\ exists ff_q_pfp_empty_add_result_witness_operationtarget. pfaa_sum_b_empty_add_result = ff_q_pfp_empty_add_result_witness_operationtarget * S ((S (pfp_index_empty_add_result_witness_operation)) * pfaa_sum_c_empty_add_result) + (pfp_value_empty_add_result_witness_operation))) /\\ ((((exists pfa_gap_empty_add_result_witness_operationoperationleft. pfa_gap_empty_add_result_witness_operationoperationleft + S (pfp_left_empty_add_result_witness_operation) = (p)) /\\ (((exists pfa_gap_empty_add_result_witness_operationoperationright. pfa_gap_empty_add_result_witness_operationoperationright + S (pfp_right_empty_add_result_witness_operation) = (p)) /\\ ((((exists pfa_gap_empty_add_result_witness_operationoperationresultbound. pfa_gap_empty_add_result_witness_operationoperationresultbound + S (pfp_value_empty_add_result_witness_operation) = (p)) /\\ ((exists pfa_offset_left_empty_add_result_witness_operationoperationresultcongruence pfa_offset_right_empty_add_result_witness_operationoperationresultcongruence. ((pfp_left_empty_add_result_witness_operation) + (pfp_right_empty_add_result_witness_operation)) + (p) * pfa_offset_left_empty_add_result_witness_operationoperationresultcongruence = (pfp_value_empty_add_result_witness_operation) + (p) * pfa_offset_right_empty_add_result_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_empty_add_result_witness_output pfrep_left_empty_add_result_witness_output pfrep_right_empty_add_result_witness_output. ((exists pfrep_position_empty_add_result_witness_outputfirst. ((pfrep_position_empty_add_result_witness_outputfirst+S (pfrep_power_empty_add_result_witness_output)=(pfaa_length_empty_add_result)) /\\ ((((exists ff_h_pfp_empty_add_result_witness_outputfirstentry. ff_h_pfp_empty_add_result_witness_outputfirstentry + S (pfrep_left_empty_add_result_witness_output) = S ((S (pfrep_position_empty_add_result_witness_outputfirst)) * pfaa_sum_c_empty_add_result)) /\\ exists ff_q_pfp_empty_add_result_witness_outputfirstentry. pfaa_sum_b_empty_add_result = ff_q_pfp_empty_add_result_witness_outputfirstentry * S ((S (pfrep_position_empty_add_result_witness_outputfirst)) * pfaa_sum_c_empty_add_result) + (pfrep_left_empty_add_result_witness_output)))))) \\/ (((exists pfrep_gap_empty_add_result_witness_outputfirstoutside. pfrep_gap_empty_add_result_witness_outputfirstoutside+(pfaa_length_empty_add_result)=(pfrep_power_empty_add_result_witness_output)) /\\ (((pfrep_left_empty_add_result_witness_output)=0))))) -> ((exists pfrep_position_empty_add_result_witness_outputsecond. ((pfrep_position_empty_add_result_witness_outputsecond+S (pfrep_power_empty_add_result_witness_output)=(A)) /\\ ((((exists ff_h_pfp_empty_add_result_witness_outputsecondentry. ff_h_pfp_empty_add_result_witness_outputsecondentry + S (pfrep_right_empty_add_result_witness_output) = S ((S (pfrep_position_empty_add_result_witness_outputsecond)) * ac)) /\\ exists ff_q_pfp_empty_add_result_witness_outputsecondentry. ab = ff_q_pfp_empty_add_result_witness_outputsecondentry * S ((S (pfrep_position_empty_add_result_witness_outputsecond)) * ac) + (pfrep_right_empty_add_result_witness_output)))))) \\/ (((exists pfrep_gap_empty_add_result_witness_outputsecondoutside. pfrep_gap_empty_add_result_witness_outputsecondoutside+(A)=(pfrep_power_empty_add_result_witness_output)) /\\ (((pfrep_right_empty_add_result_witness_output)=0))))) -> pfrep_left_empty_add_result_witness_output=pfrep_right_empty_add_result_witness_output)))))))))))))", "statement_sha256": "b2abf4a7f78ffa4b231d6a6745a6f3a47493ff1099d5857279065ee6d4a5fc75", "summary": "Construct a real zero prefix at the input length and its formal equivalence to the empty polynomial, giving an actual aligned right-zero sum for every canonical input.", "summary_sha256": "f1c9bfbd239fc574a014503f3e23a5db8dd1c4676d7b1dbd6681f8e013f3e2e1" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "11160460cec8fc37f76dd43a0dc7d520d91bf79730944e15df27d99f56246952", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 445, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "76b90226e5e29fdde3d9bb49accccf8d9b4c0cc17a4de406af253e999102533c", "test_sha256": 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hdivides_right", "cases hdivides_right_witness", "cases hdivides_right_witness_witness", "cases hdivides_right_witness_witness_witness", "cases hdivides_right_witness_witness_witness_witness", "cases hdivides_right_witness_witness_witness_witness_witness", "cases hdivides_right_witness_witness_witness_witness_witness_witness", "have hproduct_bound : forall fom_index_pfp_terminal_product_bound. (exists fom_gap_pfp_terminal_product_bound_index_bound. fom_gap_pfp_terminal_product_bound_index_bound + S (fom_index_pfp_terminal_product_bound) = x5) -> exists fom_value_pfp_terminal_product_bound. ((((exists fom_beta_height_pfp_terminal_product_bound_entry. fom_beta_height_pfp_terminal_product_bound_entry + S (fom_value_pfp_terminal_product_bound) = S ((S (fom_index_pfp_terminal_product_bound)) * x4)) /\\ exists fom_beta_quotient_pfp_terminal_product_bound_entry. x3 = fom_beta_quotient_pfp_terminal_product_bound_entry * S ((S (fom_index_pfp_terminal_product_bound)) * x4) + (fom_value_pfp_terminal_product_bound))) /\\ (exists fom_gap_pfp_terminal_product_bound_value_bound. fom_gap_pfp_terminal_product_bound_value_bound + S (fom_value_pfp_terminal_product_bound) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (x)", "specialize prime_field_polynomial_convolution_bounded (x1)", "specialize prime_field_polynomial_convolution_bounded (x2)", "specialize prime_field_polynomial_convolution_bounded (ab)", "specialize prime_field_polynomial_convolution_bounded (ac)", "specialize prime_field_polynomial_convolution_bounded (A)", "specialize prime_field_polynomial_convolution_bounded (x3)", "specialize prime_field_polynomial_convolution_bounded (x4)", "specialize prime_field_polynomial_convolution_bounded (x5)", "apply prime_field_polynomial_convolution_bounded", "exact hdivides_right_witness_witness_witness_witness_witness_witness_left", "have hempty_product : ((forall fom_index_pfp_terminal_empty_productleft. (exists fom_gap_pfp_terminal_empty_productleft_index_bound. fom_gap_pfp_terminal_empty_productleft_index_bound + S (fom_index_pfp_terminal_empty_productleft) = 0) -> exists fom_value_pfp_terminal_empty_productleft. ((((exists fom_beta_height_pfp_terminal_empty_productleft_entry. fom_beta_height_pfp_terminal_empty_productleft_entry + S (fom_value_pfp_terminal_empty_productleft) = S ((S (fom_index_pfp_terminal_empty_productleft)) * 0)) /\\ exists fom_beta_quotient_pfp_terminal_empty_productleft_entry. 0 = fom_beta_quotient_pfp_terminal_empty_productleft_entry * S ((S (fom_index_pfp_terminal_empty_productleft)) * 0) + (fom_value_pfp_terminal_empty_productleft))) /\\ (exists fom_gap_pfp_terminal_empty_productleft_value_bound. fom_gap_pfp_terminal_empty_productleft_value_bound + S (fom_value_pfp_terminal_empty_productleft) = p))) /\\ (((forall fom_index_pfp_terminal_empty_productright. (exists fom_gap_pfp_terminal_empty_productright_index_bound. fom_gap_pfp_terminal_empty_productright_index_bound + S (fom_index_pfp_terminal_empty_productright) = B) -> exists fom_value_pfp_terminal_empty_productright. ((((exists fom_beta_height_pfp_terminal_empty_productright_entry. fom_beta_height_pfp_terminal_empty_productright_entry + S (fom_value_pfp_terminal_empty_productright) = S ((S (fom_index_pfp_terminal_empty_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_terminal_empty_productright_entry. bb = fom_beta_quotient_pfp_terminal_empty_productright_entry * S ((S (fom_index_pfp_terminal_empty_productright)) * bc) + (fom_value_pfp_terminal_empty_productright))) /\\ (exists fom_gap_pfp_terminal_empty_productright_value_bound. fom_gap_pfp_terminal_empty_productright_value_bound + S (fom_value_pfp_terminal_empty_productright) = p))) /\\ (((((((0)=0 \\/ (B)=0) /\\ (((0)=0)))) \\/ (((~((0)=0)) /\\ (((~((B)=0)) /\\ (((0)+(B)=S (0)))))))) /\\ ((forall pfc_index_terminal_empty_productcoefficients. (exists pfa_gap_terminal_empty_productcoefficientsbound. pfa_gap_terminal_empty_productcoefficientsbound + S (pfc_index_terminal_empty_productcoefficients) = (0)) -> exists pfc_value_terminal_empty_productcoefficients. ((((exists ff_h_pfp_terminal_empty_productcoefficientsentry. ff_h_pfp_terminal_empty_productcoefficientsentry + S (pfc_value_terminal_empty_productcoefficients) = S ((S (pfc_index_terminal_empty_productcoefficients)) * 0)) /\\ exists ff_q_pfp_terminal_empty_productcoefficientsentry. 0 = ff_q_pfp_terminal_empty_productcoefficientsentry * S ((S (pfc_index_terminal_empty_productcoefficients)) * 0) + (pfc_value_terminal_empty_productcoefficients))) /\\ ((exists pfc_terms_code_terminal_empty_productcoefficientscoefficient pfc_terms_scale_terminal_empty_productcoefficientscoefficient pfc_natural_sum_terminal_empty_productcoefficientscoefficient. ((forall pfc_index_terminal_empty_productcoefficientscoefficientdiagonal. (exists pfa_gap_terminal_empty_productcoefficientscoefficientdiagonalbound. pfa_gap_terminal_empty_productcoefficientscoefficientdiagonalbound + S (pfc_index_terminal_empty_productcoefficientscoefficientdiagonal) = (S (pfc_index_terminal_empty_productcoefficients))) -> exists pfc_value_terminal_empty_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_terminal_empty_productcoefficientscoefficientdiagonalentry. ff_h_pfp_terminal_empty_productcoefficientscoefficientdiagonalentry + S (pfc_value_terminal_empty_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_terminal_empty_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_terminal_empty_productcoefficientscoefficient)) /\\ exists ff_q_pfp_terminal_empty_productcoefficientscoefficientdiagonalentry. pfc_terms_code_terminal_empty_productcoefficientscoefficient = ff_q_pfp_terminal_empty_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_terminal_empty_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_terminal_empty_productcoefficientscoefficient) + (pfc_value_terminal_empty_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_terminal_empty_productcoefficientscoefficientdiagonalterm pfc_left_terminal_empty_productcoefficientscoefficientdiagonalterm pfc_right_terminal_empty_productcoefficientscoefficientdiagonalterm. (((pfc_index_terminal_empty_productcoefficientscoefficientdiagonal)+pfc_complement_terminal_empty_productcoefficientscoefficientdiagonalterm=(pfc_index_terminal_empty_productcoefficients)) /\\ ((((((exists pfa_gap_terminal_empty_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_terminal_empty_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_terminal_empty_productcoefficientscoefficientdiagonal) = (0)) /\\ ((((exists ff_h_pfp_terminal_empty_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_terminal_empty_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_terminal_empty_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_terminal_empty_productcoefficientscoefficientdiagonal)) * 0)) /\\ exists ff_q_pfp_terminal_empty_productcoefficientscoefficientdiagonaltermleftentry. 0 = ff_q_pfp_terminal_empty_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_terminal_empty_productcoefficientscoefficientdiagonal)) * 0) + (pfc_left_terminal_empty_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_terminal_empty_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_terminal_empty_productcoefficientscoefficientdiagonaltermleftoutside+(0)=(pfc_index_terminal_empty_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_terminal_empty_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_terminal_empty_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_terminal_empty_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_terminal_empty_productcoefficientscoefficientdiagonalterm) = (B)) /\\ ((((exists ff_h_pfp_terminal_empty_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_terminal_empty_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_terminal_empty_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_terminal_empty_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_terminal_empty_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_terminal_empty_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_terminal_empty_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_terminal_empty_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_terminal_empty_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_terminal_empty_productcoefficientscoefficientdiagonaltermrightoutside+(B)=(pfc_complement_terminal_empty_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_terminal_empty_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_terminal_empty_productcoefficientscoefficientdiagonal)=pfc_left_terminal_empty_productcoefficientscoefficientdiagonalterm*pfc_right_terminal_empty_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_terminal_empty_productcoefficientscoefficientsum fs_v_pfc_terminal_empty_productcoefficientscoefficientsum. ((((exists fs_h_pfc_terminal_empty_productcoefficientscoefficientsum_body_start. fs_h_pfc_terminal_empty_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_terminal_empty_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_empty_productcoefficientscoefficientsum_body_start. fs_u_pfc_terminal_empty_productcoefficientscoefficientsum = fs_q_pfc_terminal_empty_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_terminal_empty_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_terminal_empty_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_terminal_empty_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_terminal_empty_productcoefficientscoefficient) = S ((S (S (pfc_index_terminal_empty_productcoefficients))) * fs_v_pfc_terminal_empty_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_empty_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_terminal_empty_productcoefficientscoefficientsum = fs_q_pfc_terminal_empty_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_terminal_empty_productcoefficients))) * fs_v_pfc_terminal_empty_productcoefficientscoefficientsum) + (pfc_natural_sum_terminal_empty_productcoefficientscoefficient))) /\\ forall fs_i_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps = S (pfc_index_terminal_empty_productcoefficients)) -> exists fs_a_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps fs_r_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps fs_s_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_terminal_empty_productcoefficientscoefficient)) /\\ exists fs_q_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_terminal_empty_productcoefficientscoefficient = fs_q_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_terminal_empty_productcoefficientscoefficient) + (fs_a_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_empty_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_terminal_empty_productcoefficientscoefficientsum = fs_q_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_empty_productcoefficientscoefficientsum) + (fs_r_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_empty_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_terminal_empty_productcoefficientscoefficientsum = fs_q_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_empty_productcoefficientscoefficientsum) + (fs_s_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps = fs_r_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps + fs_a_pfc_terminal_empty_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_terminal_empty_productcoefficientscoefficientresiduebound. pfa_gap_terminal_empty_productcoefficientscoefficientresiduebound + S (pfc_value_terminal_empty_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_terminal_empty_productcoefficientscoefficientresiduecongruence pfa_offset_right_terminal_empty_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_terminal_empty_productcoefficientscoefficient) + (p) * pfa_offset_left_terminal_empty_productcoefficientscoefficientresiduecongruence = (pfc_value_terminal_empty_productcoefficients) + (p) * pfa_offset_right_terminal_empty_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_empty (p)", "specialize prime_field_polynomial_convolution_empty (0)", "specialize prime_field_polynomial_convolution_empty (0)", "specialize prime_field_polynomial_convolution_empty (0)", "specialize prime_field_polynomial_convolution_empty (bb)", "specialize prime_field_polynomial_convolution_empty (bc)", "specialize prime_field_polynomial_convolution_empty (B)", "specialize prime_field_polynomial_convolution_empty (0)", "specialize prime_field_polynomial_convolution_empty (0)", "apply prime_field_polynomial_convolution_empty", "specialize matrix_rank_bounded_prefix_empty (0)", "specialize matrix_rank_bounded_prefix_empty (0)", "specialize matrix_rank_bounded_prefix_empty (p)", "apply matrix_rank_bounded_prefix_empty", "exact hb", "left", "refl", "have hsum : ((forall fom_index_pfp_terminal_sum_left_bounded. (exists fom_gap_pfp_terminal_sum_left_bounded_index_bound. fom_gap_pfp_terminal_sum_left_bounded_index_bound + S (fom_index_pfp_terminal_sum_left_bounded) = x5) -> exists fom_value_pfp_terminal_sum_left_bounded. ((((exists fom_beta_height_pfp_terminal_sum_left_bounded_entry. fom_beta_height_pfp_terminal_sum_left_bounded_entry + S (fom_value_pfp_terminal_sum_left_bounded) = S ((S (fom_index_pfp_terminal_sum_left_bounded)) * x4)) /\\ exists fom_beta_quotient_pfp_terminal_sum_left_bounded_entry. x3 = fom_beta_quotient_pfp_terminal_sum_left_bounded_entry * S ((S (fom_index_pfp_terminal_sum_left_bounded)) * x4) + (fom_value_pfp_terminal_sum_left_bounded))) /\\ (exists fom_gap_pfp_terminal_sum_left_bounded_value_bound. fom_gap_pfp_terminal_sum_left_bounded_value_bound + S (fom_value_pfp_terminal_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_terminal_sum_right_bounded. (exists fom_gap_pfp_terminal_sum_right_bounded_index_bound. fom_gap_pfp_terminal_sum_right_bounded_index_bound + S (fom_index_pfp_terminal_sum_right_bounded) = 0) -> exists fom_value_pfp_terminal_sum_right_bounded. ((((exists fom_beta_height_pfp_terminal_sum_right_bounded_entry. fom_beta_height_pfp_terminal_sum_right_bounded_entry + S (fom_value_pfp_terminal_sum_right_bounded) = S ((S (fom_index_pfp_terminal_sum_right_bounded)) * 0)) /\\ exists fom_beta_quotient_pfp_terminal_sum_right_bounded_entry. 0 = fom_beta_quotient_pfp_terminal_sum_right_bounded_entry * S ((S (fom_index_pfp_terminal_sum_right_bounded)) * 0) + (fom_value_pfp_terminal_sum_right_bounded))) /\\ (exists fom_gap_pfp_terminal_sum_right_bounded_value_bound. fom_gap_pfp_terminal_sum_right_bounded_value_bound + S (fom_value_pfp_terminal_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_terminal_sum_result_bounded. (exists fom_gap_pfp_terminal_sum_result_bounded_index_bound. fom_gap_pfp_terminal_sum_result_bounded_index_bound + S (fom_index_pfp_terminal_sum_result_bounded) = G) -> exists fom_value_pfp_terminal_sum_result_bounded. ((((exists fom_beta_height_pfp_terminal_sum_result_bounded_entry. fom_beta_height_pfp_terminal_sum_result_bounded_entry + S (fom_value_pfp_terminal_sum_result_bounded) = S ((S (fom_index_pfp_terminal_sum_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_terminal_sum_result_bounded_entry. gb = fom_beta_quotient_pfp_terminal_sum_result_bounded_entry * S ((S (fom_index_pfp_terminal_sum_result_bounded)) * gc) + (fom_value_pfp_terminal_sum_result_bounded))) /\\ (exists fom_gap_pfp_terminal_sum_result_bounded_value_bound. fom_gap_pfp_terminal_sum_result_bounded_value_bound + S (fom_value_pfp_terminal_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_terminal_sum pfaa_left_c_terminal_sum pfaa_right_b_terminal_sum pfaa_right_c_terminal_sum pfaa_sum_b_terminal_sum pfaa_sum_c_terminal_sum pfaa_length_terminal_sum. ((((forall pfrep_power_terminal_sum_witness_common_left pfrep_left_terminal_sum_witness_common_left pfrep_right_terminal_sum_witness_common_left. ((exists pfrep_position_terminal_sum_witness_common_leftfirst. ((pfrep_position_terminal_sum_witness_common_leftfirst+S (pfrep_power_terminal_sum_witness_common_left)=(x5)) /\\ ((((exists ff_h_pfp_terminal_sum_witness_common_leftfirstentry. ff_h_pfp_terminal_sum_witness_common_leftfirstentry + S (pfrep_left_terminal_sum_witness_common_left) = S ((S (pfrep_position_terminal_sum_witness_common_leftfirst)) * x4)) /\\ exists ff_q_pfp_terminal_sum_witness_common_leftfirstentry. x3 = ff_q_pfp_terminal_sum_witness_common_leftfirstentry * S ((S (pfrep_position_terminal_sum_witness_common_leftfirst)) * x4) + (pfrep_left_terminal_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_terminal_sum_witness_common_leftfirstoutside. pfrep_gap_terminal_sum_witness_common_leftfirstoutside+(x5)=(pfrep_power_terminal_sum_witness_common_left)) /\\ (((pfrep_left_terminal_sum_witness_common_left)=0))))) -> ((exists pfrep_position_terminal_sum_witness_common_leftsecond. ((pfrep_position_terminal_sum_witness_common_leftsecond+S (pfrep_power_terminal_sum_witness_common_left)=(pfaa_length_terminal_sum)) /\\ ((((exists ff_h_pfp_terminal_sum_witness_common_leftsecondentry. ff_h_pfp_terminal_sum_witness_common_leftsecondentry + S (pfrep_right_terminal_sum_witness_common_left) = S ((S (pfrep_position_terminal_sum_witness_common_leftsecond)) * pfaa_left_c_terminal_sum)) /\\ exists ff_q_pfp_terminal_sum_witness_common_leftsecondentry. pfaa_left_b_terminal_sum = ff_q_pfp_terminal_sum_witness_common_leftsecondentry * S ((S (pfrep_position_terminal_sum_witness_common_leftsecond)) * pfaa_left_c_terminal_sum) + (pfrep_right_terminal_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_terminal_sum_witness_common_leftsecondoutside. pfrep_gap_terminal_sum_witness_common_leftsecondoutside+(pfaa_length_terminal_sum)=(pfrep_power_terminal_sum_witness_common_left)) /\\ (((pfrep_right_terminal_sum_witness_common_left)=0))))) -> pfrep_left_terminal_sum_witness_common_left=pfrep_right_terminal_sum_witness_common_left) /\\ ((forall pfrep_power_terminal_sum_witness_common_right pfrep_left_terminal_sum_witness_common_right pfrep_right_terminal_sum_witness_common_right. ((exists pfrep_position_terminal_sum_witness_common_rightfirst. ((pfrep_position_terminal_sum_witness_common_rightfirst+S (pfrep_power_terminal_sum_witness_common_right)=(0)) /\\ ((((exists ff_h_pfp_terminal_sum_witness_common_rightfirstentry. ff_h_pfp_terminal_sum_witness_common_rightfirstentry + S (pfrep_left_terminal_sum_witness_common_right) = S ((S (pfrep_position_terminal_sum_witness_common_rightfirst)) * 0)) /\\ exists ff_q_pfp_terminal_sum_witness_common_rightfirstentry. 0 = ff_q_pfp_terminal_sum_witness_common_rightfirstentry * S ((S (pfrep_position_terminal_sum_witness_common_rightfirst)) * 0) + (pfrep_left_terminal_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_terminal_sum_witness_common_rightfirstoutside. pfrep_gap_terminal_sum_witness_common_rightfirstoutside+(0)=(pfrep_power_terminal_sum_witness_common_right)) /\\ (((pfrep_left_terminal_sum_witness_common_right)=0))))) -> ((exists pfrep_position_terminal_sum_witness_common_rightsecond. ((pfrep_position_terminal_sum_witness_common_rightsecond+S (pfrep_power_terminal_sum_witness_common_right)=(pfaa_length_terminal_sum)) /\\ ((((exists ff_h_pfp_terminal_sum_witness_common_rightsecondentry. ff_h_pfp_terminal_sum_witness_common_rightsecondentry + S (pfrep_right_terminal_sum_witness_common_right) = S ((S (pfrep_position_terminal_sum_witness_common_rightsecond)) * pfaa_right_c_terminal_sum)) /\\ exists ff_q_pfp_terminal_sum_witness_common_rightsecondentry. pfaa_right_b_terminal_sum = ff_q_pfp_terminal_sum_witness_common_rightsecondentry * S ((S (pfrep_position_terminal_sum_witness_common_rightsecond)) * pfaa_right_c_terminal_sum) + (pfrep_right_terminal_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_terminal_sum_witness_common_rightsecondoutside. pfrep_gap_terminal_sum_witness_common_rightsecondoutside+(pfaa_length_terminal_sum)=(pfrep_power_terminal_sum_witness_common_right)) /\\ (((pfrep_right_terminal_sum_witness_common_right)=0))))) -> pfrep_left_terminal_sum_witness_common_right=pfrep_right_terminal_sum_witness_common_right)))) /\\ (((forall pfp_index_terminal_sum_witness_operation. (exists pfa_gap_terminal_sum_witness_operationindex. pfa_gap_terminal_sum_witness_operationindex + S (pfp_index_terminal_sum_witness_operation) = (pfaa_length_terminal_sum)) -> exists pfp_left_terminal_sum_witness_operation pfp_right_terminal_sum_witness_operation pfp_value_terminal_sum_witness_operation. ((((exists ff_h_pfp_terminal_sum_witness_operationleft. ff_h_pfp_terminal_sum_witness_operationleft + S (pfp_left_terminal_sum_witness_operation) = S ((S (pfp_index_terminal_sum_witness_operation)) * pfaa_left_c_terminal_sum)) /\\ exists ff_q_pfp_terminal_sum_witness_operationleft. pfaa_left_b_terminal_sum = ff_q_pfp_terminal_sum_witness_operationleft * S ((S (pfp_index_terminal_sum_witness_operation)) * pfaa_left_c_terminal_sum) + (pfp_left_terminal_sum_witness_operation))) /\\ (((((exists ff_h_pfp_terminal_sum_witness_operationright. ff_h_pfp_terminal_sum_witness_operationright + S (pfp_right_terminal_sum_witness_operation) = S ((S (pfp_index_terminal_sum_witness_operation)) * pfaa_right_c_terminal_sum)) /\\ exists ff_q_pfp_terminal_sum_witness_operationright. pfaa_right_b_terminal_sum = ff_q_pfp_terminal_sum_witness_operationright * S ((S (pfp_index_terminal_sum_witness_operation)) * pfaa_right_c_terminal_sum) + (pfp_right_terminal_sum_witness_operation))) /\\ (((((exists ff_h_pfp_terminal_sum_witness_operationtarget. ff_h_pfp_terminal_sum_witness_operationtarget + S (pfp_value_terminal_sum_witness_operation) = S ((S (pfp_index_terminal_sum_witness_operation)) * pfaa_sum_c_terminal_sum)) /\\ exists ff_q_pfp_terminal_sum_witness_operationtarget. pfaa_sum_b_terminal_sum = ff_q_pfp_terminal_sum_witness_operationtarget * S ((S (pfp_index_terminal_sum_witness_operation)) * pfaa_sum_c_terminal_sum) + (pfp_value_terminal_sum_witness_operation))) /\\ ((((exists pfa_gap_terminal_sum_witness_operationoperationleft. pfa_gap_terminal_sum_witness_operationoperationleft + S (pfp_left_terminal_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_terminal_sum_witness_operationoperationright. pfa_gap_terminal_sum_witness_operationoperationright + S (pfp_right_terminal_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_terminal_sum_witness_operationoperationresultbound. pfa_gap_terminal_sum_witness_operationoperationresultbound + S (pfp_value_terminal_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_terminal_sum_witness_operationoperationresultcongruence pfa_offset_right_terminal_sum_witness_operationoperationresultcongruence. ((pfp_left_terminal_sum_witness_operation) + (pfp_right_terminal_sum_witness_operation)) + (p) * pfa_offset_left_terminal_sum_witness_operationoperationresultcongruence = (pfp_value_terminal_sum_witness_operation) + (p) * pfa_offset_right_terminal_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_terminal_sum_witness_output pfrep_left_terminal_sum_witness_output pfrep_right_terminal_sum_witness_output. ((exists pfrep_position_terminal_sum_witness_outputfirst. ((pfrep_position_terminal_sum_witness_outputfirst+S (pfrep_power_terminal_sum_witness_output)=(pfaa_length_terminal_sum)) /\\ ((((exists ff_h_pfp_terminal_sum_witness_outputfirstentry. ff_h_pfp_terminal_sum_witness_outputfirstentry + S (pfrep_left_terminal_sum_witness_output) = S ((S (pfrep_position_terminal_sum_witness_outputfirst)) * pfaa_sum_c_terminal_sum)) /\\ exists ff_q_pfp_terminal_sum_witness_outputfirstentry. pfaa_sum_b_terminal_sum = ff_q_pfp_terminal_sum_witness_outputfirstentry * S ((S (pfrep_position_terminal_sum_witness_outputfirst)) * pfaa_sum_c_terminal_sum) + (pfrep_left_terminal_sum_witness_output)))))) \\/ (((exists pfrep_gap_terminal_sum_witness_outputfirstoutside. pfrep_gap_terminal_sum_witness_outputfirstoutside+(pfaa_length_terminal_sum)=(pfrep_power_terminal_sum_witness_output)) /\\ (((pfrep_left_terminal_sum_witness_output)=0))))) -> ((exists pfrep_position_terminal_sum_witness_outputsecond. ((pfrep_position_terminal_sum_witness_outputsecond+S (pfrep_power_terminal_sum_witness_output)=(G)) /\\ ((((exists ff_h_pfp_terminal_sum_witness_outputsecondentry. ff_h_pfp_terminal_sum_witness_outputsecondentry + S (pfrep_right_terminal_sum_witness_output) = S ((S (pfrep_position_terminal_sum_witness_outputsecond)) * gc)) /\\ exists ff_q_pfp_terminal_sum_witness_outputsecondentry. gb = ff_q_pfp_terminal_sum_witness_outputsecondentry * S ((S (pfrep_position_terminal_sum_witness_outputsecond)) * gc) + (pfrep_right_terminal_sum_witness_output)))))) \\/ (((exists pfrep_gap_terminal_sum_witness_outputsecondoutside. pfrep_gap_terminal_sum_witness_outputsecondoutside+(G)=(pfrep_power_terminal_sum_witness_output)) /\\ (((pfrep_right_terminal_sum_witness_output)=0))))) -> pfrep_left_terminal_sum_witness_output=pfrep_right_terminal_sum_witness_output))))))))))))", "specialize prime_field_polynomial_aligned_add_transport (p)", "specialize prime_field_polynomial_aligned_add_transport (x3)", "specialize prime_field_polynomial_aligned_add_transport (x4)", "specialize prime_field_polynomial_aligned_add_transport (x5)", "specialize prime_field_polynomial_aligned_add_transport (0)", "specialize prime_field_polynomial_aligned_add_transport (0)", "specialize prime_field_polynomial_aligned_add_transport (0)", "specialize prime_field_polynomial_aligned_add_transport (x3)", "specialize prime_field_polynomial_aligned_add_transport (x4)", "specialize prime_field_polynomial_aligned_add_transport (x5)", "specialize prime_field_polynomial_aligned_add_transport (x3)", "specialize prime_field_polynomial_aligned_add_transport (x4)", "specialize prime_field_polynomial_aligned_add_transport (x5)", "specialize prime_field_polynomial_aligned_add_transport (0)", "specialize prime_field_polynomial_aligned_add_transport (0)", "specialize prime_field_polynomial_aligned_add_transport (0)", "specialize prime_field_polynomial_aligned_add_transport (gb)", "specialize prime_field_polynomial_aligned_add_transport (gc)", "specialize prime_field_polynomial_aligned_add_transport (G)", "apply prime_field_polynomial_aligned_add_transport", "exact hproduct_bound", "specialize matrix_rank_bounded_prefix_empty (0)", "specialize matrix_rank_bounded_prefix_empty (0)", "specialize matrix_rank_bounded_prefix_empty (p)", "apply matrix_rank_bounded_prefix_empty", "exact hdivides_left", "specialize prime_field_polynomial_power_coefficient_functional (x3)", "specialize prime_field_polynomial_power_coefficient_functional (x4)", "specialize prime_field_polynomial_power_coefficient_functional (x5)", "apply prime_field_polynomial_power_coefficient_functional", "specialize prime_field_polynomial_power_coefficient_functional (0)", "specialize prime_field_polynomial_power_coefficient_functional (0)", "specialize prime_field_polynomial_power_coefficient_functional (0)", "apply prime_field_polynomial_power_coefficient_functional", "exact hdivides_right_witness_witness_witness_witness_witness_witness_right", "specialize prime_field_polynomial_aligned_add_empty_right (p)", "specialize prime_field_polynomial_aligned_add_empty_right (x3)", "specialize prime_field_polynomial_aligned_add_empty_right (x4)", "specialize prime_field_polynomial_aligned_add_empty_right (x5)", "apply prime_field_polynomial_aligned_add_empty_right", "exact hp", "exact hproduct_bound", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "exists 0", "exists 0", "exists 0", "split", "exact hdivides_right_witness_witness_witness_witness_witness_witness_left", "split", "exact hempty_product", "exact hsum" ], "script_sha256": "8a4f2a68f01f7c3c33a0894d817a28ac8c5c64b1d500473a62c80849486852a7", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_bezout_laws_candidate.py", "sha256": "76b90226e5e29fdde3d9bb49accccf8d9b4c0cc17a4de406af253e999102533c" }, "statement": "forall p ab ac A bb bc B gb gc G. (~((p) = 1) /\\ forall pfa_factor_left_terminal_prime pfa_factor_right_terminal_prime. (p) = pfa_factor_left_terminal_prime * pfa_factor_right_terminal_prime -> pfa_factor_left_terminal_prime = 1 \\/ pfa_factor_right_terminal_prime = 1) -> (forall fom_index_pfp_terminal_other_bound. (exists fom_gap_pfp_terminal_other_bound_index_bound. fom_gap_pfp_terminal_other_bound_index_bound + S (fom_index_pfp_terminal_other_bound) = B) -> exists fom_value_pfp_terminal_other_bound. ((((exists fom_beta_height_pfp_terminal_other_bound_entry. fom_beta_height_pfp_terminal_other_bound_entry + S (fom_value_pfp_terminal_other_bound) = S ((S (fom_index_pfp_terminal_other_bound)) * bc)) /\\ exists fom_beta_quotient_pfp_terminal_other_bound_entry. bb = fom_beta_quotient_pfp_terminal_other_bound_entry * S ((S (fom_index_pfp_terminal_other_bound)) * bc) + (fom_value_pfp_terminal_other_bound))) /\\ (exists fom_gap_pfp_terminal_other_bound_value_bound. fom_gap_pfp_terminal_other_bound_value_bound + S (fom_value_pfp_terminal_other_bound) = p))) -> (((forall fom_index_pfp_terminal_input_canonical. (exists fom_gap_pfp_terminal_input_canonical_index_bound. fom_gap_pfp_terminal_input_canonical_index_bound + S (fom_index_pfp_terminal_input_canonical) = G) -> exists fom_value_pfp_terminal_input_canonical. ((((exists fom_beta_height_pfp_terminal_input_canonical_entry. fom_beta_height_pfp_terminal_input_canonical_entry + S (fom_value_pfp_terminal_input_canonical) = S ((S (fom_index_pfp_terminal_input_canonical)) * gc)) /\\ exists fom_beta_quotient_pfp_terminal_input_canonical_entry. gb = fom_beta_quotient_pfp_terminal_input_canonical_entry * S ((S (fom_index_pfp_terminal_input_canonical)) * gc) + (fom_value_pfp_terminal_input_canonical))) /\\ (exists fom_gap_pfp_terminal_input_canonical_value_bound. fom_gap_pfp_terminal_input_canonical_value_bound + S (fom_value_pfp_terminal_input_canonical) = p))) /\\ ((exists pfrd_qb_terminal_input pfrd_qc_terminal_input pfrd_qlen_terminal_input pfrd_pb_terminal_input pfrd_pc_terminal_input pfrd_plen_terminal_input. ((((forall fom_index_pfp_terminal_input_productleft. (exists fom_gap_pfp_terminal_input_productleft_index_bound. fom_gap_pfp_terminal_input_productleft_index_bound + S (fom_index_pfp_terminal_input_productleft) = pfrd_qlen_terminal_input) -> exists fom_value_pfp_terminal_input_productleft. ((((exists fom_beta_height_pfp_terminal_input_productleft_entry. fom_beta_height_pfp_terminal_input_productleft_entry + S (fom_value_pfp_terminal_input_productleft) = S ((S (fom_index_pfp_terminal_input_productleft)) * pfrd_qc_terminal_input)) /\\ exists fom_beta_quotient_pfp_terminal_input_productleft_entry. pfrd_qb_terminal_input = fom_beta_quotient_pfp_terminal_input_productleft_entry * S ((S (fom_index_pfp_terminal_input_productleft)) * pfrd_qc_terminal_input) + (fom_value_pfp_terminal_input_productleft))) /\\ (exists fom_gap_pfp_terminal_input_productleft_value_bound. fom_gap_pfp_terminal_input_productleft_value_bound + S (fom_value_pfp_terminal_input_productleft) = p))) /\\ (((forall fom_index_pfp_terminal_input_productright. (exists fom_gap_pfp_terminal_input_productright_index_bound. fom_gap_pfp_terminal_input_productright_index_bound + S (fom_index_pfp_terminal_input_productright) = A) -> exists fom_value_pfp_terminal_input_productright. ((((exists fom_beta_height_pfp_terminal_input_productright_entry. fom_beta_height_pfp_terminal_input_productright_entry + S (fom_value_pfp_terminal_input_productright) = S ((S (fom_index_pfp_terminal_input_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_terminal_input_productright_entry. ab = fom_beta_quotient_pfp_terminal_input_productright_entry * S ((S (fom_index_pfp_terminal_input_productright)) * ac) + (fom_value_pfp_terminal_input_productright))) /\\ (exists fom_gap_pfp_terminal_input_productright_value_bound. fom_gap_pfp_terminal_input_productright_value_bound + S (fom_value_pfp_terminal_input_productright) = p))) /\\ (((((((pfrd_qlen_terminal_input)=0 \\/ (A)=0) /\\ (((pfrd_plen_terminal_input)=0)))) \\/ (((~((pfrd_qlen_terminal_input)=0)) /\\ (((~((A)=0)) /\\ (((pfrd_qlen_terminal_input)+(A)=S (pfrd_plen_terminal_input)))))))) /\\ ((forall pfc_index_terminal_input_productcoefficients. (exists pfa_gap_terminal_input_productcoefficientsbound. pfa_gap_terminal_input_productcoefficientsbound + S (pfc_index_terminal_input_productcoefficients) = (pfrd_plen_terminal_input)) -> exists pfc_value_terminal_input_productcoefficients. ((((exists ff_h_pfp_terminal_input_productcoefficientsentry. ff_h_pfp_terminal_input_productcoefficientsentry + S (pfc_value_terminal_input_productcoefficients) = S ((S (pfc_index_terminal_input_productcoefficients)) * pfrd_pc_terminal_input)) /\\ exists ff_q_pfp_terminal_input_productcoefficientsentry. pfrd_pb_terminal_input = ff_q_pfp_terminal_input_productcoefficientsentry * S ((S (pfc_index_terminal_input_productcoefficients)) * pfrd_pc_terminal_input) + (pfc_value_terminal_input_productcoefficients))) /\\ ((exists pfc_terms_code_terminal_input_productcoefficientscoefficient pfc_terms_scale_terminal_input_productcoefficientscoefficient pfc_natural_sum_terminal_input_productcoefficientscoefficient. ((forall pfc_index_terminal_input_productcoefficientscoefficientdiagonal. (exists pfa_gap_terminal_input_productcoefficientscoefficientdiagonalbound. pfa_gap_terminal_input_productcoefficientscoefficientdiagonalbound + S (pfc_index_terminal_input_productcoefficientscoefficientdiagonal) = (S (pfc_index_terminal_input_productcoefficients))) -> exists pfc_value_terminal_input_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_terminal_input_productcoefficientscoefficientdiagonalentry. ff_h_pfp_terminal_input_productcoefficientscoefficientdiagonalentry + S (pfc_value_terminal_input_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_terminal_input_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_terminal_input_productcoefficientscoefficient)) /\\ exists ff_q_pfp_terminal_input_productcoefficientscoefficientdiagonalentry. pfc_terms_code_terminal_input_productcoefficientscoefficient = ff_q_pfp_terminal_input_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_terminal_input_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_terminal_input_productcoefficientscoefficient) + (pfc_value_terminal_input_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_terminal_input_productcoefficientscoefficientdiagonalterm pfc_left_terminal_input_productcoefficientscoefficientdiagonalterm pfc_right_terminal_input_productcoefficientscoefficientdiagonalterm. (((pfc_index_terminal_input_productcoefficientscoefficientdiagonal)+pfc_complement_terminal_input_productcoefficientscoefficientdiagonalterm=(pfc_index_terminal_input_productcoefficients)) /\\ ((((((exists pfa_gap_terminal_input_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_terminal_input_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_terminal_input_productcoefficientscoefficientdiagonal) = (pfrd_qlen_terminal_input)) /\\ ((((exists ff_h_pfp_terminal_input_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_terminal_input_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_terminal_input_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_terminal_input_productcoefficientscoefficientdiagonal)) * pfrd_qc_terminal_input)) /\\ exists ff_q_pfp_terminal_input_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_terminal_input = ff_q_pfp_terminal_input_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_terminal_input_productcoefficientscoefficientdiagonal)) * pfrd_qc_terminal_input) + (pfc_left_terminal_input_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_terminal_input_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_terminal_input_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_terminal_input)=(pfc_index_terminal_input_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_terminal_input_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_terminal_input_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_terminal_input_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_terminal_input_productcoefficientscoefficientdiagonalterm) = (A)) /\\ ((((exists ff_h_pfp_terminal_input_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_terminal_input_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_terminal_input_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_terminal_input_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_terminal_input_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_terminal_input_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_terminal_input_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_terminal_input_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_terminal_input_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_terminal_input_productcoefficientscoefficientdiagonaltermrightoutside+(A)=(pfc_complement_terminal_input_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_terminal_input_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_terminal_input_productcoefficientscoefficientdiagonal)=pfc_left_terminal_input_productcoefficientscoefficientdiagonalterm*pfc_right_terminal_input_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_terminal_input_productcoefficientscoefficientsum fs_v_pfc_terminal_input_productcoefficientscoefficientsum. ((((exists fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_start. fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_start. fs_u_pfc_terminal_input_productcoefficientscoefficientsum = fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_terminal_input_productcoefficientscoefficient) = S ((S (S (pfc_index_terminal_input_productcoefficients))) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_terminal_input_productcoefficientscoefficientsum = fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_terminal_input_productcoefficients))) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum) + (pfc_natural_sum_terminal_input_productcoefficientscoefficient))) /\\ forall fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps = S (pfc_index_terminal_input_productcoefficients)) -> exists fs_a_pfc_terminal_input_productcoefficientscoefficientsum_body_steps fs_r_pfc_terminal_input_productcoefficientscoefficientsum_body_steps fs_s_pfc_terminal_input_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_terminal_input_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_terminal_input_productcoefficientscoefficient)) /\\ exists fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_terminal_input_productcoefficientscoefficient = fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_terminal_input_productcoefficientscoefficient) + (fs_a_pfc_terminal_input_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_terminal_input_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_terminal_input_productcoefficientscoefficientsum = fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum) + (fs_r_pfc_terminal_input_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_terminal_input_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_terminal_input_productcoefficientscoefficientsum = fs_q_pfc_terminal_input_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_terminal_input_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_input_productcoefficientscoefficientsum) + (fs_s_pfc_terminal_input_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_terminal_input_productcoefficientscoefficientsum_body_steps = fs_r_pfc_terminal_input_productcoefficientscoefficientsum_body_steps + fs_a_pfc_terminal_input_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_terminal_input_productcoefficientscoefficientresiduebound. pfa_gap_terminal_input_productcoefficientscoefficientresiduebound + S (pfc_value_terminal_input_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_terminal_input_productcoefficientscoefficientresiduecongruence pfa_offset_right_terminal_input_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_terminal_input_productcoefficientscoefficient) + (p) * pfa_offset_left_terminal_input_productcoefficientscoefficientresiduecongruence = (pfc_value_terminal_input_productcoefficients) + (p) * pfa_offset_right_terminal_input_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_terminal_input_target pfrep_left_terminal_input_target pfrep_right_terminal_input_target. ((exists pfrep_position_terminal_input_targetfirst. ((pfrep_position_terminal_input_targetfirst+S (pfrep_power_terminal_input_target)=(pfrd_plen_terminal_input)) /\\ ((((exists ff_h_pfp_terminal_input_targetfirstentry. ff_h_pfp_terminal_input_targetfirstentry + S (pfrep_left_terminal_input_target) = S ((S (pfrep_position_terminal_input_targetfirst)) * pfrd_pc_terminal_input)) /\\ exists ff_q_pfp_terminal_input_targetfirstentry. pfrd_pb_terminal_input = ff_q_pfp_terminal_input_targetfirstentry * S ((S (pfrep_position_terminal_input_targetfirst)) * pfrd_pc_terminal_input) + (pfrep_left_terminal_input_target)))))) \\/ (((exists pfrep_gap_terminal_input_targetfirstoutside. pfrep_gap_terminal_input_targetfirstoutside+(pfrd_plen_terminal_input)=(pfrep_power_terminal_input_target)) /\\ (((pfrep_left_terminal_input_target)=0))))) -> ((exists pfrep_position_terminal_input_targetsecond. ((pfrep_position_terminal_input_targetsecond+S (pfrep_power_terminal_input_target)=(G)) /\\ ((((exists ff_h_pfp_terminal_input_targetsecondentry. ff_h_pfp_terminal_input_targetsecondentry + S (pfrep_right_terminal_input_target) = S ((S (pfrep_position_terminal_input_targetsecond)) * gc)) /\\ exists ff_q_pfp_terminal_input_targetsecondentry. gb = ff_q_pfp_terminal_input_targetsecondentry * S ((S (pfrep_position_terminal_input_targetsecond)) * gc) + (pfrep_right_terminal_input_target)))))) \\/ (((exists pfrep_gap_terminal_input_targetsecondoutside. pfrep_gap_terminal_input_targetsecondoutside+(G)=(pfrep_power_terminal_input_target)) /\\ (((pfrep_right_terminal_input_target)=0))))) -> pfrep_left_terminal_input_target=pfrep_right_terminal_input_target))))))) -> (exists ub uc U. exists pfbz_left_code_terminal_result pfbz_left_scale_terminal_result pfbz_left_length_terminal_result pfbz_right_code_terminal_result pfbz_right_scale_terminal_result pfbz_right_length_terminal_result. ((((forall fom_index_pfp_terminal_result_left_productleft. (exists fom_gap_pfp_terminal_result_left_productleft_index_bound. fom_gap_pfp_terminal_result_left_productleft_index_bound + S (fom_index_pfp_terminal_result_left_productleft) = U) -> exists fom_value_pfp_terminal_result_left_productleft. ((((exists fom_beta_height_pfp_terminal_result_left_productleft_entry. fom_beta_height_pfp_terminal_result_left_productleft_entry + S (fom_value_pfp_terminal_result_left_productleft) = S ((S (fom_index_pfp_terminal_result_left_productleft)) * uc)) /\\ exists fom_beta_quotient_pfp_terminal_result_left_productleft_entry. ub = fom_beta_quotient_pfp_terminal_result_left_productleft_entry * S ((S (fom_index_pfp_terminal_result_left_productleft)) * uc) + (fom_value_pfp_terminal_result_left_productleft))) /\\ (exists fom_gap_pfp_terminal_result_left_productleft_value_bound. fom_gap_pfp_terminal_result_left_productleft_value_bound + S (fom_value_pfp_terminal_result_left_productleft) = p))) /\\ (((forall fom_index_pfp_terminal_result_left_productright. (exists fom_gap_pfp_terminal_result_left_productright_index_bound. fom_gap_pfp_terminal_result_left_productright_index_bound + S (fom_index_pfp_terminal_result_left_productright) = A) -> exists fom_value_pfp_terminal_result_left_productright. ((((exists fom_beta_height_pfp_terminal_result_left_productright_entry. fom_beta_height_pfp_terminal_result_left_productright_entry + S (fom_value_pfp_terminal_result_left_productright) = S ((S (fom_index_pfp_terminal_result_left_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_terminal_result_left_productright_entry. ab = fom_beta_quotient_pfp_terminal_result_left_productright_entry * S ((S (fom_index_pfp_terminal_result_left_productright)) * ac) + (fom_value_pfp_terminal_result_left_productright))) /\\ (exists fom_gap_pfp_terminal_result_left_productright_value_bound. fom_gap_pfp_terminal_result_left_productright_value_bound + S (fom_value_pfp_terminal_result_left_productright) = p))) /\\ (((((((U)=0 \\/ (A)=0) /\\ (((pfbz_left_length_terminal_result)=0)))) \\/ (((~((U)=0)) /\\ (((~((A)=0)) /\\ (((U)+(A)=S (pfbz_left_length_terminal_result)))))))) /\\ ((forall pfc_index_terminal_result_left_productcoefficients. (exists pfa_gap_terminal_result_left_productcoefficientsbound. pfa_gap_terminal_result_left_productcoefficientsbound + S (pfc_index_terminal_result_left_productcoefficients) = (pfbz_left_length_terminal_result)) -> exists pfc_value_terminal_result_left_productcoefficients. ((((exists ff_h_pfp_terminal_result_left_productcoefficientsentry. ff_h_pfp_terminal_result_left_productcoefficientsentry + S (pfc_value_terminal_result_left_productcoefficients) = S ((S (pfc_index_terminal_result_left_productcoefficients)) * pfbz_left_scale_terminal_result)) /\\ exists ff_q_pfp_terminal_result_left_productcoefficientsentry. pfbz_left_code_terminal_result = ff_q_pfp_terminal_result_left_productcoefficientsentry * S ((S (pfc_index_terminal_result_left_productcoefficients)) * pfbz_left_scale_terminal_result) + (pfc_value_terminal_result_left_productcoefficients))) /\\ ((exists pfc_terms_code_terminal_result_left_productcoefficientscoefficient pfc_terms_scale_terminal_result_left_productcoefficientscoefficient pfc_natural_sum_terminal_result_left_productcoefficientscoefficient. ((forall pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_terminal_result_left_productcoefficientscoefficientdiagonalbound. pfa_gap_terminal_result_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_terminal_result_left_productcoefficients))) -> exists pfc_value_terminal_result_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_terminal_result_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_terminal_result_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_terminal_result_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_terminal_result_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_terminal_result_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_terminal_result_left_productcoefficientscoefficient = ff_q_pfp_terminal_result_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_terminal_result_left_productcoefficientscoefficient) + (pfc_value_terminal_result_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_terminal_result_left_productcoefficientscoefficientdiagonalterm pfc_left_terminal_result_left_productcoefficientscoefficientdiagonalterm pfc_right_terminal_result_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal)+pfc_complement_terminal_result_left_productcoefficientscoefficientdiagonalterm=(pfc_index_terminal_result_left_productcoefficients)) /\\ ((((((exists pfa_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal) = (U)) /\\ ((((exists ff_h_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_terminal_result_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal)) * uc) + (pfc_left_terminal_result_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermleftoutside+(U)=(pfc_index_terminal_result_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_terminal_result_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_terminal_result_left_productcoefficientscoefficientdiagonalterm) = (A)) /\\ ((((exists ff_h_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_terminal_result_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_terminal_result_left_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_terminal_result_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_terminal_result_left_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_terminal_result_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_terminal_result_left_productcoefficientscoefficientdiagonaltermrightoutside+(A)=(pfc_complement_terminal_result_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_terminal_result_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_terminal_result_left_productcoefficientscoefficientdiagonal)=pfc_left_terminal_result_left_productcoefficientscoefficientdiagonalterm*pfc_right_terminal_result_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_terminal_result_left_productcoefficientscoefficientsum fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_terminal_result_left_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_terminal_result_left_productcoefficientscoefficient) = S ((S (S (pfc_index_terminal_result_left_productcoefficients))) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_terminal_result_left_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_terminal_result_left_productcoefficients))) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum) + (pfc_natural_sum_terminal_result_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_terminal_result_left_productcoefficients)) -> exists fs_a_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_terminal_result_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_terminal_result_left_productcoefficientscoefficient = fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_terminal_result_left_productcoefficientscoefficient) + (fs_a_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_terminal_result_left_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum) + (fs_r_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_terminal_result_left_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_left_productcoefficientscoefficientsum) + (fs_s_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_terminal_result_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_terminal_result_left_productcoefficientscoefficientresiduebound. pfa_gap_terminal_result_left_productcoefficientscoefficientresiduebound + S (pfc_value_terminal_result_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_terminal_result_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_terminal_result_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_terminal_result_left_productcoefficientscoefficient) + (p) * pfa_offset_left_terminal_result_left_productcoefficientscoefficientresiduecongruence = (pfc_value_terminal_result_left_productcoefficients) + (p) * pfa_offset_right_terminal_result_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_terminal_result_right_productleft. (exists fom_gap_pfp_terminal_result_right_productleft_index_bound. fom_gap_pfp_terminal_result_right_productleft_index_bound + S (fom_index_pfp_terminal_result_right_productleft) = 0) -> exists fom_value_pfp_terminal_result_right_productleft. ((((exists fom_beta_height_pfp_terminal_result_right_productleft_entry. fom_beta_height_pfp_terminal_result_right_productleft_entry + S (fom_value_pfp_terminal_result_right_productleft) = S ((S (fom_index_pfp_terminal_result_right_productleft)) * 0)) /\\ exists fom_beta_quotient_pfp_terminal_result_right_productleft_entry. 0 = fom_beta_quotient_pfp_terminal_result_right_productleft_entry * S ((S (fom_index_pfp_terminal_result_right_productleft)) * 0) + (fom_value_pfp_terminal_result_right_productleft))) /\\ (exists fom_gap_pfp_terminal_result_right_productleft_value_bound. fom_gap_pfp_terminal_result_right_productleft_value_bound + S (fom_value_pfp_terminal_result_right_productleft) = p))) /\\ (((forall fom_index_pfp_terminal_result_right_productright. (exists fom_gap_pfp_terminal_result_right_productright_index_bound. fom_gap_pfp_terminal_result_right_productright_index_bound + S (fom_index_pfp_terminal_result_right_productright) = B) -> exists fom_value_pfp_terminal_result_right_productright. ((((exists fom_beta_height_pfp_terminal_result_right_productright_entry. fom_beta_height_pfp_terminal_result_right_productright_entry + S (fom_value_pfp_terminal_result_right_productright) = S ((S (fom_index_pfp_terminal_result_right_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_terminal_result_right_productright_entry. bb = fom_beta_quotient_pfp_terminal_result_right_productright_entry * S ((S (fom_index_pfp_terminal_result_right_productright)) * bc) + (fom_value_pfp_terminal_result_right_productright))) /\\ (exists fom_gap_pfp_terminal_result_right_productright_value_bound. fom_gap_pfp_terminal_result_right_productright_value_bound + S (fom_value_pfp_terminal_result_right_productright) = p))) /\\ (((((((0)=0 \\/ (B)=0) /\\ (((pfbz_right_length_terminal_result)=0)))) \\/ (((~((0)=0)) /\\ (((~((B)=0)) /\\ (((0)+(B)=S (pfbz_right_length_terminal_result)))))))) /\\ ((forall pfc_index_terminal_result_right_productcoefficients. (exists pfa_gap_terminal_result_right_productcoefficientsbound. pfa_gap_terminal_result_right_productcoefficientsbound + S (pfc_index_terminal_result_right_productcoefficients) = (pfbz_right_length_terminal_result)) -> exists pfc_value_terminal_result_right_productcoefficients. ((((exists ff_h_pfp_terminal_result_right_productcoefficientsentry. ff_h_pfp_terminal_result_right_productcoefficientsentry + S (pfc_value_terminal_result_right_productcoefficients) = S ((S (pfc_index_terminal_result_right_productcoefficients)) * pfbz_right_scale_terminal_result)) /\\ exists ff_q_pfp_terminal_result_right_productcoefficientsentry. pfbz_right_code_terminal_result = ff_q_pfp_terminal_result_right_productcoefficientsentry * S ((S (pfc_index_terminal_result_right_productcoefficients)) * pfbz_right_scale_terminal_result) + (pfc_value_terminal_result_right_productcoefficients))) /\\ ((exists pfc_terms_code_terminal_result_right_productcoefficientscoefficient pfc_terms_scale_terminal_result_right_productcoefficientscoefficient pfc_natural_sum_terminal_result_right_productcoefficientscoefficient. ((forall pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_terminal_result_right_productcoefficientscoefficientdiagonalbound. pfa_gap_terminal_result_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_terminal_result_right_productcoefficients))) -> exists pfc_value_terminal_result_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_terminal_result_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_terminal_result_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_terminal_result_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_terminal_result_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_terminal_result_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_terminal_result_right_productcoefficientscoefficient = ff_q_pfp_terminal_result_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_terminal_result_right_productcoefficientscoefficient) + (pfc_value_terminal_result_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_terminal_result_right_productcoefficientscoefficientdiagonalterm pfc_left_terminal_result_right_productcoefficientscoefficientdiagonalterm pfc_right_terminal_result_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal)+pfc_complement_terminal_result_right_productcoefficientscoefficientdiagonalterm=(pfc_index_terminal_result_right_productcoefficients)) /\\ ((((((exists pfa_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal) = (0)) /\\ ((((exists ff_h_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_terminal_result_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal)) * 0)) /\\ exists ff_q_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermleftentry. 0 = ff_q_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal)) * 0) + (pfc_left_terminal_result_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermleftoutside+(0)=(pfc_index_terminal_result_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_terminal_result_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_terminal_result_right_productcoefficientscoefficientdiagonalterm) = (B)) /\\ ((((exists ff_h_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_terminal_result_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_terminal_result_right_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_terminal_result_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_terminal_result_right_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_terminal_result_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_terminal_result_right_productcoefficientscoefficientdiagonaltermrightoutside+(B)=(pfc_complement_terminal_result_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_terminal_result_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_terminal_result_right_productcoefficientscoefficientdiagonal)=pfc_left_terminal_result_right_productcoefficientscoefficientdiagonalterm*pfc_right_terminal_result_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_terminal_result_right_productcoefficientscoefficientsum fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_terminal_result_right_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_terminal_result_right_productcoefficientscoefficient) = S ((S (S (pfc_index_terminal_result_right_productcoefficients))) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_terminal_result_right_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_terminal_result_right_productcoefficients))) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum) + (pfc_natural_sum_terminal_result_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_terminal_result_right_productcoefficients)) -> exists fs_a_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_terminal_result_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_terminal_result_right_productcoefficientscoefficient = fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_terminal_result_right_productcoefficientscoefficient) + (fs_a_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_terminal_result_right_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum) + (fs_r_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_terminal_result_right_productcoefficientscoefficientsum = fs_q_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_terminal_result_right_productcoefficientscoefficientsum) + (fs_s_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_terminal_result_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_terminal_result_right_productcoefficientscoefficientresiduebound. pfa_gap_terminal_result_right_productcoefficientscoefficientresiduebound + S (pfc_value_terminal_result_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_terminal_result_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_terminal_result_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_terminal_result_right_productcoefficientscoefficient) + (p) * pfa_offset_left_terminal_result_right_productcoefficientscoefficientresiduecongruence = (pfc_value_terminal_result_right_productcoefficients) + (p) * pfa_offset_right_terminal_result_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_terminal_result_sum_left_bounded. (exists fom_gap_pfp_terminal_result_sum_left_bounded_index_bound. fom_gap_pfp_terminal_result_sum_left_bounded_index_bound + S (fom_index_pfp_terminal_result_sum_left_bounded) = pfbz_left_length_terminal_result) -> exists fom_value_pfp_terminal_result_sum_left_bounded. ((((exists fom_beta_height_pfp_terminal_result_sum_left_bounded_entry. fom_beta_height_pfp_terminal_result_sum_left_bounded_entry + S (fom_value_pfp_terminal_result_sum_left_bounded) = S ((S (fom_index_pfp_terminal_result_sum_left_bounded)) * pfbz_left_scale_terminal_result)) /\\ exists fom_beta_quotient_pfp_terminal_result_sum_left_bounded_entry. pfbz_left_code_terminal_result = fom_beta_quotient_pfp_terminal_result_sum_left_bounded_entry * S ((S (fom_index_pfp_terminal_result_sum_left_bounded)) * pfbz_left_scale_terminal_result) + (fom_value_pfp_terminal_result_sum_left_bounded))) /\\ (exists fom_gap_pfp_terminal_result_sum_left_bounded_value_bound. fom_gap_pfp_terminal_result_sum_left_bounded_value_bound + S (fom_value_pfp_terminal_result_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_terminal_result_sum_right_bounded. (exists fom_gap_pfp_terminal_result_sum_right_bounded_index_bound. fom_gap_pfp_terminal_result_sum_right_bounded_index_bound + S (fom_index_pfp_terminal_result_sum_right_bounded) = pfbz_right_length_terminal_result) -> exists fom_value_pfp_terminal_result_sum_right_bounded. ((((exists fom_beta_height_pfp_terminal_result_sum_right_bounded_entry. fom_beta_height_pfp_terminal_result_sum_right_bounded_entry + S (fom_value_pfp_terminal_result_sum_right_bounded) = S ((S (fom_index_pfp_terminal_result_sum_right_bounded)) * pfbz_right_scale_terminal_result)) /\\ exists fom_beta_quotient_pfp_terminal_result_sum_right_bounded_entry. pfbz_right_code_terminal_result = fom_beta_quotient_pfp_terminal_result_sum_right_bounded_entry * S ((S (fom_index_pfp_terminal_result_sum_right_bounded)) * pfbz_right_scale_terminal_result) + (fom_value_pfp_terminal_result_sum_right_bounded))) /\\ (exists fom_gap_pfp_terminal_result_sum_right_bounded_value_bound. fom_gap_pfp_terminal_result_sum_right_bounded_value_bound + S (fom_value_pfp_terminal_result_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_terminal_result_sum_result_bounded. (exists fom_gap_pfp_terminal_result_sum_result_bounded_index_bound. fom_gap_pfp_terminal_result_sum_result_bounded_index_bound + S (fom_index_pfp_terminal_result_sum_result_bounded) = G) -> exists fom_value_pfp_terminal_result_sum_result_bounded. ((((exists fom_beta_height_pfp_terminal_result_sum_result_bounded_entry. fom_beta_height_pfp_terminal_result_sum_result_bounded_entry + S (fom_value_pfp_terminal_result_sum_result_bounded) = S ((S (fom_index_pfp_terminal_result_sum_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_terminal_result_sum_result_bounded_entry. gb = fom_beta_quotient_pfp_terminal_result_sum_result_bounded_entry * S ((S (fom_index_pfp_terminal_result_sum_result_bounded)) * gc) + (fom_value_pfp_terminal_result_sum_result_bounded))) /\\ (exists fom_gap_pfp_terminal_result_sum_result_bounded_value_bound. fom_gap_pfp_terminal_result_sum_result_bounded_value_bound + S (fom_value_pfp_terminal_result_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_terminal_result_sum pfaa_left_c_terminal_result_sum pfaa_right_b_terminal_result_sum pfaa_right_c_terminal_result_sum pfaa_sum_b_terminal_result_sum pfaa_sum_c_terminal_result_sum pfaa_length_terminal_result_sum. ((((forall pfrep_power_terminal_result_sum_witness_common_left pfrep_left_terminal_result_sum_witness_common_left pfrep_right_terminal_result_sum_witness_common_left. ((exists pfrep_position_terminal_result_sum_witness_common_leftfirst. ((pfrep_position_terminal_result_sum_witness_common_leftfirst+S (pfrep_power_terminal_result_sum_witness_common_left)=(pfbz_left_length_terminal_result)) /\\ ((((exists ff_h_pfp_terminal_result_sum_witness_common_leftfirstentry. ff_h_pfp_terminal_result_sum_witness_common_leftfirstentry + S (pfrep_left_terminal_result_sum_witness_common_left) = S ((S (pfrep_position_terminal_result_sum_witness_common_leftfirst)) * pfbz_left_scale_terminal_result)) /\\ exists ff_q_pfp_terminal_result_sum_witness_common_leftfirstentry. pfbz_left_code_terminal_result = ff_q_pfp_terminal_result_sum_witness_common_leftfirstentry * S ((S (pfrep_position_terminal_result_sum_witness_common_leftfirst)) * pfbz_left_scale_terminal_result) + (pfrep_left_terminal_result_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_terminal_result_sum_witness_common_leftfirstoutside. pfrep_gap_terminal_result_sum_witness_common_leftfirstoutside+(pfbz_left_length_terminal_result)=(pfrep_power_terminal_result_sum_witness_common_left)) /\\ (((pfrep_left_terminal_result_sum_witness_common_left)=0))))) -> ((exists pfrep_position_terminal_result_sum_witness_common_leftsecond. ((pfrep_position_terminal_result_sum_witness_common_leftsecond+S (pfrep_power_terminal_result_sum_witness_common_left)=(pfaa_length_terminal_result_sum)) /\\ ((((exists ff_h_pfp_terminal_result_sum_witness_common_leftsecondentry. ff_h_pfp_terminal_result_sum_witness_common_leftsecondentry + S (pfrep_right_terminal_result_sum_witness_common_left) = S ((S (pfrep_position_terminal_result_sum_witness_common_leftsecond)) * pfaa_left_c_terminal_result_sum)) /\\ exists ff_q_pfp_terminal_result_sum_witness_common_leftsecondentry. pfaa_left_b_terminal_result_sum = ff_q_pfp_terminal_result_sum_witness_common_leftsecondentry * S ((S (pfrep_position_terminal_result_sum_witness_common_leftsecond)) * pfaa_left_c_terminal_result_sum) + (pfrep_right_terminal_result_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_terminal_result_sum_witness_common_leftsecondoutside. pfrep_gap_terminal_result_sum_witness_common_leftsecondoutside+(pfaa_length_terminal_result_sum)=(pfrep_power_terminal_result_sum_witness_common_left)) /\\ (((pfrep_right_terminal_result_sum_witness_common_left)=0))))) -> pfrep_left_terminal_result_sum_witness_common_left=pfrep_right_terminal_result_sum_witness_common_left) /\\ ((forall pfrep_power_terminal_result_sum_witness_common_right pfrep_left_terminal_result_sum_witness_common_right pfrep_right_terminal_result_sum_witness_common_right. ((exists pfrep_position_terminal_result_sum_witness_common_rightfirst. ((pfrep_position_terminal_result_sum_witness_common_rightfirst+S (pfrep_power_terminal_result_sum_witness_common_right)=(pfbz_right_length_terminal_result)) /\\ ((((exists ff_h_pfp_terminal_result_sum_witness_common_rightfirstentry. ff_h_pfp_terminal_result_sum_witness_common_rightfirstentry + S (pfrep_left_terminal_result_sum_witness_common_right) = S ((S (pfrep_position_terminal_result_sum_witness_common_rightfirst)) * pfbz_right_scale_terminal_result)) /\\ exists ff_q_pfp_terminal_result_sum_witness_common_rightfirstentry. pfbz_right_code_terminal_result = ff_q_pfp_terminal_result_sum_witness_common_rightfirstentry * S ((S (pfrep_position_terminal_result_sum_witness_common_rightfirst)) * pfbz_right_scale_terminal_result) + (pfrep_left_terminal_result_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_terminal_result_sum_witness_common_rightfirstoutside. pfrep_gap_terminal_result_sum_witness_common_rightfirstoutside+(pfbz_right_length_terminal_result)=(pfrep_power_terminal_result_sum_witness_common_right)) /\\ (((pfrep_left_terminal_result_sum_witness_common_right)=0))))) -> ((exists pfrep_position_terminal_result_sum_witness_common_rightsecond. ((pfrep_position_terminal_result_sum_witness_common_rightsecond+S (pfrep_power_terminal_result_sum_witness_common_right)=(pfaa_length_terminal_result_sum)) /\\ ((((exists ff_h_pfp_terminal_result_sum_witness_common_rightsecondentry. ff_h_pfp_terminal_result_sum_witness_common_rightsecondentry + S (pfrep_right_terminal_result_sum_witness_common_right) = S ((S (pfrep_position_terminal_result_sum_witness_common_rightsecond)) * pfaa_right_c_terminal_result_sum)) /\\ exists ff_q_pfp_terminal_result_sum_witness_common_rightsecondentry. pfaa_right_b_terminal_result_sum = ff_q_pfp_terminal_result_sum_witness_common_rightsecondentry * S ((S (pfrep_position_terminal_result_sum_witness_common_rightsecond)) * pfaa_right_c_terminal_result_sum) + (pfrep_right_terminal_result_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_terminal_result_sum_witness_common_rightsecondoutside. pfrep_gap_terminal_result_sum_witness_common_rightsecondoutside+(pfaa_length_terminal_result_sum)=(pfrep_power_terminal_result_sum_witness_common_right)) /\\ (((pfrep_right_terminal_result_sum_witness_common_right)=0))))) -> pfrep_left_terminal_result_sum_witness_common_right=pfrep_right_terminal_result_sum_witness_common_right)))) /\\ (((forall pfp_index_terminal_result_sum_witness_operation. (exists pfa_gap_terminal_result_sum_witness_operationindex. pfa_gap_terminal_result_sum_witness_operationindex + S (pfp_index_terminal_result_sum_witness_operation) = (pfaa_length_terminal_result_sum)) -> exists pfp_left_terminal_result_sum_witness_operation pfp_right_terminal_result_sum_witness_operation pfp_value_terminal_result_sum_witness_operation. ((((exists ff_h_pfp_terminal_result_sum_witness_operationleft. ff_h_pfp_terminal_result_sum_witness_operationleft + S (pfp_left_terminal_result_sum_witness_operation) = S ((S (pfp_index_terminal_result_sum_witness_operation)) * pfaa_left_c_terminal_result_sum)) /\\ exists ff_q_pfp_terminal_result_sum_witness_operationleft. pfaa_left_b_terminal_result_sum = ff_q_pfp_terminal_result_sum_witness_operationleft * S ((S (pfp_index_terminal_result_sum_witness_operation)) * pfaa_left_c_terminal_result_sum) + (pfp_left_terminal_result_sum_witness_operation))) /\\ (((((exists ff_h_pfp_terminal_result_sum_witness_operationright. ff_h_pfp_terminal_result_sum_witness_operationright + S (pfp_right_terminal_result_sum_witness_operation) = S ((S (pfp_index_terminal_result_sum_witness_operation)) * pfaa_right_c_terminal_result_sum)) /\\ exists ff_q_pfp_terminal_result_sum_witness_operationright. pfaa_right_b_terminal_result_sum = ff_q_pfp_terminal_result_sum_witness_operationright * S ((S (pfp_index_terminal_result_sum_witness_operation)) * pfaa_right_c_terminal_result_sum) + (pfp_right_terminal_result_sum_witness_operation))) /\\ (((((exists ff_h_pfp_terminal_result_sum_witness_operationtarget. ff_h_pfp_terminal_result_sum_witness_operationtarget + S (pfp_value_terminal_result_sum_witness_operation) = S ((S (pfp_index_terminal_result_sum_witness_operation)) * pfaa_sum_c_terminal_result_sum)) /\\ exists ff_q_pfp_terminal_result_sum_witness_operationtarget. pfaa_sum_b_terminal_result_sum = ff_q_pfp_terminal_result_sum_witness_operationtarget * S ((S (pfp_index_terminal_result_sum_witness_operation)) * pfaa_sum_c_terminal_result_sum) + (pfp_value_terminal_result_sum_witness_operation))) /\\ ((((exists pfa_gap_terminal_result_sum_witness_operationoperationleft. pfa_gap_terminal_result_sum_witness_operationoperationleft + S (pfp_left_terminal_result_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_terminal_result_sum_witness_operationoperationright. pfa_gap_terminal_result_sum_witness_operationoperationright + S (pfp_right_terminal_result_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_terminal_result_sum_witness_operationoperationresultbound. pfa_gap_terminal_result_sum_witness_operationoperationresultbound + S (pfp_value_terminal_result_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_terminal_result_sum_witness_operationoperationresultcongruence pfa_offset_right_terminal_result_sum_witness_operationoperationresultcongruence. ((pfp_left_terminal_result_sum_witness_operation) + (pfp_right_terminal_result_sum_witness_operation)) + (p) * pfa_offset_left_terminal_result_sum_witness_operationoperationresultcongruence = (pfp_value_terminal_result_sum_witness_operation) + (p) * pfa_offset_right_terminal_result_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_terminal_result_sum_witness_output pfrep_left_terminal_result_sum_witness_output pfrep_right_terminal_result_sum_witness_output. ((exists pfrep_position_terminal_result_sum_witness_outputfirst. ((pfrep_position_terminal_result_sum_witness_outputfirst+S (pfrep_power_terminal_result_sum_witness_output)=(pfaa_length_terminal_result_sum)) /\\ ((((exists ff_h_pfp_terminal_result_sum_witness_outputfirstentry. ff_h_pfp_terminal_result_sum_witness_outputfirstentry + S (pfrep_left_terminal_result_sum_witness_output) = S ((S (pfrep_position_terminal_result_sum_witness_outputfirst)) * pfaa_sum_c_terminal_result_sum)) /\\ exists ff_q_pfp_terminal_result_sum_witness_outputfirstentry. pfaa_sum_b_terminal_result_sum = ff_q_pfp_terminal_result_sum_witness_outputfirstentry * S ((S (pfrep_position_terminal_result_sum_witness_outputfirst)) * pfaa_sum_c_terminal_result_sum) + (pfrep_left_terminal_result_sum_witness_output)))))) \\/ (((exists pfrep_gap_terminal_result_sum_witness_outputfirstoutside. pfrep_gap_terminal_result_sum_witness_outputfirstoutside+(pfaa_length_terminal_result_sum)=(pfrep_power_terminal_result_sum_witness_output)) /\\ (((pfrep_left_terminal_result_sum_witness_output)=0))))) -> ((exists pfrep_position_terminal_result_sum_witness_outputsecond. ((pfrep_position_terminal_result_sum_witness_outputsecond+S (pfrep_power_terminal_result_sum_witness_output)=(G)) /\\ ((((exists ff_h_pfp_terminal_result_sum_witness_outputsecondentry. ff_h_pfp_terminal_result_sum_witness_outputsecondentry + S (pfrep_right_terminal_result_sum_witness_output) = S ((S (pfrep_position_terminal_result_sum_witness_outputsecond)) * gc)) /\\ exists ff_q_pfp_terminal_result_sum_witness_outputsecondentry. gb = ff_q_pfp_terminal_result_sum_witness_outputsecondentry * S ((S (pfrep_position_terminal_result_sum_witness_outputsecond)) * gc) + (pfrep_right_terminal_result_sum_witness_output)))))) \\/ (((exists pfrep_gap_terminal_result_sum_witness_outputsecondoutside. pfrep_gap_terminal_result_sum_witness_outputsecondoutside+(G)=(pfrep_power_terminal_result_sum_witness_output)) /\\ (((pfrep_right_terminal_result_sum_witness_output)=0))))) -> pfrep_left_terminal_result_sum_witness_output=pfrep_right_terminal_result_sum_witness_output))))))))))))))))))", "statement_sha256": "a8c70d6a075d99b10b545c4f9a40df76ae4002c326108e667bed98b7b79cea1e", "summary": "An actual right multiple G of A supplies its real left quotient U. With the empty coefficient V, construct V*B and an actual aligned sum to give G=U*A+V*B. The second input only needs canonical coefficients.", "summary_sha256": "33eff0d6cad7f26284bdb3525b9fb15a8914f95a6297f4883bf4cceb6a573bb9" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "65f6ad74560d04ad415e35e2b3e73a891c26ae5c64d092e8f20e605a801e17b5", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 446, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "76b90226e5e29fdde3d9bb49accccf8d9b4c0cc17a4de406af253e999102533c", "test_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85" }, "body_checked": true, "body_receipt": { "command_count": 208, "dependency_count": 6, "dne_command_count": 0, "name": 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"families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "dacd2240a388d62dc56c5d61d9776e49c5dcabf7ead0520681c35ad61c806af1", "membership": "alpha_only", "name": "prime_field_polynomial_bezout_equivalent_transport", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro A", "intro bb", "intro bc", "intro B", "intro gb", "intro gc", "intro G", "intro ub", "intro uc", "intro U", "intro vb", "intro vc", "intro V", "intro ab2", "intro ac2", "intro A2", "intro bb2", "intro bc2", "intro B2", "intro gb2", "intro gc2", "intro G2", "intro hp0", "intro ha2", "intro hb2", "intro hg2", "intro haeq", "intro hbeq", "intro hgeq", "intro hbezout", "cases hbezout", "cases hbezout_witness", "cases hbezout_witness_witness", "cases hbezout_witness_witness_witness", "cases hbezout_witness_witness_witness_witness", "cases hbezout_witness_witness_witness_witness_witness", "cases hbezout_witness_witness_witness_witness_witness_witness", "cases hbezout_witness_witness_witness_witness_witness_witness_right", "have hub : forall fom_index_pfp_hub_bound. (exists fom_gap_pfp_hub_bound_index_bound. fom_gap_pfp_hub_bound_index_bound + S (fom_index_pfp_hub_bound) = U) -> exists fom_value_pfp_hub_bound. ((((exists fom_beta_height_pfp_hub_bound_entry. fom_beta_height_pfp_hub_bound_entry + S (fom_value_pfp_hub_bound) = S ((S (fom_index_pfp_hub_bound)) * uc)) /\\ exists fom_beta_quotient_pfp_hub_bound_entry. ub = fom_beta_quotient_pfp_hub_bound_entry * S ((S (fom_index_pfp_hub_bound)) * uc) + (fom_value_pfp_hub_bound))) /\\ (exists fom_gap_pfp_hub_bound_value_bound. fom_gap_pfp_hub_bound_value_bound + S (fom_value_pfp_hub_bound) = p))", "cases hbezout_witness_witness_witness_witness_witness_witness_left", "exact hbezout_witness_witness_witness_witness_witness_witness_left_left", "have hvb : forall fom_index_pfp_hvb_bound. (exists fom_gap_pfp_hvb_bound_index_bound. fom_gap_pfp_hvb_bound_index_bound + S (fom_index_pfp_hvb_bound) = V) -> exists fom_value_pfp_hvb_bound. ((((exists fom_beta_height_pfp_hvb_bound_entry. fom_beta_height_pfp_hvb_bound_entry + S (fom_value_pfp_hvb_bound) = S ((S (fom_index_pfp_hvb_bound)) * vc)) /\\ exists fom_beta_quotient_pfp_hvb_bound_entry. vb = fom_beta_quotient_pfp_hvb_bound_entry * S ((S (fom_index_pfp_hvb_bound)) * vc) + (fom_value_pfp_hvb_bound))) /\\ (exists fom_gap_pfp_hvb_bound_value_bound. fom_gap_pfp_hvb_bound_value_bound + S (fom_value_pfp_hvb_bound) = p))", "cases hbezout_witness_witness_witness_witness_witness_witness_right_left", "exact hbezout_witness_witness_witness_witness_witness_witness_right_left_left", "have hnew_left_length : exists n. ((((U)=0 \\/ (A2)=0) /\\ (((n)=0)))) \\/ (((~((U)=0)) /\\ (((~((A2)=0)) /\\ (((U)+(A2)=S (n)))))))", "specialize polynomial_product_length_exists (U)", "specialize polynomial_product_length_exists (A2)", "apply polynomial_product_length_exists", "cases hnew_left_length", "have hnew_left_product : exists b c. ((forall fom_index_pfp_hnew_left_productleft. (exists fom_gap_pfp_hnew_left_productleft_index_bound. fom_gap_pfp_hnew_left_productleft_index_bound + S (fom_index_pfp_hnew_left_productleft) = U) -> exists fom_value_pfp_hnew_left_productleft. ((((exists fom_beta_height_pfp_hnew_left_productleft_entry. fom_beta_height_pfp_hnew_left_productleft_entry + S (fom_value_pfp_hnew_left_productleft) = S ((S (fom_index_pfp_hnew_left_productleft)) * uc)) /\\ exists fom_beta_quotient_pfp_hnew_left_productleft_entry. ub = fom_beta_quotient_pfp_hnew_left_productleft_entry * S ((S (fom_index_pfp_hnew_left_productleft)) * uc) + (fom_value_pfp_hnew_left_productleft))) /\\ (exists fom_gap_pfp_hnew_left_productleft_value_bound. fom_gap_pfp_hnew_left_productleft_value_bound + S (fom_value_pfp_hnew_left_productleft) = p))) /\\ (((forall fom_index_pfp_hnew_left_productright. (exists fom_gap_pfp_hnew_left_productright_index_bound. fom_gap_pfp_hnew_left_productright_index_bound + S (fom_index_pfp_hnew_left_productright) = A2) -> exists fom_value_pfp_hnew_left_productright. ((((exists fom_beta_height_pfp_hnew_left_productright_entry. fom_beta_height_pfp_hnew_left_productright_entry + S (fom_value_pfp_hnew_left_productright) = S ((S (fom_index_pfp_hnew_left_productright)) * ac2)) /\\ exists fom_beta_quotient_pfp_hnew_left_productright_entry. ab2 = fom_beta_quotient_pfp_hnew_left_productright_entry * S ((S (fom_index_pfp_hnew_left_productright)) * ac2) + (fom_value_pfp_hnew_left_productright))) /\\ (exists fom_gap_pfp_hnew_left_productright_value_bound. fom_gap_pfp_hnew_left_productright_value_bound + S (fom_value_pfp_hnew_left_productright) = p))) /\\ (((((((U)=0 \\/ (A2)=0) /\\ (((x6)=0)))) \\/ (((~((U)=0)) /\\ (((~((A2)=0)) /\\ (((U)+(A2)=S (x6)))))))) /\\ ((forall pfc_index_hnew_left_productcoefficients. (exists pfa_gap_hnew_left_productcoefficientsbound. pfa_gap_hnew_left_productcoefficientsbound + S (pfc_index_hnew_left_productcoefficients) = (x6)) -> exists pfc_value_hnew_left_productcoefficients. ((((exists ff_h_pfp_hnew_left_productcoefficientsentry. ff_h_pfp_hnew_left_productcoefficientsentry + S (pfc_value_hnew_left_productcoefficients) = S ((S (pfc_index_hnew_left_productcoefficients)) * c)) /\\ exists ff_q_pfp_hnew_left_productcoefficientsentry. b = ff_q_pfp_hnew_left_productcoefficientsentry * S ((S (pfc_index_hnew_left_productcoefficients)) * c) + (pfc_value_hnew_left_productcoefficients))) /\\ ((exists pfc_terms_code_hnew_left_productcoefficientscoefficient pfc_terms_scale_hnew_left_productcoefficientscoefficient pfc_natural_sum_hnew_left_productcoefficientscoefficient. ((forall pfc_index_hnew_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_hnew_left_productcoefficientscoefficientdiagonalbound. pfa_gap_hnew_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_hnew_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_hnew_left_productcoefficients))) -> exists pfc_value_hnew_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_hnew_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_hnew_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_hnew_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_hnew_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hnew_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_hnew_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_hnew_left_productcoefficientscoefficient = ff_q_pfp_hnew_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_hnew_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hnew_left_productcoefficientscoefficient) + (pfc_value_hnew_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_hnew_left_productcoefficientscoefficientdiagonalterm pfc_left_hnew_left_productcoefficientscoefficientdiagonalterm pfc_right_hnew_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_hnew_left_productcoefficientscoefficientdiagonal)+pfc_complement_hnew_left_productcoefficientscoefficientdiagonalterm=(pfc_index_hnew_left_productcoefficients)) /\\ ((((((exists pfa_gap_hnew_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_hnew_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_hnew_left_productcoefficientscoefficientdiagonal) = (U)) /\\ ((((exists ff_h_pfp_hnew_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_hnew_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_hnew_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_hnew_left_productcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_hnew_left_productcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_hnew_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_hnew_left_productcoefficientscoefficientdiagonal)) * uc) + (pfc_left_hnew_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hnew_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_hnew_left_productcoefficientscoefficientdiagonaltermleftoutside+(U)=(pfc_index_hnew_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_hnew_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_hnew_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_hnew_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_hnew_left_productcoefficientscoefficientdiagonalterm) = (A2)) /\\ ((((exists ff_h_pfp_hnew_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_hnew_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_hnew_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_hnew_left_productcoefficientscoefficientdiagonalterm)) * ac2)) /\\ exists ff_q_pfp_hnew_left_productcoefficientscoefficientdiagonaltermrightentry. ab2 = ff_q_pfp_hnew_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_hnew_left_productcoefficientscoefficientdiagonalterm)) * ac2) + (pfc_right_hnew_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hnew_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_hnew_left_productcoefficientscoefficientdiagonaltermrightoutside+(A2)=(pfc_complement_hnew_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_hnew_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_hnew_left_productcoefficientscoefficientdiagonal)=pfc_left_hnew_left_productcoefficientscoefficientdiagonalterm*pfc_right_hnew_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_hnew_left_productcoefficientscoefficientsum fs_v_pfc_hnew_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_hnew_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_hnew_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_hnew_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnew_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_hnew_left_productcoefficientscoefficientsum = fs_q_pfc_hnew_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_hnew_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_hnew_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_hnew_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_hnew_left_productcoefficientscoefficient) = S ((S (S (pfc_index_hnew_left_productcoefficients))) * fs_v_pfc_hnew_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnew_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_hnew_left_productcoefficientscoefficientsum = fs_q_pfc_hnew_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_hnew_left_productcoefficients))) * fs_v_pfc_hnew_left_productcoefficientscoefficientsum) + (pfc_natural_sum_hnew_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_hnew_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_hnew_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_hnew_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_hnew_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_hnew_left_productcoefficients)) -> exists fs_a_pfc_hnew_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_hnew_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_hnew_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_hnew_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_hnew_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_hnew_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hnew_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hnew_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_hnew_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_hnew_left_productcoefficientscoefficient = fs_q_pfc_hnew_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_hnew_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hnew_left_productcoefficientscoefficient) + (fs_a_pfc_hnew_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hnew_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_hnew_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_hnew_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hnew_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnew_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnew_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_hnew_left_productcoefficientscoefficientsum = fs_q_pfc_hnew_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_hnew_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnew_left_productcoefficientscoefficientsum) + (fs_r_pfc_hnew_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hnew_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_hnew_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_hnew_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_hnew_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnew_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnew_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_hnew_left_productcoefficientscoefficientsum = fs_q_pfc_hnew_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_hnew_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnew_left_productcoefficientscoefficientsum) + (fs_s_pfc_hnew_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_hnew_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_hnew_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_hnew_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_hnew_left_productcoefficientscoefficientresiduebound. pfa_gap_hnew_left_productcoefficientscoefficientresiduebound + S (pfc_value_hnew_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_hnew_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_hnew_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_hnew_left_productcoefficientscoefficient) + (p) * pfa_offset_left_hnew_left_productcoefficientscoefficientresiduecongruence = (pfc_value_hnew_left_productcoefficients) + (p) * pfa_offset_right_hnew_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (ub)", "specialize prime_field_polynomial_convolution_at_length_exists (uc)", "specialize prime_field_polynomial_convolution_at_length_exists (U)", "specialize prime_field_polynomial_convolution_at_length_exists (ab2)", "specialize prime_field_polynomial_convolution_at_length_exists (ac2)", "specialize prime_field_polynomial_convolution_at_length_exists (A2)", "specialize prime_field_polynomial_convolution_at_length_exists (x6)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hub", "exact ha2", "exact hnew_left_length_witness", "cases hnew_left_product", "cases hnew_left_product_witness", "have hnew_right_length : exists n. ((((V)=0 \\/ (B2)=0) /\\ (((n)=0)))) \\/ (((~((V)=0)) /\\ (((~((B2)=0)) /\\ (((V)+(B2)=S (n)))))))", "specialize polynomial_product_length_exists (V)", "specialize polynomial_product_length_exists (B2)", "apply polynomial_product_length_exists", "cases hnew_right_length", "have hnew_right_product : exists b c. ((forall fom_index_pfp_hnew_right_productleft. (exists fom_gap_pfp_hnew_right_productleft_index_bound. fom_gap_pfp_hnew_right_productleft_index_bound + S (fom_index_pfp_hnew_right_productleft) = V) -> exists fom_value_pfp_hnew_right_productleft. ((((exists fom_beta_height_pfp_hnew_right_productleft_entry. fom_beta_height_pfp_hnew_right_productleft_entry + S (fom_value_pfp_hnew_right_productleft) = S ((S (fom_index_pfp_hnew_right_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_hnew_right_productleft_entry. vb = fom_beta_quotient_pfp_hnew_right_productleft_entry * S ((S (fom_index_pfp_hnew_right_productleft)) * vc) + (fom_value_pfp_hnew_right_productleft))) /\\ (exists fom_gap_pfp_hnew_right_productleft_value_bound. fom_gap_pfp_hnew_right_productleft_value_bound + S (fom_value_pfp_hnew_right_productleft) = p))) /\\ (((forall fom_index_pfp_hnew_right_productright. (exists fom_gap_pfp_hnew_right_productright_index_bound. fom_gap_pfp_hnew_right_productright_index_bound + S (fom_index_pfp_hnew_right_productright) = B2) -> exists fom_value_pfp_hnew_right_productright. ((((exists fom_beta_height_pfp_hnew_right_productright_entry. fom_beta_height_pfp_hnew_right_productright_entry + S (fom_value_pfp_hnew_right_productright) = S ((S (fom_index_pfp_hnew_right_productright)) * bc2)) /\\ exists fom_beta_quotient_pfp_hnew_right_productright_entry. bb2 = fom_beta_quotient_pfp_hnew_right_productright_entry * S ((S (fom_index_pfp_hnew_right_productright)) * bc2) + (fom_value_pfp_hnew_right_productright))) /\\ (exists fom_gap_pfp_hnew_right_productright_value_bound. fom_gap_pfp_hnew_right_productright_value_bound + S (fom_value_pfp_hnew_right_productright) = p))) /\\ (((((((V)=0 \\/ (B2)=0) /\\ (((x9)=0)))) \\/ (((~((V)=0)) /\\ (((~((B2)=0)) /\\ (((V)+(B2)=S (x9)))))))) /\\ ((forall pfc_index_hnew_right_productcoefficients. (exists pfa_gap_hnew_right_productcoefficientsbound. pfa_gap_hnew_right_productcoefficientsbound + S (pfc_index_hnew_right_productcoefficients) = (x9)) -> exists pfc_value_hnew_right_productcoefficients. ((((exists ff_h_pfp_hnew_right_productcoefficientsentry. ff_h_pfp_hnew_right_productcoefficientsentry + S (pfc_value_hnew_right_productcoefficients) = S ((S (pfc_index_hnew_right_productcoefficients)) * c)) /\\ exists ff_q_pfp_hnew_right_productcoefficientsentry. b = ff_q_pfp_hnew_right_productcoefficientsentry * S ((S (pfc_index_hnew_right_productcoefficients)) * c) + (pfc_value_hnew_right_productcoefficients))) /\\ ((exists pfc_terms_code_hnew_right_productcoefficientscoefficient pfc_terms_scale_hnew_right_productcoefficientscoefficient pfc_natural_sum_hnew_right_productcoefficientscoefficient. ((forall pfc_index_hnew_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_hnew_right_productcoefficientscoefficientdiagonalbound. pfa_gap_hnew_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_hnew_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_hnew_right_productcoefficients))) -> exists pfc_value_hnew_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_hnew_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_hnew_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_hnew_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_hnew_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hnew_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_hnew_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_hnew_right_productcoefficientscoefficient = ff_q_pfp_hnew_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_hnew_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_hnew_right_productcoefficientscoefficient) + (pfc_value_hnew_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_hnew_right_productcoefficientscoefficientdiagonalterm pfc_left_hnew_right_productcoefficientscoefficientdiagonalterm pfc_right_hnew_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_hnew_right_productcoefficientscoefficientdiagonal)+pfc_complement_hnew_right_productcoefficientscoefficientdiagonalterm=(pfc_index_hnew_right_productcoefficients)) /\\ ((((((exists pfa_gap_hnew_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_hnew_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_hnew_right_productcoefficientscoefficientdiagonal) = (V)) /\\ ((((exists ff_h_pfp_hnew_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_hnew_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_hnew_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_hnew_right_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_hnew_right_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_hnew_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_hnew_right_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_hnew_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hnew_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_hnew_right_productcoefficientscoefficientdiagonaltermleftoutside+(V)=(pfc_index_hnew_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_hnew_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_hnew_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_hnew_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_hnew_right_productcoefficientscoefficientdiagonalterm) = (B2)) /\\ ((((exists ff_h_pfp_hnew_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_hnew_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_hnew_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_hnew_right_productcoefficientscoefficientdiagonalterm)) * bc2)) /\\ exists ff_q_pfp_hnew_right_productcoefficientscoefficientdiagonaltermrightentry. bb2 = ff_q_pfp_hnew_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_hnew_right_productcoefficientscoefficientdiagonalterm)) * bc2) + (pfc_right_hnew_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_hnew_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_hnew_right_productcoefficientscoefficientdiagonaltermrightoutside+(B2)=(pfc_complement_hnew_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_hnew_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_hnew_right_productcoefficientscoefficientdiagonal)=pfc_left_hnew_right_productcoefficientscoefficientdiagonalterm*pfc_right_hnew_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_hnew_right_productcoefficientscoefficientsum fs_v_pfc_hnew_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_hnew_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_hnew_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_hnew_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnew_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_hnew_right_productcoefficientscoefficientsum = fs_q_pfc_hnew_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_hnew_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_hnew_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_hnew_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_hnew_right_productcoefficientscoefficient) = S ((S (S (pfc_index_hnew_right_productcoefficients))) * fs_v_pfc_hnew_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnew_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_hnew_right_productcoefficientscoefficientsum = fs_q_pfc_hnew_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_hnew_right_productcoefficients))) * fs_v_pfc_hnew_right_productcoefficientscoefficientsum) + (pfc_natural_sum_hnew_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_hnew_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_hnew_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_hnew_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_hnew_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_hnew_right_productcoefficients)) -> exists fs_a_pfc_hnew_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_hnew_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_hnew_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_hnew_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_hnew_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_hnew_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hnew_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hnew_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_hnew_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_hnew_right_productcoefficientscoefficient = fs_q_pfc_hnew_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_hnew_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_hnew_right_productcoefficientscoefficient) + (fs_a_pfc_hnew_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hnew_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_hnew_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_hnew_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_hnew_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnew_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnew_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_hnew_right_productcoefficientscoefficientsum = fs_q_pfc_hnew_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_hnew_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnew_right_productcoefficientscoefficientsum) + (fs_r_pfc_hnew_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_hnew_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_hnew_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_hnew_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_hnew_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnew_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_hnew_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_hnew_right_productcoefficientscoefficientsum = fs_q_pfc_hnew_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_hnew_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_hnew_right_productcoefficientscoefficientsum) + (fs_s_pfc_hnew_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_hnew_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_hnew_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_hnew_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_hnew_right_productcoefficientscoefficientresiduebound. pfa_gap_hnew_right_productcoefficientscoefficientresiduebound + S (pfc_value_hnew_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_hnew_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_hnew_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_hnew_right_productcoefficientscoefficient) + (p) * pfa_offset_left_hnew_right_productcoefficientscoefficientresiduecongruence = (pfc_value_hnew_right_productcoefficients) + (p) * pfa_offset_right_hnew_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (vb)", "specialize prime_field_polynomial_convolution_at_length_exists (vc)", "specialize prime_field_polynomial_convolution_at_length_exists (V)", "specialize prime_field_polynomial_convolution_at_length_exists (bb2)", "specialize prime_field_polynomial_convolution_at_length_exists (bc2)", "specialize prime_field_polynomial_convolution_at_length_exists (B2)", "specialize prime_field_polynomial_convolution_at_length_exists (x9)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hp0", "exact hvb", "exact hb2", "exact hnew_right_length_witness", "cases hnew_right_product", "cases hnew_right_product_witness", "have hpp : forall fom_index_pfp_hpp_bound. (exists fom_gap_pfp_hpp_bound_index_bound. fom_gap_pfp_hpp_bound_index_bound + S (fom_index_pfp_hpp_bound) = x6) -> exists fom_value_pfp_hpp_bound. ((((exists fom_beta_height_pfp_hpp_bound_entry. fom_beta_height_pfp_hpp_bound_entry + S (fom_value_pfp_hpp_bound) = S ((S (fom_index_pfp_hpp_bound)) * x8)) /\\ exists fom_beta_quotient_pfp_hpp_bound_entry. x7 = fom_beta_quotient_pfp_hpp_bound_entry * S ((S (fom_index_pfp_hpp_bound)) * x8) + (fom_value_pfp_hpp_bound))) /\\ (exists fom_gap_pfp_hpp_bound_value_bound. fom_gap_pfp_hpp_bound_value_bound + S (fom_value_pfp_hpp_bound) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (ub)", "specialize prime_field_polynomial_convolution_bounded (uc)", "specialize prime_field_polynomial_convolution_bounded (U)", "specialize prime_field_polynomial_convolution_bounded (ab2)", "specialize prime_field_polynomial_convolution_bounded (ac2)", "specialize prime_field_polynomial_convolution_bounded (A2)", "specialize prime_field_polynomial_convolution_bounded (x7)", "specialize prime_field_polynomial_convolution_bounded (x8)", "specialize prime_field_polynomial_convolution_bounded (x6)", "apply prime_field_polynomial_convolution_bounded", "exact hnew_left_product_witness_witness", "have hqq : forall fom_index_pfp_hqq_bound. (exists fom_gap_pfp_hqq_bound_index_bound. fom_gap_pfp_hqq_bound_index_bound + S (fom_index_pfp_hqq_bound) = x9) -> exists fom_value_pfp_hqq_bound. ((((exists fom_beta_height_pfp_hqq_bound_entry. fom_beta_height_pfp_hqq_bound_entry + S (fom_value_pfp_hqq_bound) = S ((S (fom_index_pfp_hqq_bound)) * x11)) /\\ exists fom_beta_quotient_pfp_hqq_bound_entry. x10 = fom_beta_quotient_pfp_hqq_bound_entry * S ((S (fom_index_pfp_hqq_bound)) * x11) + (fom_value_pfp_hqq_bound))) /\\ (exists fom_gap_pfp_hqq_bound_value_bound. fom_gap_pfp_hqq_bound_value_bound + S (fom_value_pfp_hqq_bound) = p))", "specialize prime_field_polynomial_convolution_bounded (p)", "specialize prime_field_polynomial_convolution_bounded (vb)", "specialize prime_field_polynomial_convolution_bounded (vc)", "specialize prime_field_polynomial_convolution_bounded (V)", "specialize prime_field_polynomial_convolution_bounded (bb2)", "specialize prime_field_polynomial_convolution_bounded (bc2)", "specialize prime_field_polynomial_convolution_bounded (B2)", "specialize prime_field_polynomial_convolution_bounded (x10)", "specialize prime_field_polynomial_convolution_bounded (x11)", "specialize prime_field_polynomial_convolution_bounded (x9)", "apply prime_field_polynomial_convolution_bounded", "exact hnew_right_product_witness_witness", "have hsum : ((forall fom_index_pfp_transport_sum_left_bounded. (exists fom_gap_pfp_transport_sum_left_bounded_index_bound. fom_gap_pfp_transport_sum_left_bounded_index_bound + S (fom_index_pfp_transport_sum_left_bounded) = x6) -> exists fom_value_pfp_transport_sum_left_bounded. ((((exists fom_beta_height_pfp_transport_sum_left_bounded_entry. fom_beta_height_pfp_transport_sum_left_bounded_entry + S (fom_value_pfp_transport_sum_left_bounded) = S ((S (fom_index_pfp_transport_sum_left_bounded)) * x8)) /\\ exists fom_beta_quotient_pfp_transport_sum_left_bounded_entry. x7 = fom_beta_quotient_pfp_transport_sum_left_bounded_entry * S ((S (fom_index_pfp_transport_sum_left_bounded)) * x8) + (fom_value_pfp_transport_sum_left_bounded))) /\\ (exists fom_gap_pfp_transport_sum_left_bounded_value_bound. fom_gap_pfp_transport_sum_left_bounded_value_bound + S (fom_value_pfp_transport_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_transport_sum_right_bounded. (exists fom_gap_pfp_transport_sum_right_bounded_index_bound. fom_gap_pfp_transport_sum_right_bounded_index_bound + S (fom_index_pfp_transport_sum_right_bounded) = x9) -> exists fom_value_pfp_transport_sum_right_bounded. ((((exists fom_beta_height_pfp_transport_sum_right_bounded_entry. fom_beta_height_pfp_transport_sum_right_bounded_entry + S (fom_value_pfp_transport_sum_right_bounded) = S ((S (fom_index_pfp_transport_sum_right_bounded)) * x11)) /\\ exists fom_beta_quotient_pfp_transport_sum_right_bounded_entry. x10 = fom_beta_quotient_pfp_transport_sum_right_bounded_entry * S ((S (fom_index_pfp_transport_sum_right_bounded)) * x11) + (fom_value_pfp_transport_sum_right_bounded))) /\\ (exists fom_gap_pfp_transport_sum_right_bounded_value_bound. fom_gap_pfp_transport_sum_right_bounded_value_bound + S (fom_value_pfp_transport_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_transport_sum_result_bounded. (exists fom_gap_pfp_transport_sum_result_bounded_index_bound. fom_gap_pfp_transport_sum_result_bounded_index_bound + S (fom_index_pfp_transport_sum_result_bounded) = G2) -> exists fom_value_pfp_transport_sum_result_bounded. ((((exists fom_beta_height_pfp_transport_sum_result_bounded_entry. fom_beta_height_pfp_transport_sum_result_bounded_entry + S (fom_value_pfp_transport_sum_result_bounded) = S ((S (fom_index_pfp_transport_sum_result_bounded)) * gc2)) /\\ exists fom_beta_quotient_pfp_transport_sum_result_bounded_entry. gb2 = fom_beta_quotient_pfp_transport_sum_result_bounded_entry * S ((S (fom_index_pfp_transport_sum_result_bounded)) * gc2) + (fom_value_pfp_transport_sum_result_bounded))) /\\ (exists fom_gap_pfp_transport_sum_result_bounded_value_bound. fom_gap_pfp_transport_sum_result_bounded_value_bound + S (fom_value_pfp_transport_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_transport_sum pfaa_left_c_transport_sum pfaa_right_b_transport_sum pfaa_right_c_transport_sum pfaa_sum_b_transport_sum pfaa_sum_c_transport_sum pfaa_length_transport_sum. ((((forall pfrep_power_transport_sum_witness_common_left pfrep_left_transport_sum_witness_common_left pfrep_right_transport_sum_witness_common_left. ((exists pfrep_position_transport_sum_witness_common_leftfirst. ((pfrep_position_transport_sum_witness_common_leftfirst+S (pfrep_power_transport_sum_witness_common_left)=(x6)) /\\ ((((exists ff_h_pfp_transport_sum_witness_common_leftfirstentry. ff_h_pfp_transport_sum_witness_common_leftfirstentry + S (pfrep_left_transport_sum_witness_common_left) = S ((S (pfrep_position_transport_sum_witness_common_leftfirst)) * x8)) /\\ exists ff_q_pfp_transport_sum_witness_common_leftfirstentry. x7 = ff_q_pfp_transport_sum_witness_common_leftfirstentry * S ((S (pfrep_position_transport_sum_witness_common_leftfirst)) * x8) + (pfrep_left_transport_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_transport_sum_witness_common_leftfirstoutside. pfrep_gap_transport_sum_witness_common_leftfirstoutside+(x6)=(pfrep_power_transport_sum_witness_common_left)) /\\ (((pfrep_left_transport_sum_witness_common_left)=0))))) -> ((exists pfrep_position_transport_sum_witness_common_leftsecond. ((pfrep_position_transport_sum_witness_common_leftsecond+S (pfrep_power_transport_sum_witness_common_left)=(pfaa_length_transport_sum)) /\\ ((((exists ff_h_pfp_transport_sum_witness_common_leftsecondentry. ff_h_pfp_transport_sum_witness_common_leftsecondentry + S (pfrep_right_transport_sum_witness_common_left) = S ((S (pfrep_position_transport_sum_witness_common_leftsecond)) * pfaa_left_c_transport_sum)) /\\ exists ff_q_pfp_transport_sum_witness_common_leftsecondentry. pfaa_left_b_transport_sum = ff_q_pfp_transport_sum_witness_common_leftsecondentry * S ((S (pfrep_position_transport_sum_witness_common_leftsecond)) * pfaa_left_c_transport_sum) + (pfrep_right_transport_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_transport_sum_witness_common_leftsecondoutside. pfrep_gap_transport_sum_witness_common_leftsecondoutside+(pfaa_length_transport_sum)=(pfrep_power_transport_sum_witness_common_left)) /\\ (((pfrep_right_transport_sum_witness_common_left)=0))))) -> pfrep_left_transport_sum_witness_common_left=pfrep_right_transport_sum_witness_common_left) /\\ ((forall pfrep_power_transport_sum_witness_common_right pfrep_left_transport_sum_witness_common_right pfrep_right_transport_sum_witness_common_right. ((exists pfrep_position_transport_sum_witness_common_rightfirst. ((pfrep_position_transport_sum_witness_common_rightfirst+S (pfrep_power_transport_sum_witness_common_right)=(x9)) /\\ ((((exists ff_h_pfp_transport_sum_witness_common_rightfirstentry. ff_h_pfp_transport_sum_witness_common_rightfirstentry + S (pfrep_left_transport_sum_witness_common_right) = S ((S (pfrep_position_transport_sum_witness_common_rightfirst)) * x11)) /\\ exists ff_q_pfp_transport_sum_witness_common_rightfirstentry. x10 = ff_q_pfp_transport_sum_witness_common_rightfirstentry * S ((S (pfrep_position_transport_sum_witness_common_rightfirst)) * x11) + (pfrep_left_transport_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_transport_sum_witness_common_rightfirstoutside. pfrep_gap_transport_sum_witness_common_rightfirstoutside+(x9)=(pfrep_power_transport_sum_witness_common_right)) /\\ (((pfrep_left_transport_sum_witness_common_right)=0))))) -> ((exists pfrep_position_transport_sum_witness_common_rightsecond. ((pfrep_position_transport_sum_witness_common_rightsecond+S (pfrep_power_transport_sum_witness_common_right)=(pfaa_length_transport_sum)) /\\ ((((exists ff_h_pfp_transport_sum_witness_common_rightsecondentry. ff_h_pfp_transport_sum_witness_common_rightsecondentry + S (pfrep_right_transport_sum_witness_common_right) = S ((S (pfrep_position_transport_sum_witness_common_rightsecond)) * pfaa_right_c_transport_sum)) /\\ exists ff_q_pfp_transport_sum_witness_common_rightsecondentry. pfaa_right_b_transport_sum = ff_q_pfp_transport_sum_witness_common_rightsecondentry * S ((S (pfrep_position_transport_sum_witness_common_rightsecond)) * pfaa_right_c_transport_sum) + (pfrep_right_transport_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_transport_sum_witness_common_rightsecondoutside. pfrep_gap_transport_sum_witness_common_rightsecondoutside+(pfaa_length_transport_sum)=(pfrep_power_transport_sum_witness_common_right)) /\\ (((pfrep_right_transport_sum_witness_common_right)=0))))) -> pfrep_left_transport_sum_witness_common_right=pfrep_right_transport_sum_witness_common_right)))) /\\ (((forall pfp_index_transport_sum_witness_operation. (exists pfa_gap_transport_sum_witness_operationindex. pfa_gap_transport_sum_witness_operationindex + S (pfp_index_transport_sum_witness_operation) = (pfaa_length_transport_sum)) -> exists pfp_left_transport_sum_witness_operation pfp_right_transport_sum_witness_operation pfp_value_transport_sum_witness_operation. ((((exists ff_h_pfp_transport_sum_witness_operationleft. ff_h_pfp_transport_sum_witness_operationleft + S (pfp_left_transport_sum_witness_operation) = S ((S (pfp_index_transport_sum_witness_operation)) * pfaa_left_c_transport_sum)) /\\ exists ff_q_pfp_transport_sum_witness_operationleft. pfaa_left_b_transport_sum = ff_q_pfp_transport_sum_witness_operationleft * S ((S (pfp_index_transport_sum_witness_operation)) * pfaa_left_c_transport_sum) + (pfp_left_transport_sum_witness_operation))) /\\ (((((exists ff_h_pfp_transport_sum_witness_operationright. ff_h_pfp_transport_sum_witness_operationright + S (pfp_right_transport_sum_witness_operation) = S ((S (pfp_index_transport_sum_witness_operation)) * pfaa_right_c_transport_sum)) /\\ exists ff_q_pfp_transport_sum_witness_operationright. pfaa_right_b_transport_sum = ff_q_pfp_transport_sum_witness_operationright * S ((S (pfp_index_transport_sum_witness_operation)) * pfaa_right_c_transport_sum) + (pfp_right_transport_sum_witness_operation))) /\\ (((((exists ff_h_pfp_transport_sum_witness_operationtarget. ff_h_pfp_transport_sum_witness_operationtarget + S (pfp_value_transport_sum_witness_operation) = S ((S (pfp_index_transport_sum_witness_operation)) * pfaa_sum_c_transport_sum)) /\\ exists ff_q_pfp_transport_sum_witness_operationtarget. pfaa_sum_b_transport_sum = ff_q_pfp_transport_sum_witness_operationtarget * S ((S (pfp_index_transport_sum_witness_operation)) * pfaa_sum_c_transport_sum) + (pfp_value_transport_sum_witness_operation))) /\\ ((((exists pfa_gap_transport_sum_witness_operationoperationleft. pfa_gap_transport_sum_witness_operationoperationleft + S (pfp_left_transport_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_transport_sum_witness_operationoperationright. pfa_gap_transport_sum_witness_operationoperationright + S (pfp_right_transport_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_transport_sum_witness_operationoperationresultbound. pfa_gap_transport_sum_witness_operationoperationresultbound + S (pfp_value_transport_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_transport_sum_witness_operationoperationresultcongruence pfa_offset_right_transport_sum_witness_operationoperationresultcongruence. ((pfp_left_transport_sum_witness_operation) + (pfp_right_transport_sum_witness_operation)) + (p) * pfa_offset_left_transport_sum_witness_operationoperationresultcongruence = (pfp_value_transport_sum_witness_operation) + (p) * pfa_offset_right_transport_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_transport_sum_witness_output pfrep_left_transport_sum_witness_output pfrep_right_transport_sum_witness_output. ((exists pfrep_position_transport_sum_witness_outputfirst. ((pfrep_position_transport_sum_witness_outputfirst+S (pfrep_power_transport_sum_witness_output)=(pfaa_length_transport_sum)) /\\ ((((exists ff_h_pfp_transport_sum_witness_outputfirstentry. ff_h_pfp_transport_sum_witness_outputfirstentry + S (pfrep_left_transport_sum_witness_output) = S ((S (pfrep_position_transport_sum_witness_outputfirst)) * pfaa_sum_c_transport_sum)) /\\ exists ff_q_pfp_transport_sum_witness_outputfirstentry. pfaa_sum_b_transport_sum = ff_q_pfp_transport_sum_witness_outputfirstentry * S ((S (pfrep_position_transport_sum_witness_outputfirst)) * pfaa_sum_c_transport_sum) + (pfrep_left_transport_sum_witness_output)))))) \\/ (((exists pfrep_gap_transport_sum_witness_outputfirstoutside. pfrep_gap_transport_sum_witness_outputfirstoutside+(pfaa_length_transport_sum)=(pfrep_power_transport_sum_witness_output)) /\\ (((pfrep_left_transport_sum_witness_output)=0))))) -> ((exists pfrep_position_transport_sum_witness_outputsecond. ((pfrep_position_transport_sum_witness_outputsecond+S (pfrep_power_transport_sum_witness_output)=(G2)) /\\ ((((exists ff_h_pfp_transport_sum_witness_outputsecondentry. ff_h_pfp_transport_sum_witness_outputsecondentry + S (pfrep_right_transport_sum_witness_output) = S ((S (pfrep_position_transport_sum_witness_outputsecond)) * gc2)) /\\ exists ff_q_pfp_transport_sum_witness_outputsecondentry. gb2 = ff_q_pfp_transport_sum_witness_outputsecondentry * S ((S (pfrep_position_transport_sum_witness_outputsecond)) * gc2) + (pfrep_right_transport_sum_witness_output)))))) \\/ (((exists pfrep_gap_transport_sum_witness_outputsecondoutside. pfrep_gap_transport_sum_witness_outputsecondoutside+(G2)=(pfrep_power_transport_sum_witness_output)) /\\ (((pfrep_right_transport_sum_witness_output)=0))))) -> pfrep_left_transport_sum_witness_output=pfrep_right_transport_sum_witness_output))))))))))))", "specialize prime_field_polynomial_aligned_add_transport (p)", "specialize prime_field_polynomial_aligned_add_transport (x)", "specialize prime_field_polynomial_aligned_add_transport (x1)", "specialize prime_field_polynomial_aligned_add_transport (x2)", "specialize prime_field_polynomial_aligned_add_transport (x3)", "specialize prime_field_polynomial_aligned_add_transport (x4)", "specialize prime_field_polynomial_aligned_add_transport (x5)", "specialize prime_field_polynomial_aligned_add_transport (gb)", "specialize prime_field_polynomial_aligned_add_transport (gc)", "specialize prime_field_polynomial_aligned_add_transport (G)", "specialize prime_field_polynomial_aligned_add_transport (x7)", "specialize prime_field_polynomial_aligned_add_transport (x8)", "specialize prime_field_polynomial_aligned_add_transport (x6)", "specialize prime_field_polynomial_aligned_add_transport (x10)", "specialize prime_field_polynomial_aligned_add_transport (x11)", "specialize prime_field_polynomial_aligned_add_transport (x9)", "specialize prime_field_polynomial_aligned_add_transport (gb2)", "specialize prime_field_polynomial_aligned_add_transport (gc2)", "specialize prime_field_polynomial_aligned_add_transport (G2)", "apply prime_field_polynomial_aligned_add_transport", "exact hpp", "exact hqq", "exact hg2", "specialize prime_field_polynomial_equivalent_symmetric (x)", "specialize prime_field_polynomial_equivalent_symmetric (x1)", "specialize prime_field_polynomial_equivalent_symmetric (x2)", "specialize prime_field_polynomial_equivalent_symmetric (x7)", "specialize prime_field_polynomial_equivalent_symmetric (x8)", "specialize prime_field_polynomial_equivalent_symmetric (x6)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (p)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (ub)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (uc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (U)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (ab)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (ac)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (A)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x1)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x2)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (ab2)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (ac2)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (A2)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x7)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x8)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x6)", "apply prime_field_polynomial_convolution_equivalent_congruent_right", "exact hp0", "exact haeq", "exact hbezout_witness_witness_witness_witness_witness_witness_left", "exact hnew_left_product_witness_witness", "specialize prime_field_polynomial_equivalent_symmetric (x3)", "specialize prime_field_polynomial_equivalent_symmetric (x4)", "specialize prime_field_polynomial_equivalent_symmetric (x5)", "specialize prime_field_polynomial_equivalent_symmetric (x10)", "specialize prime_field_polynomial_equivalent_symmetric (x11)", "specialize prime_field_polynomial_equivalent_symmetric (x9)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (p)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (vb)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (vc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (V)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (bb)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (bc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (B)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x3)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x4)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x5)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (bb2)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (bc2)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (B2)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x10)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x11)", "specialize prime_field_polynomial_convolution_equivalent_congruent_right (x9)", "apply prime_field_polynomial_convolution_equivalent_congruent_right", "exact hp0", "exact hbeq", "exact hbezout_witness_witness_witness_witness_witness_witness_right_left", "exact hnew_right_product_witness_witness", "exact hgeq", "exact hbezout_witness_witness_witness_witness_witness_witness_right_right", "exists x7", "exists x8", "exists x6", "exists x10", "exists x11", "exists x9", "split", "exact hnew_left_product_witness_witness", "split", "exact hnew_right_product_witness_witness", "exact hsum" ], "script_sha256": "d8c0f6a2b6c8970776c6b2dfd9c3dd34ad8ca9d0dacae38491a30056dc6709d0", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_bezout_laws_candidate.py", "sha256": "76b90226e5e29fdde3d9bb49accccf8d9b4c0cc17a4de406af253e999102533c" }, "statement": "forall p ab ac A bb bc B gb gc G ub uc U vb vc V ab2 ac2 A2 bb2 bc2 B2 gb2 gc2 G2. (~(p=0)) -> (forall fom_index_pfp_transport_a_bound. (exists fom_gap_pfp_transport_a_bound_index_bound. fom_gap_pfp_transport_a_bound_index_bound + S (fom_index_pfp_transport_a_bound) = A2) -> exists fom_value_pfp_transport_a_bound. ((((exists fom_beta_height_pfp_transport_a_bound_entry. fom_beta_height_pfp_transport_a_bound_entry + S (fom_value_pfp_transport_a_bound) = S ((S (fom_index_pfp_transport_a_bound)) * ac2)) /\\ exists fom_beta_quotient_pfp_transport_a_bound_entry. ab2 = fom_beta_quotient_pfp_transport_a_bound_entry * S ((S (fom_index_pfp_transport_a_bound)) * ac2) + (fom_value_pfp_transport_a_bound))) /\\ (exists fom_gap_pfp_transport_a_bound_value_bound. fom_gap_pfp_transport_a_bound_value_bound + S (fom_value_pfp_transport_a_bound) = p))) -> (forall fom_index_pfp_transport_b_bound. (exists fom_gap_pfp_transport_b_bound_index_bound. fom_gap_pfp_transport_b_bound_index_bound + S (fom_index_pfp_transport_b_bound) = B2) -> exists fom_value_pfp_transport_b_bound. ((((exists fom_beta_height_pfp_transport_b_bound_entry. fom_beta_height_pfp_transport_b_bound_entry + S (fom_value_pfp_transport_b_bound) = S ((S (fom_index_pfp_transport_b_bound)) * bc2)) /\\ exists fom_beta_quotient_pfp_transport_b_bound_entry. bb2 = fom_beta_quotient_pfp_transport_b_bound_entry * S ((S (fom_index_pfp_transport_b_bound)) * bc2) + (fom_value_pfp_transport_b_bound))) /\\ (exists fom_gap_pfp_transport_b_bound_value_bound. fom_gap_pfp_transport_b_bound_value_bound + S (fom_value_pfp_transport_b_bound) = p))) -> (forall fom_index_pfp_transport_g_bound. (exists fom_gap_pfp_transport_g_bound_index_bound. fom_gap_pfp_transport_g_bound_index_bound + S (fom_index_pfp_transport_g_bound) = G2) -> exists fom_value_pfp_transport_g_bound. ((((exists fom_beta_height_pfp_transport_g_bound_entry. fom_beta_height_pfp_transport_g_bound_entry + S (fom_value_pfp_transport_g_bound) = S ((S (fom_index_pfp_transport_g_bound)) * gc2)) /\\ exists fom_beta_quotient_pfp_transport_g_bound_entry. gb2 = fom_beta_quotient_pfp_transport_g_bound_entry * S ((S (fom_index_pfp_transport_g_bound)) * gc2) + (fom_value_pfp_transport_g_bound))) /\\ (exists fom_gap_pfp_transport_g_bound_value_bound. fom_gap_pfp_transport_g_bound_value_bound + S (fom_value_pfp_transport_g_bound) = p))) -> (forall pfrep_power_transport_a_equivalent pfrep_left_transport_a_equivalent pfrep_right_transport_a_equivalent. ((exists pfrep_position_transport_a_equivalentfirst. ((pfrep_position_transport_a_equivalentfirst+S (pfrep_power_transport_a_equivalent)=(A)) /\\ ((((exists ff_h_pfp_transport_a_equivalentfirstentry. ff_h_pfp_transport_a_equivalentfirstentry + S (pfrep_left_transport_a_equivalent) = S ((S (pfrep_position_transport_a_equivalentfirst)) * ac)) /\\ exists ff_q_pfp_transport_a_equivalentfirstentry. ab = ff_q_pfp_transport_a_equivalentfirstentry * S ((S (pfrep_position_transport_a_equivalentfirst)) * ac) + (pfrep_left_transport_a_equivalent)))))) \\/ (((exists pfrep_gap_transport_a_equivalentfirstoutside. pfrep_gap_transport_a_equivalentfirstoutside+(A)=(pfrep_power_transport_a_equivalent)) /\\ (((pfrep_left_transport_a_equivalent)=0))))) -> ((exists pfrep_position_transport_a_equivalentsecond. ((pfrep_position_transport_a_equivalentsecond+S (pfrep_power_transport_a_equivalent)=(A2)) /\\ ((((exists ff_h_pfp_transport_a_equivalentsecondentry. ff_h_pfp_transport_a_equivalentsecondentry + S (pfrep_right_transport_a_equivalent) = S ((S (pfrep_position_transport_a_equivalentsecond)) * ac2)) /\\ exists ff_q_pfp_transport_a_equivalentsecondentry. ab2 = ff_q_pfp_transport_a_equivalentsecondentry * S ((S (pfrep_position_transport_a_equivalentsecond)) * ac2) + (pfrep_right_transport_a_equivalent)))))) \\/ (((exists pfrep_gap_transport_a_equivalentsecondoutside. pfrep_gap_transport_a_equivalentsecondoutside+(A2)=(pfrep_power_transport_a_equivalent)) /\\ (((pfrep_right_transport_a_equivalent)=0))))) -> pfrep_left_transport_a_equivalent=pfrep_right_transport_a_equivalent) -> (forall pfrep_power_transport_b_equivalent pfrep_left_transport_b_equivalent pfrep_right_transport_b_equivalent. ((exists pfrep_position_transport_b_equivalentfirst. ((pfrep_position_transport_b_equivalentfirst+S (pfrep_power_transport_b_equivalent)=(B)) /\\ ((((exists ff_h_pfp_transport_b_equivalentfirstentry. ff_h_pfp_transport_b_equivalentfirstentry + S (pfrep_left_transport_b_equivalent) = S ((S (pfrep_position_transport_b_equivalentfirst)) * bc)) /\\ exists ff_q_pfp_transport_b_equivalentfirstentry. bb = ff_q_pfp_transport_b_equivalentfirstentry * S ((S (pfrep_position_transport_b_equivalentfirst)) * bc) + (pfrep_left_transport_b_equivalent)))))) \\/ (((exists pfrep_gap_transport_b_equivalentfirstoutside. pfrep_gap_transport_b_equivalentfirstoutside+(B)=(pfrep_power_transport_b_equivalent)) /\\ (((pfrep_left_transport_b_equivalent)=0))))) -> ((exists pfrep_position_transport_b_equivalentsecond. ((pfrep_position_transport_b_equivalentsecond+S (pfrep_power_transport_b_equivalent)=(B2)) /\\ ((((exists ff_h_pfp_transport_b_equivalentsecondentry. ff_h_pfp_transport_b_equivalentsecondentry + S (pfrep_right_transport_b_equivalent) = S ((S (pfrep_position_transport_b_equivalentsecond)) * bc2)) /\\ exists ff_q_pfp_transport_b_equivalentsecondentry. bb2 = ff_q_pfp_transport_b_equivalentsecondentry * S ((S (pfrep_position_transport_b_equivalentsecond)) * bc2) + (pfrep_right_transport_b_equivalent)))))) \\/ (((exists pfrep_gap_transport_b_equivalentsecondoutside. pfrep_gap_transport_b_equivalentsecondoutside+(B2)=(pfrep_power_transport_b_equivalent)) /\\ (((pfrep_right_transport_b_equivalent)=0))))) -> pfrep_left_transport_b_equivalent=pfrep_right_transport_b_equivalent) -> (forall pfrep_power_transport_g_equivalent pfrep_left_transport_g_equivalent pfrep_right_transport_g_equivalent. ((exists pfrep_position_transport_g_equivalentfirst. ((pfrep_position_transport_g_equivalentfirst+S (pfrep_power_transport_g_equivalent)=(G)) /\\ ((((exists ff_h_pfp_transport_g_equivalentfirstentry. ff_h_pfp_transport_g_equivalentfirstentry + S (pfrep_left_transport_g_equivalent) = S ((S (pfrep_position_transport_g_equivalentfirst)) * gc)) /\\ exists ff_q_pfp_transport_g_equivalentfirstentry. gb = ff_q_pfp_transport_g_equivalentfirstentry * S ((S (pfrep_position_transport_g_equivalentfirst)) * gc) + (pfrep_left_transport_g_equivalent)))))) \\/ (((exists pfrep_gap_transport_g_equivalentfirstoutside. pfrep_gap_transport_g_equivalentfirstoutside+(G)=(pfrep_power_transport_g_equivalent)) /\\ (((pfrep_left_transport_g_equivalent)=0))))) -> ((exists pfrep_position_transport_g_equivalentsecond. ((pfrep_position_transport_g_equivalentsecond+S (pfrep_power_transport_g_equivalent)=(G2)) /\\ ((((exists ff_h_pfp_transport_g_equivalentsecondentry. ff_h_pfp_transport_g_equivalentsecondentry + S (pfrep_right_transport_g_equivalent) = S ((S (pfrep_position_transport_g_equivalentsecond)) * gc2)) /\\ exists ff_q_pfp_transport_g_equivalentsecondentry. gb2 = ff_q_pfp_transport_g_equivalentsecondentry * S ((S (pfrep_position_transport_g_equivalentsecond)) * gc2) + (pfrep_right_transport_g_equivalent)))))) \\/ (((exists pfrep_gap_transport_g_equivalentsecondoutside. pfrep_gap_transport_g_equivalentsecondoutside+(G2)=(pfrep_power_transport_g_equivalent)) /\\ (((pfrep_right_transport_g_equivalent)=0))))) -> pfrep_left_transport_g_equivalent=pfrep_right_transport_g_equivalent) -> (exists pfbz_left_code_transport_old pfbz_left_scale_transport_old pfbz_left_length_transport_old pfbz_right_code_transport_old pfbz_right_scale_transport_old pfbz_right_length_transport_old. ((((forall fom_index_pfp_transport_old_left_productleft. (exists fom_gap_pfp_transport_old_left_productleft_index_bound. fom_gap_pfp_transport_old_left_productleft_index_bound + S (fom_index_pfp_transport_old_left_productleft) = U) -> exists fom_value_pfp_transport_old_left_productleft. ((((exists fom_beta_height_pfp_transport_old_left_productleft_entry. fom_beta_height_pfp_transport_old_left_productleft_entry + S (fom_value_pfp_transport_old_left_productleft) = S ((S (fom_index_pfp_transport_old_left_productleft)) * uc)) /\\ exists fom_beta_quotient_pfp_transport_old_left_productleft_entry. ub = fom_beta_quotient_pfp_transport_old_left_productleft_entry * S ((S (fom_index_pfp_transport_old_left_productleft)) * uc) + (fom_value_pfp_transport_old_left_productleft))) /\\ (exists fom_gap_pfp_transport_old_left_productleft_value_bound. fom_gap_pfp_transport_old_left_productleft_value_bound + S (fom_value_pfp_transport_old_left_productleft) = p))) /\\ (((forall fom_index_pfp_transport_old_left_productright. (exists fom_gap_pfp_transport_old_left_productright_index_bound. fom_gap_pfp_transport_old_left_productright_index_bound + S (fom_index_pfp_transport_old_left_productright) = A) -> exists fom_value_pfp_transport_old_left_productright. ((((exists fom_beta_height_pfp_transport_old_left_productright_entry. fom_beta_height_pfp_transport_old_left_productright_entry + S (fom_value_pfp_transport_old_left_productright) = S ((S (fom_index_pfp_transport_old_left_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_transport_old_left_productright_entry. ab = fom_beta_quotient_pfp_transport_old_left_productright_entry * S ((S (fom_index_pfp_transport_old_left_productright)) * ac) + (fom_value_pfp_transport_old_left_productright))) /\\ (exists fom_gap_pfp_transport_old_left_productright_value_bound. fom_gap_pfp_transport_old_left_productright_value_bound + S (fom_value_pfp_transport_old_left_productright) = p))) /\\ (((((((U)=0 \\/ (A)=0) /\\ (((pfbz_left_length_transport_old)=0)))) \\/ (((~((U)=0)) /\\ (((~((A)=0)) /\\ (((U)+(A)=S (pfbz_left_length_transport_old)))))))) /\\ ((forall pfc_index_transport_old_left_productcoefficients. (exists pfa_gap_transport_old_left_productcoefficientsbound. pfa_gap_transport_old_left_productcoefficientsbound + S (pfc_index_transport_old_left_productcoefficients) = (pfbz_left_length_transport_old)) -> exists pfc_value_transport_old_left_productcoefficients. ((((exists ff_h_pfp_transport_old_left_productcoefficientsentry. ff_h_pfp_transport_old_left_productcoefficientsentry + S (pfc_value_transport_old_left_productcoefficients) = S ((S (pfc_index_transport_old_left_productcoefficients)) * pfbz_left_scale_transport_old)) /\\ exists ff_q_pfp_transport_old_left_productcoefficientsentry. pfbz_left_code_transport_old = ff_q_pfp_transport_old_left_productcoefficientsentry * S ((S (pfc_index_transport_old_left_productcoefficients)) * pfbz_left_scale_transport_old) + (pfc_value_transport_old_left_productcoefficients))) /\\ ((exists pfc_terms_code_transport_old_left_productcoefficientscoefficient pfc_terms_scale_transport_old_left_productcoefficientscoefficient pfc_natural_sum_transport_old_left_productcoefficientscoefficient. ((forall pfc_index_transport_old_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_transport_old_left_productcoefficientscoefficientdiagonalbound. pfa_gap_transport_old_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_transport_old_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_transport_old_left_productcoefficients))) -> exists pfc_value_transport_old_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_transport_old_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_transport_old_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_transport_old_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_transport_old_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_transport_old_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_transport_old_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_transport_old_left_productcoefficientscoefficient = ff_q_pfp_transport_old_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_transport_old_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_transport_old_left_productcoefficientscoefficient) + (pfc_value_transport_old_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_transport_old_left_productcoefficientscoefficientdiagonalterm pfc_left_transport_old_left_productcoefficientscoefficientdiagonalterm pfc_right_transport_old_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_transport_old_left_productcoefficientscoefficientdiagonal)+pfc_complement_transport_old_left_productcoefficientscoefficientdiagonalterm=(pfc_index_transport_old_left_productcoefficients)) /\\ ((((((exists pfa_gap_transport_old_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_transport_old_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_transport_old_left_productcoefficientscoefficientdiagonal) = (U)) /\\ ((((exists ff_h_pfp_transport_old_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_transport_old_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_transport_old_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_transport_old_left_productcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_transport_old_left_productcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_transport_old_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_transport_old_left_productcoefficientscoefficientdiagonal)) * uc) + (pfc_left_transport_old_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_transport_old_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_transport_old_left_productcoefficientscoefficientdiagonaltermleftoutside+(U)=(pfc_index_transport_old_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_transport_old_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_transport_old_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_transport_old_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_transport_old_left_productcoefficientscoefficientdiagonalterm) = (A)) /\\ ((((exists ff_h_pfp_transport_old_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_transport_old_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_transport_old_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_transport_old_left_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_transport_old_left_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_transport_old_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_transport_old_left_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_transport_old_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_transport_old_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_transport_old_left_productcoefficientscoefficientdiagonaltermrightoutside+(A)=(pfc_complement_transport_old_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_transport_old_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_transport_old_left_productcoefficientscoefficientdiagonal)=pfc_left_transport_old_left_productcoefficientscoefficientdiagonalterm*pfc_right_transport_old_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_transport_old_left_productcoefficientscoefficientsum fs_v_pfc_transport_old_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_transport_old_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_transport_old_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_transport_old_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_old_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_transport_old_left_productcoefficientscoefficientsum = fs_q_pfc_transport_old_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_transport_old_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_transport_old_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_transport_old_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_transport_old_left_productcoefficientscoefficient) = S ((S (S (pfc_index_transport_old_left_productcoefficients))) * fs_v_pfc_transport_old_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_old_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_transport_old_left_productcoefficientscoefficientsum = fs_q_pfc_transport_old_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_transport_old_left_productcoefficients))) * fs_v_pfc_transport_old_left_productcoefficientscoefficientsum) + (pfc_natural_sum_transport_old_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_transport_old_left_productcoefficients)) -> exists fs_a_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_transport_old_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_transport_old_left_productcoefficientscoefficient = fs_q_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_transport_old_left_productcoefficientscoefficient) + (fs_a_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_old_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_transport_old_left_productcoefficientscoefficientsum = fs_q_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_old_left_productcoefficientscoefficientsum) + (fs_r_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_old_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_transport_old_left_productcoefficientscoefficientsum = fs_q_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_old_left_productcoefficientscoefficientsum) + (fs_s_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_transport_old_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_transport_old_left_productcoefficientscoefficientresiduebound. pfa_gap_transport_old_left_productcoefficientscoefficientresiduebound + S (pfc_value_transport_old_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_transport_old_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_transport_old_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_transport_old_left_productcoefficientscoefficient) + (p) * pfa_offset_left_transport_old_left_productcoefficientscoefficientresiduecongruence = (pfc_value_transport_old_left_productcoefficients) + (p) * pfa_offset_right_transport_old_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_transport_old_right_productleft. (exists fom_gap_pfp_transport_old_right_productleft_index_bound. fom_gap_pfp_transport_old_right_productleft_index_bound + S (fom_index_pfp_transport_old_right_productleft) = V) -> exists fom_value_pfp_transport_old_right_productleft. ((((exists fom_beta_height_pfp_transport_old_right_productleft_entry. fom_beta_height_pfp_transport_old_right_productleft_entry + S (fom_value_pfp_transport_old_right_productleft) = S ((S (fom_index_pfp_transport_old_right_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_transport_old_right_productleft_entry. vb = fom_beta_quotient_pfp_transport_old_right_productleft_entry * S ((S (fom_index_pfp_transport_old_right_productleft)) * vc) + (fom_value_pfp_transport_old_right_productleft))) /\\ (exists fom_gap_pfp_transport_old_right_productleft_value_bound. fom_gap_pfp_transport_old_right_productleft_value_bound + S (fom_value_pfp_transport_old_right_productleft) = p))) /\\ (((forall fom_index_pfp_transport_old_right_productright. (exists fom_gap_pfp_transport_old_right_productright_index_bound. fom_gap_pfp_transport_old_right_productright_index_bound + S (fom_index_pfp_transport_old_right_productright) = B) -> exists fom_value_pfp_transport_old_right_productright. ((((exists fom_beta_height_pfp_transport_old_right_productright_entry. fom_beta_height_pfp_transport_old_right_productright_entry + S (fom_value_pfp_transport_old_right_productright) = S ((S (fom_index_pfp_transport_old_right_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_transport_old_right_productright_entry. bb = fom_beta_quotient_pfp_transport_old_right_productright_entry * S ((S (fom_index_pfp_transport_old_right_productright)) * bc) + (fom_value_pfp_transport_old_right_productright))) /\\ (exists fom_gap_pfp_transport_old_right_productright_value_bound. fom_gap_pfp_transport_old_right_productright_value_bound + S (fom_value_pfp_transport_old_right_productright) = p))) /\\ (((((((V)=0 \\/ (B)=0) /\\ (((pfbz_right_length_transport_old)=0)))) \\/ (((~((V)=0)) /\\ (((~((B)=0)) /\\ (((V)+(B)=S (pfbz_right_length_transport_old)))))))) /\\ ((forall pfc_index_transport_old_right_productcoefficients. (exists pfa_gap_transport_old_right_productcoefficientsbound. pfa_gap_transport_old_right_productcoefficientsbound + S (pfc_index_transport_old_right_productcoefficients) = (pfbz_right_length_transport_old)) -> exists pfc_value_transport_old_right_productcoefficients. ((((exists ff_h_pfp_transport_old_right_productcoefficientsentry. ff_h_pfp_transport_old_right_productcoefficientsentry + S (pfc_value_transport_old_right_productcoefficients) = S ((S (pfc_index_transport_old_right_productcoefficients)) * pfbz_right_scale_transport_old)) /\\ exists ff_q_pfp_transport_old_right_productcoefficientsentry. pfbz_right_code_transport_old = ff_q_pfp_transport_old_right_productcoefficientsentry * S ((S (pfc_index_transport_old_right_productcoefficients)) * pfbz_right_scale_transport_old) + (pfc_value_transport_old_right_productcoefficients))) /\\ ((exists pfc_terms_code_transport_old_right_productcoefficientscoefficient pfc_terms_scale_transport_old_right_productcoefficientscoefficient pfc_natural_sum_transport_old_right_productcoefficientscoefficient. ((forall pfc_index_transport_old_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_transport_old_right_productcoefficientscoefficientdiagonalbound. pfa_gap_transport_old_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_transport_old_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_transport_old_right_productcoefficients))) -> exists pfc_value_transport_old_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_transport_old_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_transport_old_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_transport_old_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_transport_old_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_transport_old_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_transport_old_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_transport_old_right_productcoefficientscoefficient = ff_q_pfp_transport_old_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_transport_old_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_transport_old_right_productcoefficientscoefficient) + (pfc_value_transport_old_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_transport_old_right_productcoefficientscoefficientdiagonalterm pfc_left_transport_old_right_productcoefficientscoefficientdiagonalterm pfc_right_transport_old_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_transport_old_right_productcoefficientscoefficientdiagonal)+pfc_complement_transport_old_right_productcoefficientscoefficientdiagonalterm=(pfc_index_transport_old_right_productcoefficients)) /\\ ((((((exists pfa_gap_transport_old_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_transport_old_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_transport_old_right_productcoefficientscoefficientdiagonal) = (V)) /\\ ((((exists ff_h_pfp_transport_old_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_transport_old_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_transport_old_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_transport_old_right_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_transport_old_right_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_transport_old_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_transport_old_right_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_transport_old_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_transport_old_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_transport_old_right_productcoefficientscoefficientdiagonaltermleftoutside+(V)=(pfc_index_transport_old_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_transport_old_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_transport_old_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_transport_old_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_transport_old_right_productcoefficientscoefficientdiagonalterm) = (B)) /\\ ((((exists ff_h_pfp_transport_old_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_transport_old_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_transport_old_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_transport_old_right_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_transport_old_right_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_transport_old_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_transport_old_right_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_transport_old_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_transport_old_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_transport_old_right_productcoefficientscoefficientdiagonaltermrightoutside+(B)=(pfc_complement_transport_old_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_transport_old_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_transport_old_right_productcoefficientscoefficientdiagonal)=pfc_left_transport_old_right_productcoefficientscoefficientdiagonalterm*pfc_right_transport_old_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_transport_old_right_productcoefficientscoefficientsum fs_v_pfc_transport_old_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_transport_old_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_transport_old_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_transport_old_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_old_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_transport_old_right_productcoefficientscoefficientsum = fs_q_pfc_transport_old_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_transport_old_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_transport_old_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_transport_old_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_transport_old_right_productcoefficientscoefficient) = S ((S (S (pfc_index_transport_old_right_productcoefficients))) * fs_v_pfc_transport_old_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_old_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_transport_old_right_productcoefficientscoefficientsum = fs_q_pfc_transport_old_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_transport_old_right_productcoefficients))) * fs_v_pfc_transport_old_right_productcoefficientscoefficientsum) + (pfc_natural_sum_transport_old_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_transport_old_right_productcoefficients)) -> exists fs_a_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_transport_old_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_transport_old_right_productcoefficientscoefficient = fs_q_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_transport_old_right_productcoefficientscoefficient) + (fs_a_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_old_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_transport_old_right_productcoefficientscoefficientsum = fs_q_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_old_right_productcoefficientscoefficientsum) + (fs_r_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_old_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_transport_old_right_productcoefficientscoefficientsum = fs_q_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_old_right_productcoefficientscoefficientsum) + (fs_s_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_transport_old_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_transport_old_right_productcoefficientscoefficientresiduebound. pfa_gap_transport_old_right_productcoefficientscoefficientresiduebound + S (pfc_value_transport_old_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_transport_old_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_transport_old_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_transport_old_right_productcoefficientscoefficient) + (p) * pfa_offset_left_transport_old_right_productcoefficientscoefficientresiduecongruence = (pfc_value_transport_old_right_productcoefficients) + (p) * pfa_offset_right_transport_old_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_transport_old_sum_left_bounded. (exists fom_gap_pfp_transport_old_sum_left_bounded_index_bound. fom_gap_pfp_transport_old_sum_left_bounded_index_bound + S (fom_index_pfp_transport_old_sum_left_bounded) = pfbz_left_length_transport_old) -> exists fom_value_pfp_transport_old_sum_left_bounded. ((((exists fom_beta_height_pfp_transport_old_sum_left_bounded_entry. fom_beta_height_pfp_transport_old_sum_left_bounded_entry + S (fom_value_pfp_transport_old_sum_left_bounded) = S ((S (fom_index_pfp_transport_old_sum_left_bounded)) * pfbz_left_scale_transport_old)) /\\ exists fom_beta_quotient_pfp_transport_old_sum_left_bounded_entry. pfbz_left_code_transport_old = fom_beta_quotient_pfp_transport_old_sum_left_bounded_entry * S ((S (fom_index_pfp_transport_old_sum_left_bounded)) * pfbz_left_scale_transport_old) + (fom_value_pfp_transport_old_sum_left_bounded))) /\\ (exists fom_gap_pfp_transport_old_sum_left_bounded_value_bound. fom_gap_pfp_transport_old_sum_left_bounded_value_bound + S (fom_value_pfp_transport_old_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_transport_old_sum_right_bounded. (exists fom_gap_pfp_transport_old_sum_right_bounded_index_bound. fom_gap_pfp_transport_old_sum_right_bounded_index_bound + S (fom_index_pfp_transport_old_sum_right_bounded) = pfbz_right_length_transport_old) -> exists fom_value_pfp_transport_old_sum_right_bounded. ((((exists fom_beta_height_pfp_transport_old_sum_right_bounded_entry. fom_beta_height_pfp_transport_old_sum_right_bounded_entry + S (fom_value_pfp_transport_old_sum_right_bounded) = S ((S (fom_index_pfp_transport_old_sum_right_bounded)) * pfbz_right_scale_transport_old)) /\\ exists fom_beta_quotient_pfp_transport_old_sum_right_bounded_entry. pfbz_right_code_transport_old = fom_beta_quotient_pfp_transport_old_sum_right_bounded_entry * S ((S (fom_index_pfp_transport_old_sum_right_bounded)) * pfbz_right_scale_transport_old) + (fom_value_pfp_transport_old_sum_right_bounded))) /\\ (exists fom_gap_pfp_transport_old_sum_right_bounded_value_bound. fom_gap_pfp_transport_old_sum_right_bounded_value_bound + S (fom_value_pfp_transport_old_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_transport_old_sum_result_bounded. (exists fom_gap_pfp_transport_old_sum_result_bounded_index_bound. fom_gap_pfp_transport_old_sum_result_bounded_index_bound + S (fom_index_pfp_transport_old_sum_result_bounded) = G) -> exists fom_value_pfp_transport_old_sum_result_bounded. ((((exists fom_beta_height_pfp_transport_old_sum_result_bounded_entry. fom_beta_height_pfp_transport_old_sum_result_bounded_entry + S (fom_value_pfp_transport_old_sum_result_bounded) = S ((S (fom_index_pfp_transport_old_sum_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_transport_old_sum_result_bounded_entry. gb = fom_beta_quotient_pfp_transport_old_sum_result_bounded_entry * S ((S (fom_index_pfp_transport_old_sum_result_bounded)) * gc) + (fom_value_pfp_transport_old_sum_result_bounded))) /\\ (exists fom_gap_pfp_transport_old_sum_result_bounded_value_bound. fom_gap_pfp_transport_old_sum_result_bounded_value_bound + S (fom_value_pfp_transport_old_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_transport_old_sum pfaa_left_c_transport_old_sum pfaa_right_b_transport_old_sum pfaa_right_c_transport_old_sum pfaa_sum_b_transport_old_sum pfaa_sum_c_transport_old_sum pfaa_length_transport_old_sum. ((((forall pfrep_power_transport_old_sum_witness_common_left pfrep_left_transport_old_sum_witness_common_left pfrep_right_transport_old_sum_witness_common_left. ((exists pfrep_position_transport_old_sum_witness_common_leftfirst. ((pfrep_position_transport_old_sum_witness_common_leftfirst+S (pfrep_power_transport_old_sum_witness_common_left)=(pfbz_left_length_transport_old)) /\\ ((((exists ff_h_pfp_transport_old_sum_witness_common_leftfirstentry. ff_h_pfp_transport_old_sum_witness_common_leftfirstentry + S (pfrep_left_transport_old_sum_witness_common_left) = S ((S (pfrep_position_transport_old_sum_witness_common_leftfirst)) * pfbz_left_scale_transport_old)) /\\ exists ff_q_pfp_transport_old_sum_witness_common_leftfirstentry. pfbz_left_code_transport_old = ff_q_pfp_transport_old_sum_witness_common_leftfirstentry * S ((S (pfrep_position_transport_old_sum_witness_common_leftfirst)) * pfbz_left_scale_transport_old) + (pfrep_left_transport_old_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_transport_old_sum_witness_common_leftfirstoutside. pfrep_gap_transport_old_sum_witness_common_leftfirstoutside+(pfbz_left_length_transport_old)=(pfrep_power_transport_old_sum_witness_common_left)) /\\ (((pfrep_left_transport_old_sum_witness_common_left)=0))))) -> ((exists pfrep_position_transport_old_sum_witness_common_leftsecond. ((pfrep_position_transport_old_sum_witness_common_leftsecond+S (pfrep_power_transport_old_sum_witness_common_left)=(pfaa_length_transport_old_sum)) /\\ ((((exists ff_h_pfp_transport_old_sum_witness_common_leftsecondentry. ff_h_pfp_transport_old_sum_witness_common_leftsecondentry + S (pfrep_right_transport_old_sum_witness_common_left) = S ((S (pfrep_position_transport_old_sum_witness_common_leftsecond)) * pfaa_left_c_transport_old_sum)) /\\ exists ff_q_pfp_transport_old_sum_witness_common_leftsecondentry. pfaa_left_b_transport_old_sum = ff_q_pfp_transport_old_sum_witness_common_leftsecondentry * S ((S (pfrep_position_transport_old_sum_witness_common_leftsecond)) * pfaa_left_c_transport_old_sum) + (pfrep_right_transport_old_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_transport_old_sum_witness_common_leftsecondoutside. pfrep_gap_transport_old_sum_witness_common_leftsecondoutside+(pfaa_length_transport_old_sum)=(pfrep_power_transport_old_sum_witness_common_left)) /\\ (((pfrep_right_transport_old_sum_witness_common_left)=0))))) -> pfrep_left_transport_old_sum_witness_common_left=pfrep_right_transport_old_sum_witness_common_left) /\\ ((forall pfrep_power_transport_old_sum_witness_common_right pfrep_left_transport_old_sum_witness_common_right pfrep_right_transport_old_sum_witness_common_right. ((exists pfrep_position_transport_old_sum_witness_common_rightfirst. ((pfrep_position_transport_old_sum_witness_common_rightfirst+S (pfrep_power_transport_old_sum_witness_common_right)=(pfbz_right_length_transport_old)) /\\ ((((exists ff_h_pfp_transport_old_sum_witness_common_rightfirstentry. ff_h_pfp_transport_old_sum_witness_common_rightfirstentry + S (pfrep_left_transport_old_sum_witness_common_right) = S ((S (pfrep_position_transport_old_sum_witness_common_rightfirst)) * pfbz_right_scale_transport_old)) /\\ exists ff_q_pfp_transport_old_sum_witness_common_rightfirstentry. pfbz_right_code_transport_old = ff_q_pfp_transport_old_sum_witness_common_rightfirstentry * S ((S (pfrep_position_transport_old_sum_witness_common_rightfirst)) * pfbz_right_scale_transport_old) + (pfrep_left_transport_old_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_transport_old_sum_witness_common_rightfirstoutside. pfrep_gap_transport_old_sum_witness_common_rightfirstoutside+(pfbz_right_length_transport_old)=(pfrep_power_transport_old_sum_witness_common_right)) /\\ (((pfrep_left_transport_old_sum_witness_common_right)=0))))) -> ((exists pfrep_position_transport_old_sum_witness_common_rightsecond. ((pfrep_position_transport_old_sum_witness_common_rightsecond+S (pfrep_power_transport_old_sum_witness_common_right)=(pfaa_length_transport_old_sum)) /\\ ((((exists ff_h_pfp_transport_old_sum_witness_common_rightsecondentry. ff_h_pfp_transport_old_sum_witness_common_rightsecondentry + S (pfrep_right_transport_old_sum_witness_common_right) = S ((S (pfrep_position_transport_old_sum_witness_common_rightsecond)) * pfaa_right_c_transport_old_sum)) /\\ exists ff_q_pfp_transport_old_sum_witness_common_rightsecondentry. pfaa_right_b_transport_old_sum = ff_q_pfp_transport_old_sum_witness_common_rightsecondentry * S ((S (pfrep_position_transport_old_sum_witness_common_rightsecond)) * pfaa_right_c_transport_old_sum) + (pfrep_right_transport_old_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_transport_old_sum_witness_common_rightsecondoutside. pfrep_gap_transport_old_sum_witness_common_rightsecondoutside+(pfaa_length_transport_old_sum)=(pfrep_power_transport_old_sum_witness_common_right)) /\\ (((pfrep_right_transport_old_sum_witness_common_right)=0))))) -> pfrep_left_transport_old_sum_witness_common_right=pfrep_right_transport_old_sum_witness_common_right)))) /\\ (((forall pfp_index_transport_old_sum_witness_operation. (exists pfa_gap_transport_old_sum_witness_operationindex. pfa_gap_transport_old_sum_witness_operationindex + S (pfp_index_transport_old_sum_witness_operation) = (pfaa_length_transport_old_sum)) -> exists pfp_left_transport_old_sum_witness_operation pfp_right_transport_old_sum_witness_operation pfp_value_transport_old_sum_witness_operation. ((((exists ff_h_pfp_transport_old_sum_witness_operationleft. ff_h_pfp_transport_old_sum_witness_operationleft + S (pfp_left_transport_old_sum_witness_operation) = S ((S (pfp_index_transport_old_sum_witness_operation)) * pfaa_left_c_transport_old_sum)) /\\ exists ff_q_pfp_transport_old_sum_witness_operationleft. pfaa_left_b_transport_old_sum = ff_q_pfp_transport_old_sum_witness_operationleft * S ((S (pfp_index_transport_old_sum_witness_operation)) * pfaa_left_c_transport_old_sum) + (pfp_left_transport_old_sum_witness_operation))) /\\ (((((exists ff_h_pfp_transport_old_sum_witness_operationright. ff_h_pfp_transport_old_sum_witness_operationright + S (pfp_right_transport_old_sum_witness_operation) = S ((S (pfp_index_transport_old_sum_witness_operation)) * pfaa_right_c_transport_old_sum)) /\\ exists ff_q_pfp_transport_old_sum_witness_operationright. pfaa_right_b_transport_old_sum = ff_q_pfp_transport_old_sum_witness_operationright * S ((S (pfp_index_transport_old_sum_witness_operation)) * pfaa_right_c_transport_old_sum) + (pfp_right_transport_old_sum_witness_operation))) /\\ (((((exists ff_h_pfp_transport_old_sum_witness_operationtarget. ff_h_pfp_transport_old_sum_witness_operationtarget + S (pfp_value_transport_old_sum_witness_operation) = S ((S (pfp_index_transport_old_sum_witness_operation)) * pfaa_sum_c_transport_old_sum)) /\\ exists ff_q_pfp_transport_old_sum_witness_operationtarget. pfaa_sum_b_transport_old_sum = ff_q_pfp_transport_old_sum_witness_operationtarget * S ((S (pfp_index_transport_old_sum_witness_operation)) * pfaa_sum_c_transport_old_sum) + (pfp_value_transport_old_sum_witness_operation))) /\\ ((((exists pfa_gap_transport_old_sum_witness_operationoperationleft. pfa_gap_transport_old_sum_witness_operationoperationleft + S (pfp_left_transport_old_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_transport_old_sum_witness_operationoperationright. pfa_gap_transport_old_sum_witness_operationoperationright + S (pfp_right_transport_old_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_transport_old_sum_witness_operationoperationresultbound. pfa_gap_transport_old_sum_witness_operationoperationresultbound + S (pfp_value_transport_old_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_transport_old_sum_witness_operationoperationresultcongruence pfa_offset_right_transport_old_sum_witness_operationoperationresultcongruence. ((pfp_left_transport_old_sum_witness_operation) + (pfp_right_transport_old_sum_witness_operation)) + (p) * pfa_offset_left_transport_old_sum_witness_operationoperationresultcongruence = (pfp_value_transport_old_sum_witness_operation) + (p) * pfa_offset_right_transport_old_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_transport_old_sum_witness_output pfrep_left_transport_old_sum_witness_output pfrep_right_transport_old_sum_witness_output. ((exists pfrep_position_transport_old_sum_witness_outputfirst. ((pfrep_position_transport_old_sum_witness_outputfirst+S (pfrep_power_transport_old_sum_witness_output)=(pfaa_length_transport_old_sum)) /\\ ((((exists ff_h_pfp_transport_old_sum_witness_outputfirstentry. ff_h_pfp_transport_old_sum_witness_outputfirstentry + S (pfrep_left_transport_old_sum_witness_output) = S ((S (pfrep_position_transport_old_sum_witness_outputfirst)) * pfaa_sum_c_transport_old_sum)) /\\ exists ff_q_pfp_transport_old_sum_witness_outputfirstentry. pfaa_sum_b_transport_old_sum = ff_q_pfp_transport_old_sum_witness_outputfirstentry * S ((S (pfrep_position_transport_old_sum_witness_outputfirst)) * pfaa_sum_c_transport_old_sum) + (pfrep_left_transport_old_sum_witness_output)))))) \\/ (((exists pfrep_gap_transport_old_sum_witness_outputfirstoutside. pfrep_gap_transport_old_sum_witness_outputfirstoutside+(pfaa_length_transport_old_sum)=(pfrep_power_transport_old_sum_witness_output)) /\\ (((pfrep_left_transport_old_sum_witness_output)=0))))) -> ((exists pfrep_position_transport_old_sum_witness_outputsecond. ((pfrep_position_transport_old_sum_witness_outputsecond+S (pfrep_power_transport_old_sum_witness_output)=(G)) /\\ ((((exists ff_h_pfp_transport_old_sum_witness_outputsecondentry. ff_h_pfp_transport_old_sum_witness_outputsecondentry + S (pfrep_right_transport_old_sum_witness_output) = S ((S (pfrep_position_transport_old_sum_witness_outputsecond)) * gc)) /\\ exists ff_q_pfp_transport_old_sum_witness_outputsecondentry. gb = ff_q_pfp_transport_old_sum_witness_outputsecondentry * S ((S (pfrep_position_transport_old_sum_witness_outputsecond)) * gc) + (pfrep_right_transport_old_sum_witness_output)))))) \\/ (((exists pfrep_gap_transport_old_sum_witness_outputsecondoutside. pfrep_gap_transport_old_sum_witness_outputsecondoutside+(G)=(pfrep_power_transport_old_sum_witness_output)) /\\ (((pfrep_right_transport_old_sum_witness_output)=0))))) -> pfrep_left_transport_old_sum_witness_output=pfrep_right_transport_old_sum_witness_output)))))))))))))))))) -> (exists pfbz_left_code_transport_result pfbz_left_scale_transport_result pfbz_left_length_transport_result pfbz_right_code_transport_result pfbz_right_scale_transport_result pfbz_right_length_transport_result. ((((forall fom_index_pfp_transport_result_left_productleft. (exists fom_gap_pfp_transport_result_left_productleft_index_bound. fom_gap_pfp_transport_result_left_productleft_index_bound + S (fom_index_pfp_transport_result_left_productleft) = U) -> exists fom_value_pfp_transport_result_left_productleft. ((((exists fom_beta_height_pfp_transport_result_left_productleft_entry. fom_beta_height_pfp_transport_result_left_productleft_entry + S (fom_value_pfp_transport_result_left_productleft) = S ((S (fom_index_pfp_transport_result_left_productleft)) * uc)) /\\ exists fom_beta_quotient_pfp_transport_result_left_productleft_entry. ub = fom_beta_quotient_pfp_transport_result_left_productleft_entry * S ((S (fom_index_pfp_transport_result_left_productleft)) * uc) + (fom_value_pfp_transport_result_left_productleft))) /\\ (exists fom_gap_pfp_transport_result_left_productleft_value_bound. fom_gap_pfp_transport_result_left_productleft_value_bound + S (fom_value_pfp_transport_result_left_productleft) = p))) /\\ (((forall fom_index_pfp_transport_result_left_productright. (exists fom_gap_pfp_transport_result_left_productright_index_bound. fom_gap_pfp_transport_result_left_productright_index_bound + S (fom_index_pfp_transport_result_left_productright) = A2) -> exists fom_value_pfp_transport_result_left_productright. ((((exists fom_beta_height_pfp_transport_result_left_productright_entry. fom_beta_height_pfp_transport_result_left_productright_entry + S (fom_value_pfp_transport_result_left_productright) = S ((S (fom_index_pfp_transport_result_left_productright)) * ac2)) /\\ exists fom_beta_quotient_pfp_transport_result_left_productright_entry. ab2 = fom_beta_quotient_pfp_transport_result_left_productright_entry * S ((S (fom_index_pfp_transport_result_left_productright)) * ac2) + (fom_value_pfp_transport_result_left_productright))) /\\ (exists fom_gap_pfp_transport_result_left_productright_value_bound. fom_gap_pfp_transport_result_left_productright_value_bound + S (fom_value_pfp_transport_result_left_productright) = p))) /\\ (((((((U)=0 \\/ (A2)=0) /\\ (((pfbz_left_length_transport_result)=0)))) \\/ (((~((U)=0)) /\\ (((~((A2)=0)) /\\ (((U)+(A2)=S (pfbz_left_length_transport_result)))))))) /\\ ((forall pfc_index_transport_result_left_productcoefficients. (exists pfa_gap_transport_result_left_productcoefficientsbound. pfa_gap_transport_result_left_productcoefficientsbound + S (pfc_index_transport_result_left_productcoefficients) = (pfbz_left_length_transport_result)) -> exists pfc_value_transport_result_left_productcoefficients. ((((exists ff_h_pfp_transport_result_left_productcoefficientsentry. ff_h_pfp_transport_result_left_productcoefficientsentry + S (pfc_value_transport_result_left_productcoefficients) = S ((S (pfc_index_transport_result_left_productcoefficients)) * pfbz_left_scale_transport_result)) /\\ exists ff_q_pfp_transport_result_left_productcoefficientsentry. pfbz_left_code_transport_result = ff_q_pfp_transport_result_left_productcoefficientsentry * S ((S (pfc_index_transport_result_left_productcoefficients)) * pfbz_left_scale_transport_result) + (pfc_value_transport_result_left_productcoefficients))) /\\ ((exists pfc_terms_code_transport_result_left_productcoefficientscoefficient pfc_terms_scale_transport_result_left_productcoefficientscoefficient pfc_natural_sum_transport_result_left_productcoefficientscoefficient. ((forall pfc_index_transport_result_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_transport_result_left_productcoefficientscoefficientdiagonalbound. pfa_gap_transport_result_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_transport_result_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_transport_result_left_productcoefficients))) -> exists pfc_value_transport_result_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_transport_result_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_transport_result_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_transport_result_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_transport_result_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_transport_result_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_transport_result_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_transport_result_left_productcoefficientscoefficient = ff_q_pfp_transport_result_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_transport_result_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_transport_result_left_productcoefficientscoefficient) + (pfc_value_transport_result_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_transport_result_left_productcoefficientscoefficientdiagonalterm pfc_left_transport_result_left_productcoefficientscoefficientdiagonalterm pfc_right_transport_result_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_transport_result_left_productcoefficientscoefficientdiagonal)+pfc_complement_transport_result_left_productcoefficientscoefficientdiagonalterm=(pfc_index_transport_result_left_productcoefficients)) /\\ ((((((exists pfa_gap_transport_result_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_transport_result_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_transport_result_left_productcoefficientscoefficientdiagonal) = (U)) /\\ ((((exists ff_h_pfp_transport_result_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_transport_result_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_transport_result_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_transport_result_left_productcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_transport_result_left_productcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_transport_result_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_transport_result_left_productcoefficientscoefficientdiagonal)) * uc) + (pfc_left_transport_result_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_transport_result_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_transport_result_left_productcoefficientscoefficientdiagonaltermleftoutside+(U)=(pfc_index_transport_result_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_transport_result_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_transport_result_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_transport_result_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_transport_result_left_productcoefficientscoefficientdiagonalterm) = (A2)) /\\ ((((exists ff_h_pfp_transport_result_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_transport_result_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_transport_result_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_transport_result_left_productcoefficientscoefficientdiagonalterm)) * ac2)) /\\ exists ff_q_pfp_transport_result_left_productcoefficientscoefficientdiagonaltermrightentry. ab2 = ff_q_pfp_transport_result_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_transport_result_left_productcoefficientscoefficientdiagonalterm)) * ac2) + (pfc_right_transport_result_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_transport_result_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_transport_result_left_productcoefficientscoefficientdiagonaltermrightoutside+(A2)=(pfc_complement_transport_result_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_transport_result_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_transport_result_left_productcoefficientscoefficientdiagonal)=pfc_left_transport_result_left_productcoefficientscoefficientdiagonalterm*pfc_right_transport_result_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_transport_result_left_productcoefficientscoefficientsum fs_v_pfc_transport_result_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_transport_result_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_transport_result_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_transport_result_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_result_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_transport_result_left_productcoefficientscoefficientsum = fs_q_pfc_transport_result_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_transport_result_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_transport_result_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_transport_result_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_transport_result_left_productcoefficientscoefficient) = S ((S (S (pfc_index_transport_result_left_productcoefficients))) * fs_v_pfc_transport_result_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_result_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_transport_result_left_productcoefficientscoefficientsum = fs_q_pfc_transport_result_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_transport_result_left_productcoefficients))) * fs_v_pfc_transport_result_left_productcoefficientscoefficientsum) + (pfc_natural_sum_transport_result_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_transport_result_left_productcoefficients)) -> exists fs_a_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_transport_result_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_transport_result_left_productcoefficientscoefficient = fs_q_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_transport_result_left_productcoefficientscoefficient) + (fs_a_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_result_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_transport_result_left_productcoefficientscoefficientsum = fs_q_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_result_left_productcoefficientscoefficientsum) + (fs_r_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_result_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_transport_result_left_productcoefficientscoefficientsum = fs_q_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_result_left_productcoefficientscoefficientsum) + (fs_s_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_transport_result_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_transport_result_left_productcoefficientscoefficientresiduebound. pfa_gap_transport_result_left_productcoefficientscoefficientresiduebound + S (pfc_value_transport_result_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_transport_result_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_transport_result_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_transport_result_left_productcoefficientscoefficient) + (p) * pfa_offset_left_transport_result_left_productcoefficientscoefficientresiduecongruence = (pfc_value_transport_result_left_productcoefficients) + (p) * pfa_offset_right_transport_result_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_transport_result_right_productleft. (exists fom_gap_pfp_transport_result_right_productleft_index_bound. fom_gap_pfp_transport_result_right_productleft_index_bound + S (fom_index_pfp_transport_result_right_productleft) = V) -> exists fom_value_pfp_transport_result_right_productleft. ((((exists fom_beta_height_pfp_transport_result_right_productleft_entry. fom_beta_height_pfp_transport_result_right_productleft_entry + S (fom_value_pfp_transport_result_right_productleft) = S ((S (fom_index_pfp_transport_result_right_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_transport_result_right_productleft_entry. vb = fom_beta_quotient_pfp_transport_result_right_productleft_entry * S ((S (fom_index_pfp_transport_result_right_productleft)) * vc) + (fom_value_pfp_transport_result_right_productleft))) /\\ (exists fom_gap_pfp_transport_result_right_productleft_value_bound. fom_gap_pfp_transport_result_right_productleft_value_bound + S (fom_value_pfp_transport_result_right_productleft) = p))) /\\ (((forall fom_index_pfp_transport_result_right_productright. (exists fom_gap_pfp_transport_result_right_productright_index_bound. fom_gap_pfp_transport_result_right_productright_index_bound + S (fom_index_pfp_transport_result_right_productright) = B2) -> exists fom_value_pfp_transport_result_right_productright. ((((exists fom_beta_height_pfp_transport_result_right_productright_entry. fom_beta_height_pfp_transport_result_right_productright_entry + S (fom_value_pfp_transport_result_right_productright) = S ((S (fom_index_pfp_transport_result_right_productright)) * bc2)) /\\ exists fom_beta_quotient_pfp_transport_result_right_productright_entry. bb2 = fom_beta_quotient_pfp_transport_result_right_productright_entry * S ((S (fom_index_pfp_transport_result_right_productright)) * bc2) + (fom_value_pfp_transport_result_right_productright))) /\\ (exists fom_gap_pfp_transport_result_right_productright_value_bound. fom_gap_pfp_transport_result_right_productright_value_bound + S (fom_value_pfp_transport_result_right_productright) = p))) /\\ (((((((V)=0 \\/ (B2)=0) /\\ (((pfbz_right_length_transport_result)=0)))) \\/ (((~((V)=0)) /\\ (((~((B2)=0)) /\\ (((V)+(B2)=S (pfbz_right_length_transport_result)))))))) /\\ ((forall pfc_index_transport_result_right_productcoefficients. (exists pfa_gap_transport_result_right_productcoefficientsbound. pfa_gap_transport_result_right_productcoefficientsbound + S (pfc_index_transport_result_right_productcoefficients) = (pfbz_right_length_transport_result)) -> exists pfc_value_transport_result_right_productcoefficients. ((((exists ff_h_pfp_transport_result_right_productcoefficientsentry. ff_h_pfp_transport_result_right_productcoefficientsentry + S (pfc_value_transport_result_right_productcoefficients) = S ((S (pfc_index_transport_result_right_productcoefficients)) * pfbz_right_scale_transport_result)) /\\ exists ff_q_pfp_transport_result_right_productcoefficientsentry. pfbz_right_code_transport_result = ff_q_pfp_transport_result_right_productcoefficientsentry * S ((S (pfc_index_transport_result_right_productcoefficients)) * pfbz_right_scale_transport_result) + (pfc_value_transport_result_right_productcoefficients))) /\\ ((exists pfc_terms_code_transport_result_right_productcoefficientscoefficient pfc_terms_scale_transport_result_right_productcoefficientscoefficient pfc_natural_sum_transport_result_right_productcoefficientscoefficient. ((forall pfc_index_transport_result_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_transport_result_right_productcoefficientscoefficientdiagonalbound. pfa_gap_transport_result_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_transport_result_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_transport_result_right_productcoefficients))) -> exists pfc_value_transport_result_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_transport_result_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_transport_result_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_transport_result_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_transport_result_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_transport_result_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_transport_result_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_transport_result_right_productcoefficientscoefficient = ff_q_pfp_transport_result_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_transport_result_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_transport_result_right_productcoefficientscoefficient) + (pfc_value_transport_result_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_transport_result_right_productcoefficientscoefficientdiagonalterm pfc_left_transport_result_right_productcoefficientscoefficientdiagonalterm pfc_right_transport_result_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_transport_result_right_productcoefficientscoefficientdiagonal)+pfc_complement_transport_result_right_productcoefficientscoefficientdiagonalterm=(pfc_index_transport_result_right_productcoefficients)) /\\ ((((((exists pfa_gap_transport_result_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_transport_result_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_transport_result_right_productcoefficientscoefficientdiagonal) = (V)) /\\ ((((exists ff_h_pfp_transport_result_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_transport_result_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_transport_result_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_transport_result_right_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_transport_result_right_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_transport_result_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_transport_result_right_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_transport_result_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_transport_result_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_transport_result_right_productcoefficientscoefficientdiagonaltermleftoutside+(V)=(pfc_index_transport_result_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_transport_result_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_transport_result_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_transport_result_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_transport_result_right_productcoefficientscoefficientdiagonalterm) = (B2)) /\\ ((((exists ff_h_pfp_transport_result_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_transport_result_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_transport_result_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_transport_result_right_productcoefficientscoefficientdiagonalterm)) * bc2)) /\\ exists ff_q_pfp_transport_result_right_productcoefficientscoefficientdiagonaltermrightentry. bb2 = ff_q_pfp_transport_result_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_transport_result_right_productcoefficientscoefficientdiagonalterm)) * bc2) + (pfc_right_transport_result_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_transport_result_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_transport_result_right_productcoefficientscoefficientdiagonaltermrightoutside+(B2)=(pfc_complement_transport_result_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_transport_result_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_transport_result_right_productcoefficientscoefficientdiagonal)=pfc_left_transport_result_right_productcoefficientscoefficientdiagonalterm*pfc_right_transport_result_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_transport_result_right_productcoefficientscoefficientsum fs_v_pfc_transport_result_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_transport_result_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_transport_result_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_transport_result_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_result_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_transport_result_right_productcoefficientscoefficientsum = fs_q_pfc_transport_result_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_transport_result_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_transport_result_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_transport_result_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_transport_result_right_productcoefficientscoefficient) = S ((S (S (pfc_index_transport_result_right_productcoefficients))) * fs_v_pfc_transport_result_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_result_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_transport_result_right_productcoefficientscoefficientsum = fs_q_pfc_transport_result_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_transport_result_right_productcoefficients))) * fs_v_pfc_transport_result_right_productcoefficientscoefficientsum) + (pfc_natural_sum_transport_result_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_transport_result_right_productcoefficients)) -> exists fs_a_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_transport_result_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_transport_result_right_productcoefficientscoefficient = fs_q_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_transport_result_right_productcoefficientscoefficient) + (fs_a_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_result_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_transport_result_right_productcoefficientscoefficientsum = fs_q_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_result_right_productcoefficientscoefficientsum) + (fs_r_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_result_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_transport_result_right_productcoefficientscoefficientsum = fs_q_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_transport_result_right_productcoefficientscoefficientsum) + (fs_s_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_transport_result_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_transport_result_right_productcoefficientscoefficientresiduebound. pfa_gap_transport_result_right_productcoefficientscoefficientresiduebound + S (pfc_value_transport_result_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_transport_result_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_transport_result_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_transport_result_right_productcoefficientscoefficient) + (p) * pfa_offset_left_transport_result_right_productcoefficientscoefficientresiduecongruence = (pfc_value_transport_result_right_productcoefficients) + (p) * pfa_offset_right_transport_result_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_transport_result_sum_left_bounded. (exists fom_gap_pfp_transport_result_sum_left_bounded_index_bound. fom_gap_pfp_transport_result_sum_left_bounded_index_bound + S (fom_index_pfp_transport_result_sum_left_bounded) = pfbz_left_length_transport_result) -> exists fom_value_pfp_transport_result_sum_left_bounded. ((((exists fom_beta_height_pfp_transport_result_sum_left_bounded_entry. fom_beta_height_pfp_transport_result_sum_left_bounded_entry + S (fom_value_pfp_transport_result_sum_left_bounded) = S ((S (fom_index_pfp_transport_result_sum_left_bounded)) * pfbz_left_scale_transport_result)) /\\ exists fom_beta_quotient_pfp_transport_result_sum_left_bounded_entry. pfbz_left_code_transport_result = fom_beta_quotient_pfp_transport_result_sum_left_bounded_entry * S ((S (fom_index_pfp_transport_result_sum_left_bounded)) * pfbz_left_scale_transport_result) + (fom_value_pfp_transport_result_sum_left_bounded))) /\\ (exists fom_gap_pfp_transport_result_sum_left_bounded_value_bound. fom_gap_pfp_transport_result_sum_left_bounded_value_bound + S (fom_value_pfp_transport_result_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_transport_result_sum_right_bounded. (exists fom_gap_pfp_transport_result_sum_right_bounded_index_bound. fom_gap_pfp_transport_result_sum_right_bounded_index_bound + S (fom_index_pfp_transport_result_sum_right_bounded) = pfbz_right_length_transport_result) -> exists fom_value_pfp_transport_result_sum_right_bounded. ((((exists fom_beta_height_pfp_transport_result_sum_right_bounded_entry. fom_beta_height_pfp_transport_result_sum_right_bounded_entry + S (fom_value_pfp_transport_result_sum_right_bounded) = S ((S (fom_index_pfp_transport_result_sum_right_bounded)) * pfbz_right_scale_transport_result)) /\\ exists fom_beta_quotient_pfp_transport_result_sum_right_bounded_entry. pfbz_right_code_transport_result = fom_beta_quotient_pfp_transport_result_sum_right_bounded_entry * S ((S (fom_index_pfp_transport_result_sum_right_bounded)) * pfbz_right_scale_transport_result) + (fom_value_pfp_transport_result_sum_right_bounded))) /\\ (exists fom_gap_pfp_transport_result_sum_right_bounded_value_bound. fom_gap_pfp_transport_result_sum_right_bounded_value_bound + S (fom_value_pfp_transport_result_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_transport_result_sum_result_bounded. (exists fom_gap_pfp_transport_result_sum_result_bounded_index_bound. fom_gap_pfp_transport_result_sum_result_bounded_index_bound + S (fom_index_pfp_transport_result_sum_result_bounded) = G2) -> exists fom_value_pfp_transport_result_sum_result_bounded. ((((exists fom_beta_height_pfp_transport_result_sum_result_bounded_entry. fom_beta_height_pfp_transport_result_sum_result_bounded_entry + S (fom_value_pfp_transport_result_sum_result_bounded) = S ((S (fom_index_pfp_transport_result_sum_result_bounded)) * gc2)) /\\ exists fom_beta_quotient_pfp_transport_result_sum_result_bounded_entry. gb2 = fom_beta_quotient_pfp_transport_result_sum_result_bounded_entry * S ((S (fom_index_pfp_transport_result_sum_result_bounded)) * gc2) + (fom_value_pfp_transport_result_sum_result_bounded))) /\\ (exists fom_gap_pfp_transport_result_sum_result_bounded_value_bound. fom_gap_pfp_transport_result_sum_result_bounded_value_bound + S (fom_value_pfp_transport_result_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_transport_result_sum pfaa_left_c_transport_result_sum pfaa_right_b_transport_result_sum pfaa_right_c_transport_result_sum pfaa_sum_b_transport_result_sum pfaa_sum_c_transport_result_sum pfaa_length_transport_result_sum. ((((forall pfrep_power_transport_result_sum_witness_common_left pfrep_left_transport_result_sum_witness_common_left pfrep_right_transport_result_sum_witness_common_left. ((exists pfrep_position_transport_result_sum_witness_common_leftfirst. ((pfrep_position_transport_result_sum_witness_common_leftfirst+S (pfrep_power_transport_result_sum_witness_common_left)=(pfbz_left_length_transport_result)) /\\ ((((exists ff_h_pfp_transport_result_sum_witness_common_leftfirstentry. ff_h_pfp_transport_result_sum_witness_common_leftfirstentry + S (pfrep_left_transport_result_sum_witness_common_left) = S ((S (pfrep_position_transport_result_sum_witness_common_leftfirst)) * pfbz_left_scale_transport_result)) /\\ exists ff_q_pfp_transport_result_sum_witness_common_leftfirstentry. pfbz_left_code_transport_result = ff_q_pfp_transport_result_sum_witness_common_leftfirstentry * S ((S (pfrep_position_transport_result_sum_witness_common_leftfirst)) * pfbz_left_scale_transport_result) + (pfrep_left_transport_result_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_transport_result_sum_witness_common_leftfirstoutside. pfrep_gap_transport_result_sum_witness_common_leftfirstoutside+(pfbz_left_length_transport_result)=(pfrep_power_transport_result_sum_witness_common_left)) /\\ (((pfrep_left_transport_result_sum_witness_common_left)=0))))) -> ((exists pfrep_position_transport_result_sum_witness_common_leftsecond. ((pfrep_position_transport_result_sum_witness_common_leftsecond+S (pfrep_power_transport_result_sum_witness_common_left)=(pfaa_length_transport_result_sum)) /\\ ((((exists ff_h_pfp_transport_result_sum_witness_common_leftsecondentry. ff_h_pfp_transport_result_sum_witness_common_leftsecondentry + S (pfrep_right_transport_result_sum_witness_common_left) = S ((S (pfrep_position_transport_result_sum_witness_common_leftsecond)) * pfaa_left_c_transport_result_sum)) /\\ exists ff_q_pfp_transport_result_sum_witness_common_leftsecondentry. pfaa_left_b_transport_result_sum = ff_q_pfp_transport_result_sum_witness_common_leftsecondentry * S ((S (pfrep_position_transport_result_sum_witness_common_leftsecond)) * pfaa_left_c_transport_result_sum) + (pfrep_right_transport_result_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_transport_result_sum_witness_common_leftsecondoutside. pfrep_gap_transport_result_sum_witness_common_leftsecondoutside+(pfaa_length_transport_result_sum)=(pfrep_power_transport_result_sum_witness_common_left)) /\\ (((pfrep_right_transport_result_sum_witness_common_left)=0))))) -> pfrep_left_transport_result_sum_witness_common_left=pfrep_right_transport_result_sum_witness_common_left) /\\ ((forall pfrep_power_transport_result_sum_witness_common_right pfrep_left_transport_result_sum_witness_common_right pfrep_right_transport_result_sum_witness_common_right. ((exists pfrep_position_transport_result_sum_witness_common_rightfirst. ((pfrep_position_transport_result_sum_witness_common_rightfirst+S (pfrep_power_transport_result_sum_witness_common_right)=(pfbz_right_length_transport_result)) /\\ ((((exists ff_h_pfp_transport_result_sum_witness_common_rightfirstentry. ff_h_pfp_transport_result_sum_witness_common_rightfirstentry + S (pfrep_left_transport_result_sum_witness_common_right) = S ((S (pfrep_position_transport_result_sum_witness_common_rightfirst)) * pfbz_right_scale_transport_result)) /\\ exists ff_q_pfp_transport_result_sum_witness_common_rightfirstentry. pfbz_right_code_transport_result = ff_q_pfp_transport_result_sum_witness_common_rightfirstentry * S ((S (pfrep_position_transport_result_sum_witness_common_rightfirst)) * pfbz_right_scale_transport_result) + (pfrep_left_transport_result_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_transport_result_sum_witness_common_rightfirstoutside. pfrep_gap_transport_result_sum_witness_common_rightfirstoutside+(pfbz_right_length_transport_result)=(pfrep_power_transport_result_sum_witness_common_right)) /\\ (((pfrep_left_transport_result_sum_witness_common_right)=0))))) -> ((exists pfrep_position_transport_result_sum_witness_common_rightsecond. ((pfrep_position_transport_result_sum_witness_common_rightsecond+S (pfrep_power_transport_result_sum_witness_common_right)=(pfaa_length_transport_result_sum)) /\\ ((((exists ff_h_pfp_transport_result_sum_witness_common_rightsecondentry. ff_h_pfp_transport_result_sum_witness_common_rightsecondentry + S (pfrep_right_transport_result_sum_witness_common_right) = S ((S (pfrep_position_transport_result_sum_witness_common_rightsecond)) * pfaa_right_c_transport_result_sum)) /\\ exists ff_q_pfp_transport_result_sum_witness_common_rightsecondentry. pfaa_right_b_transport_result_sum = ff_q_pfp_transport_result_sum_witness_common_rightsecondentry * S ((S (pfrep_position_transport_result_sum_witness_common_rightsecond)) * pfaa_right_c_transport_result_sum) + (pfrep_right_transport_result_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_transport_result_sum_witness_common_rightsecondoutside. pfrep_gap_transport_result_sum_witness_common_rightsecondoutside+(pfaa_length_transport_result_sum)=(pfrep_power_transport_result_sum_witness_common_right)) /\\ (((pfrep_right_transport_result_sum_witness_common_right)=0))))) -> pfrep_left_transport_result_sum_witness_common_right=pfrep_right_transport_result_sum_witness_common_right)))) /\\ (((forall pfp_index_transport_result_sum_witness_operation. (exists pfa_gap_transport_result_sum_witness_operationindex. pfa_gap_transport_result_sum_witness_operationindex + S (pfp_index_transport_result_sum_witness_operation) = (pfaa_length_transport_result_sum)) -> exists pfp_left_transport_result_sum_witness_operation pfp_right_transport_result_sum_witness_operation pfp_value_transport_result_sum_witness_operation. ((((exists ff_h_pfp_transport_result_sum_witness_operationleft. ff_h_pfp_transport_result_sum_witness_operationleft + S (pfp_left_transport_result_sum_witness_operation) = S ((S (pfp_index_transport_result_sum_witness_operation)) * pfaa_left_c_transport_result_sum)) /\\ exists ff_q_pfp_transport_result_sum_witness_operationleft. pfaa_left_b_transport_result_sum = ff_q_pfp_transport_result_sum_witness_operationleft * S ((S (pfp_index_transport_result_sum_witness_operation)) * pfaa_left_c_transport_result_sum) + (pfp_left_transport_result_sum_witness_operation))) /\\ (((((exists ff_h_pfp_transport_result_sum_witness_operationright. ff_h_pfp_transport_result_sum_witness_operationright + S (pfp_right_transport_result_sum_witness_operation) = S ((S (pfp_index_transport_result_sum_witness_operation)) * pfaa_right_c_transport_result_sum)) /\\ exists ff_q_pfp_transport_result_sum_witness_operationright. pfaa_right_b_transport_result_sum = ff_q_pfp_transport_result_sum_witness_operationright * S ((S (pfp_index_transport_result_sum_witness_operation)) * pfaa_right_c_transport_result_sum) + (pfp_right_transport_result_sum_witness_operation))) /\\ (((((exists ff_h_pfp_transport_result_sum_witness_operationtarget. ff_h_pfp_transport_result_sum_witness_operationtarget + S (pfp_value_transport_result_sum_witness_operation) = S ((S (pfp_index_transport_result_sum_witness_operation)) * pfaa_sum_c_transport_result_sum)) /\\ exists ff_q_pfp_transport_result_sum_witness_operationtarget. pfaa_sum_b_transport_result_sum = ff_q_pfp_transport_result_sum_witness_operationtarget * S ((S (pfp_index_transport_result_sum_witness_operation)) * pfaa_sum_c_transport_result_sum) + (pfp_value_transport_result_sum_witness_operation))) /\\ ((((exists pfa_gap_transport_result_sum_witness_operationoperationleft. pfa_gap_transport_result_sum_witness_operationoperationleft + S (pfp_left_transport_result_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_transport_result_sum_witness_operationoperationright. pfa_gap_transport_result_sum_witness_operationoperationright + S (pfp_right_transport_result_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_transport_result_sum_witness_operationoperationresultbound. pfa_gap_transport_result_sum_witness_operationoperationresultbound + S (pfp_value_transport_result_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_transport_result_sum_witness_operationoperationresultcongruence pfa_offset_right_transport_result_sum_witness_operationoperationresultcongruence. ((pfp_left_transport_result_sum_witness_operation) + (pfp_right_transport_result_sum_witness_operation)) + (p) * pfa_offset_left_transport_result_sum_witness_operationoperationresultcongruence = (pfp_value_transport_result_sum_witness_operation) + (p) * pfa_offset_right_transport_result_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_transport_result_sum_witness_output pfrep_left_transport_result_sum_witness_output pfrep_right_transport_result_sum_witness_output. ((exists pfrep_position_transport_result_sum_witness_outputfirst. ((pfrep_position_transport_result_sum_witness_outputfirst+S (pfrep_power_transport_result_sum_witness_output)=(pfaa_length_transport_result_sum)) /\\ ((((exists ff_h_pfp_transport_result_sum_witness_outputfirstentry. ff_h_pfp_transport_result_sum_witness_outputfirstentry + S (pfrep_left_transport_result_sum_witness_output) = S ((S (pfrep_position_transport_result_sum_witness_outputfirst)) * pfaa_sum_c_transport_result_sum)) /\\ exists ff_q_pfp_transport_result_sum_witness_outputfirstentry. pfaa_sum_b_transport_result_sum = ff_q_pfp_transport_result_sum_witness_outputfirstentry * S ((S (pfrep_position_transport_result_sum_witness_outputfirst)) * pfaa_sum_c_transport_result_sum) + (pfrep_left_transport_result_sum_witness_output)))))) \\/ (((exists pfrep_gap_transport_result_sum_witness_outputfirstoutside. pfrep_gap_transport_result_sum_witness_outputfirstoutside+(pfaa_length_transport_result_sum)=(pfrep_power_transport_result_sum_witness_output)) /\\ (((pfrep_left_transport_result_sum_witness_output)=0))))) -> ((exists pfrep_position_transport_result_sum_witness_outputsecond. ((pfrep_position_transport_result_sum_witness_outputsecond+S (pfrep_power_transport_result_sum_witness_output)=(G2)) /\\ ((((exists ff_h_pfp_transport_result_sum_witness_outputsecondentry. ff_h_pfp_transport_result_sum_witness_outputsecondentry + S (pfrep_right_transport_result_sum_witness_output) = S ((S (pfrep_position_transport_result_sum_witness_outputsecond)) * gc2)) /\\ exists ff_q_pfp_transport_result_sum_witness_outputsecondentry. gb2 = ff_q_pfp_transport_result_sum_witness_outputsecondentry * S ((S (pfrep_position_transport_result_sum_witness_outputsecond)) * gc2) + (pfrep_right_transport_result_sum_witness_output)))))) \\/ (((exists pfrep_gap_transport_result_sum_witness_outputsecondoutside. pfrep_gap_transport_result_sum_witness_outputsecondoutside+(G2)=(pfrep_power_transport_result_sum_witness_output)) /\\ (((pfrep_right_transport_result_sum_witness_output)=0))))) -> pfrep_left_transport_result_sum_witness_output=pfrep_right_transport_result_sum_witness_output))))))))))))))))))", "statement_sha256": "bf4307b3128ac805daa5b0a217c1245d5f6a089526d799a7e4464cc64de3ff5b", "summary": "Independently recode both inputs and the result by formal coefficient equivalence, retaining the same Bezout coefficients. Construct both new proper products; output equivalences are proved, not supplied as premises. 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(~((p) = 1) /\\ forall pfa_factor_left_greatest_prime pfa_factor_right_greatest_prime. (p) = pfa_factor_left_greatest_prime * pfa_factor_right_greatest_prime -> pfa_factor_left_greatest_prime = 1 \\/ pfa_factor_right_greatest_prime = 1) -> (((((forall fom_index_pfp_greatest_common_left_canonical. (exists fom_gap_pfp_greatest_common_left_canonical_index_bound. fom_gap_pfp_greatest_common_left_canonical_index_bound + S (fom_index_pfp_greatest_common_left_canonical) = A) -> exists fom_value_pfp_greatest_common_left_canonical. ((((exists fom_beta_height_pfp_greatest_common_left_canonical_entry. fom_beta_height_pfp_greatest_common_left_canonical_entry + S (fom_value_pfp_greatest_common_left_canonical) = S ((S (fom_index_pfp_greatest_common_left_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_greatest_common_left_canonical_entry. ab = fom_beta_quotient_pfp_greatest_common_left_canonical_entry * S ((S (fom_index_pfp_greatest_common_left_canonical)) * ac) + (fom_value_pfp_greatest_common_left_canonical))) /\\ (exists fom_gap_pfp_greatest_common_left_canonical_value_bound. fom_gap_pfp_greatest_common_left_canonical_value_bound + S (fom_value_pfp_greatest_common_left_canonical) = p))) /\\ ((exists pfrd_qb_greatest_common_left pfrd_qc_greatest_common_left pfrd_qlen_greatest_common_left pfrd_pb_greatest_common_left pfrd_pc_greatest_common_left pfrd_plen_greatest_common_left. ((((forall fom_index_pfp_greatest_common_left_productleft. (exists fom_gap_pfp_greatest_common_left_productleft_index_bound. fom_gap_pfp_greatest_common_left_productleft_index_bound + S (fom_index_pfp_greatest_common_left_productleft) = pfrd_qlen_greatest_common_left) -> exists fom_value_pfp_greatest_common_left_productleft. ((((exists fom_beta_height_pfp_greatest_common_left_productleft_entry. fom_beta_height_pfp_greatest_common_left_productleft_entry + S (fom_value_pfp_greatest_common_left_productleft) = S ((S (fom_index_pfp_greatest_common_left_productleft)) * pfrd_qc_greatest_common_left)) /\\ exists fom_beta_quotient_pfp_greatest_common_left_productleft_entry. pfrd_qb_greatest_common_left = fom_beta_quotient_pfp_greatest_common_left_productleft_entry * S ((S (fom_index_pfp_greatest_common_left_productleft)) * pfrd_qc_greatest_common_left) + (fom_value_pfp_greatest_common_left_productleft))) /\\ (exists fom_gap_pfp_greatest_common_left_productleft_value_bound. fom_gap_pfp_greatest_common_left_productleft_value_bound + S (fom_value_pfp_greatest_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_greatest_common_left_productright. (exists fom_gap_pfp_greatest_common_left_productright_index_bound. fom_gap_pfp_greatest_common_left_productright_index_bound + S (fom_index_pfp_greatest_common_left_productright) = D) -> exists fom_value_pfp_greatest_common_left_productright. ((((exists fom_beta_height_pfp_greatest_common_left_productright_entry. fom_beta_height_pfp_greatest_common_left_productright_entry + S (fom_value_pfp_greatest_common_left_productright) = S ((S (fom_index_pfp_greatest_common_left_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_greatest_common_left_productright_entry. db = fom_beta_quotient_pfp_greatest_common_left_productright_entry * S ((S (fom_index_pfp_greatest_common_left_productright)) * dc) + (fom_value_pfp_greatest_common_left_productright))) /\\ (exists fom_gap_pfp_greatest_common_left_productright_value_bound. fom_gap_pfp_greatest_common_left_productright_value_bound + S (fom_value_pfp_greatest_common_left_productright) = p))) /\\ (((((((pfrd_qlen_greatest_common_left)=0 \\/ (D)=0) /\\ (((pfrd_plen_greatest_common_left)=0)))) \\/ (((~((pfrd_qlen_greatest_common_left)=0)) /\\ (((~((D)=0)) /\\ (((pfrd_qlen_greatest_common_left)+(D)=S (pfrd_plen_greatest_common_left)))))))) /\\ ((forall pfc_index_greatest_common_left_productcoefficients. (exists pfa_gap_greatest_common_left_productcoefficientsbound. pfa_gap_greatest_common_left_productcoefficientsbound + S (pfc_index_greatest_common_left_productcoefficients) = (pfrd_plen_greatest_common_left)) -> exists pfc_value_greatest_common_left_productcoefficients. ((((exists ff_h_pfp_greatest_common_left_productcoefficientsentry. ff_h_pfp_greatest_common_left_productcoefficientsentry + S (pfc_value_greatest_common_left_productcoefficients) = S ((S (pfc_index_greatest_common_left_productcoefficients)) * pfrd_pc_greatest_common_left)) /\\ exists ff_q_pfp_greatest_common_left_productcoefficientsentry. pfrd_pb_greatest_common_left = ff_q_pfp_greatest_common_left_productcoefficientsentry * S ((S (pfc_index_greatest_common_left_productcoefficients)) * pfrd_pc_greatest_common_left) + (pfc_value_greatest_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_greatest_common_left_productcoefficientscoefficient pfc_terms_scale_greatest_common_left_productcoefficientscoefficient pfc_natural_sum_greatest_common_left_productcoefficientscoefficient. ((forall pfc_index_greatest_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_greatest_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_greatest_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_greatest_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_greatest_common_left_productcoefficients))) -> exists pfc_value_greatest_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_greatest_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_greatest_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_greatest_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_greatest_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_greatest_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_greatest_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_greatest_common_left_productcoefficientscoefficient = ff_q_pfp_greatest_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_greatest_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_greatest_common_left_productcoefficientscoefficient) + (pfc_value_greatest_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_greatest_common_left_productcoefficientscoefficientdiagonalterm pfc_left_greatest_common_left_productcoefficientscoefficientdiagonalterm pfc_right_greatest_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_greatest_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_greatest_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_greatest_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_greatest_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_greatest_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_greatest_common_left_productcoefficientscoefficientdiagonal) = (pfrd_qlen_greatest_common_left)) /\\ ((((exists ff_h_pfp_greatest_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_greatest_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_greatest_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_greatest_common_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_greatest_common_left)) /\\ exists ff_q_pfp_greatest_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_greatest_common_left = ff_q_pfp_greatest_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_greatest_common_left_productcoefficientscoefficientdiagonal)) * pfrd_qc_greatest_common_left) + (pfc_left_greatest_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_greatest_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_greatest_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_greatest_common_left)=(pfc_index_greatest_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_greatest_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_greatest_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_greatest_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_greatest_common_left_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_greatest_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_greatest_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_greatest_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_greatest_common_left_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_greatest_common_left_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_greatest_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_greatest_common_left_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_greatest_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_greatest_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_greatest_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_greatest_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_greatest_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_greatest_common_left_productcoefficientscoefficientdiagonal)=pfc_left_greatest_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_greatest_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_greatest_common_left_productcoefficientscoefficientsum fs_v_pfc_greatest_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_greatest_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_greatest_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_greatest_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_greatest_common_left_productcoefficientscoefficientsum = fs_q_pfc_greatest_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_greatest_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_greatest_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_greatest_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_greatest_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_greatest_common_left_productcoefficients))) * fs_v_pfc_greatest_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_greatest_common_left_productcoefficientscoefficientsum = fs_q_pfc_greatest_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_greatest_common_left_productcoefficients))) * fs_v_pfc_greatest_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_greatest_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_greatest_common_left_productcoefficients)) -> exists fs_a_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_greatest_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_greatest_common_left_productcoefficientscoefficient = fs_q_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_greatest_common_left_productcoefficientscoefficient) + (fs_a_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_greatest_common_left_productcoefficientscoefficientsum = fs_q_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_greatest_common_left_productcoefficientscoefficientsum = fs_q_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_greatest_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_greatest_common_left_productcoefficientscoefficientresiduebound. pfa_gap_greatest_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_greatest_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_greatest_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_greatest_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_greatest_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_greatest_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_greatest_common_left_productcoefficients) + (p) * pfa_offset_right_greatest_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_greatest_common_left_target pfrep_left_greatest_common_left_target pfrep_right_greatest_common_left_target. ((exists pfrep_position_greatest_common_left_targetfirst. ((pfrep_position_greatest_common_left_targetfirst+S (pfrep_power_greatest_common_left_target)=(pfrd_plen_greatest_common_left)) /\\ ((((exists ff_h_pfp_greatest_common_left_targetfirstentry. ff_h_pfp_greatest_common_left_targetfirstentry + S (pfrep_left_greatest_common_left_target) = S ((S (pfrep_position_greatest_common_left_targetfirst)) * pfrd_pc_greatest_common_left)) /\\ exists ff_q_pfp_greatest_common_left_targetfirstentry. pfrd_pb_greatest_common_left = ff_q_pfp_greatest_common_left_targetfirstentry * S ((S (pfrep_position_greatest_common_left_targetfirst)) * pfrd_pc_greatest_common_left) + (pfrep_left_greatest_common_left_target)))))) \\/ (((exists pfrep_gap_greatest_common_left_targetfirstoutside. pfrep_gap_greatest_common_left_targetfirstoutside+(pfrd_plen_greatest_common_left)=(pfrep_power_greatest_common_left_target)) /\\ (((pfrep_left_greatest_common_left_target)=0))))) -> ((exists pfrep_position_greatest_common_left_targetsecond. ((pfrep_position_greatest_common_left_targetsecond+S (pfrep_power_greatest_common_left_target)=(A)) /\\ ((((exists ff_h_pfp_greatest_common_left_targetsecondentry. ff_h_pfp_greatest_common_left_targetsecondentry + S (pfrep_right_greatest_common_left_target) = S ((S (pfrep_position_greatest_common_left_targetsecond)) * ac)) /\\ exists ff_q_pfp_greatest_common_left_targetsecondentry. ab = ff_q_pfp_greatest_common_left_targetsecondentry * S ((S (pfrep_position_greatest_common_left_targetsecond)) * ac) + (pfrep_right_greatest_common_left_target)))))) \\/ (((exists pfrep_gap_greatest_common_left_targetsecondoutside. pfrep_gap_greatest_common_left_targetsecondoutside+(A)=(pfrep_power_greatest_common_left_target)) /\\ (((pfrep_right_greatest_common_left_target)=0))))) -> pfrep_left_greatest_common_left_target=pfrep_right_greatest_common_left_target))))))) /\\ ((((forall fom_index_pfp_greatest_common_right_canonical. (exists fom_gap_pfp_greatest_common_right_canonical_index_bound. fom_gap_pfp_greatest_common_right_canonical_index_bound + S (fom_index_pfp_greatest_common_right_canonical) = B) -> exists fom_value_pfp_greatest_common_right_canonical. ((((exists fom_beta_height_pfp_greatest_common_right_canonical_entry. fom_beta_height_pfp_greatest_common_right_canonical_entry + S (fom_value_pfp_greatest_common_right_canonical) = S ((S (fom_index_pfp_greatest_common_right_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_greatest_common_right_canonical_entry. bb = fom_beta_quotient_pfp_greatest_common_right_canonical_entry * S ((S (fom_index_pfp_greatest_common_right_canonical)) * bc) + (fom_value_pfp_greatest_common_right_canonical))) /\\ (exists fom_gap_pfp_greatest_common_right_canonical_value_bound. fom_gap_pfp_greatest_common_right_canonical_value_bound + S (fom_value_pfp_greatest_common_right_canonical) = p))) /\\ ((exists pfrd_qb_greatest_common_right pfrd_qc_greatest_common_right pfrd_qlen_greatest_common_right pfrd_pb_greatest_common_right pfrd_pc_greatest_common_right pfrd_plen_greatest_common_right. ((((forall fom_index_pfp_greatest_common_right_productleft. (exists fom_gap_pfp_greatest_common_right_productleft_index_bound. fom_gap_pfp_greatest_common_right_productleft_index_bound + S (fom_index_pfp_greatest_common_right_productleft) = pfrd_qlen_greatest_common_right) -> exists fom_value_pfp_greatest_common_right_productleft. ((((exists fom_beta_height_pfp_greatest_common_right_productleft_entry. fom_beta_height_pfp_greatest_common_right_productleft_entry + S (fom_value_pfp_greatest_common_right_productleft) = S ((S (fom_index_pfp_greatest_common_right_productleft)) * pfrd_qc_greatest_common_right)) /\\ exists fom_beta_quotient_pfp_greatest_common_right_productleft_entry. pfrd_qb_greatest_common_right = fom_beta_quotient_pfp_greatest_common_right_productleft_entry * S ((S (fom_index_pfp_greatest_common_right_productleft)) * pfrd_qc_greatest_common_right) + (fom_value_pfp_greatest_common_right_productleft))) /\\ (exists fom_gap_pfp_greatest_common_right_productleft_value_bound. fom_gap_pfp_greatest_common_right_productleft_value_bound + S (fom_value_pfp_greatest_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_greatest_common_right_productright. (exists fom_gap_pfp_greatest_common_right_productright_index_bound. fom_gap_pfp_greatest_common_right_productright_index_bound + S (fom_index_pfp_greatest_common_right_productright) = D) -> exists fom_value_pfp_greatest_common_right_productright. ((((exists fom_beta_height_pfp_greatest_common_right_productright_entry. fom_beta_height_pfp_greatest_common_right_productright_entry + S (fom_value_pfp_greatest_common_right_productright) = S ((S (fom_index_pfp_greatest_common_right_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_greatest_common_right_productright_entry. db = fom_beta_quotient_pfp_greatest_common_right_productright_entry * S ((S (fom_index_pfp_greatest_common_right_productright)) * dc) + (fom_value_pfp_greatest_common_right_productright))) /\\ (exists fom_gap_pfp_greatest_common_right_productright_value_bound. fom_gap_pfp_greatest_common_right_productright_value_bound + S (fom_value_pfp_greatest_common_right_productright) = p))) /\\ (((((((pfrd_qlen_greatest_common_right)=0 \\/ (D)=0) /\\ (((pfrd_plen_greatest_common_right)=0)))) \\/ (((~((pfrd_qlen_greatest_common_right)=0)) /\\ (((~((D)=0)) /\\ (((pfrd_qlen_greatest_common_right)+(D)=S (pfrd_plen_greatest_common_right)))))))) /\\ ((forall pfc_index_greatest_common_right_productcoefficients. (exists pfa_gap_greatest_common_right_productcoefficientsbound. pfa_gap_greatest_common_right_productcoefficientsbound + S (pfc_index_greatest_common_right_productcoefficients) = (pfrd_plen_greatest_common_right)) -> exists pfc_value_greatest_common_right_productcoefficients. ((((exists ff_h_pfp_greatest_common_right_productcoefficientsentry. ff_h_pfp_greatest_common_right_productcoefficientsentry + S (pfc_value_greatest_common_right_productcoefficients) = S ((S (pfc_index_greatest_common_right_productcoefficients)) * pfrd_pc_greatest_common_right)) /\\ exists ff_q_pfp_greatest_common_right_productcoefficientsentry. pfrd_pb_greatest_common_right = ff_q_pfp_greatest_common_right_productcoefficientsentry * S ((S (pfc_index_greatest_common_right_productcoefficients)) * pfrd_pc_greatest_common_right) + (pfc_value_greatest_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_greatest_common_right_productcoefficientscoefficient pfc_terms_scale_greatest_common_right_productcoefficientscoefficient pfc_natural_sum_greatest_common_right_productcoefficientscoefficient. ((forall pfc_index_greatest_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_greatest_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_greatest_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_greatest_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_greatest_common_right_productcoefficients))) -> exists pfc_value_greatest_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_greatest_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_greatest_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_greatest_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_greatest_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_greatest_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_greatest_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_greatest_common_right_productcoefficientscoefficient = ff_q_pfp_greatest_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_greatest_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_greatest_common_right_productcoefficientscoefficient) + (pfc_value_greatest_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_greatest_common_right_productcoefficientscoefficientdiagonalterm pfc_left_greatest_common_right_productcoefficientscoefficientdiagonalterm pfc_right_greatest_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_greatest_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_greatest_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_greatest_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_greatest_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_greatest_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_greatest_common_right_productcoefficientscoefficientdiagonal) = (pfrd_qlen_greatest_common_right)) /\\ ((((exists ff_h_pfp_greatest_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_greatest_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_greatest_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_greatest_common_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_greatest_common_right)) /\\ exists ff_q_pfp_greatest_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_greatest_common_right = ff_q_pfp_greatest_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_greatest_common_right_productcoefficientscoefficientdiagonal)) * pfrd_qc_greatest_common_right) + (pfc_left_greatest_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_greatest_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_greatest_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_greatest_common_right)=(pfc_index_greatest_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_greatest_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_greatest_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_greatest_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_greatest_common_right_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_greatest_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_greatest_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_greatest_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_greatest_common_right_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_greatest_common_right_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_greatest_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_greatest_common_right_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_greatest_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_greatest_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_greatest_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_greatest_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_greatest_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_greatest_common_right_productcoefficientscoefficientdiagonal)=pfc_left_greatest_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_greatest_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_greatest_common_right_productcoefficientscoefficientsum fs_v_pfc_greatest_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_greatest_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_greatest_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_greatest_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_greatest_common_right_productcoefficientscoefficientsum = fs_q_pfc_greatest_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_greatest_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_greatest_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_greatest_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_greatest_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_greatest_common_right_productcoefficients))) * fs_v_pfc_greatest_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_greatest_common_right_productcoefficientscoefficientsum = fs_q_pfc_greatest_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_greatest_common_right_productcoefficients))) * fs_v_pfc_greatest_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_greatest_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_greatest_common_right_productcoefficients)) -> exists fs_a_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_greatest_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_greatest_common_right_productcoefficientscoefficient = fs_q_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_greatest_common_right_productcoefficientscoefficient) + (fs_a_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_greatest_common_right_productcoefficientscoefficientsum = fs_q_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_greatest_common_right_productcoefficientscoefficientsum = fs_q_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_greatest_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_greatest_common_right_productcoefficientscoefficientresiduebound. pfa_gap_greatest_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_greatest_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_greatest_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_greatest_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_greatest_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_greatest_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_greatest_common_right_productcoefficients) + (p) * pfa_offset_right_greatest_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_greatest_common_right_target pfrep_left_greatest_common_right_target pfrep_right_greatest_common_right_target. ((exists pfrep_position_greatest_common_right_targetfirst. ((pfrep_position_greatest_common_right_targetfirst+S (pfrep_power_greatest_common_right_target)=(pfrd_plen_greatest_common_right)) /\\ ((((exists ff_h_pfp_greatest_common_right_targetfirstentry. ff_h_pfp_greatest_common_right_targetfirstentry + S (pfrep_left_greatest_common_right_target) = S ((S (pfrep_position_greatest_common_right_targetfirst)) * pfrd_pc_greatest_common_right)) /\\ exists ff_q_pfp_greatest_common_right_targetfirstentry. pfrd_pb_greatest_common_right = ff_q_pfp_greatest_common_right_targetfirstentry * S ((S (pfrep_position_greatest_common_right_targetfirst)) * pfrd_pc_greatest_common_right) + (pfrep_left_greatest_common_right_target)))))) \\/ (((exists pfrep_gap_greatest_common_right_targetfirstoutside. pfrep_gap_greatest_common_right_targetfirstoutside+(pfrd_plen_greatest_common_right)=(pfrep_power_greatest_common_right_target)) /\\ (((pfrep_left_greatest_common_right_target)=0))))) -> ((exists pfrep_position_greatest_common_right_targetsecond. ((pfrep_position_greatest_common_right_targetsecond+S (pfrep_power_greatest_common_right_target)=(B)) /\\ ((((exists ff_h_pfp_greatest_common_right_targetsecondentry. ff_h_pfp_greatest_common_right_targetsecondentry + S (pfrep_right_greatest_common_right_target) = S ((S (pfrep_position_greatest_common_right_targetsecond)) * bc)) /\\ exists ff_q_pfp_greatest_common_right_targetsecondentry. bb = ff_q_pfp_greatest_common_right_targetsecondentry * S ((S (pfrep_position_greatest_common_right_targetsecond)) * bc) + (pfrep_right_greatest_common_right_target)))))) \\/ (((exists pfrep_gap_greatest_common_right_targetsecondoutside. pfrep_gap_greatest_common_right_targetsecondoutside+(B)=(pfrep_power_greatest_common_right_target)) /\\ (((pfrep_right_greatest_common_right_target)=0))))) -> pfrep_left_greatest_common_right_target=pfrep_right_greatest_common_right_target)))))))))) -> (exists pfbz_left_code_greatest_bezout pfbz_left_scale_greatest_bezout pfbz_left_length_greatest_bezout pfbz_right_code_greatest_bezout pfbz_right_scale_greatest_bezout pfbz_right_length_greatest_bezout. ((((forall fom_index_pfp_greatest_bezout_left_productleft. (exists fom_gap_pfp_greatest_bezout_left_productleft_index_bound. fom_gap_pfp_greatest_bezout_left_productleft_index_bound + S (fom_index_pfp_greatest_bezout_left_productleft) = U) -> exists fom_value_pfp_greatest_bezout_left_productleft. ((((exists fom_beta_height_pfp_greatest_bezout_left_productleft_entry. fom_beta_height_pfp_greatest_bezout_left_productleft_entry + S (fom_value_pfp_greatest_bezout_left_productleft) = S ((S (fom_index_pfp_greatest_bezout_left_productleft)) * uc)) /\\ exists fom_beta_quotient_pfp_greatest_bezout_left_productleft_entry. ub = fom_beta_quotient_pfp_greatest_bezout_left_productleft_entry * S ((S (fom_index_pfp_greatest_bezout_left_productleft)) * uc) + (fom_value_pfp_greatest_bezout_left_productleft))) /\\ (exists fom_gap_pfp_greatest_bezout_left_productleft_value_bound. fom_gap_pfp_greatest_bezout_left_productleft_value_bound + S (fom_value_pfp_greatest_bezout_left_productleft) = p))) /\\ (((forall fom_index_pfp_greatest_bezout_left_productright. (exists fom_gap_pfp_greatest_bezout_left_productright_index_bound. fom_gap_pfp_greatest_bezout_left_productright_index_bound + S (fom_index_pfp_greatest_bezout_left_productright) = A) -> exists fom_value_pfp_greatest_bezout_left_productright. ((((exists fom_beta_height_pfp_greatest_bezout_left_productright_entry. fom_beta_height_pfp_greatest_bezout_left_productright_entry + S (fom_value_pfp_greatest_bezout_left_productright) = S ((S (fom_index_pfp_greatest_bezout_left_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_greatest_bezout_left_productright_entry. ab = fom_beta_quotient_pfp_greatest_bezout_left_productright_entry * S ((S (fom_index_pfp_greatest_bezout_left_productright)) * ac) + (fom_value_pfp_greatest_bezout_left_productright))) /\\ (exists fom_gap_pfp_greatest_bezout_left_productright_value_bound. fom_gap_pfp_greatest_bezout_left_productright_value_bound + S (fom_value_pfp_greatest_bezout_left_productright) = p))) /\\ (((((((U)=0 \\/ (A)=0) /\\ (((pfbz_left_length_greatest_bezout)=0)))) \\/ (((~((U)=0)) /\\ (((~((A)=0)) /\\ (((U)+(A)=S (pfbz_left_length_greatest_bezout)))))))) /\\ ((forall pfc_index_greatest_bezout_left_productcoefficients. (exists pfa_gap_greatest_bezout_left_productcoefficientsbound. pfa_gap_greatest_bezout_left_productcoefficientsbound + S (pfc_index_greatest_bezout_left_productcoefficients) = (pfbz_left_length_greatest_bezout)) -> exists pfc_value_greatest_bezout_left_productcoefficients. ((((exists ff_h_pfp_greatest_bezout_left_productcoefficientsentry. ff_h_pfp_greatest_bezout_left_productcoefficientsentry + S (pfc_value_greatest_bezout_left_productcoefficients) = S ((S (pfc_index_greatest_bezout_left_productcoefficients)) * pfbz_left_scale_greatest_bezout)) /\\ exists ff_q_pfp_greatest_bezout_left_productcoefficientsentry. pfbz_left_code_greatest_bezout = ff_q_pfp_greatest_bezout_left_productcoefficientsentry * S ((S (pfc_index_greatest_bezout_left_productcoefficients)) * pfbz_left_scale_greatest_bezout) + (pfc_value_greatest_bezout_left_productcoefficients))) /\\ ((exists pfc_terms_code_greatest_bezout_left_productcoefficientscoefficient pfc_terms_scale_greatest_bezout_left_productcoefficientscoefficient pfc_natural_sum_greatest_bezout_left_productcoefficientscoefficient. ((forall pfc_index_greatest_bezout_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_greatest_bezout_left_productcoefficientscoefficientdiagonalbound. pfa_gap_greatest_bezout_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_greatest_bezout_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_greatest_bezout_left_productcoefficients))) -> exists pfc_value_greatest_bezout_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_greatest_bezout_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_greatest_bezout_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_greatest_bezout_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_greatest_bezout_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_greatest_bezout_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_greatest_bezout_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_greatest_bezout_left_productcoefficientscoefficient = ff_q_pfp_greatest_bezout_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_greatest_bezout_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_greatest_bezout_left_productcoefficientscoefficient) + (pfc_value_greatest_bezout_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_greatest_bezout_left_productcoefficientscoefficientdiagonalterm pfc_left_greatest_bezout_left_productcoefficientscoefficientdiagonalterm pfc_right_greatest_bezout_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_greatest_bezout_left_productcoefficientscoefficientdiagonal)+pfc_complement_greatest_bezout_left_productcoefficientscoefficientdiagonalterm=(pfc_index_greatest_bezout_left_productcoefficients)) /\\ ((((((exists pfa_gap_greatest_bezout_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_greatest_bezout_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_greatest_bezout_left_productcoefficientscoefficientdiagonal) = (U)) /\\ ((((exists ff_h_pfp_greatest_bezout_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_greatest_bezout_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_greatest_bezout_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_greatest_bezout_left_productcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_greatest_bezout_left_productcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_greatest_bezout_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_greatest_bezout_left_productcoefficientscoefficientdiagonal)) * uc) + (pfc_left_greatest_bezout_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_greatest_bezout_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_greatest_bezout_left_productcoefficientscoefficientdiagonaltermleftoutside+(U)=(pfc_index_greatest_bezout_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_greatest_bezout_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_greatest_bezout_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_greatest_bezout_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_greatest_bezout_left_productcoefficientscoefficientdiagonalterm) = (A)) /\\ ((((exists ff_h_pfp_greatest_bezout_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_greatest_bezout_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_greatest_bezout_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_greatest_bezout_left_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_greatest_bezout_left_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_greatest_bezout_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_greatest_bezout_left_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_greatest_bezout_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_greatest_bezout_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_greatest_bezout_left_productcoefficientscoefficientdiagonaltermrightoutside+(A)=(pfc_complement_greatest_bezout_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_greatest_bezout_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_greatest_bezout_left_productcoefficientscoefficientdiagonal)=pfc_left_greatest_bezout_left_productcoefficientscoefficientdiagonalterm*pfc_right_greatest_bezout_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_greatest_bezout_left_productcoefficientscoefficientsum fs_v_pfc_greatest_bezout_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_greatest_bezout_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_greatest_bezout_left_productcoefficientscoefficientsum = fs_q_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_greatest_bezout_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_greatest_bezout_left_productcoefficientscoefficient) = S ((S (S (pfc_index_greatest_bezout_left_productcoefficients))) * fs_v_pfc_greatest_bezout_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_greatest_bezout_left_productcoefficientscoefficientsum = fs_q_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_greatest_bezout_left_productcoefficients))) * fs_v_pfc_greatest_bezout_left_productcoefficientscoefficientsum) + (pfc_natural_sum_greatest_bezout_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_greatest_bezout_left_productcoefficients)) -> exists fs_a_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_greatest_bezout_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_greatest_bezout_left_productcoefficientscoefficient = fs_q_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_greatest_bezout_left_productcoefficientscoefficient) + (fs_a_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_bezout_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_greatest_bezout_left_productcoefficientscoefficientsum = fs_q_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_bezout_left_productcoefficientscoefficientsum) + (fs_r_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_bezout_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_greatest_bezout_left_productcoefficientscoefficientsum = fs_q_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_bezout_left_productcoefficientscoefficientsum) + (fs_s_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_greatest_bezout_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_greatest_bezout_left_productcoefficientscoefficientresiduebound. pfa_gap_greatest_bezout_left_productcoefficientscoefficientresiduebound + S (pfc_value_greatest_bezout_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_greatest_bezout_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_greatest_bezout_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_greatest_bezout_left_productcoefficientscoefficient) + (p) * pfa_offset_left_greatest_bezout_left_productcoefficientscoefficientresiduecongruence = (pfc_value_greatest_bezout_left_productcoefficients) + (p) * pfa_offset_right_greatest_bezout_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_greatest_bezout_right_productleft. (exists fom_gap_pfp_greatest_bezout_right_productleft_index_bound. fom_gap_pfp_greatest_bezout_right_productleft_index_bound + S (fom_index_pfp_greatest_bezout_right_productleft) = V) -> exists fom_value_pfp_greatest_bezout_right_productleft. ((((exists fom_beta_height_pfp_greatest_bezout_right_productleft_entry. fom_beta_height_pfp_greatest_bezout_right_productleft_entry + S (fom_value_pfp_greatest_bezout_right_productleft) = S ((S (fom_index_pfp_greatest_bezout_right_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_greatest_bezout_right_productleft_entry. vb = fom_beta_quotient_pfp_greatest_bezout_right_productleft_entry * S ((S (fom_index_pfp_greatest_bezout_right_productleft)) * vc) + (fom_value_pfp_greatest_bezout_right_productleft))) /\\ (exists fom_gap_pfp_greatest_bezout_right_productleft_value_bound. fom_gap_pfp_greatest_bezout_right_productleft_value_bound + S (fom_value_pfp_greatest_bezout_right_productleft) = p))) /\\ (((forall fom_index_pfp_greatest_bezout_right_productright. (exists fom_gap_pfp_greatest_bezout_right_productright_index_bound. fom_gap_pfp_greatest_bezout_right_productright_index_bound + S (fom_index_pfp_greatest_bezout_right_productright) = B) -> exists fom_value_pfp_greatest_bezout_right_productright. ((((exists fom_beta_height_pfp_greatest_bezout_right_productright_entry. fom_beta_height_pfp_greatest_bezout_right_productright_entry + S (fom_value_pfp_greatest_bezout_right_productright) = S ((S (fom_index_pfp_greatest_bezout_right_productright)) * bc)) /\\ exists fom_beta_quotient_pfp_greatest_bezout_right_productright_entry. bb = fom_beta_quotient_pfp_greatest_bezout_right_productright_entry * S ((S (fom_index_pfp_greatest_bezout_right_productright)) * bc) + (fom_value_pfp_greatest_bezout_right_productright))) /\\ (exists fom_gap_pfp_greatest_bezout_right_productright_value_bound. fom_gap_pfp_greatest_bezout_right_productright_value_bound + S (fom_value_pfp_greatest_bezout_right_productright) = p))) /\\ (((((((V)=0 \\/ (B)=0) /\\ (((pfbz_right_length_greatest_bezout)=0)))) \\/ (((~((V)=0)) /\\ (((~((B)=0)) /\\ (((V)+(B)=S (pfbz_right_length_greatest_bezout)))))))) /\\ ((forall pfc_index_greatest_bezout_right_productcoefficients. (exists pfa_gap_greatest_bezout_right_productcoefficientsbound. pfa_gap_greatest_bezout_right_productcoefficientsbound + S (pfc_index_greatest_bezout_right_productcoefficients) = (pfbz_right_length_greatest_bezout)) -> exists pfc_value_greatest_bezout_right_productcoefficients. ((((exists ff_h_pfp_greatest_bezout_right_productcoefficientsentry. ff_h_pfp_greatest_bezout_right_productcoefficientsentry + S (pfc_value_greatest_bezout_right_productcoefficients) = S ((S (pfc_index_greatest_bezout_right_productcoefficients)) * pfbz_right_scale_greatest_bezout)) /\\ exists ff_q_pfp_greatest_bezout_right_productcoefficientsentry. pfbz_right_code_greatest_bezout = ff_q_pfp_greatest_bezout_right_productcoefficientsentry * S ((S (pfc_index_greatest_bezout_right_productcoefficients)) * pfbz_right_scale_greatest_bezout) + (pfc_value_greatest_bezout_right_productcoefficients))) /\\ ((exists pfc_terms_code_greatest_bezout_right_productcoefficientscoefficient pfc_terms_scale_greatest_bezout_right_productcoefficientscoefficient pfc_natural_sum_greatest_bezout_right_productcoefficientscoefficient. ((forall pfc_index_greatest_bezout_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_greatest_bezout_right_productcoefficientscoefficientdiagonalbound. pfa_gap_greatest_bezout_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_greatest_bezout_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_greatest_bezout_right_productcoefficients))) -> exists pfc_value_greatest_bezout_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_greatest_bezout_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_greatest_bezout_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_greatest_bezout_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_greatest_bezout_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_greatest_bezout_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_greatest_bezout_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_greatest_bezout_right_productcoefficientscoefficient = ff_q_pfp_greatest_bezout_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_greatest_bezout_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_greatest_bezout_right_productcoefficientscoefficient) + (pfc_value_greatest_bezout_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_greatest_bezout_right_productcoefficientscoefficientdiagonalterm pfc_left_greatest_bezout_right_productcoefficientscoefficientdiagonalterm pfc_right_greatest_bezout_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_greatest_bezout_right_productcoefficientscoefficientdiagonal)+pfc_complement_greatest_bezout_right_productcoefficientscoefficientdiagonalterm=(pfc_index_greatest_bezout_right_productcoefficients)) /\\ ((((((exists pfa_gap_greatest_bezout_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_greatest_bezout_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_greatest_bezout_right_productcoefficientscoefficientdiagonal) = (V)) /\\ ((((exists ff_h_pfp_greatest_bezout_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_greatest_bezout_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_greatest_bezout_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_greatest_bezout_right_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_greatest_bezout_right_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_greatest_bezout_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_greatest_bezout_right_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_greatest_bezout_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_greatest_bezout_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_greatest_bezout_right_productcoefficientscoefficientdiagonaltermleftoutside+(V)=(pfc_index_greatest_bezout_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_greatest_bezout_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_greatest_bezout_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_greatest_bezout_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_greatest_bezout_right_productcoefficientscoefficientdiagonalterm) = (B)) /\\ ((((exists ff_h_pfp_greatest_bezout_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_greatest_bezout_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_greatest_bezout_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_greatest_bezout_right_productcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_greatest_bezout_right_productcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_greatest_bezout_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_greatest_bezout_right_productcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_greatest_bezout_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_greatest_bezout_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_greatest_bezout_right_productcoefficientscoefficientdiagonaltermrightoutside+(B)=(pfc_complement_greatest_bezout_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_greatest_bezout_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_greatest_bezout_right_productcoefficientscoefficientdiagonal)=pfc_left_greatest_bezout_right_productcoefficientscoefficientdiagonalterm*pfc_right_greatest_bezout_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_greatest_bezout_right_productcoefficientscoefficientsum fs_v_pfc_greatest_bezout_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_greatest_bezout_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_greatest_bezout_right_productcoefficientscoefficientsum = fs_q_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_greatest_bezout_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_greatest_bezout_right_productcoefficientscoefficient) = S ((S (S (pfc_index_greatest_bezout_right_productcoefficients))) * fs_v_pfc_greatest_bezout_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_greatest_bezout_right_productcoefficientscoefficientsum = fs_q_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_greatest_bezout_right_productcoefficients))) * fs_v_pfc_greatest_bezout_right_productcoefficientscoefficientsum) + (pfc_natural_sum_greatest_bezout_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_greatest_bezout_right_productcoefficients)) -> exists fs_a_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_greatest_bezout_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_greatest_bezout_right_productcoefficientscoefficient = fs_q_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_greatest_bezout_right_productcoefficientscoefficient) + (fs_a_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_bezout_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_greatest_bezout_right_productcoefficientscoefficientsum = fs_q_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_bezout_right_productcoefficientscoefficientsum) + (fs_r_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_bezout_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_greatest_bezout_right_productcoefficientscoefficientsum = fs_q_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_bezout_right_productcoefficientscoefficientsum) + (fs_s_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_greatest_bezout_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_greatest_bezout_right_productcoefficientscoefficientresiduebound. pfa_gap_greatest_bezout_right_productcoefficientscoefficientresiduebound + S (pfc_value_greatest_bezout_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_greatest_bezout_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_greatest_bezout_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_greatest_bezout_right_productcoefficientscoefficient) + (p) * pfa_offset_left_greatest_bezout_right_productcoefficientscoefficientresiduecongruence = (pfc_value_greatest_bezout_right_productcoefficients) + (p) * pfa_offset_right_greatest_bezout_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_greatest_bezout_sum_left_bounded. (exists fom_gap_pfp_greatest_bezout_sum_left_bounded_index_bound. fom_gap_pfp_greatest_bezout_sum_left_bounded_index_bound + S (fom_index_pfp_greatest_bezout_sum_left_bounded) = pfbz_left_length_greatest_bezout) -> exists fom_value_pfp_greatest_bezout_sum_left_bounded. ((((exists fom_beta_height_pfp_greatest_bezout_sum_left_bounded_entry. fom_beta_height_pfp_greatest_bezout_sum_left_bounded_entry + S (fom_value_pfp_greatest_bezout_sum_left_bounded) = S ((S (fom_index_pfp_greatest_bezout_sum_left_bounded)) * pfbz_left_scale_greatest_bezout)) /\\ exists fom_beta_quotient_pfp_greatest_bezout_sum_left_bounded_entry. pfbz_left_code_greatest_bezout = fom_beta_quotient_pfp_greatest_bezout_sum_left_bounded_entry * S ((S (fom_index_pfp_greatest_bezout_sum_left_bounded)) * pfbz_left_scale_greatest_bezout) + (fom_value_pfp_greatest_bezout_sum_left_bounded))) /\\ (exists fom_gap_pfp_greatest_bezout_sum_left_bounded_value_bound. fom_gap_pfp_greatest_bezout_sum_left_bounded_value_bound + S (fom_value_pfp_greatest_bezout_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_greatest_bezout_sum_right_bounded. (exists fom_gap_pfp_greatest_bezout_sum_right_bounded_index_bound. fom_gap_pfp_greatest_bezout_sum_right_bounded_index_bound + S (fom_index_pfp_greatest_bezout_sum_right_bounded) = pfbz_right_length_greatest_bezout) -> exists fom_value_pfp_greatest_bezout_sum_right_bounded. ((((exists fom_beta_height_pfp_greatest_bezout_sum_right_bounded_entry. fom_beta_height_pfp_greatest_bezout_sum_right_bounded_entry + S (fom_value_pfp_greatest_bezout_sum_right_bounded) = S ((S (fom_index_pfp_greatest_bezout_sum_right_bounded)) * pfbz_right_scale_greatest_bezout)) /\\ exists fom_beta_quotient_pfp_greatest_bezout_sum_right_bounded_entry. pfbz_right_code_greatest_bezout = fom_beta_quotient_pfp_greatest_bezout_sum_right_bounded_entry * S ((S (fom_index_pfp_greatest_bezout_sum_right_bounded)) * pfbz_right_scale_greatest_bezout) + (fom_value_pfp_greatest_bezout_sum_right_bounded))) /\\ (exists fom_gap_pfp_greatest_bezout_sum_right_bounded_value_bound. fom_gap_pfp_greatest_bezout_sum_right_bounded_value_bound + S (fom_value_pfp_greatest_bezout_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_greatest_bezout_sum_result_bounded. (exists fom_gap_pfp_greatest_bezout_sum_result_bounded_index_bound. fom_gap_pfp_greatest_bezout_sum_result_bounded_index_bound + S (fom_index_pfp_greatest_bezout_sum_result_bounded) = G) -> exists fom_value_pfp_greatest_bezout_sum_result_bounded. ((((exists fom_beta_height_pfp_greatest_bezout_sum_result_bounded_entry. fom_beta_height_pfp_greatest_bezout_sum_result_bounded_entry + S (fom_value_pfp_greatest_bezout_sum_result_bounded) = S ((S (fom_index_pfp_greatest_bezout_sum_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_greatest_bezout_sum_result_bounded_entry. gb = fom_beta_quotient_pfp_greatest_bezout_sum_result_bounded_entry * S ((S (fom_index_pfp_greatest_bezout_sum_result_bounded)) * gc) + (fom_value_pfp_greatest_bezout_sum_result_bounded))) /\\ (exists fom_gap_pfp_greatest_bezout_sum_result_bounded_value_bound. fom_gap_pfp_greatest_bezout_sum_result_bounded_value_bound + S (fom_value_pfp_greatest_bezout_sum_result_bounded) = p))) /\\ ((exists pfaa_left_b_greatest_bezout_sum pfaa_left_c_greatest_bezout_sum pfaa_right_b_greatest_bezout_sum pfaa_right_c_greatest_bezout_sum pfaa_sum_b_greatest_bezout_sum pfaa_sum_c_greatest_bezout_sum pfaa_length_greatest_bezout_sum. ((((forall pfrep_power_greatest_bezout_sum_witness_common_left pfrep_left_greatest_bezout_sum_witness_common_left pfrep_right_greatest_bezout_sum_witness_common_left. ((exists pfrep_position_greatest_bezout_sum_witness_common_leftfirst. ((pfrep_position_greatest_bezout_sum_witness_common_leftfirst+S (pfrep_power_greatest_bezout_sum_witness_common_left)=(pfbz_left_length_greatest_bezout)) /\\ ((((exists ff_h_pfp_greatest_bezout_sum_witness_common_leftfirstentry. ff_h_pfp_greatest_bezout_sum_witness_common_leftfirstentry + S (pfrep_left_greatest_bezout_sum_witness_common_left) = S ((S (pfrep_position_greatest_bezout_sum_witness_common_leftfirst)) * pfbz_left_scale_greatest_bezout)) /\\ exists ff_q_pfp_greatest_bezout_sum_witness_common_leftfirstentry. pfbz_left_code_greatest_bezout = ff_q_pfp_greatest_bezout_sum_witness_common_leftfirstentry * S ((S (pfrep_position_greatest_bezout_sum_witness_common_leftfirst)) * pfbz_left_scale_greatest_bezout) + (pfrep_left_greatest_bezout_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_greatest_bezout_sum_witness_common_leftfirstoutside. pfrep_gap_greatest_bezout_sum_witness_common_leftfirstoutside+(pfbz_left_length_greatest_bezout)=(pfrep_power_greatest_bezout_sum_witness_common_left)) /\\ (((pfrep_left_greatest_bezout_sum_witness_common_left)=0))))) -> ((exists pfrep_position_greatest_bezout_sum_witness_common_leftsecond. ((pfrep_position_greatest_bezout_sum_witness_common_leftsecond+S (pfrep_power_greatest_bezout_sum_witness_common_left)=(pfaa_length_greatest_bezout_sum)) /\\ ((((exists ff_h_pfp_greatest_bezout_sum_witness_common_leftsecondentry. ff_h_pfp_greatest_bezout_sum_witness_common_leftsecondentry + S (pfrep_right_greatest_bezout_sum_witness_common_left) = S ((S (pfrep_position_greatest_bezout_sum_witness_common_leftsecond)) * pfaa_left_c_greatest_bezout_sum)) /\\ exists ff_q_pfp_greatest_bezout_sum_witness_common_leftsecondentry. pfaa_left_b_greatest_bezout_sum = ff_q_pfp_greatest_bezout_sum_witness_common_leftsecondentry * S ((S (pfrep_position_greatest_bezout_sum_witness_common_leftsecond)) * pfaa_left_c_greatest_bezout_sum) + (pfrep_right_greatest_bezout_sum_witness_common_left)))))) \\/ (((exists pfrep_gap_greatest_bezout_sum_witness_common_leftsecondoutside. pfrep_gap_greatest_bezout_sum_witness_common_leftsecondoutside+(pfaa_length_greatest_bezout_sum)=(pfrep_power_greatest_bezout_sum_witness_common_left)) /\\ (((pfrep_right_greatest_bezout_sum_witness_common_left)=0))))) -> pfrep_left_greatest_bezout_sum_witness_common_left=pfrep_right_greatest_bezout_sum_witness_common_left) /\\ ((forall pfrep_power_greatest_bezout_sum_witness_common_right pfrep_left_greatest_bezout_sum_witness_common_right pfrep_right_greatest_bezout_sum_witness_common_right. ((exists pfrep_position_greatest_bezout_sum_witness_common_rightfirst. ((pfrep_position_greatest_bezout_sum_witness_common_rightfirst+S (pfrep_power_greatest_bezout_sum_witness_common_right)=(pfbz_right_length_greatest_bezout)) /\\ ((((exists ff_h_pfp_greatest_bezout_sum_witness_common_rightfirstentry. ff_h_pfp_greatest_bezout_sum_witness_common_rightfirstentry + S (pfrep_left_greatest_bezout_sum_witness_common_right) = S ((S (pfrep_position_greatest_bezout_sum_witness_common_rightfirst)) * pfbz_right_scale_greatest_bezout)) /\\ exists ff_q_pfp_greatest_bezout_sum_witness_common_rightfirstentry. pfbz_right_code_greatest_bezout = ff_q_pfp_greatest_bezout_sum_witness_common_rightfirstentry * S ((S (pfrep_position_greatest_bezout_sum_witness_common_rightfirst)) * pfbz_right_scale_greatest_bezout) + (pfrep_left_greatest_bezout_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_greatest_bezout_sum_witness_common_rightfirstoutside. pfrep_gap_greatest_bezout_sum_witness_common_rightfirstoutside+(pfbz_right_length_greatest_bezout)=(pfrep_power_greatest_bezout_sum_witness_common_right)) /\\ (((pfrep_left_greatest_bezout_sum_witness_common_right)=0))))) -> ((exists pfrep_position_greatest_bezout_sum_witness_common_rightsecond. ((pfrep_position_greatest_bezout_sum_witness_common_rightsecond+S (pfrep_power_greatest_bezout_sum_witness_common_right)=(pfaa_length_greatest_bezout_sum)) /\\ ((((exists ff_h_pfp_greatest_bezout_sum_witness_common_rightsecondentry. ff_h_pfp_greatest_bezout_sum_witness_common_rightsecondentry + S (pfrep_right_greatest_bezout_sum_witness_common_right) = S ((S (pfrep_position_greatest_bezout_sum_witness_common_rightsecond)) * pfaa_right_c_greatest_bezout_sum)) /\\ exists ff_q_pfp_greatest_bezout_sum_witness_common_rightsecondentry. pfaa_right_b_greatest_bezout_sum = ff_q_pfp_greatest_bezout_sum_witness_common_rightsecondentry * S ((S (pfrep_position_greatest_bezout_sum_witness_common_rightsecond)) * pfaa_right_c_greatest_bezout_sum) + (pfrep_right_greatest_bezout_sum_witness_common_right)))))) \\/ (((exists pfrep_gap_greatest_bezout_sum_witness_common_rightsecondoutside. pfrep_gap_greatest_bezout_sum_witness_common_rightsecondoutside+(pfaa_length_greatest_bezout_sum)=(pfrep_power_greatest_bezout_sum_witness_common_right)) /\\ (((pfrep_right_greatest_bezout_sum_witness_common_right)=0))))) -> pfrep_left_greatest_bezout_sum_witness_common_right=pfrep_right_greatest_bezout_sum_witness_common_right)))) /\\ (((forall pfp_index_greatest_bezout_sum_witness_operation. (exists pfa_gap_greatest_bezout_sum_witness_operationindex. pfa_gap_greatest_bezout_sum_witness_operationindex + S (pfp_index_greatest_bezout_sum_witness_operation) = (pfaa_length_greatest_bezout_sum)) -> exists pfp_left_greatest_bezout_sum_witness_operation pfp_right_greatest_bezout_sum_witness_operation pfp_value_greatest_bezout_sum_witness_operation. ((((exists ff_h_pfp_greatest_bezout_sum_witness_operationleft. ff_h_pfp_greatest_bezout_sum_witness_operationleft + S (pfp_left_greatest_bezout_sum_witness_operation) = S ((S (pfp_index_greatest_bezout_sum_witness_operation)) * pfaa_left_c_greatest_bezout_sum)) /\\ exists ff_q_pfp_greatest_bezout_sum_witness_operationleft. pfaa_left_b_greatest_bezout_sum = ff_q_pfp_greatest_bezout_sum_witness_operationleft * S ((S (pfp_index_greatest_bezout_sum_witness_operation)) * pfaa_left_c_greatest_bezout_sum) + (pfp_left_greatest_bezout_sum_witness_operation))) /\\ (((((exists ff_h_pfp_greatest_bezout_sum_witness_operationright. ff_h_pfp_greatest_bezout_sum_witness_operationright + S (pfp_right_greatest_bezout_sum_witness_operation) = S ((S (pfp_index_greatest_bezout_sum_witness_operation)) * pfaa_right_c_greatest_bezout_sum)) /\\ exists ff_q_pfp_greatest_bezout_sum_witness_operationright. pfaa_right_b_greatest_bezout_sum = ff_q_pfp_greatest_bezout_sum_witness_operationright * S ((S (pfp_index_greatest_bezout_sum_witness_operation)) * pfaa_right_c_greatest_bezout_sum) + (pfp_right_greatest_bezout_sum_witness_operation))) /\\ (((((exists ff_h_pfp_greatest_bezout_sum_witness_operationtarget. ff_h_pfp_greatest_bezout_sum_witness_operationtarget + S (pfp_value_greatest_bezout_sum_witness_operation) = S ((S (pfp_index_greatest_bezout_sum_witness_operation)) * pfaa_sum_c_greatest_bezout_sum)) /\\ exists ff_q_pfp_greatest_bezout_sum_witness_operationtarget. pfaa_sum_b_greatest_bezout_sum = ff_q_pfp_greatest_bezout_sum_witness_operationtarget * S ((S (pfp_index_greatest_bezout_sum_witness_operation)) * pfaa_sum_c_greatest_bezout_sum) + (pfp_value_greatest_bezout_sum_witness_operation))) /\\ ((((exists pfa_gap_greatest_bezout_sum_witness_operationoperationleft. pfa_gap_greatest_bezout_sum_witness_operationoperationleft + S (pfp_left_greatest_bezout_sum_witness_operation) = (p)) /\\ (((exists pfa_gap_greatest_bezout_sum_witness_operationoperationright. pfa_gap_greatest_bezout_sum_witness_operationoperationright + S (pfp_right_greatest_bezout_sum_witness_operation) = (p)) /\\ ((((exists pfa_gap_greatest_bezout_sum_witness_operationoperationresultbound. pfa_gap_greatest_bezout_sum_witness_operationoperationresultbound + S (pfp_value_greatest_bezout_sum_witness_operation) = (p)) /\\ ((exists pfa_offset_left_greatest_bezout_sum_witness_operationoperationresultcongruence pfa_offset_right_greatest_bezout_sum_witness_operationoperationresultcongruence. ((pfp_left_greatest_bezout_sum_witness_operation) + (pfp_right_greatest_bezout_sum_witness_operation)) + (p) * pfa_offset_left_greatest_bezout_sum_witness_operationoperationresultcongruence = (pfp_value_greatest_bezout_sum_witness_operation) + (p) * pfa_offset_right_greatest_bezout_sum_witness_operationoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_greatest_bezout_sum_witness_output pfrep_left_greatest_bezout_sum_witness_output pfrep_right_greatest_bezout_sum_witness_output. ((exists pfrep_position_greatest_bezout_sum_witness_outputfirst. ((pfrep_position_greatest_bezout_sum_witness_outputfirst+S (pfrep_power_greatest_bezout_sum_witness_output)=(pfaa_length_greatest_bezout_sum)) /\\ ((((exists ff_h_pfp_greatest_bezout_sum_witness_outputfirstentry. ff_h_pfp_greatest_bezout_sum_witness_outputfirstentry + S (pfrep_left_greatest_bezout_sum_witness_output) = S ((S (pfrep_position_greatest_bezout_sum_witness_outputfirst)) * pfaa_sum_c_greatest_bezout_sum)) /\\ exists ff_q_pfp_greatest_bezout_sum_witness_outputfirstentry. pfaa_sum_b_greatest_bezout_sum = ff_q_pfp_greatest_bezout_sum_witness_outputfirstentry * S ((S (pfrep_position_greatest_bezout_sum_witness_outputfirst)) * pfaa_sum_c_greatest_bezout_sum) + (pfrep_left_greatest_bezout_sum_witness_output)))))) \\/ (((exists pfrep_gap_greatest_bezout_sum_witness_outputfirstoutside. pfrep_gap_greatest_bezout_sum_witness_outputfirstoutside+(pfaa_length_greatest_bezout_sum)=(pfrep_power_greatest_bezout_sum_witness_output)) /\\ (((pfrep_left_greatest_bezout_sum_witness_output)=0))))) -> ((exists pfrep_position_greatest_bezout_sum_witness_outputsecond. ((pfrep_position_greatest_bezout_sum_witness_outputsecond+S (pfrep_power_greatest_bezout_sum_witness_output)=(G)) /\\ ((((exists ff_h_pfp_greatest_bezout_sum_witness_outputsecondentry. ff_h_pfp_greatest_bezout_sum_witness_outputsecondentry + S (pfrep_right_greatest_bezout_sum_witness_output) = S ((S (pfrep_position_greatest_bezout_sum_witness_outputsecond)) * gc)) /\\ exists ff_q_pfp_greatest_bezout_sum_witness_outputsecondentry. gb = ff_q_pfp_greatest_bezout_sum_witness_outputsecondentry * S ((S (pfrep_position_greatest_bezout_sum_witness_outputsecond)) * gc) + (pfrep_right_greatest_bezout_sum_witness_output)))))) \\/ (((exists pfrep_gap_greatest_bezout_sum_witness_outputsecondoutside. pfrep_gap_greatest_bezout_sum_witness_outputsecondoutside+(G)=(pfrep_power_greatest_bezout_sum_witness_output)) /\\ (((pfrep_right_greatest_bezout_sum_witness_output)=0))))) -> pfrep_left_greatest_bezout_sum_witness_output=pfrep_right_greatest_bezout_sum_witness_output)))))))))))))))))) -> (((forall fom_index_pfp_greatest_result_canonical. (exists fom_gap_pfp_greatest_result_canonical_index_bound. fom_gap_pfp_greatest_result_canonical_index_bound + S (fom_index_pfp_greatest_result_canonical) = G) -> exists fom_value_pfp_greatest_result_canonical. ((((exists fom_beta_height_pfp_greatest_result_canonical_entry. fom_beta_height_pfp_greatest_result_canonical_entry + S (fom_value_pfp_greatest_result_canonical) = S ((S (fom_index_pfp_greatest_result_canonical)) * gc)) /\\ exists fom_beta_quotient_pfp_greatest_result_canonical_entry. gb = fom_beta_quotient_pfp_greatest_result_canonical_entry * S ((S (fom_index_pfp_greatest_result_canonical)) * gc) + (fom_value_pfp_greatest_result_canonical))) /\\ (exists fom_gap_pfp_greatest_result_canonical_value_bound. fom_gap_pfp_greatest_result_canonical_value_bound + S (fom_value_pfp_greatest_result_canonical) = p))) /\\ ((exists pfrd_qb_greatest_result pfrd_qc_greatest_result pfrd_qlen_greatest_result pfrd_pb_greatest_result pfrd_pc_greatest_result pfrd_plen_greatest_result. ((((forall fom_index_pfp_greatest_result_productleft. (exists fom_gap_pfp_greatest_result_productleft_index_bound. fom_gap_pfp_greatest_result_productleft_index_bound + S (fom_index_pfp_greatest_result_productleft) = pfrd_qlen_greatest_result) -> exists fom_value_pfp_greatest_result_productleft. ((((exists fom_beta_height_pfp_greatest_result_productleft_entry. fom_beta_height_pfp_greatest_result_productleft_entry + S (fom_value_pfp_greatest_result_productleft) = S ((S (fom_index_pfp_greatest_result_productleft)) * pfrd_qc_greatest_result)) /\\ exists fom_beta_quotient_pfp_greatest_result_productleft_entry. pfrd_qb_greatest_result = fom_beta_quotient_pfp_greatest_result_productleft_entry * S ((S (fom_index_pfp_greatest_result_productleft)) * pfrd_qc_greatest_result) + (fom_value_pfp_greatest_result_productleft))) /\\ (exists fom_gap_pfp_greatest_result_productleft_value_bound. fom_gap_pfp_greatest_result_productleft_value_bound + S (fom_value_pfp_greatest_result_productleft) = p))) /\\ (((forall fom_index_pfp_greatest_result_productright. (exists fom_gap_pfp_greatest_result_productright_index_bound. fom_gap_pfp_greatest_result_productright_index_bound + S (fom_index_pfp_greatest_result_productright) = D) -> exists fom_value_pfp_greatest_result_productright. ((((exists fom_beta_height_pfp_greatest_result_productright_entry. fom_beta_height_pfp_greatest_result_productright_entry + S (fom_value_pfp_greatest_result_productright) = S ((S (fom_index_pfp_greatest_result_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_greatest_result_productright_entry. db = fom_beta_quotient_pfp_greatest_result_productright_entry * S ((S (fom_index_pfp_greatest_result_productright)) * dc) + (fom_value_pfp_greatest_result_productright))) /\\ (exists fom_gap_pfp_greatest_result_productright_value_bound. fom_gap_pfp_greatest_result_productright_value_bound + S (fom_value_pfp_greatest_result_productright) = p))) /\\ (((((((pfrd_qlen_greatest_result)=0 \\/ (D)=0) /\\ (((pfrd_plen_greatest_result)=0)))) \\/ (((~((pfrd_qlen_greatest_result)=0)) /\\ (((~((D)=0)) /\\ (((pfrd_qlen_greatest_result)+(D)=S (pfrd_plen_greatest_result)))))))) /\\ ((forall pfc_index_greatest_result_productcoefficients. (exists pfa_gap_greatest_result_productcoefficientsbound. pfa_gap_greatest_result_productcoefficientsbound + S (pfc_index_greatest_result_productcoefficients) = (pfrd_plen_greatest_result)) -> exists pfc_value_greatest_result_productcoefficients. ((((exists ff_h_pfp_greatest_result_productcoefficientsentry. ff_h_pfp_greatest_result_productcoefficientsentry + S (pfc_value_greatest_result_productcoefficients) = S ((S (pfc_index_greatest_result_productcoefficients)) * pfrd_pc_greatest_result)) /\\ exists ff_q_pfp_greatest_result_productcoefficientsentry. pfrd_pb_greatest_result = ff_q_pfp_greatest_result_productcoefficientsentry * S ((S (pfc_index_greatest_result_productcoefficients)) * pfrd_pc_greatest_result) + (pfc_value_greatest_result_productcoefficients))) /\\ ((exists pfc_terms_code_greatest_result_productcoefficientscoefficient pfc_terms_scale_greatest_result_productcoefficientscoefficient pfc_natural_sum_greatest_result_productcoefficientscoefficient. ((forall pfc_index_greatest_result_productcoefficientscoefficientdiagonal. (exists pfa_gap_greatest_result_productcoefficientscoefficientdiagonalbound. pfa_gap_greatest_result_productcoefficientscoefficientdiagonalbound + S (pfc_index_greatest_result_productcoefficientscoefficientdiagonal) = (S (pfc_index_greatest_result_productcoefficients))) -> exists pfc_value_greatest_result_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_greatest_result_productcoefficientscoefficientdiagonalentry. ff_h_pfp_greatest_result_productcoefficientscoefficientdiagonalentry + S (pfc_value_greatest_result_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_greatest_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_greatest_result_productcoefficientscoefficient)) /\\ exists ff_q_pfp_greatest_result_productcoefficientscoefficientdiagonalentry. pfc_terms_code_greatest_result_productcoefficientscoefficient = ff_q_pfp_greatest_result_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_greatest_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_greatest_result_productcoefficientscoefficient) + (pfc_value_greatest_result_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_greatest_result_productcoefficientscoefficientdiagonalterm pfc_left_greatest_result_productcoefficientscoefficientdiagonalterm pfc_right_greatest_result_productcoefficientscoefficientdiagonalterm. (((pfc_index_greatest_result_productcoefficientscoefficientdiagonal)+pfc_complement_greatest_result_productcoefficientscoefficientdiagonalterm=(pfc_index_greatest_result_productcoefficients)) /\\ ((((((exists pfa_gap_greatest_result_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_greatest_result_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_greatest_result_productcoefficientscoefficientdiagonal) = (pfrd_qlen_greatest_result)) /\\ ((((exists ff_h_pfp_greatest_result_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_greatest_result_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_greatest_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_greatest_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_greatest_result)) /\\ exists ff_q_pfp_greatest_result_productcoefficientscoefficientdiagonaltermleftentry. pfrd_qb_greatest_result = ff_q_pfp_greatest_result_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_greatest_result_productcoefficientscoefficientdiagonal)) * pfrd_qc_greatest_result) + (pfc_left_greatest_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_greatest_result_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_greatest_result_productcoefficientscoefficientdiagonaltermleftoutside+(pfrd_qlen_greatest_result)=(pfc_index_greatest_result_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_greatest_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_greatest_result_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_greatest_result_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_greatest_result_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_greatest_result_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_greatest_result_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_greatest_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_greatest_result_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_greatest_result_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_greatest_result_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_greatest_result_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_greatest_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_greatest_result_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_greatest_result_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_greatest_result_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_greatest_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_greatest_result_productcoefficientscoefficientdiagonal)=pfc_left_greatest_result_productcoefficientscoefficientdiagonalterm*pfc_right_greatest_result_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_greatest_result_productcoefficientscoefficientsum fs_v_pfc_greatest_result_productcoefficientscoefficientsum. ((((exists fs_h_pfc_greatest_result_productcoefficientscoefficientsum_body_start. fs_h_pfc_greatest_result_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_greatest_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_result_productcoefficientscoefficientsum_body_start. fs_u_pfc_greatest_result_productcoefficientscoefficientsum = fs_q_pfc_greatest_result_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_greatest_result_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_greatest_result_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_greatest_result_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_greatest_result_productcoefficientscoefficient) = S ((S (S (pfc_index_greatest_result_productcoefficients))) * fs_v_pfc_greatest_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_result_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_greatest_result_productcoefficientscoefficientsum = fs_q_pfc_greatest_result_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_greatest_result_productcoefficients))) * fs_v_pfc_greatest_result_productcoefficientscoefficientsum) + (pfc_natural_sum_greatest_result_productcoefficientscoefficient))) /\\ forall fs_i_pfc_greatest_result_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_greatest_result_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_greatest_result_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_greatest_result_productcoefficientscoefficientsum_body_steps = S (pfc_index_greatest_result_productcoefficients)) -> exists fs_a_pfc_greatest_result_productcoefficientscoefficientsum_body_steps fs_r_pfc_greatest_result_productcoefficientscoefficientsum_body_steps fs_s_pfc_greatest_result_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_greatest_result_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_greatest_result_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_greatest_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_greatest_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_greatest_result_productcoefficientscoefficient)) /\\ exists fs_q_pfc_greatest_result_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_greatest_result_productcoefficientscoefficient = fs_q_pfc_greatest_result_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_greatest_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_greatest_result_productcoefficientscoefficient) + (fs_a_pfc_greatest_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_greatest_result_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_greatest_result_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_greatest_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_greatest_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_result_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_greatest_result_productcoefficientscoefficientsum = fs_q_pfc_greatest_result_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_greatest_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_result_productcoefficientscoefficientsum) + (fs_r_pfc_greatest_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_greatest_result_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_greatest_result_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_greatest_result_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_greatest_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_greatest_result_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_greatest_result_productcoefficientscoefficientsum = fs_q_pfc_greatest_result_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_greatest_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_greatest_result_productcoefficientscoefficientsum) + (fs_s_pfc_greatest_result_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_greatest_result_productcoefficientscoefficientsum_body_steps = fs_r_pfc_greatest_result_productcoefficientscoefficientsum_body_steps + fs_a_pfc_greatest_result_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_greatest_result_productcoefficientscoefficientresiduebound. pfa_gap_greatest_result_productcoefficientscoefficientresiduebound + S (pfc_value_greatest_result_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_greatest_result_productcoefficientscoefficientresiduecongruence pfa_offset_right_greatest_result_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_greatest_result_productcoefficientscoefficient) + (p) * pfa_offset_left_greatest_result_productcoefficientscoefficientresiduecongruence = (pfc_value_greatest_result_productcoefficients) + (p) * pfa_offset_right_greatest_result_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_greatest_result_target pfrep_left_greatest_result_target pfrep_right_greatest_result_target. ((exists pfrep_position_greatest_result_targetfirst. ((pfrep_position_greatest_result_targetfirst+S (pfrep_power_greatest_result_target)=(pfrd_plen_greatest_result)) /\\ ((((exists ff_h_pfp_greatest_result_targetfirstentry. ff_h_pfp_greatest_result_targetfirstentry + S (pfrep_left_greatest_result_target) = S ((S (pfrep_position_greatest_result_targetfirst)) * pfrd_pc_greatest_result)) /\\ exists ff_q_pfp_greatest_result_targetfirstentry. pfrd_pb_greatest_result = ff_q_pfp_greatest_result_targetfirstentry * S ((S (pfrep_position_greatest_result_targetfirst)) * pfrd_pc_greatest_result) + (pfrep_left_greatest_result_target)))))) \\/ (((exists pfrep_gap_greatest_result_targetfirstoutside. pfrep_gap_greatest_result_targetfirstoutside+(pfrd_plen_greatest_result)=(pfrep_power_greatest_result_target)) /\\ (((pfrep_left_greatest_result_target)=0))))) -> ((exists pfrep_position_greatest_result_targetsecond. ((pfrep_position_greatest_result_targetsecond+S (pfrep_power_greatest_result_target)=(G)) /\\ ((((exists ff_h_pfp_greatest_result_targetsecondentry. ff_h_pfp_greatest_result_targetsecondentry + S (pfrep_right_greatest_result_target) = S ((S (pfrep_position_greatest_result_targetsecond)) * gc)) /\\ exists ff_q_pfp_greatest_result_targetsecondentry. gb = ff_q_pfp_greatest_result_targetsecondentry * S ((S (pfrep_position_greatest_result_targetsecond)) * gc) + (pfrep_right_greatest_result_target)))))) \\/ (((exists pfrep_gap_greatest_result_targetsecondoutside. pfrep_gap_greatest_result_targetsecondoutside+(G)=(pfrep_power_greatest_result_target)) /\\ (((pfrep_right_greatest_result_target)=0))))) -> pfrep_left_greatest_result_target=pfrep_right_greatest_result_target)))))))", "statement_sha256": "2cb785ffddc3740e6d764eeb98a50cb0e09d08167dae46c669980b9901a20890", "summary": "Every actual common right divisor of A and B divides any actual Bezout representative G=U*A+V*B. 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prime_field_polynomial_trim_output_coefficients (rc)", "specialize prime_field_polynomial_trim_output_coefficients (R)", "apply prime_field_polynomial_trim_output_coefficients", "exact he_right_right_right_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right" ], "script_sha256": "138318d0d080351eb319626f6cd172fdae864157590f7291e8ee3c8a3db4b51f", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_existence_candidate.py", "sha256": "81f2f48dd2e81894c7a267453646eb6f2b6f9bd3ee320386d8c561f6b9f8b8ca" }, "statement": "forall p ab ac L bb bc d qb qc q rb rc R. (((forall fom_index_pfp_gcd_remainder_executioninput. (exists fom_gap_pfp_gcd_remainder_executioninput_index_bound. fom_gap_pfp_gcd_remainder_executioninput_index_bound + S (fom_index_pfp_gcd_remainder_executioninput) = L) -> exists fom_value_pfp_gcd_remainder_executioninput. ((((exists fom_beta_height_pfp_gcd_remainder_executioninput_entry. fom_beta_height_pfp_gcd_remainder_executioninput_entry + S (fom_value_pfp_gcd_remainder_executioninput) = S ((S (fom_index_pfp_gcd_remainder_executioninput)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_remainder_executioninput_entry. ab = fom_beta_quotient_pfp_gcd_remainder_executioninput_entry * S ((S (fom_index_pfp_gcd_remainder_executioninput)) * ac) + (fom_value_pfp_gcd_remainder_executioninput))) /\\ (exists fom_gap_pfp_gcd_remainder_executioninput_value_bound. fom_gap_pfp_gcd_remainder_executioninput_value_bound + S (fom_value_pfp_gcd_remainder_executioninput) = p))) /\\ (((forall fom_index_pfp_gcd_remainder_executiondivisor. (exists fom_gap_pfp_gcd_remainder_executiondivisor_index_bound. fom_gap_pfp_gcd_remainder_executiondivisor_index_bound + S (fom_index_pfp_gcd_remainder_executiondivisor) = S (d)) -> exists fom_value_pfp_gcd_remainder_executiondivisor. ((((exists fom_beta_height_pfp_gcd_remainder_executiondivisor_entry. fom_beta_height_pfp_gcd_remainder_executiondivisor_entry + S (fom_value_pfp_gcd_remainder_executiondivisor) = S ((S (fom_index_pfp_gcd_remainder_executiondivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_remainder_executiondivisor_entry. bb = fom_beta_quotient_pfp_gcd_remainder_executiondivisor_entry * S ((S (fom_index_pfp_gcd_remainder_executiondivisor)) * bc) + (fom_value_pfp_gcd_remainder_executiondivisor))) /\\ (exists fom_gap_pfp_gcd_remainder_executiondivisor_value_bound. fom_gap_pfp_gcd_remainder_executiondivisor_value_bound + S (fom_value_pfp_gcd_remainder_executiondivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_gcd_remainder_executionlengthshort. pfc_gap_gcd_remainder_executionlengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((exists pfd_head_gcd_remainder_execution pfd_inverse_gcd_remainder_execution pfd_product_code_gcd_remainder_execution pfd_product_scale_gcd_remainder_execution pfd_residual_code_gcd_remainder_execution pfd_residual_scale_gcd_remainder_execution pfd_cut_gcd_remainder_execution. ((((exists ff_h_pfp_gcd_remainder_executionhead. ff_h_pfp_gcd_remainder_executionhead + S (pfd_head_gcd_remainder_execution) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_gcd_remainder_executionhead. bb = ff_q_pfp_gcd_remainder_executionhead * S ((S (0)) * bc) + (pfd_head_gcd_remainder_execution))) /\\ (((((~((pfd_head_gcd_remainder_execution) = 0)) /\\ ((((exists pfa_gap_gcd_remainder_executioninversemultiplicationleft. pfa_gap_gcd_remainder_executioninversemultiplicationleft + S (pfd_head_gcd_remainder_execution) = (p)) /\\ (((exists pfa_gap_gcd_remainder_executioninversemultiplicationright. pfa_gap_gcd_remainder_executioninversemultiplicationright + S (pfd_inverse_gcd_remainder_execution) = (p)) /\\ ((((exists pfa_gap_gcd_remainder_executioninversemultiplicationresultbound. pfa_gap_gcd_remainder_executioninversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_gcd_remainder_executioninversemultiplicationresultcongruence pfa_offset_right_gcd_remainder_executioninversemultiplicationresultcongruence. ((pfd_head_gcd_remainder_execution) * (pfd_inverse_gcd_remainder_execution)) + (p) * pfa_offset_left_gcd_remainder_executioninversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_gcd_remainder_executioninversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_gcd_remainder_executionquotient. (exists pfa_gap_gcd_remainder_executionquotientbound. pfa_gap_gcd_remainder_executionquotientbound + S (pfd_index_gcd_remainder_executionquotient) = (q)) -> exists pfd_value_gcd_remainder_executionquotient. ((((exists ff_h_pfp_gcd_remainder_executionquotiententry. ff_h_pfp_gcd_remainder_executionquotiententry + S (pfd_value_gcd_remainder_executionquotient) = S ((S (pfd_index_gcd_remainder_executionquotient)) * qc)) /\\ exists ff_q_pfp_gcd_remainder_executionquotiententry. qb = ff_q_pfp_gcd_remainder_executionquotiententry * S ((S (pfd_index_gcd_remainder_executionquotient)) * qc) + (pfd_value_gcd_remainder_executionquotient))) /\\ ((exists pfd_input_gcd_remainder_executionquotientstep pfd_previous_gcd_remainder_executionquotientstep pfd_difference_gcd_remainder_executionquotientstep. ((((exists ff_h_pfp_gcd_remainder_executionquotientstepinput. ff_h_pfp_gcd_remainder_executionquotientstepinput + S (pfd_input_gcd_remainder_executionquotientstep) = S ((S (pfd_index_gcd_remainder_executionquotient)) * ac)) /\\ exists ff_q_pfp_gcd_remainder_executionquotientstepinput. ab = ff_q_pfp_gcd_remainder_executionquotientstepinput * S ((S (pfd_index_gcd_remainder_executionquotient)) * ac) + (pfd_input_gcd_remainder_executionquotientstep))) /\\ (((exists pfc_terms_code_gcd_remainder_executionquotientstepprevious pfc_terms_scale_gcd_remainder_executionquotientstepprevious pfc_natural_sum_gcd_remainder_executionquotientstepprevious. ((forall pfc_index_gcd_remainder_executionquotientsteppreviousdiagonal. (exists pfa_gap_gcd_remainder_executionquotientsteppreviousdiagonalbound. pfa_gap_gcd_remainder_executionquotientsteppreviousdiagonalbound + S (pfc_index_gcd_remainder_executionquotientsteppreviousdiagonal) = (S (pfd_index_gcd_remainder_executionquotient))) -> exists pfc_value_gcd_remainder_executionquotientsteppreviousdiagonal. ((((exists ff_h_pfp_gcd_remainder_executionquotientsteppreviousdiagonalentry. ff_h_pfp_gcd_remainder_executionquotientsteppreviousdiagonalentry + S (pfc_value_gcd_remainder_executionquotientsteppreviousdiagonal) = S ((S (pfc_index_gcd_remainder_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_gcd_remainder_executionquotientstepprevious)) /\\ exists ff_q_pfp_gcd_remainder_executionquotientsteppreviousdiagonalentry. pfc_terms_code_gcd_remainder_executionquotientstepprevious = ff_q_pfp_gcd_remainder_executionquotientsteppreviousdiagonalentry * S ((S (pfc_index_gcd_remainder_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_gcd_remainder_executionquotientstepprevious) + (pfc_value_gcd_remainder_executionquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_gcd_remainder_executionquotientsteppreviousdiagonalterm pfc_left_gcd_remainder_executionquotientsteppreviousdiagonalterm pfc_right_gcd_remainder_executionquotientsteppreviousdiagonalterm. (((pfc_index_gcd_remainder_executionquotientsteppreviousdiagonal)+pfc_complement_gcd_remainder_executionquotientsteppreviousdiagonalterm=(pfd_index_gcd_remainder_executionquotient)) /\\ ((((((exists pfa_gap_gcd_remainder_executionquotientsteppreviousdiagonaltermleftinside. pfa_gap_gcd_remainder_executionquotientsteppreviousdiagonaltermleftinside + S (pfc_index_gcd_remainder_executionquotientsteppreviousdiagonal) = (pfd_index_gcd_remainder_executionquotient)) /\\ ((((exists ff_h_pfp_gcd_remainder_executionquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_gcd_remainder_executionquotientsteppreviousdiagonaltermleftentry + S (pfc_left_gcd_remainder_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_index_gcd_remainder_executionquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_gcd_remainder_executionquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_gcd_remainder_executionquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_gcd_remainder_executionquotientsteppreviousdiagonal)) * qc) + (pfc_left_gcd_remainder_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_remainder_executionquotientsteppreviousdiagonaltermleftoutside. pfc_gap_gcd_remainder_executionquotientsteppreviousdiagonaltermleftoutside+(pfd_index_gcd_remainder_executionquotient)=(pfc_index_gcd_remainder_executionquotientsteppreviousdiagonal)) /\\ (((pfc_left_gcd_remainder_executionquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_remainder_executionquotientsteppreviousdiagonaltermrightinside. pfa_gap_gcd_remainder_executionquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_gcd_remainder_executionquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_gcd_remainder_executionquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_gcd_remainder_executionquotientsteppreviousdiagonaltermrightentry + S (pfc_right_gcd_remainder_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_gcd_remainder_executionquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_remainder_executionquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_gcd_remainder_executionquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_gcd_remainder_executionquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_gcd_remainder_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_remainder_executionquotientsteppreviousdiagonaltermrightoutside. pfc_gap_gcd_remainder_executionquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_gcd_remainder_executionquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_gcd_remainder_executionquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_gcd_remainder_executionquotientsteppreviousdiagonal)=pfc_left_gcd_remainder_executionquotientsteppreviousdiagonalterm*pfc_right_gcd_remainder_executionquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_remainder_executionquotientstepprevioussum fs_v_pfc_gcd_remainder_executionquotientstepprevioussum. ((((exists fs_h_pfc_gcd_remainder_executionquotientstepprevioussum_body_start. fs_h_pfc_gcd_remainder_executionquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_remainder_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_remainder_executionquotientstepprevioussum_body_start. fs_u_pfc_gcd_remainder_executionquotientstepprevioussum = fs_q_pfc_gcd_remainder_executionquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_gcd_remainder_executionquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_gcd_remainder_executionquotientstepprevioussum_body_terminal. fs_h_pfc_gcd_remainder_executionquotientstepprevioussum_body_terminal + S (pfc_natural_sum_gcd_remainder_executionquotientstepprevious) = S ((S (S (pfd_index_gcd_remainder_executionquotient))) * fs_v_pfc_gcd_remainder_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_remainder_executionquotientstepprevioussum_body_terminal. fs_u_pfc_gcd_remainder_executionquotientstepprevioussum = fs_q_pfc_gcd_remainder_executionquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_gcd_remainder_executionquotient))) * fs_v_pfc_gcd_remainder_executionquotientstepprevioussum) + (pfc_natural_sum_gcd_remainder_executionquotientstepprevious))) /\\ forall fs_i_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps. (exists fs_lt_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps_bound. fs_lt_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps_bound + S fs_i_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps = S (pfd_index_gcd_remainder_executionquotient)) -> exists fs_a_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps fs_r_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps fs_s_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps_summand. fs_h_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_gcd_remainder_executionquotientstepprevious)) /\\ exists fs_q_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps_summand. pfc_terms_code_gcd_remainder_executionquotientstepprevious = fs_q_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_gcd_remainder_executionquotientstepprevious) + (fs_a_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps_partial. fs_h_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_remainder_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps_partial. fs_u_pfc_gcd_remainder_executionquotientstepprevioussum = fs_q_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_remainder_executionquotientstepprevioussum) + (fs_r_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps_successor. fs_h_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_remainder_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps_successor. fs_u_pfc_gcd_remainder_executionquotientstepprevioussum = fs_q_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_remainder_executionquotientstepprevioussum) + (fs_s_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps = fs_r_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps + fs_a_pfc_gcd_remainder_executionquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_remainder_executionquotientsteppreviousresiduebound. pfa_gap_gcd_remainder_executionquotientsteppreviousresiduebound + S (pfd_previous_gcd_remainder_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_gcd_remainder_executionquotientsteppreviousresiduecongruence pfa_offset_right_gcd_remainder_executionquotientsteppreviousresiduecongruence. (pfc_natural_sum_gcd_remainder_executionquotientstepprevious) + (p) * pfa_offset_left_gcd_remainder_executionquotientsteppreviousresiduecongruence = (pfd_previous_gcd_remainder_executionquotientstep) + (p) * pfa_offset_right_gcd_remainder_executionquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_gcd_remainder_executionquotientstepsubtractleft. pfa_gap_gcd_remainder_executionquotientstepsubtractleft + S (pfd_previous_gcd_remainder_executionquotientstep) = (p)) /\\ (((exists pfa_gap_gcd_remainder_executionquotientstepsubtractright. pfa_gap_gcd_remainder_executionquotientstepsubtractright + S (pfd_difference_gcd_remainder_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_gcd_remainder_executionquotientstepsubtractresultbound. pfa_gap_gcd_remainder_executionquotientstepsubtractresultbound + S (pfd_input_gcd_remainder_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_gcd_remainder_executionquotientstepsubtractresultcongruence pfa_offset_right_gcd_remainder_executionquotientstepsubtractresultcongruence. ((pfd_previous_gcd_remainder_executionquotientstep) + (pfd_difference_gcd_remainder_executionquotientstep)) + (p) * pfa_offset_left_gcd_remainder_executionquotientstepsubtractresultcongruence = (pfd_input_gcd_remainder_executionquotientstep) + (p) * pfa_offset_right_gcd_remainder_executionquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_gcd_remainder_executionquotientstepmultiplyleft. pfa_gap_gcd_remainder_executionquotientstepmultiplyleft + S (pfd_inverse_gcd_remainder_execution) = (p)) /\\ (((exists pfa_gap_gcd_remainder_executionquotientstepmultiplyright. pfa_gap_gcd_remainder_executionquotientstepmultiplyright + S (pfd_difference_gcd_remainder_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_gcd_remainder_executionquotientstepmultiplyresultbound. pfa_gap_gcd_remainder_executionquotientstepmultiplyresultbound + S (pfd_value_gcd_remainder_executionquotient) = (p)) /\\ ((exists pfa_offset_left_gcd_remainder_executionquotientstepmultiplyresultcongruence pfa_offset_right_gcd_remainder_executionquotientstepmultiplyresultcongruence. ((pfd_inverse_gcd_remainder_execution) * (pfd_difference_gcd_remainder_executionquotientstep)) + (p) * pfa_offset_left_gcd_remainder_executionquotientstepmultiplyresultcongruence = (pfd_value_gcd_remainder_executionquotient) + (p) * pfa_offset_right_gcd_remainder_executionquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_gcd_remainder_executionproduct. (exists pfa_gap_gcd_remainder_executionproductbound. pfa_gap_gcd_remainder_executionproductbound + S (pfc_index_gcd_remainder_executionproduct) = (L)) -> exists pfc_value_gcd_remainder_executionproduct. ((((exists ff_h_pfp_gcd_remainder_executionproductentry. ff_h_pfp_gcd_remainder_executionproductentry + S (pfc_value_gcd_remainder_executionproduct) = S ((S (pfc_index_gcd_remainder_executionproduct)) * pfd_product_scale_gcd_remainder_execution)) /\\ exists ff_q_pfp_gcd_remainder_executionproductentry. pfd_product_code_gcd_remainder_execution = ff_q_pfp_gcd_remainder_executionproductentry * S ((S (pfc_index_gcd_remainder_executionproduct)) * pfd_product_scale_gcd_remainder_execution) + (pfc_value_gcd_remainder_executionproduct))) /\\ ((exists pfc_terms_code_gcd_remainder_executionproductcoefficient pfc_terms_scale_gcd_remainder_executionproductcoefficient pfc_natural_sum_gcd_remainder_executionproductcoefficient. ((forall pfc_index_gcd_remainder_executionproductcoefficientdiagonal. (exists pfa_gap_gcd_remainder_executionproductcoefficientdiagonalbound. pfa_gap_gcd_remainder_executionproductcoefficientdiagonalbound + S (pfc_index_gcd_remainder_executionproductcoefficientdiagonal) = (S (pfc_index_gcd_remainder_executionproduct))) -> exists pfc_value_gcd_remainder_executionproductcoefficientdiagonal. ((((exists ff_h_pfp_gcd_remainder_executionproductcoefficientdiagonalentry. ff_h_pfp_gcd_remainder_executionproductcoefficientdiagonalentry + S (pfc_value_gcd_remainder_executionproductcoefficientdiagonal) = S ((S (pfc_index_gcd_remainder_executionproductcoefficientdiagonal)) * pfc_terms_scale_gcd_remainder_executionproductcoefficient)) /\\ exists ff_q_pfp_gcd_remainder_executionproductcoefficientdiagonalentry. pfc_terms_code_gcd_remainder_executionproductcoefficient = ff_q_pfp_gcd_remainder_executionproductcoefficientdiagonalentry * S ((S (pfc_index_gcd_remainder_executionproductcoefficientdiagonal)) * pfc_terms_scale_gcd_remainder_executionproductcoefficient) + (pfc_value_gcd_remainder_executionproductcoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_remainder_executionproductcoefficientdiagonalterm pfc_left_gcd_remainder_executionproductcoefficientdiagonalterm pfc_right_gcd_remainder_executionproductcoefficientdiagonalterm. (((pfc_index_gcd_remainder_executionproductcoefficientdiagonal)+pfc_complement_gcd_remainder_executionproductcoefficientdiagonalterm=(pfc_index_gcd_remainder_executionproduct)) /\\ ((((((exists pfa_gap_gcd_remainder_executionproductcoefficientdiagonaltermleftinside. pfa_gap_gcd_remainder_executionproductcoefficientdiagonaltermleftinside + S (pfc_index_gcd_remainder_executionproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_gcd_remainder_executionproductcoefficientdiagonaltermleftentry. ff_h_pfp_gcd_remainder_executionproductcoefficientdiagonaltermleftentry + S (pfc_left_gcd_remainder_executionproductcoefficientdiagonalterm) = S ((S (pfc_index_gcd_remainder_executionproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_gcd_remainder_executionproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_gcd_remainder_executionproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_remainder_executionproductcoefficientdiagonal)) * qc) + (pfc_left_gcd_remainder_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_remainder_executionproductcoefficientdiagonaltermleftoutside. pfc_gap_gcd_remainder_executionproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_gcd_remainder_executionproductcoefficientdiagonal)) /\\ (((pfc_left_gcd_remainder_executionproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_remainder_executionproductcoefficientdiagonaltermrightinside. pfa_gap_gcd_remainder_executionproductcoefficientdiagonaltermrightinside + S (pfc_complement_gcd_remainder_executionproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_gcd_remainder_executionproductcoefficientdiagonaltermrightentry. ff_h_pfp_gcd_remainder_executionproductcoefficientdiagonaltermrightentry + S (pfc_right_gcd_remainder_executionproductcoefficientdiagonalterm) = S ((S (pfc_complement_gcd_remainder_executionproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_remainder_executionproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_gcd_remainder_executionproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_remainder_executionproductcoefficientdiagonalterm)) * bc) + (pfc_right_gcd_remainder_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_remainder_executionproductcoefficientdiagonaltermrightoutside. pfc_gap_gcd_remainder_executionproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_gcd_remainder_executionproductcoefficientdiagonalterm)) /\\ (((pfc_right_gcd_remainder_executionproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_remainder_executionproductcoefficientdiagonal)=pfc_left_gcd_remainder_executionproductcoefficientdiagonalterm*pfc_right_gcd_remainder_executionproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_remainder_executionproductcoefficientsum fs_v_pfc_gcd_remainder_executionproductcoefficientsum. ((((exists fs_h_pfc_gcd_remainder_executionproductcoefficientsum_body_start. fs_h_pfc_gcd_remainder_executionproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_remainder_executionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_remainder_executionproductcoefficientsum_body_start. fs_u_pfc_gcd_remainder_executionproductcoefficientsum = fs_q_pfc_gcd_remainder_executionproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_remainder_executionproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_remainder_executionproductcoefficientsum_body_terminal. fs_h_pfc_gcd_remainder_executionproductcoefficientsum_body_terminal + S (pfc_natural_sum_gcd_remainder_executionproductcoefficient) = S ((S (S (pfc_index_gcd_remainder_executionproduct))) * fs_v_pfc_gcd_remainder_executionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_remainder_executionproductcoefficientsum_body_terminal. fs_u_pfc_gcd_remainder_executionproductcoefficientsum = fs_q_pfc_gcd_remainder_executionproductcoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_remainder_executionproduct))) * fs_v_pfc_gcd_remainder_executionproductcoefficientsum) + (pfc_natural_sum_gcd_remainder_executionproductcoefficient))) /\\ forall fs_i_pfc_gcd_remainder_executionproductcoefficientsum_body_steps. (exists fs_lt_pfc_gcd_remainder_executionproductcoefficientsum_body_steps_bound. fs_lt_pfc_gcd_remainder_executionproductcoefficientsum_body_steps_bound + S fs_i_pfc_gcd_remainder_executionproductcoefficientsum_body_steps = S (pfc_index_gcd_remainder_executionproduct)) -> exists fs_a_pfc_gcd_remainder_executionproductcoefficientsum_body_steps fs_r_pfc_gcd_remainder_executionproductcoefficientsum_body_steps fs_s_pfc_gcd_remainder_executionproductcoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_remainder_executionproductcoefficientsum_body_steps_summand. fs_h_pfc_gcd_remainder_executionproductcoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_remainder_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_remainder_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_gcd_remainder_executionproductcoefficient)) /\\ exists fs_q_pfc_gcd_remainder_executionproductcoefficientsum_body_steps_summand. pfc_terms_code_gcd_remainder_executionproductcoefficient = fs_q_pfc_gcd_remainder_executionproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_remainder_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_gcd_remainder_executionproductcoefficient) + (fs_a_pfc_gcd_remainder_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_remainder_executionproductcoefficientsum_body_steps_partial. fs_h_pfc_gcd_remainder_executionproductcoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_remainder_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_remainder_executionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_remainder_executionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_remainder_executionproductcoefficientsum_body_steps_partial. fs_u_pfc_gcd_remainder_executionproductcoefficientsum = fs_q_pfc_gcd_remainder_executionproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_remainder_executionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_remainder_executionproductcoefficientsum) + (fs_r_pfc_gcd_remainder_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_remainder_executionproductcoefficientsum_body_steps_successor. fs_h_pfc_gcd_remainder_executionproductcoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_remainder_executionproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_remainder_executionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_remainder_executionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_remainder_executionproductcoefficientsum_body_steps_successor. fs_u_pfc_gcd_remainder_executionproductcoefficientsum = fs_q_pfc_gcd_remainder_executionproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_remainder_executionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_remainder_executionproductcoefficientsum) + (fs_s_pfc_gcd_remainder_executionproductcoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_remainder_executionproductcoefficientsum_body_steps = fs_r_pfc_gcd_remainder_executionproductcoefficientsum_body_steps + fs_a_pfc_gcd_remainder_executionproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_remainder_executionproductcoefficientresiduebound. pfa_gap_gcd_remainder_executionproductcoefficientresiduebound + S (pfc_value_gcd_remainder_executionproduct) = (p)) /\\ ((exists pfa_offset_left_gcd_remainder_executionproductcoefficientresiduecongruence pfa_offset_right_gcd_remainder_executionproductcoefficientresiduecongruence. (pfc_natural_sum_gcd_remainder_executionproductcoefficient) + (p) * pfa_offset_left_gcd_remainder_executionproductcoefficientresiduecongruence = (pfc_value_gcd_remainder_executionproduct) + (p) * pfa_offset_right_gcd_remainder_executionproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_gcd_remainder_executiondifference. (exists pfa_gap_gcd_remainder_executiondifferenceindex. pfa_gap_gcd_remainder_executiondifferenceindex + S (pfs_index_gcd_remainder_executiondifference) = (L)) -> exists pfs_left_gcd_remainder_executiondifference pfs_right_gcd_remainder_executiondifference pfs_result_gcd_remainder_executiondifference. ((((exists ff_h_pfp_gcd_remainder_executiondifferenceleft. ff_h_pfp_gcd_remainder_executiondifferenceleft + S (pfs_left_gcd_remainder_executiondifference) = S ((S (pfs_index_gcd_remainder_executiondifference)) * ac)) /\\ exists ff_q_pfp_gcd_remainder_executiondifferenceleft. ab = ff_q_pfp_gcd_remainder_executiondifferenceleft * S ((S (pfs_index_gcd_remainder_executiondifference)) * ac) + (pfs_left_gcd_remainder_executiondifference))) /\\ (((((exists ff_h_pfp_gcd_remainder_executiondifferenceright. ff_h_pfp_gcd_remainder_executiondifferenceright + S (pfs_right_gcd_remainder_executiondifference) = S ((S (pfs_index_gcd_remainder_executiondifference)) * pfd_product_scale_gcd_remainder_execution)) /\\ exists ff_q_pfp_gcd_remainder_executiondifferenceright. pfd_product_code_gcd_remainder_execution = ff_q_pfp_gcd_remainder_executiondifferenceright * S ((S (pfs_index_gcd_remainder_executiondifference)) * pfd_product_scale_gcd_remainder_execution) + (pfs_right_gcd_remainder_executiondifference))) /\\ (((((exists ff_h_pfp_gcd_remainder_executiondifferenceresult. ff_h_pfp_gcd_remainder_executiondifferenceresult + S (pfs_result_gcd_remainder_executiondifference) = S ((S (pfs_index_gcd_remainder_executiondifference)) * pfd_residual_scale_gcd_remainder_execution)) /\\ exists ff_q_pfp_gcd_remainder_executiondifferenceresult. pfd_residual_code_gcd_remainder_execution = ff_q_pfp_gcd_remainder_executiondifferenceresult * S ((S (pfs_index_gcd_remainder_executiondifference)) * pfd_residual_scale_gcd_remainder_execution) + (pfs_result_gcd_remainder_executiondifference))) /\\ ((((exists pfa_gap_gcd_remainder_executiondifferenceoperationleft. pfa_gap_gcd_remainder_executiondifferenceoperationleft + S (pfs_right_gcd_remainder_executiondifference) = (p)) /\\ (((exists pfa_gap_gcd_remainder_executiondifferenceoperationright. pfa_gap_gcd_remainder_executiondifferenceoperationright + S (pfs_result_gcd_remainder_executiondifference) = (p)) /\\ ((((exists pfa_gap_gcd_remainder_executiondifferenceoperationresultbound. pfa_gap_gcd_remainder_executiondifferenceoperationresultbound + S (pfs_left_gcd_remainder_executiondifference) = (p)) /\\ ((exists pfa_offset_left_gcd_remainder_executiondifferenceoperationresultcongruence pfa_offset_right_gcd_remainder_executiondifferenceoperationresultcongruence. ((pfs_right_gcd_remainder_executiondifference) + (pfs_result_gcd_remainder_executiondifference)) + (p) * pfa_offset_left_gcd_remainder_executiondifferenceoperationresultcongruence = (pfs_left_gcd_remainder_executiondifference) + (p) * pfa_offset_right_gcd_remainder_executiondifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_gcd_remainder_execution)+(R)) /\\ (((forall fom_index_pfp_gcd_remainder_executiontriminput. (exists fom_gap_pfp_gcd_remainder_executiontriminput_index_bound. fom_gap_pfp_gcd_remainder_executiontriminput_index_bound + S (fom_index_pfp_gcd_remainder_executiontriminput) = L) -> exists fom_value_pfp_gcd_remainder_executiontriminput. ((((exists fom_beta_height_pfp_gcd_remainder_executiontriminput_entry. fom_beta_height_pfp_gcd_remainder_executiontriminput_entry + S (fom_value_pfp_gcd_remainder_executiontriminput) = S ((S (fom_index_pfp_gcd_remainder_executiontriminput)) * pfd_residual_scale_gcd_remainder_execution)) /\\ exists fom_beta_quotient_pfp_gcd_remainder_executiontriminput_entry. pfd_residual_code_gcd_remainder_execution = fom_beta_quotient_pfp_gcd_remainder_executiontriminput_entry * S ((S (fom_index_pfp_gcd_remainder_executiontriminput)) * pfd_residual_scale_gcd_remainder_execution) + (fom_value_pfp_gcd_remainder_executiontriminput))) /\\ (exists fom_gap_pfp_gcd_remainder_executiontriminput_value_bound. fom_gap_pfp_gcd_remainder_executiontriminput_value_bound + S (fom_value_pfp_gcd_remainder_executiontriminput) = p))) /\\ (((forall pfp_repeat_index_gcd_remainder_executiontrimremoved. (exists pfa_gap_gcd_remainder_executiontrimremovedindex. pfa_gap_gcd_remainder_executiontrimremovedindex + S (pfp_repeat_index_gcd_remainder_executiontrimremoved) = (pfd_cut_gcd_remainder_execution)) -> (((exists ff_h_pfp_gcd_remainder_executiontrimremovedentry. ff_h_pfp_gcd_remainder_executiontrimremovedentry + S (0) = S ((S (pfp_repeat_index_gcd_remainder_executiontrimremoved)) * pfd_residual_scale_gcd_remainder_execution)) /\\ exists ff_q_pfp_gcd_remainder_executiontrimremovedentry. pfd_residual_code_gcd_remainder_execution = ff_q_pfp_gcd_remainder_executiontrimremovedentry * S ((S (pfp_repeat_index_gcd_remainder_executiontrimremoved)) * pfd_residual_scale_gcd_remainder_execution) + (0)))) /\\ (((forall pftrim_index_gcd_remainder_executiontrimsuffix pftrim_value_gcd_remainder_executiontrimsuffix. (exists pfa_gap_gcd_remainder_executiontrimsuffixbound. pfa_gap_gcd_remainder_executiontrimsuffixbound + S (pftrim_index_gcd_remainder_executiontrimsuffix) = (R)) -> (((exists ff_h_pfp_gcd_remainder_executiontrimsuffixsource. ff_h_pfp_gcd_remainder_executiontrimsuffixsource + S (pftrim_value_gcd_remainder_executiontrimsuffix) = S ((S ((pfd_cut_gcd_remainder_execution)+pftrim_index_gcd_remainder_executiontrimsuffix)) * pfd_residual_scale_gcd_remainder_execution)) /\\ exists ff_q_pfp_gcd_remainder_executiontrimsuffixsource. pfd_residual_code_gcd_remainder_execution = ff_q_pfp_gcd_remainder_executiontrimsuffixsource * S ((S ((pfd_cut_gcd_remainder_execution)+pftrim_index_gcd_remainder_executiontrimsuffix)) * pfd_residual_scale_gcd_remainder_execution) + (pftrim_value_gcd_remainder_executiontrimsuffix))) -> (((exists ff_h_pfp_gcd_remainder_executiontrimsuffixoutput. ff_h_pfp_gcd_remainder_executiontrimsuffixoutput + S (pftrim_value_gcd_remainder_executiontrimsuffix) = S ((S (pftrim_index_gcd_remainder_executiontrimsuffix)) * rc)) /\\ exists ff_q_pfp_gcd_remainder_executiontrimsuffixoutput. rb = ff_q_pfp_gcd_remainder_executiontrimsuffixoutput * S ((S (pftrim_index_gcd_remainder_executiontrimsuffix)) * rc) + (pftrim_value_gcd_remainder_executiontrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_gcd_remainder_executiontrimnormal. ((((exists ff_h_pfp_gcd_remainder_executiontrimnormalentry. ff_h_pfp_gcd_remainder_executiontrimnormalentry + S (pftrim_leading_gcd_remainder_executiontrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_gcd_remainder_executiontrimnormalentry. rb = ff_q_pfp_gcd_remainder_executiontrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_gcd_remainder_executiontrimnormal))) /\\ ((~(pftrim_leading_gcd_remainder_executiontrimnormal=0))))))))))))))))))))))))))))))))) -> (forall fom_index_pfp_gcd_remainder_bounded. (exists fom_gap_pfp_gcd_remainder_bounded_index_bound. fom_gap_pfp_gcd_remainder_bounded_index_bound + S (fom_index_pfp_gcd_remainder_bounded) = R) -> exists fom_value_pfp_gcd_remainder_bounded. ((((exists fom_beta_height_pfp_gcd_remainder_bounded_entry. fom_beta_height_pfp_gcd_remainder_bounded_entry + S (fom_value_pfp_gcd_remainder_bounded) = S ((S (fom_index_pfp_gcd_remainder_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_gcd_remainder_bounded_entry. rb = fom_beta_quotient_pfp_gcd_remainder_bounded_entry * S ((S (fom_index_pfp_gcd_remainder_bounded)) * rc) + (fom_value_pfp_gcd_remainder_bounded))) /\\ (exists fom_gap_pfp_gcd_remainder_bounded_value_bound. fom_gap_pfp_gcd_remainder_bounded_value_bound + S (fom_value_pfp_gcd_remainder_bounded) = p)))", "statement_sha256": "33705c65932790cc2139c047d667d1b772d03861294b939ef68223792f213ae3", "summary": "The actual trim inside division supplies canonical remainder coefficients, including its empty branch; no primality or degree of zero is assumed.", "summary_sha256": "9a83b4a6a3bb98d1c8fe0cf8b89677ff7111c7820bd9a8ee90c3acfdfc78bf5e" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": 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"prime_field_polynomial_trim_exists", "prime_field_polynomial_trim_output_coefficients", "prime_field_polynomial_equivalent_symmetric", "prime_field_polynomial_trim_equivalent", "prime_field_polynomial_trim_length_bounds", "eq_decidable", "prime_field_polynomial_trim_nonempty_degree_exists" ], "dependencies_sha256": "ba4785d5d73728ff1bcb5d738eb6e39d7935e46a66959d0247e49c606ff96c39", "empty_context_closure": { "body_proof_depth": 35, "body_proof_nodes": 94, "bundle_campaign": "polynomial-gcd-bezout", "bundle_dependency_edge_count": 1578, "bundle_node_count": 493, "bundle_node_id": 450, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "bundle_root_id": 492, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": 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"artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "53ad05b1dc2e1813ba676969d9927f70a8a58fb8ade28046e3350d756c554af4", "membership": "alpha_only", "name": "prime_field_polynomial_reduced_representative_exists", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro hA", "have ht : exists t tb tc K. (((L)=(t)+(K)) /\\ (((forall fom_index_pfp_gcd_reduced_triminput. (exists fom_gap_pfp_gcd_reduced_triminput_index_bound. fom_gap_pfp_gcd_reduced_triminput_index_bound + S (fom_index_pfp_gcd_reduced_triminput) = L) -> exists fom_value_pfp_gcd_reduced_triminput. ((((exists fom_beta_height_pfp_gcd_reduced_triminput_entry. fom_beta_height_pfp_gcd_reduced_triminput_entry + S (fom_value_pfp_gcd_reduced_triminput) = S ((S (fom_index_pfp_gcd_reduced_triminput)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_reduced_triminput_entry. ab = fom_beta_quotient_pfp_gcd_reduced_triminput_entry * S ((S (fom_index_pfp_gcd_reduced_triminput)) * ac) + (fom_value_pfp_gcd_reduced_triminput))) /\\ (exists fom_gap_pfp_gcd_reduced_triminput_value_bound. fom_gap_pfp_gcd_reduced_triminput_value_bound + S (fom_value_pfp_gcd_reduced_triminput) = p))) /\\ (((forall pfp_repeat_index_gcd_reduced_trimremoved. (exists pfa_gap_gcd_reduced_trimremovedindex. pfa_gap_gcd_reduced_trimremovedindex + S (pfp_repeat_index_gcd_reduced_trimremoved) = (t)) -> (((exists ff_h_pfp_gcd_reduced_trimremovedentry. ff_h_pfp_gcd_reduced_trimremovedentry + S (0) = S ((S (pfp_repeat_index_gcd_reduced_trimremoved)) * ac)) /\\ exists ff_q_pfp_gcd_reduced_trimremovedentry. ab = ff_q_pfp_gcd_reduced_trimremovedentry * S ((S (pfp_repeat_index_gcd_reduced_trimremoved)) * ac) + (0)))) /\\ (((forall pftrim_index_gcd_reduced_trimsuffix pftrim_value_gcd_reduced_trimsuffix. (exists pfa_gap_gcd_reduced_trimsuffixbound. pfa_gap_gcd_reduced_trimsuffixbound + S (pftrim_index_gcd_reduced_trimsuffix) = (K)) -> (((exists ff_h_pfp_gcd_reduced_trimsuffixsource. ff_h_pfp_gcd_reduced_trimsuffixsource + S (pftrim_value_gcd_reduced_trimsuffix) = S ((S ((t)+pftrim_index_gcd_reduced_trimsuffix)) * ac)) /\\ exists ff_q_pfp_gcd_reduced_trimsuffixsource. ab = ff_q_pfp_gcd_reduced_trimsuffixsource * S ((S ((t)+pftrim_index_gcd_reduced_trimsuffix)) * ac) + (pftrim_value_gcd_reduced_trimsuffix))) -> (((exists ff_h_pfp_gcd_reduced_trimsuffixoutput. ff_h_pfp_gcd_reduced_trimsuffixoutput + S (pftrim_value_gcd_reduced_trimsuffix) = S ((S (pftrim_index_gcd_reduced_trimsuffix)) * tc)) /\\ exists ff_q_pfp_gcd_reduced_trimsuffixoutput. tb = ff_q_pfp_gcd_reduced_trimsuffixoutput * S ((S (pftrim_index_gcd_reduced_trimsuffix)) * tc) + (pftrim_value_gcd_reduced_trimsuffix)))) /\\ (((K)=0 \\/ (exists pftrim_leading_gcd_reduced_trimnormal. ((((exists ff_h_pfp_gcd_reduced_trimnormalentry. ff_h_pfp_gcd_reduced_trimnormalentry + S (pftrim_leading_gcd_reduced_trimnormal) = S ((S (0)) * tc)) /\\ exists ff_q_pfp_gcd_reduced_trimnormalentry. tb = ff_q_pfp_gcd_reduced_trimnormalentry * S ((S (0)) * tc) + (pftrim_leading_gcd_reduced_trimnormal))) /\\ ((~(pftrim_leading_gcd_reduced_trimnormal=0))))))))))))))", "specialize prime_field_polynomial_trim_exists (p)", "specialize prime_field_polynomial_trim_exists (ab)", "specialize prime_field_polynomial_trim_exists (ac)", "specialize prime_field_polynomial_trim_exists (L)", "apply prime_field_polynomial_trim_exists", "exact hA", "cases ht", "cases ht_witness", "cases ht_witness_witness", "cases ht_witness_witness_witness", "exists x1", "exists x2", "exists x3", "split", "specialize prime_field_polynomial_trim_output_coefficients (p)", "specialize prime_field_polynomial_trim_output_coefficients (ab)", "specialize prime_field_polynomial_trim_output_coefficients (ac)", "specialize prime_field_polynomial_trim_output_coefficients (L)", "specialize prime_field_polynomial_trim_output_coefficients (x)", "specialize prime_field_polynomial_trim_output_coefficients (x1)", "specialize prime_field_polynomial_trim_output_coefficients (x2)", "specialize prime_field_polynomial_trim_output_coefficients (x3)", "apply prime_field_polynomial_trim_output_coefficients", "exact ht_witness_witness_witness_witness", "split", "specialize prime_field_polynomial_equivalent_symmetric (ab)", "specialize prime_field_polynomial_equivalent_symmetric (ac)", "specialize prime_field_polynomial_equivalent_symmetric (L)", "specialize prime_field_polynomial_equivalent_symmetric (x1)", "specialize prime_field_polynomial_equivalent_symmetric (x2)", "specialize prime_field_polynomial_equivalent_symmetric (x3)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_trim_equivalent (p)", "specialize prime_field_polynomial_trim_equivalent (ab)", "specialize prime_field_polynomial_trim_equivalent (ac)", "specialize prime_field_polynomial_trim_equivalent (L)", "specialize prime_field_polynomial_trim_equivalent (x)", "specialize prime_field_polynomial_trim_equivalent (x1)", "specialize prime_field_polynomial_trim_equivalent (x2)", "specialize prime_field_polynomial_trim_equivalent (x3)", "apply prime_field_polynomial_trim_equivalent", "exact ht_witness_witness_witness_witness", "split", "have hb : ((exists pfc_gap_gcd_reduced_removed. pfc_gap_gcd_reduced_removed+(x)=(L)) /\\ ((exists pfc_gap_gcd_reduced_retained. pfc_gap_gcd_reduced_retained+(x3)=(L))))", "specialize prime_field_polynomial_trim_length_bounds (p)", "specialize prime_field_polynomial_trim_length_bounds (ab)", "specialize prime_field_polynomial_trim_length_bounds (ac)", "specialize prime_field_polynomial_trim_length_bounds (L)", "specialize prime_field_polynomial_trim_length_bounds (x)", "specialize prime_field_polynomial_trim_length_bounds (x1)", "specialize prime_field_polynomial_trim_length_bounds (x2)", "specialize prime_field_polynomial_trim_length_bounds (x3)", "apply prime_field_polynomial_trim_length_bounds", "exact ht_witness_witness_witness_witness", "cases hb", "exact hb_right", "have hz : x3=0 \\/ ~(x3=0)", "specialize eq_decidable (x3)", "specialize eq_decidable (0)", "apply eq_decidable", "cases hz", "left", "exact hz_left", "right", "specialize prime_field_polynomial_trim_nonempty_degree_exists (p)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (ab)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (ac)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (L)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (x)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (x1)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (x2)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (x3)", "apply prime_field_polynomial_trim_nonempty_degree_exists", "exact ht_witness_witness_witness_witness", "exact hz_right" ], "script_sha256": "916d00316a3d94fdc33be7af3d421451802e7cc11d92860f82b865a5f98071c2", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_existence_candidate.py", "sha256": "81f2f48dd2e81894c7a267453646eb6f2b6f9bd3ee320386d8c561f6b9f8b8ca" }, "statement": "forall p ab ac L. (forall fom_index_pfp_gcd_reduced_input. (exists fom_gap_pfp_gcd_reduced_input_index_bound. fom_gap_pfp_gcd_reduced_input_index_bound + S (fom_index_pfp_gcd_reduced_input) = L) -> exists fom_value_pfp_gcd_reduced_input. ((((exists fom_beta_height_pfp_gcd_reduced_input_entry. fom_beta_height_pfp_gcd_reduced_input_entry + S (fom_value_pfp_gcd_reduced_input) = S ((S (fom_index_pfp_gcd_reduced_input)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_reduced_input_entry. ab = fom_beta_quotient_pfp_gcd_reduced_input_entry * S ((S (fom_index_pfp_gcd_reduced_input)) * ac) + (fom_value_pfp_gcd_reduced_input))) /\\ (exists fom_gap_pfp_gcd_reduced_input_value_bound. fom_gap_pfp_gcd_reduced_input_value_bound + S (fom_value_pfp_gcd_reduced_input) = p))) -> (exists tb tc K. ((forall fom_index_pfp_gcd_reduced_result_bounded. (exists fom_gap_pfp_gcd_reduced_result_bounded_index_bound. fom_gap_pfp_gcd_reduced_result_bounded_index_bound + S (fom_index_pfp_gcd_reduced_result_bounded) = K) -> exists fom_value_pfp_gcd_reduced_result_bounded. ((((exists fom_beta_height_pfp_gcd_reduced_result_bounded_entry. fom_beta_height_pfp_gcd_reduced_result_bounded_entry + S (fom_value_pfp_gcd_reduced_result_bounded) = S ((S (fom_index_pfp_gcd_reduced_result_bounded)) * tc)) /\\ exists fom_beta_quotient_pfp_gcd_reduced_result_bounded_entry. tb = fom_beta_quotient_pfp_gcd_reduced_result_bounded_entry * S ((S (fom_index_pfp_gcd_reduced_result_bounded)) * tc) + (fom_value_pfp_gcd_reduced_result_bounded))) /\\ (exists fom_gap_pfp_gcd_reduced_result_bounded_value_bound. fom_gap_pfp_gcd_reduced_result_bounded_value_bound + S (fom_value_pfp_gcd_reduced_result_bounded) = p))) /\\ (((forall pfrep_power_gcd_reduced_result_equivalent pfrep_left_gcd_reduced_result_equivalent pfrep_right_gcd_reduced_result_equivalent. ((exists pfrep_position_gcd_reduced_result_equivalentfirst. ((pfrep_position_gcd_reduced_result_equivalentfirst+S (pfrep_power_gcd_reduced_result_equivalent)=(K)) /\\ ((((exists ff_h_pfp_gcd_reduced_result_equivalentfirstentry. ff_h_pfp_gcd_reduced_result_equivalentfirstentry + S (pfrep_left_gcd_reduced_result_equivalent) = S ((S (pfrep_position_gcd_reduced_result_equivalentfirst)) * tc)) /\\ exists ff_q_pfp_gcd_reduced_result_equivalentfirstentry. tb = ff_q_pfp_gcd_reduced_result_equivalentfirstentry * S ((S (pfrep_position_gcd_reduced_result_equivalentfirst)) * tc) + (pfrep_left_gcd_reduced_result_equivalent)))))) \\/ (((exists pfrep_gap_gcd_reduced_result_equivalentfirstoutside. pfrep_gap_gcd_reduced_result_equivalentfirstoutside+(K)=(pfrep_power_gcd_reduced_result_equivalent)) /\\ (((pfrep_left_gcd_reduced_result_equivalent)=0))))) -> ((exists pfrep_position_gcd_reduced_result_equivalentsecond. ((pfrep_position_gcd_reduced_result_equivalentsecond+S (pfrep_power_gcd_reduced_result_equivalent)=(L)) /\\ ((((exists ff_h_pfp_gcd_reduced_result_equivalentsecondentry. ff_h_pfp_gcd_reduced_result_equivalentsecondentry + S (pfrep_right_gcd_reduced_result_equivalent) = S ((S (pfrep_position_gcd_reduced_result_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_gcd_reduced_result_equivalentsecondentry. ab = ff_q_pfp_gcd_reduced_result_equivalentsecondentry * S ((S (pfrep_position_gcd_reduced_result_equivalentsecond)) * ac) + (pfrep_right_gcd_reduced_result_equivalent)))))) \\/ (((exists pfrep_gap_gcd_reduced_result_equivalentsecondoutside. pfrep_gap_gcd_reduced_result_equivalentsecondoutside+(L)=(pfrep_power_gcd_reduced_result_equivalent)) /\\ (((pfrep_right_gcd_reduced_result_equivalent)=0))))) -> pfrep_left_gcd_reduced_result_equivalent=pfrep_right_gcd_reduced_result_equivalent) /\\ (((exists pfc_gap_gcd_reduced_result_length. pfc_gap_gcd_reduced_result_length+(K)=(L)) /\\ (((K)=0 \\/ (exists pfg_degree_gcd_reduced_result. (((K)=S (pfg_degree_gcd_reduced_result)) /\\ (((forall fom_index_pfp_gcd_reduced_result_degreecoefficients. (exists fom_gap_pfp_gcd_reduced_result_degreecoefficients_index_bound. fom_gap_pfp_gcd_reduced_result_degreecoefficients_index_bound + S (fom_index_pfp_gcd_reduced_result_degreecoefficients) = K) -> exists fom_value_pfp_gcd_reduced_result_degreecoefficients. ((((exists fom_beta_height_pfp_gcd_reduced_result_degreecoefficients_entry. fom_beta_height_pfp_gcd_reduced_result_degreecoefficients_entry + S (fom_value_pfp_gcd_reduced_result_degreecoefficients) = S ((S (fom_index_pfp_gcd_reduced_result_degreecoefficients)) * tc)) /\\ exists fom_beta_quotient_pfp_gcd_reduced_result_degreecoefficients_entry. tb = fom_beta_quotient_pfp_gcd_reduced_result_degreecoefficients_entry * S ((S (fom_index_pfp_gcd_reduced_result_degreecoefficients)) * tc) + (fom_value_pfp_gcd_reduced_result_degreecoefficients))) /\\ (exists fom_gap_pfp_gcd_reduced_result_degreecoefficients_value_bound. fom_gap_pfp_gcd_reduced_result_degreecoefficients_value_bound + S (fom_value_pfp_gcd_reduced_result_degreecoefficients) = p))) /\\ ((exists pfd_leading_gcd_reduced_result_degree. ((((exists ff_h_pfp_gcd_reduced_result_degreeentry. ff_h_pfp_gcd_reduced_result_degreeentry + S (pfd_leading_gcd_reduced_result_degree) = S ((S (0)) * tc)) /\\ exists ff_q_pfp_gcd_reduced_result_degreeentry. tb = ff_q_pfp_gcd_reduced_result_degreeentry * S ((S (0)) * tc) + (pfd_leading_gcd_reduced_result_degree))) /\\ ((~(pfd_leading_gcd_reduced_result_degree=0))))))))))))))))))", "statement_sha256": "b0241efc2826886a5d3f607c4d4ba72bdcd6ec5cc1085b19080015c08a286278", "summary": "Construct a formally equivalent canonical representative of no greater retained length, either empty or with an actual nonzero leading coefficient and represented degree. 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(((G)=0 \\/ (((~((G) = 0)) /\\ (((forall fom_index_pfp_gcd_terminal_normal_moniccoefficients. (exists fom_gap_pfp_gcd_terminal_normal_moniccoefficients_index_bound. fom_gap_pfp_gcd_terminal_normal_moniccoefficients_index_bound + S (fom_index_pfp_gcd_terminal_normal_moniccoefficients) = G) -> exists fom_value_pfp_gcd_terminal_normal_moniccoefficients. ((((exists fom_beta_height_pfp_gcd_terminal_normal_moniccoefficients_entry. fom_beta_height_pfp_gcd_terminal_normal_moniccoefficients_entry + S (fom_value_pfp_gcd_terminal_normal_moniccoefficients) = S ((S (fom_index_pfp_gcd_terminal_normal_moniccoefficients)) * gc)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_normal_moniccoefficients_entry. gb = fom_beta_quotient_pfp_gcd_terminal_normal_moniccoefficients_entry * S ((S (fom_index_pfp_gcd_terminal_normal_moniccoefficients)) * gc) + (fom_value_pfp_gcd_terminal_normal_moniccoefficients))) /\\ (exists fom_gap_pfp_gcd_terminal_normal_moniccoefficients_value_bound. fom_gap_pfp_gcd_terminal_normal_moniccoefficients_value_bound + S (fom_value_pfp_gcd_terminal_normal_moniccoefficients) = p))) /\\ ((((exists ff_h_pfp_gcd_terminal_normal_monicleading. ff_h_pfp_gcd_terminal_normal_monicleading + S (1) = S ((S (0)) * gc)) /\\ exists ff_q_pfp_gcd_terminal_normal_monicleading. gb = ff_q_pfp_gcd_terminal_normal_monicleading * S ((S (0)) * gc) + (1))))))))) /\\ (((((forall fom_index_pfp_gcd_terminal_multiple_bounded. (exists fom_gap_pfp_gcd_terminal_multiple_bounded_index_bound. fom_gap_pfp_gcd_terminal_multiple_bounded_index_bound + S (fom_index_pfp_gcd_terminal_multiple_bounded) = G) -> exists fom_value_pfp_gcd_terminal_multiple_bounded. ((((exists fom_beta_height_pfp_gcd_terminal_multiple_bounded_entry. fom_beta_height_pfp_gcd_terminal_multiple_bounded_entry + S (fom_value_pfp_gcd_terminal_multiple_bounded) = S ((S (fom_index_pfp_gcd_terminal_multiple_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_multiple_bounded_entry. gb = fom_beta_quotient_pfp_gcd_terminal_multiple_bounded_entry * S ((S (fom_index_pfp_gcd_terminal_multiple_bounded)) * gc) + (fom_value_pfp_gcd_terminal_multiple_bounded))) /\\ (exists fom_gap_pfp_gcd_terminal_multiple_bounded_value_bound. fom_gap_pfp_gcd_terminal_multiple_bounded_value_bound + S (fom_value_pfp_gcd_terminal_multiple_bounded) = p))) /\\ ((exists pfgd_qb_gcd_terminal_multiple pfgd_qc_gcd_terminal_multiple pfgd_Q_gcd_terminal_multiple pfgd_pb_gcd_terminal_multiple pfgd_pc_gcd_terminal_multiple pfgd_P_gcd_terminal_multiple. ((((forall fom_index_pfp_gcd_terminal_multiple_productleft. (exists fom_gap_pfp_gcd_terminal_multiple_productleft_index_bound. fom_gap_pfp_gcd_terminal_multiple_productleft_index_bound + S (fom_index_pfp_gcd_terminal_multiple_productleft) = pfgd_Q_gcd_terminal_multiple) -> exists fom_value_pfp_gcd_terminal_multiple_productleft. ((((exists fom_beta_height_pfp_gcd_terminal_multiple_productleft_entry. fom_beta_height_pfp_gcd_terminal_multiple_productleft_entry + S (fom_value_pfp_gcd_terminal_multiple_productleft) = S ((S (fom_index_pfp_gcd_terminal_multiple_productleft)) * pfgd_qc_gcd_terminal_multiple)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_multiple_productleft_entry. pfgd_qb_gcd_terminal_multiple = fom_beta_quotient_pfp_gcd_terminal_multiple_productleft_entry * S ((S (fom_index_pfp_gcd_terminal_multiple_productleft)) * pfgd_qc_gcd_terminal_multiple) + (fom_value_pfp_gcd_terminal_multiple_productleft))) /\\ (exists fom_gap_pfp_gcd_terminal_multiple_productleft_value_bound. fom_gap_pfp_gcd_terminal_multiple_productleft_value_bound + S (fom_value_pfp_gcd_terminal_multiple_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_terminal_multiple_productright. (exists fom_gap_pfp_gcd_terminal_multiple_productright_index_bound. fom_gap_pfp_gcd_terminal_multiple_productright_index_bound + S (fom_index_pfp_gcd_terminal_multiple_productright) = L) -> exists fom_value_pfp_gcd_terminal_multiple_productright. ((((exists fom_beta_height_pfp_gcd_terminal_multiple_productright_entry. fom_beta_height_pfp_gcd_terminal_multiple_productright_entry + S (fom_value_pfp_gcd_terminal_multiple_productright) = S ((S (fom_index_pfp_gcd_terminal_multiple_productright)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_multiple_productright_entry. ab = fom_beta_quotient_pfp_gcd_terminal_multiple_productright_entry * S ((S (fom_index_pfp_gcd_terminal_multiple_productright)) * ac) + (fom_value_pfp_gcd_terminal_multiple_productright))) /\\ (exists fom_gap_pfp_gcd_terminal_multiple_productright_value_bound. fom_gap_pfp_gcd_terminal_multiple_productright_value_bound + S (fom_value_pfp_gcd_terminal_multiple_productright) = p))) /\\ (((((((pfgd_Q_gcd_terminal_multiple)=0 \\/ (L)=0) /\\ (((pfgd_P_gcd_terminal_multiple)=0)))) \\/ (((~((pfgd_Q_gcd_terminal_multiple)=0)) /\\ (((~((L)=0)) /\\ (((pfgd_Q_gcd_terminal_multiple)+(L)=S (pfgd_P_gcd_terminal_multiple)))))))) /\\ ((forall pfc_index_gcd_terminal_multiple_productcoefficients. (exists pfa_gap_gcd_terminal_multiple_productcoefficientsbound. pfa_gap_gcd_terminal_multiple_productcoefficientsbound + S (pfc_index_gcd_terminal_multiple_productcoefficients) = (pfgd_P_gcd_terminal_multiple)) -> exists pfc_value_gcd_terminal_multiple_productcoefficients. ((((exists ff_h_pfp_gcd_terminal_multiple_productcoefficientsentry. ff_h_pfp_gcd_terminal_multiple_productcoefficientsentry + S (pfc_value_gcd_terminal_multiple_productcoefficients) = S ((S (pfc_index_gcd_terminal_multiple_productcoefficients)) * pfgd_pc_gcd_terminal_multiple)) /\\ exists ff_q_pfp_gcd_terminal_multiple_productcoefficientsentry. pfgd_pb_gcd_terminal_multiple = ff_q_pfp_gcd_terminal_multiple_productcoefficientsentry * S ((S (pfc_index_gcd_terminal_multiple_productcoefficients)) * pfgd_pc_gcd_terminal_multiple) + (pfc_value_gcd_terminal_multiple_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_terminal_multiple_productcoefficientscoefficient pfc_terms_scale_gcd_terminal_multiple_productcoefficientscoefficient pfc_natural_sum_gcd_terminal_multiple_productcoefficientscoefficient. ((forall pfc_index_gcd_terminal_multiple_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_terminal_multiple_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_terminal_multiple_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_terminal_multiple_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_terminal_multiple_productcoefficients))) -> exists pfc_value_gcd_terminal_multiple_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_terminal_multiple_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_terminal_multiple_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_terminal_multiple_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_terminal_multiple_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_multiple_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_terminal_multiple_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_terminal_multiple_productcoefficientscoefficient = ff_q_pfp_gcd_terminal_multiple_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_terminal_multiple_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_multiple_productcoefficientscoefficient) + (pfc_value_gcd_terminal_multiple_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm pfc_left_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm pfc_right_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_terminal_multiple_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_terminal_multiple_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_terminal_multiple_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_terminal_multiple)) /\\ ((((exists ff_h_pfp_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_terminal_multiple_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_terminal_multiple)) /\\ exists ff_q_pfp_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_terminal_multiple = ff_q_pfp_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_terminal_multiple_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_terminal_multiple) + (pfc_left_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_terminal_multiple)=(pfc_index_gcd_terminal_multiple_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_terminal_multiple_productcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_terminal_multiple_productcoefficientscoefficientdiagonal)=pfc_left_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_terminal_multiple_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum fs_v_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_terminal_multiple_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_terminal_multiple_productcoefficients))) * fs_v_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_terminal_multiple_productcoefficients))) * fs_v_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_terminal_multiple_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_terminal_multiple_productcoefficients)) -> exists fs_a_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_multiple_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_terminal_multiple_productcoefficientscoefficient = fs_q_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_multiple_productcoefficientscoefficient) + (fs_a_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_terminal_multiple_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_terminal_multiple_productcoefficientscoefficientresiduebound. pfa_gap_gcd_terminal_multiple_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_terminal_multiple_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_terminal_multiple_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_terminal_multiple_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_terminal_multiple_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_terminal_multiple_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_terminal_multiple_productcoefficients) + (p) * pfa_offset_right_gcd_terminal_multiple_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_terminal_multiple_equivalent pfrep_left_gcd_terminal_multiple_equivalent pfrep_right_gcd_terminal_multiple_equivalent. ((exists pfrep_position_gcd_terminal_multiple_equivalentfirst. ((pfrep_position_gcd_terminal_multiple_equivalentfirst+S (pfrep_power_gcd_terminal_multiple_equivalent)=(pfgd_P_gcd_terminal_multiple)) /\\ ((((exists ff_h_pfp_gcd_terminal_multiple_equivalentfirstentry. ff_h_pfp_gcd_terminal_multiple_equivalentfirstentry + S (pfrep_left_gcd_terminal_multiple_equivalent) = S ((S (pfrep_position_gcd_terminal_multiple_equivalentfirst)) * pfgd_pc_gcd_terminal_multiple)) /\\ exists ff_q_pfp_gcd_terminal_multiple_equivalentfirstentry. pfgd_pb_gcd_terminal_multiple = ff_q_pfp_gcd_terminal_multiple_equivalentfirstentry * S ((S (pfrep_position_gcd_terminal_multiple_equivalentfirst)) * pfgd_pc_gcd_terminal_multiple) + (pfrep_left_gcd_terminal_multiple_equivalent)))))) \\/ (((exists pfrep_gap_gcd_terminal_multiple_equivalentfirstoutside. pfrep_gap_gcd_terminal_multiple_equivalentfirstoutside+(pfgd_P_gcd_terminal_multiple)=(pfrep_power_gcd_terminal_multiple_equivalent)) /\\ (((pfrep_left_gcd_terminal_multiple_equivalent)=0))))) -> ((exists pfrep_position_gcd_terminal_multiple_equivalentsecond. ((pfrep_position_gcd_terminal_multiple_equivalentsecond+S (pfrep_power_gcd_terminal_multiple_equivalent)=(G)) /\\ ((((exists ff_h_pfp_gcd_terminal_multiple_equivalentsecondentry. ff_h_pfp_gcd_terminal_multiple_equivalentsecondentry + S (pfrep_right_gcd_terminal_multiple_equivalent) = S ((S (pfrep_position_gcd_terminal_multiple_equivalentsecond)) * gc)) /\\ exists ff_q_pfp_gcd_terminal_multiple_equivalentsecondentry. gb = ff_q_pfp_gcd_terminal_multiple_equivalentsecondentry * S ((S (pfrep_position_gcd_terminal_multiple_equivalentsecond)) * gc) + (pfrep_right_gcd_terminal_multiple_equivalent)))))) \\/ (((exists pfrep_gap_gcd_terminal_multiple_equivalentsecondoutside. pfrep_gap_gcd_terminal_multiple_equivalentsecondoutside+(G)=(pfrep_power_gcd_terminal_multiple_equivalent)) /\\ (((pfrep_right_gcd_terminal_multiple_equivalent)=0))))) -> pfrep_left_gcd_terminal_multiple_equivalent=pfrep_right_gcd_terminal_multiple_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_terminal_divisor_bounded. (exists fom_gap_pfp_gcd_terminal_divisor_bounded_index_bound. fom_gap_pfp_gcd_terminal_divisor_bounded_index_bound + S (fom_index_pfp_gcd_terminal_divisor_bounded) = L) -> exists fom_value_pfp_gcd_terminal_divisor_bounded. ((((exists fom_beta_height_pfp_gcd_terminal_divisor_bounded_entry. fom_beta_height_pfp_gcd_terminal_divisor_bounded_entry + S (fom_value_pfp_gcd_terminal_divisor_bounded) = S ((S (fom_index_pfp_gcd_terminal_divisor_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_divisor_bounded_entry. ab = fom_beta_quotient_pfp_gcd_terminal_divisor_bounded_entry * S ((S (fom_index_pfp_gcd_terminal_divisor_bounded)) * ac) + (fom_value_pfp_gcd_terminal_divisor_bounded))) /\\ (exists fom_gap_pfp_gcd_terminal_divisor_bounded_value_bound. fom_gap_pfp_gcd_terminal_divisor_bounded_value_bound + S (fom_value_pfp_gcd_terminal_divisor_bounded) = p))) /\\ ((exists pfgd_qb_gcd_terminal_divisor pfgd_qc_gcd_terminal_divisor pfgd_Q_gcd_terminal_divisor pfgd_pb_gcd_terminal_divisor pfgd_pc_gcd_terminal_divisor pfgd_P_gcd_terminal_divisor. ((((forall fom_index_pfp_gcd_terminal_divisor_productleft. (exists fom_gap_pfp_gcd_terminal_divisor_productleft_index_bound. fom_gap_pfp_gcd_terminal_divisor_productleft_index_bound + S (fom_index_pfp_gcd_terminal_divisor_productleft) = pfgd_Q_gcd_terminal_divisor) -> exists fom_value_pfp_gcd_terminal_divisor_productleft. ((((exists fom_beta_height_pfp_gcd_terminal_divisor_productleft_entry. fom_beta_height_pfp_gcd_terminal_divisor_productleft_entry + S (fom_value_pfp_gcd_terminal_divisor_productleft) = S ((S (fom_index_pfp_gcd_terminal_divisor_productleft)) * pfgd_qc_gcd_terminal_divisor)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_divisor_productleft_entry. pfgd_qb_gcd_terminal_divisor = fom_beta_quotient_pfp_gcd_terminal_divisor_productleft_entry * S ((S (fom_index_pfp_gcd_terminal_divisor_productleft)) * pfgd_qc_gcd_terminal_divisor) + (fom_value_pfp_gcd_terminal_divisor_productleft))) /\\ (exists fom_gap_pfp_gcd_terminal_divisor_productleft_value_bound. fom_gap_pfp_gcd_terminal_divisor_productleft_value_bound + S (fom_value_pfp_gcd_terminal_divisor_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_terminal_divisor_productright. (exists fom_gap_pfp_gcd_terminal_divisor_productright_index_bound. fom_gap_pfp_gcd_terminal_divisor_productright_index_bound + S (fom_index_pfp_gcd_terminal_divisor_productright) = G) -> exists fom_value_pfp_gcd_terminal_divisor_productright. ((((exists fom_beta_height_pfp_gcd_terminal_divisor_productright_entry. fom_beta_height_pfp_gcd_terminal_divisor_productright_entry + S (fom_value_pfp_gcd_terminal_divisor_productright) = S ((S (fom_index_pfp_gcd_terminal_divisor_productright)) * gc)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_divisor_productright_entry. gb = fom_beta_quotient_pfp_gcd_terminal_divisor_productright_entry * S ((S (fom_index_pfp_gcd_terminal_divisor_productright)) * gc) + (fom_value_pfp_gcd_terminal_divisor_productright))) /\\ (exists fom_gap_pfp_gcd_terminal_divisor_productright_value_bound. fom_gap_pfp_gcd_terminal_divisor_productright_value_bound + S (fom_value_pfp_gcd_terminal_divisor_productright) = p))) /\\ (((((((pfgd_Q_gcd_terminal_divisor)=0 \\/ (G)=0) /\\ (((pfgd_P_gcd_terminal_divisor)=0)))) \\/ (((~((pfgd_Q_gcd_terminal_divisor)=0)) /\\ (((~((G)=0)) /\\ (((pfgd_Q_gcd_terminal_divisor)+(G)=S (pfgd_P_gcd_terminal_divisor)))))))) /\\ ((forall pfc_index_gcd_terminal_divisor_productcoefficients. (exists pfa_gap_gcd_terminal_divisor_productcoefficientsbound. pfa_gap_gcd_terminal_divisor_productcoefficientsbound + S (pfc_index_gcd_terminal_divisor_productcoefficients) = (pfgd_P_gcd_terminal_divisor)) -> exists pfc_value_gcd_terminal_divisor_productcoefficients. ((((exists ff_h_pfp_gcd_terminal_divisor_productcoefficientsentry. ff_h_pfp_gcd_terminal_divisor_productcoefficientsentry + S (pfc_value_gcd_terminal_divisor_productcoefficients) = S ((S (pfc_index_gcd_terminal_divisor_productcoefficients)) * pfgd_pc_gcd_terminal_divisor)) /\\ exists ff_q_pfp_gcd_terminal_divisor_productcoefficientsentry. pfgd_pb_gcd_terminal_divisor = ff_q_pfp_gcd_terminal_divisor_productcoefficientsentry * S ((S (pfc_index_gcd_terminal_divisor_productcoefficients)) * pfgd_pc_gcd_terminal_divisor) + (pfc_value_gcd_terminal_divisor_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_terminal_divisor_productcoefficientscoefficient pfc_terms_scale_gcd_terminal_divisor_productcoefficientscoefficient pfc_natural_sum_gcd_terminal_divisor_productcoefficientscoefficient. ((forall pfc_index_gcd_terminal_divisor_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_terminal_divisor_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_terminal_divisor_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_terminal_divisor_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_terminal_divisor_productcoefficients))) -> exists pfc_value_gcd_terminal_divisor_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_terminal_divisor_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_terminal_divisor_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_terminal_divisor_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_terminal_divisor_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_divisor_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_terminal_divisor_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_terminal_divisor_productcoefficientscoefficient = ff_q_pfp_gcd_terminal_divisor_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_terminal_divisor_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_divisor_productcoefficientscoefficient) + (pfc_value_gcd_terminal_divisor_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm pfc_left_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm pfc_right_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_terminal_divisor_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_terminal_divisor_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_terminal_divisor_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_terminal_divisor)) /\\ ((((exists ff_h_pfp_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_terminal_divisor_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_terminal_divisor)) /\\ exists ff_q_pfp_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_terminal_divisor = ff_q_pfp_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_terminal_divisor_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_terminal_divisor) + (pfc_left_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_terminal_divisor)=(pfc_index_gcd_terminal_divisor_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm) = (G)) /\\ ((((exists ff_h_pfp_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm)) * gc)) /\\ exists ff_q_pfp_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermrightentry. gb = ff_q_pfp_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm)) * gc) + (pfc_right_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_terminal_divisor_productcoefficientscoefficientdiagonaltermrightoutside+(G)=(pfc_complement_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_terminal_divisor_productcoefficientscoefficientdiagonal)=pfc_left_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_terminal_divisor_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum fs_v_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_terminal_divisor_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_terminal_divisor_productcoefficients))) * fs_v_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_terminal_divisor_productcoefficients))) * fs_v_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_terminal_divisor_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_terminal_divisor_productcoefficients)) -> exists fs_a_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_divisor_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_terminal_divisor_productcoefficientscoefficient = fs_q_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_divisor_productcoefficientscoefficient) + (fs_a_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_terminal_divisor_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_terminal_divisor_productcoefficientscoefficientresiduebound. pfa_gap_gcd_terminal_divisor_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_terminal_divisor_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_terminal_divisor_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_terminal_divisor_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_terminal_divisor_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_terminal_divisor_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_terminal_divisor_productcoefficients) + (p) * pfa_offset_right_gcd_terminal_divisor_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_terminal_divisor_equivalent pfrep_left_gcd_terminal_divisor_equivalent pfrep_right_gcd_terminal_divisor_equivalent. ((exists pfrep_position_gcd_terminal_divisor_equivalentfirst. ((pfrep_position_gcd_terminal_divisor_equivalentfirst+S (pfrep_power_gcd_terminal_divisor_equivalent)=(pfgd_P_gcd_terminal_divisor)) /\\ ((((exists ff_h_pfp_gcd_terminal_divisor_equivalentfirstentry. ff_h_pfp_gcd_terminal_divisor_equivalentfirstentry + S (pfrep_left_gcd_terminal_divisor_equivalent) = S ((S (pfrep_position_gcd_terminal_divisor_equivalentfirst)) * pfgd_pc_gcd_terminal_divisor)) /\\ exists ff_q_pfp_gcd_terminal_divisor_equivalentfirstentry. pfgd_pb_gcd_terminal_divisor = ff_q_pfp_gcd_terminal_divisor_equivalentfirstentry * S ((S (pfrep_position_gcd_terminal_divisor_equivalentfirst)) * pfgd_pc_gcd_terminal_divisor) + (pfrep_left_gcd_terminal_divisor_equivalent)))))) \\/ (((exists pfrep_gap_gcd_terminal_divisor_equivalentfirstoutside. pfrep_gap_gcd_terminal_divisor_equivalentfirstoutside+(pfgd_P_gcd_terminal_divisor)=(pfrep_power_gcd_terminal_divisor_equivalent)) /\\ (((pfrep_left_gcd_terminal_divisor_equivalent)=0))))) -> ((exists pfrep_position_gcd_terminal_divisor_equivalentsecond. ((pfrep_position_gcd_terminal_divisor_equivalentsecond+S (pfrep_power_gcd_terminal_divisor_equivalent)=(L)) /\\ ((((exists ff_h_pfp_gcd_terminal_divisor_equivalentsecondentry. ff_h_pfp_gcd_terminal_divisor_equivalentsecondentry + S (pfrep_right_gcd_terminal_divisor_equivalent) = S ((S (pfrep_position_gcd_terminal_divisor_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_gcd_terminal_divisor_equivalentsecondentry. ab = ff_q_pfp_gcd_terminal_divisor_equivalentsecondentry * S ((S (pfrep_position_gcd_terminal_divisor_equivalentsecond)) * ac) + (pfrep_right_gcd_terminal_divisor_equivalent)))))) \\/ (((exists pfrep_gap_gcd_terminal_divisor_equivalentsecondoutside. pfrep_gap_gcd_terminal_divisor_equivalentsecondoutside+(L)=(pfrep_power_gcd_terminal_divisor_equivalent)) /\\ (((pfrep_right_gcd_terminal_divisor_equivalent)=0))))) -> pfrep_left_gcd_terminal_divisor_equivalent=pfrep_right_gcd_terminal_divisor_equivalent)))))))))))", "specialize prime_field_polynomial_normalized_right_associate_exists (p)", "specialize prime_field_polynomial_normalized_right_associate_exists (ab)", "specialize prime_field_polynomial_normalized_right_associate_exists (ac)", "specialize prime_field_polynomial_normalized_right_associate_exists (L)", "apply prime_field_polynomial_normalized_right_associate_exists", "exact hp", "exact hA", "cases hn", "cases hn_witness", "cases hn_witness_witness", "cases hn_witness_witness_witness", "cases hn_witness_witness_witness_right", "have hG : forall fom_index_pfp_gcd_terminal_bound. (exists fom_gap_pfp_gcd_terminal_bound_index_bound. fom_gap_pfp_gcd_terminal_bound_index_bound + S (fom_index_pfp_gcd_terminal_bound) = x2) -> exists fom_value_pfp_gcd_terminal_bound. ((((exists fom_beta_height_pfp_gcd_terminal_bound_entry. fom_beta_height_pfp_gcd_terminal_bound_entry + S (fom_value_pfp_gcd_terminal_bound) = S ((S (fom_index_pfp_gcd_terminal_bound)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_bound_entry. x = fom_beta_quotient_pfp_gcd_terminal_bound_entry * S ((S (fom_index_pfp_gcd_terminal_bound)) * x1) + (fom_value_pfp_gcd_terminal_bound))) /\\ (exists fom_gap_pfp_gcd_terminal_bound_value_bound. fom_gap_pfp_gcd_terminal_bound_value_bound + S (fom_value_pfp_gcd_terminal_bound) = p))", "specialize prime_field_polynomial_right_divides_divisor_bounded (p)", "specialize prime_field_polynomial_right_divides_divisor_bounded (x)", "specialize prime_field_polynomial_right_divides_divisor_bounded (x1)", "specialize prime_field_polynomial_right_divides_divisor_bounded (x2)", "specialize prime_field_polynomial_right_divides_divisor_bounded (ab)", "specialize prime_field_polynomial_right_divides_divisor_bounded (ac)", "specialize prime_field_polynomial_right_divides_divisor_bounded (L)", "apply prime_field_polynomial_right_divides_divisor_bounded", "exact hn_witness_witness_witness_right_right", "have hb : exists ub uc U. exists pfgb_pb_gcd_terminal_combination pfgb_pc_gcd_terminal_combination pfgb_P_gcd_terminal_combination pfgb_qb_gcd_terminal_combination pfgb_qc_gcd_terminal_combination pfgb_Q_gcd_terminal_combination. ((((forall fom_index_pfp_gcd_terminal_combination_leftleft. (exists fom_gap_pfp_gcd_terminal_combination_leftleft_index_bound. fom_gap_pfp_gcd_terminal_combination_leftleft_index_bound + S (fom_index_pfp_gcd_terminal_combination_leftleft) = U) -> exists fom_value_pfp_gcd_terminal_combination_leftleft. ((((exists fom_beta_height_pfp_gcd_terminal_combination_leftleft_entry. fom_beta_height_pfp_gcd_terminal_combination_leftleft_entry + S (fom_value_pfp_gcd_terminal_combination_leftleft) = S ((S (fom_index_pfp_gcd_terminal_combination_leftleft)) * uc)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_combination_leftleft_entry. ub = fom_beta_quotient_pfp_gcd_terminal_combination_leftleft_entry * S ((S (fom_index_pfp_gcd_terminal_combination_leftleft)) * uc) + (fom_value_pfp_gcd_terminal_combination_leftleft))) /\\ (exists fom_gap_pfp_gcd_terminal_combination_leftleft_value_bound. fom_gap_pfp_gcd_terminal_combination_leftleft_value_bound + S (fom_value_pfp_gcd_terminal_combination_leftleft) = p))) /\\ (((forall fom_index_pfp_gcd_terminal_combination_leftright. (exists fom_gap_pfp_gcd_terminal_combination_leftright_index_bound. fom_gap_pfp_gcd_terminal_combination_leftright_index_bound + S (fom_index_pfp_gcd_terminal_combination_leftright) = L) -> exists fom_value_pfp_gcd_terminal_combination_leftright. ((((exists fom_beta_height_pfp_gcd_terminal_combination_leftright_entry. fom_beta_height_pfp_gcd_terminal_combination_leftright_entry + S (fom_value_pfp_gcd_terminal_combination_leftright) = S ((S (fom_index_pfp_gcd_terminal_combination_leftright)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_combination_leftright_entry. ab = fom_beta_quotient_pfp_gcd_terminal_combination_leftright_entry * S ((S (fom_index_pfp_gcd_terminal_combination_leftright)) * ac) + (fom_value_pfp_gcd_terminal_combination_leftright))) /\\ (exists fom_gap_pfp_gcd_terminal_combination_leftright_value_bound. fom_gap_pfp_gcd_terminal_combination_leftright_value_bound + S (fom_value_pfp_gcd_terminal_combination_leftright) = p))) /\\ (((((((U)=0 \\/ (L)=0) /\\ (((pfgb_P_gcd_terminal_combination)=0)))) \\/ (((~((U)=0)) /\\ (((~((L)=0)) /\\ (((U)+(L)=S (pfgb_P_gcd_terminal_combination)))))))) /\\ ((forall pfc_index_gcd_terminal_combination_leftcoefficients. (exists pfa_gap_gcd_terminal_combination_leftcoefficientsbound. pfa_gap_gcd_terminal_combination_leftcoefficientsbound + S (pfc_index_gcd_terminal_combination_leftcoefficients) = (pfgb_P_gcd_terminal_combination)) -> exists pfc_value_gcd_terminal_combination_leftcoefficients. ((((exists ff_h_pfp_gcd_terminal_combination_leftcoefficientsentry. ff_h_pfp_gcd_terminal_combination_leftcoefficientsentry + S (pfc_value_gcd_terminal_combination_leftcoefficients) = S ((S (pfc_index_gcd_terminal_combination_leftcoefficients)) * pfgb_pc_gcd_terminal_combination)) /\\ exists ff_q_pfp_gcd_terminal_combination_leftcoefficientsentry. pfgb_pb_gcd_terminal_combination = ff_q_pfp_gcd_terminal_combination_leftcoefficientsentry * S ((S (pfc_index_gcd_terminal_combination_leftcoefficients)) * pfgb_pc_gcd_terminal_combination) + (pfc_value_gcd_terminal_combination_leftcoefficients))) /\\ ((exists pfc_terms_code_gcd_terminal_combination_leftcoefficientscoefficient pfc_terms_scale_gcd_terminal_combination_leftcoefficientscoefficient pfc_natural_sum_gcd_terminal_combination_leftcoefficientscoefficient. ((forall pfc_index_gcd_terminal_combination_leftcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_terminal_combination_leftcoefficientscoefficientdiagonalbound. pfa_gap_gcd_terminal_combination_leftcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_terminal_combination_leftcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_terminal_combination_leftcoefficients))) -> exists pfc_value_gcd_terminal_combination_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_terminal_combination_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_terminal_combination_leftcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_terminal_combination_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_terminal_combination_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_combination_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_terminal_combination_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_terminal_combination_leftcoefficientscoefficient = ff_q_pfp_gcd_terminal_combination_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_terminal_combination_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_combination_leftcoefficientscoefficient) + (pfc_value_gcd_terminal_combination_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm pfc_left_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm pfc_right_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_terminal_combination_leftcoefficientscoefficientdiagonal)+pfc_complement_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm=(pfc_index_gcd_terminal_combination_leftcoefficients)) /\\ ((((((exists pfa_gap_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_terminal_combination_leftcoefficientscoefficientdiagonal) = (U)) /\\ ((((exists ff_h_pfp_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_terminal_combination_leftcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_terminal_combination_leftcoefficientscoefficientdiagonal)) * uc) + (pfc_left_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermleftoutside+(U)=(pfc_index_gcd_terminal_combination_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_terminal_combination_leftcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_terminal_combination_leftcoefficientscoefficientdiagonal)=pfc_left_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm*pfc_right_gcd_terminal_combination_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum fs_v_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_terminal_combination_leftcoefficientscoefficient) = S ((S (S (pfc_index_gcd_terminal_combination_leftcoefficients))) * fs_v_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_terminal_combination_leftcoefficients))) * fs_v_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum) + (pfc_natural_sum_gcd_terminal_combination_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_terminal_combination_leftcoefficients)) -> exists fs_a_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_combination_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_terminal_combination_leftcoefficientscoefficient = fs_q_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_combination_leftcoefficientscoefficient) + (fs_a_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum) + (fs_r_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum) + (fs_s_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_terminal_combination_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_terminal_combination_leftcoefficientscoefficientresiduebound. pfa_gap_gcd_terminal_combination_leftcoefficientscoefficientresiduebound + S (pfc_value_gcd_terminal_combination_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_terminal_combination_leftcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_terminal_combination_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_terminal_combination_leftcoefficientscoefficient) + (p) * pfa_offset_left_gcd_terminal_combination_leftcoefficientscoefficientresiduecongruence = (pfc_value_gcd_terminal_combination_leftcoefficients) + (p) * pfa_offset_right_gcd_terminal_combination_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_gcd_terminal_combination_rightleft. (exists fom_gap_pfp_gcd_terminal_combination_rightleft_index_bound. fom_gap_pfp_gcd_terminal_combination_rightleft_index_bound + S (fom_index_pfp_gcd_terminal_combination_rightleft) = 0) -> exists fom_value_pfp_gcd_terminal_combination_rightleft. ((((exists fom_beta_height_pfp_gcd_terminal_combination_rightleft_entry. fom_beta_height_pfp_gcd_terminal_combination_rightleft_entry + S (fom_value_pfp_gcd_terminal_combination_rightleft) = S ((S (fom_index_pfp_gcd_terminal_combination_rightleft)) * 0)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_combination_rightleft_entry. 0 = fom_beta_quotient_pfp_gcd_terminal_combination_rightleft_entry * S ((S (fom_index_pfp_gcd_terminal_combination_rightleft)) * 0) + (fom_value_pfp_gcd_terminal_combination_rightleft))) /\\ (exists fom_gap_pfp_gcd_terminal_combination_rightleft_value_bound. fom_gap_pfp_gcd_terminal_combination_rightleft_value_bound + S (fom_value_pfp_gcd_terminal_combination_rightleft) = p))) /\\ (((forall fom_index_pfp_gcd_terminal_combination_rightright. (exists fom_gap_pfp_gcd_terminal_combination_rightright_index_bound. fom_gap_pfp_gcd_terminal_combination_rightright_index_bound + S (fom_index_pfp_gcd_terminal_combination_rightright) = 0) -> exists fom_value_pfp_gcd_terminal_combination_rightright. ((((exists fom_beta_height_pfp_gcd_terminal_combination_rightright_entry. fom_beta_height_pfp_gcd_terminal_combination_rightright_entry + S (fom_value_pfp_gcd_terminal_combination_rightright) = S ((S (fom_index_pfp_gcd_terminal_combination_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_combination_rightright_entry. bb = fom_beta_quotient_pfp_gcd_terminal_combination_rightright_entry * S ((S (fom_index_pfp_gcd_terminal_combination_rightright)) * bc) + (fom_value_pfp_gcd_terminal_combination_rightright))) /\\ (exists fom_gap_pfp_gcd_terminal_combination_rightright_value_bound. fom_gap_pfp_gcd_terminal_combination_rightright_value_bound + S (fom_value_pfp_gcd_terminal_combination_rightright) = p))) /\\ (((((((0)=0 \\/ (0)=0) /\\ (((pfgb_Q_gcd_terminal_combination)=0)))) \\/ (((~((0)=0)) /\\ (((~((0)=0)) /\\ (((0)+(0)=S (pfgb_Q_gcd_terminal_combination)))))))) /\\ ((forall pfc_index_gcd_terminal_combination_rightcoefficients. (exists pfa_gap_gcd_terminal_combination_rightcoefficientsbound. pfa_gap_gcd_terminal_combination_rightcoefficientsbound + S (pfc_index_gcd_terminal_combination_rightcoefficients) = (pfgb_Q_gcd_terminal_combination)) -> exists pfc_value_gcd_terminal_combination_rightcoefficients. ((((exists ff_h_pfp_gcd_terminal_combination_rightcoefficientsentry. ff_h_pfp_gcd_terminal_combination_rightcoefficientsentry + S (pfc_value_gcd_terminal_combination_rightcoefficients) = S ((S (pfc_index_gcd_terminal_combination_rightcoefficients)) * pfgb_qc_gcd_terminal_combination)) /\\ exists ff_q_pfp_gcd_terminal_combination_rightcoefficientsentry. pfgb_qb_gcd_terminal_combination = ff_q_pfp_gcd_terminal_combination_rightcoefficientsentry * S ((S (pfc_index_gcd_terminal_combination_rightcoefficients)) * pfgb_qc_gcd_terminal_combination) + (pfc_value_gcd_terminal_combination_rightcoefficients))) /\\ ((exists pfc_terms_code_gcd_terminal_combination_rightcoefficientscoefficient pfc_terms_scale_gcd_terminal_combination_rightcoefficientscoefficient pfc_natural_sum_gcd_terminal_combination_rightcoefficientscoefficient. ((forall pfc_index_gcd_terminal_combination_rightcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_terminal_combination_rightcoefficientscoefficientdiagonalbound. pfa_gap_gcd_terminal_combination_rightcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_terminal_combination_rightcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_terminal_combination_rightcoefficients))) -> exists pfc_value_gcd_terminal_combination_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_terminal_combination_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_terminal_combination_rightcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_terminal_combination_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_terminal_combination_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_combination_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_terminal_combination_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_terminal_combination_rightcoefficientscoefficient = ff_q_pfp_gcd_terminal_combination_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_terminal_combination_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_combination_rightcoefficientscoefficient) + (pfc_value_gcd_terminal_combination_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm pfc_left_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm pfc_right_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_terminal_combination_rightcoefficientscoefficientdiagonal)+pfc_complement_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm=(pfc_index_gcd_terminal_combination_rightcoefficients)) /\\ ((((((exists pfa_gap_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_terminal_combination_rightcoefficientscoefficientdiagonal) = (0)) /\\ ((((exists ff_h_pfp_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_terminal_combination_rightcoefficientscoefficientdiagonal)) * 0)) /\\ exists ff_q_pfp_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermleftentry. 0 = ff_q_pfp_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_terminal_combination_rightcoefficientscoefficientdiagonal)) * 0) + (pfc_left_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermleftoutside+(0)=(pfc_index_gcd_terminal_combination_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm) = (0)) /\\ ((((exists ff_h_pfp_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_terminal_combination_rightcoefficientscoefficientdiagonaltermrightoutside+(0)=(pfc_complement_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_terminal_combination_rightcoefficientscoefficientdiagonal)=pfc_left_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm*pfc_right_gcd_terminal_combination_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum fs_v_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_terminal_combination_rightcoefficientscoefficient) = S ((S (S (pfc_index_gcd_terminal_combination_rightcoefficients))) * fs_v_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_terminal_combination_rightcoefficients))) * fs_v_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum) + (pfc_natural_sum_gcd_terminal_combination_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_terminal_combination_rightcoefficients)) -> exists fs_a_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_combination_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_terminal_combination_rightcoefficientscoefficient = fs_q_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_combination_rightcoefficientscoefficient) + (fs_a_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum) + (fs_r_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum) + (fs_s_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_terminal_combination_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_terminal_combination_rightcoefficientscoefficientresiduebound. pfa_gap_gcd_terminal_combination_rightcoefficientscoefficientresiduebound + S (pfc_value_gcd_terminal_combination_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_terminal_combination_rightcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_terminal_combination_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_terminal_combination_rightcoefficientscoefficient) + (p) * pfa_offset_left_gcd_terminal_combination_rightcoefficientscoefficientresiduecongruence = (pfc_value_gcd_terminal_combination_rightcoefficients) + (p) * pfa_offset_right_gcd_terminal_combination_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_gcd_terminal_combination_sum_left_bounded. (exists fom_gap_pfp_gcd_terminal_combination_sum_left_bounded_index_bound. fom_gap_pfp_gcd_terminal_combination_sum_left_bounded_index_bound + S (fom_index_pfp_gcd_terminal_combination_sum_left_bounded) = pfgb_P_gcd_terminal_combination) -> exists fom_value_pfp_gcd_terminal_combination_sum_left_bounded. ((((exists fom_beta_height_pfp_gcd_terminal_combination_sum_left_bounded_entry. fom_beta_height_pfp_gcd_terminal_combination_sum_left_bounded_entry + S (fom_value_pfp_gcd_terminal_combination_sum_left_bounded) = S ((S (fom_index_pfp_gcd_terminal_combination_sum_left_bounded)) * pfgb_pc_gcd_terminal_combination)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_combination_sum_left_bounded_entry. pfgb_pb_gcd_terminal_combination = fom_beta_quotient_pfp_gcd_terminal_combination_sum_left_bounded_entry * S ((S (fom_index_pfp_gcd_terminal_combination_sum_left_bounded)) * pfgb_pc_gcd_terminal_combination) + (fom_value_pfp_gcd_terminal_combination_sum_left_bounded))) /\\ (exists fom_gap_pfp_gcd_terminal_combination_sum_left_bounded_value_bound. fom_gap_pfp_gcd_terminal_combination_sum_left_bounded_value_bound + S (fom_value_pfp_gcd_terminal_combination_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_terminal_combination_sum_right_bounded. (exists fom_gap_pfp_gcd_terminal_combination_sum_right_bounded_index_bound. fom_gap_pfp_gcd_terminal_combination_sum_right_bounded_index_bound + S (fom_index_pfp_gcd_terminal_combination_sum_right_bounded) = pfgb_Q_gcd_terminal_combination) -> exists fom_value_pfp_gcd_terminal_combination_sum_right_bounded. ((((exists fom_beta_height_pfp_gcd_terminal_combination_sum_right_bounded_entry. fom_beta_height_pfp_gcd_terminal_combination_sum_right_bounded_entry + S (fom_value_pfp_gcd_terminal_combination_sum_right_bounded) = S ((S (fom_index_pfp_gcd_terminal_combination_sum_right_bounded)) * pfgb_qc_gcd_terminal_combination)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_combination_sum_right_bounded_entry. pfgb_qb_gcd_terminal_combination = fom_beta_quotient_pfp_gcd_terminal_combination_sum_right_bounded_entry * S ((S (fom_index_pfp_gcd_terminal_combination_sum_right_bounded)) * pfgb_qc_gcd_terminal_combination) + (fom_value_pfp_gcd_terminal_combination_sum_right_bounded))) /\\ (exists fom_gap_pfp_gcd_terminal_combination_sum_right_bounded_value_bound. fom_gap_pfp_gcd_terminal_combination_sum_right_bounded_value_bound + S (fom_value_pfp_gcd_terminal_combination_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_terminal_combination_sum_result_bounded. (exists fom_gap_pfp_gcd_terminal_combination_sum_result_bounded_index_bound. fom_gap_pfp_gcd_terminal_combination_sum_result_bounded_index_bound + S (fom_index_pfp_gcd_terminal_combination_sum_result_bounded) = x2) -> exists fom_value_pfp_gcd_terminal_combination_sum_result_bounded. ((((exists fom_beta_height_pfp_gcd_terminal_combination_sum_result_bounded_entry. fom_beta_height_pfp_gcd_terminal_combination_sum_result_bounded_entry + S (fom_value_pfp_gcd_terminal_combination_sum_result_bounded) = S ((S (fom_index_pfp_gcd_terminal_combination_sum_result_bounded)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_combination_sum_result_bounded_entry. x = fom_beta_quotient_pfp_gcd_terminal_combination_sum_result_bounded_entry * S ((S (fom_index_pfp_gcd_terminal_combination_sum_result_bounded)) * x1) + (fom_value_pfp_gcd_terminal_combination_sum_result_bounded))) /\\ (exists fom_gap_pfp_gcd_terminal_combination_sum_result_bounded_value_bound. fom_gap_pfp_gcd_terminal_combination_sum_result_bounded_value_bound + S (fom_value_pfp_gcd_terminal_combination_sum_result_bounded) = p))) /\\ ((exists pfga_ub_gcd_terminal_combination_sum pfga_uc_gcd_terminal_combination_sum pfga_vb_gcd_terminal_combination_sum pfga_vc_gcd_terminal_combination_sum pfga_tb_gcd_terminal_combination_sum pfga_tc_gcd_terminal_combination_sum pfga_K_gcd_terminal_combination_sum. ((((forall pfrep_power_gcd_terminal_combination_sum_left pfrep_left_gcd_terminal_combination_sum_left pfrep_right_gcd_terminal_combination_sum_left. ((exists pfrep_position_gcd_terminal_combination_sum_leftfirst. ((pfrep_position_gcd_terminal_combination_sum_leftfirst+S (pfrep_power_gcd_terminal_combination_sum_left)=(pfgb_P_gcd_terminal_combination)) /\\ ((((exists ff_h_pfp_gcd_terminal_combination_sum_leftfirstentry. ff_h_pfp_gcd_terminal_combination_sum_leftfirstentry + S (pfrep_left_gcd_terminal_combination_sum_left) = S ((S (pfrep_position_gcd_terminal_combination_sum_leftfirst)) * pfgb_pc_gcd_terminal_combination)) /\\ exists ff_q_pfp_gcd_terminal_combination_sum_leftfirstentry. pfgb_pb_gcd_terminal_combination = ff_q_pfp_gcd_terminal_combination_sum_leftfirstentry * S ((S (pfrep_position_gcd_terminal_combination_sum_leftfirst)) * pfgb_pc_gcd_terminal_combination) + (pfrep_left_gcd_terminal_combination_sum_left)))))) \\/ (((exists pfrep_gap_gcd_terminal_combination_sum_leftfirstoutside. pfrep_gap_gcd_terminal_combination_sum_leftfirstoutside+(pfgb_P_gcd_terminal_combination)=(pfrep_power_gcd_terminal_combination_sum_left)) /\\ (((pfrep_left_gcd_terminal_combination_sum_left)=0))))) -> ((exists pfrep_position_gcd_terminal_combination_sum_leftsecond. ((pfrep_position_gcd_terminal_combination_sum_leftsecond+S (pfrep_power_gcd_terminal_combination_sum_left)=(pfga_K_gcd_terminal_combination_sum)) /\\ ((((exists ff_h_pfp_gcd_terminal_combination_sum_leftsecondentry. ff_h_pfp_gcd_terminal_combination_sum_leftsecondentry + S (pfrep_right_gcd_terminal_combination_sum_left) = S ((S (pfrep_position_gcd_terminal_combination_sum_leftsecond)) * pfga_uc_gcd_terminal_combination_sum)) /\\ exists ff_q_pfp_gcd_terminal_combination_sum_leftsecondentry. pfga_ub_gcd_terminal_combination_sum = ff_q_pfp_gcd_terminal_combination_sum_leftsecondentry * S ((S (pfrep_position_gcd_terminal_combination_sum_leftsecond)) * pfga_uc_gcd_terminal_combination_sum) + (pfrep_right_gcd_terminal_combination_sum_left)))))) \\/ (((exists pfrep_gap_gcd_terminal_combination_sum_leftsecondoutside. pfrep_gap_gcd_terminal_combination_sum_leftsecondoutside+(pfga_K_gcd_terminal_combination_sum)=(pfrep_power_gcd_terminal_combination_sum_left)) /\\ (((pfrep_right_gcd_terminal_combination_sum_left)=0))))) -> pfrep_left_gcd_terminal_combination_sum_left=pfrep_right_gcd_terminal_combination_sum_left) /\\ ((forall pfrep_power_gcd_terminal_combination_sum_right pfrep_left_gcd_terminal_combination_sum_right pfrep_right_gcd_terminal_combination_sum_right. ((exists pfrep_position_gcd_terminal_combination_sum_rightfirst. ((pfrep_position_gcd_terminal_combination_sum_rightfirst+S (pfrep_power_gcd_terminal_combination_sum_right)=(pfgb_Q_gcd_terminal_combination)) /\\ ((((exists ff_h_pfp_gcd_terminal_combination_sum_rightfirstentry. ff_h_pfp_gcd_terminal_combination_sum_rightfirstentry + S (pfrep_left_gcd_terminal_combination_sum_right) = S ((S (pfrep_position_gcd_terminal_combination_sum_rightfirst)) * pfgb_qc_gcd_terminal_combination)) /\\ exists ff_q_pfp_gcd_terminal_combination_sum_rightfirstentry. pfgb_qb_gcd_terminal_combination = ff_q_pfp_gcd_terminal_combination_sum_rightfirstentry * S ((S (pfrep_position_gcd_terminal_combination_sum_rightfirst)) * pfgb_qc_gcd_terminal_combination) + (pfrep_left_gcd_terminal_combination_sum_right)))))) \\/ (((exists pfrep_gap_gcd_terminal_combination_sum_rightfirstoutside. pfrep_gap_gcd_terminal_combination_sum_rightfirstoutside+(pfgb_Q_gcd_terminal_combination)=(pfrep_power_gcd_terminal_combination_sum_right)) /\\ (((pfrep_left_gcd_terminal_combination_sum_right)=0))))) -> ((exists pfrep_position_gcd_terminal_combination_sum_rightsecond. ((pfrep_position_gcd_terminal_combination_sum_rightsecond+S (pfrep_power_gcd_terminal_combination_sum_right)=(pfga_K_gcd_terminal_combination_sum)) /\\ ((((exists ff_h_pfp_gcd_terminal_combination_sum_rightsecondentry. ff_h_pfp_gcd_terminal_combination_sum_rightsecondentry + S (pfrep_right_gcd_terminal_combination_sum_right) = S ((S (pfrep_position_gcd_terminal_combination_sum_rightsecond)) * pfga_vc_gcd_terminal_combination_sum)) /\\ exists ff_q_pfp_gcd_terminal_combination_sum_rightsecondentry. pfga_vb_gcd_terminal_combination_sum = ff_q_pfp_gcd_terminal_combination_sum_rightsecondentry * S ((S (pfrep_position_gcd_terminal_combination_sum_rightsecond)) * pfga_vc_gcd_terminal_combination_sum) + (pfrep_right_gcd_terminal_combination_sum_right)))))) \\/ (((exists pfrep_gap_gcd_terminal_combination_sum_rightsecondoutside. pfrep_gap_gcd_terminal_combination_sum_rightsecondoutside+(pfga_K_gcd_terminal_combination_sum)=(pfrep_power_gcd_terminal_combination_sum_right)) /\\ (((pfrep_right_gcd_terminal_combination_sum_right)=0))))) -> pfrep_left_gcd_terminal_combination_sum_right=pfrep_right_gcd_terminal_combination_sum_right)))) /\\ (((forall pfp_index_gcd_terminal_combination_sum_add. (exists pfa_gap_gcd_terminal_combination_sum_addindex. pfa_gap_gcd_terminal_combination_sum_addindex + S (pfp_index_gcd_terminal_combination_sum_add) = (pfga_K_gcd_terminal_combination_sum)) -> exists pfp_left_gcd_terminal_combination_sum_add pfp_right_gcd_terminal_combination_sum_add pfp_value_gcd_terminal_combination_sum_add. ((((exists ff_h_pfp_gcd_terminal_combination_sum_addleft. ff_h_pfp_gcd_terminal_combination_sum_addleft + S (pfp_left_gcd_terminal_combination_sum_add) = S ((S (pfp_index_gcd_terminal_combination_sum_add)) * pfga_uc_gcd_terminal_combination_sum)) /\\ exists ff_q_pfp_gcd_terminal_combination_sum_addleft. pfga_ub_gcd_terminal_combination_sum = ff_q_pfp_gcd_terminal_combination_sum_addleft * S ((S (pfp_index_gcd_terminal_combination_sum_add)) * pfga_uc_gcd_terminal_combination_sum) + (pfp_left_gcd_terminal_combination_sum_add))) /\\ (((((exists ff_h_pfp_gcd_terminal_combination_sum_addright. ff_h_pfp_gcd_terminal_combination_sum_addright + S (pfp_right_gcd_terminal_combination_sum_add) = S ((S (pfp_index_gcd_terminal_combination_sum_add)) * pfga_vc_gcd_terminal_combination_sum)) /\\ exists ff_q_pfp_gcd_terminal_combination_sum_addright. pfga_vb_gcd_terminal_combination_sum = ff_q_pfp_gcd_terminal_combination_sum_addright * S ((S (pfp_index_gcd_terminal_combination_sum_add)) * pfga_vc_gcd_terminal_combination_sum) + (pfp_right_gcd_terminal_combination_sum_add))) /\\ (((((exists ff_h_pfp_gcd_terminal_combination_sum_addtarget. ff_h_pfp_gcd_terminal_combination_sum_addtarget + S (pfp_value_gcd_terminal_combination_sum_add) = S ((S (pfp_index_gcd_terminal_combination_sum_add)) * pfga_tc_gcd_terminal_combination_sum)) /\\ exists ff_q_pfp_gcd_terminal_combination_sum_addtarget. pfga_tb_gcd_terminal_combination_sum = ff_q_pfp_gcd_terminal_combination_sum_addtarget * S ((S (pfp_index_gcd_terminal_combination_sum_add)) * pfga_tc_gcd_terminal_combination_sum) + (pfp_value_gcd_terminal_combination_sum_add))) /\\ ((((exists pfa_gap_gcd_terminal_combination_sum_addoperationleft. pfa_gap_gcd_terminal_combination_sum_addoperationleft + S (pfp_left_gcd_terminal_combination_sum_add) = (p)) /\\ (((exists pfa_gap_gcd_terminal_combination_sum_addoperationright. pfa_gap_gcd_terminal_combination_sum_addoperationright + S (pfp_right_gcd_terminal_combination_sum_add) = (p)) /\\ ((((exists pfa_gap_gcd_terminal_combination_sum_addoperationresultbound. pfa_gap_gcd_terminal_combination_sum_addoperationresultbound + S (pfp_value_gcd_terminal_combination_sum_add) = (p)) /\\ ((exists pfa_offset_left_gcd_terminal_combination_sum_addoperationresultcongruence pfa_offset_right_gcd_terminal_combination_sum_addoperationresultcongruence. ((pfp_left_gcd_terminal_combination_sum_add) + (pfp_right_gcd_terminal_combination_sum_add)) + (p) * pfa_offset_left_gcd_terminal_combination_sum_addoperationresultcongruence = (pfp_value_gcd_terminal_combination_sum_add) + (p) * pfa_offset_right_gcd_terminal_combination_sum_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_gcd_terminal_combination_sum_result pfrep_left_gcd_terminal_combination_sum_result pfrep_right_gcd_terminal_combination_sum_result. ((exists pfrep_position_gcd_terminal_combination_sum_resultfirst. ((pfrep_position_gcd_terminal_combination_sum_resultfirst+S (pfrep_power_gcd_terminal_combination_sum_result)=(pfga_K_gcd_terminal_combination_sum)) /\\ ((((exists ff_h_pfp_gcd_terminal_combination_sum_resultfirstentry. ff_h_pfp_gcd_terminal_combination_sum_resultfirstentry + S (pfrep_left_gcd_terminal_combination_sum_result) = S ((S (pfrep_position_gcd_terminal_combination_sum_resultfirst)) * pfga_tc_gcd_terminal_combination_sum)) /\\ exists ff_q_pfp_gcd_terminal_combination_sum_resultfirstentry. pfga_tb_gcd_terminal_combination_sum = ff_q_pfp_gcd_terminal_combination_sum_resultfirstentry * S ((S (pfrep_position_gcd_terminal_combination_sum_resultfirst)) * pfga_tc_gcd_terminal_combination_sum) + (pfrep_left_gcd_terminal_combination_sum_result)))))) \\/ (((exists pfrep_gap_gcd_terminal_combination_sum_resultfirstoutside. pfrep_gap_gcd_terminal_combination_sum_resultfirstoutside+(pfga_K_gcd_terminal_combination_sum)=(pfrep_power_gcd_terminal_combination_sum_result)) /\\ (((pfrep_left_gcd_terminal_combination_sum_result)=0))))) -> ((exists pfrep_position_gcd_terminal_combination_sum_resultsecond. ((pfrep_position_gcd_terminal_combination_sum_resultsecond+S (pfrep_power_gcd_terminal_combination_sum_result)=(x2)) /\\ ((((exists ff_h_pfp_gcd_terminal_combination_sum_resultsecondentry. ff_h_pfp_gcd_terminal_combination_sum_resultsecondentry + S (pfrep_right_gcd_terminal_combination_sum_result) = S ((S (pfrep_position_gcd_terminal_combination_sum_resultsecond)) * x1)) /\\ exists ff_q_pfp_gcd_terminal_combination_sum_resultsecondentry. x = ff_q_pfp_gcd_terminal_combination_sum_resultsecondentry * S ((S (pfrep_position_gcd_terminal_combination_sum_resultsecond)) * x1) + (pfrep_right_gcd_terminal_combination_sum_result)))))) \\/ (((exists pfrep_gap_gcd_terminal_combination_sum_resultsecondoutside. pfrep_gap_gcd_terminal_combination_sum_resultsecondoutside+(x2)=(pfrep_power_gcd_terminal_combination_sum_result)) /\\ (((pfrep_right_gcd_terminal_combination_sum_result)=0))))) -> pfrep_left_gcd_terminal_combination_sum_result=pfrep_right_gcd_terminal_combination_sum_result)))))))))))))))))", "specialize prime_field_polynomial_bezout_from_right_multiple (p)", "specialize prime_field_polynomial_bezout_from_right_multiple (ab)", "specialize prime_field_polynomial_bezout_from_right_multiple (ac)", "specialize prime_field_polynomial_bezout_from_right_multiple (L)", "specialize prime_field_polynomial_bezout_from_right_multiple (bb)", "specialize prime_field_polynomial_bezout_from_right_multiple (bc)", "specialize prime_field_polynomial_bezout_from_right_multiple (0)", "specialize prime_field_polynomial_bezout_from_right_multiple (x)", "specialize prime_field_polynomial_bezout_from_right_multiple (x1)", "specialize prime_field_polynomial_bezout_from_right_multiple (x2)", "apply prime_field_polynomial_bezout_from_right_multiple", "exact hp", "specialize matrix_rank_bounded_prefix_empty (bb)", "specialize matrix_rank_bounded_prefix_empty (bc)", "specialize matrix_rank_bounded_prefix_empty (p)", "apply matrix_rank_bounded_prefix_empty", "exact hn_witness_witness_witness_right_left", "cases hb", "cases hb_witness", "cases hb_witness_witness", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "exists 0", "exists 0", "exists 0", "split", "exact hn_witness_witness_witness_left", "split", "split", "exact hn_witness_witness_witness_right_right", "specialize prime_field_polynomial_right_divides_empty (p)", "specialize prime_field_polynomial_right_divides_empty (x)", "specialize prime_field_polynomial_right_divides_empty (x1)", "specialize prime_field_polynomial_right_divides_empty (x2)", "specialize prime_field_polynomial_right_divides_empty (bb)", "specialize prime_field_polynomial_right_divides_empty (bc)", "apply prime_field_polynomial_right_divides_empty", "exact hG", "exact hb_witness_witness_witness" ], "script_sha256": "c43833cc19f1c7da3fb2b836a1a0fb5eb079e48156d6e1f649cc2dd7e31cf0e8", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_existence_candidate.py", "sha256": "81f2f48dd2e81894c7a267453646eb6f2b6f9bd3ee320386d8c561f6b9f8b8ca" }, "statement": "forall p ab ac L bb bc. (~((p) = 1) /\\ forall pfa_factor_left_gcd_terminal_prime pfa_factor_right_gcd_terminal_prime. (p) = pfa_factor_left_gcd_terminal_prime * pfa_factor_right_gcd_terminal_prime -> pfa_factor_left_gcd_terminal_prime = 1 \\/ pfa_factor_right_gcd_terminal_prime = 1) -> (forall fom_index_pfp_gcd_terminal_input. (exists fom_gap_pfp_gcd_terminal_input_index_bound. fom_gap_pfp_gcd_terminal_input_index_bound + S (fom_index_pfp_gcd_terminal_input) = L) -> exists fom_value_pfp_gcd_terminal_input. ((((exists fom_beta_height_pfp_gcd_terminal_input_entry. fom_beta_height_pfp_gcd_terminal_input_entry + S (fom_value_pfp_gcd_terminal_input) = S ((S (fom_index_pfp_gcd_terminal_input)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_input_entry. ab = fom_beta_quotient_pfp_gcd_terminal_input_entry * S ((S (fom_index_pfp_gcd_terminal_input)) * ac) + (fom_value_pfp_gcd_terminal_input))) /\\ (exists fom_gap_pfp_gcd_terminal_input_value_bound. fom_gap_pfp_gcd_terminal_input_value_bound + S (fom_value_pfp_gcd_terminal_input) = p))) -> (exists pfgs_gb_gcd_terminal_result pfgs_gc_gcd_terminal_result pfgs_G_gcd_terminal_result pfgs_ub_gcd_terminal_result pfgs_uc_gcd_terminal_result pfgs_U_gcd_terminal_result pfgs_vb_gcd_terminal_result pfgs_vc_gcd_terminal_result pfgs_V_gcd_terminal_result. (((pfgs_G_gcd_terminal_result)=0 \\/ (((~((pfgs_G_gcd_terminal_result) = 0)) /\\ (((forall fom_index_pfp_gcd_terminal_result_witness_normal_moniccoefficients. (exists fom_gap_pfp_gcd_terminal_result_witness_normal_moniccoefficients_index_bound. fom_gap_pfp_gcd_terminal_result_witness_normal_moniccoefficients_index_bound + S (fom_index_pfp_gcd_terminal_result_witness_normal_moniccoefficients) = pfgs_G_gcd_terminal_result) -> exists fom_value_pfp_gcd_terminal_result_witness_normal_moniccoefficients. ((((exists fom_beta_height_pfp_gcd_terminal_result_witness_normal_moniccoefficients_entry. fom_beta_height_pfp_gcd_terminal_result_witness_normal_moniccoefficients_entry + S (fom_value_pfp_gcd_terminal_result_witness_normal_moniccoefficients) = S ((S (fom_index_pfp_gcd_terminal_result_witness_normal_moniccoefficients)) * pfgs_gc_gcd_terminal_result)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_result_witness_normal_moniccoefficients_entry. pfgs_gb_gcd_terminal_result = fom_beta_quotient_pfp_gcd_terminal_result_witness_normal_moniccoefficients_entry * S ((S (fom_index_pfp_gcd_terminal_result_witness_normal_moniccoefficients)) * pfgs_gc_gcd_terminal_result) + (fom_value_pfp_gcd_terminal_result_witness_normal_moniccoefficients))) /\\ (exists fom_gap_pfp_gcd_terminal_result_witness_normal_moniccoefficients_value_bound. fom_gap_pfp_gcd_terminal_result_witness_normal_moniccoefficients_value_bound + S (fom_value_pfp_gcd_terminal_result_witness_normal_moniccoefficients) = p))) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_normal_monicleading. ff_h_pfp_gcd_terminal_result_witness_normal_monicleading + S (1) = S ((S (0)) * pfgs_gc_gcd_terminal_result)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_normal_monicleading. pfgs_gb_gcd_terminal_result = ff_q_pfp_gcd_terminal_result_witness_normal_monicleading * S ((S (0)) * pfgs_gc_gcd_terminal_result) + (1))))))))) /\\ (((((((forall fom_index_pfp_gcd_terminal_result_witness_common_left_bounded. (exists fom_gap_pfp_gcd_terminal_result_witness_common_left_bounded_index_bound. fom_gap_pfp_gcd_terminal_result_witness_common_left_bounded_index_bound + S (fom_index_pfp_gcd_terminal_result_witness_common_left_bounded) = L) -> exists fom_value_pfp_gcd_terminal_result_witness_common_left_bounded. ((((exists fom_beta_height_pfp_gcd_terminal_result_witness_common_left_bounded_entry. fom_beta_height_pfp_gcd_terminal_result_witness_common_left_bounded_entry + S (fom_value_pfp_gcd_terminal_result_witness_common_left_bounded) = S ((S (fom_index_pfp_gcd_terminal_result_witness_common_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_result_witness_common_left_bounded_entry. ab = fom_beta_quotient_pfp_gcd_terminal_result_witness_common_left_bounded_entry * S ((S (fom_index_pfp_gcd_terminal_result_witness_common_left_bounded)) * ac) + (fom_value_pfp_gcd_terminal_result_witness_common_left_bounded))) /\\ (exists fom_gap_pfp_gcd_terminal_result_witness_common_left_bounded_value_bound. fom_gap_pfp_gcd_terminal_result_witness_common_left_bounded_value_bound + S (fom_value_pfp_gcd_terminal_result_witness_common_left_bounded) = p))) /\\ ((exists pfgd_qb_gcd_terminal_result_witness_common_left pfgd_qc_gcd_terminal_result_witness_common_left pfgd_Q_gcd_terminal_result_witness_common_left pfgd_pb_gcd_terminal_result_witness_common_left pfgd_pc_gcd_terminal_result_witness_common_left pfgd_P_gcd_terminal_result_witness_common_left. ((((forall fom_index_pfp_gcd_terminal_result_witness_common_left_productleft. (exists fom_gap_pfp_gcd_terminal_result_witness_common_left_productleft_index_bound. fom_gap_pfp_gcd_terminal_result_witness_common_left_productleft_index_bound + S (fom_index_pfp_gcd_terminal_result_witness_common_left_productleft) = pfgd_Q_gcd_terminal_result_witness_common_left) -> exists fom_value_pfp_gcd_terminal_result_witness_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_terminal_result_witness_common_left_productleft_entry. fom_beta_height_pfp_gcd_terminal_result_witness_common_left_productleft_entry + S (fom_value_pfp_gcd_terminal_result_witness_common_left_productleft) = S ((S (fom_index_pfp_gcd_terminal_result_witness_common_left_productleft)) * pfgd_qc_gcd_terminal_result_witness_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_result_witness_common_left_productleft_entry. pfgd_qb_gcd_terminal_result_witness_common_left = fom_beta_quotient_pfp_gcd_terminal_result_witness_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_terminal_result_witness_common_left_productleft)) * pfgd_qc_gcd_terminal_result_witness_common_left) + (fom_value_pfp_gcd_terminal_result_witness_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_terminal_result_witness_common_left_productleft_value_bound. fom_gap_pfp_gcd_terminal_result_witness_common_left_productleft_value_bound + S (fom_value_pfp_gcd_terminal_result_witness_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_terminal_result_witness_common_left_productright. (exists fom_gap_pfp_gcd_terminal_result_witness_common_left_productright_index_bound. fom_gap_pfp_gcd_terminal_result_witness_common_left_productright_index_bound + S (fom_index_pfp_gcd_terminal_result_witness_common_left_productright) = pfgs_G_gcd_terminal_result) -> exists fom_value_pfp_gcd_terminal_result_witness_common_left_productright. ((((exists fom_beta_height_pfp_gcd_terminal_result_witness_common_left_productright_entry. fom_beta_height_pfp_gcd_terminal_result_witness_common_left_productright_entry + S (fom_value_pfp_gcd_terminal_result_witness_common_left_productright) = S ((S (fom_index_pfp_gcd_terminal_result_witness_common_left_productright)) * pfgs_gc_gcd_terminal_result)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_result_witness_common_left_productright_entry. pfgs_gb_gcd_terminal_result = fom_beta_quotient_pfp_gcd_terminal_result_witness_common_left_productright_entry * S ((S (fom_index_pfp_gcd_terminal_result_witness_common_left_productright)) * pfgs_gc_gcd_terminal_result) + (fom_value_pfp_gcd_terminal_result_witness_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_terminal_result_witness_common_left_productright_value_bound. fom_gap_pfp_gcd_terminal_result_witness_common_left_productright_value_bound + S (fom_value_pfp_gcd_terminal_result_witness_common_left_productright) = p))) /\\ (((((((pfgd_Q_gcd_terminal_result_witness_common_left)=0 \\/ (pfgs_G_gcd_terminal_result)=0) /\\ (((pfgd_P_gcd_terminal_result_witness_common_left)=0)))) \\/ (((~((pfgd_Q_gcd_terminal_result_witness_common_left)=0)) /\\ (((~((pfgs_G_gcd_terminal_result)=0)) /\\ (((pfgd_Q_gcd_terminal_result_witness_common_left)+(pfgs_G_gcd_terminal_result)=S (pfgd_P_gcd_terminal_result_witness_common_left)))))))) /\\ ((forall pfc_index_gcd_terminal_result_witness_common_left_productcoefficients. (exists pfa_gap_gcd_terminal_result_witness_common_left_productcoefficientsbound. pfa_gap_gcd_terminal_result_witness_common_left_productcoefficientsbound + S (pfc_index_gcd_terminal_result_witness_common_left_productcoefficients) = (pfgd_P_gcd_terminal_result_witness_common_left)) -> exists pfc_value_gcd_terminal_result_witness_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_terminal_result_witness_common_left_productcoefficientsentry. ff_h_pfp_gcd_terminal_result_witness_common_left_productcoefficientsentry + S (pfc_value_gcd_terminal_result_witness_common_left_productcoefficients) = S ((S (pfc_index_gcd_terminal_result_witness_common_left_productcoefficients)) * pfgd_pc_gcd_terminal_result_witness_common_left)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_common_left_productcoefficientsentry. pfgd_pb_gcd_terminal_result_witness_common_left = ff_q_pfp_gcd_terminal_result_witness_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_terminal_result_witness_common_left_productcoefficients)) * pfgd_pc_gcd_terminal_result_witness_common_left) + (pfc_value_gcd_terminal_result_witness_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_terminal_result_witness_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_terminal_result_witness_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_terminal_result_witness_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_terminal_result_witness_common_left_productcoefficients))) -> exists pfc_value_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_result_witness_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_terminal_result_witness_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_result_witness_common_left_productcoefficientscoefficient) + (pfc_value_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_terminal_result_witness_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_terminal_result_witness_common_left)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_terminal_result_witness_common_left)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_terminal_result_witness_common_left = ff_q_pfp_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_terminal_result_witness_common_left) + (pfc_left_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_terminal_result_witness_common_left)=(pfc_index_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm) = (pfgs_G_gcd_terminal_result)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_terminal_result)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_gcd_terminal_result = ff_q_pfp_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_terminal_result) + (pfc_right_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_gcd_terminal_result)=(pfc_complement_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_terminal_result_witness_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_terminal_result_witness_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_terminal_result_witness_common_left_productcoefficients))) * fs_v_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_terminal_result_witness_common_left_productcoefficients))) * fs_v_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_terminal_result_witness_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_terminal_result_witness_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_result_witness_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_terminal_result_witness_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_result_witness_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_terminal_result_witness_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_terminal_result_witness_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_terminal_result_witness_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_terminal_result_witness_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_terminal_result_witness_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_terminal_result_witness_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_terminal_result_witness_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_terminal_result_witness_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_terminal_result_witness_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_terminal_result_witness_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_terminal_result_witness_common_left_equivalent pfrep_left_gcd_terminal_result_witness_common_left_equivalent pfrep_right_gcd_terminal_result_witness_common_left_equivalent. ((exists pfrep_position_gcd_terminal_result_witness_common_left_equivalentfirst. ((pfrep_position_gcd_terminal_result_witness_common_left_equivalentfirst+S (pfrep_power_gcd_terminal_result_witness_common_left_equivalent)=(pfgd_P_gcd_terminal_result_witness_common_left)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_common_left_equivalentfirstentry. ff_h_pfp_gcd_terminal_result_witness_common_left_equivalentfirstentry + S (pfrep_left_gcd_terminal_result_witness_common_left_equivalent) = S ((S (pfrep_position_gcd_terminal_result_witness_common_left_equivalentfirst)) * pfgd_pc_gcd_terminal_result_witness_common_left)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_common_left_equivalentfirstentry. pfgd_pb_gcd_terminal_result_witness_common_left = ff_q_pfp_gcd_terminal_result_witness_common_left_equivalentfirstentry * S ((S (pfrep_position_gcd_terminal_result_witness_common_left_equivalentfirst)) * pfgd_pc_gcd_terminal_result_witness_common_left) + (pfrep_left_gcd_terminal_result_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_terminal_result_witness_common_left_equivalentfirstoutside. pfrep_gap_gcd_terminal_result_witness_common_left_equivalentfirstoutside+(pfgd_P_gcd_terminal_result_witness_common_left)=(pfrep_power_gcd_terminal_result_witness_common_left_equivalent)) /\\ (((pfrep_left_gcd_terminal_result_witness_common_left_equivalent)=0))))) -> ((exists pfrep_position_gcd_terminal_result_witness_common_left_equivalentsecond. ((pfrep_position_gcd_terminal_result_witness_common_left_equivalentsecond+S (pfrep_power_gcd_terminal_result_witness_common_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_common_left_equivalentsecondentry. ff_h_pfp_gcd_terminal_result_witness_common_left_equivalentsecondentry + S (pfrep_right_gcd_terminal_result_witness_common_left_equivalent) = S ((S (pfrep_position_gcd_terminal_result_witness_common_left_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_common_left_equivalentsecondentry. ab = ff_q_pfp_gcd_terminal_result_witness_common_left_equivalentsecondentry * S ((S (pfrep_position_gcd_terminal_result_witness_common_left_equivalentsecond)) * ac) + (pfrep_right_gcd_terminal_result_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_terminal_result_witness_common_left_equivalentsecondoutside. pfrep_gap_gcd_terminal_result_witness_common_left_equivalentsecondoutside+(L)=(pfrep_power_gcd_terminal_result_witness_common_left_equivalent)) /\\ (((pfrep_right_gcd_terminal_result_witness_common_left_equivalent)=0))))) -> pfrep_left_gcd_terminal_result_witness_common_left_equivalent=pfrep_right_gcd_terminal_result_witness_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_terminal_result_witness_common_right_bounded. (exists fom_gap_pfp_gcd_terminal_result_witness_common_right_bounded_index_bound. fom_gap_pfp_gcd_terminal_result_witness_common_right_bounded_index_bound + S (fom_index_pfp_gcd_terminal_result_witness_common_right_bounded) = 0) -> exists fom_value_pfp_gcd_terminal_result_witness_common_right_bounded. ((((exists fom_beta_height_pfp_gcd_terminal_result_witness_common_right_bounded_entry. fom_beta_height_pfp_gcd_terminal_result_witness_common_right_bounded_entry + S (fom_value_pfp_gcd_terminal_result_witness_common_right_bounded) = S ((S (fom_index_pfp_gcd_terminal_result_witness_common_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_result_witness_common_right_bounded_entry. bb = fom_beta_quotient_pfp_gcd_terminal_result_witness_common_right_bounded_entry * S ((S (fom_index_pfp_gcd_terminal_result_witness_common_right_bounded)) * bc) + (fom_value_pfp_gcd_terminal_result_witness_common_right_bounded))) /\\ (exists fom_gap_pfp_gcd_terminal_result_witness_common_right_bounded_value_bound. fom_gap_pfp_gcd_terminal_result_witness_common_right_bounded_value_bound + S (fom_value_pfp_gcd_terminal_result_witness_common_right_bounded) = p))) /\\ ((exists pfgd_qb_gcd_terminal_result_witness_common_right pfgd_qc_gcd_terminal_result_witness_common_right pfgd_Q_gcd_terminal_result_witness_common_right pfgd_pb_gcd_terminal_result_witness_common_right pfgd_pc_gcd_terminal_result_witness_common_right pfgd_P_gcd_terminal_result_witness_common_right. ((((forall fom_index_pfp_gcd_terminal_result_witness_common_right_productleft. (exists fom_gap_pfp_gcd_terminal_result_witness_common_right_productleft_index_bound. fom_gap_pfp_gcd_terminal_result_witness_common_right_productleft_index_bound + S (fom_index_pfp_gcd_terminal_result_witness_common_right_productleft) = pfgd_Q_gcd_terminal_result_witness_common_right) -> exists fom_value_pfp_gcd_terminal_result_witness_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_terminal_result_witness_common_right_productleft_entry. fom_beta_height_pfp_gcd_terminal_result_witness_common_right_productleft_entry + S (fom_value_pfp_gcd_terminal_result_witness_common_right_productleft) = S ((S (fom_index_pfp_gcd_terminal_result_witness_common_right_productleft)) * pfgd_qc_gcd_terminal_result_witness_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_result_witness_common_right_productleft_entry. pfgd_qb_gcd_terminal_result_witness_common_right = fom_beta_quotient_pfp_gcd_terminal_result_witness_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_terminal_result_witness_common_right_productleft)) * pfgd_qc_gcd_terminal_result_witness_common_right) + (fom_value_pfp_gcd_terminal_result_witness_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_terminal_result_witness_common_right_productleft_value_bound. fom_gap_pfp_gcd_terminal_result_witness_common_right_productleft_value_bound + S (fom_value_pfp_gcd_terminal_result_witness_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_terminal_result_witness_common_right_productright. (exists fom_gap_pfp_gcd_terminal_result_witness_common_right_productright_index_bound. fom_gap_pfp_gcd_terminal_result_witness_common_right_productright_index_bound + S (fom_index_pfp_gcd_terminal_result_witness_common_right_productright) = pfgs_G_gcd_terminal_result) -> exists fom_value_pfp_gcd_terminal_result_witness_common_right_productright. ((((exists fom_beta_height_pfp_gcd_terminal_result_witness_common_right_productright_entry. fom_beta_height_pfp_gcd_terminal_result_witness_common_right_productright_entry + S (fom_value_pfp_gcd_terminal_result_witness_common_right_productright) = S ((S (fom_index_pfp_gcd_terminal_result_witness_common_right_productright)) * pfgs_gc_gcd_terminal_result)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_result_witness_common_right_productright_entry. pfgs_gb_gcd_terminal_result = fom_beta_quotient_pfp_gcd_terminal_result_witness_common_right_productright_entry * S ((S (fom_index_pfp_gcd_terminal_result_witness_common_right_productright)) * pfgs_gc_gcd_terminal_result) + (fom_value_pfp_gcd_terminal_result_witness_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_terminal_result_witness_common_right_productright_value_bound. fom_gap_pfp_gcd_terminal_result_witness_common_right_productright_value_bound + S (fom_value_pfp_gcd_terminal_result_witness_common_right_productright) = p))) /\\ (((((((pfgd_Q_gcd_terminal_result_witness_common_right)=0 \\/ (pfgs_G_gcd_terminal_result)=0) /\\ (((pfgd_P_gcd_terminal_result_witness_common_right)=0)))) \\/ (((~((pfgd_Q_gcd_terminal_result_witness_common_right)=0)) /\\ (((~((pfgs_G_gcd_terminal_result)=0)) /\\ (((pfgd_Q_gcd_terminal_result_witness_common_right)+(pfgs_G_gcd_terminal_result)=S (pfgd_P_gcd_terminal_result_witness_common_right)))))))) /\\ ((forall pfc_index_gcd_terminal_result_witness_common_right_productcoefficients. (exists pfa_gap_gcd_terminal_result_witness_common_right_productcoefficientsbound. pfa_gap_gcd_terminal_result_witness_common_right_productcoefficientsbound + S (pfc_index_gcd_terminal_result_witness_common_right_productcoefficients) = (pfgd_P_gcd_terminal_result_witness_common_right)) -> exists pfc_value_gcd_terminal_result_witness_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_terminal_result_witness_common_right_productcoefficientsentry. ff_h_pfp_gcd_terminal_result_witness_common_right_productcoefficientsentry + S (pfc_value_gcd_terminal_result_witness_common_right_productcoefficients) = S ((S (pfc_index_gcd_terminal_result_witness_common_right_productcoefficients)) * pfgd_pc_gcd_terminal_result_witness_common_right)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_common_right_productcoefficientsentry. pfgd_pb_gcd_terminal_result_witness_common_right = ff_q_pfp_gcd_terminal_result_witness_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_terminal_result_witness_common_right_productcoefficients)) * pfgd_pc_gcd_terminal_result_witness_common_right) + (pfc_value_gcd_terminal_result_witness_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_terminal_result_witness_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_terminal_result_witness_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_terminal_result_witness_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_terminal_result_witness_common_right_productcoefficients))) -> exists pfc_value_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_result_witness_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_terminal_result_witness_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_result_witness_common_right_productcoefficientscoefficient) + (pfc_value_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_terminal_result_witness_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_terminal_result_witness_common_right)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_terminal_result_witness_common_right)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_terminal_result_witness_common_right = ff_q_pfp_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_terminal_result_witness_common_right) + (pfc_left_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_terminal_result_witness_common_right)=(pfc_index_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm) = (pfgs_G_gcd_terminal_result)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_terminal_result)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_gcd_terminal_result = ff_q_pfp_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_terminal_result) + (pfc_right_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_gcd_terminal_result)=(pfc_complement_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_terminal_result_witness_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_terminal_result_witness_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_terminal_result_witness_common_right_productcoefficients))) * fs_v_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_terminal_result_witness_common_right_productcoefficients))) * fs_v_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_terminal_result_witness_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_terminal_result_witness_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_result_witness_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_terminal_result_witness_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_result_witness_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_terminal_result_witness_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_terminal_result_witness_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_terminal_result_witness_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_terminal_result_witness_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_terminal_result_witness_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_terminal_result_witness_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_terminal_result_witness_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_terminal_result_witness_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_terminal_result_witness_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_terminal_result_witness_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_terminal_result_witness_common_right_equivalent pfrep_left_gcd_terminal_result_witness_common_right_equivalent pfrep_right_gcd_terminal_result_witness_common_right_equivalent. ((exists pfrep_position_gcd_terminal_result_witness_common_right_equivalentfirst. ((pfrep_position_gcd_terminal_result_witness_common_right_equivalentfirst+S (pfrep_power_gcd_terminal_result_witness_common_right_equivalent)=(pfgd_P_gcd_terminal_result_witness_common_right)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_common_right_equivalentfirstentry. ff_h_pfp_gcd_terminal_result_witness_common_right_equivalentfirstentry + S (pfrep_left_gcd_terminal_result_witness_common_right_equivalent) = S ((S (pfrep_position_gcd_terminal_result_witness_common_right_equivalentfirst)) * pfgd_pc_gcd_terminal_result_witness_common_right)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_common_right_equivalentfirstentry. pfgd_pb_gcd_terminal_result_witness_common_right = ff_q_pfp_gcd_terminal_result_witness_common_right_equivalentfirstentry * S ((S (pfrep_position_gcd_terminal_result_witness_common_right_equivalentfirst)) * pfgd_pc_gcd_terminal_result_witness_common_right) + (pfrep_left_gcd_terminal_result_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_terminal_result_witness_common_right_equivalentfirstoutside. pfrep_gap_gcd_terminal_result_witness_common_right_equivalentfirstoutside+(pfgd_P_gcd_terminal_result_witness_common_right)=(pfrep_power_gcd_terminal_result_witness_common_right_equivalent)) /\\ (((pfrep_left_gcd_terminal_result_witness_common_right_equivalent)=0))))) -> ((exists pfrep_position_gcd_terminal_result_witness_common_right_equivalentsecond. ((pfrep_position_gcd_terminal_result_witness_common_right_equivalentsecond+S (pfrep_power_gcd_terminal_result_witness_common_right_equivalent)=(0)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_common_right_equivalentsecondentry. ff_h_pfp_gcd_terminal_result_witness_common_right_equivalentsecondentry + S (pfrep_right_gcd_terminal_result_witness_common_right_equivalent) = S ((S (pfrep_position_gcd_terminal_result_witness_common_right_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_common_right_equivalentsecondentry. bb = ff_q_pfp_gcd_terminal_result_witness_common_right_equivalentsecondentry * S ((S (pfrep_position_gcd_terminal_result_witness_common_right_equivalentsecond)) * bc) + (pfrep_right_gcd_terminal_result_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_terminal_result_witness_common_right_equivalentsecondoutside. pfrep_gap_gcd_terminal_result_witness_common_right_equivalentsecondoutside+(0)=(pfrep_power_gcd_terminal_result_witness_common_right_equivalent)) /\\ (((pfrep_right_gcd_terminal_result_witness_common_right_equivalent)=0))))) -> pfrep_left_gcd_terminal_result_witness_common_right_equivalent=pfrep_right_gcd_terminal_result_witness_common_right_equivalent)))))))))) /\\ ((exists pfgb_pb_gcd_terminal_result_witness_bezout pfgb_pc_gcd_terminal_result_witness_bezout pfgb_P_gcd_terminal_result_witness_bezout pfgb_qb_gcd_terminal_result_witness_bezout pfgb_qc_gcd_terminal_result_witness_bezout pfgb_Q_gcd_terminal_result_witness_bezout. ((((forall fom_index_pfp_gcd_terminal_result_witness_bezout_leftleft. (exists fom_gap_pfp_gcd_terminal_result_witness_bezout_leftleft_index_bound. fom_gap_pfp_gcd_terminal_result_witness_bezout_leftleft_index_bound + S (fom_index_pfp_gcd_terminal_result_witness_bezout_leftleft) = pfgs_U_gcd_terminal_result) -> exists fom_value_pfp_gcd_terminal_result_witness_bezout_leftleft. ((((exists fom_beta_height_pfp_gcd_terminal_result_witness_bezout_leftleft_entry. fom_beta_height_pfp_gcd_terminal_result_witness_bezout_leftleft_entry + S (fom_value_pfp_gcd_terminal_result_witness_bezout_leftleft) = S ((S (fom_index_pfp_gcd_terminal_result_witness_bezout_leftleft)) * pfgs_uc_gcd_terminal_result)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_result_witness_bezout_leftleft_entry. pfgs_ub_gcd_terminal_result = fom_beta_quotient_pfp_gcd_terminal_result_witness_bezout_leftleft_entry * S ((S (fom_index_pfp_gcd_terminal_result_witness_bezout_leftleft)) * pfgs_uc_gcd_terminal_result) + (fom_value_pfp_gcd_terminal_result_witness_bezout_leftleft))) /\\ (exists fom_gap_pfp_gcd_terminal_result_witness_bezout_leftleft_value_bound. fom_gap_pfp_gcd_terminal_result_witness_bezout_leftleft_value_bound + S (fom_value_pfp_gcd_terminal_result_witness_bezout_leftleft) = p))) /\\ (((forall fom_index_pfp_gcd_terminal_result_witness_bezout_leftright. (exists fom_gap_pfp_gcd_terminal_result_witness_bezout_leftright_index_bound. fom_gap_pfp_gcd_terminal_result_witness_bezout_leftright_index_bound + S (fom_index_pfp_gcd_terminal_result_witness_bezout_leftright) = L) -> exists fom_value_pfp_gcd_terminal_result_witness_bezout_leftright. ((((exists fom_beta_height_pfp_gcd_terminal_result_witness_bezout_leftright_entry. fom_beta_height_pfp_gcd_terminal_result_witness_bezout_leftright_entry + S (fom_value_pfp_gcd_terminal_result_witness_bezout_leftright) = S ((S (fom_index_pfp_gcd_terminal_result_witness_bezout_leftright)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_result_witness_bezout_leftright_entry. ab = fom_beta_quotient_pfp_gcd_terminal_result_witness_bezout_leftright_entry * S ((S (fom_index_pfp_gcd_terminal_result_witness_bezout_leftright)) * ac) + (fom_value_pfp_gcd_terminal_result_witness_bezout_leftright))) /\\ (exists fom_gap_pfp_gcd_terminal_result_witness_bezout_leftright_value_bound. fom_gap_pfp_gcd_terminal_result_witness_bezout_leftright_value_bound + S (fom_value_pfp_gcd_terminal_result_witness_bezout_leftright) = p))) /\\ (((((((pfgs_U_gcd_terminal_result)=0 \\/ (L)=0) /\\ (((pfgb_P_gcd_terminal_result_witness_bezout)=0)))) \\/ (((~((pfgs_U_gcd_terminal_result)=0)) /\\ (((~((L)=0)) /\\ (((pfgs_U_gcd_terminal_result)+(L)=S (pfgb_P_gcd_terminal_result_witness_bezout)))))))) /\\ ((forall pfc_index_gcd_terminal_result_witness_bezout_leftcoefficients. (exists pfa_gap_gcd_terminal_result_witness_bezout_leftcoefficientsbound. pfa_gap_gcd_terminal_result_witness_bezout_leftcoefficientsbound + S (pfc_index_gcd_terminal_result_witness_bezout_leftcoefficients) = (pfgb_P_gcd_terminal_result_witness_bezout)) -> exists pfc_value_gcd_terminal_result_witness_bezout_leftcoefficients. ((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_leftcoefficientsentry. ff_h_pfp_gcd_terminal_result_witness_bezout_leftcoefficientsentry + S (pfc_value_gcd_terminal_result_witness_bezout_leftcoefficients) = S ((S (pfc_index_gcd_terminal_result_witness_bezout_leftcoefficients)) * pfgb_pc_gcd_terminal_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_leftcoefficientsentry. pfgb_pb_gcd_terminal_result_witness_bezout = ff_q_pfp_gcd_terminal_result_witness_bezout_leftcoefficientsentry * S ((S (pfc_index_gcd_terminal_result_witness_bezout_leftcoefficients)) * pfgb_pc_gcd_terminal_result_witness_bezout) + (pfc_value_gcd_terminal_result_witness_bezout_leftcoefficients))) /\\ ((exists pfc_terms_code_gcd_terminal_result_witness_bezout_leftcoefficientscoefficient pfc_terms_scale_gcd_terminal_result_witness_bezout_leftcoefficientscoefficient pfc_natural_sum_gcd_terminal_result_witness_bezout_leftcoefficientscoefficient. ((forall pfc_index_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalbound. pfa_gap_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_terminal_result_witness_bezout_leftcoefficients))) -> exists pfc_value_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_result_witness_bezout_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_terminal_result_witness_bezout_leftcoefficientscoefficient = ff_q_pfp_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_result_witness_bezout_leftcoefficientscoefficient) + (pfc_value_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_left_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_right_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonal)+pfc_complement_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm=(pfc_index_gcd_terminal_result_witness_bezout_leftcoefficients)) /\\ ((((((exists pfa_gap_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonal) = (pfgs_U_gcd_terminal_result)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_gcd_terminal_result)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. pfgs_ub_gcd_terminal_result = ff_q_pfp_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_gcd_terminal_result) + (pfc_left_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside+(pfgs_U_gcd_terminal_result)=(pfc_index_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonal)=pfc_left_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm*pfc_right_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum fs_v_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_terminal_result_witness_bezout_leftcoefficientscoefficient) = S ((S (S (pfc_index_gcd_terminal_result_witness_bezout_leftcoefficients))) * fs_v_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_terminal_result_witness_bezout_leftcoefficients))) * fs_v_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum) + (pfc_natural_sum_gcd_terminal_result_witness_bezout_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_terminal_result_witness_bezout_leftcoefficients)) -> exists fs_a_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_result_witness_bezout_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_terminal_result_witness_bezout_leftcoefficientscoefficient = fs_q_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_result_witness_bezout_leftcoefficientscoefficient) + (fs_a_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum) + (fs_r_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum) + (fs_s_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientresiduebound. pfa_gap_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientresiduebound + S (pfc_value_gcd_terminal_result_witness_bezout_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_terminal_result_witness_bezout_leftcoefficientscoefficient) + (p) * pfa_offset_left_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientresiduecongruence = (pfc_value_gcd_terminal_result_witness_bezout_leftcoefficients) + (p) * pfa_offset_right_gcd_terminal_result_witness_bezout_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_gcd_terminal_result_witness_bezout_rightleft. (exists fom_gap_pfp_gcd_terminal_result_witness_bezout_rightleft_index_bound. fom_gap_pfp_gcd_terminal_result_witness_bezout_rightleft_index_bound + S (fom_index_pfp_gcd_terminal_result_witness_bezout_rightleft) = pfgs_V_gcd_terminal_result) -> exists fom_value_pfp_gcd_terminal_result_witness_bezout_rightleft. ((((exists fom_beta_height_pfp_gcd_terminal_result_witness_bezout_rightleft_entry. fom_beta_height_pfp_gcd_terminal_result_witness_bezout_rightleft_entry + S (fom_value_pfp_gcd_terminal_result_witness_bezout_rightleft) = S ((S (fom_index_pfp_gcd_terminal_result_witness_bezout_rightleft)) * pfgs_vc_gcd_terminal_result)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_result_witness_bezout_rightleft_entry. pfgs_vb_gcd_terminal_result = fom_beta_quotient_pfp_gcd_terminal_result_witness_bezout_rightleft_entry * S ((S (fom_index_pfp_gcd_terminal_result_witness_bezout_rightleft)) * pfgs_vc_gcd_terminal_result) + (fom_value_pfp_gcd_terminal_result_witness_bezout_rightleft))) /\\ (exists fom_gap_pfp_gcd_terminal_result_witness_bezout_rightleft_value_bound. fom_gap_pfp_gcd_terminal_result_witness_bezout_rightleft_value_bound + S (fom_value_pfp_gcd_terminal_result_witness_bezout_rightleft) = p))) /\\ (((forall fom_index_pfp_gcd_terminal_result_witness_bezout_rightright. (exists fom_gap_pfp_gcd_terminal_result_witness_bezout_rightright_index_bound. fom_gap_pfp_gcd_terminal_result_witness_bezout_rightright_index_bound + S (fom_index_pfp_gcd_terminal_result_witness_bezout_rightright) = 0) -> exists fom_value_pfp_gcd_terminal_result_witness_bezout_rightright. ((((exists fom_beta_height_pfp_gcd_terminal_result_witness_bezout_rightright_entry. fom_beta_height_pfp_gcd_terminal_result_witness_bezout_rightright_entry + S (fom_value_pfp_gcd_terminal_result_witness_bezout_rightright) = S ((S (fom_index_pfp_gcd_terminal_result_witness_bezout_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_result_witness_bezout_rightright_entry. bb = fom_beta_quotient_pfp_gcd_terminal_result_witness_bezout_rightright_entry * S ((S (fom_index_pfp_gcd_terminal_result_witness_bezout_rightright)) * bc) + (fom_value_pfp_gcd_terminal_result_witness_bezout_rightright))) /\\ (exists fom_gap_pfp_gcd_terminal_result_witness_bezout_rightright_value_bound. fom_gap_pfp_gcd_terminal_result_witness_bezout_rightright_value_bound + S (fom_value_pfp_gcd_terminal_result_witness_bezout_rightright) = p))) /\\ (((((((pfgs_V_gcd_terminal_result)=0 \\/ (0)=0) /\\ (((pfgb_Q_gcd_terminal_result_witness_bezout)=0)))) \\/ (((~((pfgs_V_gcd_terminal_result)=0)) /\\ (((~((0)=0)) /\\ (((pfgs_V_gcd_terminal_result)+(0)=S (pfgb_Q_gcd_terminal_result_witness_bezout)))))))) /\\ ((forall pfc_index_gcd_terminal_result_witness_bezout_rightcoefficients. (exists pfa_gap_gcd_terminal_result_witness_bezout_rightcoefficientsbound. pfa_gap_gcd_terminal_result_witness_bezout_rightcoefficientsbound + S (pfc_index_gcd_terminal_result_witness_bezout_rightcoefficients) = (pfgb_Q_gcd_terminal_result_witness_bezout)) -> exists pfc_value_gcd_terminal_result_witness_bezout_rightcoefficients. ((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_rightcoefficientsentry. ff_h_pfp_gcd_terminal_result_witness_bezout_rightcoefficientsentry + S (pfc_value_gcd_terminal_result_witness_bezout_rightcoefficients) = S ((S (pfc_index_gcd_terminal_result_witness_bezout_rightcoefficients)) * pfgb_qc_gcd_terminal_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_rightcoefficientsentry. pfgb_qb_gcd_terminal_result_witness_bezout = ff_q_pfp_gcd_terminal_result_witness_bezout_rightcoefficientsentry * S ((S (pfc_index_gcd_terminal_result_witness_bezout_rightcoefficients)) * pfgb_qc_gcd_terminal_result_witness_bezout) + (pfc_value_gcd_terminal_result_witness_bezout_rightcoefficients))) /\\ ((exists pfc_terms_code_gcd_terminal_result_witness_bezout_rightcoefficientscoefficient pfc_terms_scale_gcd_terminal_result_witness_bezout_rightcoefficientscoefficient pfc_natural_sum_gcd_terminal_result_witness_bezout_rightcoefficientscoefficient. ((forall pfc_index_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalbound. pfa_gap_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_terminal_result_witness_bezout_rightcoefficients))) -> exists pfc_value_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_result_witness_bezout_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_terminal_result_witness_bezout_rightcoefficientscoefficient = ff_q_pfp_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_terminal_result_witness_bezout_rightcoefficientscoefficient) + (pfc_value_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_left_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_right_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonal)+pfc_complement_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm=(pfc_index_gcd_terminal_result_witness_bezout_rightcoefficients)) /\\ ((((((exists pfa_gap_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonal) = (pfgs_V_gcd_terminal_result)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_gcd_terminal_result)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. pfgs_vb_gcd_terminal_result = ff_q_pfp_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_gcd_terminal_result) + (pfc_left_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside+(pfgs_V_gcd_terminal_result)=(pfc_index_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm) = (0)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside+(0)=(pfc_complement_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonal)=pfc_left_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm*pfc_right_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum fs_v_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_terminal_result_witness_bezout_rightcoefficientscoefficient) = S ((S (S (pfc_index_gcd_terminal_result_witness_bezout_rightcoefficients))) * fs_v_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_terminal_result_witness_bezout_rightcoefficients))) * fs_v_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum) + (pfc_natural_sum_gcd_terminal_result_witness_bezout_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_terminal_result_witness_bezout_rightcoefficients)) -> exists fs_a_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_result_witness_bezout_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_terminal_result_witness_bezout_rightcoefficientscoefficient = fs_q_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_terminal_result_witness_bezout_rightcoefficientscoefficient) + (fs_a_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum) + (fs_r_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum) + (fs_s_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientresiduebound. pfa_gap_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientresiduebound + S (pfc_value_gcd_terminal_result_witness_bezout_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_terminal_result_witness_bezout_rightcoefficientscoefficient) + (p) * pfa_offset_left_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientresiduecongruence = (pfc_value_gcd_terminal_result_witness_bezout_rightcoefficients) + (p) * pfa_offset_right_gcd_terminal_result_witness_bezout_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded. (exists fom_gap_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded_index_bound. fom_gap_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded_index_bound + S (fom_index_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded) = pfgb_P_gcd_terminal_result_witness_bezout) -> exists fom_value_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded. ((((exists fom_beta_height_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded_entry. fom_beta_height_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded_entry + S (fom_value_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded) = S ((S (fom_index_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded)) * pfgb_pc_gcd_terminal_result_witness_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded_entry. pfgb_pb_gcd_terminal_result_witness_bezout = fom_beta_quotient_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded_entry * S ((S (fom_index_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded)) * pfgb_pc_gcd_terminal_result_witness_bezout) + (fom_value_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded))) /\\ (exists fom_gap_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded_value_bound. fom_gap_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded_value_bound + S (fom_value_pfp_gcd_terminal_result_witness_bezout_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded. (exists fom_gap_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded_index_bound. fom_gap_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded_index_bound + S (fom_index_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded) = pfgb_Q_gcd_terminal_result_witness_bezout) -> exists fom_value_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded. ((((exists fom_beta_height_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded_entry. fom_beta_height_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded_entry + S (fom_value_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded) = S ((S (fom_index_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded)) * pfgb_qc_gcd_terminal_result_witness_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded_entry. pfgb_qb_gcd_terminal_result_witness_bezout = fom_beta_quotient_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded_entry * S ((S (fom_index_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded)) * pfgb_qc_gcd_terminal_result_witness_bezout) + (fom_value_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded))) /\\ (exists fom_gap_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded_value_bound. fom_gap_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded_value_bound + S (fom_value_pfp_gcd_terminal_result_witness_bezout_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded. (exists fom_gap_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded_index_bound. fom_gap_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded_index_bound + S (fom_index_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded) = pfgs_G_gcd_terminal_result) -> exists fom_value_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded. ((((exists fom_beta_height_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded_entry. fom_beta_height_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded_entry + S (fom_value_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded) = S ((S (fom_index_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded)) * pfgs_gc_gcd_terminal_result)) /\\ exists fom_beta_quotient_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded_entry. pfgs_gb_gcd_terminal_result = fom_beta_quotient_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded_entry * S ((S (fom_index_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded)) * pfgs_gc_gcd_terminal_result) + (fom_value_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded))) /\\ (exists fom_gap_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded_value_bound. fom_gap_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded_value_bound + S (fom_value_pfp_gcd_terminal_result_witness_bezout_sum_result_bounded) = p))) /\\ ((exists pfga_ub_gcd_terminal_result_witness_bezout_sum pfga_uc_gcd_terminal_result_witness_bezout_sum pfga_vb_gcd_terminal_result_witness_bezout_sum pfga_vc_gcd_terminal_result_witness_bezout_sum pfga_tb_gcd_terminal_result_witness_bezout_sum pfga_tc_gcd_terminal_result_witness_bezout_sum pfga_K_gcd_terminal_result_witness_bezout_sum. ((((forall pfrep_power_gcd_terminal_result_witness_bezout_sum_left pfrep_left_gcd_terminal_result_witness_bezout_sum_left pfrep_right_gcd_terminal_result_witness_bezout_sum_left. ((exists pfrep_position_gcd_terminal_result_witness_bezout_sum_leftfirst. ((pfrep_position_gcd_terminal_result_witness_bezout_sum_leftfirst+S (pfrep_power_gcd_terminal_result_witness_bezout_sum_left)=(pfgb_P_gcd_terminal_result_witness_bezout)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_sum_leftfirstentry. ff_h_pfp_gcd_terminal_result_witness_bezout_sum_leftfirstentry + S (pfrep_left_gcd_terminal_result_witness_bezout_sum_left) = S ((S (pfrep_position_gcd_terminal_result_witness_bezout_sum_leftfirst)) * pfgb_pc_gcd_terminal_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_sum_leftfirstentry. pfgb_pb_gcd_terminal_result_witness_bezout = ff_q_pfp_gcd_terminal_result_witness_bezout_sum_leftfirstentry * S ((S (pfrep_position_gcd_terminal_result_witness_bezout_sum_leftfirst)) * pfgb_pc_gcd_terminal_result_witness_bezout) + (pfrep_left_gcd_terminal_result_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_terminal_result_witness_bezout_sum_leftfirstoutside. pfrep_gap_gcd_terminal_result_witness_bezout_sum_leftfirstoutside+(pfgb_P_gcd_terminal_result_witness_bezout)=(pfrep_power_gcd_terminal_result_witness_bezout_sum_left)) /\\ (((pfrep_left_gcd_terminal_result_witness_bezout_sum_left)=0))))) -> ((exists pfrep_position_gcd_terminal_result_witness_bezout_sum_leftsecond. ((pfrep_position_gcd_terminal_result_witness_bezout_sum_leftsecond+S (pfrep_power_gcd_terminal_result_witness_bezout_sum_left)=(pfga_K_gcd_terminal_result_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_sum_leftsecondentry. ff_h_pfp_gcd_terminal_result_witness_bezout_sum_leftsecondentry + S (pfrep_right_gcd_terminal_result_witness_bezout_sum_left) = S ((S (pfrep_position_gcd_terminal_result_witness_bezout_sum_leftsecond)) * pfga_uc_gcd_terminal_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_sum_leftsecondentry. pfga_ub_gcd_terminal_result_witness_bezout_sum = ff_q_pfp_gcd_terminal_result_witness_bezout_sum_leftsecondentry * S ((S (pfrep_position_gcd_terminal_result_witness_bezout_sum_leftsecond)) * pfga_uc_gcd_terminal_result_witness_bezout_sum) + (pfrep_right_gcd_terminal_result_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_terminal_result_witness_bezout_sum_leftsecondoutside. pfrep_gap_gcd_terminal_result_witness_bezout_sum_leftsecondoutside+(pfga_K_gcd_terminal_result_witness_bezout_sum)=(pfrep_power_gcd_terminal_result_witness_bezout_sum_left)) /\\ (((pfrep_right_gcd_terminal_result_witness_bezout_sum_left)=0))))) -> pfrep_left_gcd_terminal_result_witness_bezout_sum_left=pfrep_right_gcd_terminal_result_witness_bezout_sum_left) /\\ ((forall pfrep_power_gcd_terminal_result_witness_bezout_sum_right pfrep_left_gcd_terminal_result_witness_bezout_sum_right pfrep_right_gcd_terminal_result_witness_bezout_sum_right. ((exists pfrep_position_gcd_terminal_result_witness_bezout_sum_rightfirst. ((pfrep_position_gcd_terminal_result_witness_bezout_sum_rightfirst+S (pfrep_power_gcd_terminal_result_witness_bezout_sum_right)=(pfgb_Q_gcd_terminal_result_witness_bezout)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_sum_rightfirstentry. ff_h_pfp_gcd_terminal_result_witness_bezout_sum_rightfirstentry + S (pfrep_left_gcd_terminal_result_witness_bezout_sum_right) = S ((S (pfrep_position_gcd_terminal_result_witness_bezout_sum_rightfirst)) * pfgb_qc_gcd_terminal_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_sum_rightfirstentry. pfgb_qb_gcd_terminal_result_witness_bezout = ff_q_pfp_gcd_terminal_result_witness_bezout_sum_rightfirstentry * S ((S (pfrep_position_gcd_terminal_result_witness_bezout_sum_rightfirst)) * pfgb_qc_gcd_terminal_result_witness_bezout) + (pfrep_left_gcd_terminal_result_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_terminal_result_witness_bezout_sum_rightfirstoutside. pfrep_gap_gcd_terminal_result_witness_bezout_sum_rightfirstoutside+(pfgb_Q_gcd_terminal_result_witness_bezout)=(pfrep_power_gcd_terminal_result_witness_bezout_sum_right)) /\\ (((pfrep_left_gcd_terminal_result_witness_bezout_sum_right)=0))))) -> ((exists pfrep_position_gcd_terminal_result_witness_bezout_sum_rightsecond. ((pfrep_position_gcd_terminal_result_witness_bezout_sum_rightsecond+S (pfrep_power_gcd_terminal_result_witness_bezout_sum_right)=(pfga_K_gcd_terminal_result_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_sum_rightsecondentry. ff_h_pfp_gcd_terminal_result_witness_bezout_sum_rightsecondentry + S (pfrep_right_gcd_terminal_result_witness_bezout_sum_right) = S ((S (pfrep_position_gcd_terminal_result_witness_bezout_sum_rightsecond)) * pfga_vc_gcd_terminal_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_sum_rightsecondentry. pfga_vb_gcd_terminal_result_witness_bezout_sum = ff_q_pfp_gcd_terminal_result_witness_bezout_sum_rightsecondentry * S ((S (pfrep_position_gcd_terminal_result_witness_bezout_sum_rightsecond)) * pfga_vc_gcd_terminal_result_witness_bezout_sum) + (pfrep_right_gcd_terminal_result_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_terminal_result_witness_bezout_sum_rightsecondoutside. pfrep_gap_gcd_terminal_result_witness_bezout_sum_rightsecondoutside+(pfga_K_gcd_terminal_result_witness_bezout_sum)=(pfrep_power_gcd_terminal_result_witness_bezout_sum_right)) /\\ (((pfrep_right_gcd_terminal_result_witness_bezout_sum_right)=0))))) -> pfrep_left_gcd_terminal_result_witness_bezout_sum_right=pfrep_right_gcd_terminal_result_witness_bezout_sum_right)))) /\\ (((forall pfp_index_gcd_terminal_result_witness_bezout_sum_add. (exists pfa_gap_gcd_terminal_result_witness_bezout_sum_addindex. pfa_gap_gcd_terminal_result_witness_bezout_sum_addindex + S (pfp_index_gcd_terminal_result_witness_bezout_sum_add) = (pfga_K_gcd_terminal_result_witness_bezout_sum)) -> exists pfp_left_gcd_terminal_result_witness_bezout_sum_add pfp_right_gcd_terminal_result_witness_bezout_sum_add pfp_value_gcd_terminal_result_witness_bezout_sum_add. ((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_sum_addleft. ff_h_pfp_gcd_terminal_result_witness_bezout_sum_addleft + S (pfp_left_gcd_terminal_result_witness_bezout_sum_add) = S ((S (pfp_index_gcd_terminal_result_witness_bezout_sum_add)) * pfga_uc_gcd_terminal_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_sum_addleft. pfga_ub_gcd_terminal_result_witness_bezout_sum = ff_q_pfp_gcd_terminal_result_witness_bezout_sum_addleft * S ((S (pfp_index_gcd_terminal_result_witness_bezout_sum_add)) * pfga_uc_gcd_terminal_result_witness_bezout_sum) + (pfp_left_gcd_terminal_result_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_sum_addright. ff_h_pfp_gcd_terminal_result_witness_bezout_sum_addright + S (pfp_right_gcd_terminal_result_witness_bezout_sum_add) = S ((S (pfp_index_gcd_terminal_result_witness_bezout_sum_add)) * pfga_vc_gcd_terminal_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_sum_addright. pfga_vb_gcd_terminal_result_witness_bezout_sum = ff_q_pfp_gcd_terminal_result_witness_bezout_sum_addright * S ((S (pfp_index_gcd_terminal_result_witness_bezout_sum_add)) * pfga_vc_gcd_terminal_result_witness_bezout_sum) + (pfp_right_gcd_terminal_result_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_sum_addtarget. ff_h_pfp_gcd_terminal_result_witness_bezout_sum_addtarget + S (pfp_value_gcd_terminal_result_witness_bezout_sum_add) = S ((S (pfp_index_gcd_terminal_result_witness_bezout_sum_add)) * pfga_tc_gcd_terminal_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_sum_addtarget. pfga_tb_gcd_terminal_result_witness_bezout_sum = ff_q_pfp_gcd_terminal_result_witness_bezout_sum_addtarget * S ((S (pfp_index_gcd_terminal_result_witness_bezout_sum_add)) * pfga_tc_gcd_terminal_result_witness_bezout_sum) + (pfp_value_gcd_terminal_result_witness_bezout_sum_add))) /\\ ((((exists pfa_gap_gcd_terminal_result_witness_bezout_sum_addoperationleft. pfa_gap_gcd_terminal_result_witness_bezout_sum_addoperationleft + S (pfp_left_gcd_terminal_result_witness_bezout_sum_add) = (p)) /\\ (((exists pfa_gap_gcd_terminal_result_witness_bezout_sum_addoperationright. pfa_gap_gcd_terminal_result_witness_bezout_sum_addoperationright + S (pfp_right_gcd_terminal_result_witness_bezout_sum_add) = (p)) /\\ ((((exists pfa_gap_gcd_terminal_result_witness_bezout_sum_addoperationresultbound. pfa_gap_gcd_terminal_result_witness_bezout_sum_addoperationresultbound + S (pfp_value_gcd_terminal_result_witness_bezout_sum_add) = (p)) /\\ ((exists pfa_offset_left_gcd_terminal_result_witness_bezout_sum_addoperationresultcongruence pfa_offset_right_gcd_terminal_result_witness_bezout_sum_addoperationresultcongruence. ((pfp_left_gcd_terminal_result_witness_bezout_sum_add) + (pfp_right_gcd_terminal_result_witness_bezout_sum_add)) + (p) * pfa_offset_left_gcd_terminal_result_witness_bezout_sum_addoperationresultcongruence = (pfp_value_gcd_terminal_result_witness_bezout_sum_add) + (p) * pfa_offset_right_gcd_terminal_result_witness_bezout_sum_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_gcd_terminal_result_witness_bezout_sum_result pfrep_left_gcd_terminal_result_witness_bezout_sum_result pfrep_right_gcd_terminal_result_witness_bezout_sum_result. ((exists pfrep_position_gcd_terminal_result_witness_bezout_sum_resultfirst. ((pfrep_position_gcd_terminal_result_witness_bezout_sum_resultfirst+S (pfrep_power_gcd_terminal_result_witness_bezout_sum_result)=(pfga_K_gcd_terminal_result_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_sum_resultfirstentry. ff_h_pfp_gcd_terminal_result_witness_bezout_sum_resultfirstentry + S (pfrep_left_gcd_terminal_result_witness_bezout_sum_result) = S ((S (pfrep_position_gcd_terminal_result_witness_bezout_sum_resultfirst)) * pfga_tc_gcd_terminal_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_sum_resultfirstentry. pfga_tb_gcd_terminal_result_witness_bezout_sum = ff_q_pfp_gcd_terminal_result_witness_bezout_sum_resultfirstentry * S ((S (pfrep_position_gcd_terminal_result_witness_bezout_sum_resultfirst)) * pfga_tc_gcd_terminal_result_witness_bezout_sum) + (pfrep_left_gcd_terminal_result_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_terminal_result_witness_bezout_sum_resultfirstoutside. pfrep_gap_gcd_terminal_result_witness_bezout_sum_resultfirstoutside+(pfga_K_gcd_terminal_result_witness_bezout_sum)=(pfrep_power_gcd_terminal_result_witness_bezout_sum_result)) /\\ (((pfrep_left_gcd_terminal_result_witness_bezout_sum_result)=0))))) -> ((exists pfrep_position_gcd_terminal_result_witness_bezout_sum_resultsecond. ((pfrep_position_gcd_terminal_result_witness_bezout_sum_resultsecond+S (pfrep_power_gcd_terminal_result_witness_bezout_sum_result)=(pfgs_G_gcd_terminal_result)) /\\ ((((exists ff_h_pfp_gcd_terminal_result_witness_bezout_sum_resultsecondentry. ff_h_pfp_gcd_terminal_result_witness_bezout_sum_resultsecondentry + S (pfrep_right_gcd_terminal_result_witness_bezout_sum_result) = S ((S (pfrep_position_gcd_terminal_result_witness_bezout_sum_resultsecond)) * pfgs_gc_gcd_terminal_result)) /\\ exists ff_q_pfp_gcd_terminal_result_witness_bezout_sum_resultsecondentry. pfgs_gb_gcd_terminal_result = ff_q_pfp_gcd_terminal_result_witness_bezout_sum_resultsecondentry * S ((S (pfrep_position_gcd_terminal_result_witness_bezout_sum_resultsecond)) * pfgs_gc_gcd_terminal_result) + (pfrep_right_gcd_terminal_result_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_terminal_result_witness_bezout_sum_resultsecondoutside. pfrep_gap_gcd_terminal_result_witness_bezout_sum_resultsecondoutside+(pfgs_G_gcd_terminal_result)=(pfrep_power_gcd_terminal_result_witness_bezout_sum_result)) /\\ (((pfrep_right_gcd_terminal_result_witness_bezout_sum_result)=0))))) -> pfrep_left_gcd_terminal_result_witness_bezout_sum_result=pfrep_right_gcd_terminal_result_witness_bezout_sum_result)))))))))))))))))))))))", "statement_sha256": "281077fafb262a6c81f789b369ae704b3ded2fd73bdf666a5115490ce18bd1fe", "summary": "Construct an already zero-or-monic common divisor and actual Bezout coefficients for (A,empty), using genuine mutual right-associate witnesses. 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hs_witness_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_right", "cases hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "have hG : forall fom_index_pfp_gcd_transport_G. (exists fom_gap_pfp_gcd_transport_G_index_bound. fom_gap_pfp_gcd_transport_G_index_bound + S (fom_index_pfp_gcd_transport_G) = x2) -> exists fom_value_pfp_gcd_transport_G. ((((exists fom_beta_height_pfp_gcd_transport_G_entry. fom_beta_height_pfp_gcd_transport_G_entry + S (fom_value_pfp_gcd_transport_G) = S ((S (fom_index_pfp_gcd_transport_G)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_transport_G_entry. x = fom_beta_quotient_pfp_gcd_transport_G_entry * S ((S (fom_index_pfp_gcd_transport_G)) * x1) + (fom_value_pfp_gcd_transport_G))) /\\ (exists fom_gap_pfp_gcd_transport_G_value_bound. fom_gap_pfp_gcd_transport_G_value_bound + S (fom_value_pfp_gcd_transport_G) = p))", "specialize prime_field_polynomial_right_divides_divisor_bounded (p)", "specialize prime_field_polynomial_right_divides_divisor_bounded (x)", "specialize prime_field_polynomial_right_divides_divisor_bounded (x1)", "specialize prime_field_polynomial_right_divides_divisor_bounded (x2)", "specialize prime_field_polynomial_right_divides_divisor_bounded (ab)", "specialize prime_field_polynomial_right_divides_divisor_bounded (ac)", "specialize prime_field_polynomial_right_divides_divisor_bounded (L)", "apply prime_field_polynomial_right_divides_divisor_bounded", "exact hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_left_left", "have hb : exists pfgb_pb_gcd_transport_bezout pfgb_pc_gcd_transport_bezout pfgb_P_gcd_transport_bezout pfgb_qb_gcd_transport_bezout pfgb_qc_gcd_transport_bezout pfgb_Q_gcd_transport_bezout. ((((forall fom_index_pfp_gcd_transport_bezout_leftleft. (exists fom_gap_pfp_gcd_transport_bezout_leftleft_index_bound. fom_gap_pfp_gcd_transport_bezout_leftleft_index_bound + S (fom_index_pfp_gcd_transport_bezout_leftleft) = x5) -> exists fom_value_pfp_gcd_transport_bezout_leftleft. ((((exists fom_beta_height_pfp_gcd_transport_bezout_leftleft_entry. fom_beta_height_pfp_gcd_transport_bezout_leftleft_entry + S (fom_value_pfp_gcd_transport_bezout_leftleft) = S ((S (fom_index_pfp_gcd_transport_bezout_leftleft)) * x4)) /\\ exists fom_beta_quotient_pfp_gcd_transport_bezout_leftleft_entry. x3 = fom_beta_quotient_pfp_gcd_transport_bezout_leftleft_entry * S ((S (fom_index_pfp_gcd_transport_bezout_leftleft)) * x4) + (fom_value_pfp_gcd_transport_bezout_leftleft))) /\\ (exists fom_gap_pfp_gcd_transport_bezout_leftleft_value_bound. fom_gap_pfp_gcd_transport_bezout_leftleft_value_bound + S (fom_value_pfp_gcd_transport_bezout_leftleft) = p))) /\\ (((forall fom_index_pfp_gcd_transport_bezout_leftright. (exists fom_gap_pfp_gcd_transport_bezout_leftright_index_bound. fom_gap_pfp_gcd_transport_bezout_leftright_index_bound + S (fom_index_pfp_gcd_transport_bezout_leftright) = L) -> exists fom_value_pfp_gcd_transport_bezout_leftright. ((((exists fom_beta_height_pfp_gcd_transport_bezout_leftright_entry. fom_beta_height_pfp_gcd_transport_bezout_leftright_entry + S (fom_value_pfp_gcd_transport_bezout_leftright) = S ((S (fom_index_pfp_gcd_transport_bezout_leftright)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_transport_bezout_leftright_entry. ab = fom_beta_quotient_pfp_gcd_transport_bezout_leftright_entry * S ((S (fom_index_pfp_gcd_transport_bezout_leftright)) * ac) + (fom_value_pfp_gcd_transport_bezout_leftright))) /\\ (exists fom_gap_pfp_gcd_transport_bezout_leftright_value_bound. fom_gap_pfp_gcd_transport_bezout_leftright_value_bound + S (fom_value_pfp_gcd_transport_bezout_leftright) = p))) /\\ (((((((x5)=0 \\/ (L)=0) /\\ (((pfgb_P_gcd_transport_bezout)=0)))) \\/ (((~((x5)=0)) /\\ (((~((L)=0)) /\\ (((x5)+(L)=S (pfgb_P_gcd_transport_bezout)))))))) /\\ ((forall pfc_index_gcd_transport_bezout_leftcoefficients. (exists pfa_gap_gcd_transport_bezout_leftcoefficientsbound. pfa_gap_gcd_transport_bezout_leftcoefficientsbound + S (pfc_index_gcd_transport_bezout_leftcoefficients) = (pfgb_P_gcd_transport_bezout)) -> exists pfc_value_gcd_transport_bezout_leftcoefficients. ((((exists ff_h_pfp_gcd_transport_bezout_leftcoefficientsentry. ff_h_pfp_gcd_transport_bezout_leftcoefficientsentry + S (pfc_value_gcd_transport_bezout_leftcoefficients) = S ((S (pfc_index_gcd_transport_bezout_leftcoefficients)) * pfgb_pc_gcd_transport_bezout)) /\\ exists ff_q_pfp_gcd_transport_bezout_leftcoefficientsentry. pfgb_pb_gcd_transport_bezout = ff_q_pfp_gcd_transport_bezout_leftcoefficientsentry * S ((S (pfc_index_gcd_transport_bezout_leftcoefficients)) * pfgb_pc_gcd_transport_bezout) + (pfc_value_gcd_transport_bezout_leftcoefficients))) /\\ ((exists pfc_terms_code_gcd_transport_bezout_leftcoefficientscoefficient pfc_terms_scale_gcd_transport_bezout_leftcoefficientscoefficient pfc_natural_sum_gcd_transport_bezout_leftcoefficientscoefficient. ((forall pfc_index_gcd_transport_bezout_leftcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_transport_bezout_leftcoefficientscoefficientdiagonalbound. pfa_gap_gcd_transport_bezout_leftcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_transport_bezout_leftcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_transport_bezout_leftcoefficients))) -> exists pfc_value_gcd_transport_bezout_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_transport_bezout_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_transport_bezout_leftcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_transport_bezout_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_transport_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_bezout_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_transport_bezout_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_transport_bezout_leftcoefficientscoefficient = ff_q_pfp_gcd_transport_bezout_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_transport_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_bezout_leftcoefficientscoefficient) + (pfc_value_gcd_transport_bezout_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm pfc_left_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm pfc_right_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_transport_bezout_leftcoefficientscoefficientdiagonal)+pfc_complement_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm=(pfc_index_gcd_transport_bezout_leftcoefficients)) /\\ ((((((exists pfa_gap_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_transport_bezout_leftcoefficientscoefficientdiagonal) = (x5)) /\\ ((((exists ff_h_pfp_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_transport_bezout_leftcoefficientscoefficientdiagonal)) * x4)) /\\ exists ff_q_pfp_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermleftentry. x3 = ff_q_pfp_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_transport_bezout_leftcoefficientscoefficientdiagonal)) * x4) + (pfc_left_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermleftoutside+(x5)=(pfc_index_gcd_transport_bezout_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_transport_bezout_leftcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_transport_bezout_leftcoefficientscoefficientdiagonal)=pfc_left_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm*pfc_right_gcd_transport_bezout_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum fs_v_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_transport_bezout_leftcoefficientscoefficient) = S ((S (S (pfc_index_gcd_transport_bezout_leftcoefficients))) * fs_v_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_transport_bezout_leftcoefficients))) * fs_v_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum) + (pfc_natural_sum_gcd_transport_bezout_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_transport_bezout_leftcoefficients)) -> exists fs_a_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_bezout_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_transport_bezout_leftcoefficientscoefficient = fs_q_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_bezout_leftcoefficientscoefficient) + (fs_a_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum) + (fs_r_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum) + (fs_s_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_transport_bezout_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_transport_bezout_leftcoefficientscoefficientresiduebound. pfa_gap_gcd_transport_bezout_leftcoefficientscoefficientresiduebound + S (pfc_value_gcd_transport_bezout_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_transport_bezout_leftcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_transport_bezout_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_transport_bezout_leftcoefficientscoefficient) + (p) * pfa_offset_left_gcd_transport_bezout_leftcoefficientscoefficientresiduecongruence = (pfc_value_gcd_transport_bezout_leftcoefficients) + (p) * pfa_offset_right_gcd_transport_bezout_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_gcd_transport_bezout_rightleft. (exists fom_gap_pfp_gcd_transport_bezout_rightleft_index_bound. fom_gap_pfp_gcd_transport_bezout_rightleft_index_bound + S (fom_index_pfp_gcd_transport_bezout_rightleft) = x8) -> exists fom_value_pfp_gcd_transport_bezout_rightleft. ((((exists fom_beta_height_pfp_gcd_transport_bezout_rightleft_entry. fom_beta_height_pfp_gcd_transport_bezout_rightleft_entry + S (fom_value_pfp_gcd_transport_bezout_rightleft) = S ((S (fom_index_pfp_gcd_transport_bezout_rightleft)) * x7)) /\\ exists fom_beta_quotient_pfp_gcd_transport_bezout_rightleft_entry. x6 = fom_beta_quotient_pfp_gcd_transport_bezout_rightleft_entry * S ((S (fom_index_pfp_gcd_transport_bezout_rightleft)) * x7) + (fom_value_pfp_gcd_transport_bezout_rightleft))) /\\ (exists fom_gap_pfp_gcd_transport_bezout_rightleft_value_bound. fom_gap_pfp_gcd_transport_bezout_rightleft_value_bound + S (fom_value_pfp_gcd_transport_bezout_rightleft) = p))) /\\ (((forall fom_index_pfp_gcd_transport_bezout_rightright. (exists fom_gap_pfp_gcd_transport_bezout_rightright_index_bound. fom_gap_pfp_gcd_transport_bezout_rightright_index_bound + S (fom_index_pfp_gcd_transport_bezout_rightright) = M2) -> exists fom_value_pfp_gcd_transport_bezout_rightright. ((((exists fom_beta_height_pfp_gcd_transport_bezout_rightright_entry. fom_beta_height_pfp_gcd_transport_bezout_rightright_entry + S (fom_value_pfp_gcd_transport_bezout_rightright) = S ((S (fom_index_pfp_gcd_transport_bezout_rightright)) * bc2)) /\\ exists fom_beta_quotient_pfp_gcd_transport_bezout_rightright_entry. bb2 = fom_beta_quotient_pfp_gcd_transport_bezout_rightright_entry * S ((S (fom_index_pfp_gcd_transport_bezout_rightright)) * bc2) + (fom_value_pfp_gcd_transport_bezout_rightright))) /\\ (exists fom_gap_pfp_gcd_transport_bezout_rightright_value_bound. fom_gap_pfp_gcd_transport_bezout_rightright_value_bound + S (fom_value_pfp_gcd_transport_bezout_rightright) = p))) /\\ (((((((x8)=0 \\/ (M2)=0) /\\ (((pfgb_Q_gcd_transport_bezout)=0)))) \\/ (((~((x8)=0)) /\\ (((~((M2)=0)) /\\ (((x8)+(M2)=S (pfgb_Q_gcd_transport_bezout)))))))) /\\ ((forall pfc_index_gcd_transport_bezout_rightcoefficients. (exists pfa_gap_gcd_transport_bezout_rightcoefficientsbound. pfa_gap_gcd_transport_bezout_rightcoefficientsbound + S (pfc_index_gcd_transport_bezout_rightcoefficients) = (pfgb_Q_gcd_transport_bezout)) -> exists pfc_value_gcd_transport_bezout_rightcoefficients. ((((exists ff_h_pfp_gcd_transport_bezout_rightcoefficientsentry. ff_h_pfp_gcd_transport_bezout_rightcoefficientsentry + S (pfc_value_gcd_transport_bezout_rightcoefficients) = S ((S (pfc_index_gcd_transport_bezout_rightcoefficients)) * pfgb_qc_gcd_transport_bezout)) /\\ exists ff_q_pfp_gcd_transport_bezout_rightcoefficientsentry. pfgb_qb_gcd_transport_bezout = ff_q_pfp_gcd_transport_bezout_rightcoefficientsentry * S ((S (pfc_index_gcd_transport_bezout_rightcoefficients)) * pfgb_qc_gcd_transport_bezout) + (pfc_value_gcd_transport_bezout_rightcoefficients))) /\\ ((exists pfc_terms_code_gcd_transport_bezout_rightcoefficientscoefficient pfc_terms_scale_gcd_transport_bezout_rightcoefficientscoefficient pfc_natural_sum_gcd_transport_bezout_rightcoefficientscoefficient. ((forall pfc_index_gcd_transport_bezout_rightcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_transport_bezout_rightcoefficientscoefficientdiagonalbound. pfa_gap_gcd_transport_bezout_rightcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_transport_bezout_rightcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_transport_bezout_rightcoefficients))) -> exists pfc_value_gcd_transport_bezout_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_transport_bezout_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_transport_bezout_rightcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_transport_bezout_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_transport_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_bezout_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_transport_bezout_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_transport_bezout_rightcoefficientscoefficient = ff_q_pfp_gcd_transport_bezout_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_transport_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_bezout_rightcoefficientscoefficient) + (pfc_value_gcd_transport_bezout_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm pfc_left_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm pfc_right_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_transport_bezout_rightcoefficientscoefficientdiagonal)+pfc_complement_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm=(pfc_index_gcd_transport_bezout_rightcoefficients)) /\\ ((((((exists pfa_gap_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_transport_bezout_rightcoefficientscoefficientdiagonal) = (x8)) /\\ ((((exists ff_h_pfp_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_transport_bezout_rightcoefficientscoefficientdiagonal)) * x7)) /\\ exists ff_q_pfp_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermleftentry. x6 = ff_q_pfp_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_transport_bezout_rightcoefficientscoefficientdiagonal)) * x7) + (pfc_left_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermleftoutside+(x8)=(pfc_index_gcd_transport_bezout_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm) = (M2)) /\\ ((((exists ff_h_pfp_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm)) * bc2)) /\\ exists ff_q_pfp_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermrightentry. bb2 = ff_q_pfp_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm)) * bc2) + (pfc_right_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_transport_bezout_rightcoefficientscoefficientdiagonaltermrightoutside+(M2)=(pfc_complement_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_transport_bezout_rightcoefficientscoefficientdiagonal)=pfc_left_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm*pfc_right_gcd_transport_bezout_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum fs_v_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_transport_bezout_rightcoefficientscoefficient) = S ((S (S (pfc_index_gcd_transport_bezout_rightcoefficients))) * fs_v_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_transport_bezout_rightcoefficients))) * fs_v_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum) + (pfc_natural_sum_gcd_transport_bezout_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_transport_bezout_rightcoefficients)) -> exists fs_a_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_bezout_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_transport_bezout_rightcoefficientscoefficient = fs_q_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_bezout_rightcoefficientscoefficient) + (fs_a_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum) + (fs_r_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum) + (fs_s_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_transport_bezout_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_transport_bezout_rightcoefficientscoefficientresiduebound. pfa_gap_gcd_transport_bezout_rightcoefficientscoefficientresiduebound + S (pfc_value_gcd_transport_bezout_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_transport_bezout_rightcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_transport_bezout_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_transport_bezout_rightcoefficientscoefficient) + (p) * pfa_offset_left_gcd_transport_bezout_rightcoefficientscoefficientresiduecongruence = (pfc_value_gcd_transport_bezout_rightcoefficients) + (p) * pfa_offset_right_gcd_transport_bezout_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_gcd_transport_bezout_sum_left_bounded. (exists fom_gap_pfp_gcd_transport_bezout_sum_left_bounded_index_bound. fom_gap_pfp_gcd_transport_bezout_sum_left_bounded_index_bound + S (fom_index_pfp_gcd_transport_bezout_sum_left_bounded) = pfgb_P_gcd_transport_bezout) -> exists fom_value_pfp_gcd_transport_bezout_sum_left_bounded. ((((exists fom_beta_height_pfp_gcd_transport_bezout_sum_left_bounded_entry. fom_beta_height_pfp_gcd_transport_bezout_sum_left_bounded_entry + S (fom_value_pfp_gcd_transport_bezout_sum_left_bounded) = S ((S (fom_index_pfp_gcd_transport_bezout_sum_left_bounded)) * pfgb_pc_gcd_transport_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_transport_bezout_sum_left_bounded_entry. pfgb_pb_gcd_transport_bezout = fom_beta_quotient_pfp_gcd_transport_bezout_sum_left_bounded_entry * S ((S (fom_index_pfp_gcd_transport_bezout_sum_left_bounded)) * pfgb_pc_gcd_transport_bezout) + (fom_value_pfp_gcd_transport_bezout_sum_left_bounded))) /\\ (exists fom_gap_pfp_gcd_transport_bezout_sum_left_bounded_value_bound. fom_gap_pfp_gcd_transport_bezout_sum_left_bounded_value_bound + S (fom_value_pfp_gcd_transport_bezout_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_transport_bezout_sum_right_bounded. (exists fom_gap_pfp_gcd_transport_bezout_sum_right_bounded_index_bound. fom_gap_pfp_gcd_transport_bezout_sum_right_bounded_index_bound + S (fom_index_pfp_gcd_transport_bezout_sum_right_bounded) = pfgb_Q_gcd_transport_bezout) -> exists fom_value_pfp_gcd_transport_bezout_sum_right_bounded. ((((exists fom_beta_height_pfp_gcd_transport_bezout_sum_right_bounded_entry. fom_beta_height_pfp_gcd_transport_bezout_sum_right_bounded_entry + S (fom_value_pfp_gcd_transport_bezout_sum_right_bounded) = S ((S (fom_index_pfp_gcd_transport_bezout_sum_right_bounded)) * pfgb_qc_gcd_transport_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_transport_bezout_sum_right_bounded_entry. pfgb_qb_gcd_transport_bezout = fom_beta_quotient_pfp_gcd_transport_bezout_sum_right_bounded_entry * S ((S (fom_index_pfp_gcd_transport_bezout_sum_right_bounded)) * pfgb_qc_gcd_transport_bezout) + (fom_value_pfp_gcd_transport_bezout_sum_right_bounded))) /\\ (exists fom_gap_pfp_gcd_transport_bezout_sum_right_bounded_value_bound. fom_gap_pfp_gcd_transport_bezout_sum_right_bounded_value_bound + S (fom_value_pfp_gcd_transport_bezout_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_transport_bezout_sum_result_bounded. (exists fom_gap_pfp_gcd_transport_bezout_sum_result_bounded_index_bound. fom_gap_pfp_gcd_transport_bezout_sum_result_bounded_index_bound + S (fom_index_pfp_gcd_transport_bezout_sum_result_bounded) = x2) -> exists fom_value_pfp_gcd_transport_bezout_sum_result_bounded. ((((exists fom_beta_height_pfp_gcd_transport_bezout_sum_result_bounded_entry. fom_beta_height_pfp_gcd_transport_bezout_sum_result_bounded_entry + S (fom_value_pfp_gcd_transport_bezout_sum_result_bounded) = S ((S (fom_index_pfp_gcd_transport_bezout_sum_result_bounded)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_transport_bezout_sum_result_bounded_entry. x = fom_beta_quotient_pfp_gcd_transport_bezout_sum_result_bounded_entry * S ((S (fom_index_pfp_gcd_transport_bezout_sum_result_bounded)) * x1) + (fom_value_pfp_gcd_transport_bezout_sum_result_bounded))) /\\ (exists fom_gap_pfp_gcd_transport_bezout_sum_result_bounded_value_bound. fom_gap_pfp_gcd_transport_bezout_sum_result_bounded_value_bound + S (fom_value_pfp_gcd_transport_bezout_sum_result_bounded) = p))) /\\ ((exists pfga_ub_gcd_transport_bezout_sum pfga_uc_gcd_transport_bezout_sum pfga_vb_gcd_transport_bezout_sum pfga_vc_gcd_transport_bezout_sum pfga_tb_gcd_transport_bezout_sum pfga_tc_gcd_transport_bezout_sum pfga_K_gcd_transport_bezout_sum. ((((forall pfrep_power_gcd_transport_bezout_sum_left pfrep_left_gcd_transport_bezout_sum_left pfrep_right_gcd_transport_bezout_sum_left. ((exists pfrep_position_gcd_transport_bezout_sum_leftfirst. ((pfrep_position_gcd_transport_bezout_sum_leftfirst+S (pfrep_power_gcd_transport_bezout_sum_left)=(pfgb_P_gcd_transport_bezout)) /\\ ((((exists ff_h_pfp_gcd_transport_bezout_sum_leftfirstentry. ff_h_pfp_gcd_transport_bezout_sum_leftfirstentry + S (pfrep_left_gcd_transport_bezout_sum_left) = S ((S (pfrep_position_gcd_transport_bezout_sum_leftfirst)) * pfgb_pc_gcd_transport_bezout)) /\\ exists ff_q_pfp_gcd_transport_bezout_sum_leftfirstentry. pfgb_pb_gcd_transport_bezout = ff_q_pfp_gcd_transport_bezout_sum_leftfirstentry * S ((S (pfrep_position_gcd_transport_bezout_sum_leftfirst)) * pfgb_pc_gcd_transport_bezout) + (pfrep_left_gcd_transport_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_transport_bezout_sum_leftfirstoutside. pfrep_gap_gcd_transport_bezout_sum_leftfirstoutside+(pfgb_P_gcd_transport_bezout)=(pfrep_power_gcd_transport_bezout_sum_left)) /\\ (((pfrep_left_gcd_transport_bezout_sum_left)=0))))) -> ((exists pfrep_position_gcd_transport_bezout_sum_leftsecond. ((pfrep_position_gcd_transport_bezout_sum_leftsecond+S (pfrep_power_gcd_transport_bezout_sum_left)=(pfga_K_gcd_transport_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_transport_bezout_sum_leftsecondentry. ff_h_pfp_gcd_transport_bezout_sum_leftsecondentry + S (pfrep_right_gcd_transport_bezout_sum_left) = S ((S (pfrep_position_gcd_transport_bezout_sum_leftsecond)) * pfga_uc_gcd_transport_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_bezout_sum_leftsecondentry. pfga_ub_gcd_transport_bezout_sum = ff_q_pfp_gcd_transport_bezout_sum_leftsecondentry * S ((S (pfrep_position_gcd_transport_bezout_sum_leftsecond)) * pfga_uc_gcd_transport_bezout_sum) + (pfrep_right_gcd_transport_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_transport_bezout_sum_leftsecondoutside. pfrep_gap_gcd_transport_bezout_sum_leftsecondoutside+(pfga_K_gcd_transport_bezout_sum)=(pfrep_power_gcd_transport_bezout_sum_left)) /\\ (((pfrep_right_gcd_transport_bezout_sum_left)=0))))) -> pfrep_left_gcd_transport_bezout_sum_left=pfrep_right_gcd_transport_bezout_sum_left) /\\ ((forall pfrep_power_gcd_transport_bezout_sum_right pfrep_left_gcd_transport_bezout_sum_right pfrep_right_gcd_transport_bezout_sum_right. ((exists pfrep_position_gcd_transport_bezout_sum_rightfirst. ((pfrep_position_gcd_transport_bezout_sum_rightfirst+S (pfrep_power_gcd_transport_bezout_sum_right)=(pfgb_Q_gcd_transport_bezout)) /\\ ((((exists ff_h_pfp_gcd_transport_bezout_sum_rightfirstentry. ff_h_pfp_gcd_transport_bezout_sum_rightfirstentry + S (pfrep_left_gcd_transport_bezout_sum_right) = S ((S (pfrep_position_gcd_transport_bezout_sum_rightfirst)) * pfgb_qc_gcd_transport_bezout)) /\\ exists ff_q_pfp_gcd_transport_bezout_sum_rightfirstentry. pfgb_qb_gcd_transport_bezout = ff_q_pfp_gcd_transport_bezout_sum_rightfirstentry * S ((S (pfrep_position_gcd_transport_bezout_sum_rightfirst)) * pfgb_qc_gcd_transport_bezout) + (pfrep_left_gcd_transport_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_transport_bezout_sum_rightfirstoutside. pfrep_gap_gcd_transport_bezout_sum_rightfirstoutside+(pfgb_Q_gcd_transport_bezout)=(pfrep_power_gcd_transport_bezout_sum_right)) /\\ (((pfrep_left_gcd_transport_bezout_sum_right)=0))))) -> ((exists pfrep_position_gcd_transport_bezout_sum_rightsecond. ((pfrep_position_gcd_transport_bezout_sum_rightsecond+S (pfrep_power_gcd_transport_bezout_sum_right)=(pfga_K_gcd_transport_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_transport_bezout_sum_rightsecondentry. ff_h_pfp_gcd_transport_bezout_sum_rightsecondentry + S (pfrep_right_gcd_transport_bezout_sum_right) = S ((S (pfrep_position_gcd_transport_bezout_sum_rightsecond)) * pfga_vc_gcd_transport_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_bezout_sum_rightsecondentry. pfga_vb_gcd_transport_bezout_sum = ff_q_pfp_gcd_transport_bezout_sum_rightsecondentry * S ((S (pfrep_position_gcd_transport_bezout_sum_rightsecond)) * pfga_vc_gcd_transport_bezout_sum) + (pfrep_right_gcd_transport_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_transport_bezout_sum_rightsecondoutside. pfrep_gap_gcd_transport_bezout_sum_rightsecondoutside+(pfga_K_gcd_transport_bezout_sum)=(pfrep_power_gcd_transport_bezout_sum_right)) /\\ (((pfrep_right_gcd_transport_bezout_sum_right)=0))))) -> pfrep_left_gcd_transport_bezout_sum_right=pfrep_right_gcd_transport_bezout_sum_right)))) /\\ (((forall pfp_index_gcd_transport_bezout_sum_add. (exists pfa_gap_gcd_transport_bezout_sum_addindex. pfa_gap_gcd_transport_bezout_sum_addindex + S (pfp_index_gcd_transport_bezout_sum_add) = (pfga_K_gcd_transport_bezout_sum)) -> exists pfp_left_gcd_transport_bezout_sum_add pfp_right_gcd_transport_bezout_sum_add pfp_value_gcd_transport_bezout_sum_add. ((((exists ff_h_pfp_gcd_transport_bezout_sum_addleft. ff_h_pfp_gcd_transport_bezout_sum_addleft + S (pfp_left_gcd_transport_bezout_sum_add) = S ((S (pfp_index_gcd_transport_bezout_sum_add)) * pfga_uc_gcd_transport_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_bezout_sum_addleft. pfga_ub_gcd_transport_bezout_sum = ff_q_pfp_gcd_transport_bezout_sum_addleft * S ((S (pfp_index_gcd_transport_bezout_sum_add)) * pfga_uc_gcd_transport_bezout_sum) + (pfp_left_gcd_transport_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_transport_bezout_sum_addright. ff_h_pfp_gcd_transport_bezout_sum_addright + S (pfp_right_gcd_transport_bezout_sum_add) = S ((S (pfp_index_gcd_transport_bezout_sum_add)) * pfga_vc_gcd_transport_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_bezout_sum_addright. pfga_vb_gcd_transport_bezout_sum = ff_q_pfp_gcd_transport_bezout_sum_addright * S ((S (pfp_index_gcd_transport_bezout_sum_add)) * pfga_vc_gcd_transport_bezout_sum) + (pfp_right_gcd_transport_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_transport_bezout_sum_addtarget. ff_h_pfp_gcd_transport_bezout_sum_addtarget + S (pfp_value_gcd_transport_bezout_sum_add) = S ((S (pfp_index_gcd_transport_bezout_sum_add)) * pfga_tc_gcd_transport_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_bezout_sum_addtarget. pfga_tb_gcd_transport_bezout_sum = ff_q_pfp_gcd_transport_bezout_sum_addtarget * S ((S (pfp_index_gcd_transport_bezout_sum_add)) * pfga_tc_gcd_transport_bezout_sum) + (pfp_value_gcd_transport_bezout_sum_add))) /\\ ((((exists pfa_gap_gcd_transport_bezout_sum_addoperationleft. pfa_gap_gcd_transport_bezout_sum_addoperationleft + S (pfp_left_gcd_transport_bezout_sum_add) = (p)) /\\ (((exists pfa_gap_gcd_transport_bezout_sum_addoperationright. pfa_gap_gcd_transport_bezout_sum_addoperationright + S (pfp_right_gcd_transport_bezout_sum_add) = (p)) /\\ ((((exists pfa_gap_gcd_transport_bezout_sum_addoperationresultbound. pfa_gap_gcd_transport_bezout_sum_addoperationresultbound + S (pfp_value_gcd_transport_bezout_sum_add) = (p)) /\\ ((exists pfa_offset_left_gcd_transport_bezout_sum_addoperationresultcongruence pfa_offset_right_gcd_transport_bezout_sum_addoperationresultcongruence. ((pfp_left_gcd_transport_bezout_sum_add) + (pfp_right_gcd_transport_bezout_sum_add)) + (p) * pfa_offset_left_gcd_transport_bezout_sum_addoperationresultcongruence = (pfp_value_gcd_transport_bezout_sum_add) + (p) * pfa_offset_right_gcd_transport_bezout_sum_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_gcd_transport_bezout_sum_result pfrep_left_gcd_transport_bezout_sum_result pfrep_right_gcd_transport_bezout_sum_result. ((exists pfrep_position_gcd_transport_bezout_sum_resultfirst. ((pfrep_position_gcd_transport_bezout_sum_resultfirst+S (pfrep_power_gcd_transport_bezout_sum_result)=(pfga_K_gcd_transport_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_transport_bezout_sum_resultfirstentry. ff_h_pfp_gcd_transport_bezout_sum_resultfirstentry + S (pfrep_left_gcd_transport_bezout_sum_result) = S ((S (pfrep_position_gcd_transport_bezout_sum_resultfirst)) * pfga_tc_gcd_transport_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_bezout_sum_resultfirstentry. pfga_tb_gcd_transport_bezout_sum = ff_q_pfp_gcd_transport_bezout_sum_resultfirstentry * S ((S (pfrep_position_gcd_transport_bezout_sum_resultfirst)) * pfga_tc_gcd_transport_bezout_sum) + (pfrep_left_gcd_transport_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_transport_bezout_sum_resultfirstoutside. pfrep_gap_gcd_transport_bezout_sum_resultfirstoutside+(pfga_K_gcd_transport_bezout_sum)=(pfrep_power_gcd_transport_bezout_sum_result)) /\\ (((pfrep_left_gcd_transport_bezout_sum_result)=0))))) -> ((exists pfrep_position_gcd_transport_bezout_sum_resultsecond. ((pfrep_position_gcd_transport_bezout_sum_resultsecond+S (pfrep_power_gcd_transport_bezout_sum_result)=(x2)) /\\ ((((exists ff_h_pfp_gcd_transport_bezout_sum_resultsecondentry. ff_h_pfp_gcd_transport_bezout_sum_resultsecondentry + S (pfrep_right_gcd_transport_bezout_sum_result) = S ((S (pfrep_position_gcd_transport_bezout_sum_resultsecond)) * x1)) /\\ exists ff_q_pfp_gcd_transport_bezout_sum_resultsecondentry. x = ff_q_pfp_gcd_transport_bezout_sum_resultsecondentry * S ((S (pfrep_position_gcd_transport_bezout_sum_resultsecond)) * x1) + (pfrep_right_gcd_transport_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_transport_bezout_sum_resultsecondoutside. pfrep_gap_gcd_transport_bezout_sum_resultsecondoutside+(x2)=(pfrep_power_gcd_transport_bezout_sum_result)) /\\ (((pfrep_right_gcd_transport_bezout_sum_result)=0))))) -> pfrep_left_gcd_transport_bezout_sum_result=pfrep_right_gcd_transport_bezout_sum_result)))))))))))))))))", "specialize prime_field_polynomial_bezout_equivalent_transport (p)", "specialize prime_field_polynomial_bezout_equivalent_transport (ab)", "specialize prime_field_polynomial_bezout_equivalent_transport (ac)", "specialize prime_field_polynomial_bezout_equivalent_transport (L)", "specialize prime_field_polynomial_bezout_equivalent_transport (bb)", "specialize prime_field_polynomial_bezout_equivalent_transport (bc)", "specialize prime_field_polynomial_bezout_equivalent_transport (M)", "specialize prime_field_polynomial_bezout_equivalent_transport (x)", "specialize prime_field_polynomial_bezout_equivalent_transport (x1)", "specialize prime_field_polynomial_bezout_equivalent_transport (x2)", "specialize prime_field_polynomial_bezout_equivalent_transport (x3)", "specialize prime_field_polynomial_bezout_equivalent_transport (x4)", "specialize prime_field_polynomial_bezout_equivalent_transport (x5)", "specialize prime_field_polynomial_bezout_equivalent_transport (x6)", "specialize prime_field_polynomial_bezout_equivalent_transport (x7)", "specialize prime_field_polynomial_bezout_equivalent_transport (x8)", "specialize prime_field_polynomial_bezout_equivalent_transport (ab)", "specialize prime_field_polynomial_bezout_equivalent_transport (ac)", "specialize prime_field_polynomial_bezout_equivalent_transport (L)", "specialize prime_field_polynomial_bezout_equivalent_transport (bb2)", "specialize prime_field_polynomial_bezout_equivalent_transport (bc2)", "specialize prime_field_polynomial_bezout_equivalent_transport (M2)", "specialize prime_field_polynomial_bezout_equivalent_transport (x)", "specialize prime_field_polynomial_bezout_equivalent_transport (x1)", "specialize prime_field_polynomial_bezout_equivalent_transport (x2)", "apply prime_field_polynomial_bezout_equivalent_transport", "exact hp0", "exact hA", "exact hBB", "exact hG", "specialize prime_field_polynomial_power_coefficient_functional (ab)", "specialize prime_field_polynomial_power_coefficient_functional (ac)", "specialize prime_field_polynomial_power_coefficient_functional (L)", "apply prime_field_polynomial_power_coefficient_functional", "exact heq", "specialize prime_field_polynomial_power_coefficient_functional (x)", "specialize prime_field_polynomial_power_coefficient_functional (x1)", "specialize prime_field_polynomial_power_coefficient_functional (x2)", "apply prime_field_polynomial_power_coefficient_functional", "exact hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "exists x6", "exists x7", "exists x8", "split", "exact hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_left", "split", "split", "exact hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_left_left", "specialize prime_field_polynomial_right_divides_equivalent_target (p)", "specialize prime_field_polynomial_right_divides_equivalent_target (x)", "specialize prime_field_polynomial_right_divides_equivalent_target (x1)", "specialize prime_field_polynomial_right_divides_equivalent_target (x2)", "specialize prime_field_polynomial_right_divides_equivalent_target (bb)", "specialize prime_field_polynomial_right_divides_equivalent_target (bc)", "specialize prime_field_polynomial_right_divides_equivalent_target (M)", "specialize prime_field_polynomial_right_divides_equivalent_target (bb2)", "specialize prime_field_polynomial_right_divides_equivalent_target (bc2)", "specialize prime_field_polynomial_right_divides_equivalent_target (M2)", "apply prime_field_polynomial_right_divides_equivalent_target", "exact hBB", "exact hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_left_right", "exact heq", "exact hb" ], "script_sha256": "de5c354fc90ce79664ba17d79467b400692843f2b9ae64e8f46050e05a6f6c60", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_existence_candidate.py", "sha256": "81f2f48dd2e81894c7a267453646eb6f2b6f9bd3ee320386d8c561f6b9f8b8ca" }, "statement": "forall p ab ac L bb bc M bb2 bc2 M2. (~((p) = 1) /\\ forall pfa_factor_left_gcd_transport_prime pfa_factor_right_gcd_transport_prime. (p) = pfa_factor_left_gcd_transport_prime * pfa_factor_right_gcd_transport_prime -> pfa_factor_left_gcd_transport_prime = 1 \\/ pfa_factor_right_gcd_transport_prime = 1) -> (forall fom_index_pfp_gcd_transport_A. (exists fom_gap_pfp_gcd_transport_A_index_bound. fom_gap_pfp_gcd_transport_A_index_bound + S (fom_index_pfp_gcd_transport_A) = L) -> exists fom_value_pfp_gcd_transport_A. ((((exists fom_beta_height_pfp_gcd_transport_A_entry. fom_beta_height_pfp_gcd_transport_A_entry + S (fom_value_pfp_gcd_transport_A) = S ((S (fom_index_pfp_gcd_transport_A)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_transport_A_entry. ab = fom_beta_quotient_pfp_gcd_transport_A_entry * S ((S (fom_index_pfp_gcd_transport_A)) * ac) + (fom_value_pfp_gcd_transport_A))) /\\ (exists fom_gap_pfp_gcd_transport_A_value_bound. fom_gap_pfp_gcd_transport_A_value_bound + S (fom_value_pfp_gcd_transport_A) = p))) -> (forall fom_index_pfp_gcd_transport_B. (exists fom_gap_pfp_gcd_transport_B_index_bound. fom_gap_pfp_gcd_transport_B_index_bound + S (fom_index_pfp_gcd_transport_B) = M2) -> exists fom_value_pfp_gcd_transport_B. ((((exists fom_beta_height_pfp_gcd_transport_B_entry. fom_beta_height_pfp_gcd_transport_B_entry + S (fom_value_pfp_gcd_transport_B) = S ((S (fom_index_pfp_gcd_transport_B)) * bc2)) /\\ exists fom_beta_quotient_pfp_gcd_transport_B_entry. bb2 = fom_beta_quotient_pfp_gcd_transport_B_entry * S ((S (fom_index_pfp_gcd_transport_B)) * bc2) + (fom_value_pfp_gcd_transport_B))) /\\ (exists fom_gap_pfp_gcd_transport_B_value_bound. fom_gap_pfp_gcd_transport_B_value_bound + S (fom_value_pfp_gcd_transport_B) = p))) -> (forall pfrep_power_gcd_transport_equivalent pfrep_left_gcd_transport_equivalent pfrep_right_gcd_transport_equivalent. ((exists pfrep_position_gcd_transport_equivalentfirst. ((pfrep_position_gcd_transport_equivalentfirst+S (pfrep_power_gcd_transport_equivalent)=(M)) /\\ ((((exists ff_h_pfp_gcd_transport_equivalentfirstentry. ff_h_pfp_gcd_transport_equivalentfirstentry + S (pfrep_left_gcd_transport_equivalent) = S ((S (pfrep_position_gcd_transport_equivalentfirst)) * bc)) /\\ exists ff_q_pfp_gcd_transport_equivalentfirstentry. bb = ff_q_pfp_gcd_transport_equivalentfirstentry * S ((S (pfrep_position_gcd_transport_equivalentfirst)) * bc) + (pfrep_left_gcd_transport_equivalent)))))) \\/ (((exists pfrep_gap_gcd_transport_equivalentfirstoutside. pfrep_gap_gcd_transport_equivalentfirstoutside+(M)=(pfrep_power_gcd_transport_equivalent)) /\\ (((pfrep_left_gcd_transport_equivalent)=0))))) -> ((exists pfrep_position_gcd_transport_equivalentsecond. ((pfrep_position_gcd_transport_equivalentsecond+S (pfrep_power_gcd_transport_equivalent)=(M2)) /\\ ((((exists ff_h_pfp_gcd_transport_equivalentsecondentry. ff_h_pfp_gcd_transport_equivalentsecondentry + S (pfrep_right_gcd_transport_equivalent) = S ((S (pfrep_position_gcd_transport_equivalentsecond)) * bc2)) /\\ exists ff_q_pfp_gcd_transport_equivalentsecondentry. bb2 = ff_q_pfp_gcd_transport_equivalentsecondentry * S ((S (pfrep_position_gcd_transport_equivalentsecond)) * bc2) + (pfrep_right_gcd_transport_equivalent)))))) \\/ (((exists pfrep_gap_gcd_transport_equivalentsecondoutside. pfrep_gap_gcd_transport_equivalentsecondoutside+(M2)=(pfrep_power_gcd_transport_equivalent)) /\\ (((pfrep_right_gcd_transport_equivalent)=0))))) -> pfrep_left_gcd_transport_equivalent=pfrep_right_gcd_transport_equivalent) -> (exists pfgs_gb_gcd_transport_source pfgs_gc_gcd_transport_source pfgs_G_gcd_transport_source pfgs_ub_gcd_transport_source pfgs_uc_gcd_transport_source pfgs_U_gcd_transport_source pfgs_vb_gcd_transport_source pfgs_vc_gcd_transport_source pfgs_V_gcd_transport_source. (((pfgs_G_gcd_transport_source)=0 \\/ (((~((pfgs_G_gcd_transport_source) = 0)) /\\ (((forall fom_index_pfp_gcd_transport_source_witness_normal_moniccoefficients. (exists fom_gap_pfp_gcd_transport_source_witness_normal_moniccoefficients_index_bound. fom_gap_pfp_gcd_transport_source_witness_normal_moniccoefficients_index_bound + S (fom_index_pfp_gcd_transport_source_witness_normal_moniccoefficients) = pfgs_G_gcd_transport_source) -> exists fom_value_pfp_gcd_transport_source_witness_normal_moniccoefficients. ((((exists fom_beta_height_pfp_gcd_transport_source_witness_normal_moniccoefficients_entry. fom_beta_height_pfp_gcd_transport_source_witness_normal_moniccoefficients_entry + S (fom_value_pfp_gcd_transport_source_witness_normal_moniccoefficients) = S ((S (fom_index_pfp_gcd_transport_source_witness_normal_moniccoefficients)) * pfgs_gc_gcd_transport_source)) /\\ exists fom_beta_quotient_pfp_gcd_transport_source_witness_normal_moniccoefficients_entry. pfgs_gb_gcd_transport_source = fom_beta_quotient_pfp_gcd_transport_source_witness_normal_moniccoefficients_entry * S ((S (fom_index_pfp_gcd_transport_source_witness_normal_moniccoefficients)) * pfgs_gc_gcd_transport_source) + (fom_value_pfp_gcd_transport_source_witness_normal_moniccoefficients))) /\\ (exists fom_gap_pfp_gcd_transport_source_witness_normal_moniccoefficients_value_bound. fom_gap_pfp_gcd_transport_source_witness_normal_moniccoefficients_value_bound + S (fom_value_pfp_gcd_transport_source_witness_normal_moniccoefficients) = p))) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_normal_monicleading. ff_h_pfp_gcd_transport_source_witness_normal_monicleading + S (1) = S ((S (0)) * pfgs_gc_gcd_transport_source)) /\\ exists ff_q_pfp_gcd_transport_source_witness_normal_monicleading. pfgs_gb_gcd_transport_source = ff_q_pfp_gcd_transport_source_witness_normal_monicleading * S ((S (0)) * pfgs_gc_gcd_transport_source) + (1))))))))) /\\ (((((((forall fom_index_pfp_gcd_transport_source_witness_common_left_bounded. (exists fom_gap_pfp_gcd_transport_source_witness_common_left_bounded_index_bound. fom_gap_pfp_gcd_transport_source_witness_common_left_bounded_index_bound + S (fom_index_pfp_gcd_transport_source_witness_common_left_bounded) = L) -> exists fom_value_pfp_gcd_transport_source_witness_common_left_bounded. ((((exists fom_beta_height_pfp_gcd_transport_source_witness_common_left_bounded_entry. fom_beta_height_pfp_gcd_transport_source_witness_common_left_bounded_entry + S (fom_value_pfp_gcd_transport_source_witness_common_left_bounded) = S ((S (fom_index_pfp_gcd_transport_source_witness_common_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_transport_source_witness_common_left_bounded_entry. ab = fom_beta_quotient_pfp_gcd_transport_source_witness_common_left_bounded_entry * S ((S (fom_index_pfp_gcd_transport_source_witness_common_left_bounded)) * ac) + (fom_value_pfp_gcd_transport_source_witness_common_left_bounded))) /\\ (exists fom_gap_pfp_gcd_transport_source_witness_common_left_bounded_value_bound. fom_gap_pfp_gcd_transport_source_witness_common_left_bounded_value_bound + S (fom_value_pfp_gcd_transport_source_witness_common_left_bounded) = p))) /\\ ((exists pfgd_qb_gcd_transport_source_witness_common_left pfgd_qc_gcd_transport_source_witness_common_left pfgd_Q_gcd_transport_source_witness_common_left pfgd_pb_gcd_transport_source_witness_common_left pfgd_pc_gcd_transport_source_witness_common_left pfgd_P_gcd_transport_source_witness_common_left. ((((forall fom_index_pfp_gcd_transport_source_witness_common_left_productleft. (exists fom_gap_pfp_gcd_transport_source_witness_common_left_productleft_index_bound. fom_gap_pfp_gcd_transport_source_witness_common_left_productleft_index_bound + S (fom_index_pfp_gcd_transport_source_witness_common_left_productleft) = pfgd_Q_gcd_transport_source_witness_common_left) -> exists fom_value_pfp_gcd_transport_source_witness_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_transport_source_witness_common_left_productleft_entry. fom_beta_height_pfp_gcd_transport_source_witness_common_left_productleft_entry + S (fom_value_pfp_gcd_transport_source_witness_common_left_productleft) = S ((S (fom_index_pfp_gcd_transport_source_witness_common_left_productleft)) * pfgd_qc_gcd_transport_source_witness_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_transport_source_witness_common_left_productleft_entry. pfgd_qb_gcd_transport_source_witness_common_left = fom_beta_quotient_pfp_gcd_transport_source_witness_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_transport_source_witness_common_left_productleft)) * pfgd_qc_gcd_transport_source_witness_common_left) + (fom_value_pfp_gcd_transport_source_witness_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_transport_source_witness_common_left_productleft_value_bound. fom_gap_pfp_gcd_transport_source_witness_common_left_productleft_value_bound + S (fom_value_pfp_gcd_transport_source_witness_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_transport_source_witness_common_left_productright. (exists fom_gap_pfp_gcd_transport_source_witness_common_left_productright_index_bound. fom_gap_pfp_gcd_transport_source_witness_common_left_productright_index_bound + S (fom_index_pfp_gcd_transport_source_witness_common_left_productright) = pfgs_G_gcd_transport_source) -> exists fom_value_pfp_gcd_transport_source_witness_common_left_productright. ((((exists fom_beta_height_pfp_gcd_transport_source_witness_common_left_productright_entry. fom_beta_height_pfp_gcd_transport_source_witness_common_left_productright_entry + S (fom_value_pfp_gcd_transport_source_witness_common_left_productright) = S ((S (fom_index_pfp_gcd_transport_source_witness_common_left_productright)) * pfgs_gc_gcd_transport_source)) /\\ exists fom_beta_quotient_pfp_gcd_transport_source_witness_common_left_productright_entry. pfgs_gb_gcd_transport_source = fom_beta_quotient_pfp_gcd_transport_source_witness_common_left_productright_entry * S ((S (fom_index_pfp_gcd_transport_source_witness_common_left_productright)) * pfgs_gc_gcd_transport_source) + (fom_value_pfp_gcd_transport_source_witness_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_transport_source_witness_common_left_productright_value_bound. fom_gap_pfp_gcd_transport_source_witness_common_left_productright_value_bound + S (fom_value_pfp_gcd_transport_source_witness_common_left_productright) = p))) /\\ (((((((pfgd_Q_gcd_transport_source_witness_common_left)=0 \\/ (pfgs_G_gcd_transport_source)=0) /\\ (((pfgd_P_gcd_transport_source_witness_common_left)=0)))) \\/ (((~((pfgd_Q_gcd_transport_source_witness_common_left)=0)) /\\ (((~((pfgs_G_gcd_transport_source)=0)) /\\ (((pfgd_Q_gcd_transport_source_witness_common_left)+(pfgs_G_gcd_transport_source)=S (pfgd_P_gcd_transport_source_witness_common_left)))))))) /\\ ((forall pfc_index_gcd_transport_source_witness_common_left_productcoefficients. (exists pfa_gap_gcd_transport_source_witness_common_left_productcoefficientsbound. pfa_gap_gcd_transport_source_witness_common_left_productcoefficientsbound + S (pfc_index_gcd_transport_source_witness_common_left_productcoefficients) = (pfgd_P_gcd_transport_source_witness_common_left)) -> exists pfc_value_gcd_transport_source_witness_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_transport_source_witness_common_left_productcoefficientsentry. ff_h_pfp_gcd_transport_source_witness_common_left_productcoefficientsentry + S (pfc_value_gcd_transport_source_witness_common_left_productcoefficients) = S ((S (pfc_index_gcd_transport_source_witness_common_left_productcoefficients)) * pfgd_pc_gcd_transport_source_witness_common_left)) /\\ exists ff_q_pfp_gcd_transport_source_witness_common_left_productcoefficientsentry. pfgd_pb_gcd_transport_source_witness_common_left = ff_q_pfp_gcd_transport_source_witness_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_transport_source_witness_common_left_productcoefficients)) * pfgd_pc_gcd_transport_source_witness_common_left) + (pfc_value_gcd_transport_source_witness_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_transport_source_witness_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_transport_source_witness_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_transport_source_witness_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_transport_source_witness_common_left_productcoefficients))) -> exists pfc_value_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_source_witness_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_transport_source_witness_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_source_witness_common_left_productcoefficientscoefficient) + (pfc_value_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_transport_source_witness_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_transport_source_witness_common_left)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_transport_source_witness_common_left)) /\\ exists ff_q_pfp_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_transport_source_witness_common_left = ff_q_pfp_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_transport_source_witness_common_left) + (pfc_left_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_transport_source_witness_common_left)=(pfc_index_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm) = (pfgs_G_gcd_transport_source)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_transport_source)) /\\ exists ff_q_pfp_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_gcd_transport_source = ff_q_pfp_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_transport_source) + (pfc_right_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_gcd_transport_source)=(pfc_complement_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_transport_source_witness_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_transport_source_witness_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_transport_source_witness_common_left_productcoefficients))) * fs_v_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_transport_source_witness_common_left_productcoefficients))) * fs_v_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_transport_source_witness_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_transport_source_witness_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_source_witness_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_transport_source_witness_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_source_witness_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_transport_source_witness_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_transport_source_witness_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_transport_source_witness_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_transport_source_witness_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_transport_source_witness_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_transport_source_witness_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_transport_source_witness_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_transport_source_witness_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_transport_source_witness_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_transport_source_witness_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_transport_source_witness_common_left_equivalent pfrep_left_gcd_transport_source_witness_common_left_equivalent pfrep_right_gcd_transport_source_witness_common_left_equivalent. ((exists pfrep_position_gcd_transport_source_witness_common_left_equivalentfirst. ((pfrep_position_gcd_transport_source_witness_common_left_equivalentfirst+S (pfrep_power_gcd_transport_source_witness_common_left_equivalent)=(pfgd_P_gcd_transport_source_witness_common_left)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_common_left_equivalentfirstentry. ff_h_pfp_gcd_transport_source_witness_common_left_equivalentfirstentry + S (pfrep_left_gcd_transport_source_witness_common_left_equivalent) = S ((S (pfrep_position_gcd_transport_source_witness_common_left_equivalentfirst)) * pfgd_pc_gcd_transport_source_witness_common_left)) /\\ exists ff_q_pfp_gcd_transport_source_witness_common_left_equivalentfirstentry. pfgd_pb_gcd_transport_source_witness_common_left = ff_q_pfp_gcd_transport_source_witness_common_left_equivalentfirstentry * S ((S (pfrep_position_gcd_transport_source_witness_common_left_equivalentfirst)) * pfgd_pc_gcd_transport_source_witness_common_left) + (pfrep_left_gcd_transport_source_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_transport_source_witness_common_left_equivalentfirstoutside. pfrep_gap_gcd_transport_source_witness_common_left_equivalentfirstoutside+(pfgd_P_gcd_transport_source_witness_common_left)=(pfrep_power_gcd_transport_source_witness_common_left_equivalent)) /\\ (((pfrep_left_gcd_transport_source_witness_common_left_equivalent)=0))))) -> ((exists pfrep_position_gcd_transport_source_witness_common_left_equivalentsecond. ((pfrep_position_gcd_transport_source_witness_common_left_equivalentsecond+S (pfrep_power_gcd_transport_source_witness_common_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_common_left_equivalentsecondentry. ff_h_pfp_gcd_transport_source_witness_common_left_equivalentsecondentry + S (pfrep_right_gcd_transport_source_witness_common_left_equivalent) = S ((S (pfrep_position_gcd_transport_source_witness_common_left_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_gcd_transport_source_witness_common_left_equivalentsecondentry. ab = ff_q_pfp_gcd_transport_source_witness_common_left_equivalentsecondentry * S ((S (pfrep_position_gcd_transport_source_witness_common_left_equivalentsecond)) * ac) + (pfrep_right_gcd_transport_source_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_transport_source_witness_common_left_equivalentsecondoutside. pfrep_gap_gcd_transport_source_witness_common_left_equivalentsecondoutside+(L)=(pfrep_power_gcd_transport_source_witness_common_left_equivalent)) /\\ (((pfrep_right_gcd_transport_source_witness_common_left_equivalent)=0))))) -> pfrep_left_gcd_transport_source_witness_common_left_equivalent=pfrep_right_gcd_transport_source_witness_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_transport_source_witness_common_right_bounded. (exists fom_gap_pfp_gcd_transport_source_witness_common_right_bounded_index_bound. fom_gap_pfp_gcd_transport_source_witness_common_right_bounded_index_bound + S (fom_index_pfp_gcd_transport_source_witness_common_right_bounded) = M) -> exists fom_value_pfp_gcd_transport_source_witness_common_right_bounded. ((((exists fom_beta_height_pfp_gcd_transport_source_witness_common_right_bounded_entry. fom_beta_height_pfp_gcd_transport_source_witness_common_right_bounded_entry + S (fom_value_pfp_gcd_transport_source_witness_common_right_bounded) = S ((S (fom_index_pfp_gcd_transport_source_witness_common_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_transport_source_witness_common_right_bounded_entry. bb = fom_beta_quotient_pfp_gcd_transport_source_witness_common_right_bounded_entry * S ((S (fom_index_pfp_gcd_transport_source_witness_common_right_bounded)) * bc) + (fom_value_pfp_gcd_transport_source_witness_common_right_bounded))) /\\ (exists fom_gap_pfp_gcd_transport_source_witness_common_right_bounded_value_bound. fom_gap_pfp_gcd_transport_source_witness_common_right_bounded_value_bound + S (fom_value_pfp_gcd_transport_source_witness_common_right_bounded) = p))) /\\ ((exists pfgd_qb_gcd_transport_source_witness_common_right pfgd_qc_gcd_transport_source_witness_common_right pfgd_Q_gcd_transport_source_witness_common_right pfgd_pb_gcd_transport_source_witness_common_right pfgd_pc_gcd_transport_source_witness_common_right pfgd_P_gcd_transport_source_witness_common_right. ((((forall fom_index_pfp_gcd_transport_source_witness_common_right_productleft. (exists fom_gap_pfp_gcd_transport_source_witness_common_right_productleft_index_bound. fom_gap_pfp_gcd_transport_source_witness_common_right_productleft_index_bound + S (fom_index_pfp_gcd_transport_source_witness_common_right_productleft) = pfgd_Q_gcd_transport_source_witness_common_right) -> exists fom_value_pfp_gcd_transport_source_witness_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_transport_source_witness_common_right_productleft_entry. fom_beta_height_pfp_gcd_transport_source_witness_common_right_productleft_entry + S (fom_value_pfp_gcd_transport_source_witness_common_right_productleft) = S ((S (fom_index_pfp_gcd_transport_source_witness_common_right_productleft)) * pfgd_qc_gcd_transport_source_witness_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_transport_source_witness_common_right_productleft_entry. pfgd_qb_gcd_transport_source_witness_common_right = fom_beta_quotient_pfp_gcd_transport_source_witness_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_transport_source_witness_common_right_productleft)) * pfgd_qc_gcd_transport_source_witness_common_right) + (fom_value_pfp_gcd_transport_source_witness_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_transport_source_witness_common_right_productleft_value_bound. fom_gap_pfp_gcd_transport_source_witness_common_right_productleft_value_bound + S (fom_value_pfp_gcd_transport_source_witness_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_transport_source_witness_common_right_productright. (exists fom_gap_pfp_gcd_transport_source_witness_common_right_productright_index_bound. fom_gap_pfp_gcd_transport_source_witness_common_right_productright_index_bound + S (fom_index_pfp_gcd_transport_source_witness_common_right_productright) = pfgs_G_gcd_transport_source) -> exists fom_value_pfp_gcd_transport_source_witness_common_right_productright. ((((exists fom_beta_height_pfp_gcd_transport_source_witness_common_right_productright_entry. fom_beta_height_pfp_gcd_transport_source_witness_common_right_productright_entry + S (fom_value_pfp_gcd_transport_source_witness_common_right_productright) = S ((S (fom_index_pfp_gcd_transport_source_witness_common_right_productright)) * pfgs_gc_gcd_transport_source)) /\\ exists fom_beta_quotient_pfp_gcd_transport_source_witness_common_right_productright_entry. pfgs_gb_gcd_transport_source = fom_beta_quotient_pfp_gcd_transport_source_witness_common_right_productright_entry * S ((S (fom_index_pfp_gcd_transport_source_witness_common_right_productright)) * pfgs_gc_gcd_transport_source) + (fom_value_pfp_gcd_transport_source_witness_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_transport_source_witness_common_right_productright_value_bound. fom_gap_pfp_gcd_transport_source_witness_common_right_productright_value_bound + S (fom_value_pfp_gcd_transport_source_witness_common_right_productright) = p))) /\\ (((((((pfgd_Q_gcd_transport_source_witness_common_right)=0 \\/ (pfgs_G_gcd_transport_source)=0) /\\ (((pfgd_P_gcd_transport_source_witness_common_right)=0)))) \\/ (((~((pfgd_Q_gcd_transport_source_witness_common_right)=0)) /\\ (((~((pfgs_G_gcd_transport_source)=0)) /\\ (((pfgd_Q_gcd_transport_source_witness_common_right)+(pfgs_G_gcd_transport_source)=S (pfgd_P_gcd_transport_source_witness_common_right)))))))) /\\ ((forall pfc_index_gcd_transport_source_witness_common_right_productcoefficients. (exists pfa_gap_gcd_transport_source_witness_common_right_productcoefficientsbound. pfa_gap_gcd_transport_source_witness_common_right_productcoefficientsbound + S (pfc_index_gcd_transport_source_witness_common_right_productcoefficients) = (pfgd_P_gcd_transport_source_witness_common_right)) -> exists pfc_value_gcd_transport_source_witness_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_transport_source_witness_common_right_productcoefficientsentry. ff_h_pfp_gcd_transport_source_witness_common_right_productcoefficientsentry + S (pfc_value_gcd_transport_source_witness_common_right_productcoefficients) = S ((S (pfc_index_gcd_transport_source_witness_common_right_productcoefficients)) * pfgd_pc_gcd_transport_source_witness_common_right)) /\\ exists ff_q_pfp_gcd_transport_source_witness_common_right_productcoefficientsentry. pfgd_pb_gcd_transport_source_witness_common_right = ff_q_pfp_gcd_transport_source_witness_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_transport_source_witness_common_right_productcoefficients)) * pfgd_pc_gcd_transport_source_witness_common_right) + (pfc_value_gcd_transport_source_witness_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_transport_source_witness_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_transport_source_witness_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_transport_source_witness_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_transport_source_witness_common_right_productcoefficients))) -> exists pfc_value_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_source_witness_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_transport_source_witness_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_source_witness_common_right_productcoefficientscoefficient) + (pfc_value_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_transport_source_witness_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_transport_source_witness_common_right)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_transport_source_witness_common_right)) /\\ exists ff_q_pfp_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_transport_source_witness_common_right = ff_q_pfp_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_transport_source_witness_common_right) + (pfc_left_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_transport_source_witness_common_right)=(pfc_index_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm) = (pfgs_G_gcd_transport_source)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_transport_source)) /\\ exists ff_q_pfp_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_gcd_transport_source = ff_q_pfp_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_transport_source) + (pfc_right_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_gcd_transport_source)=(pfc_complement_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_transport_source_witness_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_transport_source_witness_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_transport_source_witness_common_right_productcoefficients))) * fs_v_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_transport_source_witness_common_right_productcoefficients))) * fs_v_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_transport_source_witness_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_transport_source_witness_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_source_witness_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_transport_source_witness_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_source_witness_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_transport_source_witness_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_transport_source_witness_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_transport_source_witness_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_transport_source_witness_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_transport_source_witness_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_transport_source_witness_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_transport_source_witness_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_transport_source_witness_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_transport_source_witness_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_transport_source_witness_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_transport_source_witness_common_right_equivalent pfrep_left_gcd_transport_source_witness_common_right_equivalent pfrep_right_gcd_transport_source_witness_common_right_equivalent. ((exists pfrep_position_gcd_transport_source_witness_common_right_equivalentfirst. ((pfrep_position_gcd_transport_source_witness_common_right_equivalentfirst+S (pfrep_power_gcd_transport_source_witness_common_right_equivalent)=(pfgd_P_gcd_transport_source_witness_common_right)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_common_right_equivalentfirstentry. ff_h_pfp_gcd_transport_source_witness_common_right_equivalentfirstentry + S (pfrep_left_gcd_transport_source_witness_common_right_equivalent) = S ((S (pfrep_position_gcd_transport_source_witness_common_right_equivalentfirst)) * pfgd_pc_gcd_transport_source_witness_common_right)) /\\ exists ff_q_pfp_gcd_transport_source_witness_common_right_equivalentfirstentry. pfgd_pb_gcd_transport_source_witness_common_right = ff_q_pfp_gcd_transport_source_witness_common_right_equivalentfirstentry * S ((S (pfrep_position_gcd_transport_source_witness_common_right_equivalentfirst)) * pfgd_pc_gcd_transport_source_witness_common_right) + (pfrep_left_gcd_transport_source_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_transport_source_witness_common_right_equivalentfirstoutside. pfrep_gap_gcd_transport_source_witness_common_right_equivalentfirstoutside+(pfgd_P_gcd_transport_source_witness_common_right)=(pfrep_power_gcd_transport_source_witness_common_right_equivalent)) /\\ (((pfrep_left_gcd_transport_source_witness_common_right_equivalent)=0))))) -> ((exists pfrep_position_gcd_transport_source_witness_common_right_equivalentsecond. ((pfrep_position_gcd_transport_source_witness_common_right_equivalentsecond+S (pfrep_power_gcd_transport_source_witness_common_right_equivalent)=(M)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_common_right_equivalentsecondentry. ff_h_pfp_gcd_transport_source_witness_common_right_equivalentsecondentry + S (pfrep_right_gcd_transport_source_witness_common_right_equivalent) = S ((S (pfrep_position_gcd_transport_source_witness_common_right_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_gcd_transport_source_witness_common_right_equivalentsecondentry. bb = ff_q_pfp_gcd_transport_source_witness_common_right_equivalentsecondentry * S ((S (pfrep_position_gcd_transport_source_witness_common_right_equivalentsecond)) * bc) + (pfrep_right_gcd_transport_source_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_transport_source_witness_common_right_equivalentsecondoutside. pfrep_gap_gcd_transport_source_witness_common_right_equivalentsecondoutside+(M)=(pfrep_power_gcd_transport_source_witness_common_right_equivalent)) /\\ (((pfrep_right_gcd_transport_source_witness_common_right_equivalent)=0))))) -> pfrep_left_gcd_transport_source_witness_common_right_equivalent=pfrep_right_gcd_transport_source_witness_common_right_equivalent)))))))))) /\\ ((exists pfgb_pb_gcd_transport_source_witness_bezout pfgb_pc_gcd_transport_source_witness_bezout pfgb_P_gcd_transport_source_witness_bezout pfgb_qb_gcd_transport_source_witness_bezout pfgb_qc_gcd_transport_source_witness_bezout pfgb_Q_gcd_transport_source_witness_bezout. ((((forall fom_index_pfp_gcd_transport_source_witness_bezout_leftleft. (exists fom_gap_pfp_gcd_transport_source_witness_bezout_leftleft_index_bound. fom_gap_pfp_gcd_transport_source_witness_bezout_leftleft_index_bound + S (fom_index_pfp_gcd_transport_source_witness_bezout_leftleft) = pfgs_U_gcd_transport_source) -> exists fom_value_pfp_gcd_transport_source_witness_bezout_leftleft. ((((exists fom_beta_height_pfp_gcd_transport_source_witness_bezout_leftleft_entry. fom_beta_height_pfp_gcd_transport_source_witness_bezout_leftleft_entry + S (fom_value_pfp_gcd_transport_source_witness_bezout_leftleft) = S ((S (fom_index_pfp_gcd_transport_source_witness_bezout_leftleft)) * pfgs_uc_gcd_transport_source)) /\\ exists fom_beta_quotient_pfp_gcd_transport_source_witness_bezout_leftleft_entry. pfgs_ub_gcd_transport_source = fom_beta_quotient_pfp_gcd_transport_source_witness_bezout_leftleft_entry * S ((S (fom_index_pfp_gcd_transport_source_witness_bezout_leftleft)) * pfgs_uc_gcd_transport_source) + (fom_value_pfp_gcd_transport_source_witness_bezout_leftleft))) /\\ (exists fom_gap_pfp_gcd_transport_source_witness_bezout_leftleft_value_bound. fom_gap_pfp_gcd_transport_source_witness_bezout_leftleft_value_bound + S (fom_value_pfp_gcd_transport_source_witness_bezout_leftleft) = p))) /\\ (((forall fom_index_pfp_gcd_transport_source_witness_bezout_leftright. (exists fom_gap_pfp_gcd_transport_source_witness_bezout_leftright_index_bound. fom_gap_pfp_gcd_transport_source_witness_bezout_leftright_index_bound + S (fom_index_pfp_gcd_transport_source_witness_bezout_leftright) = L) -> exists fom_value_pfp_gcd_transport_source_witness_bezout_leftright. ((((exists fom_beta_height_pfp_gcd_transport_source_witness_bezout_leftright_entry. fom_beta_height_pfp_gcd_transport_source_witness_bezout_leftright_entry + S (fom_value_pfp_gcd_transport_source_witness_bezout_leftright) = S ((S (fom_index_pfp_gcd_transport_source_witness_bezout_leftright)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_transport_source_witness_bezout_leftright_entry. ab = fom_beta_quotient_pfp_gcd_transport_source_witness_bezout_leftright_entry * S ((S (fom_index_pfp_gcd_transport_source_witness_bezout_leftright)) * ac) + (fom_value_pfp_gcd_transport_source_witness_bezout_leftright))) /\\ (exists fom_gap_pfp_gcd_transport_source_witness_bezout_leftright_value_bound. fom_gap_pfp_gcd_transport_source_witness_bezout_leftright_value_bound + S (fom_value_pfp_gcd_transport_source_witness_bezout_leftright) = p))) /\\ (((((((pfgs_U_gcd_transport_source)=0 \\/ (L)=0) /\\ (((pfgb_P_gcd_transport_source_witness_bezout)=0)))) \\/ (((~((pfgs_U_gcd_transport_source)=0)) /\\ (((~((L)=0)) /\\ (((pfgs_U_gcd_transport_source)+(L)=S (pfgb_P_gcd_transport_source_witness_bezout)))))))) /\\ ((forall pfc_index_gcd_transport_source_witness_bezout_leftcoefficients. (exists pfa_gap_gcd_transport_source_witness_bezout_leftcoefficientsbound. pfa_gap_gcd_transport_source_witness_bezout_leftcoefficientsbound + S (pfc_index_gcd_transport_source_witness_bezout_leftcoefficients) = (pfgb_P_gcd_transport_source_witness_bezout)) -> exists pfc_value_gcd_transport_source_witness_bezout_leftcoefficients. ((((exists ff_h_pfp_gcd_transport_source_witness_bezout_leftcoefficientsentry. ff_h_pfp_gcd_transport_source_witness_bezout_leftcoefficientsentry + S (pfc_value_gcd_transport_source_witness_bezout_leftcoefficients) = S ((S (pfc_index_gcd_transport_source_witness_bezout_leftcoefficients)) * pfgb_pc_gcd_transport_source_witness_bezout)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_leftcoefficientsentry. pfgb_pb_gcd_transport_source_witness_bezout = ff_q_pfp_gcd_transport_source_witness_bezout_leftcoefficientsentry * S ((S (pfc_index_gcd_transport_source_witness_bezout_leftcoefficients)) * pfgb_pc_gcd_transport_source_witness_bezout) + (pfc_value_gcd_transport_source_witness_bezout_leftcoefficients))) /\\ ((exists pfc_terms_code_gcd_transport_source_witness_bezout_leftcoefficientscoefficient pfc_terms_scale_gcd_transport_source_witness_bezout_leftcoefficientscoefficient pfc_natural_sum_gcd_transport_source_witness_bezout_leftcoefficientscoefficient. ((forall pfc_index_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalbound. pfa_gap_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_transport_source_witness_bezout_leftcoefficients))) -> exists pfc_value_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_source_witness_bezout_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_transport_source_witness_bezout_leftcoefficientscoefficient = ff_q_pfp_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_source_witness_bezout_leftcoefficientscoefficient) + (pfc_value_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_left_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_right_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonal)+pfc_complement_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm=(pfc_index_gcd_transport_source_witness_bezout_leftcoefficients)) /\\ ((((((exists pfa_gap_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonal) = (pfgs_U_gcd_transport_source)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_gcd_transport_source)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. pfgs_ub_gcd_transport_source = ff_q_pfp_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_gcd_transport_source) + (pfc_left_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside+(pfgs_U_gcd_transport_source)=(pfc_index_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonal)=pfc_left_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm*pfc_right_gcd_transport_source_witness_bezout_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum fs_v_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_transport_source_witness_bezout_leftcoefficientscoefficient) = S ((S (S (pfc_index_gcd_transport_source_witness_bezout_leftcoefficients))) * fs_v_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_transport_source_witness_bezout_leftcoefficients))) * fs_v_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum) + (pfc_natural_sum_gcd_transport_source_witness_bezout_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_transport_source_witness_bezout_leftcoefficients)) -> exists fs_a_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_source_witness_bezout_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_transport_source_witness_bezout_leftcoefficientscoefficient = fs_q_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_source_witness_bezout_leftcoefficientscoefficient) + (fs_a_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum) + (fs_r_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum) + (fs_s_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_transport_source_witness_bezout_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_transport_source_witness_bezout_leftcoefficientscoefficientresiduebound. pfa_gap_gcd_transport_source_witness_bezout_leftcoefficientscoefficientresiduebound + S (pfc_value_gcd_transport_source_witness_bezout_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_transport_source_witness_bezout_leftcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_transport_source_witness_bezout_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_transport_source_witness_bezout_leftcoefficientscoefficient) + (p) * pfa_offset_left_gcd_transport_source_witness_bezout_leftcoefficientscoefficientresiduecongruence = (pfc_value_gcd_transport_source_witness_bezout_leftcoefficients) + (p) * pfa_offset_right_gcd_transport_source_witness_bezout_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_gcd_transport_source_witness_bezout_rightleft. (exists fom_gap_pfp_gcd_transport_source_witness_bezout_rightleft_index_bound. fom_gap_pfp_gcd_transport_source_witness_bezout_rightleft_index_bound + S (fom_index_pfp_gcd_transport_source_witness_bezout_rightleft) = pfgs_V_gcd_transport_source) -> exists fom_value_pfp_gcd_transport_source_witness_bezout_rightleft. ((((exists fom_beta_height_pfp_gcd_transport_source_witness_bezout_rightleft_entry. fom_beta_height_pfp_gcd_transport_source_witness_bezout_rightleft_entry + S (fom_value_pfp_gcd_transport_source_witness_bezout_rightleft) = S ((S (fom_index_pfp_gcd_transport_source_witness_bezout_rightleft)) * pfgs_vc_gcd_transport_source)) /\\ exists fom_beta_quotient_pfp_gcd_transport_source_witness_bezout_rightleft_entry. pfgs_vb_gcd_transport_source = fom_beta_quotient_pfp_gcd_transport_source_witness_bezout_rightleft_entry * S ((S (fom_index_pfp_gcd_transport_source_witness_bezout_rightleft)) * pfgs_vc_gcd_transport_source) + (fom_value_pfp_gcd_transport_source_witness_bezout_rightleft))) /\\ (exists fom_gap_pfp_gcd_transport_source_witness_bezout_rightleft_value_bound. fom_gap_pfp_gcd_transport_source_witness_bezout_rightleft_value_bound + S (fom_value_pfp_gcd_transport_source_witness_bezout_rightleft) = p))) /\\ (((forall fom_index_pfp_gcd_transport_source_witness_bezout_rightright. (exists fom_gap_pfp_gcd_transport_source_witness_bezout_rightright_index_bound. fom_gap_pfp_gcd_transport_source_witness_bezout_rightright_index_bound + S (fom_index_pfp_gcd_transport_source_witness_bezout_rightright) = M) -> exists fom_value_pfp_gcd_transport_source_witness_bezout_rightright. ((((exists fom_beta_height_pfp_gcd_transport_source_witness_bezout_rightright_entry. fom_beta_height_pfp_gcd_transport_source_witness_bezout_rightright_entry + S (fom_value_pfp_gcd_transport_source_witness_bezout_rightright) = S ((S (fom_index_pfp_gcd_transport_source_witness_bezout_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_transport_source_witness_bezout_rightright_entry. bb = fom_beta_quotient_pfp_gcd_transport_source_witness_bezout_rightright_entry * S ((S (fom_index_pfp_gcd_transport_source_witness_bezout_rightright)) * bc) + (fom_value_pfp_gcd_transport_source_witness_bezout_rightright))) /\\ (exists fom_gap_pfp_gcd_transport_source_witness_bezout_rightright_value_bound. fom_gap_pfp_gcd_transport_source_witness_bezout_rightright_value_bound + S (fom_value_pfp_gcd_transport_source_witness_bezout_rightright) = p))) /\\ (((((((pfgs_V_gcd_transport_source)=0 \\/ (M)=0) /\\ (((pfgb_Q_gcd_transport_source_witness_bezout)=0)))) \\/ (((~((pfgs_V_gcd_transport_source)=0)) /\\ (((~((M)=0)) /\\ (((pfgs_V_gcd_transport_source)+(M)=S (pfgb_Q_gcd_transport_source_witness_bezout)))))))) /\\ ((forall pfc_index_gcd_transport_source_witness_bezout_rightcoefficients. (exists pfa_gap_gcd_transport_source_witness_bezout_rightcoefficientsbound. pfa_gap_gcd_transport_source_witness_bezout_rightcoefficientsbound + S (pfc_index_gcd_transport_source_witness_bezout_rightcoefficients) = (pfgb_Q_gcd_transport_source_witness_bezout)) -> exists pfc_value_gcd_transport_source_witness_bezout_rightcoefficients. ((((exists ff_h_pfp_gcd_transport_source_witness_bezout_rightcoefficientsentry. ff_h_pfp_gcd_transport_source_witness_bezout_rightcoefficientsentry + S (pfc_value_gcd_transport_source_witness_bezout_rightcoefficients) = S ((S (pfc_index_gcd_transport_source_witness_bezout_rightcoefficients)) * pfgb_qc_gcd_transport_source_witness_bezout)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_rightcoefficientsentry. pfgb_qb_gcd_transport_source_witness_bezout = ff_q_pfp_gcd_transport_source_witness_bezout_rightcoefficientsentry * S ((S (pfc_index_gcd_transport_source_witness_bezout_rightcoefficients)) * pfgb_qc_gcd_transport_source_witness_bezout) + (pfc_value_gcd_transport_source_witness_bezout_rightcoefficients))) /\\ ((exists pfc_terms_code_gcd_transport_source_witness_bezout_rightcoefficientscoefficient pfc_terms_scale_gcd_transport_source_witness_bezout_rightcoefficientscoefficient pfc_natural_sum_gcd_transport_source_witness_bezout_rightcoefficientscoefficient. ((forall pfc_index_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalbound. pfa_gap_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_transport_source_witness_bezout_rightcoefficients))) -> exists pfc_value_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_source_witness_bezout_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_transport_source_witness_bezout_rightcoefficientscoefficient = ff_q_pfp_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_source_witness_bezout_rightcoefficientscoefficient) + (pfc_value_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_left_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_right_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonal)+pfc_complement_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm=(pfc_index_gcd_transport_source_witness_bezout_rightcoefficients)) /\\ ((((((exists pfa_gap_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonal) = (pfgs_V_gcd_transport_source)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_gcd_transport_source)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. pfgs_vb_gcd_transport_source = ff_q_pfp_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_gcd_transport_source) + (pfc_left_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside+(pfgs_V_gcd_transport_source)=(pfc_index_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonal)=pfc_left_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm*pfc_right_gcd_transport_source_witness_bezout_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum fs_v_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_transport_source_witness_bezout_rightcoefficientscoefficient) = S ((S (S (pfc_index_gcd_transport_source_witness_bezout_rightcoefficients))) * fs_v_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_transport_source_witness_bezout_rightcoefficients))) * fs_v_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum) + (pfc_natural_sum_gcd_transport_source_witness_bezout_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_transport_source_witness_bezout_rightcoefficients)) -> exists fs_a_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_source_witness_bezout_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_transport_source_witness_bezout_rightcoefficientscoefficient = fs_q_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_source_witness_bezout_rightcoefficientscoefficient) + (fs_a_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum) + (fs_r_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum) + (fs_s_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_transport_source_witness_bezout_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_transport_source_witness_bezout_rightcoefficientscoefficientresiduebound. pfa_gap_gcd_transport_source_witness_bezout_rightcoefficientscoefficientresiduebound + S (pfc_value_gcd_transport_source_witness_bezout_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_transport_source_witness_bezout_rightcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_transport_source_witness_bezout_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_transport_source_witness_bezout_rightcoefficientscoefficient) + (p) * pfa_offset_left_gcd_transport_source_witness_bezout_rightcoefficientscoefficientresiduecongruence = (pfc_value_gcd_transport_source_witness_bezout_rightcoefficients) + (p) * pfa_offset_right_gcd_transport_source_witness_bezout_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_gcd_transport_source_witness_bezout_sum_left_bounded. (exists fom_gap_pfp_gcd_transport_source_witness_bezout_sum_left_bounded_index_bound. fom_gap_pfp_gcd_transport_source_witness_bezout_sum_left_bounded_index_bound + S (fom_index_pfp_gcd_transport_source_witness_bezout_sum_left_bounded) = pfgb_P_gcd_transport_source_witness_bezout) -> exists fom_value_pfp_gcd_transport_source_witness_bezout_sum_left_bounded. ((((exists fom_beta_height_pfp_gcd_transport_source_witness_bezout_sum_left_bounded_entry. fom_beta_height_pfp_gcd_transport_source_witness_bezout_sum_left_bounded_entry + S (fom_value_pfp_gcd_transport_source_witness_bezout_sum_left_bounded) = S ((S (fom_index_pfp_gcd_transport_source_witness_bezout_sum_left_bounded)) * pfgb_pc_gcd_transport_source_witness_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_transport_source_witness_bezout_sum_left_bounded_entry. pfgb_pb_gcd_transport_source_witness_bezout = fom_beta_quotient_pfp_gcd_transport_source_witness_bezout_sum_left_bounded_entry * S ((S (fom_index_pfp_gcd_transport_source_witness_bezout_sum_left_bounded)) * pfgb_pc_gcd_transport_source_witness_bezout) + (fom_value_pfp_gcd_transport_source_witness_bezout_sum_left_bounded))) /\\ (exists fom_gap_pfp_gcd_transport_source_witness_bezout_sum_left_bounded_value_bound. fom_gap_pfp_gcd_transport_source_witness_bezout_sum_left_bounded_value_bound + S (fom_value_pfp_gcd_transport_source_witness_bezout_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_transport_source_witness_bezout_sum_right_bounded. (exists fom_gap_pfp_gcd_transport_source_witness_bezout_sum_right_bounded_index_bound. fom_gap_pfp_gcd_transport_source_witness_bezout_sum_right_bounded_index_bound + S (fom_index_pfp_gcd_transport_source_witness_bezout_sum_right_bounded) = pfgb_Q_gcd_transport_source_witness_bezout) -> exists fom_value_pfp_gcd_transport_source_witness_bezout_sum_right_bounded. ((((exists fom_beta_height_pfp_gcd_transport_source_witness_bezout_sum_right_bounded_entry. fom_beta_height_pfp_gcd_transport_source_witness_bezout_sum_right_bounded_entry + S (fom_value_pfp_gcd_transport_source_witness_bezout_sum_right_bounded) = S ((S (fom_index_pfp_gcd_transport_source_witness_bezout_sum_right_bounded)) * pfgb_qc_gcd_transport_source_witness_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_transport_source_witness_bezout_sum_right_bounded_entry. pfgb_qb_gcd_transport_source_witness_bezout = fom_beta_quotient_pfp_gcd_transport_source_witness_bezout_sum_right_bounded_entry * S ((S (fom_index_pfp_gcd_transport_source_witness_bezout_sum_right_bounded)) * pfgb_qc_gcd_transport_source_witness_bezout) + (fom_value_pfp_gcd_transport_source_witness_bezout_sum_right_bounded))) /\\ (exists fom_gap_pfp_gcd_transport_source_witness_bezout_sum_right_bounded_value_bound. fom_gap_pfp_gcd_transport_source_witness_bezout_sum_right_bounded_value_bound + S (fom_value_pfp_gcd_transport_source_witness_bezout_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_transport_source_witness_bezout_sum_result_bounded. (exists fom_gap_pfp_gcd_transport_source_witness_bezout_sum_result_bounded_index_bound. fom_gap_pfp_gcd_transport_source_witness_bezout_sum_result_bounded_index_bound + S (fom_index_pfp_gcd_transport_source_witness_bezout_sum_result_bounded) = pfgs_G_gcd_transport_source) -> exists fom_value_pfp_gcd_transport_source_witness_bezout_sum_result_bounded. ((((exists fom_beta_height_pfp_gcd_transport_source_witness_bezout_sum_result_bounded_entry. fom_beta_height_pfp_gcd_transport_source_witness_bezout_sum_result_bounded_entry + S (fom_value_pfp_gcd_transport_source_witness_bezout_sum_result_bounded) = S ((S (fom_index_pfp_gcd_transport_source_witness_bezout_sum_result_bounded)) * pfgs_gc_gcd_transport_source)) /\\ exists fom_beta_quotient_pfp_gcd_transport_source_witness_bezout_sum_result_bounded_entry. pfgs_gb_gcd_transport_source = fom_beta_quotient_pfp_gcd_transport_source_witness_bezout_sum_result_bounded_entry * S ((S (fom_index_pfp_gcd_transport_source_witness_bezout_sum_result_bounded)) * pfgs_gc_gcd_transport_source) + (fom_value_pfp_gcd_transport_source_witness_bezout_sum_result_bounded))) /\\ (exists fom_gap_pfp_gcd_transport_source_witness_bezout_sum_result_bounded_value_bound. fom_gap_pfp_gcd_transport_source_witness_bezout_sum_result_bounded_value_bound + S (fom_value_pfp_gcd_transport_source_witness_bezout_sum_result_bounded) = p))) /\\ ((exists pfga_ub_gcd_transport_source_witness_bezout_sum pfga_uc_gcd_transport_source_witness_bezout_sum pfga_vb_gcd_transport_source_witness_bezout_sum pfga_vc_gcd_transport_source_witness_bezout_sum pfga_tb_gcd_transport_source_witness_bezout_sum pfga_tc_gcd_transport_source_witness_bezout_sum pfga_K_gcd_transport_source_witness_bezout_sum. ((((forall pfrep_power_gcd_transport_source_witness_bezout_sum_left pfrep_left_gcd_transport_source_witness_bezout_sum_left pfrep_right_gcd_transport_source_witness_bezout_sum_left. ((exists pfrep_position_gcd_transport_source_witness_bezout_sum_leftfirst. ((pfrep_position_gcd_transport_source_witness_bezout_sum_leftfirst+S (pfrep_power_gcd_transport_source_witness_bezout_sum_left)=(pfgb_P_gcd_transport_source_witness_bezout)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_bezout_sum_leftfirstentry. ff_h_pfp_gcd_transport_source_witness_bezout_sum_leftfirstentry + S (pfrep_left_gcd_transport_source_witness_bezout_sum_left) = S ((S (pfrep_position_gcd_transport_source_witness_bezout_sum_leftfirst)) * pfgb_pc_gcd_transport_source_witness_bezout)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_sum_leftfirstentry. pfgb_pb_gcd_transport_source_witness_bezout = ff_q_pfp_gcd_transport_source_witness_bezout_sum_leftfirstentry * S ((S (pfrep_position_gcd_transport_source_witness_bezout_sum_leftfirst)) * pfgb_pc_gcd_transport_source_witness_bezout) + (pfrep_left_gcd_transport_source_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_transport_source_witness_bezout_sum_leftfirstoutside. pfrep_gap_gcd_transport_source_witness_bezout_sum_leftfirstoutside+(pfgb_P_gcd_transport_source_witness_bezout)=(pfrep_power_gcd_transport_source_witness_bezout_sum_left)) /\\ (((pfrep_left_gcd_transport_source_witness_bezout_sum_left)=0))))) -> ((exists pfrep_position_gcd_transport_source_witness_bezout_sum_leftsecond. ((pfrep_position_gcd_transport_source_witness_bezout_sum_leftsecond+S (pfrep_power_gcd_transport_source_witness_bezout_sum_left)=(pfga_K_gcd_transport_source_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_bezout_sum_leftsecondentry. ff_h_pfp_gcd_transport_source_witness_bezout_sum_leftsecondentry + S (pfrep_right_gcd_transport_source_witness_bezout_sum_left) = S ((S (pfrep_position_gcd_transport_source_witness_bezout_sum_leftsecond)) * pfga_uc_gcd_transport_source_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_sum_leftsecondentry. pfga_ub_gcd_transport_source_witness_bezout_sum = ff_q_pfp_gcd_transport_source_witness_bezout_sum_leftsecondentry * S ((S (pfrep_position_gcd_transport_source_witness_bezout_sum_leftsecond)) * pfga_uc_gcd_transport_source_witness_bezout_sum) + (pfrep_right_gcd_transport_source_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_transport_source_witness_bezout_sum_leftsecondoutside. pfrep_gap_gcd_transport_source_witness_bezout_sum_leftsecondoutside+(pfga_K_gcd_transport_source_witness_bezout_sum)=(pfrep_power_gcd_transport_source_witness_bezout_sum_left)) /\\ (((pfrep_right_gcd_transport_source_witness_bezout_sum_left)=0))))) -> pfrep_left_gcd_transport_source_witness_bezout_sum_left=pfrep_right_gcd_transport_source_witness_bezout_sum_left) /\\ ((forall pfrep_power_gcd_transport_source_witness_bezout_sum_right pfrep_left_gcd_transport_source_witness_bezout_sum_right pfrep_right_gcd_transport_source_witness_bezout_sum_right. ((exists pfrep_position_gcd_transport_source_witness_bezout_sum_rightfirst. ((pfrep_position_gcd_transport_source_witness_bezout_sum_rightfirst+S (pfrep_power_gcd_transport_source_witness_bezout_sum_right)=(pfgb_Q_gcd_transport_source_witness_bezout)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_bezout_sum_rightfirstentry. ff_h_pfp_gcd_transport_source_witness_bezout_sum_rightfirstentry + S (pfrep_left_gcd_transport_source_witness_bezout_sum_right) = S ((S (pfrep_position_gcd_transport_source_witness_bezout_sum_rightfirst)) * pfgb_qc_gcd_transport_source_witness_bezout)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_sum_rightfirstentry. pfgb_qb_gcd_transport_source_witness_bezout = ff_q_pfp_gcd_transport_source_witness_bezout_sum_rightfirstentry * S ((S (pfrep_position_gcd_transport_source_witness_bezout_sum_rightfirst)) * pfgb_qc_gcd_transport_source_witness_bezout) + (pfrep_left_gcd_transport_source_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_transport_source_witness_bezout_sum_rightfirstoutside. pfrep_gap_gcd_transport_source_witness_bezout_sum_rightfirstoutside+(pfgb_Q_gcd_transport_source_witness_bezout)=(pfrep_power_gcd_transport_source_witness_bezout_sum_right)) /\\ (((pfrep_left_gcd_transport_source_witness_bezout_sum_right)=0))))) -> ((exists pfrep_position_gcd_transport_source_witness_bezout_sum_rightsecond. ((pfrep_position_gcd_transport_source_witness_bezout_sum_rightsecond+S (pfrep_power_gcd_transport_source_witness_bezout_sum_right)=(pfga_K_gcd_transport_source_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_bezout_sum_rightsecondentry. ff_h_pfp_gcd_transport_source_witness_bezout_sum_rightsecondentry + S (pfrep_right_gcd_transport_source_witness_bezout_sum_right) = S ((S (pfrep_position_gcd_transport_source_witness_bezout_sum_rightsecond)) * pfga_vc_gcd_transport_source_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_sum_rightsecondentry. pfga_vb_gcd_transport_source_witness_bezout_sum = ff_q_pfp_gcd_transport_source_witness_bezout_sum_rightsecondentry * S ((S (pfrep_position_gcd_transport_source_witness_bezout_sum_rightsecond)) * pfga_vc_gcd_transport_source_witness_bezout_sum) + (pfrep_right_gcd_transport_source_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_transport_source_witness_bezout_sum_rightsecondoutside. pfrep_gap_gcd_transport_source_witness_bezout_sum_rightsecondoutside+(pfga_K_gcd_transport_source_witness_bezout_sum)=(pfrep_power_gcd_transport_source_witness_bezout_sum_right)) /\\ (((pfrep_right_gcd_transport_source_witness_bezout_sum_right)=0))))) -> pfrep_left_gcd_transport_source_witness_bezout_sum_right=pfrep_right_gcd_transport_source_witness_bezout_sum_right)))) /\\ (((forall pfp_index_gcd_transport_source_witness_bezout_sum_add. (exists pfa_gap_gcd_transport_source_witness_bezout_sum_addindex. pfa_gap_gcd_transport_source_witness_bezout_sum_addindex + S (pfp_index_gcd_transport_source_witness_bezout_sum_add) = (pfga_K_gcd_transport_source_witness_bezout_sum)) -> exists pfp_left_gcd_transport_source_witness_bezout_sum_add pfp_right_gcd_transport_source_witness_bezout_sum_add pfp_value_gcd_transport_source_witness_bezout_sum_add. ((((exists ff_h_pfp_gcd_transport_source_witness_bezout_sum_addleft. ff_h_pfp_gcd_transport_source_witness_bezout_sum_addleft + S (pfp_left_gcd_transport_source_witness_bezout_sum_add) = S ((S (pfp_index_gcd_transport_source_witness_bezout_sum_add)) * pfga_uc_gcd_transport_source_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_sum_addleft. pfga_ub_gcd_transport_source_witness_bezout_sum = ff_q_pfp_gcd_transport_source_witness_bezout_sum_addleft * S ((S (pfp_index_gcd_transport_source_witness_bezout_sum_add)) * pfga_uc_gcd_transport_source_witness_bezout_sum) + (pfp_left_gcd_transport_source_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_transport_source_witness_bezout_sum_addright. ff_h_pfp_gcd_transport_source_witness_bezout_sum_addright + S (pfp_right_gcd_transport_source_witness_bezout_sum_add) = S ((S (pfp_index_gcd_transport_source_witness_bezout_sum_add)) * pfga_vc_gcd_transport_source_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_sum_addright. pfga_vb_gcd_transport_source_witness_bezout_sum = ff_q_pfp_gcd_transport_source_witness_bezout_sum_addright * S ((S (pfp_index_gcd_transport_source_witness_bezout_sum_add)) * pfga_vc_gcd_transport_source_witness_bezout_sum) + (pfp_right_gcd_transport_source_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_transport_source_witness_bezout_sum_addtarget. ff_h_pfp_gcd_transport_source_witness_bezout_sum_addtarget + S (pfp_value_gcd_transport_source_witness_bezout_sum_add) = S ((S (pfp_index_gcd_transport_source_witness_bezout_sum_add)) * pfga_tc_gcd_transport_source_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_sum_addtarget. pfga_tb_gcd_transport_source_witness_bezout_sum = ff_q_pfp_gcd_transport_source_witness_bezout_sum_addtarget * S ((S (pfp_index_gcd_transport_source_witness_bezout_sum_add)) * pfga_tc_gcd_transport_source_witness_bezout_sum) + (pfp_value_gcd_transport_source_witness_bezout_sum_add))) /\\ ((((exists pfa_gap_gcd_transport_source_witness_bezout_sum_addoperationleft. pfa_gap_gcd_transport_source_witness_bezout_sum_addoperationleft + S (pfp_left_gcd_transport_source_witness_bezout_sum_add) = (p)) /\\ (((exists pfa_gap_gcd_transport_source_witness_bezout_sum_addoperationright. pfa_gap_gcd_transport_source_witness_bezout_sum_addoperationright + S (pfp_right_gcd_transport_source_witness_bezout_sum_add) = (p)) /\\ ((((exists pfa_gap_gcd_transport_source_witness_bezout_sum_addoperationresultbound. pfa_gap_gcd_transport_source_witness_bezout_sum_addoperationresultbound + S (pfp_value_gcd_transport_source_witness_bezout_sum_add) = (p)) /\\ ((exists pfa_offset_left_gcd_transport_source_witness_bezout_sum_addoperationresultcongruence pfa_offset_right_gcd_transport_source_witness_bezout_sum_addoperationresultcongruence. ((pfp_left_gcd_transport_source_witness_bezout_sum_add) + (pfp_right_gcd_transport_source_witness_bezout_sum_add)) + (p) * pfa_offset_left_gcd_transport_source_witness_bezout_sum_addoperationresultcongruence = (pfp_value_gcd_transport_source_witness_bezout_sum_add) + (p) * pfa_offset_right_gcd_transport_source_witness_bezout_sum_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_gcd_transport_source_witness_bezout_sum_result pfrep_left_gcd_transport_source_witness_bezout_sum_result pfrep_right_gcd_transport_source_witness_bezout_sum_result. ((exists pfrep_position_gcd_transport_source_witness_bezout_sum_resultfirst. ((pfrep_position_gcd_transport_source_witness_bezout_sum_resultfirst+S (pfrep_power_gcd_transport_source_witness_bezout_sum_result)=(pfga_K_gcd_transport_source_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_bezout_sum_resultfirstentry. ff_h_pfp_gcd_transport_source_witness_bezout_sum_resultfirstentry + S (pfrep_left_gcd_transport_source_witness_bezout_sum_result) = S ((S (pfrep_position_gcd_transport_source_witness_bezout_sum_resultfirst)) * pfga_tc_gcd_transport_source_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_sum_resultfirstentry. pfga_tb_gcd_transport_source_witness_bezout_sum = ff_q_pfp_gcd_transport_source_witness_bezout_sum_resultfirstentry * S ((S (pfrep_position_gcd_transport_source_witness_bezout_sum_resultfirst)) * pfga_tc_gcd_transport_source_witness_bezout_sum) + (pfrep_left_gcd_transport_source_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_transport_source_witness_bezout_sum_resultfirstoutside. pfrep_gap_gcd_transport_source_witness_bezout_sum_resultfirstoutside+(pfga_K_gcd_transport_source_witness_bezout_sum)=(pfrep_power_gcd_transport_source_witness_bezout_sum_result)) /\\ (((pfrep_left_gcd_transport_source_witness_bezout_sum_result)=0))))) -> ((exists pfrep_position_gcd_transport_source_witness_bezout_sum_resultsecond. ((pfrep_position_gcd_transport_source_witness_bezout_sum_resultsecond+S (pfrep_power_gcd_transport_source_witness_bezout_sum_result)=(pfgs_G_gcd_transport_source)) /\\ ((((exists ff_h_pfp_gcd_transport_source_witness_bezout_sum_resultsecondentry. ff_h_pfp_gcd_transport_source_witness_bezout_sum_resultsecondentry + S (pfrep_right_gcd_transport_source_witness_bezout_sum_result) = S ((S (pfrep_position_gcd_transport_source_witness_bezout_sum_resultsecond)) * pfgs_gc_gcd_transport_source)) /\\ exists ff_q_pfp_gcd_transport_source_witness_bezout_sum_resultsecondentry. pfgs_gb_gcd_transport_source = ff_q_pfp_gcd_transport_source_witness_bezout_sum_resultsecondentry * S ((S (pfrep_position_gcd_transport_source_witness_bezout_sum_resultsecond)) * pfgs_gc_gcd_transport_source) + (pfrep_right_gcd_transport_source_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_transport_source_witness_bezout_sum_resultsecondoutside. pfrep_gap_gcd_transport_source_witness_bezout_sum_resultsecondoutside+(pfgs_G_gcd_transport_source)=(pfrep_power_gcd_transport_source_witness_bezout_sum_result)) /\\ (((pfrep_right_gcd_transport_source_witness_bezout_sum_result)=0))))) -> pfrep_left_gcd_transport_source_witness_bezout_sum_result=pfrep_right_gcd_transport_source_witness_bezout_sum_result))))))))))))))))))))))) -> (exists pfgs_gb_gcd_transport_result pfgs_gc_gcd_transport_result pfgs_G_gcd_transport_result pfgs_ub_gcd_transport_result pfgs_uc_gcd_transport_result pfgs_U_gcd_transport_result pfgs_vb_gcd_transport_result pfgs_vc_gcd_transport_result pfgs_V_gcd_transport_result. (((pfgs_G_gcd_transport_result)=0 \\/ (((~((pfgs_G_gcd_transport_result) = 0)) /\\ (((forall fom_index_pfp_gcd_transport_result_witness_normal_moniccoefficients. (exists fom_gap_pfp_gcd_transport_result_witness_normal_moniccoefficients_index_bound. fom_gap_pfp_gcd_transport_result_witness_normal_moniccoefficients_index_bound + S (fom_index_pfp_gcd_transport_result_witness_normal_moniccoefficients) = pfgs_G_gcd_transport_result) -> exists fom_value_pfp_gcd_transport_result_witness_normal_moniccoefficients. ((((exists fom_beta_height_pfp_gcd_transport_result_witness_normal_moniccoefficients_entry. fom_beta_height_pfp_gcd_transport_result_witness_normal_moniccoefficients_entry + S (fom_value_pfp_gcd_transport_result_witness_normal_moniccoefficients) = S ((S (fom_index_pfp_gcd_transport_result_witness_normal_moniccoefficients)) * pfgs_gc_gcd_transport_result)) /\\ exists fom_beta_quotient_pfp_gcd_transport_result_witness_normal_moniccoefficients_entry. pfgs_gb_gcd_transport_result = fom_beta_quotient_pfp_gcd_transport_result_witness_normal_moniccoefficients_entry * S ((S (fom_index_pfp_gcd_transport_result_witness_normal_moniccoefficients)) * pfgs_gc_gcd_transport_result) + (fom_value_pfp_gcd_transport_result_witness_normal_moniccoefficients))) /\\ (exists fom_gap_pfp_gcd_transport_result_witness_normal_moniccoefficients_value_bound. fom_gap_pfp_gcd_transport_result_witness_normal_moniccoefficients_value_bound + S (fom_value_pfp_gcd_transport_result_witness_normal_moniccoefficients) = p))) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_normal_monicleading. ff_h_pfp_gcd_transport_result_witness_normal_monicleading + S (1) = S ((S (0)) * pfgs_gc_gcd_transport_result)) /\\ exists ff_q_pfp_gcd_transport_result_witness_normal_monicleading. pfgs_gb_gcd_transport_result = ff_q_pfp_gcd_transport_result_witness_normal_monicleading * S ((S (0)) * pfgs_gc_gcd_transport_result) + (1))))))))) /\\ (((((((forall fom_index_pfp_gcd_transport_result_witness_common_left_bounded. (exists fom_gap_pfp_gcd_transport_result_witness_common_left_bounded_index_bound. fom_gap_pfp_gcd_transport_result_witness_common_left_bounded_index_bound + S (fom_index_pfp_gcd_transport_result_witness_common_left_bounded) = L) -> exists fom_value_pfp_gcd_transport_result_witness_common_left_bounded. ((((exists fom_beta_height_pfp_gcd_transport_result_witness_common_left_bounded_entry. fom_beta_height_pfp_gcd_transport_result_witness_common_left_bounded_entry + S (fom_value_pfp_gcd_transport_result_witness_common_left_bounded) = S ((S (fom_index_pfp_gcd_transport_result_witness_common_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_transport_result_witness_common_left_bounded_entry. ab = fom_beta_quotient_pfp_gcd_transport_result_witness_common_left_bounded_entry * S ((S (fom_index_pfp_gcd_transport_result_witness_common_left_bounded)) * ac) + (fom_value_pfp_gcd_transport_result_witness_common_left_bounded))) /\\ (exists fom_gap_pfp_gcd_transport_result_witness_common_left_bounded_value_bound. fom_gap_pfp_gcd_transport_result_witness_common_left_bounded_value_bound + S (fom_value_pfp_gcd_transport_result_witness_common_left_bounded) = p))) /\\ ((exists pfgd_qb_gcd_transport_result_witness_common_left pfgd_qc_gcd_transport_result_witness_common_left pfgd_Q_gcd_transport_result_witness_common_left pfgd_pb_gcd_transport_result_witness_common_left pfgd_pc_gcd_transport_result_witness_common_left pfgd_P_gcd_transport_result_witness_common_left. ((((forall fom_index_pfp_gcd_transport_result_witness_common_left_productleft. (exists fom_gap_pfp_gcd_transport_result_witness_common_left_productleft_index_bound. fom_gap_pfp_gcd_transport_result_witness_common_left_productleft_index_bound + S (fom_index_pfp_gcd_transport_result_witness_common_left_productleft) = pfgd_Q_gcd_transport_result_witness_common_left) -> exists fom_value_pfp_gcd_transport_result_witness_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_transport_result_witness_common_left_productleft_entry. fom_beta_height_pfp_gcd_transport_result_witness_common_left_productleft_entry + S (fom_value_pfp_gcd_transport_result_witness_common_left_productleft) = S ((S (fom_index_pfp_gcd_transport_result_witness_common_left_productleft)) * pfgd_qc_gcd_transport_result_witness_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_transport_result_witness_common_left_productleft_entry. pfgd_qb_gcd_transport_result_witness_common_left = fom_beta_quotient_pfp_gcd_transport_result_witness_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_transport_result_witness_common_left_productleft)) * pfgd_qc_gcd_transport_result_witness_common_left) + (fom_value_pfp_gcd_transport_result_witness_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_transport_result_witness_common_left_productleft_value_bound. fom_gap_pfp_gcd_transport_result_witness_common_left_productleft_value_bound + S (fom_value_pfp_gcd_transport_result_witness_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_transport_result_witness_common_left_productright. (exists fom_gap_pfp_gcd_transport_result_witness_common_left_productright_index_bound. fom_gap_pfp_gcd_transport_result_witness_common_left_productright_index_bound + S (fom_index_pfp_gcd_transport_result_witness_common_left_productright) = pfgs_G_gcd_transport_result) -> exists fom_value_pfp_gcd_transport_result_witness_common_left_productright. ((((exists fom_beta_height_pfp_gcd_transport_result_witness_common_left_productright_entry. fom_beta_height_pfp_gcd_transport_result_witness_common_left_productright_entry + S (fom_value_pfp_gcd_transport_result_witness_common_left_productright) = S ((S (fom_index_pfp_gcd_transport_result_witness_common_left_productright)) * pfgs_gc_gcd_transport_result)) /\\ exists fom_beta_quotient_pfp_gcd_transport_result_witness_common_left_productright_entry. pfgs_gb_gcd_transport_result = fom_beta_quotient_pfp_gcd_transport_result_witness_common_left_productright_entry * S ((S (fom_index_pfp_gcd_transport_result_witness_common_left_productright)) * pfgs_gc_gcd_transport_result) + (fom_value_pfp_gcd_transport_result_witness_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_transport_result_witness_common_left_productright_value_bound. fom_gap_pfp_gcd_transport_result_witness_common_left_productright_value_bound + S (fom_value_pfp_gcd_transport_result_witness_common_left_productright) = p))) /\\ (((((((pfgd_Q_gcd_transport_result_witness_common_left)=0 \\/ (pfgs_G_gcd_transport_result)=0) /\\ (((pfgd_P_gcd_transport_result_witness_common_left)=0)))) \\/ (((~((pfgd_Q_gcd_transport_result_witness_common_left)=0)) /\\ (((~((pfgs_G_gcd_transport_result)=0)) /\\ (((pfgd_Q_gcd_transport_result_witness_common_left)+(pfgs_G_gcd_transport_result)=S (pfgd_P_gcd_transport_result_witness_common_left)))))))) /\\ ((forall pfc_index_gcd_transport_result_witness_common_left_productcoefficients. (exists pfa_gap_gcd_transport_result_witness_common_left_productcoefficientsbound. pfa_gap_gcd_transport_result_witness_common_left_productcoefficientsbound + S (pfc_index_gcd_transport_result_witness_common_left_productcoefficients) = (pfgd_P_gcd_transport_result_witness_common_left)) -> exists pfc_value_gcd_transport_result_witness_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_transport_result_witness_common_left_productcoefficientsentry. ff_h_pfp_gcd_transport_result_witness_common_left_productcoefficientsentry + S (pfc_value_gcd_transport_result_witness_common_left_productcoefficients) = S ((S (pfc_index_gcd_transport_result_witness_common_left_productcoefficients)) * pfgd_pc_gcd_transport_result_witness_common_left)) /\\ exists ff_q_pfp_gcd_transport_result_witness_common_left_productcoefficientsentry. pfgd_pb_gcd_transport_result_witness_common_left = ff_q_pfp_gcd_transport_result_witness_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_transport_result_witness_common_left_productcoefficients)) * pfgd_pc_gcd_transport_result_witness_common_left) + (pfc_value_gcd_transport_result_witness_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_transport_result_witness_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_transport_result_witness_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_transport_result_witness_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_transport_result_witness_common_left_productcoefficients))) -> exists pfc_value_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_result_witness_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_transport_result_witness_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_result_witness_common_left_productcoefficientscoefficient) + (pfc_value_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_transport_result_witness_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_transport_result_witness_common_left)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_transport_result_witness_common_left)) /\\ exists ff_q_pfp_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_transport_result_witness_common_left = ff_q_pfp_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_transport_result_witness_common_left) + (pfc_left_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_transport_result_witness_common_left)=(pfc_index_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm) = (pfgs_G_gcd_transport_result)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_transport_result)) /\\ exists ff_q_pfp_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_gcd_transport_result = ff_q_pfp_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_transport_result) + (pfc_right_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_gcd_transport_result)=(pfc_complement_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_transport_result_witness_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_transport_result_witness_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_transport_result_witness_common_left_productcoefficients))) * fs_v_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_transport_result_witness_common_left_productcoefficients))) * fs_v_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_transport_result_witness_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_transport_result_witness_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_result_witness_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_transport_result_witness_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_result_witness_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_transport_result_witness_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_transport_result_witness_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_transport_result_witness_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_transport_result_witness_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_transport_result_witness_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_transport_result_witness_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_transport_result_witness_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_transport_result_witness_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_transport_result_witness_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_transport_result_witness_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_transport_result_witness_common_left_equivalent pfrep_left_gcd_transport_result_witness_common_left_equivalent pfrep_right_gcd_transport_result_witness_common_left_equivalent. ((exists pfrep_position_gcd_transport_result_witness_common_left_equivalentfirst. ((pfrep_position_gcd_transport_result_witness_common_left_equivalentfirst+S (pfrep_power_gcd_transport_result_witness_common_left_equivalent)=(pfgd_P_gcd_transport_result_witness_common_left)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_common_left_equivalentfirstentry. ff_h_pfp_gcd_transport_result_witness_common_left_equivalentfirstentry + S (pfrep_left_gcd_transport_result_witness_common_left_equivalent) = S ((S (pfrep_position_gcd_transport_result_witness_common_left_equivalentfirst)) * pfgd_pc_gcd_transport_result_witness_common_left)) /\\ exists ff_q_pfp_gcd_transport_result_witness_common_left_equivalentfirstentry. pfgd_pb_gcd_transport_result_witness_common_left = ff_q_pfp_gcd_transport_result_witness_common_left_equivalentfirstentry * S ((S (pfrep_position_gcd_transport_result_witness_common_left_equivalentfirst)) * pfgd_pc_gcd_transport_result_witness_common_left) + (pfrep_left_gcd_transport_result_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_transport_result_witness_common_left_equivalentfirstoutside. pfrep_gap_gcd_transport_result_witness_common_left_equivalentfirstoutside+(pfgd_P_gcd_transport_result_witness_common_left)=(pfrep_power_gcd_transport_result_witness_common_left_equivalent)) /\\ (((pfrep_left_gcd_transport_result_witness_common_left_equivalent)=0))))) -> ((exists pfrep_position_gcd_transport_result_witness_common_left_equivalentsecond. ((pfrep_position_gcd_transport_result_witness_common_left_equivalentsecond+S (pfrep_power_gcd_transport_result_witness_common_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_common_left_equivalentsecondentry. ff_h_pfp_gcd_transport_result_witness_common_left_equivalentsecondentry + S (pfrep_right_gcd_transport_result_witness_common_left_equivalent) = S ((S (pfrep_position_gcd_transport_result_witness_common_left_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_gcd_transport_result_witness_common_left_equivalentsecondentry. ab = ff_q_pfp_gcd_transport_result_witness_common_left_equivalentsecondentry * S ((S (pfrep_position_gcd_transport_result_witness_common_left_equivalentsecond)) * ac) + (pfrep_right_gcd_transport_result_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_transport_result_witness_common_left_equivalentsecondoutside. pfrep_gap_gcd_transport_result_witness_common_left_equivalentsecondoutside+(L)=(pfrep_power_gcd_transport_result_witness_common_left_equivalent)) /\\ (((pfrep_right_gcd_transport_result_witness_common_left_equivalent)=0))))) -> pfrep_left_gcd_transport_result_witness_common_left_equivalent=pfrep_right_gcd_transport_result_witness_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_transport_result_witness_common_right_bounded. (exists fom_gap_pfp_gcd_transport_result_witness_common_right_bounded_index_bound. fom_gap_pfp_gcd_transport_result_witness_common_right_bounded_index_bound + S (fom_index_pfp_gcd_transport_result_witness_common_right_bounded) = M2) -> exists fom_value_pfp_gcd_transport_result_witness_common_right_bounded. ((((exists fom_beta_height_pfp_gcd_transport_result_witness_common_right_bounded_entry. fom_beta_height_pfp_gcd_transport_result_witness_common_right_bounded_entry + S (fom_value_pfp_gcd_transport_result_witness_common_right_bounded) = S ((S (fom_index_pfp_gcd_transport_result_witness_common_right_bounded)) * bc2)) /\\ exists fom_beta_quotient_pfp_gcd_transport_result_witness_common_right_bounded_entry. bb2 = fom_beta_quotient_pfp_gcd_transport_result_witness_common_right_bounded_entry * S ((S (fom_index_pfp_gcd_transport_result_witness_common_right_bounded)) * bc2) + (fom_value_pfp_gcd_transport_result_witness_common_right_bounded))) /\\ (exists fom_gap_pfp_gcd_transport_result_witness_common_right_bounded_value_bound. fom_gap_pfp_gcd_transport_result_witness_common_right_bounded_value_bound + S (fom_value_pfp_gcd_transport_result_witness_common_right_bounded) = p))) /\\ ((exists pfgd_qb_gcd_transport_result_witness_common_right pfgd_qc_gcd_transport_result_witness_common_right pfgd_Q_gcd_transport_result_witness_common_right pfgd_pb_gcd_transport_result_witness_common_right pfgd_pc_gcd_transport_result_witness_common_right pfgd_P_gcd_transport_result_witness_common_right. ((((forall fom_index_pfp_gcd_transport_result_witness_common_right_productleft. (exists fom_gap_pfp_gcd_transport_result_witness_common_right_productleft_index_bound. fom_gap_pfp_gcd_transport_result_witness_common_right_productleft_index_bound + S (fom_index_pfp_gcd_transport_result_witness_common_right_productleft) = pfgd_Q_gcd_transport_result_witness_common_right) -> exists fom_value_pfp_gcd_transport_result_witness_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_transport_result_witness_common_right_productleft_entry. fom_beta_height_pfp_gcd_transport_result_witness_common_right_productleft_entry + S (fom_value_pfp_gcd_transport_result_witness_common_right_productleft) = S ((S (fom_index_pfp_gcd_transport_result_witness_common_right_productleft)) * pfgd_qc_gcd_transport_result_witness_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_transport_result_witness_common_right_productleft_entry. pfgd_qb_gcd_transport_result_witness_common_right = fom_beta_quotient_pfp_gcd_transport_result_witness_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_transport_result_witness_common_right_productleft)) * pfgd_qc_gcd_transport_result_witness_common_right) + (fom_value_pfp_gcd_transport_result_witness_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_transport_result_witness_common_right_productleft_value_bound. fom_gap_pfp_gcd_transport_result_witness_common_right_productleft_value_bound + S (fom_value_pfp_gcd_transport_result_witness_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_transport_result_witness_common_right_productright. (exists fom_gap_pfp_gcd_transport_result_witness_common_right_productright_index_bound. fom_gap_pfp_gcd_transport_result_witness_common_right_productright_index_bound + S (fom_index_pfp_gcd_transport_result_witness_common_right_productright) = pfgs_G_gcd_transport_result) -> exists fom_value_pfp_gcd_transport_result_witness_common_right_productright. ((((exists fom_beta_height_pfp_gcd_transport_result_witness_common_right_productright_entry. fom_beta_height_pfp_gcd_transport_result_witness_common_right_productright_entry + S (fom_value_pfp_gcd_transport_result_witness_common_right_productright) = S ((S (fom_index_pfp_gcd_transport_result_witness_common_right_productright)) * pfgs_gc_gcd_transport_result)) /\\ exists fom_beta_quotient_pfp_gcd_transport_result_witness_common_right_productright_entry. pfgs_gb_gcd_transport_result = fom_beta_quotient_pfp_gcd_transport_result_witness_common_right_productright_entry * S ((S (fom_index_pfp_gcd_transport_result_witness_common_right_productright)) * pfgs_gc_gcd_transport_result) + (fom_value_pfp_gcd_transport_result_witness_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_transport_result_witness_common_right_productright_value_bound. fom_gap_pfp_gcd_transport_result_witness_common_right_productright_value_bound + S (fom_value_pfp_gcd_transport_result_witness_common_right_productright) = p))) /\\ (((((((pfgd_Q_gcd_transport_result_witness_common_right)=0 \\/ (pfgs_G_gcd_transport_result)=0) /\\ (((pfgd_P_gcd_transport_result_witness_common_right)=0)))) \\/ (((~((pfgd_Q_gcd_transport_result_witness_common_right)=0)) /\\ (((~((pfgs_G_gcd_transport_result)=0)) /\\ (((pfgd_Q_gcd_transport_result_witness_common_right)+(pfgs_G_gcd_transport_result)=S (pfgd_P_gcd_transport_result_witness_common_right)))))))) /\\ ((forall pfc_index_gcd_transport_result_witness_common_right_productcoefficients. (exists pfa_gap_gcd_transport_result_witness_common_right_productcoefficientsbound. pfa_gap_gcd_transport_result_witness_common_right_productcoefficientsbound + S (pfc_index_gcd_transport_result_witness_common_right_productcoefficients) = (pfgd_P_gcd_transport_result_witness_common_right)) -> exists pfc_value_gcd_transport_result_witness_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_transport_result_witness_common_right_productcoefficientsentry. ff_h_pfp_gcd_transport_result_witness_common_right_productcoefficientsentry + S (pfc_value_gcd_transport_result_witness_common_right_productcoefficients) = S ((S (pfc_index_gcd_transport_result_witness_common_right_productcoefficients)) * pfgd_pc_gcd_transport_result_witness_common_right)) /\\ exists ff_q_pfp_gcd_transport_result_witness_common_right_productcoefficientsentry. pfgd_pb_gcd_transport_result_witness_common_right = ff_q_pfp_gcd_transport_result_witness_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_transport_result_witness_common_right_productcoefficients)) * pfgd_pc_gcd_transport_result_witness_common_right) + (pfc_value_gcd_transport_result_witness_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_transport_result_witness_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_transport_result_witness_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_transport_result_witness_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_transport_result_witness_common_right_productcoefficients))) -> exists pfc_value_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_result_witness_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_transport_result_witness_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_result_witness_common_right_productcoefficientscoefficient) + (pfc_value_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_transport_result_witness_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_transport_result_witness_common_right)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_transport_result_witness_common_right)) /\\ exists ff_q_pfp_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_transport_result_witness_common_right = ff_q_pfp_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_transport_result_witness_common_right) + (pfc_left_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_transport_result_witness_common_right)=(pfc_index_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm) = (pfgs_G_gcd_transport_result)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_transport_result)) /\\ exists ff_q_pfp_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_gcd_transport_result = ff_q_pfp_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_transport_result) + (pfc_right_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_gcd_transport_result)=(pfc_complement_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_transport_result_witness_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_transport_result_witness_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_transport_result_witness_common_right_productcoefficients))) * fs_v_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_transport_result_witness_common_right_productcoefficients))) * fs_v_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_transport_result_witness_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_transport_result_witness_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_result_witness_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_transport_result_witness_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_result_witness_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_transport_result_witness_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_transport_result_witness_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_transport_result_witness_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_transport_result_witness_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_transport_result_witness_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_transport_result_witness_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_transport_result_witness_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_transport_result_witness_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_transport_result_witness_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_transport_result_witness_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_transport_result_witness_common_right_equivalent pfrep_left_gcd_transport_result_witness_common_right_equivalent pfrep_right_gcd_transport_result_witness_common_right_equivalent. ((exists pfrep_position_gcd_transport_result_witness_common_right_equivalentfirst. ((pfrep_position_gcd_transport_result_witness_common_right_equivalentfirst+S (pfrep_power_gcd_transport_result_witness_common_right_equivalent)=(pfgd_P_gcd_transport_result_witness_common_right)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_common_right_equivalentfirstentry. ff_h_pfp_gcd_transport_result_witness_common_right_equivalentfirstentry + S (pfrep_left_gcd_transport_result_witness_common_right_equivalent) = S ((S (pfrep_position_gcd_transport_result_witness_common_right_equivalentfirst)) * pfgd_pc_gcd_transport_result_witness_common_right)) /\\ exists ff_q_pfp_gcd_transport_result_witness_common_right_equivalentfirstentry. pfgd_pb_gcd_transport_result_witness_common_right = ff_q_pfp_gcd_transport_result_witness_common_right_equivalentfirstentry * S ((S (pfrep_position_gcd_transport_result_witness_common_right_equivalentfirst)) * pfgd_pc_gcd_transport_result_witness_common_right) + (pfrep_left_gcd_transport_result_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_transport_result_witness_common_right_equivalentfirstoutside. pfrep_gap_gcd_transport_result_witness_common_right_equivalentfirstoutside+(pfgd_P_gcd_transport_result_witness_common_right)=(pfrep_power_gcd_transport_result_witness_common_right_equivalent)) /\\ (((pfrep_left_gcd_transport_result_witness_common_right_equivalent)=0))))) -> ((exists pfrep_position_gcd_transport_result_witness_common_right_equivalentsecond. ((pfrep_position_gcd_transport_result_witness_common_right_equivalentsecond+S (pfrep_power_gcd_transport_result_witness_common_right_equivalent)=(M2)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_common_right_equivalentsecondentry. ff_h_pfp_gcd_transport_result_witness_common_right_equivalentsecondentry + S (pfrep_right_gcd_transport_result_witness_common_right_equivalent) = S ((S (pfrep_position_gcd_transport_result_witness_common_right_equivalentsecond)) * bc2)) /\\ exists ff_q_pfp_gcd_transport_result_witness_common_right_equivalentsecondentry. bb2 = ff_q_pfp_gcd_transport_result_witness_common_right_equivalentsecondentry * S ((S (pfrep_position_gcd_transport_result_witness_common_right_equivalentsecond)) * bc2) + (pfrep_right_gcd_transport_result_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_transport_result_witness_common_right_equivalentsecondoutside. pfrep_gap_gcd_transport_result_witness_common_right_equivalentsecondoutside+(M2)=(pfrep_power_gcd_transport_result_witness_common_right_equivalent)) /\\ (((pfrep_right_gcd_transport_result_witness_common_right_equivalent)=0))))) -> pfrep_left_gcd_transport_result_witness_common_right_equivalent=pfrep_right_gcd_transport_result_witness_common_right_equivalent)))))))))) /\\ ((exists pfgb_pb_gcd_transport_result_witness_bezout pfgb_pc_gcd_transport_result_witness_bezout pfgb_P_gcd_transport_result_witness_bezout pfgb_qb_gcd_transport_result_witness_bezout pfgb_qc_gcd_transport_result_witness_bezout pfgb_Q_gcd_transport_result_witness_bezout. ((((forall fom_index_pfp_gcd_transport_result_witness_bezout_leftleft. (exists fom_gap_pfp_gcd_transport_result_witness_bezout_leftleft_index_bound. fom_gap_pfp_gcd_transport_result_witness_bezout_leftleft_index_bound + S (fom_index_pfp_gcd_transport_result_witness_bezout_leftleft) = pfgs_U_gcd_transport_result) -> exists fom_value_pfp_gcd_transport_result_witness_bezout_leftleft. ((((exists fom_beta_height_pfp_gcd_transport_result_witness_bezout_leftleft_entry. fom_beta_height_pfp_gcd_transport_result_witness_bezout_leftleft_entry + S (fom_value_pfp_gcd_transport_result_witness_bezout_leftleft) = S ((S (fom_index_pfp_gcd_transport_result_witness_bezout_leftleft)) * pfgs_uc_gcd_transport_result)) /\\ exists fom_beta_quotient_pfp_gcd_transport_result_witness_bezout_leftleft_entry. pfgs_ub_gcd_transport_result = fom_beta_quotient_pfp_gcd_transport_result_witness_bezout_leftleft_entry * S ((S (fom_index_pfp_gcd_transport_result_witness_bezout_leftleft)) * pfgs_uc_gcd_transport_result) + (fom_value_pfp_gcd_transport_result_witness_bezout_leftleft))) /\\ (exists fom_gap_pfp_gcd_transport_result_witness_bezout_leftleft_value_bound. fom_gap_pfp_gcd_transport_result_witness_bezout_leftleft_value_bound + S (fom_value_pfp_gcd_transport_result_witness_bezout_leftleft) = p))) /\\ (((forall fom_index_pfp_gcd_transport_result_witness_bezout_leftright. (exists fom_gap_pfp_gcd_transport_result_witness_bezout_leftright_index_bound. fom_gap_pfp_gcd_transport_result_witness_bezout_leftright_index_bound + S (fom_index_pfp_gcd_transport_result_witness_bezout_leftright) = L) -> exists fom_value_pfp_gcd_transport_result_witness_bezout_leftright. ((((exists fom_beta_height_pfp_gcd_transport_result_witness_bezout_leftright_entry. fom_beta_height_pfp_gcd_transport_result_witness_bezout_leftright_entry + S (fom_value_pfp_gcd_transport_result_witness_bezout_leftright) = S ((S (fom_index_pfp_gcd_transport_result_witness_bezout_leftright)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_transport_result_witness_bezout_leftright_entry. ab = fom_beta_quotient_pfp_gcd_transport_result_witness_bezout_leftright_entry * S ((S (fom_index_pfp_gcd_transport_result_witness_bezout_leftright)) * ac) + (fom_value_pfp_gcd_transport_result_witness_bezout_leftright))) /\\ (exists fom_gap_pfp_gcd_transport_result_witness_bezout_leftright_value_bound. fom_gap_pfp_gcd_transport_result_witness_bezout_leftright_value_bound + S (fom_value_pfp_gcd_transport_result_witness_bezout_leftright) = p))) /\\ (((((((pfgs_U_gcd_transport_result)=0 \\/ (L)=0) /\\ (((pfgb_P_gcd_transport_result_witness_bezout)=0)))) \\/ (((~((pfgs_U_gcd_transport_result)=0)) /\\ (((~((L)=0)) /\\ (((pfgs_U_gcd_transport_result)+(L)=S (pfgb_P_gcd_transport_result_witness_bezout)))))))) /\\ ((forall pfc_index_gcd_transport_result_witness_bezout_leftcoefficients. (exists pfa_gap_gcd_transport_result_witness_bezout_leftcoefficientsbound. pfa_gap_gcd_transport_result_witness_bezout_leftcoefficientsbound + S (pfc_index_gcd_transport_result_witness_bezout_leftcoefficients) = (pfgb_P_gcd_transport_result_witness_bezout)) -> exists pfc_value_gcd_transport_result_witness_bezout_leftcoefficients. ((((exists ff_h_pfp_gcd_transport_result_witness_bezout_leftcoefficientsentry. ff_h_pfp_gcd_transport_result_witness_bezout_leftcoefficientsentry + S (pfc_value_gcd_transport_result_witness_bezout_leftcoefficients) = S ((S (pfc_index_gcd_transport_result_witness_bezout_leftcoefficients)) * pfgb_pc_gcd_transport_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_leftcoefficientsentry. pfgb_pb_gcd_transport_result_witness_bezout = ff_q_pfp_gcd_transport_result_witness_bezout_leftcoefficientsentry * S ((S (pfc_index_gcd_transport_result_witness_bezout_leftcoefficients)) * pfgb_pc_gcd_transport_result_witness_bezout) + (pfc_value_gcd_transport_result_witness_bezout_leftcoefficients))) /\\ ((exists pfc_terms_code_gcd_transport_result_witness_bezout_leftcoefficientscoefficient pfc_terms_scale_gcd_transport_result_witness_bezout_leftcoefficientscoefficient pfc_natural_sum_gcd_transport_result_witness_bezout_leftcoefficientscoefficient. ((forall pfc_index_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalbound. pfa_gap_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_transport_result_witness_bezout_leftcoefficients))) -> exists pfc_value_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_result_witness_bezout_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_transport_result_witness_bezout_leftcoefficientscoefficient = ff_q_pfp_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_result_witness_bezout_leftcoefficientscoefficient) + (pfc_value_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_left_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_right_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonal)+pfc_complement_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm=(pfc_index_gcd_transport_result_witness_bezout_leftcoefficients)) /\\ ((((((exists pfa_gap_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonal) = (pfgs_U_gcd_transport_result)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_gcd_transport_result)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. pfgs_ub_gcd_transport_result = ff_q_pfp_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_gcd_transport_result) + (pfc_left_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside+(pfgs_U_gcd_transport_result)=(pfc_index_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonal)=pfc_left_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm*pfc_right_gcd_transport_result_witness_bezout_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum fs_v_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_transport_result_witness_bezout_leftcoefficientscoefficient) = S ((S (S (pfc_index_gcd_transport_result_witness_bezout_leftcoefficients))) * fs_v_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_transport_result_witness_bezout_leftcoefficients))) * fs_v_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum) + (pfc_natural_sum_gcd_transport_result_witness_bezout_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_transport_result_witness_bezout_leftcoefficients)) -> exists fs_a_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_result_witness_bezout_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_transport_result_witness_bezout_leftcoefficientscoefficient = fs_q_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_result_witness_bezout_leftcoefficientscoefficient) + (fs_a_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum) + (fs_r_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum) + (fs_s_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_transport_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_transport_result_witness_bezout_leftcoefficientscoefficientresiduebound. pfa_gap_gcd_transport_result_witness_bezout_leftcoefficientscoefficientresiduebound + S (pfc_value_gcd_transport_result_witness_bezout_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_transport_result_witness_bezout_leftcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_transport_result_witness_bezout_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_transport_result_witness_bezout_leftcoefficientscoefficient) + (p) * pfa_offset_left_gcd_transport_result_witness_bezout_leftcoefficientscoefficientresiduecongruence = (pfc_value_gcd_transport_result_witness_bezout_leftcoefficients) + (p) * pfa_offset_right_gcd_transport_result_witness_bezout_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_gcd_transport_result_witness_bezout_rightleft. (exists fom_gap_pfp_gcd_transport_result_witness_bezout_rightleft_index_bound. fom_gap_pfp_gcd_transport_result_witness_bezout_rightleft_index_bound + S (fom_index_pfp_gcd_transport_result_witness_bezout_rightleft) = pfgs_V_gcd_transport_result) -> exists fom_value_pfp_gcd_transport_result_witness_bezout_rightleft. ((((exists fom_beta_height_pfp_gcd_transport_result_witness_bezout_rightleft_entry. fom_beta_height_pfp_gcd_transport_result_witness_bezout_rightleft_entry + S (fom_value_pfp_gcd_transport_result_witness_bezout_rightleft) = S ((S (fom_index_pfp_gcd_transport_result_witness_bezout_rightleft)) * pfgs_vc_gcd_transport_result)) /\\ exists fom_beta_quotient_pfp_gcd_transport_result_witness_bezout_rightleft_entry. pfgs_vb_gcd_transport_result = fom_beta_quotient_pfp_gcd_transport_result_witness_bezout_rightleft_entry * S ((S (fom_index_pfp_gcd_transport_result_witness_bezout_rightleft)) * pfgs_vc_gcd_transport_result) + (fom_value_pfp_gcd_transport_result_witness_bezout_rightleft))) /\\ (exists fom_gap_pfp_gcd_transport_result_witness_bezout_rightleft_value_bound. fom_gap_pfp_gcd_transport_result_witness_bezout_rightleft_value_bound + S (fom_value_pfp_gcd_transport_result_witness_bezout_rightleft) = p))) /\\ (((forall fom_index_pfp_gcd_transport_result_witness_bezout_rightright. (exists fom_gap_pfp_gcd_transport_result_witness_bezout_rightright_index_bound. fom_gap_pfp_gcd_transport_result_witness_bezout_rightright_index_bound + S (fom_index_pfp_gcd_transport_result_witness_bezout_rightright) = M2) -> exists fom_value_pfp_gcd_transport_result_witness_bezout_rightright. ((((exists fom_beta_height_pfp_gcd_transport_result_witness_bezout_rightright_entry. fom_beta_height_pfp_gcd_transport_result_witness_bezout_rightright_entry + S (fom_value_pfp_gcd_transport_result_witness_bezout_rightright) = S ((S (fom_index_pfp_gcd_transport_result_witness_bezout_rightright)) * bc2)) /\\ exists fom_beta_quotient_pfp_gcd_transport_result_witness_bezout_rightright_entry. bb2 = fom_beta_quotient_pfp_gcd_transport_result_witness_bezout_rightright_entry * S ((S (fom_index_pfp_gcd_transport_result_witness_bezout_rightright)) * bc2) + (fom_value_pfp_gcd_transport_result_witness_bezout_rightright))) /\\ (exists fom_gap_pfp_gcd_transport_result_witness_bezout_rightright_value_bound. fom_gap_pfp_gcd_transport_result_witness_bezout_rightright_value_bound + S (fom_value_pfp_gcd_transport_result_witness_bezout_rightright) = p))) /\\ (((((((pfgs_V_gcd_transport_result)=0 \\/ (M2)=0) /\\ (((pfgb_Q_gcd_transport_result_witness_bezout)=0)))) \\/ (((~((pfgs_V_gcd_transport_result)=0)) /\\ (((~((M2)=0)) /\\ (((pfgs_V_gcd_transport_result)+(M2)=S (pfgb_Q_gcd_transport_result_witness_bezout)))))))) /\\ ((forall pfc_index_gcd_transport_result_witness_bezout_rightcoefficients. (exists pfa_gap_gcd_transport_result_witness_bezout_rightcoefficientsbound. pfa_gap_gcd_transport_result_witness_bezout_rightcoefficientsbound + S (pfc_index_gcd_transport_result_witness_bezout_rightcoefficients) = (pfgb_Q_gcd_transport_result_witness_bezout)) -> exists pfc_value_gcd_transport_result_witness_bezout_rightcoefficients. ((((exists ff_h_pfp_gcd_transport_result_witness_bezout_rightcoefficientsentry. ff_h_pfp_gcd_transport_result_witness_bezout_rightcoefficientsentry + S (pfc_value_gcd_transport_result_witness_bezout_rightcoefficients) = S ((S (pfc_index_gcd_transport_result_witness_bezout_rightcoefficients)) * pfgb_qc_gcd_transport_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_rightcoefficientsentry. pfgb_qb_gcd_transport_result_witness_bezout = ff_q_pfp_gcd_transport_result_witness_bezout_rightcoefficientsentry * S ((S (pfc_index_gcd_transport_result_witness_bezout_rightcoefficients)) * pfgb_qc_gcd_transport_result_witness_bezout) + (pfc_value_gcd_transport_result_witness_bezout_rightcoefficients))) /\\ ((exists pfc_terms_code_gcd_transport_result_witness_bezout_rightcoefficientscoefficient pfc_terms_scale_gcd_transport_result_witness_bezout_rightcoefficientscoefficient pfc_natural_sum_gcd_transport_result_witness_bezout_rightcoefficientscoefficient. ((forall pfc_index_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalbound. pfa_gap_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_transport_result_witness_bezout_rightcoefficients))) -> exists pfc_value_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_result_witness_bezout_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_transport_result_witness_bezout_rightcoefficientscoefficient = ff_q_pfp_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_transport_result_witness_bezout_rightcoefficientscoefficient) + (pfc_value_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_left_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_right_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonal)+pfc_complement_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm=(pfc_index_gcd_transport_result_witness_bezout_rightcoefficients)) /\\ ((((((exists pfa_gap_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonal) = (pfgs_V_gcd_transport_result)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_gcd_transport_result)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. pfgs_vb_gcd_transport_result = ff_q_pfp_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_gcd_transport_result) + (pfc_left_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside+(pfgs_V_gcd_transport_result)=(pfc_index_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm) = (M2)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * bc2)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. bb2 = ff_q_pfp_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * bc2) + (pfc_right_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside+(M2)=(pfc_complement_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonal)=pfc_left_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm*pfc_right_gcd_transport_result_witness_bezout_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum fs_v_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_transport_result_witness_bezout_rightcoefficientscoefficient) = S ((S (S (pfc_index_gcd_transport_result_witness_bezout_rightcoefficients))) * fs_v_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_transport_result_witness_bezout_rightcoefficients))) * fs_v_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum) + (pfc_natural_sum_gcd_transport_result_witness_bezout_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_transport_result_witness_bezout_rightcoefficients)) -> exists fs_a_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_result_witness_bezout_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_transport_result_witness_bezout_rightcoefficientscoefficient = fs_q_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_transport_result_witness_bezout_rightcoefficientscoefficient) + (fs_a_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum) + (fs_r_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum) + (fs_s_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_transport_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_transport_result_witness_bezout_rightcoefficientscoefficientresiduebound. pfa_gap_gcd_transport_result_witness_bezout_rightcoefficientscoefficientresiduebound + S (pfc_value_gcd_transport_result_witness_bezout_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_transport_result_witness_bezout_rightcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_transport_result_witness_bezout_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_transport_result_witness_bezout_rightcoefficientscoefficient) + (p) * pfa_offset_left_gcd_transport_result_witness_bezout_rightcoefficientscoefficientresiduecongruence = (pfc_value_gcd_transport_result_witness_bezout_rightcoefficients) + (p) * pfa_offset_right_gcd_transport_result_witness_bezout_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_gcd_transport_result_witness_bezout_sum_left_bounded. (exists fom_gap_pfp_gcd_transport_result_witness_bezout_sum_left_bounded_index_bound. fom_gap_pfp_gcd_transport_result_witness_bezout_sum_left_bounded_index_bound + S (fom_index_pfp_gcd_transport_result_witness_bezout_sum_left_bounded) = pfgb_P_gcd_transport_result_witness_bezout) -> exists fom_value_pfp_gcd_transport_result_witness_bezout_sum_left_bounded. ((((exists fom_beta_height_pfp_gcd_transport_result_witness_bezout_sum_left_bounded_entry. fom_beta_height_pfp_gcd_transport_result_witness_bezout_sum_left_bounded_entry + S (fom_value_pfp_gcd_transport_result_witness_bezout_sum_left_bounded) = S ((S (fom_index_pfp_gcd_transport_result_witness_bezout_sum_left_bounded)) * pfgb_pc_gcd_transport_result_witness_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_transport_result_witness_bezout_sum_left_bounded_entry. pfgb_pb_gcd_transport_result_witness_bezout = fom_beta_quotient_pfp_gcd_transport_result_witness_bezout_sum_left_bounded_entry * S ((S (fom_index_pfp_gcd_transport_result_witness_bezout_sum_left_bounded)) * pfgb_pc_gcd_transport_result_witness_bezout) + (fom_value_pfp_gcd_transport_result_witness_bezout_sum_left_bounded))) /\\ (exists fom_gap_pfp_gcd_transport_result_witness_bezout_sum_left_bounded_value_bound. fom_gap_pfp_gcd_transport_result_witness_bezout_sum_left_bounded_value_bound + S (fom_value_pfp_gcd_transport_result_witness_bezout_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_transport_result_witness_bezout_sum_right_bounded. (exists fom_gap_pfp_gcd_transport_result_witness_bezout_sum_right_bounded_index_bound. fom_gap_pfp_gcd_transport_result_witness_bezout_sum_right_bounded_index_bound + S (fom_index_pfp_gcd_transport_result_witness_bezout_sum_right_bounded) = pfgb_Q_gcd_transport_result_witness_bezout) -> exists fom_value_pfp_gcd_transport_result_witness_bezout_sum_right_bounded. ((((exists fom_beta_height_pfp_gcd_transport_result_witness_bezout_sum_right_bounded_entry. fom_beta_height_pfp_gcd_transport_result_witness_bezout_sum_right_bounded_entry + S (fom_value_pfp_gcd_transport_result_witness_bezout_sum_right_bounded) = S ((S (fom_index_pfp_gcd_transport_result_witness_bezout_sum_right_bounded)) * pfgb_qc_gcd_transport_result_witness_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_transport_result_witness_bezout_sum_right_bounded_entry. pfgb_qb_gcd_transport_result_witness_bezout = fom_beta_quotient_pfp_gcd_transport_result_witness_bezout_sum_right_bounded_entry * S ((S (fom_index_pfp_gcd_transport_result_witness_bezout_sum_right_bounded)) * pfgb_qc_gcd_transport_result_witness_bezout) + (fom_value_pfp_gcd_transport_result_witness_bezout_sum_right_bounded))) /\\ (exists fom_gap_pfp_gcd_transport_result_witness_bezout_sum_right_bounded_value_bound. fom_gap_pfp_gcd_transport_result_witness_bezout_sum_right_bounded_value_bound + S (fom_value_pfp_gcd_transport_result_witness_bezout_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_transport_result_witness_bezout_sum_result_bounded. (exists fom_gap_pfp_gcd_transport_result_witness_bezout_sum_result_bounded_index_bound. fom_gap_pfp_gcd_transport_result_witness_bezout_sum_result_bounded_index_bound + S (fom_index_pfp_gcd_transport_result_witness_bezout_sum_result_bounded) = pfgs_G_gcd_transport_result) -> exists fom_value_pfp_gcd_transport_result_witness_bezout_sum_result_bounded. ((((exists fom_beta_height_pfp_gcd_transport_result_witness_bezout_sum_result_bounded_entry. fom_beta_height_pfp_gcd_transport_result_witness_bezout_sum_result_bounded_entry + S (fom_value_pfp_gcd_transport_result_witness_bezout_sum_result_bounded) = S ((S (fom_index_pfp_gcd_transport_result_witness_bezout_sum_result_bounded)) * pfgs_gc_gcd_transport_result)) /\\ exists fom_beta_quotient_pfp_gcd_transport_result_witness_bezout_sum_result_bounded_entry. pfgs_gb_gcd_transport_result = fom_beta_quotient_pfp_gcd_transport_result_witness_bezout_sum_result_bounded_entry * S ((S (fom_index_pfp_gcd_transport_result_witness_bezout_sum_result_bounded)) * pfgs_gc_gcd_transport_result) + (fom_value_pfp_gcd_transport_result_witness_bezout_sum_result_bounded))) /\\ (exists fom_gap_pfp_gcd_transport_result_witness_bezout_sum_result_bounded_value_bound. fom_gap_pfp_gcd_transport_result_witness_bezout_sum_result_bounded_value_bound + S (fom_value_pfp_gcd_transport_result_witness_bezout_sum_result_bounded) = p))) /\\ ((exists pfga_ub_gcd_transport_result_witness_bezout_sum pfga_uc_gcd_transport_result_witness_bezout_sum pfga_vb_gcd_transport_result_witness_bezout_sum pfga_vc_gcd_transport_result_witness_bezout_sum pfga_tb_gcd_transport_result_witness_bezout_sum pfga_tc_gcd_transport_result_witness_bezout_sum pfga_K_gcd_transport_result_witness_bezout_sum. ((((forall pfrep_power_gcd_transport_result_witness_bezout_sum_left pfrep_left_gcd_transport_result_witness_bezout_sum_left pfrep_right_gcd_transport_result_witness_bezout_sum_left. ((exists pfrep_position_gcd_transport_result_witness_bezout_sum_leftfirst. ((pfrep_position_gcd_transport_result_witness_bezout_sum_leftfirst+S (pfrep_power_gcd_transport_result_witness_bezout_sum_left)=(pfgb_P_gcd_transport_result_witness_bezout)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_bezout_sum_leftfirstentry. ff_h_pfp_gcd_transport_result_witness_bezout_sum_leftfirstentry + S (pfrep_left_gcd_transport_result_witness_bezout_sum_left) = S ((S (pfrep_position_gcd_transport_result_witness_bezout_sum_leftfirst)) * pfgb_pc_gcd_transport_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_sum_leftfirstentry. pfgb_pb_gcd_transport_result_witness_bezout = ff_q_pfp_gcd_transport_result_witness_bezout_sum_leftfirstentry * S ((S (pfrep_position_gcd_transport_result_witness_bezout_sum_leftfirst)) * pfgb_pc_gcd_transport_result_witness_bezout) + (pfrep_left_gcd_transport_result_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_transport_result_witness_bezout_sum_leftfirstoutside. pfrep_gap_gcd_transport_result_witness_bezout_sum_leftfirstoutside+(pfgb_P_gcd_transport_result_witness_bezout)=(pfrep_power_gcd_transport_result_witness_bezout_sum_left)) /\\ (((pfrep_left_gcd_transport_result_witness_bezout_sum_left)=0))))) -> ((exists pfrep_position_gcd_transport_result_witness_bezout_sum_leftsecond. ((pfrep_position_gcd_transport_result_witness_bezout_sum_leftsecond+S (pfrep_power_gcd_transport_result_witness_bezout_sum_left)=(pfga_K_gcd_transport_result_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_bezout_sum_leftsecondentry. ff_h_pfp_gcd_transport_result_witness_bezout_sum_leftsecondentry + S (pfrep_right_gcd_transport_result_witness_bezout_sum_left) = S ((S (pfrep_position_gcd_transport_result_witness_bezout_sum_leftsecond)) * pfga_uc_gcd_transport_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_sum_leftsecondentry. pfga_ub_gcd_transport_result_witness_bezout_sum = ff_q_pfp_gcd_transport_result_witness_bezout_sum_leftsecondentry * S ((S (pfrep_position_gcd_transport_result_witness_bezout_sum_leftsecond)) * pfga_uc_gcd_transport_result_witness_bezout_sum) + (pfrep_right_gcd_transport_result_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_transport_result_witness_bezout_sum_leftsecondoutside. pfrep_gap_gcd_transport_result_witness_bezout_sum_leftsecondoutside+(pfga_K_gcd_transport_result_witness_bezout_sum)=(pfrep_power_gcd_transport_result_witness_bezout_sum_left)) /\\ (((pfrep_right_gcd_transport_result_witness_bezout_sum_left)=0))))) -> pfrep_left_gcd_transport_result_witness_bezout_sum_left=pfrep_right_gcd_transport_result_witness_bezout_sum_left) /\\ ((forall pfrep_power_gcd_transport_result_witness_bezout_sum_right pfrep_left_gcd_transport_result_witness_bezout_sum_right pfrep_right_gcd_transport_result_witness_bezout_sum_right. ((exists pfrep_position_gcd_transport_result_witness_bezout_sum_rightfirst. ((pfrep_position_gcd_transport_result_witness_bezout_sum_rightfirst+S (pfrep_power_gcd_transport_result_witness_bezout_sum_right)=(pfgb_Q_gcd_transport_result_witness_bezout)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_bezout_sum_rightfirstentry. ff_h_pfp_gcd_transport_result_witness_bezout_sum_rightfirstentry + S (pfrep_left_gcd_transport_result_witness_bezout_sum_right) = S ((S (pfrep_position_gcd_transport_result_witness_bezout_sum_rightfirst)) * pfgb_qc_gcd_transport_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_sum_rightfirstentry. pfgb_qb_gcd_transport_result_witness_bezout = ff_q_pfp_gcd_transport_result_witness_bezout_sum_rightfirstentry * S ((S (pfrep_position_gcd_transport_result_witness_bezout_sum_rightfirst)) * pfgb_qc_gcd_transport_result_witness_bezout) + (pfrep_left_gcd_transport_result_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_transport_result_witness_bezout_sum_rightfirstoutside. pfrep_gap_gcd_transport_result_witness_bezout_sum_rightfirstoutside+(pfgb_Q_gcd_transport_result_witness_bezout)=(pfrep_power_gcd_transport_result_witness_bezout_sum_right)) /\\ (((pfrep_left_gcd_transport_result_witness_bezout_sum_right)=0))))) -> ((exists pfrep_position_gcd_transport_result_witness_bezout_sum_rightsecond. ((pfrep_position_gcd_transport_result_witness_bezout_sum_rightsecond+S (pfrep_power_gcd_transport_result_witness_bezout_sum_right)=(pfga_K_gcd_transport_result_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_bezout_sum_rightsecondentry. ff_h_pfp_gcd_transport_result_witness_bezout_sum_rightsecondentry + S (pfrep_right_gcd_transport_result_witness_bezout_sum_right) = S ((S (pfrep_position_gcd_transport_result_witness_bezout_sum_rightsecond)) * pfga_vc_gcd_transport_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_sum_rightsecondentry. pfga_vb_gcd_transport_result_witness_bezout_sum = ff_q_pfp_gcd_transport_result_witness_bezout_sum_rightsecondentry * S ((S (pfrep_position_gcd_transport_result_witness_bezout_sum_rightsecond)) * pfga_vc_gcd_transport_result_witness_bezout_sum) + (pfrep_right_gcd_transport_result_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_transport_result_witness_bezout_sum_rightsecondoutside. pfrep_gap_gcd_transport_result_witness_bezout_sum_rightsecondoutside+(pfga_K_gcd_transport_result_witness_bezout_sum)=(pfrep_power_gcd_transport_result_witness_bezout_sum_right)) /\\ (((pfrep_right_gcd_transport_result_witness_bezout_sum_right)=0))))) -> pfrep_left_gcd_transport_result_witness_bezout_sum_right=pfrep_right_gcd_transport_result_witness_bezout_sum_right)))) /\\ (((forall pfp_index_gcd_transport_result_witness_bezout_sum_add. (exists pfa_gap_gcd_transport_result_witness_bezout_sum_addindex. pfa_gap_gcd_transport_result_witness_bezout_sum_addindex + S (pfp_index_gcd_transport_result_witness_bezout_sum_add) = (pfga_K_gcd_transport_result_witness_bezout_sum)) -> exists pfp_left_gcd_transport_result_witness_bezout_sum_add pfp_right_gcd_transport_result_witness_bezout_sum_add pfp_value_gcd_transport_result_witness_bezout_sum_add. ((((exists ff_h_pfp_gcd_transport_result_witness_bezout_sum_addleft. ff_h_pfp_gcd_transport_result_witness_bezout_sum_addleft + S (pfp_left_gcd_transport_result_witness_bezout_sum_add) = S ((S (pfp_index_gcd_transport_result_witness_bezout_sum_add)) * pfga_uc_gcd_transport_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_sum_addleft. pfga_ub_gcd_transport_result_witness_bezout_sum = ff_q_pfp_gcd_transport_result_witness_bezout_sum_addleft * S ((S (pfp_index_gcd_transport_result_witness_bezout_sum_add)) * pfga_uc_gcd_transport_result_witness_bezout_sum) + (pfp_left_gcd_transport_result_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_transport_result_witness_bezout_sum_addright. ff_h_pfp_gcd_transport_result_witness_bezout_sum_addright + S (pfp_right_gcd_transport_result_witness_bezout_sum_add) = S ((S (pfp_index_gcd_transport_result_witness_bezout_sum_add)) * pfga_vc_gcd_transport_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_sum_addright. pfga_vb_gcd_transport_result_witness_bezout_sum = ff_q_pfp_gcd_transport_result_witness_bezout_sum_addright * S ((S (pfp_index_gcd_transport_result_witness_bezout_sum_add)) * pfga_vc_gcd_transport_result_witness_bezout_sum) + (pfp_right_gcd_transport_result_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_transport_result_witness_bezout_sum_addtarget. ff_h_pfp_gcd_transport_result_witness_bezout_sum_addtarget + S (pfp_value_gcd_transport_result_witness_bezout_sum_add) = S ((S (pfp_index_gcd_transport_result_witness_bezout_sum_add)) * pfga_tc_gcd_transport_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_sum_addtarget. pfga_tb_gcd_transport_result_witness_bezout_sum = ff_q_pfp_gcd_transport_result_witness_bezout_sum_addtarget * S ((S (pfp_index_gcd_transport_result_witness_bezout_sum_add)) * pfga_tc_gcd_transport_result_witness_bezout_sum) + (pfp_value_gcd_transport_result_witness_bezout_sum_add))) /\\ ((((exists pfa_gap_gcd_transport_result_witness_bezout_sum_addoperationleft. pfa_gap_gcd_transport_result_witness_bezout_sum_addoperationleft + S (pfp_left_gcd_transport_result_witness_bezout_sum_add) = (p)) /\\ (((exists pfa_gap_gcd_transport_result_witness_bezout_sum_addoperationright. pfa_gap_gcd_transport_result_witness_bezout_sum_addoperationright + S (pfp_right_gcd_transport_result_witness_bezout_sum_add) = (p)) /\\ ((((exists pfa_gap_gcd_transport_result_witness_bezout_sum_addoperationresultbound. pfa_gap_gcd_transport_result_witness_bezout_sum_addoperationresultbound + S (pfp_value_gcd_transport_result_witness_bezout_sum_add) = (p)) /\\ ((exists pfa_offset_left_gcd_transport_result_witness_bezout_sum_addoperationresultcongruence pfa_offset_right_gcd_transport_result_witness_bezout_sum_addoperationresultcongruence. ((pfp_left_gcd_transport_result_witness_bezout_sum_add) + (pfp_right_gcd_transport_result_witness_bezout_sum_add)) + (p) * pfa_offset_left_gcd_transport_result_witness_bezout_sum_addoperationresultcongruence = (pfp_value_gcd_transport_result_witness_bezout_sum_add) + (p) * pfa_offset_right_gcd_transport_result_witness_bezout_sum_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_gcd_transport_result_witness_bezout_sum_result pfrep_left_gcd_transport_result_witness_bezout_sum_result pfrep_right_gcd_transport_result_witness_bezout_sum_result. ((exists pfrep_position_gcd_transport_result_witness_bezout_sum_resultfirst. ((pfrep_position_gcd_transport_result_witness_bezout_sum_resultfirst+S (pfrep_power_gcd_transport_result_witness_bezout_sum_result)=(pfga_K_gcd_transport_result_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_bezout_sum_resultfirstentry. ff_h_pfp_gcd_transport_result_witness_bezout_sum_resultfirstentry + S (pfrep_left_gcd_transport_result_witness_bezout_sum_result) = S ((S (pfrep_position_gcd_transport_result_witness_bezout_sum_resultfirst)) * pfga_tc_gcd_transport_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_sum_resultfirstentry. pfga_tb_gcd_transport_result_witness_bezout_sum = ff_q_pfp_gcd_transport_result_witness_bezout_sum_resultfirstentry * S ((S (pfrep_position_gcd_transport_result_witness_bezout_sum_resultfirst)) * pfga_tc_gcd_transport_result_witness_bezout_sum) + (pfrep_left_gcd_transport_result_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_transport_result_witness_bezout_sum_resultfirstoutside. pfrep_gap_gcd_transport_result_witness_bezout_sum_resultfirstoutside+(pfga_K_gcd_transport_result_witness_bezout_sum)=(pfrep_power_gcd_transport_result_witness_bezout_sum_result)) /\\ (((pfrep_left_gcd_transport_result_witness_bezout_sum_result)=0))))) -> ((exists pfrep_position_gcd_transport_result_witness_bezout_sum_resultsecond. ((pfrep_position_gcd_transport_result_witness_bezout_sum_resultsecond+S (pfrep_power_gcd_transport_result_witness_bezout_sum_result)=(pfgs_G_gcd_transport_result)) /\\ ((((exists ff_h_pfp_gcd_transport_result_witness_bezout_sum_resultsecondentry. ff_h_pfp_gcd_transport_result_witness_bezout_sum_resultsecondentry + S (pfrep_right_gcd_transport_result_witness_bezout_sum_result) = S ((S (pfrep_position_gcd_transport_result_witness_bezout_sum_resultsecond)) * pfgs_gc_gcd_transport_result)) /\\ exists ff_q_pfp_gcd_transport_result_witness_bezout_sum_resultsecondentry. pfgs_gb_gcd_transport_result = ff_q_pfp_gcd_transport_result_witness_bezout_sum_resultsecondentry * S ((S (pfrep_position_gcd_transport_result_witness_bezout_sum_resultsecond)) * pfgs_gc_gcd_transport_result) + (pfrep_right_gcd_transport_result_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_transport_result_witness_bezout_sum_resultsecondoutside. pfrep_gap_gcd_transport_result_witness_bezout_sum_resultsecondoutside+(pfgs_G_gcd_transport_result)=(pfrep_power_gcd_transport_result_witness_bezout_sum_result)) /\\ (((pfrep_right_gcd_transport_result_witness_bezout_sum_result)=0))))) -> pfrep_left_gcd_transport_result_witness_bezout_sum_result=pfrep_right_gcd_transport_result_witness_bezout_sum_result)))))))))))))))))))))))", "statement_sha256": "fe2d469dcf0ef5702134afc474733d776dcb9a65051981c541a8ea155b3a74d5", "summary": "Replace the second input by any formally equivalent canonical representation while keeping the same zero-or-monic common divisor and the same Bezout coefficients; 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(~((p) = 1) /\\ forall pfa_factor_left_gcd_backward_local_prime pfa_factor_right_gcd_backward_local_prime. (p) = pfa_factor_left_gcd_backward_local_prime * pfa_factor_right_gcd_backward_local_prime -> pfa_factor_left_gcd_backward_local_prime = 1 \\/ pfa_factor_right_gcd_backward_local_prime = 1) -> (((forall fom_index_pfp_gcd_backward_executioninput. (exists fom_gap_pfp_gcd_backward_executioninput_index_bound. fom_gap_pfp_gcd_backward_executioninput_index_bound + S (fom_index_pfp_gcd_backward_executioninput) = L) -> exists fom_value_pfp_gcd_backward_executioninput. ((((exists fom_beta_height_pfp_gcd_backward_executioninput_entry. fom_beta_height_pfp_gcd_backward_executioninput_entry + S (fom_value_pfp_gcd_backward_executioninput) = S ((S (fom_index_pfp_gcd_backward_executioninput)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_backward_executioninput_entry. ab = fom_beta_quotient_pfp_gcd_backward_executioninput_entry * S ((S (fom_index_pfp_gcd_backward_executioninput)) * ac) + (fom_value_pfp_gcd_backward_executioninput))) /\\ (exists fom_gap_pfp_gcd_backward_executioninput_value_bound. fom_gap_pfp_gcd_backward_executioninput_value_bound + S (fom_value_pfp_gcd_backward_executioninput) = p))) /\\ (((forall fom_index_pfp_gcd_backward_executiondivisor. (exists fom_gap_pfp_gcd_backward_executiondivisor_index_bound. fom_gap_pfp_gcd_backward_executiondivisor_index_bound + S (fom_index_pfp_gcd_backward_executiondivisor) = S (d)) -> exists fom_value_pfp_gcd_backward_executiondivisor. ((((exists fom_beta_height_pfp_gcd_backward_executiondivisor_entry. fom_beta_height_pfp_gcd_backward_executiondivisor_entry + S (fom_value_pfp_gcd_backward_executiondivisor) = S ((S (fom_index_pfp_gcd_backward_executiondivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_executiondivisor_entry. bb = fom_beta_quotient_pfp_gcd_backward_executiondivisor_entry * S ((S (fom_index_pfp_gcd_backward_executiondivisor)) * bc) + (fom_value_pfp_gcd_backward_executiondivisor))) /\\ (exists fom_gap_pfp_gcd_backward_executiondivisor_value_bound. fom_gap_pfp_gcd_backward_executiondivisor_value_bound + S (fom_value_pfp_gcd_backward_executiondivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_gcd_backward_executionlengthshort. pfc_gap_gcd_backward_executionlengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((exists pfd_head_gcd_backward_execution pfd_inverse_gcd_backward_execution pfd_product_code_gcd_backward_execution pfd_product_scale_gcd_backward_execution pfd_residual_code_gcd_backward_execution pfd_residual_scale_gcd_backward_execution pfd_cut_gcd_backward_execution. ((((exists ff_h_pfp_gcd_backward_executionhead. ff_h_pfp_gcd_backward_executionhead + S (pfd_head_gcd_backward_execution) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_gcd_backward_executionhead. bb = ff_q_pfp_gcd_backward_executionhead * S ((S (0)) * bc) + (pfd_head_gcd_backward_execution))) /\\ (((((~((pfd_head_gcd_backward_execution) = 0)) /\\ ((((exists pfa_gap_gcd_backward_executioninversemultiplicationleft. pfa_gap_gcd_backward_executioninversemultiplicationleft + S (pfd_head_gcd_backward_execution) = (p)) /\\ (((exists pfa_gap_gcd_backward_executioninversemultiplicationright. pfa_gap_gcd_backward_executioninversemultiplicationright + S (pfd_inverse_gcd_backward_execution) = (p)) /\\ ((((exists pfa_gap_gcd_backward_executioninversemultiplicationresultbound. pfa_gap_gcd_backward_executioninversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_executioninversemultiplicationresultcongruence pfa_offset_right_gcd_backward_executioninversemultiplicationresultcongruence. ((pfd_head_gcd_backward_execution) * (pfd_inverse_gcd_backward_execution)) + (p) * pfa_offset_left_gcd_backward_executioninversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_gcd_backward_executioninversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_gcd_backward_executionquotient. (exists pfa_gap_gcd_backward_executionquotientbound. pfa_gap_gcd_backward_executionquotientbound + S (pfd_index_gcd_backward_executionquotient) = (q)) -> exists pfd_value_gcd_backward_executionquotient. ((((exists ff_h_pfp_gcd_backward_executionquotiententry. ff_h_pfp_gcd_backward_executionquotiententry + S (pfd_value_gcd_backward_executionquotient) = S ((S (pfd_index_gcd_backward_executionquotient)) * qc)) /\\ exists ff_q_pfp_gcd_backward_executionquotiententry. qb = ff_q_pfp_gcd_backward_executionquotiententry * S ((S (pfd_index_gcd_backward_executionquotient)) * qc) + (pfd_value_gcd_backward_executionquotient))) /\\ ((exists pfd_input_gcd_backward_executionquotientstep pfd_previous_gcd_backward_executionquotientstep pfd_difference_gcd_backward_executionquotientstep. ((((exists ff_h_pfp_gcd_backward_executionquotientstepinput. ff_h_pfp_gcd_backward_executionquotientstepinput + S (pfd_input_gcd_backward_executionquotientstep) = S ((S (pfd_index_gcd_backward_executionquotient)) * ac)) /\\ exists ff_q_pfp_gcd_backward_executionquotientstepinput. ab = ff_q_pfp_gcd_backward_executionquotientstepinput * S ((S (pfd_index_gcd_backward_executionquotient)) * ac) + (pfd_input_gcd_backward_executionquotientstep))) /\\ (((exists pfc_terms_code_gcd_backward_executionquotientstepprevious pfc_terms_scale_gcd_backward_executionquotientstepprevious pfc_natural_sum_gcd_backward_executionquotientstepprevious. ((forall pfc_index_gcd_backward_executionquotientsteppreviousdiagonal. (exists pfa_gap_gcd_backward_executionquotientsteppreviousdiagonalbound. pfa_gap_gcd_backward_executionquotientsteppreviousdiagonalbound + S (pfc_index_gcd_backward_executionquotientsteppreviousdiagonal) = (S (pfd_index_gcd_backward_executionquotient))) -> exists pfc_value_gcd_backward_executionquotientsteppreviousdiagonal. ((((exists ff_h_pfp_gcd_backward_executionquotientsteppreviousdiagonalentry. ff_h_pfp_gcd_backward_executionquotientsteppreviousdiagonalentry + S (pfc_value_gcd_backward_executionquotientsteppreviousdiagonal) = S ((S (pfc_index_gcd_backward_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_gcd_backward_executionquotientstepprevious)) /\\ exists ff_q_pfp_gcd_backward_executionquotientsteppreviousdiagonalentry. pfc_terms_code_gcd_backward_executionquotientstepprevious = ff_q_pfp_gcd_backward_executionquotientsteppreviousdiagonalentry * S ((S (pfc_index_gcd_backward_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_gcd_backward_executionquotientstepprevious) + (pfc_value_gcd_backward_executionquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_gcd_backward_executionquotientsteppreviousdiagonalterm pfc_left_gcd_backward_executionquotientsteppreviousdiagonalterm pfc_right_gcd_backward_executionquotientsteppreviousdiagonalterm. (((pfc_index_gcd_backward_executionquotientsteppreviousdiagonal)+pfc_complement_gcd_backward_executionquotientsteppreviousdiagonalterm=(pfd_index_gcd_backward_executionquotient)) /\\ ((((((exists pfa_gap_gcd_backward_executionquotientsteppreviousdiagonaltermleftinside. pfa_gap_gcd_backward_executionquotientsteppreviousdiagonaltermleftinside + S (pfc_index_gcd_backward_executionquotientsteppreviousdiagonal) = (pfd_index_gcd_backward_executionquotient)) /\\ ((((exists ff_h_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermleftentry + S (pfc_left_gcd_backward_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_index_gcd_backward_executionquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_executionquotientsteppreviousdiagonal)) * qc) + (pfc_left_gcd_backward_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_executionquotientsteppreviousdiagonaltermleftoutside. pfc_gap_gcd_backward_executionquotientsteppreviousdiagonaltermleftoutside+(pfd_index_gcd_backward_executionquotient)=(pfc_index_gcd_backward_executionquotientsteppreviousdiagonal)) /\\ (((pfc_left_gcd_backward_executionquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_executionquotientsteppreviousdiagonaltermrightinside. pfa_gap_gcd_backward_executionquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_gcd_backward_executionquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermrightentry + S (pfc_right_gcd_backward_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_gcd_backward_executionquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_executionquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_gcd_backward_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_executionquotientsteppreviousdiagonaltermrightoutside. pfc_gap_gcd_backward_executionquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_gcd_backward_executionquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_gcd_backward_executionquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_executionquotientsteppreviousdiagonal)=pfc_left_gcd_backward_executionquotientsteppreviousdiagonalterm*pfc_right_gcd_backward_executionquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_executionquotientstepprevioussum fs_v_pfc_gcd_backward_executionquotientstepprevioussum. ((((exists fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_start. fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_start. fs_u_pfc_gcd_backward_executionquotientstepprevioussum = fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_terminal. fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_terminal + S (pfc_natural_sum_gcd_backward_executionquotientstepprevious) = S ((S (S (pfd_index_gcd_backward_executionquotient))) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_terminal. fs_u_pfc_gcd_backward_executionquotientstepprevioussum = fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_gcd_backward_executionquotient))) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum) + (pfc_natural_sum_gcd_backward_executionquotientstepprevious))) /\\ forall fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps. (exists fs_lt_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_bound. fs_lt_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_bound + S fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps = S (pfd_index_gcd_backward_executionquotient)) -> exists fs_a_pfc_gcd_backward_executionquotientstepprevioussum_body_steps fs_r_pfc_gcd_backward_executionquotientstepprevioussum_body_steps fs_s_pfc_gcd_backward_executionquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_summand. fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_gcd_backward_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_gcd_backward_executionquotientstepprevious)) /\\ exists fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_summand. pfc_terms_code_gcd_backward_executionquotientstepprevious = fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_gcd_backward_executionquotientstepprevious) + (fs_a_pfc_gcd_backward_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_partial. fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_gcd_backward_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_partial. fs_u_pfc_gcd_backward_executionquotientstepprevioussum = fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum) + (fs_r_pfc_gcd_backward_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_successor. fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_gcd_backward_executionquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_successor. fs_u_pfc_gcd_backward_executionquotientstepprevioussum = fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum) + (fs_s_pfc_gcd_backward_executionquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_gcd_backward_executionquotientstepprevioussum_body_steps = fs_r_pfc_gcd_backward_executionquotientstepprevioussum_body_steps + fs_a_pfc_gcd_backward_executionquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_executionquotientsteppreviousresiduebound. pfa_gap_gcd_backward_executionquotientsteppreviousresiduebound + S (pfd_previous_gcd_backward_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_executionquotientsteppreviousresiduecongruence pfa_offset_right_gcd_backward_executionquotientsteppreviousresiduecongruence. (pfc_natural_sum_gcd_backward_executionquotientstepprevious) + (p) * pfa_offset_left_gcd_backward_executionquotientsteppreviousresiduecongruence = (pfd_previous_gcd_backward_executionquotientstep) + (p) * pfa_offset_right_gcd_backward_executionquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_gcd_backward_executionquotientstepsubtractleft. pfa_gap_gcd_backward_executionquotientstepsubtractleft + S (pfd_previous_gcd_backward_executionquotientstep) = (p)) /\\ (((exists pfa_gap_gcd_backward_executionquotientstepsubtractright. pfa_gap_gcd_backward_executionquotientstepsubtractright + S (pfd_difference_gcd_backward_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_gcd_backward_executionquotientstepsubtractresultbound. pfa_gap_gcd_backward_executionquotientstepsubtractresultbound + S (pfd_input_gcd_backward_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_executionquotientstepsubtractresultcongruence pfa_offset_right_gcd_backward_executionquotientstepsubtractresultcongruence. ((pfd_previous_gcd_backward_executionquotientstep) + (pfd_difference_gcd_backward_executionquotientstep)) + (p) * pfa_offset_left_gcd_backward_executionquotientstepsubtractresultcongruence = (pfd_input_gcd_backward_executionquotientstep) + (p) * pfa_offset_right_gcd_backward_executionquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_gcd_backward_executionquotientstepmultiplyleft. pfa_gap_gcd_backward_executionquotientstepmultiplyleft + S (pfd_inverse_gcd_backward_execution) = (p)) /\\ (((exists pfa_gap_gcd_backward_executionquotientstepmultiplyright. pfa_gap_gcd_backward_executionquotientstepmultiplyright + S (pfd_difference_gcd_backward_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_gcd_backward_executionquotientstepmultiplyresultbound. pfa_gap_gcd_backward_executionquotientstepmultiplyresultbound + S (pfd_value_gcd_backward_executionquotient) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_executionquotientstepmultiplyresultcongruence pfa_offset_right_gcd_backward_executionquotientstepmultiplyresultcongruence. ((pfd_inverse_gcd_backward_execution) * (pfd_difference_gcd_backward_executionquotientstep)) + (p) * pfa_offset_left_gcd_backward_executionquotientstepmultiplyresultcongruence = (pfd_value_gcd_backward_executionquotient) + (p) * pfa_offset_right_gcd_backward_executionquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_gcd_backward_executionproduct. (exists pfa_gap_gcd_backward_executionproductbound. pfa_gap_gcd_backward_executionproductbound + S (pfc_index_gcd_backward_executionproduct) = (L)) -> exists pfc_value_gcd_backward_executionproduct. ((((exists ff_h_pfp_gcd_backward_executionproductentry. ff_h_pfp_gcd_backward_executionproductentry + S (pfc_value_gcd_backward_executionproduct) = S ((S (pfc_index_gcd_backward_executionproduct)) * pfd_product_scale_gcd_backward_execution)) /\\ exists ff_q_pfp_gcd_backward_executionproductentry. pfd_product_code_gcd_backward_execution = ff_q_pfp_gcd_backward_executionproductentry * S ((S (pfc_index_gcd_backward_executionproduct)) * pfd_product_scale_gcd_backward_execution) + (pfc_value_gcd_backward_executionproduct))) /\\ ((exists pfc_terms_code_gcd_backward_executionproductcoefficient pfc_terms_scale_gcd_backward_executionproductcoefficient pfc_natural_sum_gcd_backward_executionproductcoefficient. ((forall pfc_index_gcd_backward_executionproductcoefficientdiagonal. (exists pfa_gap_gcd_backward_executionproductcoefficientdiagonalbound. pfa_gap_gcd_backward_executionproductcoefficientdiagonalbound + S (pfc_index_gcd_backward_executionproductcoefficientdiagonal) = (S (pfc_index_gcd_backward_executionproduct))) -> exists pfc_value_gcd_backward_executionproductcoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_executionproductcoefficientdiagonalentry. ff_h_pfp_gcd_backward_executionproductcoefficientdiagonalentry + S (pfc_value_gcd_backward_executionproductcoefficientdiagonal) = S ((S (pfc_index_gcd_backward_executionproductcoefficientdiagonal)) * pfc_terms_scale_gcd_backward_executionproductcoefficient)) /\\ exists ff_q_pfp_gcd_backward_executionproductcoefficientdiagonalentry. pfc_terms_code_gcd_backward_executionproductcoefficient = ff_q_pfp_gcd_backward_executionproductcoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_executionproductcoefficientdiagonal)) * pfc_terms_scale_gcd_backward_executionproductcoefficient) + (pfc_value_gcd_backward_executionproductcoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_executionproductcoefficientdiagonalterm pfc_left_gcd_backward_executionproductcoefficientdiagonalterm pfc_right_gcd_backward_executionproductcoefficientdiagonalterm. (((pfc_index_gcd_backward_executionproductcoefficientdiagonal)+pfc_complement_gcd_backward_executionproductcoefficientdiagonalterm=(pfc_index_gcd_backward_executionproduct)) /\\ ((((((exists pfa_gap_gcd_backward_executionproductcoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_executionproductcoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_executionproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_gcd_backward_executionproductcoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_executionproductcoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_executionproductcoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_executionproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_gcd_backward_executionproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_gcd_backward_executionproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_executionproductcoefficientdiagonal)) * qc) + (pfc_left_gcd_backward_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_executionproductcoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_executionproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_gcd_backward_executionproductcoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_executionproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_executionproductcoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_executionproductcoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_executionproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_gcd_backward_executionproductcoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_executionproductcoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_executionproductcoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_executionproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_backward_executionproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_gcd_backward_executionproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_executionproductcoefficientdiagonalterm)) * bc) + (pfc_right_gcd_backward_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_executionproductcoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_executionproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_gcd_backward_executionproductcoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_executionproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_executionproductcoefficientdiagonal)=pfc_left_gcd_backward_executionproductcoefficientdiagonalterm*pfc_right_gcd_backward_executionproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_executionproductcoefficientsum fs_v_pfc_gcd_backward_executionproductcoefficientsum. ((((exists fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_start. fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_executionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_start. fs_u_pfc_gcd_backward_executionproductcoefficientsum = fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_executionproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_terminal. fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_executionproductcoefficient) = S ((S (S (pfc_index_gcd_backward_executionproduct))) * fs_v_pfc_gcd_backward_executionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_terminal. fs_u_pfc_gcd_backward_executionproductcoefficientsum = fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_executionproduct))) * fs_v_pfc_gcd_backward_executionproductcoefficientsum) + (pfc_natural_sum_gcd_backward_executionproductcoefficient))) /\\ forall fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_executionproductcoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_executionproductcoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps = S (pfc_index_gcd_backward_executionproduct)) -> exists fs_a_pfc_gcd_backward_executionproductcoefficientsum_body_steps fs_r_pfc_gcd_backward_executionproductcoefficientsum_body_steps fs_s_pfc_gcd_backward_executionproductcoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_executionproductcoefficient)) /\\ exists fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_executionproductcoefficient = fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_executionproductcoefficient) + (fs_a_pfc_gcd_backward_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_executionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_executionproductcoefficientsum = fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_executionproductcoefficientsum) + (fs_r_pfc_gcd_backward_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_executionproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_executionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_executionproductcoefficientsum = fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_executionproductcoefficientsum) + (fs_s_pfc_gcd_backward_executionproductcoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_executionproductcoefficientsum_body_steps = fs_r_pfc_gcd_backward_executionproductcoefficientsum_body_steps + fs_a_pfc_gcd_backward_executionproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_executionproductcoefficientresiduebound. pfa_gap_gcd_backward_executionproductcoefficientresiduebound + S (pfc_value_gcd_backward_executionproduct) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_executionproductcoefficientresiduecongruence pfa_offset_right_gcd_backward_executionproductcoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_executionproductcoefficient) + (p) * pfa_offset_left_gcd_backward_executionproductcoefficientresiduecongruence = (pfc_value_gcd_backward_executionproduct) + (p) * pfa_offset_right_gcd_backward_executionproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_gcd_backward_executiondifference. (exists pfa_gap_gcd_backward_executiondifferenceindex. pfa_gap_gcd_backward_executiondifferenceindex + S (pfs_index_gcd_backward_executiondifference) = (L)) -> exists pfs_left_gcd_backward_executiondifference pfs_right_gcd_backward_executiondifference pfs_result_gcd_backward_executiondifference. ((((exists ff_h_pfp_gcd_backward_executiondifferenceleft. ff_h_pfp_gcd_backward_executiondifferenceleft + S (pfs_left_gcd_backward_executiondifference) = S ((S (pfs_index_gcd_backward_executiondifference)) * ac)) /\\ exists ff_q_pfp_gcd_backward_executiondifferenceleft. ab = ff_q_pfp_gcd_backward_executiondifferenceleft * S ((S (pfs_index_gcd_backward_executiondifference)) * ac) + (pfs_left_gcd_backward_executiondifference))) /\\ (((((exists ff_h_pfp_gcd_backward_executiondifferenceright. ff_h_pfp_gcd_backward_executiondifferenceright + S (pfs_right_gcd_backward_executiondifference) = S ((S (pfs_index_gcd_backward_executiondifference)) * pfd_product_scale_gcd_backward_execution)) /\\ exists ff_q_pfp_gcd_backward_executiondifferenceright. pfd_product_code_gcd_backward_execution = ff_q_pfp_gcd_backward_executiondifferenceright * S ((S (pfs_index_gcd_backward_executiondifference)) * pfd_product_scale_gcd_backward_execution) + (pfs_right_gcd_backward_executiondifference))) /\\ (((((exists ff_h_pfp_gcd_backward_executiondifferenceresult. ff_h_pfp_gcd_backward_executiondifferenceresult + S (pfs_result_gcd_backward_executiondifference) = S ((S (pfs_index_gcd_backward_executiondifference)) * pfd_residual_scale_gcd_backward_execution)) /\\ exists ff_q_pfp_gcd_backward_executiondifferenceresult. pfd_residual_code_gcd_backward_execution = ff_q_pfp_gcd_backward_executiondifferenceresult * S ((S (pfs_index_gcd_backward_executiondifference)) * pfd_residual_scale_gcd_backward_execution) + (pfs_result_gcd_backward_executiondifference))) /\\ ((((exists pfa_gap_gcd_backward_executiondifferenceoperationleft. pfa_gap_gcd_backward_executiondifferenceoperationleft + S (pfs_right_gcd_backward_executiondifference) = (p)) /\\ (((exists pfa_gap_gcd_backward_executiondifferenceoperationright. pfa_gap_gcd_backward_executiondifferenceoperationright + S (pfs_result_gcd_backward_executiondifference) = (p)) /\\ ((((exists pfa_gap_gcd_backward_executiondifferenceoperationresultbound. pfa_gap_gcd_backward_executiondifferenceoperationresultbound + S (pfs_left_gcd_backward_executiondifference) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_executiondifferenceoperationresultcongruence pfa_offset_right_gcd_backward_executiondifferenceoperationresultcongruence. ((pfs_right_gcd_backward_executiondifference) + (pfs_result_gcd_backward_executiondifference)) + (p) * pfa_offset_left_gcd_backward_executiondifferenceoperationresultcongruence = (pfs_left_gcd_backward_executiondifference) + (p) * pfa_offset_right_gcd_backward_executiondifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_gcd_backward_execution)+(R)) /\\ (((forall fom_index_pfp_gcd_backward_executiontriminput. (exists fom_gap_pfp_gcd_backward_executiontriminput_index_bound. fom_gap_pfp_gcd_backward_executiontriminput_index_bound + S (fom_index_pfp_gcd_backward_executiontriminput) = L) -> exists fom_value_pfp_gcd_backward_executiontriminput. ((((exists fom_beta_height_pfp_gcd_backward_executiontriminput_entry. fom_beta_height_pfp_gcd_backward_executiontriminput_entry + S (fom_value_pfp_gcd_backward_executiontriminput) = S ((S (fom_index_pfp_gcd_backward_executiontriminput)) * pfd_residual_scale_gcd_backward_execution)) /\\ exists fom_beta_quotient_pfp_gcd_backward_executiontriminput_entry. pfd_residual_code_gcd_backward_execution = fom_beta_quotient_pfp_gcd_backward_executiontriminput_entry * S ((S (fom_index_pfp_gcd_backward_executiontriminput)) * pfd_residual_scale_gcd_backward_execution) + (fom_value_pfp_gcd_backward_executiontriminput))) /\\ (exists fom_gap_pfp_gcd_backward_executiontriminput_value_bound. fom_gap_pfp_gcd_backward_executiontriminput_value_bound + S (fom_value_pfp_gcd_backward_executiontriminput) = p))) /\\ (((forall pfp_repeat_index_gcd_backward_executiontrimremoved. (exists pfa_gap_gcd_backward_executiontrimremovedindex. pfa_gap_gcd_backward_executiontrimremovedindex + S (pfp_repeat_index_gcd_backward_executiontrimremoved) = (pfd_cut_gcd_backward_execution)) -> (((exists ff_h_pfp_gcd_backward_executiontrimremovedentry. ff_h_pfp_gcd_backward_executiontrimremovedentry + S (0) = S ((S (pfp_repeat_index_gcd_backward_executiontrimremoved)) * pfd_residual_scale_gcd_backward_execution)) /\\ exists ff_q_pfp_gcd_backward_executiontrimremovedentry. pfd_residual_code_gcd_backward_execution = ff_q_pfp_gcd_backward_executiontrimremovedentry * S ((S (pfp_repeat_index_gcd_backward_executiontrimremoved)) * pfd_residual_scale_gcd_backward_execution) + (0)))) /\\ (((forall pftrim_index_gcd_backward_executiontrimsuffix pftrim_value_gcd_backward_executiontrimsuffix. (exists pfa_gap_gcd_backward_executiontrimsuffixbound. pfa_gap_gcd_backward_executiontrimsuffixbound + S (pftrim_index_gcd_backward_executiontrimsuffix) = (R)) -> (((exists ff_h_pfp_gcd_backward_executiontrimsuffixsource. ff_h_pfp_gcd_backward_executiontrimsuffixsource + S (pftrim_value_gcd_backward_executiontrimsuffix) = S ((S ((pfd_cut_gcd_backward_execution)+pftrim_index_gcd_backward_executiontrimsuffix)) * pfd_residual_scale_gcd_backward_execution)) /\\ exists ff_q_pfp_gcd_backward_executiontrimsuffixsource. pfd_residual_code_gcd_backward_execution = ff_q_pfp_gcd_backward_executiontrimsuffixsource * S ((S ((pfd_cut_gcd_backward_execution)+pftrim_index_gcd_backward_executiontrimsuffix)) * pfd_residual_scale_gcd_backward_execution) + (pftrim_value_gcd_backward_executiontrimsuffix))) -> (((exists ff_h_pfp_gcd_backward_executiontrimsuffixoutput. ff_h_pfp_gcd_backward_executiontrimsuffixoutput + S (pftrim_value_gcd_backward_executiontrimsuffix) = S ((S (pftrim_index_gcd_backward_executiontrimsuffix)) * rc)) /\\ exists ff_q_pfp_gcd_backward_executiontrimsuffixoutput. rb = ff_q_pfp_gcd_backward_executiontrimsuffixoutput * S ((S (pftrim_index_gcd_backward_executiontrimsuffix)) * rc) + (pftrim_value_gcd_backward_executiontrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_gcd_backward_executiontrimnormal. ((((exists ff_h_pfp_gcd_backward_executiontrimnormalentry. ff_h_pfp_gcd_backward_executiontrimnormalentry + S (pftrim_leading_gcd_backward_executiontrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_gcd_backward_executiontrimnormalentry. rb = ff_q_pfp_gcd_backward_executiontrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_gcd_backward_executiontrimnormal))) /\\ ((~(pftrim_leading_gcd_backward_executiontrimnormal=0))))))))))))))))))))))))))))))))) -> (exists pfgb_pb_gcd_backward_local_old pfgb_pc_gcd_backward_local_old pfgb_P_gcd_backward_local_old pfgb_qb_gcd_backward_local_old pfgb_qc_gcd_backward_local_old pfgb_Q_gcd_backward_local_old. ((((forall fom_index_pfp_gcd_backward_local_old_leftleft. (exists fom_gap_pfp_gcd_backward_local_old_leftleft_index_bound. fom_gap_pfp_gcd_backward_local_old_leftleft_index_bound + S (fom_index_pfp_gcd_backward_local_old_leftleft) = U) -> exists fom_value_pfp_gcd_backward_local_old_leftleft. ((((exists fom_beta_height_pfp_gcd_backward_local_old_leftleft_entry. fom_beta_height_pfp_gcd_backward_local_old_leftleft_entry + S (fom_value_pfp_gcd_backward_local_old_leftleft) = S ((S (fom_index_pfp_gcd_backward_local_old_leftleft)) * uc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_old_leftleft_entry. ub = fom_beta_quotient_pfp_gcd_backward_local_old_leftleft_entry * S ((S (fom_index_pfp_gcd_backward_local_old_leftleft)) * uc) + (fom_value_pfp_gcd_backward_local_old_leftleft))) /\\ (exists fom_gap_pfp_gcd_backward_local_old_leftleft_value_bound. fom_gap_pfp_gcd_backward_local_old_leftleft_value_bound + S (fom_value_pfp_gcd_backward_local_old_leftleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_local_old_leftright. (exists fom_gap_pfp_gcd_backward_local_old_leftright_index_bound. fom_gap_pfp_gcd_backward_local_old_leftright_index_bound + S (fom_index_pfp_gcd_backward_local_old_leftright) = S d) -> exists fom_value_pfp_gcd_backward_local_old_leftright. ((((exists fom_beta_height_pfp_gcd_backward_local_old_leftright_entry. fom_beta_height_pfp_gcd_backward_local_old_leftright_entry + S (fom_value_pfp_gcd_backward_local_old_leftright) = S ((S (fom_index_pfp_gcd_backward_local_old_leftright)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_old_leftright_entry. bb = fom_beta_quotient_pfp_gcd_backward_local_old_leftright_entry * S ((S (fom_index_pfp_gcd_backward_local_old_leftright)) * bc) + (fom_value_pfp_gcd_backward_local_old_leftright))) /\\ (exists fom_gap_pfp_gcd_backward_local_old_leftright_value_bound. fom_gap_pfp_gcd_backward_local_old_leftright_value_bound + S (fom_value_pfp_gcd_backward_local_old_leftright) = p))) /\\ (((((((U)=0 \\/ (S d)=0) /\\ (((pfgb_P_gcd_backward_local_old)=0)))) \\/ (((~((U)=0)) /\\ (((~((S d)=0)) /\\ (((U)+(S d)=S (pfgb_P_gcd_backward_local_old)))))))) /\\ ((forall pfc_index_gcd_backward_local_old_leftcoefficients. (exists pfa_gap_gcd_backward_local_old_leftcoefficientsbound. pfa_gap_gcd_backward_local_old_leftcoefficientsbound + S (pfc_index_gcd_backward_local_old_leftcoefficients) = (pfgb_P_gcd_backward_local_old)) -> exists pfc_value_gcd_backward_local_old_leftcoefficients. ((((exists ff_h_pfp_gcd_backward_local_old_leftcoefficientsentry. ff_h_pfp_gcd_backward_local_old_leftcoefficientsentry + S (pfc_value_gcd_backward_local_old_leftcoefficients) = S ((S (pfc_index_gcd_backward_local_old_leftcoefficients)) * pfgb_pc_gcd_backward_local_old)) /\\ exists ff_q_pfp_gcd_backward_local_old_leftcoefficientsentry. pfgb_pb_gcd_backward_local_old = ff_q_pfp_gcd_backward_local_old_leftcoefficientsentry * S ((S (pfc_index_gcd_backward_local_old_leftcoefficients)) * pfgb_pc_gcd_backward_local_old) + (pfc_value_gcd_backward_local_old_leftcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_local_old_leftcoefficientscoefficient pfc_terms_scale_gcd_backward_local_old_leftcoefficientscoefficient pfc_natural_sum_gcd_backward_local_old_leftcoefficientscoefficient. ((forall pfc_index_gcd_backward_local_old_leftcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_local_old_leftcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_local_old_leftcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_local_old_leftcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_local_old_leftcoefficients))) -> exists pfc_value_gcd_backward_local_old_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_local_old_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_local_old_leftcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_local_old_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_local_old_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_local_old_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_local_old_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_local_old_leftcoefficientscoefficient = ff_q_pfp_gcd_backward_local_old_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_local_old_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_local_old_leftcoefficientscoefficient) + (pfc_value_gcd_backward_local_old_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_local_old_leftcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_local_old_leftcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_local_old_leftcoefficientscoefficientdiagonal) = (U)) /\\ ((((exists ff_h_pfp_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_local_old_leftcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_local_old_leftcoefficientscoefficientdiagonal)) * uc) + (pfc_left_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermleftoutside+(U)=(pfc_index_gcd_backward_local_old_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_local_old_leftcoefficientscoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_local_old_leftcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_local_old_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum fs_v_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_local_old_leftcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_local_old_leftcoefficients))) * fs_v_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_local_old_leftcoefficients))) * fs_v_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_local_old_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_local_old_leftcoefficients)) -> exists fs_a_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_local_old_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_local_old_leftcoefficientscoefficient = fs_q_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_local_old_leftcoefficientscoefficient) + (fs_a_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_local_old_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_local_old_leftcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_local_old_leftcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_local_old_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_local_old_leftcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_local_old_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_local_old_leftcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_local_old_leftcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_local_old_leftcoefficients) + (p) * pfa_offset_right_gcd_backward_local_old_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_gcd_backward_local_old_rightleft. (exists fom_gap_pfp_gcd_backward_local_old_rightleft_index_bound. fom_gap_pfp_gcd_backward_local_old_rightleft_index_bound + S (fom_index_pfp_gcd_backward_local_old_rightleft) = V) -> exists fom_value_pfp_gcd_backward_local_old_rightleft. ((((exists fom_beta_height_pfp_gcd_backward_local_old_rightleft_entry. fom_beta_height_pfp_gcd_backward_local_old_rightleft_entry + S (fom_value_pfp_gcd_backward_local_old_rightleft) = S ((S (fom_index_pfp_gcd_backward_local_old_rightleft)) * vc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_old_rightleft_entry. vb = fom_beta_quotient_pfp_gcd_backward_local_old_rightleft_entry * S ((S (fom_index_pfp_gcd_backward_local_old_rightleft)) * vc) + (fom_value_pfp_gcd_backward_local_old_rightleft))) /\\ (exists fom_gap_pfp_gcd_backward_local_old_rightleft_value_bound. fom_gap_pfp_gcd_backward_local_old_rightleft_value_bound + S (fom_value_pfp_gcd_backward_local_old_rightleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_local_old_rightright. (exists fom_gap_pfp_gcd_backward_local_old_rightright_index_bound. fom_gap_pfp_gcd_backward_local_old_rightright_index_bound + S (fom_index_pfp_gcd_backward_local_old_rightright) = R) -> exists fom_value_pfp_gcd_backward_local_old_rightright. ((((exists fom_beta_height_pfp_gcd_backward_local_old_rightright_entry. fom_beta_height_pfp_gcd_backward_local_old_rightright_entry + S (fom_value_pfp_gcd_backward_local_old_rightright) = S ((S (fom_index_pfp_gcd_backward_local_old_rightright)) * rc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_old_rightright_entry. rb = fom_beta_quotient_pfp_gcd_backward_local_old_rightright_entry * S ((S (fom_index_pfp_gcd_backward_local_old_rightright)) * rc) + (fom_value_pfp_gcd_backward_local_old_rightright))) /\\ (exists fom_gap_pfp_gcd_backward_local_old_rightright_value_bound. fom_gap_pfp_gcd_backward_local_old_rightright_value_bound + S (fom_value_pfp_gcd_backward_local_old_rightright) = p))) /\\ (((((((V)=0 \\/ (R)=0) /\\ (((pfgb_Q_gcd_backward_local_old)=0)))) \\/ (((~((V)=0)) /\\ (((~((R)=0)) /\\ (((V)+(R)=S (pfgb_Q_gcd_backward_local_old)))))))) /\\ ((forall pfc_index_gcd_backward_local_old_rightcoefficients. (exists pfa_gap_gcd_backward_local_old_rightcoefficientsbound. pfa_gap_gcd_backward_local_old_rightcoefficientsbound + S (pfc_index_gcd_backward_local_old_rightcoefficients) = (pfgb_Q_gcd_backward_local_old)) -> exists pfc_value_gcd_backward_local_old_rightcoefficients. ((((exists ff_h_pfp_gcd_backward_local_old_rightcoefficientsentry. ff_h_pfp_gcd_backward_local_old_rightcoefficientsentry + S (pfc_value_gcd_backward_local_old_rightcoefficients) = S ((S (pfc_index_gcd_backward_local_old_rightcoefficients)) * pfgb_qc_gcd_backward_local_old)) /\\ exists ff_q_pfp_gcd_backward_local_old_rightcoefficientsentry. pfgb_qb_gcd_backward_local_old = ff_q_pfp_gcd_backward_local_old_rightcoefficientsentry * S ((S (pfc_index_gcd_backward_local_old_rightcoefficients)) * pfgb_qc_gcd_backward_local_old) + (pfc_value_gcd_backward_local_old_rightcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_local_old_rightcoefficientscoefficient pfc_terms_scale_gcd_backward_local_old_rightcoefficientscoefficient pfc_natural_sum_gcd_backward_local_old_rightcoefficientscoefficient. ((forall pfc_index_gcd_backward_local_old_rightcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_local_old_rightcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_local_old_rightcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_local_old_rightcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_local_old_rightcoefficients))) -> exists pfc_value_gcd_backward_local_old_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_local_old_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_local_old_rightcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_local_old_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_local_old_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_local_old_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_local_old_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_local_old_rightcoefficientscoefficient = ff_q_pfp_gcd_backward_local_old_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_local_old_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_local_old_rightcoefficientscoefficient) + (pfc_value_gcd_backward_local_old_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_local_old_rightcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_local_old_rightcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_local_old_rightcoefficientscoefficientdiagonal) = (V)) /\\ ((((exists ff_h_pfp_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_local_old_rightcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_local_old_rightcoefficientscoefficientdiagonal)) * vc) + (pfc_left_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermleftoutside+(V)=(pfc_index_gcd_backward_local_old_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm) = (R)) /\\ ((((exists ff_h_pfp_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm)) * rc)) /\\ exists ff_q_pfp_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermrightentry. rb = ff_q_pfp_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm)) * rc) + (pfc_right_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_local_old_rightcoefficientscoefficientdiagonaltermrightoutside+(R)=(pfc_complement_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_local_old_rightcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_local_old_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum fs_v_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_local_old_rightcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_local_old_rightcoefficients))) * fs_v_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_local_old_rightcoefficients))) * fs_v_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_local_old_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_local_old_rightcoefficients)) -> exists fs_a_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_local_old_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_local_old_rightcoefficientscoefficient = fs_q_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_local_old_rightcoefficientscoefficient) + (fs_a_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_local_old_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_local_old_rightcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_local_old_rightcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_local_old_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_local_old_rightcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_local_old_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_local_old_rightcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_local_old_rightcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_local_old_rightcoefficients) + (p) * pfa_offset_right_gcd_backward_local_old_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_gcd_backward_local_old_sum_left_bounded. (exists fom_gap_pfp_gcd_backward_local_old_sum_left_bounded_index_bound. fom_gap_pfp_gcd_backward_local_old_sum_left_bounded_index_bound + S (fom_index_pfp_gcd_backward_local_old_sum_left_bounded) = pfgb_P_gcd_backward_local_old) -> exists fom_value_pfp_gcd_backward_local_old_sum_left_bounded. ((((exists fom_beta_height_pfp_gcd_backward_local_old_sum_left_bounded_entry. fom_beta_height_pfp_gcd_backward_local_old_sum_left_bounded_entry + S (fom_value_pfp_gcd_backward_local_old_sum_left_bounded) = S ((S (fom_index_pfp_gcd_backward_local_old_sum_left_bounded)) * pfgb_pc_gcd_backward_local_old)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_old_sum_left_bounded_entry. pfgb_pb_gcd_backward_local_old = fom_beta_quotient_pfp_gcd_backward_local_old_sum_left_bounded_entry * S ((S (fom_index_pfp_gcd_backward_local_old_sum_left_bounded)) * pfgb_pc_gcd_backward_local_old) + (fom_value_pfp_gcd_backward_local_old_sum_left_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_local_old_sum_left_bounded_value_bound. fom_gap_pfp_gcd_backward_local_old_sum_left_bounded_value_bound + S (fom_value_pfp_gcd_backward_local_old_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_backward_local_old_sum_right_bounded. (exists fom_gap_pfp_gcd_backward_local_old_sum_right_bounded_index_bound. fom_gap_pfp_gcd_backward_local_old_sum_right_bounded_index_bound + S (fom_index_pfp_gcd_backward_local_old_sum_right_bounded) = pfgb_Q_gcd_backward_local_old) -> exists fom_value_pfp_gcd_backward_local_old_sum_right_bounded. ((((exists fom_beta_height_pfp_gcd_backward_local_old_sum_right_bounded_entry. fom_beta_height_pfp_gcd_backward_local_old_sum_right_bounded_entry + S (fom_value_pfp_gcd_backward_local_old_sum_right_bounded) = S ((S (fom_index_pfp_gcd_backward_local_old_sum_right_bounded)) * pfgb_qc_gcd_backward_local_old)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_old_sum_right_bounded_entry. pfgb_qb_gcd_backward_local_old = fom_beta_quotient_pfp_gcd_backward_local_old_sum_right_bounded_entry * S ((S (fom_index_pfp_gcd_backward_local_old_sum_right_bounded)) * pfgb_qc_gcd_backward_local_old) + (fom_value_pfp_gcd_backward_local_old_sum_right_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_local_old_sum_right_bounded_value_bound. fom_gap_pfp_gcd_backward_local_old_sum_right_bounded_value_bound + S (fom_value_pfp_gcd_backward_local_old_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_backward_local_old_sum_result_bounded. (exists fom_gap_pfp_gcd_backward_local_old_sum_result_bounded_index_bound. fom_gap_pfp_gcd_backward_local_old_sum_result_bounded_index_bound + S (fom_index_pfp_gcd_backward_local_old_sum_result_bounded) = G) -> exists fom_value_pfp_gcd_backward_local_old_sum_result_bounded. ((((exists fom_beta_height_pfp_gcd_backward_local_old_sum_result_bounded_entry. fom_beta_height_pfp_gcd_backward_local_old_sum_result_bounded_entry + S (fom_value_pfp_gcd_backward_local_old_sum_result_bounded) = S ((S (fom_index_pfp_gcd_backward_local_old_sum_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_old_sum_result_bounded_entry. gb = fom_beta_quotient_pfp_gcd_backward_local_old_sum_result_bounded_entry * S ((S (fom_index_pfp_gcd_backward_local_old_sum_result_bounded)) * gc) + (fom_value_pfp_gcd_backward_local_old_sum_result_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_local_old_sum_result_bounded_value_bound. fom_gap_pfp_gcd_backward_local_old_sum_result_bounded_value_bound + S (fom_value_pfp_gcd_backward_local_old_sum_result_bounded) = p))) /\\ ((exists pfga_ub_gcd_backward_local_old_sum pfga_uc_gcd_backward_local_old_sum pfga_vb_gcd_backward_local_old_sum pfga_vc_gcd_backward_local_old_sum pfga_tb_gcd_backward_local_old_sum pfga_tc_gcd_backward_local_old_sum pfga_K_gcd_backward_local_old_sum. ((((forall pfrep_power_gcd_backward_local_old_sum_left pfrep_left_gcd_backward_local_old_sum_left pfrep_right_gcd_backward_local_old_sum_left. ((exists pfrep_position_gcd_backward_local_old_sum_leftfirst. ((pfrep_position_gcd_backward_local_old_sum_leftfirst+S (pfrep_power_gcd_backward_local_old_sum_left)=(pfgb_P_gcd_backward_local_old)) /\\ ((((exists ff_h_pfp_gcd_backward_local_old_sum_leftfirstentry. ff_h_pfp_gcd_backward_local_old_sum_leftfirstentry + S (pfrep_left_gcd_backward_local_old_sum_left) = S ((S (pfrep_position_gcd_backward_local_old_sum_leftfirst)) * pfgb_pc_gcd_backward_local_old)) /\\ exists ff_q_pfp_gcd_backward_local_old_sum_leftfirstentry. pfgb_pb_gcd_backward_local_old = ff_q_pfp_gcd_backward_local_old_sum_leftfirstentry * S ((S (pfrep_position_gcd_backward_local_old_sum_leftfirst)) * pfgb_pc_gcd_backward_local_old) + (pfrep_left_gcd_backward_local_old_sum_left)))))) \\/ (((exists pfrep_gap_gcd_backward_local_old_sum_leftfirstoutside. pfrep_gap_gcd_backward_local_old_sum_leftfirstoutside+(pfgb_P_gcd_backward_local_old)=(pfrep_power_gcd_backward_local_old_sum_left)) /\\ (((pfrep_left_gcd_backward_local_old_sum_left)=0))))) -> ((exists pfrep_position_gcd_backward_local_old_sum_leftsecond. ((pfrep_position_gcd_backward_local_old_sum_leftsecond+S (pfrep_power_gcd_backward_local_old_sum_left)=(pfga_K_gcd_backward_local_old_sum)) /\\ ((((exists ff_h_pfp_gcd_backward_local_old_sum_leftsecondentry. ff_h_pfp_gcd_backward_local_old_sum_leftsecondentry + S (pfrep_right_gcd_backward_local_old_sum_left) = S ((S (pfrep_position_gcd_backward_local_old_sum_leftsecond)) * pfga_uc_gcd_backward_local_old_sum)) /\\ exists ff_q_pfp_gcd_backward_local_old_sum_leftsecondentry. pfga_ub_gcd_backward_local_old_sum = ff_q_pfp_gcd_backward_local_old_sum_leftsecondentry * S ((S (pfrep_position_gcd_backward_local_old_sum_leftsecond)) * pfga_uc_gcd_backward_local_old_sum) + (pfrep_right_gcd_backward_local_old_sum_left)))))) \\/ (((exists pfrep_gap_gcd_backward_local_old_sum_leftsecondoutside. pfrep_gap_gcd_backward_local_old_sum_leftsecondoutside+(pfga_K_gcd_backward_local_old_sum)=(pfrep_power_gcd_backward_local_old_sum_left)) /\\ (((pfrep_right_gcd_backward_local_old_sum_left)=0))))) -> pfrep_left_gcd_backward_local_old_sum_left=pfrep_right_gcd_backward_local_old_sum_left) /\\ ((forall pfrep_power_gcd_backward_local_old_sum_right pfrep_left_gcd_backward_local_old_sum_right pfrep_right_gcd_backward_local_old_sum_right. ((exists pfrep_position_gcd_backward_local_old_sum_rightfirst. ((pfrep_position_gcd_backward_local_old_sum_rightfirst+S (pfrep_power_gcd_backward_local_old_sum_right)=(pfgb_Q_gcd_backward_local_old)) /\\ ((((exists ff_h_pfp_gcd_backward_local_old_sum_rightfirstentry. ff_h_pfp_gcd_backward_local_old_sum_rightfirstentry + S (pfrep_left_gcd_backward_local_old_sum_right) = S ((S (pfrep_position_gcd_backward_local_old_sum_rightfirst)) * pfgb_qc_gcd_backward_local_old)) /\\ exists ff_q_pfp_gcd_backward_local_old_sum_rightfirstentry. pfgb_qb_gcd_backward_local_old = ff_q_pfp_gcd_backward_local_old_sum_rightfirstentry * S ((S (pfrep_position_gcd_backward_local_old_sum_rightfirst)) * pfgb_qc_gcd_backward_local_old) + (pfrep_left_gcd_backward_local_old_sum_right)))))) \\/ (((exists pfrep_gap_gcd_backward_local_old_sum_rightfirstoutside. pfrep_gap_gcd_backward_local_old_sum_rightfirstoutside+(pfgb_Q_gcd_backward_local_old)=(pfrep_power_gcd_backward_local_old_sum_right)) /\\ (((pfrep_left_gcd_backward_local_old_sum_right)=0))))) -> ((exists pfrep_position_gcd_backward_local_old_sum_rightsecond. ((pfrep_position_gcd_backward_local_old_sum_rightsecond+S (pfrep_power_gcd_backward_local_old_sum_right)=(pfga_K_gcd_backward_local_old_sum)) /\\ ((((exists ff_h_pfp_gcd_backward_local_old_sum_rightsecondentry. ff_h_pfp_gcd_backward_local_old_sum_rightsecondentry + S (pfrep_right_gcd_backward_local_old_sum_right) = S ((S (pfrep_position_gcd_backward_local_old_sum_rightsecond)) * pfga_vc_gcd_backward_local_old_sum)) /\\ exists ff_q_pfp_gcd_backward_local_old_sum_rightsecondentry. pfga_vb_gcd_backward_local_old_sum = ff_q_pfp_gcd_backward_local_old_sum_rightsecondentry * S ((S (pfrep_position_gcd_backward_local_old_sum_rightsecond)) * pfga_vc_gcd_backward_local_old_sum) + (pfrep_right_gcd_backward_local_old_sum_right)))))) \\/ (((exists pfrep_gap_gcd_backward_local_old_sum_rightsecondoutside. pfrep_gap_gcd_backward_local_old_sum_rightsecondoutside+(pfga_K_gcd_backward_local_old_sum)=(pfrep_power_gcd_backward_local_old_sum_right)) /\\ (((pfrep_right_gcd_backward_local_old_sum_right)=0))))) -> pfrep_left_gcd_backward_local_old_sum_right=pfrep_right_gcd_backward_local_old_sum_right)))) /\\ (((forall pfp_index_gcd_backward_local_old_sum_add. (exists pfa_gap_gcd_backward_local_old_sum_addindex. pfa_gap_gcd_backward_local_old_sum_addindex + S (pfp_index_gcd_backward_local_old_sum_add) = (pfga_K_gcd_backward_local_old_sum)) -> exists pfp_left_gcd_backward_local_old_sum_add pfp_right_gcd_backward_local_old_sum_add pfp_value_gcd_backward_local_old_sum_add. ((((exists ff_h_pfp_gcd_backward_local_old_sum_addleft. ff_h_pfp_gcd_backward_local_old_sum_addleft + S (pfp_left_gcd_backward_local_old_sum_add) = S ((S (pfp_index_gcd_backward_local_old_sum_add)) * pfga_uc_gcd_backward_local_old_sum)) /\\ exists ff_q_pfp_gcd_backward_local_old_sum_addleft. pfga_ub_gcd_backward_local_old_sum = ff_q_pfp_gcd_backward_local_old_sum_addleft * S ((S (pfp_index_gcd_backward_local_old_sum_add)) * pfga_uc_gcd_backward_local_old_sum) + (pfp_left_gcd_backward_local_old_sum_add))) /\\ (((((exists ff_h_pfp_gcd_backward_local_old_sum_addright. ff_h_pfp_gcd_backward_local_old_sum_addright + S (pfp_right_gcd_backward_local_old_sum_add) = S ((S (pfp_index_gcd_backward_local_old_sum_add)) * pfga_vc_gcd_backward_local_old_sum)) /\\ exists ff_q_pfp_gcd_backward_local_old_sum_addright. pfga_vb_gcd_backward_local_old_sum = ff_q_pfp_gcd_backward_local_old_sum_addright * S ((S (pfp_index_gcd_backward_local_old_sum_add)) * pfga_vc_gcd_backward_local_old_sum) + (pfp_right_gcd_backward_local_old_sum_add))) /\\ (((((exists ff_h_pfp_gcd_backward_local_old_sum_addtarget. ff_h_pfp_gcd_backward_local_old_sum_addtarget + S (pfp_value_gcd_backward_local_old_sum_add) = S ((S (pfp_index_gcd_backward_local_old_sum_add)) * pfga_tc_gcd_backward_local_old_sum)) /\\ exists ff_q_pfp_gcd_backward_local_old_sum_addtarget. pfga_tb_gcd_backward_local_old_sum = ff_q_pfp_gcd_backward_local_old_sum_addtarget * S ((S (pfp_index_gcd_backward_local_old_sum_add)) * pfga_tc_gcd_backward_local_old_sum) + (pfp_value_gcd_backward_local_old_sum_add))) /\\ ((((exists pfa_gap_gcd_backward_local_old_sum_addoperationleft. pfa_gap_gcd_backward_local_old_sum_addoperationleft + S (pfp_left_gcd_backward_local_old_sum_add) = (p)) /\\ (((exists pfa_gap_gcd_backward_local_old_sum_addoperationright. pfa_gap_gcd_backward_local_old_sum_addoperationright + S (pfp_right_gcd_backward_local_old_sum_add) = (p)) /\\ ((((exists pfa_gap_gcd_backward_local_old_sum_addoperationresultbound. pfa_gap_gcd_backward_local_old_sum_addoperationresultbound + S (pfp_value_gcd_backward_local_old_sum_add) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_local_old_sum_addoperationresultcongruence pfa_offset_right_gcd_backward_local_old_sum_addoperationresultcongruence. ((pfp_left_gcd_backward_local_old_sum_add) + (pfp_right_gcd_backward_local_old_sum_add)) + (p) * pfa_offset_left_gcd_backward_local_old_sum_addoperationresultcongruence = (pfp_value_gcd_backward_local_old_sum_add) + (p) * pfa_offset_right_gcd_backward_local_old_sum_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_local_old_sum_result pfrep_left_gcd_backward_local_old_sum_result pfrep_right_gcd_backward_local_old_sum_result. ((exists pfrep_position_gcd_backward_local_old_sum_resultfirst. ((pfrep_position_gcd_backward_local_old_sum_resultfirst+S (pfrep_power_gcd_backward_local_old_sum_result)=(pfga_K_gcd_backward_local_old_sum)) /\\ ((((exists ff_h_pfp_gcd_backward_local_old_sum_resultfirstentry. ff_h_pfp_gcd_backward_local_old_sum_resultfirstentry + S (pfrep_left_gcd_backward_local_old_sum_result) = S ((S (pfrep_position_gcd_backward_local_old_sum_resultfirst)) * pfga_tc_gcd_backward_local_old_sum)) /\\ exists ff_q_pfp_gcd_backward_local_old_sum_resultfirstentry. pfga_tb_gcd_backward_local_old_sum = ff_q_pfp_gcd_backward_local_old_sum_resultfirstentry * S ((S (pfrep_position_gcd_backward_local_old_sum_resultfirst)) * pfga_tc_gcd_backward_local_old_sum) + (pfrep_left_gcd_backward_local_old_sum_result)))))) \\/ (((exists pfrep_gap_gcd_backward_local_old_sum_resultfirstoutside. pfrep_gap_gcd_backward_local_old_sum_resultfirstoutside+(pfga_K_gcd_backward_local_old_sum)=(pfrep_power_gcd_backward_local_old_sum_result)) /\\ (((pfrep_left_gcd_backward_local_old_sum_result)=0))))) -> ((exists pfrep_position_gcd_backward_local_old_sum_resultsecond. ((pfrep_position_gcd_backward_local_old_sum_resultsecond+S (pfrep_power_gcd_backward_local_old_sum_result)=(G)) /\\ ((((exists ff_h_pfp_gcd_backward_local_old_sum_resultsecondentry. ff_h_pfp_gcd_backward_local_old_sum_resultsecondentry + S (pfrep_right_gcd_backward_local_old_sum_result) = S ((S (pfrep_position_gcd_backward_local_old_sum_resultsecond)) * gc)) /\\ exists ff_q_pfp_gcd_backward_local_old_sum_resultsecondentry. gb = ff_q_pfp_gcd_backward_local_old_sum_resultsecondentry * S ((S (pfrep_position_gcd_backward_local_old_sum_resultsecond)) * gc) + (pfrep_right_gcd_backward_local_old_sum_result)))))) \\/ (((exists pfrep_gap_gcd_backward_local_old_sum_resultsecondoutside. pfrep_gap_gcd_backward_local_old_sum_resultsecondoutside+(G)=(pfrep_power_gcd_backward_local_old_sum_result)) /\\ (((pfrep_right_gcd_backward_local_old_sum_result)=0))))) -> pfrep_left_gcd_backward_local_old_sum_result=pfrep_right_gcd_backward_local_old_sum_result)))))))))))))))))) -> (exists pfg_update_wb_gcd_backward_local_result pfg_update_wc_gcd_backward_local_result pfg_update_W_gcd_backward_local_result pfg_update_tb_gcd_backward_local_result pfg_update_tc_gcd_backward_local_result pfg_update_T_gcd_backward_local_result. ((((forall fom_index_pfp_gcd_backward_local_result_productleft. (exists fom_gap_pfp_gcd_backward_local_result_productleft_index_bound. fom_gap_pfp_gcd_backward_local_result_productleft_index_bound + S (fom_index_pfp_gcd_backward_local_result_productleft) = V) -> exists fom_value_pfp_gcd_backward_local_result_productleft. ((((exists fom_beta_height_pfp_gcd_backward_local_result_productleft_entry. fom_beta_height_pfp_gcd_backward_local_result_productleft_entry + S (fom_value_pfp_gcd_backward_local_result_productleft) = S ((S (fom_index_pfp_gcd_backward_local_result_productleft)) * vc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_result_productleft_entry. vb = fom_beta_quotient_pfp_gcd_backward_local_result_productleft_entry * S ((S (fom_index_pfp_gcd_backward_local_result_productleft)) * vc) + (fom_value_pfp_gcd_backward_local_result_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_local_result_productleft_value_bound. fom_gap_pfp_gcd_backward_local_result_productleft_value_bound + S (fom_value_pfp_gcd_backward_local_result_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_local_result_productright. (exists fom_gap_pfp_gcd_backward_local_result_productright_index_bound. fom_gap_pfp_gcd_backward_local_result_productright_index_bound + S (fom_index_pfp_gcd_backward_local_result_productright) = q) -> exists fom_value_pfp_gcd_backward_local_result_productright. ((((exists fom_beta_height_pfp_gcd_backward_local_result_productright_entry. fom_beta_height_pfp_gcd_backward_local_result_productright_entry + S (fom_value_pfp_gcd_backward_local_result_productright) = S ((S (fom_index_pfp_gcd_backward_local_result_productright)) * qc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_result_productright_entry. qb = fom_beta_quotient_pfp_gcd_backward_local_result_productright_entry * S ((S (fom_index_pfp_gcd_backward_local_result_productright)) * qc) + (fom_value_pfp_gcd_backward_local_result_productright))) /\\ (exists fom_gap_pfp_gcd_backward_local_result_productright_value_bound. fom_gap_pfp_gcd_backward_local_result_productright_value_bound + S (fom_value_pfp_gcd_backward_local_result_productright) = p))) /\\ (((((((V)=0 \\/ (q)=0) /\\ (((pfg_update_W_gcd_backward_local_result)=0)))) \\/ (((~((V)=0)) /\\ (((~((q)=0)) /\\ (((V)+(q)=S (pfg_update_W_gcd_backward_local_result)))))))) /\\ ((forall pfc_index_gcd_backward_local_result_productcoefficients. (exists pfa_gap_gcd_backward_local_result_productcoefficientsbound. pfa_gap_gcd_backward_local_result_productcoefficientsbound + S (pfc_index_gcd_backward_local_result_productcoefficients) = (pfg_update_W_gcd_backward_local_result)) -> exists pfc_value_gcd_backward_local_result_productcoefficients. ((((exists ff_h_pfp_gcd_backward_local_result_productcoefficientsentry. ff_h_pfp_gcd_backward_local_result_productcoefficientsentry + S (pfc_value_gcd_backward_local_result_productcoefficients) = S ((S (pfc_index_gcd_backward_local_result_productcoefficients)) * pfg_update_wc_gcd_backward_local_result)) /\\ exists ff_q_pfp_gcd_backward_local_result_productcoefficientsentry. pfg_update_wb_gcd_backward_local_result = ff_q_pfp_gcd_backward_local_result_productcoefficientsentry * S ((S (pfc_index_gcd_backward_local_result_productcoefficients)) * pfg_update_wc_gcd_backward_local_result) + (pfc_value_gcd_backward_local_result_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_local_result_productcoefficientscoefficient pfc_terms_scale_gcd_backward_local_result_productcoefficientscoefficient pfc_natural_sum_gcd_backward_local_result_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_local_result_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_local_result_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_local_result_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_local_result_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_local_result_productcoefficients))) -> exists pfc_value_gcd_backward_local_result_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_local_result_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_local_result_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_local_result_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_local_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_local_result_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_local_result_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_local_result_productcoefficientscoefficient = ff_q_pfp_gcd_backward_local_result_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_local_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_local_result_productcoefficientscoefficient) + (pfc_value_gcd_backward_local_result_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_local_result_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_local_result_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_local_result_productcoefficientscoefficientdiagonal) = (V)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_local_result_productcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_local_result_productcoefficientscoefficientdiagonal)) * vc) + (pfc_left_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermleftoutside+(V)=(pfc_index_gcd_backward_local_result_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm) = (q)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm)) * qc)) /\\ exists ff_q_pfp_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermrightentry. qb = ff_q_pfp_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm)) * qc) + (pfc_right_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_local_result_productcoefficientscoefficientdiagonaltermrightoutside+(q)=(pfc_complement_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_local_result_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_local_result_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_local_result_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_local_result_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_local_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_local_result_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_local_result_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_local_result_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_local_result_productcoefficients))) * fs_v_pfc_gcd_backward_local_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_local_result_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_local_result_productcoefficients))) * fs_v_pfc_gcd_backward_local_result_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_local_result_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_local_result_productcoefficients)) -> exists fs_a_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_local_result_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_local_result_productcoefficientscoefficient = fs_q_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_local_result_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_local_result_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_result_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_local_result_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_result_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_local_result_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_local_result_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_local_result_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_local_result_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_local_result_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_local_result_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_local_result_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_local_result_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_local_result_productcoefficients) + (p) * pfa_offset_right_gcd_backward_local_result_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_gcd_backward_local_result_difference_left_bounded. (exists fom_gap_pfp_gcd_backward_local_result_difference_left_bounded_index_bound. fom_gap_pfp_gcd_backward_local_result_difference_left_bounded_index_bound + S (fom_index_pfp_gcd_backward_local_result_difference_left_bounded) = pfg_update_W_gcd_backward_local_result) -> exists fom_value_pfp_gcd_backward_local_result_difference_left_bounded. ((((exists fom_beta_height_pfp_gcd_backward_local_result_difference_left_bounded_entry. fom_beta_height_pfp_gcd_backward_local_result_difference_left_bounded_entry + S (fom_value_pfp_gcd_backward_local_result_difference_left_bounded) = S ((S (fom_index_pfp_gcd_backward_local_result_difference_left_bounded)) * pfg_update_wc_gcd_backward_local_result)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_result_difference_left_bounded_entry. pfg_update_wb_gcd_backward_local_result = fom_beta_quotient_pfp_gcd_backward_local_result_difference_left_bounded_entry * S ((S (fom_index_pfp_gcd_backward_local_result_difference_left_bounded)) * pfg_update_wc_gcd_backward_local_result) + (fom_value_pfp_gcd_backward_local_result_difference_left_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_local_result_difference_left_bounded_value_bound. fom_gap_pfp_gcd_backward_local_result_difference_left_bounded_value_bound + S (fom_value_pfp_gcd_backward_local_result_difference_left_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_backward_local_result_difference_right_bounded. (exists fom_gap_pfp_gcd_backward_local_result_difference_right_bounded_index_bound. fom_gap_pfp_gcd_backward_local_result_difference_right_bounded_index_bound + S (fom_index_pfp_gcd_backward_local_result_difference_right_bounded) = pfg_update_T_gcd_backward_local_result) -> exists fom_value_pfp_gcd_backward_local_result_difference_right_bounded. ((((exists fom_beta_height_pfp_gcd_backward_local_result_difference_right_bounded_entry. fom_beta_height_pfp_gcd_backward_local_result_difference_right_bounded_entry + S (fom_value_pfp_gcd_backward_local_result_difference_right_bounded) = S ((S (fom_index_pfp_gcd_backward_local_result_difference_right_bounded)) * pfg_update_tc_gcd_backward_local_result)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_result_difference_right_bounded_entry. pfg_update_tb_gcd_backward_local_result = fom_beta_quotient_pfp_gcd_backward_local_result_difference_right_bounded_entry * S ((S (fom_index_pfp_gcd_backward_local_result_difference_right_bounded)) * pfg_update_tc_gcd_backward_local_result) + (fom_value_pfp_gcd_backward_local_result_difference_right_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_local_result_difference_right_bounded_value_bound. fom_gap_pfp_gcd_backward_local_result_difference_right_bounded_value_bound + S (fom_value_pfp_gcd_backward_local_result_difference_right_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_backward_local_result_difference_result_bounded. (exists fom_gap_pfp_gcd_backward_local_result_difference_result_bounded_index_bound. fom_gap_pfp_gcd_backward_local_result_difference_result_bounded_index_bound + S (fom_index_pfp_gcd_backward_local_result_difference_result_bounded) = U) -> exists fom_value_pfp_gcd_backward_local_result_difference_result_bounded. ((((exists fom_beta_height_pfp_gcd_backward_local_result_difference_result_bounded_entry. fom_beta_height_pfp_gcd_backward_local_result_difference_result_bounded_entry + S (fom_value_pfp_gcd_backward_local_result_difference_result_bounded) = S ((S (fom_index_pfp_gcd_backward_local_result_difference_result_bounded)) * uc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_result_difference_result_bounded_entry. ub = fom_beta_quotient_pfp_gcd_backward_local_result_difference_result_bounded_entry * S ((S (fom_index_pfp_gcd_backward_local_result_difference_result_bounded)) * uc) + (fom_value_pfp_gcd_backward_local_result_difference_result_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_local_result_difference_result_bounded_value_bound. fom_gap_pfp_gcd_backward_local_result_difference_result_bounded_value_bound + S (fom_value_pfp_gcd_backward_local_result_difference_result_bounded) = p))) /\\ ((exists pfga_ub_gcd_backward_local_result_difference pfga_uc_gcd_backward_local_result_difference pfga_vb_gcd_backward_local_result_difference pfga_vc_gcd_backward_local_result_difference pfga_tb_gcd_backward_local_result_difference pfga_tc_gcd_backward_local_result_difference pfga_K_gcd_backward_local_result_difference. ((((forall pfrep_power_gcd_backward_local_result_difference_left pfrep_left_gcd_backward_local_result_difference_left pfrep_right_gcd_backward_local_result_difference_left. ((exists pfrep_position_gcd_backward_local_result_difference_leftfirst. ((pfrep_position_gcd_backward_local_result_difference_leftfirst+S (pfrep_power_gcd_backward_local_result_difference_left)=(pfg_update_W_gcd_backward_local_result)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_difference_leftfirstentry. ff_h_pfp_gcd_backward_local_result_difference_leftfirstentry + S (pfrep_left_gcd_backward_local_result_difference_left) = S ((S (pfrep_position_gcd_backward_local_result_difference_leftfirst)) * pfg_update_wc_gcd_backward_local_result)) /\\ exists ff_q_pfp_gcd_backward_local_result_difference_leftfirstentry. pfg_update_wb_gcd_backward_local_result = ff_q_pfp_gcd_backward_local_result_difference_leftfirstentry * S ((S (pfrep_position_gcd_backward_local_result_difference_leftfirst)) * pfg_update_wc_gcd_backward_local_result) + (pfrep_left_gcd_backward_local_result_difference_left)))))) \\/ (((exists pfrep_gap_gcd_backward_local_result_difference_leftfirstoutside. pfrep_gap_gcd_backward_local_result_difference_leftfirstoutside+(pfg_update_W_gcd_backward_local_result)=(pfrep_power_gcd_backward_local_result_difference_left)) /\\ (((pfrep_left_gcd_backward_local_result_difference_left)=0))))) -> ((exists pfrep_position_gcd_backward_local_result_difference_leftsecond. ((pfrep_position_gcd_backward_local_result_difference_leftsecond+S (pfrep_power_gcd_backward_local_result_difference_left)=(pfga_K_gcd_backward_local_result_difference)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_difference_leftsecondentry. ff_h_pfp_gcd_backward_local_result_difference_leftsecondentry + S (pfrep_right_gcd_backward_local_result_difference_left) = S ((S (pfrep_position_gcd_backward_local_result_difference_leftsecond)) * pfga_uc_gcd_backward_local_result_difference)) /\\ exists ff_q_pfp_gcd_backward_local_result_difference_leftsecondentry. pfga_ub_gcd_backward_local_result_difference = ff_q_pfp_gcd_backward_local_result_difference_leftsecondentry * S ((S (pfrep_position_gcd_backward_local_result_difference_leftsecond)) * pfga_uc_gcd_backward_local_result_difference) + (pfrep_right_gcd_backward_local_result_difference_left)))))) \\/ (((exists pfrep_gap_gcd_backward_local_result_difference_leftsecondoutside. pfrep_gap_gcd_backward_local_result_difference_leftsecondoutside+(pfga_K_gcd_backward_local_result_difference)=(pfrep_power_gcd_backward_local_result_difference_left)) /\\ (((pfrep_right_gcd_backward_local_result_difference_left)=0))))) -> pfrep_left_gcd_backward_local_result_difference_left=pfrep_right_gcd_backward_local_result_difference_left) /\\ ((forall pfrep_power_gcd_backward_local_result_difference_right pfrep_left_gcd_backward_local_result_difference_right pfrep_right_gcd_backward_local_result_difference_right. ((exists pfrep_position_gcd_backward_local_result_difference_rightfirst. ((pfrep_position_gcd_backward_local_result_difference_rightfirst+S (pfrep_power_gcd_backward_local_result_difference_right)=(pfg_update_T_gcd_backward_local_result)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_difference_rightfirstentry. ff_h_pfp_gcd_backward_local_result_difference_rightfirstentry + S (pfrep_left_gcd_backward_local_result_difference_right) = S ((S (pfrep_position_gcd_backward_local_result_difference_rightfirst)) * pfg_update_tc_gcd_backward_local_result)) /\\ exists ff_q_pfp_gcd_backward_local_result_difference_rightfirstentry. pfg_update_tb_gcd_backward_local_result = ff_q_pfp_gcd_backward_local_result_difference_rightfirstentry * S ((S (pfrep_position_gcd_backward_local_result_difference_rightfirst)) * pfg_update_tc_gcd_backward_local_result) + (pfrep_left_gcd_backward_local_result_difference_right)))))) \\/ (((exists pfrep_gap_gcd_backward_local_result_difference_rightfirstoutside. pfrep_gap_gcd_backward_local_result_difference_rightfirstoutside+(pfg_update_T_gcd_backward_local_result)=(pfrep_power_gcd_backward_local_result_difference_right)) /\\ (((pfrep_left_gcd_backward_local_result_difference_right)=0))))) -> ((exists pfrep_position_gcd_backward_local_result_difference_rightsecond. ((pfrep_position_gcd_backward_local_result_difference_rightsecond+S (pfrep_power_gcd_backward_local_result_difference_right)=(pfga_K_gcd_backward_local_result_difference)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_difference_rightsecondentry. ff_h_pfp_gcd_backward_local_result_difference_rightsecondentry + S (pfrep_right_gcd_backward_local_result_difference_right) = S ((S (pfrep_position_gcd_backward_local_result_difference_rightsecond)) * pfga_vc_gcd_backward_local_result_difference)) /\\ exists ff_q_pfp_gcd_backward_local_result_difference_rightsecondentry. pfga_vb_gcd_backward_local_result_difference = ff_q_pfp_gcd_backward_local_result_difference_rightsecondentry * S ((S (pfrep_position_gcd_backward_local_result_difference_rightsecond)) * pfga_vc_gcd_backward_local_result_difference) + (pfrep_right_gcd_backward_local_result_difference_right)))))) \\/ (((exists pfrep_gap_gcd_backward_local_result_difference_rightsecondoutside. pfrep_gap_gcd_backward_local_result_difference_rightsecondoutside+(pfga_K_gcd_backward_local_result_difference)=(pfrep_power_gcd_backward_local_result_difference_right)) /\\ (((pfrep_right_gcd_backward_local_result_difference_right)=0))))) -> pfrep_left_gcd_backward_local_result_difference_right=pfrep_right_gcd_backward_local_result_difference_right)))) /\\ (((forall pfp_index_gcd_backward_local_result_difference_add. (exists pfa_gap_gcd_backward_local_result_difference_addindex. pfa_gap_gcd_backward_local_result_difference_addindex + S (pfp_index_gcd_backward_local_result_difference_add) = (pfga_K_gcd_backward_local_result_difference)) -> exists pfp_left_gcd_backward_local_result_difference_add pfp_right_gcd_backward_local_result_difference_add pfp_value_gcd_backward_local_result_difference_add. ((((exists ff_h_pfp_gcd_backward_local_result_difference_addleft. ff_h_pfp_gcd_backward_local_result_difference_addleft + S (pfp_left_gcd_backward_local_result_difference_add) = S ((S (pfp_index_gcd_backward_local_result_difference_add)) * pfga_uc_gcd_backward_local_result_difference)) /\\ exists ff_q_pfp_gcd_backward_local_result_difference_addleft. pfga_ub_gcd_backward_local_result_difference = ff_q_pfp_gcd_backward_local_result_difference_addleft * S ((S (pfp_index_gcd_backward_local_result_difference_add)) * pfga_uc_gcd_backward_local_result_difference) + (pfp_left_gcd_backward_local_result_difference_add))) /\\ (((((exists ff_h_pfp_gcd_backward_local_result_difference_addright. ff_h_pfp_gcd_backward_local_result_difference_addright + S (pfp_right_gcd_backward_local_result_difference_add) = S ((S (pfp_index_gcd_backward_local_result_difference_add)) * pfga_vc_gcd_backward_local_result_difference)) /\\ exists ff_q_pfp_gcd_backward_local_result_difference_addright. pfga_vb_gcd_backward_local_result_difference = ff_q_pfp_gcd_backward_local_result_difference_addright * S ((S (pfp_index_gcd_backward_local_result_difference_add)) * pfga_vc_gcd_backward_local_result_difference) + (pfp_right_gcd_backward_local_result_difference_add))) /\\ (((((exists ff_h_pfp_gcd_backward_local_result_difference_addtarget. ff_h_pfp_gcd_backward_local_result_difference_addtarget + S (pfp_value_gcd_backward_local_result_difference_add) = S ((S (pfp_index_gcd_backward_local_result_difference_add)) * pfga_tc_gcd_backward_local_result_difference)) /\\ exists ff_q_pfp_gcd_backward_local_result_difference_addtarget. pfga_tb_gcd_backward_local_result_difference = ff_q_pfp_gcd_backward_local_result_difference_addtarget * S ((S (pfp_index_gcd_backward_local_result_difference_add)) * pfga_tc_gcd_backward_local_result_difference) + (pfp_value_gcd_backward_local_result_difference_add))) /\\ ((((exists pfa_gap_gcd_backward_local_result_difference_addoperationleft. pfa_gap_gcd_backward_local_result_difference_addoperationleft + S (pfp_left_gcd_backward_local_result_difference_add) = (p)) /\\ (((exists pfa_gap_gcd_backward_local_result_difference_addoperationright. pfa_gap_gcd_backward_local_result_difference_addoperationright + S (pfp_right_gcd_backward_local_result_difference_add) = (p)) /\\ ((((exists pfa_gap_gcd_backward_local_result_difference_addoperationresultbound. pfa_gap_gcd_backward_local_result_difference_addoperationresultbound + S (pfp_value_gcd_backward_local_result_difference_add) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_local_result_difference_addoperationresultcongruence pfa_offset_right_gcd_backward_local_result_difference_addoperationresultcongruence. ((pfp_left_gcd_backward_local_result_difference_add) + (pfp_right_gcd_backward_local_result_difference_add)) + (p) * pfa_offset_left_gcd_backward_local_result_difference_addoperationresultcongruence = (pfp_value_gcd_backward_local_result_difference_add) + (p) * pfa_offset_right_gcd_backward_local_result_difference_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_local_result_difference_result pfrep_left_gcd_backward_local_result_difference_result pfrep_right_gcd_backward_local_result_difference_result. ((exists pfrep_position_gcd_backward_local_result_difference_resultfirst. ((pfrep_position_gcd_backward_local_result_difference_resultfirst+S (pfrep_power_gcd_backward_local_result_difference_result)=(pfga_K_gcd_backward_local_result_difference)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_difference_resultfirstentry. ff_h_pfp_gcd_backward_local_result_difference_resultfirstentry + S (pfrep_left_gcd_backward_local_result_difference_result) = S ((S (pfrep_position_gcd_backward_local_result_difference_resultfirst)) * pfga_tc_gcd_backward_local_result_difference)) /\\ exists ff_q_pfp_gcd_backward_local_result_difference_resultfirstentry. pfga_tb_gcd_backward_local_result_difference = ff_q_pfp_gcd_backward_local_result_difference_resultfirstentry * S ((S (pfrep_position_gcd_backward_local_result_difference_resultfirst)) * pfga_tc_gcd_backward_local_result_difference) + (pfrep_left_gcd_backward_local_result_difference_result)))))) \\/ (((exists pfrep_gap_gcd_backward_local_result_difference_resultfirstoutside. pfrep_gap_gcd_backward_local_result_difference_resultfirstoutside+(pfga_K_gcd_backward_local_result_difference)=(pfrep_power_gcd_backward_local_result_difference_result)) /\\ (((pfrep_left_gcd_backward_local_result_difference_result)=0))))) -> ((exists pfrep_position_gcd_backward_local_result_difference_resultsecond. ((pfrep_position_gcd_backward_local_result_difference_resultsecond+S (pfrep_power_gcd_backward_local_result_difference_result)=(U)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_difference_resultsecondentry. ff_h_pfp_gcd_backward_local_result_difference_resultsecondentry + S (pfrep_right_gcd_backward_local_result_difference_result) = S ((S (pfrep_position_gcd_backward_local_result_difference_resultsecond)) * uc)) /\\ exists ff_q_pfp_gcd_backward_local_result_difference_resultsecondentry. ub = ff_q_pfp_gcd_backward_local_result_difference_resultsecondentry * S ((S (pfrep_position_gcd_backward_local_result_difference_resultsecond)) * uc) + (pfrep_right_gcd_backward_local_result_difference_result)))))) \\/ (((exists pfrep_gap_gcd_backward_local_result_difference_resultsecondoutside. pfrep_gap_gcd_backward_local_result_difference_resultsecondoutside+(U)=(pfrep_power_gcd_backward_local_result_difference_result)) /\\ (((pfrep_right_gcd_backward_local_result_difference_result)=0))))) -> pfrep_left_gcd_backward_local_result_difference_result=pfrep_right_gcd_backward_local_result_difference_result))))))))))))) /\\ ((exists pfgb_pb_gcd_backward_local_result_combination pfgb_pc_gcd_backward_local_result_combination pfgb_P_gcd_backward_local_result_combination pfgb_qb_gcd_backward_local_result_combination pfgb_qc_gcd_backward_local_result_combination pfgb_Q_gcd_backward_local_result_combination. ((((forall fom_index_pfp_gcd_backward_local_result_combination_leftleft. (exists fom_gap_pfp_gcd_backward_local_result_combination_leftleft_index_bound. fom_gap_pfp_gcd_backward_local_result_combination_leftleft_index_bound + S (fom_index_pfp_gcd_backward_local_result_combination_leftleft) = V) -> exists fom_value_pfp_gcd_backward_local_result_combination_leftleft. ((((exists fom_beta_height_pfp_gcd_backward_local_result_combination_leftleft_entry. fom_beta_height_pfp_gcd_backward_local_result_combination_leftleft_entry + S (fom_value_pfp_gcd_backward_local_result_combination_leftleft) = S ((S (fom_index_pfp_gcd_backward_local_result_combination_leftleft)) * vc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_result_combination_leftleft_entry. vb = fom_beta_quotient_pfp_gcd_backward_local_result_combination_leftleft_entry * S ((S (fom_index_pfp_gcd_backward_local_result_combination_leftleft)) * vc) + (fom_value_pfp_gcd_backward_local_result_combination_leftleft))) /\\ (exists fom_gap_pfp_gcd_backward_local_result_combination_leftleft_value_bound. fom_gap_pfp_gcd_backward_local_result_combination_leftleft_value_bound + S (fom_value_pfp_gcd_backward_local_result_combination_leftleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_local_result_combination_leftright. (exists fom_gap_pfp_gcd_backward_local_result_combination_leftright_index_bound. fom_gap_pfp_gcd_backward_local_result_combination_leftright_index_bound + S (fom_index_pfp_gcd_backward_local_result_combination_leftright) = L) -> exists fom_value_pfp_gcd_backward_local_result_combination_leftright. ((((exists fom_beta_height_pfp_gcd_backward_local_result_combination_leftright_entry. fom_beta_height_pfp_gcd_backward_local_result_combination_leftright_entry + S (fom_value_pfp_gcd_backward_local_result_combination_leftright) = S ((S (fom_index_pfp_gcd_backward_local_result_combination_leftright)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_result_combination_leftright_entry. ab = fom_beta_quotient_pfp_gcd_backward_local_result_combination_leftright_entry * S ((S (fom_index_pfp_gcd_backward_local_result_combination_leftright)) * ac) + (fom_value_pfp_gcd_backward_local_result_combination_leftright))) /\\ (exists fom_gap_pfp_gcd_backward_local_result_combination_leftright_value_bound. fom_gap_pfp_gcd_backward_local_result_combination_leftright_value_bound + S (fom_value_pfp_gcd_backward_local_result_combination_leftright) = p))) /\\ (((((((V)=0 \\/ (L)=0) /\\ (((pfgb_P_gcd_backward_local_result_combination)=0)))) \\/ (((~((V)=0)) /\\ (((~((L)=0)) /\\ (((V)+(L)=S (pfgb_P_gcd_backward_local_result_combination)))))))) /\\ ((forall pfc_index_gcd_backward_local_result_combination_leftcoefficients. (exists pfa_gap_gcd_backward_local_result_combination_leftcoefficientsbound. pfa_gap_gcd_backward_local_result_combination_leftcoefficientsbound + S (pfc_index_gcd_backward_local_result_combination_leftcoefficients) = (pfgb_P_gcd_backward_local_result_combination)) -> exists pfc_value_gcd_backward_local_result_combination_leftcoefficients. ((((exists ff_h_pfp_gcd_backward_local_result_combination_leftcoefficientsentry. ff_h_pfp_gcd_backward_local_result_combination_leftcoefficientsentry + S (pfc_value_gcd_backward_local_result_combination_leftcoefficients) = S ((S (pfc_index_gcd_backward_local_result_combination_leftcoefficients)) * pfgb_pc_gcd_backward_local_result_combination)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_leftcoefficientsentry. pfgb_pb_gcd_backward_local_result_combination = ff_q_pfp_gcd_backward_local_result_combination_leftcoefficientsentry * S ((S (pfc_index_gcd_backward_local_result_combination_leftcoefficients)) * pfgb_pc_gcd_backward_local_result_combination) + (pfc_value_gcd_backward_local_result_combination_leftcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_local_result_combination_leftcoefficientscoefficient pfc_terms_scale_gcd_backward_local_result_combination_leftcoefficientscoefficient pfc_natural_sum_gcd_backward_local_result_combination_leftcoefficientscoefficient. ((forall pfc_index_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_local_result_combination_leftcoefficients))) -> exists pfc_value_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_local_result_combination_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_local_result_combination_leftcoefficientscoefficient = ff_q_pfp_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_local_result_combination_leftcoefficientscoefficient) + (pfc_value_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_local_result_combination_leftcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonal) = (V)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonal)) * vc) + (pfc_left_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermleftoutside+(V)=(pfc_index_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_local_result_combination_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum fs_v_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_local_result_combination_leftcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_local_result_combination_leftcoefficients))) * fs_v_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_local_result_combination_leftcoefficients))) * fs_v_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_local_result_combination_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_local_result_combination_leftcoefficients)) -> exists fs_a_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_local_result_combination_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_local_result_combination_leftcoefficientscoefficient = fs_q_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_local_result_combination_leftcoefficientscoefficient) + (fs_a_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_local_result_combination_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_local_result_combination_leftcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_local_result_combination_leftcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_local_result_combination_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_local_result_combination_leftcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_local_result_combination_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_local_result_combination_leftcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_local_result_combination_leftcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_local_result_combination_leftcoefficients) + (p) * pfa_offset_right_gcd_backward_local_result_combination_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_gcd_backward_local_result_combination_rightleft. (exists fom_gap_pfp_gcd_backward_local_result_combination_rightleft_index_bound. fom_gap_pfp_gcd_backward_local_result_combination_rightleft_index_bound + S (fom_index_pfp_gcd_backward_local_result_combination_rightleft) = pfg_update_T_gcd_backward_local_result) -> exists fom_value_pfp_gcd_backward_local_result_combination_rightleft. ((((exists fom_beta_height_pfp_gcd_backward_local_result_combination_rightleft_entry. fom_beta_height_pfp_gcd_backward_local_result_combination_rightleft_entry + S (fom_value_pfp_gcd_backward_local_result_combination_rightleft) = S ((S (fom_index_pfp_gcd_backward_local_result_combination_rightleft)) * pfg_update_tc_gcd_backward_local_result)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_result_combination_rightleft_entry. pfg_update_tb_gcd_backward_local_result = fom_beta_quotient_pfp_gcd_backward_local_result_combination_rightleft_entry * S ((S (fom_index_pfp_gcd_backward_local_result_combination_rightleft)) * pfg_update_tc_gcd_backward_local_result) + (fom_value_pfp_gcd_backward_local_result_combination_rightleft))) /\\ (exists fom_gap_pfp_gcd_backward_local_result_combination_rightleft_value_bound. fom_gap_pfp_gcd_backward_local_result_combination_rightleft_value_bound + S (fom_value_pfp_gcd_backward_local_result_combination_rightleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_local_result_combination_rightright. (exists fom_gap_pfp_gcd_backward_local_result_combination_rightright_index_bound. fom_gap_pfp_gcd_backward_local_result_combination_rightright_index_bound + S (fom_index_pfp_gcd_backward_local_result_combination_rightright) = S d) -> exists fom_value_pfp_gcd_backward_local_result_combination_rightright. ((((exists fom_beta_height_pfp_gcd_backward_local_result_combination_rightright_entry. fom_beta_height_pfp_gcd_backward_local_result_combination_rightright_entry + S (fom_value_pfp_gcd_backward_local_result_combination_rightright) = S ((S (fom_index_pfp_gcd_backward_local_result_combination_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_result_combination_rightright_entry. bb = fom_beta_quotient_pfp_gcd_backward_local_result_combination_rightright_entry * S ((S (fom_index_pfp_gcd_backward_local_result_combination_rightright)) * bc) + (fom_value_pfp_gcd_backward_local_result_combination_rightright))) /\\ (exists fom_gap_pfp_gcd_backward_local_result_combination_rightright_value_bound. fom_gap_pfp_gcd_backward_local_result_combination_rightright_value_bound + S (fom_value_pfp_gcd_backward_local_result_combination_rightright) = p))) /\\ (((((((pfg_update_T_gcd_backward_local_result)=0 \\/ (S d)=0) /\\ (((pfgb_Q_gcd_backward_local_result_combination)=0)))) \\/ (((~((pfg_update_T_gcd_backward_local_result)=0)) /\\ (((~((S d)=0)) /\\ (((pfg_update_T_gcd_backward_local_result)+(S d)=S (pfgb_Q_gcd_backward_local_result_combination)))))))) /\\ ((forall pfc_index_gcd_backward_local_result_combination_rightcoefficients. (exists pfa_gap_gcd_backward_local_result_combination_rightcoefficientsbound. pfa_gap_gcd_backward_local_result_combination_rightcoefficientsbound + S (pfc_index_gcd_backward_local_result_combination_rightcoefficients) = (pfgb_Q_gcd_backward_local_result_combination)) -> exists pfc_value_gcd_backward_local_result_combination_rightcoefficients. ((((exists ff_h_pfp_gcd_backward_local_result_combination_rightcoefficientsentry. ff_h_pfp_gcd_backward_local_result_combination_rightcoefficientsentry + S (pfc_value_gcd_backward_local_result_combination_rightcoefficients) = S ((S (pfc_index_gcd_backward_local_result_combination_rightcoefficients)) * pfgb_qc_gcd_backward_local_result_combination)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_rightcoefficientsentry. pfgb_qb_gcd_backward_local_result_combination = ff_q_pfp_gcd_backward_local_result_combination_rightcoefficientsentry * S ((S (pfc_index_gcd_backward_local_result_combination_rightcoefficients)) * pfgb_qc_gcd_backward_local_result_combination) + (pfc_value_gcd_backward_local_result_combination_rightcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_local_result_combination_rightcoefficientscoefficient pfc_terms_scale_gcd_backward_local_result_combination_rightcoefficientscoefficient pfc_natural_sum_gcd_backward_local_result_combination_rightcoefficientscoefficient. ((forall pfc_index_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_local_result_combination_rightcoefficients))) -> exists pfc_value_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_local_result_combination_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_local_result_combination_rightcoefficientscoefficient = ff_q_pfp_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_local_result_combination_rightcoefficientscoefficient) + (pfc_value_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_local_result_combination_rightcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonal) = (pfg_update_T_gcd_backward_local_result)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonal)) * pfg_update_tc_gcd_backward_local_result)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermleftentry. pfg_update_tb_gcd_backward_local_result = ff_q_pfp_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonal)) * pfg_update_tc_gcd_backward_local_result) + (pfc_left_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermleftoutside+(pfg_update_T_gcd_backward_local_result)=(pfc_index_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_local_result_combination_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum fs_v_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_local_result_combination_rightcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_local_result_combination_rightcoefficients))) * fs_v_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_local_result_combination_rightcoefficients))) * fs_v_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_local_result_combination_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_local_result_combination_rightcoefficients)) -> exists fs_a_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_local_result_combination_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_local_result_combination_rightcoefficientscoefficient = fs_q_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_local_result_combination_rightcoefficientscoefficient) + (fs_a_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_local_result_combination_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_local_result_combination_rightcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_local_result_combination_rightcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_local_result_combination_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_local_result_combination_rightcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_local_result_combination_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_local_result_combination_rightcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_local_result_combination_rightcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_local_result_combination_rightcoefficients) + (p) * pfa_offset_right_gcd_backward_local_result_combination_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_gcd_backward_local_result_combination_sum_left_bounded. (exists fom_gap_pfp_gcd_backward_local_result_combination_sum_left_bounded_index_bound. fom_gap_pfp_gcd_backward_local_result_combination_sum_left_bounded_index_bound + S (fom_index_pfp_gcd_backward_local_result_combination_sum_left_bounded) = pfgb_P_gcd_backward_local_result_combination) -> exists fom_value_pfp_gcd_backward_local_result_combination_sum_left_bounded. ((((exists fom_beta_height_pfp_gcd_backward_local_result_combination_sum_left_bounded_entry. fom_beta_height_pfp_gcd_backward_local_result_combination_sum_left_bounded_entry + S (fom_value_pfp_gcd_backward_local_result_combination_sum_left_bounded) = S ((S (fom_index_pfp_gcd_backward_local_result_combination_sum_left_bounded)) * pfgb_pc_gcd_backward_local_result_combination)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_result_combination_sum_left_bounded_entry. pfgb_pb_gcd_backward_local_result_combination = fom_beta_quotient_pfp_gcd_backward_local_result_combination_sum_left_bounded_entry * S ((S (fom_index_pfp_gcd_backward_local_result_combination_sum_left_bounded)) * pfgb_pc_gcd_backward_local_result_combination) + (fom_value_pfp_gcd_backward_local_result_combination_sum_left_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_local_result_combination_sum_left_bounded_value_bound. fom_gap_pfp_gcd_backward_local_result_combination_sum_left_bounded_value_bound + S (fom_value_pfp_gcd_backward_local_result_combination_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_backward_local_result_combination_sum_right_bounded. (exists fom_gap_pfp_gcd_backward_local_result_combination_sum_right_bounded_index_bound. fom_gap_pfp_gcd_backward_local_result_combination_sum_right_bounded_index_bound + S (fom_index_pfp_gcd_backward_local_result_combination_sum_right_bounded) = pfgb_Q_gcd_backward_local_result_combination) -> exists fom_value_pfp_gcd_backward_local_result_combination_sum_right_bounded. ((((exists fom_beta_height_pfp_gcd_backward_local_result_combination_sum_right_bounded_entry. fom_beta_height_pfp_gcd_backward_local_result_combination_sum_right_bounded_entry + S (fom_value_pfp_gcd_backward_local_result_combination_sum_right_bounded) = S ((S (fom_index_pfp_gcd_backward_local_result_combination_sum_right_bounded)) * pfgb_qc_gcd_backward_local_result_combination)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_result_combination_sum_right_bounded_entry. pfgb_qb_gcd_backward_local_result_combination = fom_beta_quotient_pfp_gcd_backward_local_result_combination_sum_right_bounded_entry * S ((S (fom_index_pfp_gcd_backward_local_result_combination_sum_right_bounded)) * pfgb_qc_gcd_backward_local_result_combination) + (fom_value_pfp_gcd_backward_local_result_combination_sum_right_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_local_result_combination_sum_right_bounded_value_bound. fom_gap_pfp_gcd_backward_local_result_combination_sum_right_bounded_value_bound + S (fom_value_pfp_gcd_backward_local_result_combination_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_backward_local_result_combination_sum_result_bounded. (exists fom_gap_pfp_gcd_backward_local_result_combination_sum_result_bounded_index_bound. fom_gap_pfp_gcd_backward_local_result_combination_sum_result_bounded_index_bound + S (fom_index_pfp_gcd_backward_local_result_combination_sum_result_bounded) = G) -> exists fom_value_pfp_gcd_backward_local_result_combination_sum_result_bounded. ((((exists fom_beta_height_pfp_gcd_backward_local_result_combination_sum_result_bounded_entry. fom_beta_height_pfp_gcd_backward_local_result_combination_sum_result_bounded_entry + S (fom_value_pfp_gcd_backward_local_result_combination_sum_result_bounded) = S ((S (fom_index_pfp_gcd_backward_local_result_combination_sum_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_local_result_combination_sum_result_bounded_entry. gb = fom_beta_quotient_pfp_gcd_backward_local_result_combination_sum_result_bounded_entry * S ((S (fom_index_pfp_gcd_backward_local_result_combination_sum_result_bounded)) * gc) + (fom_value_pfp_gcd_backward_local_result_combination_sum_result_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_local_result_combination_sum_result_bounded_value_bound. fom_gap_pfp_gcd_backward_local_result_combination_sum_result_bounded_value_bound + S (fom_value_pfp_gcd_backward_local_result_combination_sum_result_bounded) = p))) /\\ ((exists pfga_ub_gcd_backward_local_result_combination_sum pfga_uc_gcd_backward_local_result_combination_sum pfga_vb_gcd_backward_local_result_combination_sum pfga_vc_gcd_backward_local_result_combination_sum pfga_tb_gcd_backward_local_result_combination_sum pfga_tc_gcd_backward_local_result_combination_sum pfga_K_gcd_backward_local_result_combination_sum. ((((forall pfrep_power_gcd_backward_local_result_combination_sum_left pfrep_left_gcd_backward_local_result_combination_sum_left pfrep_right_gcd_backward_local_result_combination_sum_left. ((exists pfrep_position_gcd_backward_local_result_combination_sum_leftfirst. ((pfrep_position_gcd_backward_local_result_combination_sum_leftfirst+S (pfrep_power_gcd_backward_local_result_combination_sum_left)=(pfgb_P_gcd_backward_local_result_combination)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_combination_sum_leftfirstentry. ff_h_pfp_gcd_backward_local_result_combination_sum_leftfirstentry + S (pfrep_left_gcd_backward_local_result_combination_sum_left) = S ((S (pfrep_position_gcd_backward_local_result_combination_sum_leftfirst)) * pfgb_pc_gcd_backward_local_result_combination)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_sum_leftfirstentry. pfgb_pb_gcd_backward_local_result_combination = ff_q_pfp_gcd_backward_local_result_combination_sum_leftfirstentry * S ((S (pfrep_position_gcd_backward_local_result_combination_sum_leftfirst)) * pfgb_pc_gcd_backward_local_result_combination) + (pfrep_left_gcd_backward_local_result_combination_sum_left)))))) \\/ (((exists pfrep_gap_gcd_backward_local_result_combination_sum_leftfirstoutside. pfrep_gap_gcd_backward_local_result_combination_sum_leftfirstoutside+(pfgb_P_gcd_backward_local_result_combination)=(pfrep_power_gcd_backward_local_result_combination_sum_left)) /\\ (((pfrep_left_gcd_backward_local_result_combination_sum_left)=0))))) -> ((exists pfrep_position_gcd_backward_local_result_combination_sum_leftsecond. ((pfrep_position_gcd_backward_local_result_combination_sum_leftsecond+S (pfrep_power_gcd_backward_local_result_combination_sum_left)=(pfga_K_gcd_backward_local_result_combination_sum)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_combination_sum_leftsecondentry. ff_h_pfp_gcd_backward_local_result_combination_sum_leftsecondentry + S (pfrep_right_gcd_backward_local_result_combination_sum_left) = S ((S (pfrep_position_gcd_backward_local_result_combination_sum_leftsecond)) * pfga_uc_gcd_backward_local_result_combination_sum)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_sum_leftsecondentry. pfga_ub_gcd_backward_local_result_combination_sum = ff_q_pfp_gcd_backward_local_result_combination_sum_leftsecondentry * S ((S (pfrep_position_gcd_backward_local_result_combination_sum_leftsecond)) * pfga_uc_gcd_backward_local_result_combination_sum) + (pfrep_right_gcd_backward_local_result_combination_sum_left)))))) \\/ (((exists pfrep_gap_gcd_backward_local_result_combination_sum_leftsecondoutside. pfrep_gap_gcd_backward_local_result_combination_sum_leftsecondoutside+(pfga_K_gcd_backward_local_result_combination_sum)=(pfrep_power_gcd_backward_local_result_combination_sum_left)) /\\ (((pfrep_right_gcd_backward_local_result_combination_sum_left)=0))))) -> pfrep_left_gcd_backward_local_result_combination_sum_left=pfrep_right_gcd_backward_local_result_combination_sum_left) /\\ ((forall pfrep_power_gcd_backward_local_result_combination_sum_right pfrep_left_gcd_backward_local_result_combination_sum_right pfrep_right_gcd_backward_local_result_combination_sum_right. ((exists pfrep_position_gcd_backward_local_result_combination_sum_rightfirst. ((pfrep_position_gcd_backward_local_result_combination_sum_rightfirst+S (pfrep_power_gcd_backward_local_result_combination_sum_right)=(pfgb_Q_gcd_backward_local_result_combination)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_combination_sum_rightfirstentry. ff_h_pfp_gcd_backward_local_result_combination_sum_rightfirstentry + S (pfrep_left_gcd_backward_local_result_combination_sum_right) = S ((S (pfrep_position_gcd_backward_local_result_combination_sum_rightfirst)) * pfgb_qc_gcd_backward_local_result_combination)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_sum_rightfirstentry. pfgb_qb_gcd_backward_local_result_combination = ff_q_pfp_gcd_backward_local_result_combination_sum_rightfirstentry * S ((S (pfrep_position_gcd_backward_local_result_combination_sum_rightfirst)) * pfgb_qc_gcd_backward_local_result_combination) + (pfrep_left_gcd_backward_local_result_combination_sum_right)))))) \\/ (((exists pfrep_gap_gcd_backward_local_result_combination_sum_rightfirstoutside. pfrep_gap_gcd_backward_local_result_combination_sum_rightfirstoutside+(pfgb_Q_gcd_backward_local_result_combination)=(pfrep_power_gcd_backward_local_result_combination_sum_right)) /\\ (((pfrep_left_gcd_backward_local_result_combination_sum_right)=0))))) -> ((exists pfrep_position_gcd_backward_local_result_combination_sum_rightsecond. ((pfrep_position_gcd_backward_local_result_combination_sum_rightsecond+S (pfrep_power_gcd_backward_local_result_combination_sum_right)=(pfga_K_gcd_backward_local_result_combination_sum)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_combination_sum_rightsecondentry. ff_h_pfp_gcd_backward_local_result_combination_sum_rightsecondentry + S (pfrep_right_gcd_backward_local_result_combination_sum_right) = S ((S (pfrep_position_gcd_backward_local_result_combination_sum_rightsecond)) * pfga_vc_gcd_backward_local_result_combination_sum)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_sum_rightsecondentry. pfga_vb_gcd_backward_local_result_combination_sum = ff_q_pfp_gcd_backward_local_result_combination_sum_rightsecondentry * S ((S (pfrep_position_gcd_backward_local_result_combination_sum_rightsecond)) * pfga_vc_gcd_backward_local_result_combination_sum) + (pfrep_right_gcd_backward_local_result_combination_sum_right)))))) \\/ (((exists pfrep_gap_gcd_backward_local_result_combination_sum_rightsecondoutside. pfrep_gap_gcd_backward_local_result_combination_sum_rightsecondoutside+(pfga_K_gcd_backward_local_result_combination_sum)=(pfrep_power_gcd_backward_local_result_combination_sum_right)) /\\ (((pfrep_right_gcd_backward_local_result_combination_sum_right)=0))))) -> pfrep_left_gcd_backward_local_result_combination_sum_right=pfrep_right_gcd_backward_local_result_combination_sum_right)))) /\\ (((forall pfp_index_gcd_backward_local_result_combination_sum_add. (exists pfa_gap_gcd_backward_local_result_combination_sum_addindex. pfa_gap_gcd_backward_local_result_combination_sum_addindex + S (pfp_index_gcd_backward_local_result_combination_sum_add) = (pfga_K_gcd_backward_local_result_combination_sum)) -> exists pfp_left_gcd_backward_local_result_combination_sum_add pfp_right_gcd_backward_local_result_combination_sum_add pfp_value_gcd_backward_local_result_combination_sum_add. ((((exists ff_h_pfp_gcd_backward_local_result_combination_sum_addleft. ff_h_pfp_gcd_backward_local_result_combination_sum_addleft + S (pfp_left_gcd_backward_local_result_combination_sum_add) = S ((S (pfp_index_gcd_backward_local_result_combination_sum_add)) * pfga_uc_gcd_backward_local_result_combination_sum)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_sum_addleft. pfga_ub_gcd_backward_local_result_combination_sum = ff_q_pfp_gcd_backward_local_result_combination_sum_addleft * S ((S (pfp_index_gcd_backward_local_result_combination_sum_add)) * pfga_uc_gcd_backward_local_result_combination_sum) + (pfp_left_gcd_backward_local_result_combination_sum_add))) /\\ (((((exists ff_h_pfp_gcd_backward_local_result_combination_sum_addright. ff_h_pfp_gcd_backward_local_result_combination_sum_addright + S (pfp_right_gcd_backward_local_result_combination_sum_add) = S ((S (pfp_index_gcd_backward_local_result_combination_sum_add)) * pfga_vc_gcd_backward_local_result_combination_sum)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_sum_addright. pfga_vb_gcd_backward_local_result_combination_sum = ff_q_pfp_gcd_backward_local_result_combination_sum_addright * S ((S (pfp_index_gcd_backward_local_result_combination_sum_add)) * pfga_vc_gcd_backward_local_result_combination_sum) + (pfp_right_gcd_backward_local_result_combination_sum_add))) /\\ (((((exists ff_h_pfp_gcd_backward_local_result_combination_sum_addtarget. ff_h_pfp_gcd_backward_local_result_combination_sum_addtarget + S (pfp_value_gcd_backward_local_result_combination_sum_add) = S ((S (pfp_index_gcd_backward_local_result_combination_sum_add)) * pfga_tc_gcd_backward_local_result_combination_sum)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_sum_addtarget. pfga_tb_gcd_backward_local_result_combination_sum = ff_q_pfp_gcd_backward_local_result_combination_sum_addtarget * S ((S (pfp_index_gcd_backward_local_result_combination_sum_add)) * pfga_tc_gcd_backward_local_result_combination_sum) + (pfp_value_gcd_backward_local_result_combination_sum_add))) /\\ ((((exists pfa_gap_gcd_backward_local_result_combination_sum_addoperationleft. pfa_gap_gcd_backward_local_result_combination_sum_addoperationleft + S (pfp_left_gcd_backward_local_result_combination_sum_add) = (p)) /\\ (((exists pfa_gap_gcd_backward_local_result_combination_sum_addoperationright. pfa_gap_gcd_backward_local_result_combination_sum_addoperationright + S (pfp_right_gcd_backward_local_result_combination_sum_add) = (p)) /\\ ((((exists pfa_gap_gcd_backward_local_result_combination_sum_addoperationresultbound. pfa_gap_gcd_backward_local_result_combination_sum_addoperationresultbound + S (pfp_value_gcd_backward_local_result_combination_sum_add) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_local_result_combination_sum_addoperationresultcongruence pfa_offset_right_gcd_backward_local_result_combination_sum_addoperationresultcongruence. ((pfp_left_gcd_backward_local_result_combination_sum_add) + (pfp_right_gcd_backward_local_result_combination_sum_add)) + (p) * pfa_offset_left_gcd_backward_local_result_combination_sum_addoperationresultcongruence = (pfp_value_gcd_backward_local_result_combination_sum_add) + (p) * pfa_offset_right_gcd_backward_local_result_combination_sum_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_local_result_combination_sum_result pfrep_left_gcd_backward_local_result_combination_sum_result pfrep_right_gcd_backward_local_result_combination_sum_result. ((exists pfrep_position_gcd_backward_local_result_combination_sum_resultfirst. ((pfrep_position_gcd_backward_local_result_combination_sum_resultfirst+S (pfrep_power_gcd_backward_local_result_combination_sum_result)=(pfga_K_gcd_backward_local_result_combination_sum)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_combination_sum_resultfirstentry. ff_h_pfp_gcd_backward_local_result_combination_sum_resultfirstentry + S (pfrep_left_gcd_backward_local_result_combination_sum_result) = S ((S (pfrep_position_gcd_backward_local_result_combination_sum_resultfirst)) * pfga_tc_gcd_backward_local_result_combination_sum)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_sum_resultfirstentry. pfga_tb_gcd_backward_local_result_combination_sum = ff_q_pfp_gcd_backward_local_result_combination_sum_resultfirstentry * S ((S (pfrep_position_gcd_backward_local_result_combination_sum_resultfirst)) * pfga_tc_gcd_backward_local_result_combination_sum) + (pfrep_left_gcd_backward_local_result_combination_sum_result)))))) \\/ (((exists pfrep_gap_gcd_backward_local_result_combination_sum_resultfirstoutside. pfrep_gap_gcd_backward_local_result_combination_sum_resultfirstoutside+(pfga_K_gcd_backward_local_result_combination_sum)=(pfrep_power_gcd_backward_local_result_combination_sum_result)) /\\ (((pfrep_left_gcd_backward_local_result_combination_sum_result)=0))))) -> ((exists pfrep_position_gcd_backward_local_result_combination_sum_resultsecond. ((pfrep_position_gcd_backward_local_result_combination_sum_resultsecond+S (pfrep_power_gcd_backward_local_result_combination_sum_result)=(G)) /\\ ((((exists ff_h_pfp_gcd_backward_local_result_combination_sum_resultsecondentry. ff_h_pfp_gcd_backward_local_result_combination_sum_resultsecondentry + S (pfrep_right_gcd_backward_local_result_combination_sum_result) = S ((S (pfrep_position_gcd_backward_local_result_combination_sum_resultsecond)) * gc)) /\\ exists ff_q_pfp_gcd_backward_local_result_combination_sum_resultsecondentry. gb = ff_q_pfp_gcd_backward_local_result_combination_sum_resultsecondentry * S ((S (pfrep_position_gcd_backward_local_result_combination_sum_resultsecond)) * gc) + (pfrep_right_gcd_backward_local_result_combination_sum_result)))))) \\/ (((exists pfrep_gap_gcd_backward_local_result_combination_sum_resultsecondoutside. pfrep_gap_gcd_backward_local_result_combination_sum_resultsecondoutside+(G)=(pfrep_power_gcd_backward_local_result_combination_sum_result)) /\\ (((pfrep_right_gcd_backward_local_result_combination_sum_result)=0))))) -> pfrep_left_gcd_backward_local_result_combination_sum_result=pfrep_right_gcd_backward_local_result_combination_sum_result)))))))))))))))))))))))", "specialize prime_field_polynomial_division_execution_bezout_backward (p)", "specialize prime_field_polynomial_division_execution_bezout_backward (ab)", "specialize prime_field_polynomial_division_execution_bezout_backward (ac)", "specialize prime_field_polynomial_division_execution_bezout_backward (L)", "specialize prime_field_polynomial_division_execution_bezout_backward (bb)", "specialize prime_field_polynomial_division_execution_bezout_backward (bc)", "specialize prime_field_polynomial_division_execution_bezout_backward (d)", "specialize prime_field_polynomial_division_execution_bezout_backward (qb)", "specialize prime_field_polynomial_division_execution_bezout_backward (qc)", "specialize prime_field_polynomial_division_execution_bezout_backward (q)", "specialize prime_field_polynomial_division_execution_bezout_backward (rb)", "specialize prime_field_polynomial_division_execution_bezout_backward (rc)", "specialize prime_field_polynomial_division_execution_bezout_backward (R)", "exact prime_field_polynomial_division_execution_bezout_backward", "intro hp", "intro he", "intro hs", "cases hs", "cases hs_witness", "cases hs_witness_witness", "cases hs_witness_witness_witness", "cases hs_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_right", "have hc : ((((forall fom_index_pfp_gcd_backward_new_common_left_bounded. (exists fom_gap_pfp_gcd_backward_new_common_left_bounded_index_bound. fom_gap_pfp_gcd_backward_new_common_left_bounded_index_bound + S (fom_index_pfp_gcd_backward_new_common_left_bounded) = L) -> exists fom_value_pfp_gcd_backward_new_common_left_bounded. ((((exists fom_beta_height_pfp_gcd_backward_new_common_left_bounded_entry. fom_beta_height_pfp_gcd_backward_new_common_left_bounded_entry + S (fom_value_pfp_gcd_backward_new_common_left_bounded) = S ((S (fom_index_pfp_gcd_backward_new_common_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_left_bounded_entry. ab = fom_beta_quotient_pfp_gcd_backward_new_common_left_bounded_entry * S ((S (fom_index_pfp_gcd_backward_new_common_left_bounded)) * ac) + (fom_value_pfp_gcd_backward_new_common_left_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_left_bounded_value_bound. fom_gap_pfp_gcd_backward_new_common_left_bounded_value_bound + S (fom_value_pfp_gcd_backward_new_common_left_bounded) = p))) /\\ ((exists pfgd_qb_gcd_backward_new_common_left pfgd_qc_gcd_backward_new_common_left pfgd_Q_gcd_backward_new_common_left pfgd_pb_gcd_backward_new_common_left pfgd_pc_gcd_backward_new_common_left pfgd_P_gcd_backward_new_common_left. ((((forall fom_index_pfp_gcd_backward_new_common_left_productleft. (exists fom_gap_pfp_gcd_backward_new_common_left_productleft_index_bound. fom_gap_pfp_gcd_backward_new_common_left_productleft_index_bound + S (fom_index_pfp_gcd_backward_new_common_left_productleft) = pfgd_Q_gcd_backward_new_common_left) -> exists fom_value_pfp_gcd_backward_new_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_backward_new_common_left_productleft_entry. fom_beta_height_pfp_gcd_backward_new_common_left_productleft_entry + S (fom_value_pfp_gcd_backward_new_common_left_productleft) = S ((S (fom_index_pfp_gcd_backward_new_common_left_productleft)) * pfgd_qc_gcd_backward_new_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_left_productleft_entry. pfgd_qb_gcd_backward_new_common_left = fom_beta_quotient_pfp_gcd_backward_new_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_backward_new_common_left_productleft)) * pfgd_qc_gcd_backward_new_common_left) + (fom_value_pfp_gcd_backward_new_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_left_productleft_value_bound. fom_gap_pfp_gcd_backward_new_common_left_productleft_value_bound + S (fom_value_pfp_gcd_backward_new_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_new_common_left_productright. (exists fom_gap_pfp_gcd_backward_new_common_left_productright_index_bound. fom_gap_pfp_gcd_backward_new_common_left_productright_index_bound + S (fom_index_pfp_gcd_backward_new_common_left_productright) = x2) -> exists fom_value_pfp_gcd_backward_new_common_left_productright. ((((exists fom_beta_height_pfp_gcd_backward_new_common_left_productright_entry. fom_beta_height_pfp_gcd_backward_new_common_left_productright_entry + S (fom_value_pfp_gcd_backward_new_common_left_productright) = S ((S (fom_index_pfp_gcd_backward_new_common_left_productright)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_left_productright_entry. x = fom_beta_quotient_pfp_gcd_backward_new_common_left_productright_entry * S ((S (fom_index_pfp_gcd_backward_new_common_left_productright)) * x1) + (fom_value_pfp_gcd_backward_new_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_left_productright_value_bound. fom_gap_pfp_gcd_backward_new_common_left_productright_value_bound + S (fom_value_pfp_gcd_backward_new_common_left_productright) = p))) /\\ (((((((pfgd_Q_gcd_backward_new_common_left)=0 \\/ (x2)=0) /\\ (((pfgd_P_gcd_backward_new_common_left)=0)))) \\/ (((~((pfgd_Q_gcd_backward_new_common_left)=0)) /\\ (((~((x2)=0)) /\\ (((pfgd_Q_gcd_backward_new_common_left)+(x2)=S (pfgd_P_gcd_backward_new_common_left)))))))) /\\ ((forall pfc_index_gcd_backward_new_common_left_productcoefficients. (exists pfa_gap_gcd_backward_new_common_left_productcoefficientsbound. pfa_gap_gcd_backward_new_common_left_productcoefficientsbound + S (pfc_index_gcd_backward_new_common_left_productcoefficients) = (pfgd_P_gcd_backward_new_common_left)) -> exists pfc_value_gcd_backward_new_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_backward_new_common_left_productcoefficientsentry. ff_h_pfp_gcd_backward_new_common_left_productcoefficientsentry + S (pfc_value_gcd_backward_new_common_left_productcoefficients) = S ((S (pfc_index_gcd_backward_new_common_left_productcoefficients)) * pfgd_pc_gcd_backward_new_common_left)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_productcoefficientsentry. pfgd_pb_gcd_backward_new_common_left = ff_q_pfp_gcd_backward_new_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_backward_new_common_left_productcoefficients)) * pfgd_pc_gcd_backward_new_common_left) + (pfc_value_gcd_backward_new_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_new_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_backward_new_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_backward_new_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_new_common_left_productcoefficients))) -> exists pfc_value_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_new_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_new_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_new_common_left_productcoefficientscoefficient) + (pfc_value_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_new_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_backward_new_common_left)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_new_common_left)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_backward_new_common_left = ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_new_common_left) + (pfc_left_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_backward_new_common_left)=(pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm) = (x2)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightentry. x = ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)) * x1) + (pfc_right_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(x2)=(pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_new_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_new_common_left_productcoefficients))) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_new_common_left_productcoefficients))) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_new_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_new_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_new_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_new_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_new_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_new_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_new_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_new_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_new_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_new_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_new_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_backward_new_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_new_common_left_equivalent pfrep_left_gcd_backward_new_common_left_equivalent pfrep_right_gcd_backward_new_common_left_equivalent. ((exists pfrep_position_gcd_backward_new_common_left_equivalentfirst. ((pfrep_position_gcd_backward_new_common_left_equivalentfirst+S (pfrep_power_gcd_backward_new_common_left_equivalent)=(pfgd_P_gcd_backward_new_common_left)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_left_equivalentfirstentry. ff_h_pfp_gcd_backward_new_common_left_equivalentfirstentry + S (pfrep_left_gcd_backward_new_common_left_equivalent) = S ((S (pfrep_position_gcd_backward_new_common_left_equivalentfirst)) * pfgd_pc_gcd_backward_new_common_left)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_equivalentfirstentry. pfgd_pb_gcd_backward_new_common_left = ff_q_pfp_gcd_backward_new_common_left_equivalentfirstentry * S ((S (pfrep_position_gcd_backward_new_common_left_equivalentfirst)) * pfgd_pc_gcd_backward_new_common_left) + (pfrep_left_gcd_backward_new_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_new_common_left_equivalentfirstoutside. pfrep_gap_gcd_backward_new_common_left_equivalentfirstoutside+(pfgd_P_gcd_backward_new_common_left)=(pfrep_power_gcd_backward_new_common_left_equivalent)) /\\ (((pfrep_left_gcd_backward_new_common_left_equivalent)=0))))) -> ((exists pfrep_position_gcd_backward_new_common_left_equivalentsecond. ((pfrep_position_gcd_backward_new_common_left_equivalentsecond+S (pfrep_power_gcd_backward_new_common_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_left_equivalentsecondentry. ff_h_pfp_gcd_backward_new_common_left_equivalentsecondentry + S (pfrep_right_gcd_backward_new_common_left_equivalent) = S ((S (pfrep_position_gcd_backward_new_common_left_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_equivalentsecondentry. ab = ff_q_pfp_gcd_backward_new_common_left_equivalentsecondentry * S ((S (pfrep_position_gcd_backward_new_common_left_equivalentsecond)) * ac) + (pfrep_right_gcd_backward_new_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_new_common_left_equivalentsecondoutside. pfrep_gap_gcd_backward_new_common_left_equivalentsecondoutside+(L)=(pfrep_power_gcd_backward_new_common_left_equivalent)) /\\ (((pfrep_right_gcd_backward_new_common_left_equivalent)=0))))) -> pfrep_left_gcd_backward_new_common_left_equivalent=pfrep_right_gcd_backward_new_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_backward_new_common_right_bounded. (exists fom_gap_pfp_gcd_backward_new_common_right_bounded_index_bound. fom_gap_pfp_gcd_backward_new_common_right_bounded_index_bound + S (fom_index_pfp_gcd_backward_new_common_right_bounded) = S d) -> exists fom_value_pfp_gcd_backward_new_common_right_bounded. ((((exists fom_beta_height_pfp_gcd_backward_new_common_right_bounded_entry. fom_beta_height_pfp_gcd_backward_new_common_right_bounded_entry + S (fom_value_pfp_gcd_backward_new_common_right_bounded) = S ((S (fom_index_pfp_gcd_backward_new_common_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_right_bounded_entry. bb = fom_beta_quotient_pfp_gcd_backward_new_common_right_bounded_entry * S ((S (fom_index_pfp_gcd_backward_new_common_right_bounded)) * bc) + (fom_value_pfp_gcd_backward_new_common_right_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_right_bounded_value_bound. fom_gap_pfp_gcd_backward_new_common_right_bounded_value_bound + S (fom_value_pfp_gcd_backward_new_common_right_bounded) = p))) /\\ ((exists pfgd_qb_gcd_backward_new_common_right pfgd_qc_gcd_backward_new_common_right pfgd_Q_gcd_backward_new_common_right pfgd_pb_gcd_backward_new_common_right pfgd_pc_gcd_backward_new_common_right pfgd_P_gcd_backward_new_common_right. ((((forall fom_index_pfp_gcd_backward_new_common_right_productleft. (exists fom_gap_pfp_gcd_backward_new_common_right_productleft_index_bound. fom_gap_pfp_gcd_backward_new_common_right_productleft_index_bound + S (fom_index_pfp_gcd_backward_new_common_right_productleft) = pfgd_Q_gcd_backward_new_common_right) -> exists fom_value_pfp_gcd_backward_new_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_backward_new_common_right_productleft_entry. fom_beta_height_pfp_gcd_backward_new_common_right_productleft_entry + S (fom_value_pfp_gcd_backward_new_common_right_productleft) = S ((S (fom_index_pfp_gcd_backward_new_common_right_productleft)) * pfgd_qc_gcd_backward_new_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_right_productleft_entry. pfgd_qb_gcd_backward_new_common_right = fom_beta_quotient_pfp_gcd_backward_new_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_backward_new_common_right_productleft)) * pfgd_qc_gcd_backward_new_common_right) + (fom_value_pfp_gcd_backward_new_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_right_productleft_value_bound. fom_gap_pfp_gcd_backward_new_common_right_productleft_value_bound + S (fom_value_pfp_gcd_backward_new_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_new_common_right_productright. (exists fom_gap_pfp_gcd_backward_new_common_right_productright_index_bound. fom_gap_pfp_gcd_backward_new_common_right_productright_index_bound + S (fom_index_pfp_gcd_backward_new_common_right_productright) = x2) -> exists fom_value_pfp_gcd_backward_new_common_right_productright. ((((exists fom_beta_height_pfp_gcd_backward_new_common_right_productright_entry. fom_beta_height_pfp_gcd_backward_new_common_right_productright_entry + S (fom_value_pfp_gcd_backward_new_common_right_productright) = S ((S (fom_index_pfp_gcd_backward_new_common_right_productright)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_right_productright_entry. x = fom_beta_quotient_pfp_gcd_backward_new_common_right_productright_entry * S ((S (fom_index_pfp_gcd_backward_new_common_right_productright)) * x1) + (fom_value_pfp_gcd_backward_new_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_right_productright_value_bound. fom_gap_pfp_gcd_backward_new_common_right_productright_value_bound + S (fom_value_pfp_gcd_backward_new_common_right_productright) = p))) /\\ (((((((pfgd_Q_gcd_backward_new_common_right)=0 \\/ (x2)=0) /\\ (((pfgd_P_gcd_backward_new_common_right)=0)))) \\/ (((~((pfgd_Q_gcd_backward_new_common_right)=0)) /\\ (((~((x2)=0)) /\\ (((pfgd_Q_gcd_backward_new_common_right)+(x2)=S (pfgd_P_gcd_backward_new_common_right)))))))) /\\ ((forall pfc_index_gcd_backward_new_common_right_productcoefficients. (exists pfa_gap_gcd_backward_new_common_right_productcoefficientsbound. pfa_gap_gcd_backward_new_common_right_productcoefficientsbound + S (pfc_index_gcd_backward_new_common_right_productcoefficients) = (pfgd_P_gcd_backward_new_common_right)) -> exists pfc_value_gcd_backward_new_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_backward_new_common_right_productcoefficientsentry. ff_h_pfp_gcd_backward_new_common_right_productcoefficientsentry + S (pfc_value_gcd_backward_new_common_right_productcoefficients) = S ((S (pfc_index_gcd_backward_new_common_right_productcoefficients)) * pfgd_pc_gcd_backward_new_common_right)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_productcoefficientsentry. pfgd_pb_gcd_backward_new_common_right = ff_q_pfp_gcd_backward_new_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_backward_new_common_right_productcoefficients)) * pfgd_pc_gcd_backward_new_common_right) + (pfc_value_gcd_backward_new_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_new_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_backward_new_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_backward_new_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_new_common_right_productcoefficients))) -> exists pfc_value_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_new_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_new_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_new_common_right_productcoefficientscoefficient) + (pfc_value_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_new_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_backward_new_common_right)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_new_common_right)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_backward_new_common_right = ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_new_common_right) + (pfc_left_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_backward_new_common_right)=(pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm) = (x2)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightentry. x = ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)) * x1) + (pfc_right_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(x2)=(pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_new_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_new_common_right_productcoefficients))) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_new_common_right_productcoefficients))) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_new_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_new_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_new_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_new_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_new_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_new_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_new_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_new_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_new_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_new_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_new_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_backward_new_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_new_common_right_equivalent pfrep_left_gcd_backward_new_common_right_equivalent pfrep_right_gcd_backward_new_common_right_equivalent. ((exists pfrep_position_gcd_backward_new_common_right_equivalentfirst. ((pfrep_position_gcd_backward_new_common_right_equivalentfirst+S (pfrep_power_gcd_backward_new_common_right_equivalent)=(pfgd_P_gcd_backward_new_common_right)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_right_equivalentfirstentry. ff_h_pfp_gcd_backward_new_common_right_equivalentfirstentry + S (pfrep_left_gcd_backward_new_common_right_equivalent) = S ((S (pfrep_position_gcd_backward_new_common_right_equivalentfirst)) * pfgd_pc_gcd_backward_new_common_right)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_equivalentfirstentry. pfgd_pb_gcd_backward_new_common_right = ff_q_pfp_gcd_backward_new_common_right_equivalentfirstentry * S ((S (pfrep_position_gcd_backward_new_common_right_equivalentfirst)) * pfgd_pc_gcd_backward_new_common_right) + (pfrep_left_gcd_backward_new_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_new_common_right_equivalentfirstoutside. pfrep_gap_gcd_backward_new_common_right_equivalentfirstoutside+(pfgd_P_gcd_backward_new_common_right)=(pfrep_power_gcd_backward_new_common_right_equivalent)) /\\ (((pfrep_left_gcd_backward_new_common_right_equivalent)=0))))) -> ((exists pfrep_position_gcd_backward_new_common_right_equivalentsecond. ((pfrep_position_gcd_backward_new_common_right_equivalentsecond+S (pfrep_power_gcd_backward_new_common_right_equivalent)=(S d)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_right_equivalentsecondentry. ff_h_pfp_gcd_backward_new_common_right_equivalentsecondentry + S (pfrep_right_gcd_backward_new_common_right_equivalent) = S ((S (pfrep_position_gcd_backward_new_common_right_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_equivalentsecondentry. bb = ff_q_pfp_gcd_backward_new_common_right_equivalentsecondentry * S ((S (pfrep_position_gcd_backward_new_common_right_equivalentsecond)) * bc) + (pfrep_right_gcd_backward_new_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_new_common_right_equivalentsecondoutside. pfrep_gap_gcd_backward_new_common_right_equivalentsecondoutside+(S d)=(pfrep_power_gcd_backward_new_common_right_equivalent)) /\\ (((pfrep_right_gcd_backward_new_common_right_equivalent)=0))))) -> pfrep_left_gcd_backward_new_common_right_equivalent=pfrep_right_gcd_backward_new_common_right_equivalent)))))))))", "have hm : (((((((forall fom_index_pfp_gcd_backward_new_common_left_bounded. (exists fom_gap_pfp_gcd_backward_new_common_left_bounded_index_bound. fom_gap_pfp_gcd_backward_new_common_left_bounded_index_bound + S (fom_index_pfp_gcd_backward_new_common_left_bounded) = L) -> exists fom_value_pfp_gcd_backward_new_common_left_bounded. ((((exists fom_beta_height_pfp_gcd_backward_new_common_left_bounded_entry. fom_beta_height_pfp_gcd_backward_new_common_left_bounded_entry + S (fom_value_pfp_gcd_backward_new_common_left_bounded) = S ((S (fom_index_pfp_gcd_backward_new_common_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_left_bounded_entry. ab = fom_beta_quotient_pfp_gcd_backward_new_common_left_bounded_entry * S ((S (fom_index_pfp_gcd_backward_new_common_left_bounded)) * ac) + (fom_value_pfp_gcd_backward_new_common_left_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_left_bounded_value_bound. fom_gap_pfp_gcd_backward_new_common_left_bounded_value_bound + S (fom_value_pfp_gcd_backward_new_common_left_bounded) = p))) /\\ ((exists pfgd_qb_gcd_backward_new_common_left pfgd_qc_gcd_backward_new_common_left pfgd_Q_gcd_backward_new_common_left pfgd_pb_gcd_backward_new_common_left pfgd_pc_gcd_backward_new_common_left pfgd_P_gcd_backward_new_common_left. ((((forall fom_index_pfp_gcd_backward_new_common_left_productleft. (exists fom_gap_pfp_gcd_backward_new_common_left_productleft_index_bound. fom_gap_pfp_gcd_backward_new_common_left_productleft_index_bound + S (fom_index_pfp_gcd_backward_new_common_left_productleft) = pfgd_Q_gcd_backward_new_common_left) -> exists fom_value_pfp_gcd_backward_new_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_backward_new_common_left_productleft_entry. fom_beta_height_pfp_gcd_backward_new_common_left_productleft_entry + S (fom_value_pfp_gcd_backward_new_common_left_productleft) = S ((S (fom_index_pfp_gcd_backward_new_common_left_productleft)) * pfgd_qc_gcd_backward_new_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_left_productleft_entry. pfgd_qb_gcd_backward_new_common_left = fom_beta_quotient_pfp_gcd_backward_new_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_backward_new_common_left_productleft)) * pfgd_qc_gcd_backward_new_common_left) + (fom_value_pfp_gcd_backward_new_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_left_productleft_value_bound. fom_gap_pfp_gcd_backward_new_common_left_productleft_value_bound + S (fom_value_pfp_gcd_backward_new_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_new_common_left_productright. (exists fom_gap_pfp_gcd_backward_new_common_left_productright_index_bound. fom_gap_pfp_gcd_backward_new_common_left_productright_index_bound + S (fom_index_pfp_gcd_backward_new_common_left_productright) = x2) -> exists fom_value_pfp_gcd_backward_new_common_left_productright. ((((exists fom_beta_height_pfp_gcd_backward_new_common_left_productright_entry. fom_beta_height_pfp_gcd_backward_new_common_left_productright_entry + S (fom_value_pfp_gcd_backward_new_common_left_productright) = S ((S (fom_index_pfp_gcd_backward_new_common_left_productright)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_left_productright_entry. x = fom_beta_quotient_pfp_gcd_backward_new_common_left_productright_entry * S ((S (fom_index_pfp_gcd_backward_new_common_left_productright)) * x1) + (fom_value_pfp_gcd_backward_new_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_left_productright_value_bound. fom_gap_pfp_gcd_backward_new_common_left_productright_value_bound + S (fom_value_pfp_gcd_backward_new_common_left_productright) = p))) /\\ (((((((pfgd_Q_gcd_backward_new_common_left)=0 \\/ (x2)=0) /\\ (((pfgd_P_gcd_backward_new_common_left)=0)))) \\/ (((~((pfgd_Q_gcd_backward_new_common_left)=0)) /\\ (((~((x2)=0)) /\\ (((pfgd_Q_gcd_backward_new_common_left)+(x2)=S (pfgd_P_gcd_backward_new_common_left)))))))) /\\ ((forall pfc_index_gcd_backward_new_common_left_productcoefficients. (exists pfa_gap_gcd_backward_new_common_left_productcoefficientsbound. pfa_gap_gcd_backward_new_common_left_productcoefficientsbound + S (pfc_index_gcd_backward_new_common_left_productcoefficients) = (pfgd_P_gcd_backward_new_common_left)) -> exists pfc_value_gcd_backward_new_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_backward_new_common_left_productcoefficientsentry. ff_h_pfp_gcd_backward_new_common_left_productcoefficientsentry + S (pfc_value_gcd_backward_new_common_left_productcoefficients) = S ((S (pfc_index_gcd_backward_new_common_left_productcoefficients)) * pfgd_pc_gcd_backward_new_common_left)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_productcoefficientsentry. pfgd_pb_gcd_backward_new_common_left = ff_q_pfp_gcd_backward_new_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_backward_new_common_left_productcoefficients)) * pfgd_pc_gcd_backward_new_common_left) + (pfc_value_gcd_backward_new_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_new_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_backward_new_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_backward_new_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_new_common_left_productcoefficients))) -> exists pfc_value_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_new_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_new_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_new_common_left_productcoefficientscoefficient) + (pfc_value_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_new_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_backward_new_common_left)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_new_common_left)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_backward_new_common_left = ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_new_common_left) + (pfc_left_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_backward_new_common_left)=(pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm) = (x2)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightentry. x = ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)) * x1) + (pfc_right_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(x2)=(pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_new_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_new_common_left_productcoefficients))) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_new_common_left_productcoefficients))) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_new_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_new_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_new_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_new_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_new_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_new_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_new_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_new_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_new_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_new_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_new_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_backward_new_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_new_common_left_equivalent pfrep_left_gcd_backward_new_common_left_equivalent pfrep_right_gcd_backward_new_common_left_equivalent. ((exists pfrep_position_gcd_backward_new_common_left_equivalentfirst. ((pfrep_position_gcd_backward_new_common_left_equivalentfirst+S (pfrep_power_gcd_backward_new_common_left_equivalent)=(pfgd_P_gcd_backward_new_common_left)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_left_equivalentfirstentry. ff_h_pfp_gcd_backward_new_common_left_equivalentfirstentry + S (pfrep_left_gcd_backward_new_common_left_equivalent) = S ((S (pfrep_position_gcd_backward_new_common_left_equivalentfirst)) * pfgd_pc_gcd_backward_new_common_left)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_equivalentfirstentry. pfgd_pb_gcd_backward_new_common_left = ff_q_pfp_gcd_backward_new_common_left_equivalentfirstentry * S ((S (pfrep_position_gcd_backward_new_common_left_equivalentfirst)) * pfgd_pc_gcd_backward_new_common_left) + (pfrep_left_gcd_backward_new_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_new_common_left_equivalentfirstoutside. pfrep_gap_gcd_backward_new_common_left_equivalentfirstoutside+(pfgd_P_gcd_backward_new_common_left)=(pfrep_power_gcd_backward_new_common_left_equivalent)) /\\ (((pfrep_left_gcd_backward_new_common_left_equivalent)=0))))) -> ((exists pfrep_position_gcd_backward_new_common_left_equivalentsecond. ((pfrep_position_gcd_backward_new_common_left_equivalentsecond+S (pfrep_power_gcd_backward_new_common_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_left_equivalentsecondentry. ff_h_pfp_gcd_backward_new_common_left_equivalentsecondentry + S (pfrep_right_gcd_backward_new_common_left_equivalent) = S ((S (pfrep_position_gcd_backward_new_common_left_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_equivalentsecondentry. ab = ff_q_pfp_gcd_backward_new_common_left_equivalentsecondentry * S ((S (pfrep_position_gcd_backward_new_common_left_equivalentsecond)) * ac) + (pfrep_right_gcd_backward_new_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_new_common_left_equivalentsecondoutside. pfrep_gap_gcd_backward_new_common_left_equivalentsecondoutside+(L)=(pfrep_power_gcd_backward_new_common_left_equivalent)) /\\ (((pfrep_right_gcd_backward_new_common_left_equivalent)=0))))) -> pfrep_left_gcd_backward_new_common_left_equivalent=pfrep_right_gcd_backward_new_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_backward_new_common_right_bounded. (exists fom_gap_pfp_gcd_backward_new_common_right_bounded_index_bound. fom_gap_pfp_gcd_backward_new_common_right_bounded_index_bound + S (fom_index_pfp_gcd_backward_new_common_right_bounded) = S d) -> exists fom_value_pfp_gcd_backward_new_common_right_bounded. ((((exists fom_beta_height_pfp_gcd_backward_new_common_right_bounded_entry. fom_beta_height_pfp_gcd_backward_new_common_right_bounded_entry + S (fom_value_pfp_gcd_backward_new_common_right_bounded) = S ((S (fom_index_pfp_gcd_backward_new_common_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_right_bounded_entry. bb = fom_beta_quotient_pfp_gcd_backward_new_common_right_bounded_entry * S ((S (fom_index_pfp_gcd_backward_new_common_right_bounded)) * bc) + (fom_value_pfp_gcd_backward_new_common_right_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_right_bounded_value_bound. fom_gap_pfp_gcd_backward_new_common_right_bounded_value_bound + S (fom_value_pfp_gcd_backward_new_common_right_bounded) = p))) /\\ ((exists pfgd_qb_gcd_backward_new_common_right pfgd_qc_gcd_backward_new_common_right pfgd_Q_gcd_backward_new_common_right pfgd_pb_gcd_backward_new_common_right pfgd_pc_gcd_backward_new_common_right pfgd_P_gcd_backward_new_common_right. ((((forall fom_index_pfp_gcd_backward_new_common_right_productleft. (exists fom_gap_pfp_gcd_backward_new_common_right_productleft_index_bound. fom_gap_pfp_gcd_backward_new_common_right_productleft_index_bound + S (fom_index_pfp_gcd_backward_new_common_right_productleft) = pfgd_Q_gcd_backward_new_common_right) -> exists fom_value_pfp_gcd_backward_new_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_backward_new_common_right_productleft_entry. fom_beta_height_pfp_gcd_backward_new_common_right_productleft_entry + S (fom_value_pfp_gcd_backward_new_common_right_productleft) = S ((S (fom_index_pfp_gcd_backward_new_common_right_productleft)) * pfgd_qc_gcd_backward_new_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_right_productleft_entry. pfgd_qb_gcd_backward_new_common_right = fom_beta_quotient_pfp_gcd_backward_new_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_backward_new_common_right_productleft)) * pfgd_qc_gcd_backward_new_common_right) + (fom_value_pfp_gcd_backward_new_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_right_productleft_value_bound. fom_gap_pfp_gcd_backward_new_common_right_productleft_value_bound + S (fom_value_pfp_gcd_backward_new_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_new_common_right_productright. (exists fom_gap_pfp_gcd_backward_new_common_right_productright_index_bound. fom_gap_pfp_gcd_backward_new_common_right_productright_index_bound + S (fom_index_pfp_gcd_backward_new_common_right_productright) = x2) -> exists fom_value_pfp_gcd_backward_new_common_right_productright. ((((exists fom_beta_height_pfp_gcd_backward_new_common_right_productright_entry. fom_beta_height_pfp_gcd_backward_new_common_right_productright_entry + S (fom_value_pfp_gcd_backward_new_common_right_productright) = S ((S (fom_index_pfp_gcd_backward_new_common_right_productright)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_right_productright_entry. x = fom_beta_quotient_pfp_gcd_backward_new_common_right_productright_entry * S ((S (fom_index_pfp_gcd_backward_new_common_right_productright)) * x1) + (fom_value_pfp_gcd_backward_new_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_right_productright_value_bound. fom_gap_pfp_gcd_backward_new_common_right_productright_value_bound + S (fom_value_pfp_gcd_backward_new_common_right_productright) = p))) /\\ (((((((pfgd_Q_gcd_backward_new_common_right)=0 \\/ (x2)=0) /\\ (((pfgd_P_gcd_backward_new_common_right)=0)))) \\/ (((~((pfgd_Q_gcd_backward_new_common_right)=0)) /\\ (((~((x2)=0)) /\\ (((pfgd_Q_gcd_backward_new_common_right)+(x2)=S (pfgd_P_gcd_backward_new_common_right)))))))) /\\ ((forall pfc_index_gcd_backward_new_common_right_productcoefficients. (exists pfa_gap_gcd_backward_new_common_right_productcoefficientsbound. pfa_gap_gcd_backward_new_common_right_productcoefficientsbound + S (pfc_index_gcd_backward_new_common_right_productcoefficients) = (pfgd_P_gcd_backward_new_common_right)) -> exists pfc_value_gcd_backward_new_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_backward_new_common_right_productcoefficientsentry. ff_h_pfp_gcd_backward_new_common_right_productcoefficientsentry + S (pfc_value_gcd_backward_new_common_right_productcoefficients) = S ((S (pfc_index_gcd_backward_new_common_right_productcoefficients)) * pfgd_pc_gcd_backward_new_common_right)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_productcoefficientsentry. pfgd_pb_gcd_backward_new_common_right = ff_q_pfp_gcd_backward_new_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_backward_new_common_right_productcoefficients)) * pfgd_pc_gcd_backward_new_common_right) + (pfc_value_gcd_backward_new_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_new_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_backward_new_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_backward_new_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_new_common_right_productcoefficients))) -> exists pfc_value_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_new_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_new_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_new_common_right_productcoefficientscoefficient) + (pfc_value_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_new_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_backward_new_common_right)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_new_common_right)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_backward_new_common_right = ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_new_common_right) + (pfc_left_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_backward_new_common_right)=(pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm) = (x2)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightentry. x = ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)) * x1) + (pfc_right_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(x2)=(pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_new_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_new_common_right_productcoefficients))) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_new_common_right_productcoefficients))) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_new_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_new_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_new_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_new_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_new_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_new_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_new_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_new_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_new_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_new_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_new_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_backward_new_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_new_common_right_equivalent pfrep_left_gcd_backward_new_common_right_equivalent pfrep_right_gcd_backward_new_common_right_equivalent. ((exists pfrep_position_gcd_backward_new_common_right_equivalentfirst. ((pfrep_position_gcd_backward_new_common_right_equivalentfirst+S (pfrep_power_gcd_backward_new_common_right_equivalent)=(pfgd_P_gcd_backward_new_common_right)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_right_equivalentfirstentry. ff_h_pfp_gcd_backward_new_common_right_equivalentfirstentry + S (pfrep_left_gcd_backward_new_common_right_equivalent) = S ((S (pfrep_position_gcd_backward_new_common_right_equivalentfirst)) * pfgd_pc_gcd_backward_new_common_right)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_equivalentfirstentry. pfgd_pb_gcd_backward_new_common_right = ff_q_pfp_gcd_backward_new_common_right_equivalentfirstentry * S ((S (pfrep_position_gcd_backward_new_common_right_equivalentfirst)) * pfgd_pc_gcd_backward_new_common_right) + (pfrep_left_gcd_backward_new_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_new_common_right_equivalentfirstoutside. pfrep_gap_gcd_backward_new_common_right_equivalentfirstoutside+(pfgd_P_gcd_backward_new_common_right)=(pfrep_power_gcd_backward_new_common_right_equivalent)) /\\ (((pfrep_left_gcd_backward_new_common_right_equivalent)=0))))) -> ((exists pfrep_position_gcd_backward_new_common_right_equivalentsecond. ((pfrep_position_gcd_backward_new_common_right_equivalentsecond+S (pfrep_power_gcd_backward_new_common_right_equivalent)=(S d)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_right_equivalentsecondentry. ff_h_pfp_gcd_backward_new_common_right_equivalentsecondentry + S (pfrep_right_gcd_backward_new_common_right_equivalent) = S ((S (pfrep_position_gcd_backward_new_common_right_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_equivalentsecondentry. bb = ff_q_pfp_gcd_backward_new_common_right_equivalentsecondentry * S ((S (pfrep_position_gcd_backward_new_common_right_equivalentsecond)) * bc) + (pfrep_right_gcd_backward_new_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_new_common_right_equivalentsecondoutside. pfrep_gap_gcd_backward_new_common_right_equivalentsecondoutside+(S d)=(pfrep_power_gcd_backward_new_common_right_equivalent)) /\\ (((pfrep_right_gcd_backward_new_common_right_equivalent)=0))))) -> pfrep_left_gcd_backward_new_common_right_equivalent=pfrep_right_gcd_backward_new_common_right_equivalent)))))))))) -> (((((forall fom_index_pfp_gcd_backward_old_common_left_bounded. (exists fom_gap_pfp_gcd_backward_old_common_left_bounded_index_bound. fom_gap_pfp_gcd_backward_old_common_left_bounded_index_bound + S (fom_index_pfp_gcd_backward_old_common_left_bounded) = S d) -> exists fom_value_pfp_gcd_backward_old_common_left_bounded. ((((exists fom_beta_height_pfp_gcd_backward_old_common_left_bounded_entry. fom_beta_height_pfp_gcd_backward_old_common_left_bounded_entry + S (fom_value_pfp_gcd_backward_old_common_left_bounded) = S ((S (fom_index_pfp_gcd_backward_old_common_left_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_old_common_left_bounded_entry. bb = fom_beta_quotient_pfp_gcd_backward_old_common_left_bounded_entry * S ((S (fom_index_pfp_gcd_backward_old_common_left_bounded)) * bc) + (fom_value_pfp_gcd_backward_old_common_left_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_old_common_left_bounded_value_bound. fom_gap_pfp_gcd_backward_old_common_left_bounded_value_bound + S (fom_value_pfp_gcd_backward_old_common_left_bounded) = p))) /\\ ((exists pfgd_qb_gcd_backward_old_common_left pfgd_qc_gcd_backward_old_common_left pfgd_Q_gcd_backward_old_common_left pfgd_pb_gcd_backward_old_common_left pfgd_pc_gcd_backward_old_common_left pfgd_P_gcd_backward_old_common_left. ((((forall fom_index_pfp_gcd_backward_old_common_left_productleft. (exists fom_gap_pfp_gcd_backward_old_common_left_productleft_index_bound. fom_gap_pfp_gcd_backward_old_common_left_productleft_index_bound + S (fom_index_pfp_gcd_backward_old_common_left_productleft) = pfgd_Q_gcd_backward_old_common_left) -> exists fom_value_pfp_gcd_backward_old_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_backward_old_common_left_productleft_entry. fom_beta_height_pfp_gcd_backward_old_common_left_productleft_entry + S (fom_value_pfp_gcd_backward_old_common_left_productleft) = S ((S (fom_index_pfp_gcd_backward_old_common_left_productleft)) * pfgd_qc_gcd_backward_old_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_backward_old_common_left_productleft_entry. pfgd_qb_gcd_backward_old_common_left = fom_beta_quotient_pfp_gcd_backward_old_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_backward_old_common_left_productleft)) * pfgd_qc_gcd_backward_old_common_left) + (fom_value_pfp_gcd_backward_old_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_old_common_left_productleft_value_bound. fom_gap_pfp_gcd_backward_old_common_left_productleft_value_bound + S (fom_value_pfp_gcd_backward_old_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_old_common_left_productright. (exists fom_gap_pfp_gcd_backward_old_common_left_productright_index_bound. fom_gap_pfp_gcd_backward_old_common_left_productright_index_bound + S (fom_index_pfp_gcd_backward_old_common_left_productright) = x2) -> exists fom_value_pfp_gcd_backward_old_common_left_productright. ((((exists fom_beta_height_pfp_gcd_backward_old_common_left_productright_entry. fom_beta_height_pfp_gcd_backward_old_common_left_productright_entry + S (fom_value_pfp_gcd_backward_old_common_left_productright) = S ((S (fom_index_pfp_gcd_backward_old_common_left_productright)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_backward_old_common_left_productright_entry. x = fom_beta_quotient_pfp_gcd_backward_old_common_left_productright_entry * S ((S (fom_index_pfp_gcd_backward_old_common_left_productright)) * x1) + (fom_value_pfp_gcd_backward_old_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_backward_old_common_left_productright_value_bound. fom_gap_pfp_gcd_backward_old_common_left_productright_value_bound + S (fom_value_pfp_gcd_backward_old_common_left_productright) = p))) /\\ (((((((pfgd_Q_gcd_backward_old_common_left)=0 \\/ (x2)=0) /\\ (((pfgd_P_gcd_backward_old_common_left)=0)))) \\/ (((~((pfgd_Q_gcd_backward_old_common_left)=0)) /\\ (((~((x2)=0)) /\\ (((pfgd_Q_gcd_backward_old_common_left)+(x2)=S (pfgd_P_gcd_backward_old_common_left)))))))) /\\ ((forall pfc_index_gcd_backward_old_common_left_productcoefficients. (exists pfa_gap_gcd_backward_old_common_left_productcoefficientsbound. pfa_gap_gcd_backward_old_common_left_productcoefficientsbound + S (pfc_index_gcd_backward_old_common_left_productcoefficients) = (pfgd_P_gcd_backward_old_common_left)) -> exists pfc_value_gcd_backward_old_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_backward_old_common_left_productcoefficientsentry. ff_h_pfp_gcd_backward_old_common_left_productcoefficientsentry + S (pfc_value_gcd_backward_old_common_left_productcoefficients) = S ((S (pfc_index_gcd_backward_old_common_left_productcoefficients)) * pfgd_pc_gcd_backward_old_common_left)) /\\ exists ff_q_pfp_gcd_backward_old_common_left_productcoefficientsentry. pfgd_pb_gcd_backward_old_common_left = ff_q_pfp_gcd_backward_old_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_backward_old_common_left_productcoefficients)) * pfgd_pc_gcd_backward_old_common_left) + (pfc_value_gcd_backward_old_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_old_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_backward_old_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_backward_old_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_old_common_left_productcoefficients))) -> exists pfc_value_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_old_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_old_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_old_common_left_productcoefficientscoefficient) + (pfc_value_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_old_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_backward_old_common_left)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_old_common_left)) /\\ exists ff_q_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_backward_old_common_left = ff_q_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_old_common_left) + (pfc_left_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_backward_old_common_left)=(pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm) = (x2)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightentry. x = ff_q_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm)) * x1) + (pfc_right_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(x2)=(pfc_complement_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_old_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_old_common_left_productcoefficients))) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_old_common_left_productcoefficients))) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_old_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_old_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_old_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_old_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_old_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_old_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_old_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_old_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_old_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_old_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_old_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_backward_old_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_old_common_left_equivalent pfrep_left_gcd_backward_old_common_left_equivalent pfrep_right_gcd_backward_old_common_left_equivalent. ((exists pfrep_position_gcd_backward_old_common_left_equivalentfirst. ((pfrep_position_gcd_backward_old_common_left_equivalentfirst+S (pfrep_power_gcd_backward_old_common_left_equivalent)=(pfgd_P_gcd_backward_old_common_left)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_left_equivalentfirstentry. ff_h_pfp_gcd_backward_old_common_left_equivalentfirstentry + S (pfrep_left_gcd_backward_old_common_left_equivalent) = S ((S (pfrep_position_gcd_backward_old_common_left_equivalentfirst)) * pfgd_pc_gcd_backward_old_common_left)) /\\ exists ff_q_pfp_gcd_backward_old_common_left_equivalentfirstentry. pfgd_pb_gcd_backward_old_common_left = ff_q_pfp_gcd_backward_old_common_left_equivalentfirstentry * S ((S (pfrep_position_gcd_backward_old_common_left_equivalentfirst)) * pfgd_pc_gcd_backward_old_common_left) + (pfrep_left_gcd_backward_old_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_old_common_left_equivalentfirstoutside. pfrep_gap_gcd_backward_old_common_left_equivalentfirstoutside+(pfgd_P_gcd_backward_old_common_left)=(pfrep_power_gcd_backward_old_common_left_equivalent)) /\\ (((pfrep_left_gcd_backward_old_common_left_equivalent)=0))))) -> ((exists pfrep_position_gcd_backward_old_common_left_equivalentsecond. ((pfrep_position_gcd_backward_old_common_left_equivalentsecond+S (pfrep_power_gcd_backward_old_common_left_equivalent)=(S d)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_left_equivalentsecondentry. ff_h_pfp_gcd_backward_old_common_left_equivalentsecondentry + S (pfrep_right_gcd_backward_old_common_left_equivalent) = S ((S (pfrep_position_gcd_backward_old_common_left_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_gcd_backward_old_common_left_equivalentsecondentry. bb = ff_q_pfp_gcd_backward_old_common_left_equivalentsecondentry * S ((S (pfrep_position_gcd_backward_old_common_left_equivalentsecond)) * bc) + (pfrep_right_gcd_backward_old_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_old_common_left_equivalentsecondoutside. pfrep_gap_gcd_backward_old_common_left_equivalentsecondoutside+(S d)=(pfrep_power_gcd_backward_old_common_left_equivalent)) /\\ (((pfrep_right_gcd_backward_old_common_left_equivalent)=0))))) -> pfrep_left_gcd_backward_old_common_left_equivalent=pfrep_right_gcd_backward_old_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_backward_old_common_right_bounded. (exists fom_gap_pfp_gcd_backward_old_common_right_bounded_index_bound. fom_gap_pfp_gcd_backward_old_common_right_bounded_index_bound + S (fom_index_pfp_gcd_backward_old_common_right_bounded) = R) -> exists fom_value_pfp_gcd_backward_old_common_right_bounded. ((((exists fom_beta_height_pfp_gcd_backward_old_common_right_bounded_entry. fom_beta_height_pfp_gcd_backward_old_common_right_bounded_entry + S (fom_value_pfp_gcd_backward_old_common_right_bounded) = S ((S (fom_index_pfp_gcd_backward_old_common_right_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_old_common_right_bounded_entry. rb = fom_beta_quotient_pfp_gcd_backward_old_common_right_bounded_entry * S ((S (fom_index_pfp_gcd_backward_old_common_right_bounded)) * rc) + (fom_value_pfp_gcd_backward_old_common_right_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_old_common_right_bounded_value_bound. fom_gap_pfp_gcd_backward_old_common_right_bounded_value_bound + S (fom_value_pfp_gcd_backward_old_common_right_bounded) = p))) /\\ ((exists pfgd_qb_gcd_backward_old_common_right pfgd_qc_gcd_backward_old_common_right pfgd_Q_gcd_backward_old_common_right pfgd_pb_gcd_backward_old_common_right pfgd_pc_gcd_backward_old_common_right pfgd_P_gcd_backward_old_common_right. ((((forall fom_index_pfp_gcd_backward_old_common_right_productleft. (exists fom_gap_pfp_gcd_backward_old_common_right_productleft_index_bound. fom_gap_pfp_gcd_backward_old_common_right_productleft_index_bound + S (fom_index_pfp_gcd_backward_old_common_right_productleft) = pfgd_Q_gcd_backward_old_common_right) -> exists fom_value_pfp_gcd_backward_old_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_backward_old_common_right_productleft_entry. fom_beta_height_pfp_gcd_backward_old_common_right_productleft_entry + S (fom_value_pfp_gcd_backward_old_common_right_productleft) = S ((S (fom_index_pfp_gcd_backward_old_common_right_productleft)) * pfgd_qc_gcd_backward_old_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_backward_old_common_right_productleft_entry. pfgd_qb_gcd_backward_old_common_right = fom_beta_quotient_pfp_gcd_backward_old_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_backward_old_common_right_productleft)) * pfgd_qc_gcd_backward_old_common_right) + (fom_value_pfp_gcd_backward_old_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_old_common_right_productleft_value_bound. fom_gap_pfp_gcd_backward_old_common_right_productleft_value_bound + S (fom_value_pfp_gcd_backward_old_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_old_common_right_productright. (exists fom_gap_pfp_gcd_backward_old_common_right_productright_index_bound. fom_gap_pfp_gcd_backward_old_common_right_productright_index_bound + S (fom_index_pfp_gcd_backward_old_common_right_productright) = x2) -> exists fom_value_pfp_gcd_backward_old_common_right_productright. ((((exists fom_beta_height_pfp_gcd_backward_old_common_right_productright_entry. fom_beta_height_pfp_gcd_backward_old_common_right_productright_entry + S (fom_value_pfp_gcd_backward_old_common_right_productright) = S ((S (fom_index_pfp_gcd_backward_old_common_right_productright)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_backward_old_common_right_productright_entry. x = fom_beta_quotient_pfp_gcd_backward_old_common_right_productright_entry * S ((S (fom_index_pfp_gcd_backward_old_common_right_productright)) * x1) + (fom_value_pfp_gcd_backward_old_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_backward_old_common_right_productright_value_bound. fom_gap_pfp_gcd_backward_old_common_right_productright_value_bound + S (fom_value_pfp_gcd_backward_old_common_right_productright) = p))) /\\ (((((((pfgd_Q_gcd_backward_old_common_right)=0 \\/ (x2)=0) /\\ (((pfgd_P_gcd_backward_old_common_right)=0)))) \\/ (((~((pfgd_Q_gcd_backward_old_common_right)=0)) /\\ (((~((x2)=0)) /\\ (((pfgd_Q_gcd_backward_old_common_right)+(x2)=S (pfgd_P_gcd_backward_old_common_right)))))))) /\\ ((forall pfc_index_gcd_backward_old_common_right_productcoefficients. (exists pfa_gap_gcd_backward_old_common_right_productcoefficientsbound. pfa_gap_gcd_backward_old_common_right_productcoefficientsbound + S (pfc_index_gcd_backward_old_common_right_productcoefficients) = (pfgd_P_gcd_backward_old_common_right)) -> exists pfc_value_gcd_backward_old_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_backward_old_common_right_productcoefficientsentry. ff_h_pfp_gcd_backward_old_common_right_productcoefficientsentry + S (pfc_value_gcd_backward_old_common_right_productcoefficients) = S ((S (pfc_index_gcd_backward_old_common_right_productcoefficients)) * pfgd_pc_gcd_backward_old_common_right)) /\\ exists ff_q_pfp_gcd_backward_old_common_right_productcoefficientsentry. pfgd_pb_gcd_backward_old_common_right = ff_q_pfp_gcd_backward_old_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_backward_old_common_right_productcoefficients)) * pfgd_pc_gcd_backward_old_common_right) + (pfc_value_gcd_backward_old_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_old_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_backward_old_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_backward_old_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_old_common_right_productcoefficients))) -> exists pfc_value_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_old_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_old_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_old_common_right_productcoefficientscoefficient) + (pfc_value_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_old_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_backward_old_common_right)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_old_common_right)) /\\ exists ff_q_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_backward_old_common_right = ff_q_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_old_common_right) + (pfc_left_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_backward_old_common_right)=(pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm) = (x2)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightentry. x = ff_q_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm)) * x1) + (pfc_right_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(x2)=(pfc_complement_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_old_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_old_common_right_productcoefficients))) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_old_common_right_productcoefficients))) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_old_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_old_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_old_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_old_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_old_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_old_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_old_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_old_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_old_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_old_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_old_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_backward_old_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_old_common_right_equivalent pfrep_left_gcd_backward_old_common_right_equivalent pfrep_right_gcd_backward_old_common_right_equivalent. ((exists pfrep_position_gcd_backward_old_common_right_equivalentfirst. ((pfrep_position_gcd_backward_old_common_right_equivalentfirst+S (pfrep_power_gcd_backward_old_common_right_equivalent)=(pfgd_P_gcd_backward_old_common_right)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_right_equivalentfirstentry. ff_h_pfp_gcd_backward_old_common_right_equivalentfirstentry + S (pfrep_left_gcd_backward_old_common_right_equivalent) = S ((S (pfrep_position_gcd_backward_old_common_right_equivalentfirst)) * pfgd_pc_gcd_backward_old_common_right)) /\\ exists ff_q_pfp_gcd_backward_old_common_right_equivalentfirstentry. pfgd_pb_gcd_backward_old_common_right = ff_q_pfp_gcd_backward_old_common_right_equivalentfirstentry * S ((S (pfrep_position_gcd_backward_old_common_right_equivalentfirst)) * pfgd_pc_gcd_backward_old_common_right) + (pfrep_left_gcd_backward_old_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_old_common_right_equivalentfirstoutside. pfrep_gap_gcd_backward_old_common_right_equivalentfirstoutside+(pfgd_P_gcd_backward_old_common_right)=(pfrep_power_gcd_backward_old_common_right_equivalent)) /\\ (((pfrep_left_gcd_backward_old_common_right_equivalent)=0))))) -> ((exists pfrep_position_gcd_backward_old_common_right_equivalentsecond. ((pfrep_position_gcd_backward_old_common_right_equivalentsecond+S (pfrep_power_gcd_backward_old_common_right_equivalent)=(R)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_right_equivalentsecondentry. ff_h_pfp_gcd_backward_old_common_right_equivalentsecondentry + S (pfrep_right_gcd_backward_old_common_right_equivalent) = S ((S (pfrep_position_gcd_backward_old_common_right_equivalentsecond)) * rc)) /\\ exists ff_q_pfp_gcd_backward_old_common_right_equivalentsecondentry. rb = ff_q_pfp_gcd_backward_old_common_right_equivalentsecondentry * S ((S (pfrep_position_gcd_backward_old_common_right_equivalentsecond)) * rc) + (pfrep_right_gcd_backward_old_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_old_common_right_equivalentsecondoutside. pfrep_gap_gcd_backward_old_common_right_equivalentsecondoutside+(R)=(pfrep_power_gcd_backward_old_common_right_equivalent)) /\\ (((pfrep_right_gcd_backward_old_common_right_equivalent)=0))))) -> pfrep_left_gcd_backward_old_common_right_equivalent=pfrep_right_gcd_backward_old_common_right_equivalent))))))))))) /\\ (((((((forall fom_index_pfp_gcd_backward_old_common_left_bounded. (exists fom_gap_pfp_gcd_backward_old_common_left_bounded_index_bound. fom_gap_pfp_gcd_backward_old_common_left_bounded_index_bound + S (fom_index_pfp_gcd_backward_old_common_left_bounded) = S d) -> exists fom_value_pfp_gcd_backward_old_common_left_bounded. ((((exists fom_beta_height_pfp_gcd_backward_old_common_left_bounded_entry. fom_beta_height_pfp_gcd_backward_old_common_left_bounded_entry + S (fom_value_pfp_gcd_backward_old_common_left_bounded) = S ((S (fom_index_pfp_gcd_backward_old_common_left_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_old_common_left_bounded_entry. bb = fom_beta_quotient_pfp_gcd_backward_old_common_left_bounded_entry * S ((S (fom_index_pfp_gcd_backward_old_common_left_bounded)) * bc) + (fom_value_pfp_gcd_backward_old_common_left_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_old_common_left_bounded_value_bound. fom_gap_pfp_gcd_backward_old_common_left_bounded_value_bound + S (fom_value_pfp_gcd_backward_old_common_left_bounded) = p))) /\\ ((exists pfgd_qb_gcd_backward_old_common_left pfgd_qc_gcd_backward_old_common_left pfgd_Q_gcd_backward_old_common_left pfgd_pb_gcd_backward_old_common_left pfgd_pc_gcd_backward_old_common_left pfgd_P_gcd_backward_old_common_left. ((((forall fom_index_pfp_gcd_backward_old_common_left_productleft. (exists fom_gap_pfp_gcd_backward_old_common_left_productleft_index_bound. fom_gap_pfp_gcd_backward_old_common_left_productleft_index_bound + S (fom_index_pfp_gcd_backward_old_common_left_productleft) = pfgd_Q_gcd_backward_old_common_left) -> exists fom_value_pfp_gcd_backward_old_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_backward_old_common_left_productleft_entry. fom_beta_height_pfp_gcd_backward_old_common_left_productleft_entry + S (fom_value_pfp_gcd_backward_old_common_left_productleft) = S ((S (fom_index_pfp_gcd_backward_old_common_left_productleft)) * pfgd_qc_gcd_backward_old_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_backward_old_common_left_productleft_entry. pfgd_qb_gcd_backward_old_common_left = fom_beta_quotient_pfp_gcd_backward_old_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_backward_old_common_left_productleft)) * pfgd_qc_gcd_backward_old_common_left) + (fom_value_pfp_gcd_backward_old_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_old_common_left_productleft_value_bound. fom_gap_pfp_gcd_backward_old_common_left_productleft_value_bound + S (fom_value_pfp_gcd_backward_old_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_old_common_left_productright. (exists fom_gap_pfp_gcd_backward_old_common_left_productright_index_bound. fom_gap_pfp_gcd_backward_old_common_left_productright_index_bound + S (fom_index_pfp_gcd_backward_old_common_left_productright) = x2) -> exists fom_value_pfp_gcd_backward_old_common_left_productright. ((((exists fom_beta_height_pfp_gcd_backward_old_common_left_productright_entry. fom_beta_height_pfp_gcd_backward_old_common_left_productright_entry + S (fom_value_pfp_gcd_backward_old_common_left_productright) = S ((S (fom_index_pfp_gcd_backward_old_common_left_productright)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_backward_old_common_left_productright_entry. x = fom_beta_quotient_pfp_gcd_backward_old_common_left_productright_entry * S ((S (fom_index_pfp_gcd_backward_old_common_left_productright)) * x1) + (fom_value_pfp_gcd_backward_old_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_backward_old_common_left_productright_value_bound. fom_gap_pfp_gcd_backward_old_common_left_productright_value_bound + S (fom_value_pfp_gcd_backward_old_common_left_productright) = p))) /\\ (((((((pfgd_Q_gcd_backward_old_common_left)=0 \\/ (x2)=0) /\\ (((pfgd_P_gcd_backward_old_common_left)=0)))) \\/ (((~((pfgd_Q_gcd_backward_old_common_left)=0)) /\\ (((~((x2)=0)) /\\ (((pfgd_Q_gcd_backward_old_common_left)+(x2)=S (pfgd_P_gcd_backward_old_common_left)))))))) /\\ ((forall pfc_index_gcd_backward_old_common_left_productcoefficients. (exists pfa_gap_gcd_backward_old_common_left_productcoefficientsbound. pfa_gap_gcd_backward_old_common_left_productcoefficientsbound + S (pfc_index_gcd_backward_old_common_left_productcoefficients) = (pfgd_P_gcd_backward_old_common_left)) -> exists pfc_value_gcd_backward_old_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_backward_old_common_left_productcoefficientsentry. ff_h_pfp_gcd_backward_old_common_left_productcoefficientsentry + S (pfc_value_gcd_backward_old_common_left_productcoefficients) = S ((S (pfc_index_gcd_backward_old_common_left_productcoefficients)) * pfgd_pc_gcd_backward_old_common_left)) /\\ exists ff_q_pfp_gcd_backward_old_common_left_productcoefficientsentry. pfgd_pb_gcd_backward_old_common_left = ff_q_pfp_gcd_backward_old_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_backward_old_common_left_productcoefficients)) * pfgd_pc_gcd_backward_old_common_left) + (pfc_value_gcd_backward_old_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_old_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_backward_old_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_backward_old_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_old_common_left_productcoefficients))) -> exists pfc_value_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_old_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_old_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_old_common_left_productcoefficientscoefficient) + (pfc_value_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_old_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_backward_old_common_left)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_old_common_left)) /\\ exists ff_q_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_backward_old_common_left = ff_q_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_old_common_left) + (pfc_left_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_backward_old_common_left)=(pfc_index_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm) = (x2)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightentry. x = ff_q_pfp_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm)) * x1) + (pfc_right_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_old_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(x2)=(pfc_complement_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_old_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_old_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_old_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_old_common_left_productcoefficients))) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_old_common_left_productcoefficients))) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_old_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_old_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_old_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_old_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_old_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_old_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_old_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_old_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_old_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_old_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_old_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_old_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_old_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_backward_old_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_old_common_left_equivalent pfrep_left_gcd_backward_old_common_left_equivalent pfrep_right_gcd_backward_old_common_left_equivalent. ((exists pfrep_position_gcd_backward_old_common_left_equivalentfirst. ((pfrep_position_gcd_backward_old_common_left_equivalentfirst+S (pfrep_power_gcd_backward_old_common_left_equivalent)=(pfgd_P_gcd_backward_old_common_left)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_left_equivalentfirstentry. ff_h_pfp_gcd_backward_old_common_left_equivalentfirstentry + S (pfrep_left_gcd_backward_old_common_left_equivalent) = S ((S (pfrep_position_gcd_backward_old_common_left_equivalentfirst)) * pfgd_pc_gcd_backward_old_common_left)) /\\ exists ff_q_pfp_gcd_backward_old_common_left_equivalentfirstentry. pfgd_pb_gcd_backward_old_common_left = ff_q_pfp_gcd_backward_old_common_left_equivalentfirstentry * S ((S (pfrep_position_gcd_backward_old_common_left_equivalentfirst)) * pfgd_pc_gcd_backward_old_common_left) + (pfrep_left_gcd_backward_old_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_old_common_left_equivalentfirstoutside. pfrep_gap_gcd_backward_old_common_left_equivalentfirstoutside+(pfgd_P_gcd_backward_old_common_left)=(pfrep_power_gcd_backward_old_common_left_equivalent)) /\\ (((pfrep_left_gcd_backward_old_common_left_equivalent)=0))))) -> ((exists pfrep_position_gcd_backward_old_common_left_equivalentsecond. ((pfrep_position_gcd_backward_old_common_left_equivalentsecond+S (pfrep_power_gcd_backward_old_common_left_equivalent)=(S d)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_left_equivalentsecondentry. ff_h_pfp_gcd_backward_old_common_left_equivalentsecondentry + S (pfrep_right_gcd_backward_old_common_left_equivalent) = S ((S (pfrep_position_gcd_backward_old_common_left_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_gcd_backward_old_common_left_equivalentsecondentry. bb = ff_q_pfp_gcd_backward_old_common_left_equivalentsecondentry * S ((S (pfrep_position_gcd_backward_old_common_left_equivalentsecond)) * bc) + (pfrep_right_gcd_backward_old_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_old_common_left_equivalentsecondoutside. pfrep_gap_gcd_backward_old_common_left_equivalentsecondoutside+(S d)=(pfrep_power_gcd_backward_old_common_left_equivalent)) /\\ (((pfrep_right_gcd_backward_old_common_left_equivalent)=0))))) -> pfrep_left_gcd_backward_old_common_left_equivalent=pfrep_right_gcd_backward_old_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_backward_old_common_right_bounded. (exists fom_gap_pfp_gcd_backward_old_common_right_bounded_index_bound. fom_gap_pfp_gcd_backward_old_common_right_bounded_index_bound + S (fom_index_pfp_gcd_backward_old_common_right_bounded) = R) -> exists fom_value_pfp_gcd_backward_old_common_right_bounded. ((((exists fom_beta_height_pfp_gcd_backward_old_common_right_bounded_entry. fom_beta_height_pfp_gcd_backward_old_common_right_bounded_entry + S (fom_value_pfp_gcd_backward_old_common_right_bounded) = S ((S (fom_index_pfp_gcd_backward_old_common_right_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_old_common_right_bounded_entry. rb = fom_beta_quotient_pfp_gcd_backward_old_common_right_bounded_entry * S ((S (fom_index_pfp_gcd_backward_old_common_right_bounded)) * rc) + (fom_value_pfp_gcd_backward_old_common_right_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_old_common_right_bounded_value_bound. fom_gap_pfp_gcd_backward_old_common_right_bounded_value_bound + S (fom_value_pfp_gcd_backward_old_common_right_bounded) = p))) /\\ ((exists pfgd_qb_gcd_backward_old_common_right pfgd_qc_gcd_backward_old_common_right pfgd_Q_gcd_backward_old_common_right pfgd_pb_gcd_backward_old_common_right pfgd_pc_gcd_backward_old_common_right pfgd_P_gcd_backward_old_common_right. ((((forall fom_index_pfp_gcd_backward_old_common_right_productleft. (exists fom_gap_pfp_gcd_backward_old_common_right_productleft_index_bound. fom_gap_pfp_gcd_backward_old_common_right_productleft_index_bound + S (fom_index_pfp_gcd_backward_old_common_right_productleft) = pfgd_Q_gcd_backward_old_common_right) -> exists fom_value_pfp_gcd_backward_old_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_backward_old_common_right_productleft_entry. fom_beta_height_pfp_gcd_backward_old_common_right_productleft_entry + S (fom_value_pfp_gcd_backward_old_common_right_productleft) = S ((S (fom_index_pfp_gcd_backward_old_common_right_productleft)) * pfgd_qc_gcd_backward_old_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_backward_old_common_right_productleft_entry. pfgd_qb_gcd_backward_old_common_right = fom_beta_quotient_pfp_gcd_backward_old_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_backward_old_common_right_productleft)) * pfgd_qc_gcd_backward_old_common_right) + (fom_value_pfp_gcd_backward_old_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_old_common_right_productleft_value_bound. fom_gap_pfp_gcd_backward_old_common_right_productleft_value_bound + S (fom_value_pfp_gcd_backward_old_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_old_common_right_productright. (exists fom_gap_pfp_gcd_backward_old_common_right_productright_index_bound. fom_gap_pfp_gcd_backward_old_common_right_productright_index_bound + S (fom_index_pfp_gcd_backward_old_common_right_productright) = x2) -> exists fom_value_pfp_gcd_backward_old_common_right_productright. ((((exists fom_beta_height_pfp_gcd_backward_old_common_right_productright_entry. fom_beta_height_pfp_gcd_backward_old_common_right_productright_entry + S (fom_value_pfp_gcd_backward_old_common_right_productright) = S ((S (fom_index_pfp_gcd_backward_old_common_right_productright)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_backward_old_common_right_productright_entry. x = fom_beta_quotient_pfp_gcd_backward_old_common_right_productright_entry * S ((S (fom_index_pfp_gcd_backward_old_common_right_productright)) * x1) + (fom_value_pfp_gcd_backward_old_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_backward_old_common_right_productright_value_bound. fom_gap_pfp_gcd_backward_old_common_right_productright_value_bound + S (fom_value_pfp_gcd_backward_old_common_right_productright) = p))) /\\ (((((((pfgd_Q_gcd_backward_old_common_right)=0 \\/ (x2)=0) /\\ (((pfgd_P_gcd_backward_old_common_right)=0)))) \\/ (((~((pfgd_Q_gcd_backward_old_common_right)=0)) /\\ (((~((x2)=0)) /\\ (((pfgd_Q_gcd_backward_old_common_right)+(x2)=S (pfgd_P_gcd_backward_old_common_right)))))))) /\\ ((forall pfc_index_gcd_backward_old_common_right_productcoefficients. (exists pfa_gap_gcd_backward_old_common_right_productcoefficientsbound. pfa_gap_gcd_backward_old_common_right_productcoefficientsbound + S (pfc_index_gcd_backward_old_common_right_productcoefficients) = (pfgd_P_gcd_backward_old_common_right)) -> exists pfc_value_gcd_backward_old_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_backward_old_common_right_productcoefficientsentry. ff_h_pfp_gcd_backward_old_common_right_productcoefficientsentry + S (pfc_value_gcd_backward_old_common_right_productcoefficients) = S ((S (pfc_index_gcd_backward_old_common_right_productcoefficients)) * pfgd_pc_gcd_backward_old_common_right)) /\\ exists ff_q_pfp_gcd_backward_old_common_right_productcoefficientsentry. pfgd_pb_gcd_backward_old_common_right = ff_q_pfp_gcd_backward_old_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_backward_old_common_right_productcoefficients)) * pfgd_pc_gcd_backward_old_common_right) + (pfc_value_gcd_backward_old_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_old_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_backward_old_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_backward_old_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_old_common_right_productcoefficients))) -> exists pfc_value_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_old_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_old_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_old_common_right_productcoefficientscoefficient) + (pfc_value_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_old_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_backward_old_common_right)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_old_common_right)) /\\ exists ff_q_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_backward_old_common_right = ff_q_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_old_common_right) + (pfc_left_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_backward_old_common_right)=(pfc_index_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm) = (x2)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightentry. x = ff_q_pfp_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm)) * x1) + (pfc_right_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_old_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(x2)=(pfc_complement_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_old_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_old_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_old_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_old_common_right_productcoefficients))) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_old_common_right_productcoefficients))) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_old_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_old_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_old_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_old_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_old_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_old_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_old_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_old_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_old_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_old_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_old_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_old_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_old_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_backward_old_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_old_common_right_equivalent pfrep_left_gcd_backward_old_common_right_equivalent pfrep_right_gcd_backward_old_common_right_equivalent. ((exists pfrep_position_gcd_backward_old_common_right_equivalentfirst. ((pfrep_position_gcd_backward_old_common_right_equivalentfirst+S (pfrep_power_gcd_backward_old_common_right_equivalent)=(pfgd_P_gcd_backward_old_common_right)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_right_equivalentfirstentry. ff_h_pfp_gcd_backward_old_common_right_equivalentfirstentry + S (pfrep_left_gcd_backward_old_common_right_equivalent) = S ((S (pfrep_position_gcd_backward_old_common_right_equivalentfirst)) * pfgd_pc_gcd_backward_old_common_right)) /\\ exists ff_q_pfp_gcd_backward_old_common_right_equivalentfirstentry. pfgd_pb_gcd_backward_old_common_right = ff_q_pfp_gcd_backward_old_common_right_equivalentfirstentry * S ((S (pfrep_position_gcd_backward_old_common_right_equivalentfirst)) * pfgd_pc_gcd_backward_old_common_right) + (pfrep_left_gcd_backward_old_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_old_common_right_equivalentfirstoutside. pfrep_gap_gcd_backward_old_common_right_equivalentfirstoutside+(pfgd_P_gcd_backward_old_common_right)=(pfrep_power_gcd_backward_old_common_right_equivalent)) /\\ (((pfrep_left_gcd_backward_old_common_right_equivalent)=0))))) -> ((exists pfrep_position_gcd_backward_old_common_right_equivalentsecond. ((pfrep_position_gcd_backward_old_common_right_equivalentsecond+S (pfrep_power_gcd_backward_old_common_right_equivalent)=(R)) /\\ ((((exists ff_h_pfp_gcd_backward_old_common_right_equivalentsecondentry. ff_h_pfp_gcd_backward_old_common_right_equivalentsecondentry + S (pfrep_right_gcd_backward_old_common_right_equivalent) = S ((S (pfrep_position_gcd_backward_old_common_right_equivalentsecond)) * rc)) /\\ exists ff_q_pfp_gcd_backward_old_common_right_equivalentsecondentry. rb = ff_q_pfp_gcd_backward_old_common_right_equivalentsecondentry * S ((S (pfrep_position_gcd_backward_old_common_right_equivalentsecond)) * rc) + (pfrep_right_gcd_backward_old_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_old_common_right_equivalentsecondoutside. pfrep_gap_gcd_backward_old_common_right_equivalentsecondoutside+(R)=(pfrep_power_gcd_backward_old_common_right_equivalent)) /\\ (((pfrep_right_gcd_backward_old_common_right_equivalent)=0))))) -> pfrep_left_gcd_backward_old_common_right_equivalent=pfrep_right_gcd_backward_old_common_right_equivalent)))))))))) -> (((((forall fom_index_pfp_gcd_backward_new_common_left_bounded. (exists fom_gap_pfp_gcd_backward_new_common_left_bounded_index_bound. fom_gap_pfp_gcd_backward_new_common_left_bounded_index_bound + S (fom_index_pfp_gcd_backward_new_common_left_bounded) = L) -> exists fom_value_pfp_gcd_backward_new_common_left_bounded. ((((exists fom_beta_height_pfp_gcd_backward_new_common_left_bounded_entry. fom_beta_height_pfp_gcd_backward_new_common_left_bounded_entry + S (fom_value_pfp_gcd_backward_new_common_left_bounded) = S ((S (fom_index_pfp_gcd_backward_new_common_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_left_bounded_entry. ab = fom_beta_quotient_pfp_gcd_backward_new_common_left_bounded_entry * S ((S (fom_index_pfp_gcd_backward_new_common_left_bounded)) * ac) + (fom_value_pfp_gcd_backward_new_common_left_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_left_bounded_value_bound. fom_gap_pfp_gcd_backward_new_common_left_bounded_value_bound + S (fom_value_pfp_gcd_backward_new_common_left_bounded) = p))) /\\ ((exists pfgd_qb_gcd_backward_new_common_left pfgd_qc_gcd_backward_new_common_left pfgd_Q_gcd_backward_new_common_left pfgd_pb_gcd_backward_new_common_left pfgd_pc_gcd_backward_new_common_left pfgd_P_gcd_backward_new_common_left. ((((forall fom_index_pfp_gcd_backward_new_common_left_productleft. (exists fom_gap_pfp_gcd_backward_new_common_left_productleft_index_bound. fom_gap_pfp_gcd_backward_new_common_left_productleft_index_bound + S (fom_index_pfp_gcd_backward_new_common_left_productleft) = pfgd_Q_gcd_backward_new_common_left) -> exists fom_value_pfp_gcd_backward_new_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_backward_new_common_left_productleft_entry. fom_beta_height_pfp_gcd_backward_new_common_left_productleft_entry + S (fom_value_pfp_gcd_backward_new_common_left_productleft) = S ((S (fom_index_pfp_gcd_backward_new_common_left_productleft)) * pfgd_qc_gcd_backward_new_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_left_productleft_entry. pfgd_qb_gcd_backward_new_common_left = fom_beta_quotient_pfp_gcd_backward_new_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_backward_new_common_left_productleft)) * pfgd_qc_gcd_backward_new_common_left) + (fom_value_pfp_gcd_backward_new_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_left_productleft_value_bound. fom_gap_pfp_gcd_backward_new_common_left_productleft_value_bound + S (fom_value_pfp_gcd_backward_new_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_new_common_left_productright. (exists fom_gap_pfp_gcd_backward_new_common_left_productright_index_bound. fom_gap_pfp_gcd_backward_new_common_left_productright_index_bound + S (fom_index_pfp_gcd_backward_new_common_left_productright) = x2) -> exists fom_value_pfp_gcd_backward_new_common_left_productright. ((((exists fom_beta_height_pfp_gcd_backward_new_common_left_productright_entry. fom_beta_height_pfp_gcd_backward_new_common_left_productright_entry + S (fom_value_pfp_gcd_backward_new_common_left_productright) = S ((S (fom_index_pfp_gcd_backward_new_common_left_productright)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_left_productright_entry. x = fom_beta_quotient_pfp_gcd_backward_new_common_left_productright_entry * S ((S (fom_index_pfp_gcd_backward_new_common_left_productright)) * x1) + (fom_value_pfp_gcd_backward_new_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_left_productright_value_bound. fom_gap_pfp_gcd_backward_new_common_left_productright_value_bound + S (fom_value_pfp_gcd_backward_new_common_left_productright) = p))) /\\ (((((((pfgd_Q_gcd_backward_new_common_left)=0 \\/ (x2)=0) /\\ (((pfgd_P_gcd_backward_new_common_left)=0)))) \\/ (((~((pfgd_Q_gcd_backward_new_common_left)=0)) /\\ (((~((x2)=0)) /\\ (((pfgd_Q_gcd_backward_new_common_left)+(x2)=S (pfgd_P_gcd_backward_new_common_left)))))))) /\\ ((forall pfc_index_gcd_backward_new_common_left_productcoefficients. (exists pfa_gap_gcd_backward_new_common_left_productcoefficientsbound. pfa_gap_gcd_backward_new_common_left_productcoefficientsbound + S (pfc_index_gcd_backward_new_common_left_productcoefficients) = (pfgd_P_gcd_backward_new_common_left)) -> exists pfc_value_gcd_backward_new_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_backward_new_common_left_productcoefficientsentry. ff_h_pfp_gcd_backward_new_common_left_productcoefficientsentry + S (pfc_value_gcd_backward_new_common_left_productcoefficients) = S ((S (pfc_index_gcd_backward_new_common_left_productcoefficients)) * pfgd_pc_gcd_backward_new_common_left)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_productcoefficientsentry. pfgd_pb_gcd_backward_new_common_left = ff_q_pfp_gcd_backward_new_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_backward_new_common_left_productcoefficients)) * pfgd_pc_gcd_backward_new_common_left) + (pfc_value_gcd_backward_new_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_new_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_backward_new_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_backward_new_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_new_common_left_productcoefficients))) -> exists pfc_value_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_new_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_new_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_new_common_left_productcoefficientscoefficient) + (pfc_value_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_new_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_backward_new_common_left)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_new_common_left)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_backward_new_common_left = ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_new_common_left) + (pfc_left_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_backward_new_common_left)=(pfc_index_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm) = (x2)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightentry. x = ff_q_pfp_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)) * x1) + (pfc_right_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_new_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(x2)=(pfc_complement_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_new_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_new_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_new_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_new_common_left_productcoefficients))) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_new_common_left_productcoefficients))) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_new_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_new_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_new_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_new_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_new_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_new_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_new_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_new_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_new_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_new_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_new_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_new_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_new_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_backward_new_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_new_common_left_equivalent pfrep_left_gcd_backward_new_common_left_equivalent pfrep_right_gcd_backward_new_common_left_equivalent. ((exists pfrep_position_gcd_backward_new_common_left_equivalentfirst. ((pfrep_position_gcd_backward_new_common_left_equivalentfirst+S (pfrep_power_gcd_backward_new_common_left_equivalent)=(pfgd_P_gcd_backward_new_common_left)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_left_equivalentfirstentry. ff_h_pfp_gcd_backward_new_common_left_equivalentfirstentry + S (pfrep_left_gcd_backward_new_common_left_equivalent) = S ((S (pfrep_position_gcd_backward_new_common_left_equivalentfirst)) * pfgd_pc_gcd_backward_new_common_left)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_equivalentfirstentry. pfgd_pb_gcd_backward_new_common_left = ff_q_pfp_gcd_backward_new_common_left_equivalentfirstentry * S ((S (pfrep_position_gcd_backward_new_common_left_equivalentfirst)) * pfgd_pc_gcd_backward_new_common_left) + (pfrep_left_gcd_backward_new_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_new_common_left_equivalentfirstoutside. pfrep_gap_gcd_backward_new_common_left_equivalentfirstoutside+(pfgd_P_gcd_backward_new_common_left)=(pfrep_power_gcd_backward_new_common_left_equivalent)) /\\ (((pfrep_left_gcd_backward_new_common_left_equivalent)=0))))) -> ((exists pfrep_position_gcd_backward_new_common_left_equivalentsecond. ((pfrep_position_gcd_backward_new_common_left_equivalentsecond+S (pfrep_power_gcd_backward_new_common_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_left_equivalentsecondentry. ff_h_pfp_gcd_backward_new_common_left_equivalentsecondentry + S (pfrep_right_gcd_backward_new_common_left_equivalent) = S ((S (pfrep_position_gcd_backward_new_common_left_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_gcd_backward_new_common_left_equivalentsecondentry. ab = ff_q_pfp_gcd_backward_new_common_left_equivalentsecondentry * S ((S (pfrep_position_gcd_backward_new_common_left_equivalentsecond)) * ac) + (pfrep_right_gcd_backward_new_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_new_common_left_equivalentsecondoutside. pfrep_gap_gcd_backward_new_common_left_equivalentsecondoutside+(L)=(pfrep_power_gcd_backward_new_common_left_equivalent)) /\\ (((pfrep_right_gcd_backward_new_common_left_equivalent)=0))))) -> pfrep_left_gcd_backward_new_common_left_equivalent=pfrep_right_gcd_backward_new_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_backward_new_common_right_bounded. (exists fom_gap_pfp_gcd_backward_new_common_right_bounded_index_bound. fom_gap_pfp_gcd_backward_new_common_right_bounded_index_bound + S (fom_index_pfp_gcd_backward_new_common_right_bounded) = S d) -> exists fom_value_pfp_gcd_backward_new_common_right_bounded. ((((exists fom_beta_height_pfp_gcd_backward_new_common_right_bounded_entry. fom_beta_height_pfp_gcd_backward_new_common_right_bounded_entry + S (fom_value_pfp_gcd_backward_new_common_right_bounded) = S ((S (fom_index_pfp_gcd_backward_new_common_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_right_bounded_entry. bb = fom_beta_quotient_pfp_gcd_backward_new_common_right_bounded_entry * S ((S (fom_index_pfp_gcd_backward_new_common_right_bounded)) * bc) + (fom_value_pfp_gcd_backward_new_common_right_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_right_bounded_value_bound. fom_gap_pfp_gcd_backward_new_common_right_bounded_value_bound + S (fom_value_pfp_gcd_backward_new_common_right_bounded) = p))) /\\ ((exists pfgd_qb_gcd_backward_new_common_right pfgd_qc_gcd_backward_new_common_right pfgd_Q_gcd_backward_new_common_right pfgd_pb_gcd_backward_new_common_right pfgd_pc_gcd_backward_new_common_right pfgd_P_gcd_backward_new_common_right. ((((forall fom_index_pfp_gcd_backward_new_common_right_productleft. (exists fom_gap_pfp_gcd_backward_new_common_right_productleft_index_bound. fom_gap_pfp_gcd_backward_new_common_right_productleft_index_bound + S (fom_index_pfp_gcd_backward_new_common_right_productleft) = pfgd_Q_gcd_backward_new_common_right) -> exists fom_value_pfp_gcd_backward_new_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_backward_new_common_right_productleft_entry. fom_beta_height_pfp_gcd_backward_new_common_right_productleft_entry + S (fom_value_pfp_gcd_backward_new_common_right_productleft) = S ((S (fom_index_pfp_gcd_backward_new_common_right_productleft)) * pfgd_qc_gcd_backward_new_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_right_productleft_entry. pfgd_qb_gcd_backward_new_common_right = fom_beta_quotient_pfp_gcd_backward_new_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_backward_new_common_right_productleft)) * pfgd_qc_gcd_backward_new_common_right) + (fom_value_pfp_gcd_backward_new_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_right_productleft_value_bound. fom_gap_pfp_gcd_backward_new_common_right_productleft_value_bound + S (fom_value_pfp_gcd_backward_new_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_new_common_right_productright. (exists fom_gap_pfp_gcd_backward_new_common_right_productright_index_bound. fom_gap_pfp_gcd_backward_new_common_right_productright_index_bound + S (fom_index_pfp_gcd_backward_new_common_right_productright) = x2) -> exists fom_value_pfp_gcd_backward_new_common_right_productright. ((((exists fom_beta_height_pfp_gcd_backward_new_common_right_productright_entry. fom_beta_height_pfp_gcd_backward_new_common_right_productright_entry + S (fom_value_pfp_gcd_backward_new_common_right_productright) = S ((S (fom_index_pfp_gcd_backward_new_common_right_productright)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_backward_new_common_right_productright_entry. x = fom_beta_quotient_pfp_gcd_backward_new_common_right_productright_entry * S ((S (fom_index_pfp_gcd_backward_new_common_right_productright)) * x1) + (fom_value_pfp_gcd_backward_new_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_backward_new_common_right_productright_value_bound. fom_gap_pfp_gcd_backward_new_common_right_productright_value_bound + S (fom_value_pfp_gcd_backward_new_common_right_productright) = p))) /\\ (((((((pfgd_Q_gcd_backward_new_common_right)=0 \\/ (x2)=0) /\\ (((pfgd_P_gcd_backward_new_common_right)=0)))) \\/ (((~((pfgd_Q_gcd_backward_new_common_right)=0)) /\\ (((~((x2)=0)) /\\ (((pfgd_Q_gcd_backward_new_common_right)+(x2)=S (pfgd_P_gcd_backward_new_common_right)))))))) /\\ ((forall pfc_index_gcd_backward_new_common_right_productcoefficients. (exists pfa_gap_gcd_backward_new_common_right_productcoefficientsbound. pfa_gap_gcd_backward_new_common_right_productcoefficientsbound + S (pfc_index_gcd_backward_new_common_right_productcoefficients) = (pfgd_P_gcd_backward_new_common_right)) -> exists pfc_value_gcd_backward_new_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_backward_new_common_right_productcoefficientsentry. ff_h_pfp_gcd_backward_new_common_right_productcoefficientsentry + S (pfc_value_gcd_backward_new_common_right_productcoefficients) = S ((S (pfc_index_gcd_backward_new_common_right_productcoefficients)) * pfgd_pc_gcd_backward_new_common_right)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_productcoefficientsentry. pfgd_pb_gcd_backward_new_common_right = ff_q_pfp_gcd_backward_new_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_backward_new_common_right_productcoefficients)) * pfgd_pc_gcd_backward_new_common_right) + (pfc_value_gcd_backward_new_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_new_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_backward_new_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_backward_new_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_new_common_right_productcoefficients))) -> exists pfc_value_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_new_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_new_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_new_common_right_productcoefficientscoefficient) + (pfc_value_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_new_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_backward_new_common_right)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_new_common_right)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_backward_new_common_right = ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_new_common_right) + (pfc_left_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_backward_new_common_right)=(pfc_index_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm) = (x2)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightentry. x = ff_q_pfp_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)) * x1) + (pfc_right_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_new_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(x2)=(pfc_complement_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_new_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_new_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_new_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_new_common_right_productcoefficients))) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_new_common_right_productcoefficients))) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_new_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_new_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_new_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_new_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_new_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_new_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_new_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_new_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_new_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_new_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_new_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_new_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_new_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_backward_new_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_new_common_right_equivalent pfrep_left_gcd_backward_new_common_right_equivalent pfrep_right_gcd_backward_new_common_right_equivalent. ((exists pfrep_position_gcd_backward_new_common_right_equivalentfirst. ((pfrep_position_gcd_backward_new_common_right_equivalentfirst+S (pfrep_power_gcd_backward_new_common_right_equivalent)=(pfgd_P_gcd_backward_new_common_right)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_right_equivalentfirstentry. ff_h_pfp_gcd_backward_new_common_right_equivalentfirstentry + S (pfrep_left_gcd_backward_new_common_right_equivalent) = S ((S (pfrep_position_gcd_backward_new_common_right_equivalentfirst)) * pfgd_pc_gcd_backward_new_common_right)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_equivalentfirstentry. pfgd_pb_gcd_backward_new_common_right = ff_q_pfp_gcd_backward_new_common_right_equivalentfirstentry * S ((S (pfrep_position_gcd_backward_new_common_right_equivalentfirst)) * pfgd_pc_gcd_backward_new_common_right) + (pfrep_left_gcd_backward_new_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_new_common_right_equivalentfirstoutside. pfrep_gap_gcd_backward_new_common_right_equivalentfirstoutside+(pfgd_P_gcd_backward_new_common_right)=(pfrep_power_gcd_backward_new_common_right_equivalent)) /\\ (((pfrep_left_gcd_backward_new_common_right_equivalent)=0))))) -> ((exists pfrep_position_gcd_backward_new_common_right_equivalentsecond. ((pfrep_position_gcd_backward_new_common_right_equivalentsecond+S (pfrep_power_gcd_backward_new_common_right_equivalent)=(S d)) /\\ ((((exists ff_h_pfp_gcd_backward_new_common_right_equivalentsecondentry. ff_h_pfp_gcd_backward_new_common_right_equivalentsecondentry + S (pfrep_right_gcd_backward_new_common_right_equivalent) = S ((S (pfrep_position_gcd_backward_new_common_right_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_gcd_backward_new_common_right_equivalentsecondentry. bb = ff_q_pfp_gcd_backward_new_common_right_equivalentsecondentry * S ((S (pfrep_position_gcd_backward_new_common_right_equivalentsecond)) * bc) + (pfrep_right_gcd_backward_new_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_new_common_right_equivalentsecondoutside. pfrep_gap_gcd_backward_new_common_right_equivalentsecondoutside+(S d)=(pfrep_power_gcd_backward_new_common_right_equivalent)) /\\ (((pfrep_right_gcd_backward_new_common_right_equivalent)=0))))) -> pfrep_left_gcd_backward_new_common_right_equivalent=pfrep_right_gcd_backward_new_common_right_equivalent)))))))))))))", "specialize prime_field_polynomial_division_execution_common_right_divisors (p)", "specialize prime_field_polynomial_division_execution_common_right_divisors (ab)", "specialize prime_field_polynomial_division_execution_common_right_divisors (ac)", "specialize prime_field_polynomial_division_execution_common_right_divisors (L)", "specialize prime_field_polynomial_division_execution_common_right_divisors (bb)", "specialize prime_field_polynomial_division_execution_common_right_divisors (bc)", "specialize prime_field_polynomial_division_execution_common_right_divisors (d)", "specialize prime_field_polynomial_division_execution_common_right_divisors (qb)", "specialize prime_field_polynomial_division_execution_common_right_divisors (qc)", "specialize prime_field_polynomial_division_execution_common_right_divisors (q)", "specialize prime_field_polynomial_division_execution_common_right_divisors (rb)", "specialize prime_field_polynomial_division_execution_common_right_divisors (rc)", "specialize prime_field_polynomial_division_execution_common_right_divisors (R)", "specialize prime_field_polynomial_division_execution_common_right_divisors (x)", "specialize prime_field_polynomial_division_execution_common_right_divisors (x1)", "specialize prime_field_polynomial_division_execution_common_right_divisors (x2)", "apply prime_field_polynomial_division_execution_common_right_divisors", "exact hp", "exact he", "cases hm", "apply hm_right", "exact hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "have hb : exists pfg_update_wb_gcd_backward_update pfg_update_wc_gcd_backward_update pfg_update_W_gcd_backward_update pfg_update_tb_gcd_backward_update pfg_update_tc_gcd_backward_update pfg_update_T_gcd_backward_update. ((((forall fom_index_pfp_gcd_backward_update_productleft. (exists fom_gap_pfp_gcd_backward_update_productleft_index_bound. fom_gap_pfp_gcd_backward_update_productleft_index_bound + S (fom_index_pfp_gcd_backward_update_productleft) = x8) -> exists fom_value_pfp_gcd_backward_update_productleft. ((((exists fom_beta_height_pfp_gcd_backward_update_productleft_entry. fom_beta_height_pfp_gcd_backward_update_productleft_entry + S (fom_value_pfp_gcd_backward_update_productleft) = S ((S (fom_index_pfp_gcd_backward_update_productleft)) * x7)) /\\ exists fom_beta_quotient_pfp_gcd_backward_update_productleft_entry. x6 = fom_beta_quotient_pfp_gcd_backward_update_productleft_entry * S ((S (fom_index_pfp_gcd_backward_update_productleft)) * x7) + (fom_value_pfp_gcd_backward_update_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_update_productleft_value_bound. fom_gap_pfp_gcd_backward_update_productleft_value_bound + S (fom_value_pfp_gcd_backward_update_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_update_productright. (exists fom_gap_pfp_gcd_backward_update_productright_index_bound. fom_gap_pfp_gcd_backward_update_productright_index_bound + S (fom_index_pfp_gcd_backward_update_productright) = q) -> exists fom_value_pfp_gcd_backward_update_productright. ((((exists fom_beta_height_pfp_gcd_backward_update_productright_entry. fom_beta_height_pfp_gcd_backward_update_productright_entry + S (fom_value_pfp_gcd_backward_update_productright) = S ((S (fom_index_pfp_gcd_backward_update_productright)) * qc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_update_productright_entry. qb = fom_beta_quotient_pfp_gcd_backward_update_productright_entry * S ((S (fom_index_pfp_gcd_backward_update_productright)) * qc) + (fom_value_pfp_gcd_backward_update_productright))) /\\ (exists fom_gap_pfp_gcd_backward_update_productright_value_bound. fom_gap_pfp_gcd_backward_update_productright_value_bound + S (fom_value_pfp_gcd_backward_update_productright) = p))) /\\ (((((((x8)=0 \\/ (q)=0) /\\ (((pfg_update_W_gcd_backward_update)=0)))) \\/ (((~((x8)=0)) /\\ (((~((q)=0)) /\\ (((x8)+(q)=S (pfg_update_W_gcd_backward_update)))))))) /\\ ((forall pfc_index_gcd_backward_update_productcoefficients. (exists pfa_gap_gcd_backward_update_productcoefficientsbound. pfa_gap_gcd_backward_update_productcoefficientsbound + S (pfc_index_gcd_backward_update_productcoefficients) = (pfg_update_W_gcd_backward_update)) -> exists pfc_value_gcd_backward_update_productcoefficients. ((((exists ff_h_pfp_gcd_backward_update_productcoefficientsentry. ff_h_pfp_gcd_backward_update_productcoefficientsentry + S (pfc_value_gcd_backward_update_productcoefficients) = S ((S (pfc_index_gcd_backward_update_productcoefficients)) * pfg_update_wc_gcd_backward_update)) /\\ exists ff_q_pfp_gcd_backward_update_productcoefficientsentry. pfg_update_wb_gcd_backward_update = ff_q_pfp_gcd_backward_update_productcoefficientsentry * S ((S (pfc_index_gcd_backward_update_productcoefficients)) * pfg_update_wc_gcd_backward_update) + (pfc_value_gcd_backward_update_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_update_productcoefficientscoefficient pfc_terms_scale_gcd_backward_update_productcoefficientscoefficient pfc_natural_sum_gcd_backward_update_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_update_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_update_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_update_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_update_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_update_productcoefficients))) -> exists pfc_value_gcd_backward_update_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_update_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_update_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_update_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_update_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_update_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_update_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_update_productcoefficientscoefficient = ff_q_pfp_gcd_backward_update_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_update_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_update_productcoefficientscoefficient) + (pfc_value_gcd_backward_update_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_update_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_update_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_update_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_update_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_update_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_update_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_update_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_update_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_update_productcoefficientscoefficientdiagonal) = (x8)) /\\ ((((exists ff_h_pfp_gcd_backward_update_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_update_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_update_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_update_productcoefficientscoefficientdiagonal)) * x7)) /\\ exists ff_q_pfp_gcd_backward_update_productcoefficientscoefficientdiagonaltermleftentry. x6 = ff_q_pfp_gcd_backward_update_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_update_productcoefficientscoefficientdiagonal)) * x7) + (pfc_left_gcd_backward_update_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_update_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_update_productcoefficientscoefficientdiagonaltermleftoutside+(x8)=(pfc_index_gcd_backward_update_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_update_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_update_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_update_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_update_productcoefficientscoefficientdiagonalterm) = (q)) /\\ ((((exists ff_h_pfp_gcd_backward_update_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_update_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_update_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_update_productcoefficientscoefficientdiagonalterm)) * qc)) /\\ exists ff_q_pfp_gcd_backward_update_productcoefficientscoefficientdiagonaltermrightentry. qb = ff_q_pfp_gcd_backward_update_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_update_productcoefficientscoefficientdiagonalterm)) * qc) + (pfc_right_gcd_backward_update_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_update_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_update_productcoefficientscoefficientdiagonaltermrightoutside+(q)=(pfc_complement_gcd_backward_update_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_update_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_update_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_update_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_update_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_update_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_update_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_update_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_update_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_update_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_update_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_update_productcoefficients))) * fs_v_pfc_gcd_backward_update_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_update_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_update_productcoefficients))) * fs_v_pfc_gcd_backward_update_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_update_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_update_productcoefficients)) -> exists fs_a_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_update_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_update_productcoefficientscoefficient = fs_q_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_update_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_update_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_update_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_update_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_update_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_update_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_update_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_update_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_update_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_update_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_update_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_update_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_update_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_update_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_update_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_update_productcoefficients) + (p) * pfa_offset_right_gcd_backward_update_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_gcd_backward_update_difference_left_bounded. (exists fom_gap_pfp_gcd_backward_update_difference_left_bounded_index_bound. fom_gap_pfp_gcd_backward_update_difference_left_bounded_index_bound + S (fom_index_pfp_gcd_backward_update_difference_left_bounded) = pfg_update_W_gcd_backward_update) -> exists fom_value_pfp_gcd_backward_update_difference_left_bounded. ((((exists fom_beta_height_pfp_gcd_backward_update_difference_left_bounded_entry. fom_beta_height_pfp_gcd_backward_update_difference_left_bounded_entry + S (fom_value_pfp_gcd_backward_update_difference_left_bounded) = S ((S (fom_index_pfp_gcd_backward_update_difference_left_bounded)) * pfg_update_wc_gcd_backward_update)) /\\ exists fom_beta_quotient_pfp_gcd_backward_update_difference_left_bounded_entry. pfg_update_wb_gcd_backward_update = fom_beta_quotient_pfp_gcd_backward_update_difference_left_bounded_entry * S ((S (fom_index_pfp_gcd_backward_update_difference_left_bounded)) * pfg_update_wc_gcd_backward_update) + (fom_value_pfp_gcd_backward_update_difference_left_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_update_difference_left_bounded_value_bound. fom_gap_pfp_gcd_backward_update_difference_left_bounded_value_bound + S (fom_value_pfp_gcd_backward_update_difference_left_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_backward_update_difference_right_bounded. (exists fom_gap_pfp_gcd_backward_update_difference_right_bounded_index_bound. fom_gap_pfp_gcd_backward_update_difference_right_bounded_index_bound + S (fom_index_pfp_gcd_backward_update_difference_right_bounded) = pfg_update_T_gcd_backward_update) -> exists fom_value_pfp_gcd_backward_update_difference_right_bounded. ((((exists fom_beta_height_pfp_gcd_backward_update_difference_right_bounded_entry. fom_beta_height_pfp_gcd_backward_update_difference_right_bounded_entry + S (fom_value_pfp_gcd_backward_update_difference_right_bounded) = S ((S (fom_index_pfp_gcd_backward_update_difference_right_bounded)) * pfg_update_tc_gcd_backward_update)) /\\ exists fom_beta_quotient_pfp_gcd_backward_update_difference_right_bounded_entry. pfg_update_tb_gcd_backward_update = fom_beta_quotient_pfp_gcd_backward_update_difference_right_bounded_entry * S ((S (fom_index_pfp_gcd_backward_update_difference_right_bounded)) * pfg_update_tc_gcd_backward_update) + (fom_value_pfp_gcd_backward_update_difference_right_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_update_difference_right_bounded_value_bound. fom_gap_pfp_gcd_backward_update_difference_right_bounded_value_bound + S (fom_value_pfp_gcd_backward_update_difference_right_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_backward_update_difference_result_bounded. (exists fom_gap_pfp_gcd_backward_update_difference_result_bounded_index_bound. fom_gap_pfp_gcd_backward_update_difference_result_bounded_index_bound + S (fom_index_pfp_gcd_backward_update_difference_result_bounded) = x5) -> exists fom_value_pfp_gcd_backward_update_difference_result_bounded. ((((exists fom_beta_height_pfp_gcd_backward_update_difference_result_bounded_entry. fom_beta_height_pfp_gcd_backward_update_difference_result_bounded_entry + S (fom_value_pfp_gcd_backward_update_difference_result_bounded) = S ((S (fom_index_pfp_gcd_backward_update_difference_result_bounded)) * x4)) /\\ exists fom_beta_quotient_pfp_gcd_backward_update_difference_result_bounded_entry. x3 = fom_beta_quotient_pfp_gcd_backward_update_difference_result_bounded_entry * S ((S (fom_index_pfp_gcd_backward_update_difference_result_bounded)) * x4) + (fom_value_pfp_gcd_backward_update_difference_result_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_update_difference_result_bounded_value_bound. fom_gap_pfp_gcd_backward_update_difference_result_bounded_value_bound + S (fom_value_pfp_gcd_backward_update_difference_result_bounded) = p))) /\\ ((exists pfga_ub_gcd_backward_update_difference pfga_uc_gcd_backward_update_difference pfga_vb_gcd_backward_update_difference pfga_vc_gcd_backward_update_difference pfga_tb_gcd_backward_update_difference pfga_tc_gcd_backward_update_difference pfga_K_gcd_backward_update_difference. ((((forall pfrep_power_gcd_backward_update_difference_left pfrep_left_gcd_backward_update_difference_left pfrep_right_gcd_backward_update_difference_left. ((exists pfrep_position_gcd_backward_update_difference_leftfirst. ((pfrep_position_gcd_backward_update_difference_leftfirst+S (pfrep_power_gcd_backward_update_difference_left)=(pfg_update_W_gcd_backward_update)) /\\ ((((exists ff_h_pfp_gcd_backward_update_difference_leftfirstentry. ff_h_pfp_gcd_backward_update_difference_leftfirstentry + S (pfrep_left_gcd_backward_update_difference_left) = S ((S (pfrep_position_gcd_backward_update_difference_leftfirst)) * pfg_update_wc_gcd_backward_update)) /\\ exists ff_q_pfp_gcd_backward_update_difference_leftfirstentry. pfg_update_wb_gcd_backward_update = ff_q_pfp_gcd_backward_update_difference_leftfirstentry * S ((S (pfrep_position_gcd_backward_update_difference_leftfirst)) * pfg_update_wc_gcd_backward_update) + (pfrep_left_gcd_backward_update_difference_left)))))) \\/ (((exists pfrep_gap_gcd_backward_update_difference_leftfirstoutside. pfrep_gap_gcd_backward_update_difference_leftfirstoutside+(pfg_update_W_gcd_backward_update)=(pfrep_power_gcd_backward_update_difference_left)) /\\ (((pfrep_left_gcd_backward_update_difference_left)=0))))) -> ((exists pfrep_position_gcd_backward_update_difference_leftsecond. ((pfrep_position_gcd_backward_update_difference_leftsecond+S (pfrep_power_gcd_backward_update_difference_left)=(pfga_K_gcd_backward_update_difference)) /\\ ((((exists ff_h_pfp_gcd_backward_update_difference_leftsecondentry. ff_h_pfp_gcd_backward_update_difference_leftsecondentry + S (pfrep_right_gcd_backward_update_difference_left) = S ((S (pfrep_position_gcd_backward_update_difference_leftsecond)) * pfga_uc_gcd_backward_update_difference)) /\\ exists ff_q_pfp_gcd_backward_update_difference_leftsecondentry. pfga_ub_gcd_backward_update_difference = ff_q_pfp_gcd_backward_update_difference_leftsecondentry * S ((S (pfrep_position_gcd_backward_update_difference_leftsecond)) * pfga_uc_gcd_backward_update_difference) + (pfrep_right_gcd_backward_update_difference_left)))))) \\/ (((exists pfrep_gap_gcd_backward_update_difference_leftsecondoutside. pfrep_gap_gcd_backward_update_difference_leftsecondoutside+(pfga_K_gcd_backward_update_difference)=(pfrep_power_gcd_backward_update_difference_left)) /\\ (((pfrep_right_gcd_backward_update_difference_left)=0))))) -> pfrep_left_gcd_backward_update_difference_left=pfrep_right_gcd_backward_update_difference_left) /\\ ((forall pfrep_power_gcd_backward_update_difference_right pfrep_left_gcd_backward_update_difference_right pfrep_right_gcd_backward_update_difference_right. ((exists pfrep_position_gcd_backward_update_difference_rightfirst. ((pfrep_position_gcd_backward_update_difference_rightfirst+S (pfrep_power_gcd_backward_update_difference_right)=(pfg_update_T_gcd_backward_update)) /\\ ((((exists ff_h_pfp_gcd_backward_update_difference_rightfirstentry. ff_h_pfp_gcd_backward_update_difference_rightfirstentry + S (pfrep_left_gcd_backward_update_difference_right) = S ((S (pfrep_position_gcd_backward_update_difference_rightfirst)) * pfg_update_tc_gcd_backward_update)) /\\ exists ff_q_pfp_gcd_backward_update_difference_rightfirstentry. pfg_update_tb_gcd_backward_update = ff_q_pfp_gcd_backward_update_difference_rightfirstentry * S ((S (pfrep_position_gcd_backward_update_difference_rightfirst)) * pfg_update_tc_gcd_backward_update) + (pfrep_left_gcd_backward_update_difference_right)))))) \\/ (((exists pfrep_gap_gcd_backward_update_difference_rightfirstoutside. pfrep_gap_gcd_backward_update_difference_rightfirstoutside+(pfg_update_T_gcd_backward_update)=(pfrep_power_gcd_backward_update_difference_right)) /\\ (((pfrep_left_gcd_backward_update_difference_right)=0))))) -> ((exists pfrep_position_gcd_backward_update_difference_rightsecond. ((pfrep_position_gcd_backward_update_difference_rightsecond+S (pfrep_power_gcd_backward_update_difference_right)=(pfga_K_gcd_backward_update_difference)) /\\ ((((exists ff_h_pfp_gcd_backward_update_difference_rightsecondentry. ff_h_pfp_gcd_backward_update_difference_rightsecondentry + S (pfrep_right_gcd_backward_update_difference_right) = S ((S (pfrep_position_gcd_backward_update_difference_rightsecond)) * pfga_vc_gcd_backward_update_difference)) /\\ exists ff_q_pfp_gcd_backward_update_difference_rightsecondentry. pfga_vb_gcd_backward_update_difference = ff_q_pfp_gcd_backward_update_difference_rightsecondentry * S ((S (pfrep_position_gcd_backward_update_difference_rightsecond)) * pfga_vc_gcd_backward_update_difference) + (pfrep_right_gcd_backward_update_difference_right)))))) \\/ (((exists pfrep_gap_gcd_backward_update_difference_rightsecondoutside. pfrep_gap_gcd_backward_update_difference_rightsecondoutside+(pfga_K_gcd_backward_update_difference)=(pfrep_power_gcd_backward_update_difference_right)) /\\ (((pfrep_right_gcd_backward_update_difference_right)=0))))) -> pfrep_left_gcd_backward_update_difference_right=pfrep_right_gcd_backward_update_difference_right)))) /\\ (((forall pfp_index_gcd_backward_update_difference_add. (exists pfa_gap_gcd_backward_update_difference_addindex. pfa_gap_gcd_backward_update_difference_addindex + S (pfp_index_gcd_backward_update_difference_add) = (pfga_K_gcd_backward_update_difference)) -> exists pfp_left_gcd_backward_update_difference_add pfp_right_gcd_backward_update_difference_add pfp_value_gcd_backward_update_difference_add. ((((exists ff_h_pfp_gcd_backward_update_difference_addleft. ff_h_pfp_gcd_backward_update_difference_addleft + S (pfp_left_gcd_backward_update_difference_add) = S ((S (pfp_index_gcd_backward_update_difference_add)) * pfga_uc_gcd_backward_update_difference)) /\\ exists ff_q_pfp_gcd_backward_update_difference_addleft. pfga_ub_gcd_backward_update_difference = ff_q_pfp_gcd_backward_update_difference_addleft * S ((S (pfp_index_gcd_backward_update_difference_add)) * pfga_uc_gcd_backward_update_difference) + (pfp_left_gcd_backward_update_difference_add))) /\\ (((((exists ff_h_pfp_gcd_backward_update_difference_addright. ff_h_pfp_gcd_backward_update_difference_addright + S (pfp_right_gcd_backward_update_difference_add) = S ((S (pfp_index_gcd_backward_update_difference_add)) * pfga_vc_gcd_backward_update_difference)) /\\ exists ff_q_pfp_gcd_backward_update_difference_addright. pfga_vb_gcd_backward_update_difference = ff_q_pfp_gcd_backward_update_difference_addright * S ((S (pfp_index_gcd_backward_update_difference_add)) * pfga_vc_gcd_backward_update_difference) + (pfp_right_gcd_backward_update_difference_add))) /\\ (((((exists ff_h_pfp_gcd_backward_update_difference_addtarget. ff_h_pfp_gcd_backward_update_difference_addtarget + S (pfp_value_gcd_backward_update_difference_add) = S ((S (pfp_index_gcd_backward_update_difference_add)) * pfga_tc_gcd_backward_update_difference)) /\\ exists ff_q_pfp_gcd_backward_update_difference_addtarget. pfga_tb_gcd_backward_update_difference = ff_q_pfp_gcd_backward_update_difference_addtarget * S ((S (pfp_index_gcd_backward_update_difference_add)) * pfga_tc_gcd_backward_update_difference) + (pfp_value_gcd_backward_update_difference_add))) /\\ ((((exists pfa_gap_gcd_backward_update_difference_addoperationleft. pfa_gap_gcd_backward_update_difference_addoperationleft + S (pfp_left_gcd_backward_update_difference_add) = (p)) /\\ (((exists pfa_gap_gcd_backward_update_difference_addoperationright. pfa_gap_gcd_backward_update_difference_addoperationright + S (pfp_right_gcd_backward_update_difference_add) = (p)) /\\ ((((exists pfa_gap_gcd_backward_update_difference_addoperationresultbound. pfa_gap_gcd_backward_update_difference_addoperationresultbound + S (pfp_value_gcd_backward_update_difference_add) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_update_difference_addoperationresultcongruence pfa_offset_right_gcd_backward_update_difference_addoperationresultcongruence. ((pfp_left_gcd_backward_update_difference_add) + (pfp_right_gcd_backward_update_difference_add)) + (p) * pfa_offset_left_gcd_backward_update_difference_addoperationresultcongruence = (pfp_value_gcd_backward_update_difference_add) + (p) * pfa_offset_right_gcd_backward_update_difference_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_update_difference_result pfrep_left_gcd_backward_update_difference_result pfrep_right_gcd_backward_update_difference_result. ((exists pfrep_position_gcd_backward_update_difference_resultfirst. ((pfrep_position_gcd_backward_update_difference_resultfirst+S (pfrep_power_gcd_backward_update_difference_result)=(pfga_K_gcd_backward_update_difference)) /\\ ((((exists ff_h_pfp_gcd_backward_update_difference_resultfirstentry. ff_h_pfp_gcd_backward_update_difference_resultfirstentry + S (pfrep_left_gcd_backward_update_difference_result) = S ((S (pfrep_position_gcd_backward_update_difference_resultfirst)) * pfga_tc_gcd_backward_update_difference)) /\\ exists ff_q_pfp_gcd_backward_update_difference_resultfirstentry. pfga_tb_gcd_backward_update_difference = ff_q_pfp_gcd_backward_update_difference_resultfirstentry * S ((S (pfrep_position_gcd_backward_update_difference_resultfirst)) * pfga_tc_gcd_backward_update_difference) + (pfrep_left_gcd_backward_update_difference_result)))))) \\/ (((exists pfrep_gap_gcd_backward_update_difference_resultfirstoutside. pfrep_gap_gcd_backward_update_difference_resultfirstoutside+(pfga_K_gcd_backward_update_difference)=(pfrep_power_gcd_backward_update_difference_result)) /\\ (((pfrep_left_gcd_backward_update_difference_result)=0))))) -> ((exists pfrep_position_gcd_backward_update_difference_resultsecond. ((pfrep_position_gcd_backward_update_difference_resultsecond+S (pfrep_power_gcd_backward_update_difference_result)=(x5)) /\\ ((((exists ff_h_pfp_gcd_backward_update_difference_resultsecondentry. ff_h_pfp_gcd_backward_update_difference_resultsecondentry + S (pfrep_right_gcd_backward_update_difference_result) = S ((S (pfrep_position_gcd_backward_update_difference_resultsecond)) * x4)) /\\ exists ff_q_pfp_gcd_backward_update_difference_resultsecondentry. x3 = ff_q_pfp_gcd_backward_update_difference_resultsecondentry * S ((S (pfrep_position_gcd_backward_update_difference_resultsecond)) * x4) + (pfrep_right_gcd_backward_update_difference_result)))))) \\/ (((exists pfrep_gap_gcd_backward_update_difference_resultsecondoutside. pfrep_gap_gcd_backward_update_difference_resultsecondoutside+(x5)=(pfrep_power_gcd_backward_update_difference_result)) /\\ (((pfrep_right_gcd_backward_update_difference_result)=0))))) -> pfrep_left_gcd_backward_update_difference_result=pfrep_right_gcd_backward_update_difference_result))))))))))))) /\\ ((exists pfgb_pb_gcd_backward_update_combination pfgb_pc_gcd_backward_update_combination pfgb_P_gcd_backward_update_combination pfgb_qb_gcd_backward_update_combination pfgb_qc_gcd_backward_update_combination pfgb_Q_gcd_backward_update_combination. ((((forall fom_index_pfp_gcd_backward_update_combination_leftleft. (exists fom_gap_pfp_gcd_backward_update_combination_leftleft_index_bound. fom_gap_pfp_gcd_backward_update_combination_leftleft_index_bound + S (fom_index_pfp_gcd_backward_update_combination_leftleft) = x8) -> exists fom_value_pfp_gcd_backward_update_combination_leftleft. ((((exists fom_beta_height_pfp_gcd_backward_update_combination_leftleft_entry. fom_beta_height_pfp_gcd_backward_update_combination_leftleft_entry + S (fom_value_pfp_gcd_backward_update_combination_leftleft) = S ((S (fom_index_pfp_gcd_backward_update_combination_leftleft)) * x7)) /\\ exists fom_beta_quotient_pfp_gcd_backward_update_combination_leftleft_entry. x6 = fom_beta_quotient_pfp_gcd_backward_update_combination_leftleft_entry * S ((S (fom_index_pfp_gcd_backward_update_combination_leftleft)) * x7) + (fom_value_pfp_gcd_backward_update_combination_leftleft))) /\\ (exists fom_gap_pfp_gcd_backward_update_combination_leftleft_value_bound. fom_gap_pfp_gcd_backward_update_combination_leftleft_value_bound + S (fom_value_pfp_gcd_backward_update_combination_leftleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_update_combination_leftright. (exists fom_gap_pfp_gcd_backward_update_combination_leftright_index_bound. fom_gap_pfp_gcd_backward_update_combination_leftright_index_bound + S (fom_index_pfp_gcd_backward_update_combination_leftright) = L) -> exists fom_value_pfp_gcd_backward_update_combination_leftright. ((((exists fom_beta_height_pfp_gcd_backward_update_combination_leftright_entry. fom_beta_height_pfp_gcd_backward_update_combination_leftright_entry + S (fom_value_pfp_gcd_backward_update_combination_leftright) = S ((S (fom_index_pfp_gcd_backward_update_combination_leftright)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_backward_update_combination_leftright_entry. ab = fom_beta_quotient_pfp_gcd_backward_update_combination_leftright_entry * S ((S (fom_index_pfp_gcd_backward_update_combination_leftright)) * ac) + (fom_value_pfp_gcd_backward_update_combination_leftright))) /\\ (exists fom_gap_pfp_gcd_backward_update_combination_leftright_value_bound. fom_gap_pfp_gcd_backward_update_combination_leftright_value_bound + S (fom_value_pfp_gcd_backward_update_combination_leftright) = p))) /\\ (((((((x8)=0 \\/ (L)=0) /\\ (((pfgb_P_gcd_backward_update_combination)=0)))) \\/ (((~((x8)=0)) /\\ (((~((L)=0)) /\\ (((x8)+(L)=S (pfgb_P_gcd_backward_update_combination)))))))) /\\ ((forall pfc_index_gcd_backward_update_combination_leftcoefficients. (exists pfa_gap_gcd_backward_update_combination_leftcoefficientsbound. pfa_gap_gcd_backward_update_combination_leftcoefficientsbound + S (pfc_index_gcd_backward_update_combination_leftcoefficients) = (pfgb_P_gcd_backward_update_combination)) -> exists pfc_value_gcd_backward_update_combination_leftcoefficients. ((((exists ff_h_pfp_gcd_backward_update_combination_leftcoefficientsentry. ff_h_pfp_gcd_backward_update_combination_leftcoefficientsentry + S (pfc_value_gcd_backward_update_combination_leftcoefficients) = S ((S (pfc_index_gcd_backward_update_combination_leftcoefficients)) * pfgb_pc_gcd_backward_update_combination)) /\\ exists ff_q_pfp_gcd_backward_update_combination_leftcoefficientsentry. pfgb_pb_gcd_backward_update_combination = ff_q_pfp_gcd_backward_update_combination_leftcoefficientsentry * S ((S (pfc_index_gcd_backward_update_combination_leftcoefficients)) * pfgb_pc_gcd_backward_update_combination) + (pfc_value_gcd_backward_update_combination_leftcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_update_combination_leftcoefficientscoefficient pfc_terms_scale_gcd_backward_update_combination_leftcoefficientscoefficient pfc_natural_sum_gcd_backward_update_combination_leftcoefficientscoefficient. ((forall pfc_index_gcd_backward_update_combination_leftcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_update_combination_leftcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_update_combination_leftcoefficients))) -> exists pfc_value_gcd_backward_update_combination_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_update_combination_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_update_combination_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_update_combination_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_update_combination_leftcoefficientscoefficient = ff_q_pfp_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_update_combination_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_update_combination_leftcoefficientscoefficient) + (pfc_value_gcd_backward_update_combination_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_update_combination_leftcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_update_combination_leftcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_update_combination_leftcoefficientscoefficientdiagonal) = (x8)) /\\ ((((exists ff_h_pfp_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_update_combination_leftcoefficientscoefficientdiagonal)) * x7)) /\\ exists ff_q_pfp_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermleftentry. x6 = ff_q_pfp_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_update_combination_leftcoefficientscoefficientdiagonal)) * x7) + (pfc_left_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermleftoutside+(x8)=(pfc_index_gcd_backward_update_combination_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_update_combination_leftcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_update_combination_leftcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_update_combination_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum fs_v_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_update_combination_leftcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_update_combination_leftcoefficients))) * fs_v_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_update_combination_leftcoefficients))) * fs_v_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_update_combination_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_update_combination_leftcoefficients)) -> exists fs_a_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_update_combination_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_update_combination_leftcoefficientscoefficient = fs_q_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_update_combination_leftcoefficientscoefficient) + (fs_a_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_update_combination_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_update_combination_leftcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_update_combination_leftcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_update_combination_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_update_combination_leftcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_update_combination_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_update_combination_leftcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_update_combination_leftcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_update_combination_leftcoefficients) + (p) * pfa_offset_right_gcd_backward_update_combination_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_gcd_backward_update_combination_rightleft. (exists fom_gap_pfp_gcd_backward_update_combination_rightleft_index_bound. fom_gap_pfp_gcd_backward_update_combination_rightleft_index_bound + S (fom_index_pfp_gcd_backward_update_combination_rightleft) = pfg_update_T_gcd_backward_update) -> exists fom_value_pfp_gcd_backward_update_combination_rightleft. ((((exists fom_beta_height_pfp_gcd_backward_update_combination_rightleft_entry. fom_beta_height_pfp_gcd_backward_update_combination_rightleft_entry + S (fom_value_pfp_gcd_backward_update_combination_rightleft) = S ((S (fom_index_pfp_gcd_backward_update_combination_rightleft)) * pfg_update_tc_gcd_backward_update)) /\\ exists fom_beta_quotient_pfp_gcd_backward_update_combination_rightleft_entry. pfg_update_tb_gcd_backward_update = fom_beta_quotient_pfp_gcd_backward_update_combination_rightleft_entry * S ((S (fom_index_pfp_gcd_backward_update_combination_rightleft)) * pfg_update_tc_gcd_backward_update) + (fom_value_pfp_gcd_backward_update_combination_rightleft))) /\\ (exists fom_gap_pfp_gcd_backward_update_combination_rightleft_value_bound. fom_gap_pfp_gcd_backward_update_combination_rightleft_value_bound + S (fom_value_pfp_gcd_backward_update_combination_rightleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_update_combination_rightright. (exists fom_gap_pfp_gcd_backward_update_combination_rightright_index_bound. fom_gap_pfp_gcd_backward_update_combination_rightright_index_bound + S (fom_index_pfp_gcd_backward_update_combination_rightright) = S d) -> exists fom_value_pfp_gcd_backward_update_combination_rightright. ((((exists fom_beta_height_pfp_gcd_backward_update_combination_rightright_entry. fom_beta_height_pfp_gcd_backward_update_combination_rightright_entry + S (fom_value_pfp_gcd_backward_update_combination_rightright) = S ((S (fom_index_pfp_gcd_backward_update_combination_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_update_combination_rightright_entry. bb = fom_beta_quotient_pfp_gcd_backward_update_combination_rightright_entry * S ((S (fom_index_pfp_gcd_backward_update_combination_rightright)) * bc) + (fom_value_pfp_gcd_backward_update_combination_rightright))) /\\ (exists fom_gap_pfp_gcd_backward_update_combination_rightright_value_bound. fom_gap_pfp_gcd_backward_update_combination_rightright_value_bound + S (fom_value_pfp_gcd_backward_update_combination_rightright) = p))) /\\ (((((((pfg_update_T_gcd_backward_update)=0 \\/ (S d)=0) /\\ (((pfgb_Q_gcd_backward_update_combination)=0)))) \\/ (((~((pfg_update_T_gcd_backward_update)=0)) /\\ (((~((S d)=0)) /\\ (((pfg_update_T_gcd_backward_update)+(S d)=S (pfgb_Q_gcd_backward_update_combination)))))))) /\\ ((forall pfc_index_gcd_backward_update_combination_rightcoefficients. (exists pfa_gap_gcd_backward_update_combination_rightcoefficientsbound. pfa_gap_gcd_backward_update_combination_rightcoefficientsbound + S (pfc_index_gcd_backward_update_combination_rightcoefficients) = (pfgb_Q_gcd_backward_update_combination)) -> exists pfc_value_gcd_backward_update_combination_rightcoefficients. ((((exists ff_h_pfp_gcd_backward_update_combination_rightcoefficientsentry. ff_h_pfp_gcd_backward_update_combination_rightcoefficientsentry + S (pfc_value_gcd_backward_update_combination_rightcoefficients) = S ((S (pfc_index_gcd_backward_update_combination_rightcoefficients)) * pfgb_qc_gcd_backward_update_combination)) /\\ exists ff_q_pfp_gcd_backward_update_combination_rightcoefficientsentry. pfgb_qb_gcd_backward_update_combination = ff_q_pfp_gcd_backward_update_combination_rightcoefficientsentry * S ((S (pfc_index_gcd_backward_update_combination_rightcoefficients)) * pfgb_qc_gcd_backward_update_combination) + (pfc_value_gcd_backward_update_combination_rightcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_update_combination_rightcoefficientscoefficient pfc_terms_scale_gcd_backward_update_combination_rightcoefficientscoefficient pfc_natural_sum_gcd_backward_update_combination_rightcoefficientscoefficient. ((forall pfc_index_gcd_backward_update_combination_rightcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_update_combination_rightcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_update_combination_rightcoefficients))) -> exists pfc_value_gcd_backward_update_combination_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_update_combination_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_update_combination_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_update_combination_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_update_combination_rightcoefficientscoefficient = ff_q_pfp_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_update_combination_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_update_combination_rightcoefficientscoefficient) + (pfc_value_gcd_backward_update_combination_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_update_combination_rightcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_update_combination_rightcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_update_combination_rightcoefficientscoefficientdiagonal) = (pfg_update_T_gcd_backward_update)) /\\ ((((exists ff_h_pfp_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_update_combination_rightcoefficientscoefficientdiagonal)) * pfg_update_tc_gcd_backward_update)) /\\ exists ff_q_pfp_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermleftentry. pfg_update_tb_gcd_backward_update = ff_q_pfp_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_update_combination_rightcoefficientscoefficientdiagonal)) * pfg_update_tc_gcd_backward_update) + (pfc_left_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermleftoutside+(pfg_update_T_gcd_backward_update)=(pfc_index_gcd_backward_update_combination_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_update_combination_rightcoefficientscoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_update_combination_rightcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_update_combination_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum fs_v_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_update_combination_rightcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_update_combination_rightcoefficients))) * fs_v_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_update_combination_rightcoefficients))) * fs_v_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_update_combination_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_update_combination_rightcoefficients)) -> exists fs_a_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_update_combination_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_update_combination_rightcoefficientscoefficient = fs_q_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_update_combination_rightcoefficientscoefficient) + (fs_a_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_update_combination_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_update_combination_rightcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_update_combination_rightcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_update_combination_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_update_combination_rightcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_update_combination_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_update_combination_rightcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_update_combination_rightcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_update_combination_rightcoefficients) + (p) * pfa_offset_right_gcd_backward_update_combination_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_gcd_backward_update_combination_sum_left_bounded. (exists fom_gap_pfp_gcd_backward_update_combination_sum_left_bounded_index_bound. fom_gap_pfp_gcd_backward_update_combination_sum_left_bounded_index_bound + S (fom_index_pfp_gcd_backward_update_combination_sum_left_bounded) = pfgb_P_gcd_backward_update_combination) -> exists fom_value_pfp_gcd_backward_update_combination_sum_left_bounded. ((((exists fom_beta_height_pfp_gcd_backward_update_combination_sum_left_bounded_entry. fom_beta_height_pfp_gcd_backward_update_combination_sum_left_bounded_entry + S (fom_value_pfp_gcd_backward_update_combination_sum_left_bounded) = S ((S (fom_index_pfp_gcd_backward_update_combination_sum_left_bounded)) * pfgb_pc_gcd_backward_update_combination)) /\\ exists fom_beta_quotient_pfp_gcd_backward_update_combination_sum_left_bounded_entry. pfgb_pb_gcd_backward_update_combination = fom_beta_quotient_pfp_gcd_backward_update_combination_sum_left_bounded_entry * S ((S (fom_index_pfp_gcd_backward_update_combination_sum_left_bounded)) * pfgb_pc_gcd_backward_update_combination) + (fom_value_pfp_gcd_backward_update_combination_sum_left_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_update_combination_sum_left_bounded_value_bound. fom_gap_pfp_gcd_backward_update_combination_sum_left_bounded_value_bound + S (fom_value_pfp_gcd_backward_update_combination_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_backward_update_combination_sum_right_bounded. (exists fom_gap_pfp_gcd_backward_update_combination_sum_right_bounded_index_bound. fom_gap_pfp_gcd_backward_update_combination_sum_right_bounded_index_bound + S (fom_index_pfp_gcd_backward_update_combination_sum_right_bounded) = pfgb_Q_gcd_backward_update_combination) -> exists fom_value_pfp_gcd_backward_update_combination_sum_right_bounded. ((((exists fom_beta_height_pfp_gcd_backward_update_combination_sum_right_bounded_entry. fom_beta_height_pfp_gcd_backward_update_combination_sum_right_bounded_entry + S (fom_value_pfp_gcd_backward_update_combination_sum_right_bounded) = S ((S (fom_index_pfp_gcd_backward_update_combination_sum_right_bounded)) * pfgb_qc_gcd_backward_update_combination)) /\\ exists fom_beta_quotient_pfp_gcd_backward_update_combination_sum_right_bounded_entry. pfgb_qb_gcd_backward_update_combination = fom_beta_quotient_pfp_gcd_backward_update_combination_sum_right_bounded_entry * S ((S (fom_index_pfp_gcd_backward_update_combination_sum_right_bounded)) * pfgb_qc_gcd_backward_update_combination) + (fom_value_pfp_gcd_backward_update_combination_sum_right_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_update_combination_sum_right_bounded_value_bound. fom_gap_pfp_gcd_backward_update_combination_sum_right_bounded_value_bound + S (fom_value_pfp_gcd_backward_update_combination_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_backward_update_combination_sum_result_bounded. (exists fom_gap_pfp_gcd_backward_update_combination_sum_result_bounded_index_bound. fom_gap_pfp_gcd_backward_update_combination_sum_result_bounded_index_bound + S (fom_index_pfp_gcd_backward_update_combination_sum_result_bounded) = x2) -> exists fom_value_pfp_gcd_backward_update_combination_sum_result_bounded. ((((exists fom_beta_height_pfp_gcd_backward_update_combination_sum_result_bounded_entry. fom_beta_height_pfp_gcd_backward_update_combination_sum_result_bounded_entry + S (fom_value_pfp_gcd_backward_update_combination_sum_result_bounded) = S ((S (fom_index_pfp_gcd_backward_update_combination_sum_result_bounded)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_backward_update_combination_sum_result_bounded_entry. x = fom_beta_quotient_pfp_gcd_backward_update_combination_sum_result_bounded_entry * S ((S (fom_index_pfp_gcd_backward_update_combination_sum_result_bounded)) * x1) + (fom_value_pfp_gcd_backward_update_combination_sum_result_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_update_combination_sum_result_bounded_value_bound. fom_gap_pfp_gcd_backward_update_combination_sum_result_bounded_value_bound + S (fom_value_pfp_gcd_backward_update_combination_sum_result_bounded) = p))) /\\ ((exists pfga_ub_gcd_backward_update_combination_sum pfga_uc_gcd_backward_update_combination_sum pfga_vb_gcd_backward_update_combination_sum pfga_vc_gcd_backward_update_combination_sum pfga_tb_gcd_backward_update_combination_sum pfga_tc_gcd_backward_update_combination_sum pfga_K_gcd_backward_update_combination_sum. ((((forall pfrep_power_gcd_backward_update_combination_sum_left pfrep_left_gcd_backward_update_combination_sum_left pfrep_right_gcd_backward_update_combination_sum_left. ((exists pfrep_position_gcd_backward_update_combination_sum_leftfirst. ((pfrep_position_gcd_backward_update_combination_sum_leftfirst+S (pfrep_power_gcd_backward_update_combination_sum_left)=(pfgb_P_gcd_backward_update_combination)) /\\ ((((exists ff_h_pfp_gcd_backward_update_combination_sum_leftfirstentry. ff_h_pfp_gcd_backward_update_combination_sum_leftfirstentry + S (pfrep_left_gcd_backward_update_combination_sum_left) = S ((S (pfrep_position_gcd_backward_update_combination_sum_leftfirst)) * pfgb_pc_gcd_backward_update_combination)) /\\ exists ff_q_pfp_gcd_backward_update_combination_sum_leftfirstentry. pfgb_pb_gcd_backward_update_combination = ff_q_pfp_gcd_backward_update_combination_sum_leftfirstentry * S ((S (pfrep_position_gcd_backward_update_combination_sum_leftfirst)) * pfgb_pc_gcd_backward_update_combination) + (pfrep_left_gcd_backward_update_combination_sum_left)))))) \\/ (((exists pfrep_gap_gcd_backward_update_combination_sum_leftfirstoutside. pfrep_gap_gcd_backward_update_combination_sum_leftfirstoutside+(pfgb_P_gcd_backward_update_combination)=(pfrep_power_gcd_backward_update_combination_sum_left)) /\\ (((pfrep_left_gcd_backward_update_combination_sum_left)=0))))) -> ((exists pfrep_position_gcd_backward_update_combination_sum_leftsecond. ((pfrep_position_gcd_backward_update_combination_sum_leftsecond+S (pfrep_power_gcd_backward_update_combination_sum_left)=(pfga_K_gcd_backward_update_combination_sum)) /\\ ((((exists ff_h_pfp_gcd_backward_update_combination_sum_leftsecondentry. ff_h_pfp_gcd_backward_update_combination_sum_leftsecondentry + S (pfrep_right_gcd_backward_update_combination_sum_left) = S ((S (pfrep_position_gcd_backward_update_combination_sum_leftsecond)) * pfga_uc_gcd_backward_update_combination_sum)) /\\ exists ff_q_pfp_gcd_backward_update_combination_sum_leftsecondentry. pfga_ub_gcd_backward_update_combination_sum = ff_q_pfp_gcd_backward_update_combination_sum_leftsecondentry * S ((S (pfrep_position_gcd_backward_update_combination_sum_leftsecond)) * pfga_uc_gcd_backward_update_combination_sum) + (pfrep_right_gcd_backward_update_combination_sum_left)))))) \\/ (((exists pfrep_gap_gcd_backward_update_combination_sum_leftsecondoutside. pfrep_gap_gcd_backward_update_combination_sum_leftsecondoutside+(pfga_K_gcd_backward_update_combination_sum)=(pfrep_power_gcd_backward_update_combination_sum_left)) /\\ (((pfrep_right_gcd_backward_update_combination_sum_left)=0))))) -> pfrep_left_gcd_backward_update_combination_sum_left=pfrep_right_gcd_backward_update_combination_sum_left) /\\ ((forall pfrep_power_gcd_backward_update_combination_sum_right pfrep_left_gcd_backward_update_combination_sum_right pfrep_right_gcd_backward_update_combination_sum_right. ((exists pfrep_position_gcd_backward_update_combination_sum_rightfirst. ((pfrep_position_gcd_backward_update_combination_sum_rightfirst+S (pfrep_power_gcd_backward_update_combination_sum_right)=(pfgb_Q_gcd_backward_update_combination)) /\\ ((((exists ff_h_pfp_gcd_backward_update_combination_sum_rightfirstentry. ff_h_pfp_gcd_backward_update_combination_sum_rightfirstentry + S (pfrep_left_gcd_backward_update_combination_sum_right) = S ((S (pfrep_position_gcd_backward_update_combination_sum_rightfirst)) * pfgb_qc_gcd_backward_update_combination)) /\\ exists ff_q_pfp_gcd_backward_update_combination_sum_rightfirstentry. pfgb_qb_gcd_backward_update_combination = ff_q_pfp_gcd_backward_update_combination_sum_rightfirstentry * S ((S (pfrep_position_gcd_backward_update_combination_sum_rightfirst)) * pfgb_qc_gcd_backward_update_combination) + (pfrep_left_gcd_backward_update_combination_sum_right)))))) \\/ (((exists pfrep_gap_gcd_backward_update_combination_sum_rightfirstoutside. pfrep_gap_gcd_backward_update_combination_sum_rightfirstoutside+(pfgb_Q_gcd_backward_update_combination)=(pfrep_power_gcd_backward_update_combination_sum_right)) /\\ (((pfrep_left_gcd_backward_update_combination_sum_right)=0))))) -> ((exists pfrep_position_gcd_backward_update_combination_sum_rightsecond. ((pfrep_position_gcd_backward_update_combination_sum_rightsecond+S (pfrep_power_gcd_backward_update_combination_sum_right)=(pfga_K_gcd_backward_update_combination_sum)) /\\ ((((exists ff_h_pfp_gcd_backward_update_combination_sum_rightsecondentry. ff_h_pfp_gcd_backward_update_combination_sum_rightsecondentry + S (pfrep_right_gcd_backward_update_combination_sum_right) = S ((S (pfrep_position_gcd_backward_update_combination_sum_rightsecond)) * pfga_vc_gcd_backward_update_combination_sum)) /\\ exists ff_q_pfp_gcd_backward_update_combination_sum_rightsecondentry. pfga_vb_gcd_backward_update_combination_sum = ff_q_pfp_gcd_backward_update_combination_sum_rightsecondentry * S ((S (pfrep_position_gcd_backward_update_combination_sum_rightsecond)) * pfga_vc_gcd_backward_update_combination_sum) + (pfrep_right_gcd_backward_update_combination_sum_right)))))) \\/ (((exists pfrep_gap_gcd_backward_update_combination_sum_rightsecondoutside. pfrep_gap_gcd_backward_update_combination_sum_rightsecondoutside+(pfga_K_gcd_backward_update_combination_sum)=(pfrep_power_gcd_backward_update_combination_sum_right)) /\\ (((pfrep_right_gcd_backward_update_combination_sum_right)=0))))) -> pfrep_left_gcd_backward_update_combination_sum_right=pfrep_right_gcd_backward_update_combination_sum_right)))) /\\ (((forall pfp_index_gcd_backward_update_combination_sum_add. (exists pfa_gap_gcd_backward_update_combination_sum_addindex. pfa_gap_gcd_backward_update_combination_sum_addindex + S (pfp_index_gcd_backward_update_combination_sum_add) = (pfga_K_gcd_backward_update_combination_sum)) -> exists pfp_left_gcd_backward_update_combination_sum_add pfp_right_gcd_backward_update_combination_sum_add pfp_value_gcd_backward_update_combination_sum_add. ((((exists ff_h_pfp_gcd_backward_update_combination_sum_addleft. ff_h_pfp_gcd_backward_update_combination_sum_addleft + S (pfp_left_gcd_backward_update_combination_sum_add) = S ((S (pfp_index_gcd_backward_update_combination_sum_add)) * pfga_uc_gcd_backward_update_combination_sum)) /\\ exists ff_q_pfp_gcd_backward_update_combination_sum_addleft. pfga_ub_gcd_backward_update_combination_sum = ff_q_pfp_gcd_backward_update_combination_sum_addleft * S ((S (pfp_index_gcd_backward_update_combination_sum_add)) * pfga_uc_gcd_backward_update_combination_sum) + (pfp_left_gcd_backward_update_combination_sum_add))) /\\ (((((exists ff_h_pfp_gcd_backward_update_combination_sum_addright. ff_h_pfp_gcd_backward_update_combination_sum_addright + S (pfp_right_gcd_backward_update_combination_sum_add) = S ((S (pfp_index_gcd_backward_update_combination_sum_add)) * pfga_vc_gcd_backward_update_combination_sum)) /\\ exists ff_q_pfp_gcd_backward_update_combination_sum_addright. pfga_vb_gcd_backward_update_combination_sum = ff_q_pfp_gcd_backward_update_combination_sum_addright * S ((S (pfp_index_gcd_backward_update_combination_sum_add)) * pfga_vc_gcd_backward_update_combination_sum) + (pfp_right_gcd_backward_update_combination_sum_add))) /\\ (((((exists ff_h_pfp_gcd_backward_update_combination_sum_addtarget. ff_h_pfp_gcd_backward_update_combination_sum_addtarget + S (pfp_value_gcd_backward_update_combination_sum_add) = S ((S (pfp_index_gcd_backward_update_combination_sum_add)) * pfga_tc_gcd_backward_update_combination_sum)) /\\ exists ff_q_pfp_gcd_backward_update_combination_sum_addtarget. pfga_tb_gcd_backward_update_combination_sum = ff_q_pfp_gcd_backward_update_combination_sum_addtarget * S ((S (pfp_index_gcd_backward_update_combination_sum_add)) * pfga_tc_gcd_backward_update_combination_sum) + (pfp_value_gcd_backward_update_combination_sum_add))) /\\ ((((exists pfa_gap_gcd_backward_update_combination_sum_addoperationleft. pfa_gap_gcd_backward_update_combination_sum_addoperationleft + S (pfp_left_gcd_backward_update_combination_sum_add) = (p)) /\\ (((exists pfa_gap_gcd_backward_update_combination_sum_addoperationright. pfa_gap_gcd_backward_update_combination_sum_addoperationright + S (pfp_right_gcd_backward_update_combination_sum_add) = (p)) /\\ ((((exists pfa_gap_gcd_backward_update_combination_sum_addoperationresultbound. pfa_gap_gcd_backward_update_combination_sum_addoperationresultbound + S (pfp_value_gcd_backward_update_combination_sum_add) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_update_combination_sum_addoperationresultcongruence pfa_offset_right_gcd_backward_update_combination_sum_addoperationresultcongruence. ((pfp_left_gcd_backward_update_combination_sum_add) + (pfp_right_gcd_backward_update_combination_sum_add)) + (p) * pfa_offset_left_gcd_backward_update_combination_sum_addoperationresultcongruence = (pfp_value_gcd_backward_update_combination_sum_add) + (p) * pfa_offset_right_gcd_backward_update_combination_sum_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_update_combination_sum_result pfrep_left_gcd_backward_update_combination_sum_result pfrep_right_gcd_backward_update_combination_sum_result. ((exists pfrep_position_gcd_backward_update_combination_sum_resultfirst. ((pfrep_position_gcd_backward_update_combination_sum_resultfirst+S (pfrep_power_gcd_backward_update_combination_sum_result)=(pfga_K_gcd_backward_update_combination_sum)) /\\ ((((exists ff_h_pfp_gcd_backward_update_combination_sum_resultfirstentry. ff_h_pfp_gcd_backward_update_combination_sum_resultfirstentry + S (pfrep_left_gcd_backward_update_combination_sum_result) = S ((S (pfrep_position_gcd_backward_update_combination_sum_resultfirst)) * pfga_tc_gcd_backward_update_combination_sum)) /\\ exists ff_q_pfp_gcd_backward_update_combination_sum_resultfirstentry. pfga_tb_gcd_backward_update_combination_sum = ff_q_pfp_gcd_backward_update_combination_sum_resultfirstentry * S ((S (pfrep_position_gcd_backward_update_combination_sum_resultfirst)) * pfga_tc_gcd_backward_update_combination_sum) + (pfrep_left_gcd_backward_update_combination_sum_result)))))) \\/ (((exists pfrep_gap_gcd_backward_update_combination_sum_resultfirstoutside. pfrep_gap_gcd_backward_update_combination_sum_resultfirstoutside+(pfga_K_gcd_backward_update_combination_sum)=(pfrep_power_gcd_backward_update_combination_sum_result)) /\\ (((pfrep_left_gcd_backward_update_combination_sum_result)=0))))) -> ((exists pfrep_position_gcd_backward_update_combination_sum_resultsecond. ((pfrep_position_gcd_backward_update_combination_sum_resultsecond+S (pfrep_power_gcd_backward_update_combination_sum_result)=(x2)) /\\ ((((exists ff_h_pfp_gcd_backward_update_combination_sum_resultsecondentry. ff_h_pfp_gcd_backward_update_combination_sum_resultsecondentry + S (pfrep_right_gcd_backward_update_combination_sum_result) = S ((S (pfrep_position_gcd_backward_update_combination_sum_resultsecond)) * x1)) /\\ exists ff_q_pfp_gcd_backward_update_combination_sum_resultsecondentry. x = ff_q_pfp_gcd_backward_update_combination_sum_resultsecondentry * S ((S (pfrep_position_gcd_backward_update_combination_sum_resultsecond)) * x1) + (pfrep_right_gcd_backward_update_combination_sum_result)))))) \\/ (((exists pfrep_gap_gcd_backward_update_combination_sum_resultsecondoutside. pfrep_gap_gcd_backward_update_combination_sum_resultsecondoutside+(x2)=(pfrep_power_gcd_backward_update_combination_sum_result)) /\\ (((pfrep_right_gcd_backward_update_combination_sum_result)=0))))) -> pfrep_left_gcd_backward_update_combination_sum_result=pfrep_right_gcd_backward_update_combination_sum_result))))))))))))))))))))))", "specialize hupdate (x)", "specialize hupdate (x1)", "specialize hupdate (x2)", "specialize hupdate (x3)", "specialize hupdate (x4)", "specialize hupdate (x5)", "specialize hupdate (x6)", "specialize hupdate (x7)", "specialize hupdate (x8)", "apply hupdate", "exact hp", "exact he", "exact hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right", "cases hb", "cases hb_witness", "cases hb_witness_witness", "cases hb_witness_witness_witness", "cases hb_witness_witness_witness_witness", "cases hb_witness_witness_witness_witness_witness", "cases hb_witness_witness_witness_witness_witness_witness", "cases hb_witness_witness_witness_witness_witness_witness_right", "exists x", "exists x1", "exists x2", "exists x6", "exists x7", "exists x8", "exists x12", "exists x13", "exists x14", "split", "exact hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_left", "split", "exact hc", "exact hb_witness_witness_witness_witness_witness_witness_right_right" ], "script_sha256": "241b983d2748fdc842ccc99487344c9280692064c1dd7b6995a31994f8c908a1", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_existence_candidate.py", "sha256": "81f2f48dd2e81894c7a267453646eb6f2b6f9bd3ee320386d8c561f6b9f8b8ca" }, "statement": "forall p ab ac L bb bc d qb qc q rb rc R. (~((p) = 1) /\\ forall pfa_factor_left_gcd_backward_prime pfa_factor_right_gcd_backward_prime. (p) = pfa_factor_left_gcd_backward_prime * pfa_factor_right_gcd_backward_prime -> pfa_factor_left_gcd_backward_prime = 1 \\/ pfa_factor_right_gcd_backward_prime = 1) -> (((forall fom_index_pfp_gcd_backward_executioninput. (exists fom_gap_pfp_gcd_backward_executioninput_index_bound. fom_gap_pfp_gcd_backward_executioninput_index_bound + S (fom_index_pfp_gcd_backward_executioninput) = L) -> exists fom_value_pfp_gcd_backward_executioninput. ((((exists fom_beta_height_pfp_gcd_backward_executioninput_entry. fom_beta_height_pfp_gcd_backward_executioninput_entry + S (fom_value_pfp_gcd_backward_executioninput) = S ((S (fom_index_pfp_gcd_backward_executioninput)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_backward_executioninput_entry. ab = fom_beta_quotient_pfp_gcd_backward_executioninput_entry * S ((S (fom_index_pfp_gcd_backward_executioninput)) * ac) + (fom_value_pfp_gcd_backward_executioninput))) /\\ (exists fom_gap_pfp_gcd_backward_executioninput_value_bound. fom_gap_pfp_gcd_backward_executioninput_value_bound + S (fom_value_pfp_gcd_backward_executioninput) = p))) /\\ (((forall fom_index_pfp_gcd_backward_executiondivisor. (exists fom_gap_pfp_gcd_backward_executiondivisor_index_bound. fom_gap_pfp_gcd_backward_executiondivisor_index_bound + S (fom_index_pfp_gcd_backward_executiondivisor) = S (d)) -> exists fom_value_pfp_gcd_backward_executiondivisor. ((((exists fom_beta_height_pfp_gcd_backward_executiondivisor_entry. fom_beta_height_pfp_gcd_backward_executiondivisor_entry + S (fom_value_pfp_gcd_backward_executiondivisor) = S ((S (fom_index_pfp_gcd_backward_executiondivisor)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_executiondivisor_entry. bb = fom_beta_quotient_pfp_gcd_backward_executiondivisor_entry * S ((S (fom_index_pfp_gcd_backward_executiondivisor)) * bc) + (fom_value_pfp_gcd_backward_executiondivisor))) /\\ (exists fom_gap_pfp_gcd_backward_executiondivisor_value_bound. fom_gap_pfp_gcd_backward_executiondivisor_value_bound + S (fom_value_pfp_gcd_backward_executiondivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_gcd_backward_executionlengthshort. pfc_gap_gcd_backward_executionlengthshort+(L)=(d))))) \\/ (((~((q)=0)) /\\ (((q)+(d)=(L)))))) /\\ ((exists pfd_head_gcd_backward_execution pfd_inverse_gcd_backward_execution pfd_product_code_gcd_backward_execution pfd_product_scale_gcd_backward_execution pfd_residual_code_gcd_backward_execution pfd_residual_scale_gcd_backward_execution pfd_cut_gcd_backward_execution. ((((exists ff_h_pfp_gcd_backward_executionhead. ff_h_pfp_gcd_backward_executionhead + S (pfd_head_gcd_backward_execution) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_gcd_backward_executionhead. bb = ff_q_pfp_gcd_backward_executionhead * S ((S (0)) * bc) + (pfd_head_gcd_backward_execution))) /\\ (((((~((pfd_head_gcd_backward_execution) = 0)) /\\ ((((exists pfa_gap_gcd_backward_executioninversemultiplicationleft. pfa_gap_gcd_backward_executioninversemultiplicationleft + S (pfd_head_gcd_backward_execution) = (p)) /\\ (((exists pfa_gap_gcd_backward_executioninversemultiplicationright. pfa_gap_gcd_backward_executioninversemultiplicationright + S (pfd_inverse_gcd_backward_execution) = (p)) /\\ ((((exists pfa_gap_gcd_backward_executioninversemultiplicationresultbound. pfa_gap_gcd_backward_executioninversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_executioninversemultiplicationresultcongruence pfa_offset_right_gcd_backward_executioninversemultiplicationresultcongruence. ((pfd_head_gcd_backward_execution) * (pfd_inverse_gcd_backward_execution)) + (p) * pfa_offset_left_gcd_backward_executioninversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_gcd_backward_executioninversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_gcd_backward_executionquotient. (exists pfa_gap_gcd_backward_executionquotientbound. pfa_gap_gcd_backward_executionquotientbound + S (pfd_index_gcd_backward_executionquotient) = (q)) -> exists pfd_value_gcd_backward_executionquotient. ((((exists ff_h_pfp_gcd_backward_executionquotiententry. ff_h_pfp_gcd_backward_executionquotiententry + S (pfd_value_gcd_backward_executionquotient) = S ((S (pfd_index_gcd_backward_executionquotient)) * qc)) /\\ exists ff_q_pfp_gcd_backward_executionquotiententry. qb = ff_q_pfp_gcd_backward_executionquotiententry * S ((S (pfd_index_gcd_backward_executionquotient)) * qc) + (pfd_value_gcd_backward_executionquotient))) /\\ ((exists pfd_input_gcd_backward_executionquotientstep pfd_previous_gcd_backward_executionquotientstep pfd_difference_gcd_backward_executionquotientstep. ((((exists ff_h_pfp_gcd_backward_executionquotientstepinput. ff_h_pfp_gcd_backward_executionquotientstepinput + S (pfd_input_gcd_backward_executionquotientstep) = S ((S (pfd_index_gcd_backward_executionquotient)) * ac)) /\\ exists ff_q_pfp_gcd_backward_executionquotientstepinput. ab = ff_q_pfp_gcd_backward_executionquotientstepinput * S ((S (pfd_index_gcd_backward_executionquotient)) * ac) + (pfd_input_gcd_backward_executionquotientstep))) /\\ (((exists pfc_terms_code_gcd_backward_executionquotientstepprevious pfc_terms_scale_gcd_backward_executionquotientstepprevious pfc_natural_sum_gcd_backward_executionquotientstepprevious. ((forall pfc_index_gcd_backward_executionquotientsteppreviousdiagonal. (exists pfa_gap_gcd_backward_executionquotientsteppreviousdiagonalbound. pfa_gap_gcd_backward_executionquotientsteppreviousdiagonalbound + S (pfc_index_gcd_backward_executionquotientsteppreviousdiagonal) = (S (pfd_index_gcd_backward_executionquotient))) -> exists pfc_value_gcd_backward_executionquotientsteppreviousdiagonal. ((((exists ff_h_pfp_gcd_backward_executionquotientsteppreviousdiagonalentry. ff_h_pfp_gcd_backward_executionquotientsteppreviousdiagonalentry + S (pfc_value_gcd_backward_executionquotientsteppreviousdiagonal) = S ((S (pfc_index_gcd_backward_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_gcd_backward_executionquotientstepprevious)) /\\ exists ff_q_pfp_gcd_backward_executionquotientsteppreviousdiagonalentry. pfc_terms_code_gcd_backward_executionquotientstepprevious = ff_q_pfp_gcd_backward_executionquotientsteppreviousdiagonalentry * S ((S (pfc_index_gcd_backward_executionquotientsteppreviousdiagonal)) * pfc_terms_scale_gcd_backward_executionquotientstepprevious) + (pfc_value_gcd_backward_executionquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_gcd_backward_executionquotientsteppreviousdiagonalterm pfc_left_gcd_backward_executionquotientsteppreviousdiagonalterm pfc_right_gcd_backward_executionquotientsteppreviousdiagonalterm. (((pfc_index_gcd_backward_executionquotientsteppreviousdiagonal)+pfc_complement_gcd_backward_executionquotientsteppreviousdiagonalterm=(pfd_index_gcd_backward_executionquotient)) /\\ ((((((exists pfa_gap_gcd_backward_executionquotientsteppreviousdiagonaltermleftinside. pfa_gap_gcd_backward_executionquotientsteppreviousdiagonaltermleftinside + S (pfc_index_gcd_backward_executionquotientsteppreviousdiagonal) = (pfd_index_gcd_backward_executionquotient)) /\\ ((((exists ff_h_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermleftentry + S (pfc_left_gcd_backward_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_index_gcd_backward_executionquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_executionquotientsteppreviousdiagonal)) * qc) + (pfc_left_gcd_backward_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_executionquotientsteppreviousdiagonaltermleftoutside. pfc_gap_gcd_backward_executionquotientsteppreviousdiagonaltermleftoutside+(pfd_index_gcd_backward_executionquotient)=(pfc_index_gcd_backward_executionquotientsteppreviousdiagonal)) /\\ (((pfc_left_gcd_backward_executionquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_executionquotientsteppreviousdiagonaltermrightinside. pfa_gap_gcd_backward_executionquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_gcd_backward_executionquotientsteppreviousdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermrightentry + S (pfc_right_gcd_backward_executionquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_gcd_backward_executionquotientsteppreviousdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermrightentry. bb = ff_q_pfp_gcd_backward_executionquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_executionquotientsteppreviousdiagonalterm)) * bc) + (pfc_right_gcd_backward_executionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_executionquotientsteppreviousdiagonaltermrightoutside. pfc_gap_gcd_backward_executionquotientsteppreviousdiagonaltermrightoutside+(S (d))=(pfc_complement_gcd_backward_executionquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_gcd_backward_executionquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_executionquotientsteppreviousdiagonal)=pfc_left_gcd_backward_executionquotientsteppreviousdiagonalterm*pfc_right_gcd_backward_executionquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_executionquotientstepprevioussum fs_v_pfc_gcd_backward_executionquotientstepprevioussum. ((((exists fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_start. fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_start. fs_u_pfc_gcd_backward_executionquotientstepprevioussum = fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_terminal. fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_terminal + S (pfc_natural_sum_gcd_backward_executionquotientstepprevious) = S ((S (S (pfd_index_gcd_backward_executionquotient))) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_terminal. fs_u_pfc_gcd_backward_executionquotientstepprevioussum = fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_gcd_backward_executionquotient))) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum) + (pfc_natural_sum_gcd_backward_executionquotientstepprevious))) /\\ forall fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps. (exists fs_lt_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_bound. fs_lt_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_bound + S fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps = S (pfd_index_gcd_backward_executionquotient)) -> exists fs_a_pfc_gcd_backward_executionquotientstepprevioussum_body_steps fs_r_pfc_gcd_backward_executionquotientstepprevioussum_body_steps fs_s_pfc_gcd_backward_executionquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_summand. fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_gcd_backward_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_gcd_backward_executionquotientstepprevious)) /\\ exists fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_summand. pfc_terms_code_gcd_backward_executionquotientstepprevious = fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps)) * pfc_terms_scale_gcd_backward_executionquotientstepprevious) + (fs_a_pfc_gcd_backward_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_partial. fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_gcd_backward_executionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_partial. fs_u_pfc_gcd_backward_executionquotientstepprevioussum = fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum) + (fs_r_pfc_gcd_backward_executionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_successor. fs_h_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_gcd_backward_executionquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_successor. fs_u_pfc_gcd_backward_executionquotientstepprevioussum = fs_q_pfc_gcd_backward_executionquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_executionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_backward_executionquotientstepprevioussum) + (fs_s_pfc_gcd_backward_executionquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_gcd_backward_executionquotientstepprevioussum_body_steps = fs_r_pfc_gcd_backward_executionquotientstepprevioussum_body_steps + fs_a_pfc_gcd_backward_executionquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_executionquotientsteppreviousresiduebound. pfa_gap_gcd_backward_executionquotientsteppreviousresiduebound + S (pfd_previous_gcd_backward_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_executionquotientsteppreviousresiduecongruence pfa_offset_right_gcd_backward_executionquotientsteppreviousresiduecongruence. (pfc_natural_sum_gcd_backward_executionquotientstepprevious) + (p) * pfa_offset_left_gcd_backward_executionquotientsteppreviousresiduecongruence = (pfd_previous_gcd_backward_executionquotientstep) + (p) * pfa_offset_right_gcd_backward_executionquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_gcd_backward_executionquotientstepsubtractleft. pfa_gap_gcd_backward_executionquotientstepsubtractleft + S (pfd_previous_gcd_backward_executionquotientstep) = (p)) /\\ (((exists pfa_gap_gcd_backward_executionquotientstepsubtractright. pfa_gap_gcd_backward_executionquotientstepsubtractright + S (pfd_difference_gcd_backward_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_gcd_backward_executionquotientstepsubtractresultbound. pfa_gap_gcd_backward_executionquotientstepsubtractresultbound + S (pfd_input_gcd_backward_executionquotientstep) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_executionquotientstepsubtractresultcongruence pfa_offset_right_gcd_backward_executionquotientstepsubtractresultcongruence. ((pfd_previous_gcd_backward_executionquotientstep) + (pfd_difference_gcd_backward_executionquotientstep)) + (p) * pfa_offset_left_gcd_backward_executionquotientstepsubtractresultcongruence = (pfd_input_gcd_backward_executionquotientstep) + (p) * pfa_offset_right_gcd_backward_executionquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_gcd_backward_executionquotientstepmultiplyleft. pfa_gap_gcd_backward_executionquotientstepmultiplyleft + S (pfd_inverse_gcd_backward_execution) = (p)) /\\ (((exists pfa_gap_gcd_backward_executionquotientstepmultiplyright. pfa_gap_gcd_backward_executionquotientstepmultiplyright + S (pfd_difference_gcd_backward_executionquotientstep) = (p)) /\\ ((((exists pfa_gap_gcd_backward_executionquotientstepmultiplyresultbound. pfa_gap_gcd_backward_executionquotientstepmultiplyresultbound + S (pfd_value_gcd_backward_executionquotient) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_executionquotientstepmultiplyresultcongruence pfa_offset_right_gcd_backward_executionquotientstepmultiplyresultcongruence. ((pfd_inverse_gcd_backward_execution) * (pfd_difference_gcd_backward_executionquotientstep)) + (p) * pfa_offset_left_gcd_backward_executionquotientstepmultiplyresultcongruence = (pfd_value_gcd_backward_executionquotient) + (p) * pfa_offset_right_gcd_backward_executionquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_gcd_backward_executionproduct. (exists pfa_gap_gcd_backward_executionproductbound. pfa_gap_gcd_backward_executionproductbound + S (pfc_index_gcd_backward_executionproduct) = (L)) -> exists pfc_value_gcd_backward_executionproduct. ((((exists ff_h_pfp_gcd_backward_executionproductentry. ff_h_pfp_gcd_backward_executionproductentry + S (pfc_value_gcd_backward_executionproduct) = S ((S (pfc_index_gcd_backward_executionproduct)) * pfd_product_scale_gcd_backward_execution)) /\\ exists ff_q_pfp_gcd_backward_executionproductentry. pfd_product_code_gcd_backward_execution = ff_q_pfp_gcd_backward_executionproductentry * S ((S (pfc_index_gcd_backward_executionproduct)) * pfd_product_scale_gcd_backward_execution) + (pfc_value_gcd_backward_executionproduct))) /\\ ((exists pfc_terms_code_gcd_backward_executionproductcoefficient pfc_terms_scale_gcd_backward_executionproductcoefficient pfc_natural_sum_gcd_backward_executionproductcoefficient. ((forall pfc_index_gcd_backward_executionproductcoefficientdiagonal. (exists pfa_gap_gcd_backward_executionproductcoefficientdiagonalbound. pfa_gap_gcd_backward_executionproductcoefficientdiagonalbound + S (pfc_index_gcd_backward_executionproductcoefficientdiagonal) = (S (pfc_index_gcd_backward_executionproduct))) -> exists pfc_value_gcd_backward_executionproductcoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_executionproductcoefficientdiagonalentry. ff_h_pfp_gcd_backward_executionproductcoefficientdiagonalentry + S (pfc_value_gcd_backward_executionproductcoefficientdiagonal) = S ((S (pfc_index_gcd_backward_executionproductcoefficientdiagonal)) * pfc_terms_scale_gcd_backward_executionproductcoefficient)) /\\ exists ff_q_pfp_gcd_backward_executionproductcoefficientdiagonalentry. pfc_terms_code_gcd_backward_executionproductcoefficient = ff_q_pfp_gcd_backward_executionproductcoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_executionproductcoefficientdiagonal)) * pfc_terms_scale_gcd_backward_executionproductcoefficient) + (pfc_value_gcd_backward_executionproductcoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_executionproductcoefficientdiagonalterm pfc_left_gcd_backward_executionproductcoefficientdiagonalterm pfc_right_gcd_backward_executionproductcoefficientdiagonalterm. (((pfc_index_gcd_backward_executionproductcoefficientdiagonal)+pfc_complement_gcd_backward_executionproductcoefficientdiagonalterm=(pfc_index_gcd_backward_executionproduct)) /\\ ((((((exists pfa_gap_gcd_backward_executionproductcoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_executionproductcoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_executionproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_gcd_backward_executionproductcoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_executionproductcoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_executionproductcoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_executionproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_gcd_backward_executionproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_gcd_backward_executionproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_executionproductcoefficientdiagonal)) * qc) + (pfc_left_gcd_backward_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_executionproductcoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_executionproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_gcd_backward_executionproductcoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_executionproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_executionproductcoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_executionproductcoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_executionproductcoefficientdiagonalterm) = (S (d))) /\\ ((((exists ff_h_pfp_gcd_backward_executionproductcoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_executionproductcoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_executionproductcoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_executionproductcoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_backward_executionproductcoefficientdiagonaltermrightentry. bb = ff_q_pfp_gcd_backward_executionproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_executionproductcoefficientdiagonalterm)) * bc) + (pfc_right_gcd_backward_executionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_executionproductcoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_executionproductcoefficientdiagonaltermrightoutside+(S (d))=(pfc_complement_gcd_backward_executionproductcoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_executionproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_executionproductcoefficientdiagonal)=pfc_left_gcd_backward_executionproductcoefficientdiagonalterm*pfc_right_gcd_backward_executionproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_executionproductcoefficientsum fs_v_pfc_gcd_backward_executionproductcoefficientsum. ((((exists fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_start. fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_executionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_start. fs_u_pfc_gcd_backward_executionproductcoefficientsum = fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_executionproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_terminal. fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_executionproductcoefficient) = S ((S (S (pfc_index_gcd_backward_executionproduct))) * fs_v_pfc_gcd_backward_executionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_terminal. fs_u_pfc_gcd_backward_executionproductcoefficientsum = fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_executionproduct))) * fs_v_pfc_gcd_backward_executionproductcoefficientsum) + (pfc_natural_sum_gcd_backward_executionproductcoefficient))) /\\ forall fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_executionproductcoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_executionproductcoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps = S (pfc_index_gcd_backward_executionproduct)) -> exists fs_a_pfc_gcd_backward_executionproductcoefficientsum_body_steps fs_r_pfc_gcd_backward_executionproductcoefficientsum_body_steps fs_s_pfc_gcd_backward_executionproductcoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_executionproductcoefficient)) /\\ exists fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_executionproductcoefficient = fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_executionproductcoefficient) + (fs_a_pfc_gcd_backward_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_executionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_executionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_executionproductcoefficientsum = fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_executionproductcoefficientsum) + (fs_r_pfc_gcd_backward_executionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_executionproductcoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_executionproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_executionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_executionproductcoefficientsum = fs_q_pfc_gcd_backward_executionproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_executionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_executionproductcoefficientsum) + (fs_s_pfc_gcd_backward_executionproductcoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_executionproductcoefficientsum_body_steps = fs_r_pfc_gcd_backward_executionproductcoefficientsum_body_steps + fs_a_pfc_gcd_backward_executionproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_executionproductcoefficientresiduebound. pfa_gap_gcd_backward_executionproductcoefficientresiduebound + S (pfc_value_gcd_backward_executionproduct) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_executionproductcoefficientresiduecongruence pfa_offset_right_gcd_backward_executionproductcoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_executionproductcoefficient) + (p) * pfa_offset_left_gcd_backward_executionproductcoefficientresiduecongruence = (pfc_value_gcd_backward_executionproduct) + (p) * pfa_offset_right_gcd_backward_executionproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_gcd_backward_executiondifference. (exists pfa_gap_gcd_backward_executiondifferenceindex. pfa_gap_gcd_backward_executiondifferenceindex + S (pfs_index_gcd_backward_executiondifference) = (L)) -> exists pfs_left_gcd_backward_executiondifference pfs_right_gcd_backward_executiondifference pfs_result_gcd_backward_executiondifference. ((((exists ff_h_pfp_gcd_backward_executiondifferenceleft. ff_h_pfp_gcd_backward_executiondifferenceleft + S (pfs_left_gcd_backward_executiondifference) = S ((S (pfs_index_gcd_backward_executiondifference)) * ac)) /\\ exists ff_q_pfp_gcd_backward_executiondifferenceleft. ab = ff_q_pfp_gcd_backward_executiondifferenceleft * S ((S (pfs_index_gcd_backward_executiondifference)) * ac) + (pfs_left_gcd_backward_executiondifference))) /\\ (((((exists ff_h_pfp_gcd_backward_executiondifferenceright. ff_h_pfp_gcd_backward_executiondifferenceright + S (pfs_right_gcd_backward_executiondifference) = S ((S (pfs_index_gcd_backward_executiondifference)) * pfd_product_scale_gcd_backward_execution)) /\\ exists ff_q_pfp_gcd_backward_executiondifferenceright. pfd_product_code_gcd_backward_execution = ff_q_pfp_gcd_backward_executiondifferenceright * S ((S (pfs_index_gcd_backward_executiondifference)) * pfd_product_scale_gcd_backward_execution) + (pfs_right_gcd_backward_executiondifference))) /\\ (((((exists ff_h_pfp_gcd_backward_executiondifferenceresult. ff_h_pfp_gcd_backward_executiondifferenceresult + S (pfs_result_gcd_backward_executiondifference) = S ((S (pfs_index_gcd_backward_executiondifference)) * pfd_residual_scale_gcd_backward_execution)) /\\ exists ff_q_pfp_gcd_backward_executiondifferenceresult. pfd_residual_code_gcd_backward_execution = ff_q_pfp_gcd_backward_executiondifferenceresult * S ((S (pfs_index_gcd_backward_executiondifference)) * pfd_residual_scale_gcd_backward_execution) + (pfs_result_gcd_backward_executiondifference))) /\\ ((((exists pfa_gap_gcd_backward_executiondifferenceoperationleft. pfa_gap_gcd_backward_executiondifferenceoperationleft + S (pfs_right_gcd_backward_executiondifference) = (p)) /\\ (((exists pfa_gap_gcd_backward_executiondifferenceoperationright. pfa_gap_gcd_backward_executiondifferenceoperationright + S (pfs_result_gcd_backward_executiondifference) = (p)) /\\ ((((exists pfa_gap_gcd_backward_executiondifferenceoperationresultbound. pfa_gap_gcd_backward_executiondifferenceoperationresultbound + S (pfs_left_gcd_backward_executiondifference) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_executiondifferenceoperationresultcongruence pfa_offset_right_gcd_backward_executiondifferenceoperationresultcongruence. ((pfs_right_gcd_backward_executiondifference) + (pfs_result_gcd_backward_executiondifference)) + (p) * pfa_offset_left_gcd_backward_executiondifferenceoperationresultcongruence = (pfs_left_gcd_backward_executiondifference) + (p) * pfa_offset_right_gcd_backward_executiondifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_gcd_backward_execution)+(R)) /\\ (((forall fom_index_pfp_gcd_backward_executiontriminput. (exists fom_gap_pfp_gcd_backward_executiontriminput_index_bound. fom_gap_pfp_gcd_backward_executiontriminput_index_bound + S (fom_index_pfp_gcd_backward_executiontriminput) = L) -> exists fom_value_pfp_gcd_backward_executiontriminput. ((((exists fom_beta_height_pfp_gcd_backward_executiontriminput_entry. fom_beta_height_pfp_gcd_backward_executiontriminput_entry + S (fom_value_pfp_gcd_backward_executiontriminput) = S ((S (fom_index_pfp_gcd_backward_executiontriminput)) * pfd_residual_scale_gcd_backward_execution)) /\\ exists fom_beta_quotient_pfp_gcd_backward_executiontriminput_entry. pfd_residual_code_gcd_backward_execution = fom_beta_quotient_pfp_gcd_backward_executiontriminput_entry * S ((S (fom_index_pfp_gcd_backward_executiontriminput)) * pfd_residual_scale_gcd_backward_execution) + (fom_value_pfp_gcd_backward_executiontriminput))) /\\ (exists fom_gap_pfp_gcd_backward_executiontriminput_value_bound. fom_gap_pfp_gcd_backward_executiontriminput_value_bound + S (fom_value_pfp_gcd_backward_executiontriminput) = p))) /\\ (((forall pfp_repeat_index_gcd_backward_executiontrimremoved. (exists pfa_gap_gcd_backward_executiontrimremovedindex. pfa_gap_gcd_backward_executiontrimremovedindex + S (pfp_repeat_index_gcd_backward_executiontrimremoved) = (pfd_cut_gcd_backward_execution)) -> (((exists ff_h_pfp_gcd_backward_executiontrimremovedentry. ff_h_pfp_gcd_backward_executiontrimremovedentry + S (0) = S ((S (pfp_repeat_index_gcd_backward_executiontrimremoved)) * pfd_residual_scale_gcd_backward_execution)) /\\ exists ff_q_pfp_gcd_backward_executiontrimremovedentry. pfd_residual_code_gcd_backward_execution = ff_q_pfp_gcd_backward_executiontrimremovedentry * S ((S (pfp_repeat_index_gcd_backward_executiontrimremoved)) * pfd_residual_scale_gcd_backward_execution) + (0)))) /\\ (((forall pftrim_index_gcd_backward_executiontrimsuffix pftrim_value_gcd_backward_executiontrimsuffix. (exists pfa_gap_gcd_backward_executiontrimsuffixbound. pfa_gap_gcd_backward_executiontrimsuffixbound + S (pftrim_index_gcd_backward_executiontrimsuffix) = (R)) -> (((exists ff_h_pfp_gcd_backward_executiontrimsuffixsource. ff_h_pfp_gcd_backward_executiontrimsuffixsource + S (pftrim_value_gcd_backward_executiontrimsuffix) = S ((S ((pfd_cut_gcd_backward_execution)+pftrim_index_gcd_backward_executiontrimsuffix)) * pfd_residual_scale_gcd_backward_execution)) /\\ exists ff_q_pfp_gcd_backward_executiontrimsuffixsource. pfd_residual_code_gcd_backward_execution = ff_q_pfp_gcd_backward_executiontrimsuffixsource * S ((S ((pfd_cut_gcd_backward_execution)+pftrim_index_gcd_backward_executiontrimsuffix)) * pfd_residual_scale_gcd_backward_execution) + (pftrim_value_gcd_backward_executiontrimsuffix))) -> (((exists ff_h_pfp_gcd_backward_executiontrimsuffixoutput. ff_h_pfp_gcd_backward_executiontrimsuffixoutput + S (pftrim_value_gcd_backward_executiontrimsuffix) = S ((S (pftrim_index_gcd_backward_executiontrimsuffix)) * rc)) /\\ exists ff_q_pfp_gcd_backward_executiontrimsuffixoutput. rb = ff_q_pfp_gcd_backward_executiontrimsuffixoutput * S ((S (pftrim_index_gcd_backward_executiontrimsuffix)) * rc) + (pftrim_value_gcd_backward_executiontrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_gcd_backward_executiontrimnormal. ((((exists ff_h_pfp_gcd_backward_executiontrimnormalentry. ff_h_pfp_gcd_backward_executiontrimnormalentry + S (pftrim_leading_gcd_backward_executiontrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_gcd_backward_executiontrimnormalentry. rb = ff_q_pfp_gcd_backward_executiontrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_gcd_backward_executiontrimnormal))) /\\ ((~(pftrim_leading_gcd_backward_executiontrimnormal=0))))))))))))))))))))))))))))))))) -> (exists pfgs_gb_gcd_backward_small pfgs_gc_gcd_backward_small pfgs_G_gcd_backward_small pfgs_ub_gcd_backward_small pfgs_uc_gcd_backward_small pfgs_U_gcd_backward_small pfgs_vb_gcd_backward_small pfgs_vc_gcd_backward_small pfgs_V_gcd_backward_small. (((pfgs_G_gcd_backward_small)=0 \\/ (((~((pfgs_G_gcd_backward_small) = 0)) /\\ (((forall fom_index_pfp_gcd_backward_small_witness_normal_moniccoefficients. (exists fom_gap_pfp_gcd_backward_small_witness_normal_moniccoefficients_index_bound. fom_gap_pfp_gcd_backward_small_witness_normal_moniccoefficients_index_bound + S (fom_index_pfp_gcd_backward_small_witness_normal_moniccoefficients) = pfgs_G_gcd_backward_small) -> exists fom_value_pfp_gcd_backward_small_witness_normal_moniccoefficients. ((((exists fom_beta_height_pfp_gcd_backward_small_witness_normal_moniccoefficients_entry. fom_beta_height_pfp_gcd_backward_small_witness_normal_moniccoefficients_entry + S (fom_value_pfp_gcd_backward_small_witness_normal_moniccoefficients) = S ((S (fom_index_pfp_gcd_backward_small_witness_normal_moniccoefficients)) * pfgs_gc_gcd_backward_small)) /\\ exists fom_beta_quotient_pfp_gcd_backward_small_witness_normal_moniccoefficients_entry. pfgs_gb_gcd_backward_small = fom_beta_quotient_pfp_gcd_backward_small_witness_normal_moniccoefficients_entry * S ((S (fom_index_pfp_gcd_backward_small_witness_normal_moniccoefficients)) * pfgs_gc_gcd_backward_small) + (fom_value_pfp_gcd_backward_small_witness_normal_moniccoefficients))) /\\ (exists fom_gap_pfp_gcd_backward_small_witness_normal_moniccoefficients_value_bound. fom_gap_pfp_gcd_backward_small_witness_normal_moniccoefficients_value_bound + S (fom_value_pfp_gcd_backward_small_witness_normal_moniccoefficients) = p))) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_normal_monicleading. ff_h_pfp_gcd_backward_small_witness_normal_monicleading + S (1) = S ((S (0)) * pfgs_gc_gcd_backward_small)) /\\ exists ff_q_pfp_gcd_backward_small_witness_normal_monicleading. pfgs_gb_gcd_backward_small = ff_q_pfp_gcd_backward_small_witness_normal_monicleading * S ((S (0)) * pfgs_gc_gcd_backward_small) + (1))))))))) /\\ (((((((forall fom_index_pfp_gcd_backward_small_witness_common_left_bounded. (exists fom_gap_pfp_gcd_backward_small_witness_common_left_bounded_index_bound. fom_gap_pfp_gcd_backward_small_witness_common_left_bounded_index_bound + S (fom_index_pfp_gcd_backward_small_witness_common_left_bounded) = S d) -> exists fom_value_pfp_gcd_backward_small_witness_common_left_bounded. ((((exists fom_beta_height_pfp_gcd_backward_small_witness_common_left_bounded_entry. fom_beta_height_pfp_gcd_backward_small_witness_common_left_bounded_entry + S (fom_value_pfp_gcd_backward_small_witness_common_left_bounded) = S ((S (fom_index_pfp_gcd_backward_small_witness_common_left_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_small_witness_common_left_bounded_entry. bb = fom_beta_quotient_pfp_gcd_backward_small_witness_common_left_bounded_entry * S ((S (fom_index_pfp_gcd_backward_small_witness_common_left_bounded)) * bc) + (fom_value_pfp_gcd_backward_small_witness_common_left_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_small_witness_common_left_bounded_value_bound. fom_gap_pfp_gcd_backward_small_witness_common_left_bounded_value_bound + S (fom_value_pfp_gcd_backward_small_witness_common_left_bounded) = p))) /\\ ((exists pfgd_qb_gcd_backward_small_witness_common_left pfgd_qc_gcd_backward_small_witness_common_left pfgd_Q_gcd_backward_small_witness_common_left pfgd_pb_gcd_backward_small_witness_common_left pfgd_pc_gcd_backward_small_witness_common_left pfgd_P_gcd_backward_small_witness_common_left. ((((forall fom_index_pfp_gcd_backward_small_witness_common_left_productleft. (exists fom_gap_pfp_gcd_backward_small_witness_common_left_productleft_index_bound. fom_gap_pfp_gcd_backward_small_witness_common_left_productleft_index_bound + S (fom_index_pfp_gcd_backward_small_witness_common_left_productleft) = pfgd_Q_gcd_backward_small_witness_common_left) -> exists fom_value_pfp_gcd_backward_small_witness_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_backward_small_witness_common_left_productleft_entry. fom_beta_height_pfp_gcd_backward_small_witness_common_left_productleft_entry + S (fom_value_pfp_gcd_backward_small_witness_common_left_productleft) = S ((S (fom_index_pfp_gcd_backward_small_witness_common_left_productleft)) * pfgd_qc_gcd_backward_small_witness_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_backward_small_witness_common_left_productleft_entry. pfgd_qb_gcd_backward_small_witness_common_left = fom_beta_quotient_pfp_gcd_backward_small_witness_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_backward_small_witness_common_left_productleft)) * pfgd_qc_gcd_backward_small_witness_common_left) + (fom_value_pfp_gcd_backward_small_witness_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_small_witness_common_left_productleft_value_bound. fom_gap_pfp_gcd_backward_small_witness_common_left_productleft_value_bound + S (fom_value_pfp_gcd_backward_small_witness_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_small_witness_common_left_productright. (exists fom_gap_pfp_gcd_backward_small_witness_common_left_productright_index_bound. fom_gap_pfp_gcd_backward_small_witness_common_left_productright_index_bound + S (fom_index_pfp_gcd_backward_small_witness_common_left_productright) = pfgs_G_gcd_backward_small) -> exists fom_value_pfp_gcd_backward_small_witness_common_left_productright. ((((exists fom_beta_height_pfp_gcd_backward_small_witness_common_left_productright_entry. fom_beta_height_pfp_gcd_backward_small_witness_common_left_productright_entry + S (fom_value_pfp_gcd_backward_small_witness_common_left_productright) = S ((S (fom_index_pfp_gcd_backward_small_witness_common_left_productright)) * pfgs_gc_gcd_backward_small)) /\\ exists fom_beta_quotient_pfp_gcd_backward_small_witness_common_left_productright_entry. pfgs_gb_gcd_backward_small = fom_beta_quotient_pfp_gcd_backward_small_witness_common_left_productright_entry * S ((S (fom_index_pfp_gcd_backward_small_witness_common_left_productright)) * pfgs_gc_gcd_backward_small) + (fom_value_pfp_gcd_backward_small_witness_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_backward_small_witness_common_left_productright_value_bound. fom_gap_pfp_gcd_backward_small_witness_common_left_productright_value_bound + S (fom_value_pfp_gcd_backward_small_witness_common_left_productright) = p))) /\\ (((((((pfgd_Q_gcd_backward_small_witness_common_left)=0 \\/ (pfgs_G_gcd_backward_small)=0) /\\ (((pfgd_P_gcd_backward_small_witness_common_left)=0)))) \\/ (((~((pfgd_Q_gcd_backward_small_witness_common_left)=0)) /\\ (((~((pfgs_G_gcd_backward_small)=0)) /\\ (((pfgd_Q_gcd_backward_small_witness_common_left)+(pfgs_G_gcd_backward_small)=S (pfgd_P_gcd_backward_small_witness_common_left)))))))) /\\ ((forall pfc_index_gcd_backward_small_witness_common_left_productcoefficients. (exists pfa_gap_gcd_backward_small_witness_common_left_productcoefficientsbound. pfa_gap_gcd_backward_small_witness_common_left_productcoefficientsbound + S (pfc_index_gcd_backward_small_witness_common_left_productcoefficients) = (pfgd_P_gcd_backward_small_witness_common_left)) -> exists pfc_value_gcd_backward_small_witness_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_backward_small_witness_common_left_productcoefficientsentry. ff_h_pfp_gcd_backward_small_witness_common_left_productcoefficientsentry + S (pfc_value_gcd_backward_small_witness_common_left_productcoefficients) = S ((S (pfc_index_gcd_backward_small_witness_common_left_productcoefficients)) * pfgd_pc_gcd_backward_small_witness_common_left)) /\\ exists ff_q_pfp_gcd_backward_small_witness_common_left_productcoefficientsentry. pfgd_pb_gcd_backward_small_witness_common_left = ff_q_pfp_gcd_backward_small_witness_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_backward_small_witness_common_left_productcoefficients)) * pfgd_pc_gcd_backward_small_witness_common_left) + (pfc_value_gcd_backward_small_witness_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_small_witness_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_backward_small_witness_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_backward_small_witness_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_small_witness_common_left_productcoefficients))) -> exists pfc_value_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_small_witness_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_small_witness_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_small_witness_common_left_productcoefficientscoefficient) + (pfc_value_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_small_witness_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_backward_small_witness_common_left)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_small_witness_common_left)) /\\ exists ff_q_pfp_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_backward_small_witness_common_left = ff_q_pfp_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_small_witness_common_left) + (pfc_left_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_backward_small_witness_common_left)=(pfc_index_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm) = (pfgs_G_gcd_backward_small)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_backward_small)) /\\ exists ff_q_pfp_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_gcd_backward_small = ff_q_pfp_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_backward_small) + (pfc_right_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_gcd_backward_small)=(pfc_complement_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_small_witness_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_small_witness_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_small_witness_common_left_productcoefficients))) * fs_v_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_small_witness_common_left_productcoefficients))) * fs_v_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_small_witness_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_small_witness_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_small_witness_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_small_witness_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_small_witness_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_small_witness_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_small_witness_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_small_witness_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_small_witness_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_small_witness_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_small_witness_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_small_witness_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_small_witness_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_small_witness_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_backward_small_witness_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_small_witness_common_left_equivalent pfrep_left_gcd_backward_small_witness_common_left_equivalent pfrep_right_gcd_backward_small_witness_common_left_equivalent. ((exists pfrep_position_gcd_backward_small_witness_common_left_equivalentfirst. ((pfrep_position_gcd_backward_small_witness_common_left_equivalentfirst+S (pfrep_power_gcd_backward_small_witness_common_left_equivalent)=(pfgd_P_gcd_backward_small_witness_common_left)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_common_left_equivalentfirstentry. ff_h_pfp_gcd_backward_small_witness_common_left_equivalentfirstentry + S (pfrep_left_gcd_backward_small_witness_common_left_equivalent) = S ((S (pfrep_position_gcd_backward_small_witness_common_left_equivalentfirst)) * pfgd_pc_gcd_backward_small_witness_common_left)) /\\ exists ff_q_pfp_gcd_backward_small_witness_common_left_equivalentfirstentry. pfgd_pb_gcd_backward_small_witness_common_left = ff_q_pfp_gcd_backward_small_witness_common_left_equivalentfirstentry * S ((S (pfrep_position_gcd_backward_small_witness_common_left_equivalentfirst)) * pfgd_pc_gcd_backward_small_witness_common_left) + (pfrep_left_gcd_backward_small_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_small_witness_common_left_equivalentfirstoutside. pfrep_gap_gcd_backward_small_witness_common_left_equivalentfirstoutside+(pfgd_P_gcd_backward_small_witness_common_left)=(pfrep_power_gcd_backward_small_witness_common_left_equivalent)) /\\ (((pfrep_left_gcd_backward_small_witness_common_left_equivalent)=0))))) -> ((exists pfrep_position_gcd_backward_small_witness_common_left_equivalentsecond. ((pfrep_position_gcd_backward_small_witness_common_left_equivalentsecond+S (pfrep_power_gcd_backward_small_witness_common_left_equivalent)=(S d)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_common_left_equivalentsecondentry. ff_h_pfp_gcd_backward_small_witness_common_left_equivalentsecondentry + S (pfrep_right_gcd_backward_small_witness_common_left_equivalent) = S ((S (pfrep_position_gcd_backward_small_witness_common_left_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_gcd_backward_small_witness_common_left_equivalentsecondentry. bb = ff_q_pfp_gcd_backward_small_witness_common_left_equivalentsecondentry * S ((S (pfrep_position_gcd_backward_small_witness_common_left_equivalentsecond)) * bc) + (pfrep_right_gcd_backward_small_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_small_witness_common_left_equivalentsecondoutside. pfrep_gap_gcd_backward_small_witness_common_left_equivalentsecondoutside+(S d)=(pfrep_power_gcd_backward_small_witness_common_left_equivalent)) /\\ (((pfrep_right_gcd_backward_small_witness_common_left_equivalent)=0))))) -> pfrep_left_gcd_backward_small_witness_common_left_equivalent=pfrep_right_gcd_backward_small_witness_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_backward_small_witness_common_right_bounded. (exists fom_gap_pfp_gcd_backward_small_witness_common_right_bounded_index_bound. fom_gap_pfp_gcd_backward_small_witness_common_right_bounded_index_bound + S (fom_index_pfp_gcd_backward_small_witness_common_right_bounded) = R) -> exists fom_value_pfp_gcd_backward_small_witness_common_right_bounded. ((((exists fom_beta_height_pfp_gcd_backward_small_witness_common_right_bounded_entry. fom_beta_height_pfp_gcd_backward_small_witness_common_right_bounded_entry + S (fom_value_pfp_gcd_backward_small_witness_common_right_bounded) = S ((S (fom_index_pfp_gcd_backward_small_witness_common_right_bounded)) * rc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_small_witness_common_right_bounded_entry. rb = fom_beta_quotient_pfp_gcd_backward_small_witness_common_right_bounded_entry * S ((S (fom_index_pfp_gcd_backward_small_witness_common_right_bounded)) * rc) + (fom_value_pfp_gcd_backward_small_witness_common_right_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_small_witness_common_right_bounded_value_bound. fom_gap_pfp_gcd_backward_small_witness_common_right_bounded_value_bound + S (fom_value_pfp_gcd_backward_small_witness_common_right_bounded) = p))) /\\ ((exists pfgd_qb_gcd_backward_small_witness_common_right pfgd_qc_gcd_backward_small_witness_common_right pfgd_Q_gcd_backward_small_witness_common_right pfgd_pb_gcd_backward_small_witness_common_right pfgd_pc_gcd_backward_small_witness_common_right pfgd_P_gcd_backward_small_witness_common_right. ((((forall fom_index_pfp_gcd_backward_small_witness_common_right_productleft. (exists fom_gap_pfp_gcd_backward_small_witness_common_right_productleft_index_bound. fom_gap_pfp_gcd_backward_small_witness_common_right_productleft_index_bound + S (fom_index_pfp_gcd_backward_small_witness_common_right_productleft) = pfgd_Q_gcd_backward_small_witness_common_right) -> exists fom_value_pfp_gcd_backward_small_witness_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_backward_small_witness_common_right_productleft_entry. fom_beta_height_pfp_gcd_backward_small_witness_common_right_productleft_entry + S (fom_value_pfp_gcd_backward_small_witness_common_right_productleft) = S ((S (fom_index_pfp_gcd_backward_small_witness_common_right_productleft)) * pfgd_qc_gcd_backward_small_witness_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_backward_small_witness_common_right_productleft_entry. pfgd_qb_gcd_backward_small_witness_common_right = fom_beta_quotient_pfp_gcd_backward_small_witness_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_backward_small_witness_common_right_productleft)) * pfgd_qc_gcd_backward_small_witness_common_right) + (fom_value_pfp_gcd_backward_small_witness_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_small_witness_common_right_productleft_value_bound. fom_gap_pfp_gcd_backward_small_witness_common_right_productleft_value_bound + S (fom_value_pfp_gcd_backward_small_witness_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_small_witness_common_right_productright. (exists fom_gap_pfp_gcd_backward_small_witness_common_right_productright_index_bound. fom_gap_pfp_gcd_backward_small_witness_common_right_productright_index_bound + S (fom_index_pfp_gcd_backward_small_witness_common_right_productright) = pfgs_G_gcd_backward_small) -> exists fom_value_pfp_gcd_backward_small_witness_common_right_productright. ((((exists fom_beta_height_pfp_gcd_backward_small_witness_common_right_productright_entry. fom_beta_height_pfp_gcd_backward_small_witness_common_right_productright_entry + S (fom_value_pfp_gcd_backward_small_witness_common_right_productright) = S ((S (fom_index_pfp_gcd_backward_small_witness_common_right_productright)) * pfgs_gc_gcd_backward_small)) /\\ exists fom_beta_quotient_pfp_gcd_backward_small_witness_common_right_productright_entry. pfgs_gb_gcd_backward_small = fom_beta_quotient_pfp_gcd_backward_small_witness_common_right_productright_entry * S ((S (fom_index_pfp_gcd_backward_small_witness_common_right_productright)) * pfgs_gc_gcd_backward_small) + (fom_value_pfp_gcd_backward_small_witness_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_backward_small_witness_common_right_productright_value_bound. fom_gap_pfp_gcd_backward_small_witness_common_right_productright_value_bound + S (fom_value_pfp_gcd_backward_small_witness_common_right_productright) = p))) /\\ (((((((pfgd_Q_gcd_backward_small_witness_common_right)=0 \\/ (pfgs_G_gcd_backward_small)=0) /\\ (((pfgd_P_gcd_backward_small_witness_common_right)=0)))) \\/ (((~((pfgd_Q_gcd_backward_small_witness_common_right)=0)) /\\ (((~((pfgs_G_gcd_backward_small)=0)) /\\ (((pfgd_Q_gcd_backward_small_witness_common_right)+(pfgs_G_gcd_backward_small)=S (pfgd_P_gcd_backward_small_witness_common_right)))))))) /\\ ((forall pfc_index_gcd_backward_small_witness_common_right_productcoefficients. (exists pfa_gap_gcd_backward_small_witness_common_right_productcoefficientsbound. pfa_gap_gcd_backward_small_witness_common_right_productcoefficientsbound + S (pfc_index_gcd_backward_small_witness_common_right_productcoefficients) = (pfgd_P_gcd_backward_small_witness_common_right)) -> exists pfc_value_gcd_backward_small_witness_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_backward_small_witness_common_right_productcoefficientsentry. ff_h_pfp_gcd_backward_small_witness_common_right_productcoefficientsentry + S (pfc_value_gcd_backward_small_witness_common_right_productcoefficients) = S ((S (pfc_index_gcd_backward_small_witness_common_right_productcoefficients)) * pfgd_pc_gcd_backward_small_witness_common_right)) /\\ exists ff_q_pfp_gcd_backward_small_witness_common_right_productcoefficientsentry. pfgd_pb_gcd_backward_small_witness_common_right = ff_q_pfp_gcd_backward_small_witness_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_backward_small_witness_common_right_productcoefficients)) * pfgd_pc_gcd_backward_small_witness_common_right) + (pfc_value_gcd_backward_small_witness_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_small_witness_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_backward_small_witness_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_backward_small_witness_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_small_witness_common_right_productcoefficients))) -> exists pfc_value_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_small_witness_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_small_witness_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_small_witness_common_right_productcoefficientscoefficient) + (pfc_value_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_small_witness_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_backward_small_witness_common_right)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_small_witness_common_right)) /\\ exists ff_q_pfp_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_backward_small_witness_common_right = ff_q_pfp_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_small_witness_common_right) + (pfc_left_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_backward_small_witness_common_right)=(pfc_index_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm) = (pfgs_G_gcd_backward_small)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_backward_small)) /\\ exists ff_q_pfp_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_gcd_backward_small = ff_q_pfp_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_backward_small) + (pfc_right_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_gcd_backward_small)=(pfc_complement_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_small_witness_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_small_witness_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_small_witness_common_right_productcoefficients))) * fs_v_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_small_witness_common_right_productcoefficients))) * fs_v_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_small_witness_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_small_witness_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_small_witness_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_small_witness_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_small_witness_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_small_witness_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_small_witness_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_small_witness_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_small_witness_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_small_witness_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_small_witness_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_small_witness_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_small_witness_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_small_witness_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_backward_small_witness_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_small_witness_common_right_equivalent pfrep_left_gcd_backward_small_witness_common_right_equivalent pfrep_right_gcd_backward_small_witness_common_right_equivalent. ((exists pfrep_position_gcd_backward_small_witness_common_right_equivalentfirst. ((pfrep_position_gcd_backward_small_witness_common_right_equivalentfirst+S (pfrep_power_gcd_backward_small_witness_common_right_equivalent)=(pfgd_P_gcd_backward_small_witness_common_right)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_common_right_equivalentfirstentry. ff_h_pfp_gcd_backward_small_witness_common_right_equivalentfirstentry + S (pfrep_left_gcd_backward_small_witness_common_right_equivalent) = S ((S (pfrep_position_gcd_backward_small_witness_common_right_equivalentfirst)) * pfgd_pc_gcd_backward_small_witness_common_right)) /\\ exists ff_q_pfp_gcd_backward_small_witness_common_right_equivalentfirstentry. pfgd_pb_gcd_backward_small_witness_common_right = ff_q_pfp_gcd_backward_small_witness_common_right_equivalentfirstentry * S ((S (pfrep_position_gcd_backward_small_witness_common_right_equivalentfirst)) * pfgd_pc_gcd_backward_small_witness_common_right) + (pfrep_left_gcd_backward_small_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_small_witness_common_right_equivalentfirstoutside. pfrep_gap_gcd_backward_small_witness_common_right_equivalentfirstoutside+(pfgd_P_gcd_backward_small_witness_common_right)=(pfrep_power_gcd_backward_small_witness_common_right_equivalent)) /\\ (((pfrep_left_gcd_backward_small_witness_common_right_equivalent)=0))))) -> ((exists pfrep_position_gcd_backward_small_witness_common_right_equivalentsecond. ((pfrep_position_gcd_backward_small_witness_common_right_equivalentsecond+S (pfrep_power_gcd_backward_small_witness_common_right_equivalent)=(R)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_common_right_equivalentsecondentry. ff_h_pfp_gcd_backward_small_witness_common_right_equivalentsecondentry + S (pfrep_right_gcd_backward_small_witness_common_right_equivalent) = S ((S (pfrep_position_gcd_backward_small_witness_common_right_equivalentsecond)) * rc)) /\\ exists ff_q_pfp_gcd_backward_small_witness_common_right_equivalentsecondentry. rb = ff_q_pfp_gcd_backward_small_witness_common_right_equivalentsecondentry * S ((S (pfrep_position_gcd_backward_small_witness_common_right_equivalentsecond)) * rc) + (pfrep_right_gcd_backward_small_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_small_witness_common_right_equivalentsecondoutside. pfrep_gap_gcd_backward_small_witness_common_right_equivalentsecondoutside+(R)=(pfrep_power_gcd_backward_small_witness_common_right_equivalent)) /\\ (((pfrep_right_gcd_backward_small_witness_common_right_equivalent)=0))))) -> pfrep_left_gcd_backward_small_witness_common_right_equivalent=pfrep_right_gcd_backward_small_witness_common_right_equivalent)))))))))) /\\ ((exists pfgb_pb_gcd_backward_small_witness_bezout pfgb_pc_gcd_backward_small_witness_bezout pfgb_P_gcd_backward_small_witness_bezout pfgb_qb_gcd_backward_small_witness_bezout pfgb_qc_gcd_backward_small_witness_bezout pfgb_Q_gcd_backward_small_witness_bezout. ((((forall fom_index_pfp_gcd_backward_small_witness_bezout_leftleft. (exists fom_gap_pfp_gcd_backward_small_witness_bezout_leftleft_index_bound. fom_gap_pfp_gcd_backward_small_witness_bezout_leftleft_index_bound + S (fom_index_pfp_gcd_backward_small_witness_bezout_leftleft) = pfgs_U_gcd_backward_small) -> exists fom_value_pfp_gcd_backward_small_witness_bezout_leftleft. ((((exists fom_beta_height_pfp_gcd_backward_small_witness_bezout_leftleft_entry. fom_beta_height_pfp_gcd_backward_small_witness_bezout_leftleft_entry + S (fom_value_pfp_gcd_backward_small_witness_bezout_leftleft) = S ((S (fom_index_pfp_gcd_backward_small_witness_bezout_leftleft)) * pfgs_uc_gcd_backward_small)) /\\ exists fom_beta_quotient_pfp_gcd_backward_small_witness_bezout_leftleft_entry. pfgs_ub_gcd_backward_small = fom_beta_quotient_pfp_gcd_backward_small_witness_bezout_leftleft_entry * S ((S (fom_index_pfp_gcd_backward_small_witness_bezout_leftleft)) * pfgs_uc_gcd_backward_small) + (fom_value_pfp_gcd_backward_small_witness_bezout_leftleft))) /\\ (exists fom_gap_pfp_gcd_backward_small_witness_bezout_leftleft_value_bound. fom_gap_pfp_gcd_backward_small_witness_bezout_leftleft_value_bound + S (fom_value_pfp_gcd_backward_small_witness_bezout_leftleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_small_witness_bezout_leftright. (exists fom_gap_pfp_gcd_backward_small_witness_bezout_leftright_index_bound. fom_gap_pfp_gcd_backward_small_witness_bezout_leftright_index_bound + S (fom_index_pfp_gcd_backward_small_witness_bezout_leftright) = S d) -> exists fom_value_pfp_gcd_backward_small_witness_bezout_leftright. ((((exists fom_beta_height_pfp_gcd_backward_small_witness_bezout_leftright_entry. fom_beta_height_pfp_gcd_backward_small_witness_bezout_leftright_entry + S (fom_value_pfp_gcd_backward_small_witness_bezout_leftright) = S ((S (fom_index_pfp_gcd_backward_small_witness_bezout_leftright)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_small_witness_bezout_leftright_entry. bb = fom_beta_quotient_pfp_gcd_backward_small_witness_bezout_leftright_entry * S ((S (fom_index_pfp_gcd_backward_small_witness_bezout_leftright)) * bc) + (fom_value_pfp_gcd_backward_small_witness_bezout_leftright))) /\\ (exists fom_gap_pfp_gcd_backward_small_witness_bezout_leftright_value_bound. fom_gap_pfp_gcd_backward_small_witness_bezout_leftright_value_bound + S (fom_value_pfp_gcd_backward_small_witness_bezout_leftright) = p))) /\\ (((((((pfgs_U_gcd_backward_small)=0 \\/ (S d)=0) /\\ (((pfgb_P_gcd_backward_small_witness_bezout)=0)))) \\/ (((~((pfgs_U_gcd_backward_small)=0)) /\\ (((~((S d)=0)) /\\ (((pfgs_U_gcd_backward_small)+(S d)=S (pfgb_P_gcd_backward_small_witness_bezout)))))))) /\\ ((forall pfc_index_gcd_backward_small_witness_bezout_leftcoefficients. (exists pfa_gap_gcd_backward_small_witness_bezout_leftcoefficientsbound. pfa_gap_gcd_backward_small_witness_bezout_leftcoefficientsbound + S (pfc_index_gcd_backward_small_witness_bezout_leftcoefficients) = (pfgb_P_gcd_backward_small_witness_bezout)) -> exists pfc_value_gcd_backward_small_witness_bezout_leftcoefficients. ((((exists ff_h_pfp_gcd_backward_small_witness_bezout_leftcoefficientsentry. ff_h_pfp_gcd_backward_small_witness_bezout_leftcoefficientsentry + S (pfc_value_gcd_backward_small_witness_bezout_leftcoefficients) = S ((S (pfc_index_gcd_backward_small_witness_bezout_leftcoefficients)) * pfgb_pc_gcd_backward_small_witness_bezout)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_leftcoefficientsentry. pfgb_pb_gcd_backward_small_witness_bezout = ff_q_pfp_gcd_backward_small_witness_bezout_leftcoefficientsentry * S ((S (pfc_index_gcd_backward_small_witness_bezout_leftcoefficients)) * pfgb_pc_gcd_backward_small_witness_bezout) + (pfc_value_gcd_backward_small_witness_bezout_leftcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_small_witness_bezout_leftcoefficientscoefficient pfc_terms_scale_gcd_backward_small_witness_bezout_leftcoefficientscoefficient pfc_natural_sum_gcd_backward_small_witness_bezout_leftcoefficientscoefficient. ((forall pfc_index_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_small_witness_bezout_leftcoefficients))) -> exists pfc_value_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_small_witness_bezout_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_small_witness_bezout_leftcoefficientscoefficient = ff_q_pfp_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_small_witness_bezout_leftcoefficientscoefficient) + (pfc_value_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_small_witness_bezout_leftcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonal) = (pfgs_U_gcd_backward_small)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_gcd_backward_small)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. pfgs_ub_gcd_backward_small = ff_q_pfp_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_gcd_backward_small) + (pfc_left_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside+(pfgs_U_gcd_backward_small)=(pfc_index_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_small_witness_bezout_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum fs_v_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_small_witness_bezout_leftcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_small_witness_bezout_leftcoefficients))) * fs_v_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_small_witness_bezout_leftcoefficients))) * fs_v_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_small_witness_bezout_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_small_witness_bezout_leftcoefficients)) -> exists fs_a_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_small_witness_bezout_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_small_witness_bezout_leftcoefficientscoefficient = fs_q_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_small_witness_bezout_leftcoefficientscoefficient) + (fs_a_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_small_witness_bezout_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_small_witness_bezout_leftcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_small_witness_bezout_leftcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_small_witness_bezout_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_small_witness_bezout_leftcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_small_witness_bezout_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_small_witness_bezout_leftcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_small_witness_bezout_leftcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_small_witness_bezout_leftcoefficients) + (p) * pfa_offset_right_gcd_backward_small_witness_bezout_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_gcd_backward_small_witness_bezout_rightleft. (exists fom_gap_pfp_gcd_backward_small_witness_bezout_rightleft_index_bound. fom_gap_pfp_gcd_backward_small_witness_bezout_rightleft_index_bound + S (fom_index_pfp_gcd_backward_small_witness_bezout_rightleft) = pfgs_V_gcd_backward_small) -> exists fom_value_pfp_gcd_backward_small_witness_bezout_rightleft. ((((exists fom_beta_height_pfp_gcd_backward_small_witness_bezout_rightleft_entry. fom_beta_height_pfp_gcd_backward_small_witness_bezout_rightleft_entry + S (fom_value_pfp_gcd_backward_small_witness_bezout_rightleft) = S ((S (fom_index_pfp_gcd_backward_small_witness_bezout_rightleft)) * pfgs_vc_gcd_backward_small)) /\\ exists fom_beta_quotient_pfp_gcd_backward_small_witness_bezout_rightleft_entry. pfgs_vb_gcd_backward_small = fom_beta_quotient_pfp_gcd_backward_small_witness_bezout_rightleft_entry * S ((S (fom_index_pfp_gcd_backward_small_witness_bezout_rightleft)) * pfgs_vc_gcd_backward_small) + (fom_value_pfp_gcd_backward_small_witness_bezout_rightleft))) /\\ (exists fom_gap_pfp_gcd_backward_small_witness_bezout_rightleft_value_bound. fom_gap_pfp_gcd_backward_small_witness_bezout_rightleft_value_bound + S (fom_value_pfp_gcd_backward_small_witness_bezout_rightleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_small_witness_bezout_rightright. (exists fom_gap_pfp_gcd_backward_small_witness_bezout_rightright_index_bound. fom_gap_pfp_gcd_backward_small_witness_bezout_rightright_index_bound + S (fom_index_pfp_gcd_backward_small_witness_bezout_rightright) = R) -> exists fom_value_pfp_gcd_backward_small_witness_bezout_rightright. ((((exists fom_beta_height_pfp_gcd_backward_small_witness_bezout_rightright_entry. fom_beta_height_pfp_gcd_backward_small_witness_bezout_rightright_entry + S (fom_value_pfp_gcd_backward_small_witness_bezout_rightright) = S ((S (fom_index_pfp_gcd_backward_small_witness_bezout_rightright)) * rc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_small_witness_bezout_rightright_entry. rb = fom_beta_quotient_pfp_gcd_backward_small_witness_bezout_rightright_entry * S ((S (fom_index_pfp_gcd_backward_small_witness_bezout_rightright)) * rc) + (fom_value_pfp_gcd_backward_small_witness_bezout_rightright))) /\\ (exists fom_gap_pfp_gcd_backward_small_witness_bezout_rightright_value_bound. fom_gap_pfp_gcd_backward_small_witness_bezout_rightright_value_bound + S (fom_value_pfp_gcd_backward_small_witness_bezout_rightright) = p))) /\\ (((((((pfgs_V_gcd_backward_small)=0 \\/ (R)=0) /\\ (((pfgb_Q_gcd_backward_small_witness_bezout)=0)))) \\/ (((~((pfgs_V_gcd_backward_small)=0)) /\\ (((~((R)=0)) /\\ (((pfgs_V_gcd_backward_small)+(R)=S (pfgb_Q_gcd_backward_small_witness_bezout)))))))) /\\ ((forall pfc_index_gcd_backward_small_witness_bezout_rightcoefficients. (exists pfa_gap_gcd_backward_small_witness_bezout_rightcoefficientsbound. pfa_gap_gcd_backward_small_witness_bezout_rightcoefficientsbound + S (pfc_index_gcd_backward_small_witness_bezout_rightcoefficients) = (pfgb_Q_gcd_backward_small_witness_bezout)) -> exists pfc_value_gcd_backward_small_witness_bezout_rightcoefficients. ((((exists ff_h_pfp_gcd_backward_small_witness_bezout_rightcoefficientsentry. ff_h_pfp_gcd_backward_small_witness_bezout_rightcoefficientsentry + S (pfc_value_gcd_backward_small_witness_bezout_rightcoefficients) = S ((S (pfc_index_gcd_backward_small_witness_bezout_rightcoefficients)) * pfgb_qc_gcd_backward_small_witness_bezout)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_rightcoefficientsentry. pfgb_qb_gcd_backward_small_witness_bezout = ff_q_pfp_gcd_backward_small_witness_bezout_rightcoefficientsentry * S ((S (pfc_index_gcd_backward_small_witness_bezout_rightcoefficients)) * pfgb_qc_gcd_backward_small_witness_bezout) + (pfc_value_gcd_backward_small_witness_bezout_rightcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_small_witness_bezout_rightcoefficientscoefficient pfc_terms_scale_gcd_backward_small_witness_bezout_rightcoefficientscoefficient pfc_natural_sum_gcd_backward_small_witness_bezout_rightcoefficientscoefficient. ((forall pfc_index_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_small_witness_bezout_rightcoefficients))) -> exists pfc_value_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_small_witness_bezout_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_small_witness_bezout_rightcoefficientscoefficient = ff_q_pfp_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_small_witness_bezout_rightcoefficientscoefficient) + (pfc_value_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_small_witness_bezout_rightcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonal) = (pfgs_V_gcd_backward_small)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_gcd_backward_small)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. pfgs_vb_gcd_backward_small = ff_q_pfp_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_gcd_backward_small) + (pfc_left_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside+(pfgs_V_gcd_backward_small)=(pfc_index_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm) = (R)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * rc)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. rb = ff_q_pfp_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * rc) + (pfc_right_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside+(R)=(pfc_complement_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_small_witness_bezout_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum fs_v_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_small_witness_bezout_rightcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_small_witness_bezout_rightcoefficients))) * fs_v_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_small_witness_bezout_rightcoefficients))) * fs_v_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_small_witness_bezout_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_small_witness_bezout_rightcoefficients)) -> exists fs_a_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_small_witness_bezout_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_small_witness_bezout_rightcoefficientscoefficient = fs_q_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_small_witness_bezout_rightcoefficientscoefficient) + (fs_a_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_small_witness_bezout_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_small_witness_bezout_rightcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_small_witness_bezout_rightcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_small_witness_bezout_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_small_witness_bezout_rightcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_small_witness_bezout_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_small_witness_bezout_rightcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_small_witness_bezout_rightcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_small_witness_bezout_rightcoefficients) + (p) * pfa_offset_right_gcd_backward_small_witness_bezout_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_gcd_backward_small_witness_bezout_sum_left_bounded. (exists fom_gap_pfp_gcd_backward_small_witness_bezout_sum_left_bounded_index_bound. fom_gap_pfp_gcd_backward_small_witness_bezout_sum_left_bounded_index_bound + S (fom_index_pfp_gcd_backward_small_witness_bezout_sum_left_bounded) = pfgb_P_gcd_backward_small_witness_bezout) -> exists fom_value_pfp_gcd_backward_small_witness_bezout_sum_left_bounded. ((((exists fom_beta_height_pfp_gcd_backward_small_witness_bezout_sum_left_bounded_entry. fom_beta_height_pfp_gcd_backward_small_witness_bezout_sum_left_bounded_entry + S (fom_value_pfp_gcd_backward_small_witness_bezout_sum_left_bounded) = S ((S (fom_index_pfp_gcd_backward_small_witness_bezout_sum_left_bounded)) * pfgb_pc_gcd_backward_small_witness_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_backward_small_witness_bezout_sum_left_bounded_entry. pfgb_pb_gcd_backward_small_witness_bezout = fom_beta_quotient_pfp_gcd_backward_small_witness_bezout_sum_left_bounded_entry * S ((S (fom_index_pfp_gcd_backward_small_witness_bezout_sum_left_bounded)) * pfgb_pc_gcd_backward_small_witness_bezout) + (fom_value_pfp_gcd_backward_small_witness_bezout_sum_left_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_small_witness_bezout_sum_left_bounded_value_bound. fom_gap_pfp_gcd_backward_small_witness_bezout_sum_left_bounded_value_bound + S (fom_value_pfp_gcd_backward_small_witness_bezout_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_backward_small_witness_bezout_sum_right_bounded. (exists fom_gap_pfp_gcd_backward_small_witness_bezout_sum_right_bounded_index_bound. fom_gap_pfp_gcd_backward_small_witness_bezout_sum_right_bounded_index_bound + S (fom_index_pfp_gcd_backward_small_witness_bezout_sum_right_bounded) = pfgb_Q_gcd_backward_small_witness_bezout) -> exists fom_value_pfp_gcd_backward_small_witness_bezout_sum_right_bounded. ((((exists fom_beta_height_pfp_gcd_backward_small_witness_bezout_sum_right_bounded_entry. fom_beta_height_pfp_gcd_backward_small_witness_bezout_sum_right_bounded_entry + S (fom_value_pfp_gcd_backward_small_witness_bezout_sum_right_bounded) = S ((S (fom_index_pfp_gcd_backward_small_witness_bezout_sum_right_bounded)) * pfgb_qc_gcd_backward_small_witness_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_backward_small_witness_bezout_sum_right_bounded_entry. pfgb_qb_gcd_backward_small_witness_bezout = fom_beta_quotient_pfp_gcd_backward_small_witness_bezout_sum_right_bounded_entry * S ((S (fom_index_pfp_gcd_backward_small_witness_bezout_sum_right_bounded)) * pfgb_qc_gcd_backward_small_witness_bezout) + (fom_value_pfp_gcd_backward_small_witness_bezout_sum_right_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_small_witness_bezout_sum_right_bounded_value_bound. fom_gap_pfp_gcd_backward_small_witness_bezout_sum_right_bounded_value_bound + S (fom_value_pfp_gcd_backward_small_witness_bezout_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_backward_small_witness_bezout_sum_result_bounded. (exists fom_gap_pfp_gcd_backward_small_witness_bezout_sum_result_bounded_index_bound. fom_gap_pfp_gcd_backward_small_witness_bezout_sum_result_bounded_index_bound + S (fom_index_pfp_gcd_backward_small_witness_bezout_sum_result_bounded) = pfgs_G_gcd_backward_small) -> exists fom_value_pfp_gcd_backward_small_witness_bezout_sum_result_bounded. ((((exists fom_beta_height_pfp_gcd_backward_small_witness_bezout_sum_result_bounded_entry. fom_beta_height_pfp_gcd_backward_small_witness_bezout_sum_result_bounded_entry + S (fom_value_pfp_gcd_backward_small_witness_bezout_sum_result_bounded) = S ((S (fom_index_pfp_gcd_backward_small_witness_bezout_sum_result_bounded)) * pfgs_gc_gcd_backward_small)) /\\ exists fom_beta_quotient_pfp_gcd_backward_small_witness_bezout_sum_result_bounded_entry. pfgs_gb_gcd_backward_small = fom_beta_quotient_pfp_gcd_backward_small_witness_bezout_sum_result_bounded_entry * S ((S (fom_index_pfp_gcd_backward_small_witness_bezout_sum_result_bounded)) * pfgs_gc_gcd_backward_small) + (fom_value_pfp_gcd_backward_small_witness_bezout_sum_result_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_small_witness_bezout_sum_result_bounded_value_bound. fom_gap_pfp_gcd_backward_small_witness_bezout_sum_result_bounded_value_bound + S (fom_value_pfp_gcd_backward_small_witness_bezout_sum_result_bounded) = p))) /\\ ((exists pfga_ub_gcd_backward_small_witness_bezout_sum pfga_uc_gcd_backward_small_witness_bezout_sum pfga_vb_gcd_backward_small_witness_bezout_sum pfga_vc_gcd_backward_small_witness_bezout_sum pfga_tb_gcd_backward_small_witness_bezout_sum pfga_tc_gcd_backward_small_witness_bezout_sum pfga_K_gcd_backward_small_witness_bezout_sum. ((((forall pfrep_power_gcd_backward_small_witness_bezout_sum_left pfrep_left_gcd_backward_small_witness_bezout_sum_left pfrep_right_gcd_backward_small_witness_bezout_sum_left. ((exists pfrep_position_gcd_backward_small_witness_bezout_sum_leftfirst. ((pfrep_position_gcd_backward_small_witness_bezout_sum_leftfirst+S (pfrep_power_gcd_backward_small_witness_bezout_sum_left)=(pfgb_P_gcd_backward_small_witness_bezout)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_bezout_sum_leftfirstentry. ff_h_pfp_gcd_backward_small_witness_bezout_sum_leftfirstentry + S (pfrep_left_gcd_backward_small_witness_bezout_sum_left) = S ((S (pfrep_position_gcd_backward_small_witness_bezout_sum_leftfirst)) * pfgb_pc_gcd_backward_small_witness_bezout)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_sum_leftfirstentry. pfgb_pb_gcd_backward_small_witness_bezout = ff_q_pfp_gcd_backward_small_witness_bezout_sum_leftfirstentry * S ((S (pfrep_position_gcd_backward_small_witness_bezout_sum_leftfirst)) * pfgb_pc_gcd_backward_small_witness_bezout) + (pfrep_left_gcd_backward_small_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_backward_small_witness_bezout_sum_leftfirstoutside. pfrep_gap_gcd_backward_small_witness_bezout_sum_leftfirstoutside+(pfgb_P_gcd_backward_small_witness_bezout)=(pfrep_power_gcd_backward_small_witness_bezout_sum_left)) /\\ (((pfrep_left_gcd_backward_small_witness_bezout_sum_left)=0))))) -> ((exists pfrep_position_gcd_backward_small_witness_bezout_sum_leftsecond. ((pfrep_position_gcd_backward_small_witness_bezout_sum_leftsecond+S (pfrep_power_gcd_backward_small_witness_bezout_sum_left)=(pfga_K_gcd_backward_small_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_bezout_sum_leftsecondentry. ff_h_pfp_gcd_backward_small_witness_bezout_sum_leftsecondentry + S (pfrep_right_gcd_backward_small_witness_bezout_sum_left) = S ((S (pfrep_position_gcd_backward_small_witness_bezout_sum_leftsecond)) * pfga_uc_gcd_backward_small_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_sum_leftsecondentry. pfga_ub_gcd_backward_small_witness_bezout_sum = ff_q_pfp_gcd_backward_small_witness_bezout_sum_leftsecondentry * S ((S (pfrep_position_gcd_backward_small_witness_bezout_sum_leftsecond)) * pfga_uc_gcd_backward_small_witness_bezout_sum) + (pfrep_right_gcd_backward_small_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_backward_small_witness_bezout_sum_leftsecondoutside. pfrep_gap_gcd_backward_small_witness_bezout_sum_leftsecondoutside+(pfga_K_gcd_backward_small_witness_bezout_sum)=(pfrep_power_gcd_backward_small_witness_bezout_sum_left)) /\\ (((pfrep_right_gcd_backward_small_witness_bezout_sum_left)=0))))) -> pfrep_left_gcd_backward_small_witness_bezout_sum_left=pfrep_right_gcd_backward_small_witness_bezout_sum_left) /\\ ((forall pfrep_power_gcd_backward_small_witness_bezout_sum_right pfrep_left_gcd_backward_small_witness_bezout_sum_right pfrep_right_gcd_backward_small_witness_bezout_sum_right. ((exists pfrep_position_gcd_backward_small_witness_bezout_sum_rightfirst. ((pfrep_position_gcd_backward_small_witness_bezout_sum_rightfirst+S (pfrep_power_gcd_backward_small_witness_bezout_sum_right)=(pfgb_Q_gcd_backward_small_witness_bezout)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_bezout_sum_rightfirstentry. ff_h_pfp_gcd_backward_small_witness_bezout_sum_rightfirstentry + S (pfrep_left_gcd_backward_small_witness_bezout_sum_right) = S ((S (pfrep_position_gcd_backward_small_witness_bezout_sum_rightfirst)) * pfgb_qc_gcd_backward_small_witness_bezout)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_sum_rightfirstentry. pfgb_qb_gcd_backward_small_witness_bezout = ff_q_pfp_gcd_backward_small_witness_bezout_sum_rightfirstentry * S ((S (pfrep_position_gcd_backward_small_witness_bezout_sum_rightfirst)) * pfgb_qc_gcd_backward_small_witness_bezout) + (pfrep_left_gcd_backward_small_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_backward_small_witness_bezout_sum_rightfirstoutside. pfrep_gap_gcd_backward_small_witness_bezout_sum_rightfirstoutside+(pfgb_Q_gcd_backward_small_witness_bezout)=(pfrep_power_gcd_backward_small_witness_bezout_sum_right)) /\\ (((pfrep_left_gcd_backward_small_witness_bezout_sum_right)=0))))) -> ((exists pfrep_position_gcd_backward_small_witness_bezout_sum_rightsecond. ((pfrep_position_gcd_backward_small_witness_bezout_sum_rightsecond+S (pfrep_power_gcd_backward_small_witness_bezout_sum_right)=(pfga_K_gcd_backward_small_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_bezout_sum_rightsecondentry. ff_h_pfp_gcd_backward_small_witness_bezout_sum_rightsecondentry + S (pfrep_right_gcd_backward_small_witness_bezout_sum_right) = S ((S (pfrep_position_gcd_backward_small_witness_bezout_sum_rightsecond)) * pfga_vc_gcd_backward_small_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_sum_rightsecondentry. pfga_vb_gcd_backward_small_witness_bezout_sum = ff_q_pfp_gcd_backward_small_witness_bezout_sum_rightsecondentry * S ((S (pfrep_position_gcd_backward_small_witness_bezout_sum_rightsecond)) * pfga_vc_gcd_backward_small_witness_bezout_sum) + (pfrep_right_gcd_backward_small_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_backward_small_witness_bezout_sum_rightsecondoutside. pfrep_gap_gcd_backward_small_witness_bezout_sum_rightsecondoutside+(pfga_K_gcd_backward_small_witness_bezout_sum)=(pfrep_power_gcd_backward_small_witness_bezout_sum_right)) /\\ (((pfrep_right_gcd_backward_small_witness_bezout_sum_right)=0))))) -> pfrep_left_gcd_backward_small_witness_bezout_sum_right=pfrep_right_gcd_backward_small_witness_bezout_sum_right)))) /\\ (((forall pfp_index_gcd_backward_small_witness_bezout_sum_add. (exists pfa_gap_gcd_backward_small_witness_bezout_sum_addindex. pfa_gap_gcd_backward_small_witness_bezout_sum_addindex + S (pfp_index_gcd_backward_small_witness_bezout_sum_add) = (pfga_K_gcd_backward_small_witness_bezout_sum)) -> exists pfp_left_gcd_backward_small_witness_bezout_sum_add pfp_right_gcd_backward_small_witness_bezout_sum_add pfp_value_gcd_backward_small_witness_bezout_sum_add. ((((exists ff_h_pfp_gcd_backward_small_witness_bezout_sum_addleft. ff_h_pfp_gcd_backward_small_witness_bezout_sum_addleft + S (pfp_left_gcd_backward_small_witness_bezout_sum_add) = S ((S (pfp_index_gcd_backward_small_witness_bezout_sum_add)) * pfga_uc_gcd_backward_small_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_sum_addleft. pfga_ub_gcd_backward_small_witness_bezout_sum = ff_q_pfp_gcd_backward_small_witness_bezout_sum_addleft * S ((S (pfp_index_gcd_backward_small_witness_bezout_sum_add)) * pfga_uc_gcd_backward_small_witness_bezout_sum) + (pfp_left_gcd_backward_small_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_backward_small_witness_bezout_sum_addright. ff_h_pfp_gcd_backward_small_witness_bezout_sum_addright + S (pfp_right_gcd_backward_small_witness_bezout_sum_add) = S ((S (pfp_index_gcd_backward_small_witness_bezout_sum_add)) * pfga_vc_gcd_backward_small_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_sum_addright. pfga_vb_gcd_backward_small_witness_bezout_sum = ff_q_pfp_gcd_backward_small_witness_bezout_sum_addright * S ((S (pfp_index_gcd_backward_small_witness_bezout_sum_add)) * pfga_vc_gcd_backward_small_witness_bezout_sum) + (pfp_right_gcd_backward_small_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_backward_small_witness_bezout_sum_addtarget. ff_h_pfp_gcd_backward_small_witness_bezout_sum_addtarget + S (pfp_value_gcd_backward_small_witness_bezout_sum_add) = S ((S (pfp_index_gcd_backward_small_witness_bezout_sum_add)) * pfga_tc_gcd_backward_small_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_sum_addtarget. pfga_tb_gcd_backward_small_witness_bezout_sum = ff_q_pfp_gcd_backward_small_witness_bezout_sum_addtarget * S ((S (pfp_index_gcd_backward_small_witness_bezout_sum_add)) * pfga_tc_gcd_backward_small_witness_bezout_sum) + (pfp_value_gcd_backward_small_witness_bezout_sum_add))) /\\ ((((exists pfa_gap_gcd_backward_small_witness_bezout_sum_addoperationleft. pfa_gap_gcd_backward_small_witness_bezout_sum_addoperationleft + S (pfp_left_gcd_backward_small_witness_bezout_sum_add) = (p)) /\\ (((exists pfa_gap_gcd_backward_small_witness_bezout_sum_addoperationright. pfa_gap_gcd_backward_small_witness_bezout_sum_addoperationright + S (pfp_right_gcd_backward_small_witness_bezout_sum_add) = (p)) /\\ ((((exists pfa_gap_gcd_backward_small_witness_bezout_sum_addoperationresultbound. pfa_gap_gcd_backward_small_witness_bezout_sum_addoperationresultbound + S (pfp_value_gcd_backward_small_witness_bezout_sum_add) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_small_witness_bezout_sum_addoperationresultcongruence pfa_offset_right_gcd_backward_small_witness_bezout_sum_addoperationresultcongruence. ((pfp_left_gcd_backward_small_witness_bezout_sum_add) + (pfp_right_gcd_backward_small_witness_bezout_sum_add)) + (p) * pfa_offset_left_gcd_backward_small_witness_bezout_sum_addoperationresultcongruence = (pfp_value_gcd_backward_small_witness_bezout_sum_add) + (p) * pfa_offset_right_gcd_backward_small_witness_bezout_sum_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_small_witness_bezout_sum_result pfrep_left_gcd_backward_small_witness_bezout_sum_result pfrep_right_gcd_backward_small_witness_bezout_sum_result. ((exists pfrep_position_gcd_backward_small_witness_bezout_sum_resultfirst. ((pfrep_position_gcd_backward_small_witness_bezout_sum_resultfirst+S (pfrep_power_gcd_backward_small_witness_bezout_sum_result)=(pfga_K_gcd_backward_small_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_bezout_sum_resultfirstentry. ff_h_pfp_gcd_backward_small_witness_bezout_sum_resultfirstentry + S (pfrep_left_gcd_backward_small_witness_bezout_sum_result) = S ((S (pfrep_position_gcd_backward_small_witness_bezout_sum_resultfirst)) * pfga_tc_gcd_backward_small_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_sum_resultfirstentry. pfga_tb_gcd_backward_small_witness_bezout_sum = ff_q_pfp_gcd_backward_small_witness_bezout_sum_resultfirstentry * S ((S (pfrep_position_gcd_backward_small_witness_bezout_sum_resultfirst)) * pfga_tc_gcd_backward_small_witness_bezout_sum) + (pfrep_left_gcd_backward_small_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_backward_small_witness_bezout_sum_resultfirstoutside. pfrep_gap_gcd_backward_small_witness_bezout_sum_resultfirstoutside+(pfga_K_gcd_backward_small_witness_bezout_sum)=(pfrep_power_gcd_backward_small_witness_bezout_sum_result)) /\\ (((pfrep_left_gcd_backward_small_witness_bezout_sum_result)=0))))) -> ((exists pfrep_position_gcd_backward_small_witness_bezout_sum_resultsecond. ((pfrep_position_gcd_backward_small_witness_bezout_sum_resultsecond+S (pfrep_power_gcd_backward_small_witness_bezout_sum_result)=(pfgs_G_gcd_backward_small)) /\\ ((((exists ff_h_pfp_gcd_backward_small_witness_bezout_sum_resultsecondentry. ff_h_pfp_gcd_backward_small_witness_bezout_sum_resultsecondentry + S (pfrep_right_gcd_backward_small_witness_bezout_sum_result) = S ((S (pfrep_position_gcd_backward_small_witness_bezout_sum_resultsecond)) * pfgs_gc_gcd_backward_small)) /\\ exists ff_q_pfp_gcd_backward_small_witness_bezout_sum_resultsecondentry. pfgs_gb_gcd_backward_small = ff_q_pfp_gcd_backward_small_witness_bezout_sum_resultsecondentry * S ((S (pfrep_position_gcd_backward_small_witness_bezout_sum_resultsecond)) * pfgs_gc_gcd_backward_small) + (pfrep_right_gcd_backward_small_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_backward_small_witness_bezout_sum_resultsecondoutside. pfrep_gap_gcd_backward_small_witness_bezout_sum_resultsecondoutside+(pfgs_G_gcd_backward_small)=(pfrep_power_gcd_backward_small_witness_bezout_sum_result)) /\\ (((pfrep_right_gcd_backward_small_witness_bezout_sum_result)=0))))) -> pfrep_left_gcd_backward_small_witness_bezout_sum_result=pfrep_right_gcd_backward_small_witness_bezout_sum_result))))))))))))))))))))))) -> (exists pfgs_gb_gcd_backward_result pfgs_gc_gcd_backward_result pfgs_G_gcd_backward_result pfgs_ub_gcd_backward_result pfgs_uc_gcd_backward_result pfgs_U_gcd_backward_result pfgs_vb_gcd_backward_result pfgs_vc_gcd_backward_result pfgs_V_gcd_backward_result. (((pfgs_G_gcd_backward_result)=0 \\/ (((~((pfgs_G_gcd_backward_result) = 0)) /\\ (((forall fom_index_pfp_gcd_backward_result_witness_normal_moniccoefficients. (exists fom_gap_pfp_gcd_backward_result_witness_normal_moniccoefficients_index_bound. fom_gap_pfp_gcd_backward_result_witness_normal_moniccoefficients_index_bound + S (fom_index_pfp_gcd_backward_result_witness_normal_moniccoefficients) = pfgs_G_gcd_backward_result) -> exists fom_value_pfp_gcd_backward_result_witness_normal_moniccoefficients. ((((exists fom_beta_height_pfp_gcd_backward_result_witness_normal_moniccoefficients_entry. fom_beta_height_pfp_gcd_backward_result_witness_normal_moniccoefficients_entry + S (fom_value_pfp_gcd_backward_result_witness_normal_moniccoefficients) = S ((S (fom_index_pfp_gcd_backward_result_witness_normal_moniccoefficients)) * pfgs_gc_gcd_backward_result)) /\\ exists fom_beta_quotient_pfp_gcd_backward_result_witness_normal_moniccoefficients_entry. pfgs_gb_gcd_backward_result = fom_beta_quotient_pfp_gcd_backward_result_witness_normal_moniccoefficients_entry * S ((S (fom_index_pfp_gcd_backward_result_witness_normal_moniccoefficients)) * pfgs_gc_gcd_backward_result) + (fom_value_pfp_gcd_backward_result_witness_normal_moniccoefficients))) /\\ (exists fom_gap_pfp_gcd_backward_result_witness_normal_moniccoefficients_value_bound. fom_gap_pfp_gcd_backward_result_witness_normal_moniccoefficients_value_bound + S (fom_value_pfp_gcd_backward_result_witness_normal_moniccoefficients) = p))) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_normal_monicleading. ff_h_pfp_gcd_backward_result_witness_normal_monicleading + S (1) = S ((S (0)) * pfgs_gc_gcd_backward_result)) /\\ exists ff_q_pfp_gcd_backward_result_witness_normal_monicleading. pfgs_gb_gcd_backward_result = ff_q_pfp_gcd_backward_result_witness_normal_monicleading * S ((S (0)) * pfgs_gc_gcd_backward_result) + (1))))))))) /\\ (((((((forall fom_index_pfp_gcd_backward_result_witness_common_left_bounded. (exists fom_gap_pfp_gcd_backward_result_witness_common_left_bounded_index_bound. fom_gap_pfp_gcd_backward_result_witness_common_left_bounded_index_bound + S (fom_index_pfp_gcd_backward_result_witness_common_left_bounded) = L) -> exists fom_value_pfp_gcd_backward_result_witness_common_left_bounded. ((((exists fom_beta_height_pfp_gcd_backward_result_witness_common_left_bounded_entry. fom_beta_height_pfp_gcd_backward_result_witness_common_left_bounded_entry + S (fom_value_pfp_gcd_backward_result_witness_common_left_bounded) = S ((S (fom_index_pfp_gcd_backward_result_witness_common_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_backward_result_witness_common_left_bounded_entry. ab = fom_beta_quotient_pfp_gcd_backward_result_witness_common_left_bounded_entry * S ((S (fom_index_pfp_gcd_backward_result_witness_common_left_bounded)) * ac) + (fom_value_pfp_gcd_backward_result_witness_common_left_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_result_witness_common_left_bounded_value_bound. fom_gap_pfp_gcd_backward_result_witness_common_left_bounded_value_bound + S (fom_value_pfp_gcd_backward_result_witness_common_left_bounded) = p))) /\\ ((exists pfgd_qb_gcd_backward_result_witness_common_left pfgd_qc_gcd_backward_result_witness_common_left pfgd_Q_gcd_backward_result_witness_common_left pfgd_pb_gcd_backward_result_witness_common_left pfgd_pc_gcd_backward_result_witness_common_left pfgd_P_gcd_backward_result_witness_common_left. ((((forall fom_index_pfp_gcd_backward_result_witness_common_left_productleft. (exists fom_gap_pfp_gcd_backward_result_witness_common_left_productleft_index_bound. fom_gap_pfp_gcd_backward_result_witness_common_left_productleft_index_bound + S (fom_index_pfp_gcd_backward_result_witness_common_left_productleft) = pfgd_Q_gcd_backward_result_witness_common_left) -> exists fom_value_pfp_gcd_backward_result_witness_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_backward_result_witness_common_left_productleft_entry. fom_beta_height_pfp_gcd_backward_result_witness_common_left_productleft_entry + S (fom_value_pfp_gcd_backward_result_witness_common_left_productleft) = S ((S (fom_index_pfp_gcd_backward_result_witness_common_left_productleft)) * pfgd_qc_gcd_backward_result_witness_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_backward_result_witness_common_left_productleft_entry. pfgd_qb_gcd_backward_result_witness_common_left = fom_beta_quotient_pfp_gcd_backward_result_witness_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_backward_result_witness_common_left_productleft)) * pfgd_qc_gcd_backward_result_witness_common_left) + (fom_value_pfp_gcd_backward_result_witness_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_result_witness_common_left_productleft_value_bound. fom_gap_pfp_gcd_backward_result_witness_common_left_productleft_value_bound + S (fom_value_pfp_gcd_backward_result_witness_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_result_witness_common_left_productright. (exists fom_gap_pfp_gcd_backward_result_witness_common_left_productright_index_bound. fom_gap_pfp_gcd_backward_result_witness_common_left_productright_index_bound + S (fom_index_pfp_gcd_backward_result_witness_common_left_productright) = pfgs_G_gcd_backward_result) -> exists fom_value_pfp_gcd_backward_result_witness_common_left_productright. ((((exists fom_beta_height_pfp_gcd_backward_result_witness_common_left_productright_entry. fom_beta_height_pfp_gcd_backward_result_witness_common_left_productright_entry + S (fom_value_pfp_gcd_backward_result_witness_common_left_productright) = S ((S (fom_index_pfp_gcd_backward_result_witness_common_left_productright)) * pfgs_gc_gcd_backward_result)) /\\ exists fom_beta_quotient_pfp_gcd_backward_result_witness_common_left_productright_entry. pfgs_gb_gcd_backward_result = fom_beta_quotient_pfp_gcd_backward_result_witness_common_left_productright_entry * S ((S (fom_index_pfp_gcd_backward_result_witness_common_left_productright)) * pfgs_gc_gcd_backward_result) + (fom_value_pfp_gcd_backward_result_witness_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_backward_result_witness_common_left_productright_value_bound. fom_gap_pfp_gcd_backward_result_witness_common_left_productright_value_bound + S (fom_value_pfp_gcd_backward_result_witness_common_left_productright) = p))) /\\ (((((((pfgd_Q_gcd_backward_result_witness_common_left)=0 \\/ (pfgs_G_gcd_backward_result)=0) /\\ (((pfgd_P_gcd_backward_result_witness_common_left)=0)))) \\/ (((~((pfgd_Q_gcd_backward_result_witness_common_left)=0)) /\\ (((~((pfgs_G_gcd_backward_result)=0)) /\\ (((pfgd_Q_gcd_backward_result_witness_common_left)+(pfgs_G_gcd_backward_result)=S (pfgd_P_gcd_backward_result_witness_common_left)))))))) /\\ ((forall pfc_index_gcd_backward_result_witness_common_left_productcoefficients. (exists pfa_gap_gcd_backward_result_witness_common_left_productcoefficientsbound. pfa_gap_gcd_backward_result_witness_common_left_productcoefficientsbound + S (pfc_index_gcd_backward_result_witness_common_left_productcoefficients) = (pfgd_P_gcd_backward_result_witness_common_left)) -> exists pfc_value_gcd_backward_result_witness_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_backward_result_witness_common_left_productcoefficientsentry. ff_h_pfp_gcd_backward_result_witness_common_left_productcoefficientsentry + S (pfc_value_gcd_backward_result_witness_common_left_productcoefficients) = S ((S (pfc_index_gcd_backward_result_witness_common_left_productcoefficients)) * pfgd_pc_gcd_backward_result_witness_common_left)) /\\ exists ff_q_pfp_gcd_backward_result_witness_common_left_productcoefficientsentry. pfgd_pb_gcd_backward_result_witness_common_left = ff_q_pfp_gcd_backward_result_witness_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_backward_result_witness_common_left_productcoefficients)) * pfgd_pc_gcd_backward_result_witness_common_left) + (pfc_value_gcd_backward_result_witness_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_result_witness_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_backward_result_witness_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_backward_result_witness_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_result_witness_common_left_productcoefficients))) -> exists pfc_value_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_result_witness_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_result_witness_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_result_witness_common_left_productcoefficientscoefficient) + (pfc_value_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_result_witness_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_backward_result_witness_common_left)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_result_witness_common_left)) /\\ exists ff_q_pfp_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_backward_result_witness_common_left = ff_q_pfp_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_result_witness_common_left) + (pfc_left_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_backward_result_witness_common_left)=(pfc_index_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm) = (pfgs_G_gcd_backward_result)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_backward_result)) /\\ exists ff_q_pfp_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_gcd_backward_result = ff_q_pfp_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_backward_result) + (pfc_right_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_gcd_backward_result)=(pfc_complement_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_result_witness_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_result_witness_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_result_witness_common_left_productcoefficients))) * fs_v_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_result_witness_common_left_productcoefficients))) * fs_v_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_result_witness_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_result_witness_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_result_witness_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_result_witness_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_result_witness_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_result_witness_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_result_witness_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_result_witness_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_result_witness_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_result_witness_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_result_witness_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_result_witness_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_result_witness_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_result_witness_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_backward_result_witness_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_result_witness_common_left_equivalent pfrep_left_gcd_backward_result_witness_common_left_equivalent pfrep_right_gcd_backward_result_witness_common_left_equivalent. ((exists pfrep_position_gcd_backward_result_witness_common_left_equivalentfirst. ((pfrep_position_gcd_backward_result_witness_common_left_equivalentfirst+S (pfrep_power_gcd_backward_result_witness_common_left_equivalent)=(pfgd_P_gcd_backward_result_witness_common_left)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_common_left_equivalentfirstentry. ff_h_pfp_gcd_backward_result_witness_common_left_equivalentfirstentry + S (pfrep_left_gcd_backward_result_witness_common_left_equivalent) = S ((S (pfrep_position_gcd_backward_result_witness_common_left_equivalentfirst)) * pfgd_pc_gcd_backward_result_witness_common_left)) /\\ exists ff_q_pfp_gcd_backward_result_witness_common_left_equivalentfirstentry. pfgd_pb_gcd_backward_result_witness_common_left = ff_q_pfp_gcd_backward_result_witness_common_left_equivalentfirstentry * S ((S (pfrep_position_gcd_backward_result_witness_common_left_equivalentfirst)) * pfgd_pc_gcd_backward_result_witness_common_left) + (pfrep_left_gcd_backward_result_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_result_witness_common_left_equivalentfirstoutside. pfrep_gap_gcd_backward_result_witness_common_left_equivalentfirstoutside+(pfgd_P_gcd_backward_result_witness_common_left)=(pfrep_power_gcd_backward_result_witness_common_left_equivalent)) /\\ (((pfrep_left_gcd_backward_result_witness_common_left_equivalent)=0))))) -> ((exists pfrep_position_gcd_backward_result_witness_common_left_equivalentsecond. ((pfrep_position_gcd_backward_result_witness_common_left_equivalentsecond+S (pfrep_power_gcd_backward_result_witness_common_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_common_left_equivalentsecondentry. ff_h_pfp_gcd_backward_result_witness_common_left_equivalentsecondentry + S (pfrep_right_gcd_backward_result_witness_common_left_equivalent) = S ((S (pfrep_position_gcd_backward_result_witness_common_left_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_gcd_backward_result_witness_common_left_equivalentsecondentry. ab = ff_q_pfp_gcd_backward_result_witness_common_left_equivalentsecondentry * S ((S (pfrep_position_gcd_backward_result_witness_common_left_equivalentsecond)) * ac) + (pfrep_right_gcd_backward_result_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_result_witness_common_left_equivalentsecondoutside. pfrep_gap_gcd_backward_result_witness_common_left_equivalentsecondoutside+(L)=(pfrep_power_gcd_backward_result_witness_common_left_equivalent)) /\\ (((pfrep_right_gcd_backward_result_witness_common_left_equivalent)=0))))) -> pfrep_left_gcd_backward_result_witness_common_left_equivalent=pfrep_right_gcd_backward_result_witness_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_backward_result_witness_common_right_bounded. (exists fom_gap_pfp_gcd_backward_result_witness_common_right_bounded_index_bound. fom_gap_pfp_gcd_backward_result_witness_common_right_bounded_index_bound + S (fom_index_pfp_gcd_backward_result_witness_common_right_bounded) = S d) -> exists fom_value_pfp_gcd_backward_result_witness_common_right_bounded. ((((exists fom_beta_height_pfp_gcd_backward_result_witness_common_right_bounded_entry. fom_beta_height_pfp_gcd_backward_result_witness_common_right_bounded_entry + S (fom_value_pfp_gcd_backward_result_witness_common_right_bounded) = S ((S (fom_index_pfp_gcd_backward_result_witness_common_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_result_witness_common_right_bounded_entry. bb = fom_beta_quotient_pfp_gcd_backward_result_witness_common_right_bounded_entry * S ((S (fom_index_pfp_gcd_backward_result_witness_common_right_bounded)) * bc) + (fom_value_pfp_gcd_backward_result_witness_common_right_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_result_witness_common_right_bounded_value_bound. fom_gap_pfp_gcd_backward_result_witness_common_right_bounded_value_bound + S (fom_value_pfp_gcd_backward_result_witness_common_right_bounded) = p))) /\\ ((exists pfgd_qb_gcd_backward_result_witness_common_right pfgd_qc_gcd_backward_result_witness_common_right pfgd_Q_gcd_backward_result_witness_common_right pfgd_pb_gcd_backward_result_witness_common_right pfgd_pc_gcd_backward_result_witness_common_right pfgd_P_gcd_backward_result_witness_common_right. ((((forall fom_index_pfp_gcd_backward_result_witness_common_right_productleft. (exists fom_gap_pfp_gcd_backward_result_witness_common_right_productleft_index_bound. fom_gap_pfp_gcd_backward_result_witness_common_right_productleft_index_bound + S (fom_index_pfp_gcd_backward_result_witness_common_right_productleft) = pfgd_Q_gcd_backward_result_witness_common_right) -> exists fom_value_pfp_gcd_backward_result_witness_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_backward_result_witness_common_right_productleft_entry. fom_beta_height_pfp_gcd_backward_result_witness_common_right_productleft_entry + S (fom_value_pfp_gcd_backward_result_witness_common_right_productleft) = S ((S (fom_index_pfp_gcd_backward_result_witness_common_right_productleft)) * pfgd_qc_gcd_backward_result_witness_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_backward_result_witness_common_right_productleft_entry. pfgd_qb_gcd_backward_result_witness_common_right = fom_beta_quotient_pfp_gcd_backward_result_witness_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_backward_result_witness_common_right_productleft)) * pfgd_qc_gcd_backward_result_witness_common_right) + (fom_value_pfp_gcd_backward_result_witness_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_backward_result_witness_common_right_productleft_value_bound. fom_gap_pfp_gcd_backward_result_witness_common_right_productleft_value_bound + S (fom_value_pfp_gcd_backward_result_witness_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_result_witness_common_right_productright. (exists fom_gap_pfp_gcd_backward_result_witness_common_right_productright_index_bound. fom_gap_pfp_gcd_backward_result_witness_common_right_productright_index_bound + S (fom_index_pfp_gcd_backward_result_witness_common_right_productright) = pfgs_G_gcd_backward_result) -> exists fom_value_pfp_gcd_backward_result_witness_common_right_productright. ((((exists fom_beta_height_pfp_gcd_backward_result_witness_common_right_productright_entry. fom_beta_height_pfp_gcd_backward_result_witness_common_right_productright_entry + S (fom_value_pfp_gcd_backward_result_witness_common_right_productright) = S ((S (fom_index_pfp_gcd_backward_result_witness_common_right_productright)) * pfgs_gc_gcd_backward_result)) /\\ exists fom_beta_quotient_pfp_gcd_backward_result_witness_common_right_productright_entry. pfgs_gb_gcd_backward_result = fom_beta_quotient_pfp_gcd_backward_result_witness_common_right_productright_entry * S ((S (fom_index_pfp_gcd_backward_result_witness_common_right_productright)) * pfgs_gc_gcd_backward_result) + (fom_value_pfp_gcd_backward_result_witness_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_backward_result_witness_common_right_productright_value_bound. fom_gap_pfp_gcd_backward_result_witness_common_right_productright_value_bound + S (fom_value_pfp_gcd_backward_result_witness_common_right_productright) = p))) /\\ (((((((pfgd_Q_gcd_backward_result_witness_common_right)=0 \\/ (pfgs_G_gcd_backward_result)=0) /\\ (((pfgd_P_gcd_backward_result_witness_common_right)=0)))) \\/ (((~((pfgd_Q_gcd_backward_result_witness_common_right)=0)) /\\ (((~((pfgs_G_gcd_backward_result)=0)) /\\ (((pfgd_Q_gcd_backward_result_witness_common_right)+(pfgs_G_gcd_backward_result)=S (pfgd_P_gcd_backward_result_witness_common_right)))))))) /\\ ((forall pfc_index_gcd_backward_result_witness_common_right_productcoefficients. (exists pfa_gap_gcd_backward_result_witness_common_right_productcoefficientsbound. pfa_gap_gcd_backward_result_witness_common_right_productcoefficientsbound + S (pfc_index_gcd_backward_result_witness_common_right_productcoefficients) = (pfgd_P_gcd_backward_result_witness_common_right)) -> exists pfc_value_gcd_backward_result_witness_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_backward_result_witness_common_right_productcoefficientsentry. ff_h_pfp_gcd_backward_result_witness_common_right_productcoefficientsentry + S (pfc_value_gcd_backward_result_witness_common_right_productcoefficients) = S ((S (pfc_index_gcd_backward_result_witness_common_right_productcoefficients)) * pfgd_pc_gcd_backward_result_witness_common_right)) /\\ exists ff_q_pfp_gcd_backward_result_witness_common_right_productcoefficientsentry. pfgd_pb_gcd_backward_result_witness_common_right = ff_q_pfp_gcd_backward_result_witness_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_backward_result_witness_common_right_productcoefficients)) * pfgd_pc_gcd_backward_result_witness_common_right) + (pfc_value_gcd_backward_result_witness_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_result_witness_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_backward_result_witness_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_backward_result_witness_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_result_witness_common_right_productcoefficients))) -> exists pfc_value_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_result_witness_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_result_witness_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_result_witness_common_right_productcoefficientscoefficient) + (pfc_value_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_result_witness_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_backward_result_witness_common_right)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_result_witness_common_right)) /\\ exists ff_q_pfp_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_backward_result_witness_common_right = ff_q_pfp_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_backward_result_witness_common_right) + (pfc_left_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_backward_result_witness_common_right)=(pfc_index_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm) = (pfgs_G_gcd_backward_result)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_backward_result)) /\\ exists ff_q_pfp_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_gcd_backward_result = ff_q_pfp_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_backward_result) + (pfc_right_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_gcd_backward_result)=(pfc_complement_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_result_witness_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_result_witness_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_result_witness_common_right_productcoefficients))) * fs_v_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_result_witness_common_right_productcoefficients))) * fs_v_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_result_witness_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_result_witness_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_result_witness_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_result_witness_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_result_witness_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_result_witness_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_result_witness_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_result_witness_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_result_witness_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_result_witness_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_result_witness_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_result_witness_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_result_witness_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_result_witness_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_backward_result_witness_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_result_witness_common_right_equivalent pfrep_left_gcd_backward_result_witness_common_right_equivalent pfrep_right_gcd_backward_result_witness_common_right_equivalent. ((exists pfrep_position_gcd_backward_result_witness_common_right_equivalentfirst. ((pfrep_position_gcd_backward_result_witness_common_right_equivalentfirst+S (pfrep_power_gcd_backward_result_witness_common_right_equivalent)=(pfgd_P_gcd_backward_result_witness_common_right)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_common_right_equivalentfirstentry. ff_h_pfp_gcd_backward_result_witness_common_right_equivalentfirstentry + S (pfrep_left_gcd_backward_result_witness_common_right_equivalent) = S ((S (pfrep_position_gcd_backward_result_witness_common_right_equivalentfirst)) * pfgd_pc_gcd_backward_result_witness_common_right)) /\\ exists ff_q_pfp_gcd_backward_result_witness_common_right_equivalentfirstentry. pfgd_pb_gcd_backward_result_witness_common_right = ff_q_pfp_gcd_backward_result_witness_common_right_equivalentfirstentry * S ((S (pfrep_position_gcd_backward_result_witness_common_right_equivalentfirst)) * pfgd_pc_gcd_backward_result_witness_common_right) + (pfrep_left_gcd_backward_result_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_result_witness_common_right_equivalentfirstoutside. pfrep_gap_gcd_backward_result_witness_common_right_equivalentfirstoutside+(pfgd_P_gcd_backward_result_witness_common_right)=(pfrep_power_gcd_backward_result_witness_common_right_equivalent)) /\\ (((pfrep_left_gcd_backward_result_witness_common_right_equivalent)=0))))) -> ((exists pfrep_position_gcd_backward_result_witness_common_right_equivalentsecond. ((pfrep_position_gcd_backward_result_witness_common_right_equivalentsecond+S (pfrep_power_gcd_backward_result_witness_common_right_equivalent)=(S d)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_common_right_equivalentsecondentry. ff_h_pfp_gcd_backward_result_witness_common_right_equivalentsecondentry + S (pfrep_right_gcd_backward_result_witness_common_right_equivalent) = S ((S (pfrep_position_gcd_backward_result_witness_common_right_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_gcd_backward_result_witness_common_right_equivalentsecondentry. bb = ff_q_pfp_gcd_backward_result_witness_common_right_equivalentsecondentry * S ((S (pfrep_position_gcd_backward_result_witness_common_right_equivalentsecond)) * bc) + (pfrep_right_gcd_backward_result_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_backward_result_witness_common_right_equivalentsecondoutside. pfrep_gap_gcd_backward_result_witness_common_right_equivalentsecondoutside+(S d)=(pfrep_power_gcd_backward_result_witness_common_right_equivalent)) /\\ (((pfrep_right_gcd_backward_result_witness_common_right_equivalent)=0))))) -> pfrep_left_gcd_backward_result_witness_common_right_equivalent=pfrep_right_gcd_backward_result_witness_common_right_equivalent)))))))))) /\\ ((exists pfgb_pb_gcd_backward_result_witness_bezout pfgb_pc_gcd_backward_result_witness_bezout pfgb_P_gcd_backward_result_witness_bezout pfgb_qb_gcd_backward_result_witness_bezout pfgb_qc_gcd_backward_result_witness_bezout pfgb_Q_gcd_backward_result_witness_bezout. ((((forall fom_index_pfp_gcd_backward_result_witness_bezout_leftleft. (exists fom_gap_pfp_gcd_backward_result_witness_bezout_leftleft_index_bound. fom_gap_pfp_gcd_backward_result_witness_bezout_leftleft_index_bound + S (fom_index_pfp_gcd_backward_result_witness_bezout_leftleft) = pfgs_U_gcd_backward_result) -> exists fom_value_pfp_gcd_backward_result_witness_bezout_leftleft. ((((exists fom_beta_height_pfp_gcd_backward_result_witness_bezout_leftleft_entry. fom_beta_height_pfp_gcd_backward_result_witness_bezout_leftleft_entry + S (fom_value_pfp_gcd_backward_result_witness_bezout_leftleft) = S ((S (fom_index_pfp_gcd_backward_result_witness_bezout_leftleft)) * pfgs_uc_gcd_backward_result)) /\\ exists fom_beta_quotient_pfp_gcd_backward_result_witness_bezout_leftleft_entry. pfgs_ub_gcd_backward_result = fom_beta_quotient_pfp_gcd_backward_result_witness_bezout_leftleft_entry * S ((S (fom_index_pfp_gcd_backward_result_witness_bezout_leftleft)) * pfgs_uc_gcd_backward_result) + (fom_value_pfp_gcd_backward_result_witness_bezout_leftleft))) /\\ (exists fom_gap_pfp_gcd_backward_result_witness_bezout_leftleft_value_bound. fom_gap_pfp_gcd_backward_result_witness_bezout_leftleft_value_bound + S (fom_value_pfp_gcd_backward_result_witness_bezout_leftleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_result_witness_bezout_leftright. (exists fom_gap_pfp_gcd_backward_result_witness_bezout_leftright_index_bound. fom_gap_pfp_gcd_backward_result_witness_bezout_leftright_index_bound + S (fom_index_pfp_gcd_backward_result_witness_bezout_leftright) = L) -> exists fom_value_pfp_gcd_backward_result_witness_bezout_leftright. ((((exists fom_beta_height_pfp_gcd_backward_result_witness_bezout_leftright_entry. fom_beta_height_pfp_gcd_backward_result_witness_bezout_leftright_entry + S (fom_value_pfp_gcd_backward_result_witness_bezout_leftright) = S ((S (fom_index_pfp_gcd_backward_result_witness_bezout_leftright)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_backward_result_witness_bezout_leftright_entry. ab = fom_beta_quotient_pfp_gcd_backward_result_witness_bezout_leftright_entry * S ((S (fom_index_pfp_gcd_backward_result_witness_bezout_leftright)) * ac) + (fom_value_pfp_gcd_backward_result_witness_bezout_leftright))) /\\ (exists fom_gap_pfp_gcd_backward_result_witness_bezout_leftright_value_bound. fom_gap_pfp_gcd_backward_result_witness_bezout_leftright_value_bound + S (fom_value_pfp_gcd_backward_result_witness_bezout_leftright) = p))) /\\ (((((((pfgs_U_gcd_backward_result)=0 \\/ (L)=0) /\\ (((pfgb_P_gcd_backward_result_witness_bezout)=0)))) \\/ (((~((pfgs_U_gcd_backward_result)=0)) /\\ (((~((L)=0)) /\\ (((pfgs_U_gcd_backward_result)+(L)=S (pfgb_P_gcd_backward_result_witness_bezout)))))))) /\\ ((forall pfc_index_gcd_backward_result_witness_bezout_leftcoefficients. (exists pfa_gap_gcd_backward_result_witness_bezout_leftcoefficientsbound. pfa_gap_gcd_backward_result_witness_bezout_leftcoefficientsbound + S (pfc_index_gcd_backward_result_witness_bezout_leftcoefficients) = (pfgb_P_gcd_backward_result_witness_bezout)) -> exists pfc_value_gcd_backward_result_witness_bezout_leftcoefficients. ((((exists ff_h_pfp_gcd_backward_result_witness_bezout_leftcoefficientsentry. ff_h_pfp_gcd_backward_result_witness_bezout_leftcoefficientsentry + S (pfc_value_gcd_backward_result_witness_bezout_leftcoefficients) = S ((S (pfc_index_gcd_backward_result_witness_bezout_leftcoefficients)) * pfgb_pc_gcd_backward_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_leftcoefficientsentry. pfgb_pb_gcd_backward_result_witness_bezout = ff_q_pfp_gcd_backward_result_witness_bezout_leftcoefficientsentry * S ((S (pfc_index_gcd_backward_result_witness_bezout_leftcoefficients)) * pfgb_pc_gcd_backward_result_witness_bezout) + (pfc_value_gcd_backward_result_witness_bezout_leftcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_result_witness_bezout_leftcoefficientscoefficient pfc_terms_scale_gcd_backward_result_witness_bezout_leftcoefficientscoefficient pfc_natural_sum_gcd_backward_result_witness_bezout_leftcoefficientscoefficient. ((forall pfc_index_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_result_witness_bezout_leftcoefficients))) -> exists pfc_value_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_result_witness_bezout_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_result_witness_bezout_leftcoefficientscoefficient = ff_q_pfp_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_result_witness_bezout_leftcoefficientscoefficient) + (pfc_value_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_result_witness_bezout_leftcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonal) = (pfgs_U_gcd_backward_result)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_gcd_backward_result)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. pfgs_ub_gcd_backward_result = ff_q_pfp_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_gcd_backward_result) + (pfc_left_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside+(pfgs_U_gcd_backward_result)=(pfc_index_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_result_witness_bezout_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum fs_v_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_result_witness_bezout_leftcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_result_witness_bezout_leftcoefficients))) * fs_v_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_result_witness_bezout_leftcoefficients))) * fs_v_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_result_witness_bezout_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_result_witness_bezout_leftcoefficients)) -> exists fs_a_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_result_witness_bezout_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_result_witness_bezout_leftcoefficientscoefficient = fs_q_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_result_witness_bezout_leftcoefficientscoefficient) + (fs_a_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_result_witness_bezout_leftcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_result_witness_bezout_leftcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_result_witness_bezout_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_result_witness_bezout_leftcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_result_witness_bezout_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_result_witness_bezout_leftcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_result_witness_bezout_leftcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_result_witness_bezout_leftcoefficients) + (p) * pfa_offset_right_gcd_backward_result_witness_bezout_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_gcd_backward_result_witness_bezout_rightleft. (exists fom_gap_pfp_gcd_backward_result_witness_bezout_rightleft_index_bound. fom_gap_pfp_gcd_backward_result_witness_bezout_rightleft_index_bound + S (fom_index_pfp_gcd_backward_result_witness_bezout_rightleft) = pfgs_V_gcd_backward_result) -> exists fom_value_pfp_gcd_backward_result_witness_bezout_rightleft. ((((exists fom_beta_height_pfp_gcd_backward_result_witness_bezout_rightleft_entry. fom_beta_height_pfp_gcd_backward_result_witness_bezout_rightleft_entry + S (fom_value_pfp_gcd_backward_result_witness_bezout_rightleft) = S ((S (fom_index_pfp_gcd_backward_result_witness_bezout_rightleft)) * pfgs_vc_gcd_backward_result)) /\\ exists fom_beta_quotient_pfp_gcd_backward_result_witness_bezout_rightleft_entry. pfgs_vb_gcd_backward_result = fom_beta_quotient_pfp_gcd_backward_result_witness_bezout_rightleft_entry * S ((S (fom_index_pfp_gcd_backward_result_witness_bezout_rightleft)) * pfgs_vc_gcd_backward_result) + (fom_value_pfp_gcd_backward_result_witness_bezout_rightleft))) /\\ (exists fom_gap_pfp_gcd_backward_result_witness_bezout_rightleft_value_bound. fom_gap_pfp_gcd_backward_result_witness_bezout_rightleft_value_bound + S (fom_value_pfp_gcd_backward_result_witness_bezout_rightleft) = p))) /\\ (((forall fom_index_pfp_gcd_backward_result_witness_bezout_rightright. (exists fom_gap_pfp_gcd_backward_result_witness_bezout_rightright_index_bound. fom_gap_pfp_gcd_backward_result_witness_bezout_rightright_index_bound + S (fom_index_pfp_gcd_backward_result_witness_bezout_rightright) = S d) -> exists fom_value_pfp_gcd_backward_result_witness_bezout_rightright. ((((exists fom_beta_height_pfp_gcd_backward_result_witness_bezout_rightright_entry. fom_beta_height_pfp_gcd_backward_result_witness_bezout_rightright_entry + S (fom_value_pfp_gcd_backward_result_witness_bezout_rightright) = S ((S (fom_index_pfp_gcd_backward_result_witness_bezout_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_backward_result_witness_bezout_rightright_entry. bb = fom_beta_quotient_pfp_gcd_backward_result_witness_bezout_rightright_entry * S ((S (fom_index_pfp_gcd_backward_result_witness_bezout_rightright)) * bc) + (fom_value_pfp_gcd_backward_result_witness_bezout_rightright))) /\\ (exists fom_gap_pfp_gcd_backward_result_witness_bezout_rightright_value_bound. fom_gap_pfp_gcd_backward_result_witness_bezout_rightright_value_bound + S (fom_value_pfp_gcd_backward_result_witness_bezout_rightright) = p))) /\\ (((((((pfgs_V_gcd_backward_result)=0 \\/ (S d)=0) /\\ (((pfgb_Q_gcd_backward_result_witness_bezout)=0)))) \\/ (((~((pfgs_V_gcd_backward_result)=0)) /\\ (((~((S d)=0)) /\\ (((pfgs_V_gcd_backward_result)+(S d)=S (pfgb_Q_gcd_backward_result_witness_bezout)))))))) /\\ ((forall pfc_index_gcd_backward_result_witness_bezout_rightcoefficients. (exists pfa_gap_gcd_backward_result_witness_bezout_rightcoefficientsbound. pfa_gap_gcd_backward_result_witness_bezout_rightcoefficientsbound + S (pfc_index_gcd_backward_result_witness_bezout_rightcoefficients) = (pfgb_Q_gcd_backward_result_witness_bezout)) -> exists pfc_value_gcd_backward_result_witness_bezout_rightcoefficients. ((((exists ff_h_pfp_gcd_backward_result_witness_bezout_rightcoefficientsentry. ff_h_pfp_gcd_backward_result_witness_bezout_rightcoefficientsentry + S (pfc_value_gcd_backward_result_witness_bezout_rightcoefficients) = S ((S (pfc_index_gcd_backward_result_witness_bezout_rightcoefficients)) * pfgb_qc_gcd_backward_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_rightcoefficientsentry. pfgb_qb_gcd_backward_result_witness_bezout = ff_q_pfp_gcd_backward_result_witness_bezout_rightcoefficientsentry * S ((S (pfc_index_gcd_backward_result_witness_bezout_rightcoefficients)) * pfgb_qc_gcd_backward_result_witness_bezout) + (pfc_value_gcd_backward_result_witness_bezout_rightcoefficients))) /\\ ((exists pfc_terms_code_gcd_backward_result_witness_bezout_rightcoefficientscoefficient pfc_terms_scale_gcd_backward_result_witness_bezout_rightcoefficientscoefficient pfc_natural_sum_gcd_backward_result_witness_bezout_rightcoefficientscoefficient. ((forall pfc_index_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalbound. pfa_gap_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_backward_result_witness_bezout_rightcoefficients))) -> exists pfc_value_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_result_witness_bezout_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_backward_result_witness_bezout_rightcoefficientscoefficient = ff_q_pfp_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_backward_result_witness_bezout_rightcoefficientscoefficient) + (pfc_value_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_left_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_right_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonal)+pfc_complement_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm=(pfc_index_gcd_backward_result_witness_bezout_rightcoefficients)) /\\ ((((((exists pfa_gap_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonal) = (pfgs_V_gcd_backward_result)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_gcd_backward_result)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. pfgs_vb_gcd_backward_result = ff_q_pfp_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_gcd_backward_result) + (pfc_left_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside+(pfgs_V_gcd_backward_result)=(pfc_index_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonal)=pfc_left_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm*pfc_right_gcd_backward_result_witness_bezout_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum fs_v_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_backward_result_witness_bezout_rightcoefficientscoefficient) = S ((S (S (pfc_index_gcd_backward_result_witness_bezout_rightcoefficients))) * fs_v_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_backward_result_witness_bezout_rightcoefficients))) * fs_v_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum) + (pfc_natural_sum_gcd_backward_result_witness_bezout_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_backward_result_witness_bezout_rightcoefficients)) -> exists fs_a_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_result_witness_bezout_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_backward_result_witness_bezout_rightcoefficientscoefficient = fs_q_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_backward_result_witness_bezout_rightcoefficientscoefficient) + (fs_a_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum) + (fs_r_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum) + (fs_s_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_backward_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_backward_result_witness_bezout_rightcoefficientscoefficientresiduebound. pfa_gap_gcd_backward_result_witness_bezout_rightcoefficientscoefficientresiduebound + S (pfc_value_gcd_backward_result_witness_bezout_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_result_witness_bezout_rightcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_backward_result_witness_bezout_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_backward_result_witness_bezout_rightcoefficientscoefficient) + (p) * pfa_offset_left_gcd_backward_result_witness_bezout_rightcoefficientscoefficientresiduecongruence = (pfc_value_gcd_backward_result_witness_bezout_rightcoefficients) + (p) * pfa_offset_right_gcd_backward_result_witness_bezout_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_gcd_backward_result_witness_bezout_sum_left_bounded. (exists fom_gap_pfp_gcd_backward_result_witness_bezout_sum_left_bounded_index_bound. fom_gap_pfp_gcd_backward_result_witness_bezout_sum_left_bounded_index_bound + S (fom_index_pfp_gcd_backward_result_witness_bezout_sum_left_bounded) = pfgb_P_gcd_backward_result_witness_bezout) -> exists fom_value_pfp_gcd_backward_result_witness_bezout_sum_left_bounded. ((((exists fom_beta_height_pfp_gcd_backward_result_witness_bezout_sum_left_bounded_entry. fom_beta_height_pfp_gcd_backward_result_witness_bezout_sum_left_bounded_entry + S (fom_value_pfp_gcd_backward_result_witness_bezout_sum_left_bounded) = S ((S (fom_index_pfp_gcd_backward_result_witness_bezout_sum_left_bounded)) * pfgb_pc_gcd_backward_result_witness_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_backward_result_witness_bezout_sum_left_bounded_entry. pfgb_pb_gcd_backward_result_witness_bezout = fom_beta_quotient_pfp_gcd_backward_result_witness_bezout_sum_left_bounded_entry * S ((S (fom_index_pfp_gcd_backward_result_witness_bezout_sum_left_bounded)) * pfgb_pc_gcd_backward_result_witness_bezout) + (fom_value_pfp_gcd_backward_result_witness_bezout_sum_left_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_result_witness_bezout_sum_left_bounded_value_bound. fom_gap_pfp_gcd_backward_result_witness_bezout_sum_left_bounded_value_bound + S (fom_value_pfp_gcd_backward_result_witness_bezout_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_backward_result_witness_bezout_sum_right_bounded. (exists fom_gap_pfp_gcd_backward_result_witness_bezout_sum_right_bounded_index_bound. fom_gap_pfp_gcd_backward_result_witness_bezout_sum_right_bounded_index_bound + S (fom_index_pfp_gcd_backward_result_witness_bezout_sum_right_bounded) = pfgb_Q_gcd_backward_result_witness_bezout) -> exists fom_value_pfp_gcd_backward_result_witness_bezout_sum_right_bounded. ((((exists fom_beta_height_pfp_gcd_backward_result_witness_bezout_sum_right_bounded_entry. fom_beta_height_pfp_gcd_backward_result_witness_bezout_sum_right_bounded_entry + S (fom_value_pfp_gcd_backward_result_witness_bezout_sum_right_bounded) = S ((S (fom_index_pfp_gcd_backward_result_witness_bezout_sum_right_bounded)) * pfgb_qc_gcd_backward_result_witness_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_backward_result_witness_bezout_sum_right_bounded_entry. pfgb_qb_gcd_backward_result_witness_bezout = fom_beta_quotient_pfp_gcd_backward_result_witness_bezout_sum_right_bounded_entry * S ((S (fom_index_pfp_gcd_backward_result_witness_bezout_sum_right_bounded)) * pfgb_qc_gcd_backward_result_witness_bezout) + (fom_value_pfp_gcd_backward_result_witness_bezout_sum_right_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_result_witness_bezout_sum_right_bounded_value_bound. fom_gap_pfp_gcd_backward_result_witness_bezout_sum_right_bounded_value_bound + S (fom_value_pfp_gcd_backward_result_witness_bezout_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_backward_result_witness_bezout_sum_result_bounded. (exists fom_gap_pfp_gcd_backward_result_witness_bezout_sum_result_bounded_index_bound. fom_gap_pfp_gcd_backward_result_witness_bezout_sum_result_bounded_index_bound + S (fom_index_pfp_gcd_backward_result_witness_bezout_sum_result_bounded) = pfgs_G_gcd_backward_result) -> exists fom_value_pfp_gcd_backward_result_witness_bezout_sum_result_bounded. ((((exists fom_beta_height_pfp_gcd_backward_result_witness_bezout_sum_result_bounded_entry. fom_beta_height_pfp_gcd_backward_result_witness_bezout_sum_result_bounded_entry + S (fom_value_pfp_gcd_backward_result_witness_bezout_sum_result_bounded) = S ((S (fom_index_pfp_gcd_backward_result_witness_bezout_sum_result_bounded)) * pfgs_gc_gcd_backward_result)) /\\ exists fom_beta_quotient_pfp_gcd_backward_result_witness_bezout_sum_result_bounded_entry. pfgs_gb_gcd_backward_result = fom_beta_quotient_pfp_gcd_backward_result_witness_bezout_sum_result_bounded_entry * S ((S (fom_index_pfp_gcd_backward_result_witness_bezout_sum_result_bounded)) * pfgs_gc_gcd_backward_result) + (fom_value_pfp_gcd_backward_result_witness_bezout_sum_result_bounded))) /\\ (exists fom_gap_pfp_gcd_backward_result_witness_bezout_sum_result_bounded_value_bound. fom_gap_pfp_gcd_backward_result_witness_bezout_sum_result_bounded_value_bound + S (fom_value_pfp_gcd_backward_result_witness_bezout_sum_result_bounded) = p))) /\\ ((exists pfga_ub_gcd_backward_result_witness_bezout_sum pfga_uc_gcd_backward_result_witness_bezout_sum pfga_vb_gcd_backward_result_witness_bezout_sum pfga_vc_gcd_backward_result_witness_bezout_sum pfga_tb_gcd_backward_result_witness_bezout_sum pfga_tc_gcd_backward_result_witness_bezout_sum pfga_K_gcd_backward_result_witness_bezout_sum. ((((forall pfrep_power_gcd_backward_result_witness_bezout_sum_left pfrep_left_gcd_backward_result_witness_bezout_sum_left pfrep_right_gcd_backward_result_witness_bezout_sum_left. ((exists pfrep_position_gcd_backward_result_witness_bezout_sum_leftfirst. ((pfrep_position_gcd_backward_result_witness_bezout_sum_leftfirst+S (pfrep_power_gcd_backward_result_witness_bezout_sum_left)=(pfgb_P_gcd_backward_result_witness_bezout)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_bezout_sum_leftfirstentry. ff_h_pfp_gcd_backward_result_witness_bezout_sum_leftfirstentry + S (pfrep_left_gcd_backward_result_witness_bezout_sum_left) = S ((S (pfrep_position_gcd_backward_result_witness_bezout_sum_leftfirst)) * pfgb_pc_gcd_backward_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_sum_leftfirstentry. pfgb_pb_gcd_backward_result_witness_bezout = ff_q_pfp_gcd_backward_result_witness_bezout_sum_leftfirstentry * S ((S (pfrep_position_gcd_backward_result_witness_bezout_sum_leftfirst)) * pfgb_pc_gcd_backward_result_witness_bezout) + (pfrep_left_gcd_backward_result_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_backward_result_witness_bezout_sum_leftfirstoutside. pfrep_gap_gcd_backward_result_witness_bezout_sum_leftfirstoutside+(pfgb_P_gcd_backward_result_witness_bezout)=(pfrep_power_gcd_backward_result_witness_bezout_sum_left)) /\\ (((pfrep_left_gcd_backward_result_witness_bezout_sum_left)=0))))) -> ((exists pfrep_position_gcd_backward_result_witness_bezout_sum_leftsecond. ((pfrep_position_gcd_backward_result_witness_bezout_sum_leftsecond+S (pfrep_power_gcd_backward_result_witness_bezout_sum_left)=(pfga_K_gcd_backward_result_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_bezout_sum_leftsecondentry. ff_h_pfp_gcd_backward_result_witness_bezout_sum_leftsecondentry + S (pfrep_right_gcd_backward_result_witness_bezout_sum_left) = S ((S (pfrep_position_gcd_backward_result_witness_bezout_sum_leftsecond)) * pfga_uc_gcd_backward_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_sum_leftsecondentry. pfga_ub_gcd_backward_result_witness_bezout_sum = ff_q_pfp_gcd_backward_result_witness_bezout_sum_leftsecondentry * S ((S (pfrep_position_gcd_backward_result_witness_bezout_sum_leftsecond)) * pfga_uc_gcd_backward_result_witness_bezout_sum) + (pfrep_right_gcd_backward_result_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_backward_result_witness_bezout_sum_leftsecondoutside. pfrep_gap_gcd_backward_result_witness_bezout_sum_leftsecondoutside+(pfga_K_gcd_backward_result_witness_bezout_sum)=(pfrep_power_gcd_backward_result_witness_bezout_sum_left)) /\\ (((pfrep_right_gcd_backward_result_witness_bezout_sum_left)=0))))) -> pfrep_left_gcd_backward_result_witness_bezout_sum_left=pfrep_right_gcd_backward_result_witness_bezout_sum_left) /\\ ((forall pfrep_power_gcd_backward_result_witness_bezout_sum_right pfrep_left_gcd_backward_result_witness_bezout_sum_right pfrep_right_gcd_backward_result_witness_bezout_sum_right. ((exists pfrep_position_gcd_backward_result_witness_bezout_sum_rightfirst. ((pfrep_position_gcd_backward_result_witness_bezout_sum_rightfirst+S (pfrep_power_gcd_backward_result_witness_bezout_sum_right)=(pfgb_Q_gcd_backward_result_witness_bezout)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_bezout_sum_rightfirstentry. ff_h_pfp_gcd_backward_result_witness_bezout_sum_rightfirstentry + S (pfrep_left_gcd_backward_result_witness_bezout_sum_right) = S ((S (pfrep_position_gcd_backward_result_witness_bezout_sum_rightfirst)) * pfgb_qc_gcd_backward_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_sum_rightfirstentry. pfgb_qb_gcd_backward_result_witness_bezout = ff_q_pfp_gcd_backward_result_witness_bezout_sum_rightfirstentry * S ((S (pfrep_position_gcd_backward_result_witness_bezout_sum_rightfirst)) * pfgb_qc_gcd_backward_result_witness_bezout) + (pfrep_left_gcd_backward_result_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_backward_result_witness_bezout_sum_rightfirstoutside. pfrep_gap_gcd_backward_result_witness_bezout_sum_rightfirstoutside+(pfgb_Q_gcd_backward_result_witness_bezout)=(pfrep_power_gcd_backward_result_witness_bezout_sum_right)) /\\ (((pfrep_left_gcd_backward_result_witness_bezout_sum_right)=0))))) -> ((exists pfrep_position_gcd_backward_result_witness_bezout_sum_rightsecond. ((pfrep_position_gcd_backward_result_witness_bezout_sum_rightsecond+S (pfrep_power_gcd_backward_result_witness_bezout_sum_right)=(pfga_K_gcd_backward_result_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_bezout_sum_rightsecondentry. ff_h_pfp_gcd_backward_result_witness_bezout_sum_rightsecondentry + S (pfrep_right_gcd_backward_result_witness_bezout_sum_right) = S ((S (pfrep_position_gcd_backward_result_witness_bezout_sum_rightsecond)) * pfga_vc_gcd_backward_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_sum_rightsecondentry. pfga_vb_gcd_backward_result_witness_bezout_sum = ff_q_pfp_gcd_backward_result_witness_bezout_sum_rightsecondentry * S ((S (pfrep_position_gcd_backward_result_witness_bezout_sum_rightsecond)) * pfga_vc_gcd_backward_result_witness_bezout_sum) + (pfrep_right_gcd_backward_result_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_backward_result_witness_bezout_sum_rightsecondoutside. pfrep_gap_gcd_backward_result_witness_bezout_sum_rightsecondoutside+(pfga_K_gcd_backward_result_witness_bezout_sum)=(pfrep_power_gcd_backward_result_witness_bezout_sum_right)) /\\ (((pfrep_right_gcd_backward_result_witness_bezout_sum_right)=0))))) -> pfrep_left_gcd_backward_result_witness_bezout_sum_right=pfrep_right_gcd_backward_result_witness_bezout_sum_right)))) /\\ (((forall pfp_index_gcd_backward_result_witness_bezout_sum_add. (exists pfa_gap_gcd_backward_result_witness_bezout_sum_addindex. pfa_gap_gcd_backward_result_witness_bezout_sum_addindex + S (pfp_index_gcd_backward_result_witness_bezout_sum_add) = (pfga_K_gcd_backward_result_witness_bezout_sum)) -> exists pfp_left_gcd_backward_result_witness_bezout_sum_add pfp_right_gcd_backward_result_witness_bezout_sum_add pfp_value_gcd_backward_result_witness_bezout_sum_add. ((((exists ff_h_pfp_gcd_backward_result_witness_bezout_sum_addleft. ff_h_pfp_gcd_backward_result_witness_bezout_sum_addleft + S (pfp_left_gcd_backward_result_witness_bezout_sum_add) = S ((S (pfp_index_gcd_backward_result_witness_bezout_sum_add)) * pfga_uc_gcd_backward_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_sum_addleft. pfga_ub_gcd_backward_result_witness_bezout_sum = ff_q_pfp_gcd_backward_result_witness_bezout_sum_addleft * S ((S (pfp_index_gcd_backward_result_witness_bezout_sum_add)) * pfga_uc_gcd_backward_result_witness_bezout_sum) + (pfp_left_gcd_backward_result_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_backward_result_witness_bezout_sum_addright. ff_h_pfp_gcd_backward_result_witness_bezout_sum_addright + S (pfp_right_gcd_backward_result_witness_bezout_sum_add) = S ((S (pfp_index_gcd_backward_result_witness_bezout_sum_add)) * pfga_vc_gcd_backward_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_sum_addright. pfga_vb_gcd_backward_result_witness_bezout_sum = ff_q_pfp_gcd_backward_result_witness_bezout_sum_addright * S ((S (pfp_index_gcd_backward_result_witness_bezout_sum_add)) * pfga_vc_gcd_backward_result_witness_bezout_sum) + (pfp_right_gcd_backward_result_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_backward_result_witness_bezout_sum_addtarget. ff_h_pfp_gcd_backward_result_witness_bezout_sum_addtarget + S (pfp_value_gcd_backward_result_witness_bezout_sum_add) = S ((S (pfp_index_gcd_backward_result_witness_bezout_sum_add)) * pfga_tc_gcd_backward_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_sum_addtarget. pfga_tb_gcd_backward_result_witness_bezout_sum = ff_q_pfp_gcd_backward_result_witness_bezout_sum_addtarget * S ((S (pfp_index_gcd_backward_result_witness_bezout_sum_add)) * pfga_tc_gcd_backward_result_witness_bezout_sum) + (pfp_value_gcd_backward_result_witness_bezout_sum_add))) /\\ ((((exists pfa_gap_gcd_backward_result_witness_bezout_sum_addoperationleft. pfa_gap_gcd_backward_result_witness_bezout_sum_addoperationleft + S (pfp_left_gcd_backward_result_witness_bezout_sum_add) = (p)) /\\ (((exists pfa_gap_gcd_backward_result_witness_bezout_sum_addoperationright. pfa_gap_gcd_backward_result_witness_bezout_sum_addoperationright + S (pfp_right_gcd_backward_result_witness_bezout_sum_add) = (p)) /\\ ((((exists pfa_gap_gcd_backward_result_witness_bezout_sum_addoperationresultbound. pfa_gap_gcd_backward_result_witness_bezout_sum_addoperationresultbound + S (pfp_value_gcd_backward_result_witness_bezout_sum_add) = (p)) /\\ ((exists pfa_offset_left_gcd_backward_result_witness_bezout_sum_addoperationresultcongruence pfa_offset_right_gcd_backward_result_witness_bezout_sum_addoperationresultcongruence. ((pfp_left_gcd_backward_result_witness_bezout_sum_add) + (pfp_right_gcd_backward_result_witness_bezout_sum_add)) + (p) * pfa_offset_left_gcd_backward_result_witness_bezout_sum_addoperationresultcongruence = (pfp_value_gcd_backward_result_witness_bezout_sum_add) + (p) * pfa_offset_right_gcd_backward_result_witness_bezout_sum_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_gcd_backward_result_witness_bezout_sum_result pfrep_left_gcd_backward_result_witness_bezout_sum_result pfrep_right_gcd_backward_result_witness_bezout_sum_result. ((exists pfrep_position_gcd_backward_result_witness_bezout_sum_resultfirst. ((pfrep_position_gcd_backward_result_witness_bezout_sum_resultfirst+S (pfrep_power_gcd_backward_result_witness_bezout_sum_result)=(pfga_K_gcd_backward_result_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_bezout_sum_resultfirstentry. ff_h_pfp_gcd_backward_result_witness_bezout_sum_resultfirstentry + S (pfrep_left_gcd_backward_result_witness_bezout_sum_result) = S ((S (pfrep_position_gcd_backward_result_witness_bezout_sum_resultfirst)) * pfga_tc_gcd_backward_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_sum_resultfirstentry. pfga_tb_gcd_backward_result_witness_bezout_sum = ff_q_pfp_gcd_backward_result_witness_bezout_sum_resultfirstentry * S ((S (pfrep_position_gcd_backward_result_witness_bezout_sum_resultfirst)) * pfga_tc_gcd_backward_result_witness_bezout_sum) + (pfrep_left_gcd_backward_result_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_backward_result_witness_bezout_sum_resultfirstoutside. pfrep_gap_gcd_backward_result_witness_bezout_sum_resultfirstoutside+(pfga_K_gcd_backward_result_witness_bezout_sum)=(pfrep_power_gcd_backward_result_witness_bezout_sum_result)) /\\ (((pfrep_left_gcd_backward_result_witness_bezout_sum_result)=0))))) -> ((exists pfrep_position_gcd_backward_result_witness_bezout_sum_resultsecond. ((pfrep_position_gcd_backward_result_witness_bezout_sum_resultsecond+S (pfrep_power_gcd_backward_result_witness_bezout_sum_result)=(pfgs_G_gcd_backward_result)) /\\ ((((exists ff_h_pfp_gcd_backward_result_witness_bezout_sum_resultsecondentry. ff_h_pfp_gcd_backward_result_witness_bezout_sum_resultsecondentry + S (pfrep_right_gcd_backward_result_witness_bezout_sum_result) = S ((S (pfrep_position_gcd_backward_result_witness_bezout_sum_resultsecond)) * pfgs_gc_gcd_backward_result)) /\\ exists ff_q_pfp_gcd_backward_result_witness_bezout_sum_resultsecondentry. pfgs_gb_gcd_backward_result = ff_q_pfp_gcd_backward_result_witness_bezout_sum_resultsecondentry * S ((S (pfrep_position_gcd_backward_result_witness_bezout_sum_resultsecond)) * pfgs_gc_gcd_backward_result) + (pfrep_right_gcd_backward_result_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_backward_result_witness_bezout_sum_resultsecondoutside. pfrep_gap_gcd_backward_result_witness_bezout_sum_resultsecondoutside+(pfgs_G_gcd_backward_result)=(pfrep_power_gcd_backward_result_witness_bezout_sum_result)) /\\ (((pfrep_right_gcd_backward_result_witness_bezout_sum_result)=0))))) -> pfrep_left_gcd_backward_result_witness_bezout_sum_result=pfrep_right_gcd_backward_result_witness_bezout_sum_result)))))))))))))))))))))))", "statement_sha256": "47effc74c7288edebd59bc86780ba39667f6cdd1d02b487d26c3daf86a340ed0", "summary": "Carry an already normalized common divisor through an actual Euclidean step. 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((forall fom_index_pfp_gcd_induction_trim_bounded. (exists fom_gap_pfp_gcd_induction_trim_bounded_index_bound. fom_gap_pfp_gcd_induction_trim_bounded_index_bound + S (fom_index_pfp_gcd_induction_trim_bounded) = K) -> exists fom_value_pfp_gcd_induction_trim_bounded. ((((exists fom_beta_height_pfp_gcd_induction_trim_bounded_entry. fom_beta_height_pfp_gcd_induction_trim_bounded_entry + S (fom_value_pfp_gcd_induction_trim_bounded) = S ((S (fom_index_pfp_gcd_induction_trim_bounded)) * tc)) /\\ exists fom_beta_quotient_pfp_gcd_induction_trim_bounded_entry. tb = fom_beta_quotient_pfp_gcd_induction_trim_bounded_entry * S ((S (fom_index_pfp_gcd_induction_trim_bounded)) * tc) + (fom_value_pfp_gcd_induction_trim_bounded))) /\\ (exists fom_gap_pfp_gcd_induction_trim_bounded_value_bound. fom_gap_pfp_gcd_induction_trim_bounded_value_bound + S (fom_value_pfp_gcd_induction_trim_bounded) = p))) /\\ (((forall pfrep_power_gcd_induction_trim_equivalent pfrep_left_gcd_induction_trim_equivalent pfrep_right_gcd_induction_trim_equivalent. ((exists pfrep_position_gcd_induction_trim_equivalentfirst. ((pfrep_position_gcd_induction_trim_equivalentfirst+S (pfrep_power_gcd_induction_trim_equivalent)=(K)) /\\ ((((exists ff_h_pfp_gcd_induction_trim_equivalentfirstentry. ff_h_pfp_gcd_induction_trim_equivalentfirstentry + S (pfrep_left_gcd_induction_trim_equivalent) = S ((S (pfrep_position_gcd_induction_trim_equivalentfirst)) * tc)) /\\ exists ff_q_pfp_gcd_induction_trim_equivalentfirstentry. tb = ff_q_pfp_gcd_induction_trim_equivalentfirstentry * S ((S (pfrep_position_gcd_induction_trim_equivalentfirst)) * tc) + (pfrep_left_gcd_induction_trim_equivalent)))))) \\/ (((exists pfrep_gap_gcd_induction_trim_equivalentfirstoutside. pfrep_gap_gcd_induction_trim_equivalentfirstoutside+(K)=(pfrep_power_gcd_induction_trim_equivalent)) /\\ (((pfrep_left_gcd_induction_trim_equivalent)=0))))) -> ((exists pfrep_position_gcd_induction_trim_equivalentsecond. ((pfrep_position_gcd_induction_trim_equivalentsecond+S (pfrep_power_gcd_induction_trim_equivalent)=(M)) /\\ ((((exists ff_h_pfp_gcd_induction_trim_equivalentsecondentry. ff_h_pfp_gcd_induction_trim_equivalentsecondentry + S (pfrep_right_gcd_induction_trim_equivalent) = S ((S (pfrep_position_gcd_induction_trim_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_gcd_induction_trim_equivalentsecondentry. bb = ff_q_pfp_gcd_induction_trim_equivalentsecondentry * S ((S (pfrep_position_gcd_induction_trim_equivalentsecond)) * bc) + (pfrep_right_gcd_induction_trim_equivalent)))))) \\/ (((exists pfrep_gap_gcd_induction_trim_equivalentsecondoutside. pfrep_gap_gcd_induction_trim_equivalentsecondoutside+(M)=(pfrep_power_gcd_induction_trim_equivalent)) /\\ (((pfrep_right_gcd_induction_trim_equivalent)=0))))) -> pfrep_left_gcd_induction_trim_equivalent=pfrep_right_gcd_induction_trim_equivalent) /\\ (((exists pfc_gap_gcd_induction_trim_length. pfc_gap_gcd_induction_trim_length+(K)=(M)) /\\ (((K)=0 \\/ (exists pfg_degree_gcd_induction_trim. (((K)=S (pfg_degree_gcd_induction_trim)) /\\ (((forall fom_index_pfp_gcd_induction_trim_degreecoefficients. (exists fom_gap_pfp_gcd_induction_trim_degreecoefficients_index_bound. fom_gap_pfp_gcd_induction_trim_degreecoefficients_index_bound + S (fom_index_pfp_gcd_induction_trim_degreecoefficients) = K) -> exists fom_value_pfp_gcd_induction_trim_degreecoefficients. ((((exists fom_beta_height_pfp_gcd_induction_trim_degreecoefficients_entry. fom_beta_height_pfp_gcd_induction_trim_degreecoefficients_entry + S (fom_value_pfp_gcd_induction_trim_degreecoefficients) = S ((S (fom_index_pfp_gcd_induction_trim_degreecoefficients)) * tc)) /\\ exists fom_beta_quotient_pfp_gcd_induction_trim_degreecoefficients_entry. tb = fom_beta_quotient_pfp_gcd_induction_trim_degreecoefficients_entry * S ((S (fom_index_pfp_gcd_induction_trim_degreecoefficients)) * tc) + (fom_value_pfp_gcd_induction_trim_degreecoefficients))) /\\ (exists fom_gap_pfp_gcd_induction_trim_degreecoefficients_value_bound. fom_gap_pfp_gcd_induction_trim_degreecoefficients_value_bound + S (fom_value_pfp_gcd_induction_trim_degreecoefficients) = p))) /\\ ((exists pfd_leading_gcd_induction_trim_degree. ((((exists ff_h_pfp_gcd_induction_trim_degreeentry. ff_h_pfp_gcd_induction_trim_degreeentry + S (pfd_leading_gcd_induction_trim_degree) = S ((S (0)) * tc)) /\\ exists ff_q_pfp_gcd_induction_trim_degreeentry. tb = ff_q_pfp_gcd_induction_trim_degreeentry * S ((S (0)) * tc) + (pfd_leading_gcd_induction_trim_degree))) /\\ ((~(pfd_leading_gcd_induction_trim_degree=0)))))))))))))))))", "specialize prime_field_polynomial_reduced_representative_exists (p)", "specialize prime_field_polynomial_reduced_representative_exists (bb)", "specialize prime_field_polynomial_reduced_representative_exists (bc)", "specialize prime_field_polynomial_reduced_representative_exists (M)", "apply prime_field_polynomial_reduced_representative_exists", "exact hB", "cases ht", "cases ht_witness", "cases ht_witness_witness", "cases ht_witness_witness_witness", "cases ht_witness_witness_witness_right", "cases ht_witness_witness_witness_right_right", "specialize prime_field_polynomial_gcd_bezout_equivalent_second (p)", "specialize prime_field_polynomial_gcd_bezout_equivalent_second (ab)", "specialize prime_field_polynomial_gcd_bezout_equivalent_second (ac)", "specialize prime_field_polynomial_gcd_bezout_equivalent_second (L)", "specialize prime_field_polynomial_gcd_bezout_equivalent_second (x)", "specialize prime_field_polynomial_gcd_bezout_equivalent_second (x1)", "specialize prime_field_polynomial_gcd_bezout_equivalent_second (x2)", "specialize prime_field_polynomial_gcd_bezout_equivalent_second (bb)", "specialize prime_field_polynomial_gcd_bezout_equivalent_second (bc)", "specialize prime_field_polynomial_gcd_bezout_equivalent_second (M)", "apply prime_field_polynomial_gcd_bezout_equivalent_second", "exact hp", "exact hA", "exact hB", "exact ht_witness_witness_witness_right_left", "cases ht_witness_witness_witness_right_right_right", "rewrite ht_witness_witness_witness_right_right_right_left", "rewrite ht_witness_witness_witness_right_right_right_left", "rewrite ht_witness_witness_witness_right_right_right_left", "rewrite ht_witness_witness_witness_right_right_right_left", "rewrite ht_witness_witness_witness_right_right_right_left", "rewrite ht_witness_witness_witness_right_right_right_left", "rewrite ht_witness_witness_witness_right_right_right_left", "rewrite ht_witness_witness_witness_right_right_right_left", "rewrite ht_witness_witness_witness_right_right_right_left", "specialize prime_field_polynomial_gcd_bezout_empty_second (p)", "specialize prime_field_polynomial_gcd_bezout_empty_second (ab)", "specialize prime_field_polynomial_gcd_bezout_empty_second (ac)", "specialize prime_field_polynomial_gcd_bezout_empty_second (L)", "specialize prime_field_polynomial_gcd_bezout_empty_second (x)", "specialize prime_field_polynomial_gcd_bezout_empty_second (x1)", "apply prime_field_polynomial_gcd_bezout_empty_second", "exact hp", "exact hA", "cases ht_witness_witness_witness_right_right_right_right", "have hlength : x2=S x3", "cases ht_witness_witness_witness_right_right_right_right_witness", "exact ht_witness_witness_witness_right_right_right_right_witness_left", "have hdBound : exists pfc_gap_gcd_induction_degree_bound. pfc_gap_gcd_induction_degree_bound+(x3)=(n)", "specialize le_of_succ_le_succ (x3)", "specialize le_of_succ_le_succ (n)", "apply le_of_succ_le_succ", "have hlengthBound : exists pfc_gap_gcd_induction_length_bound. pfc_gap_gcd_induction_length_bound+(x2)=(S n)", "specialize le_trans (x2)", "specialize le_trans (M)", "specialize le_trans (S n)", "apply le_trans", "exact ht_witness_witness_witness_right_right_left", "exact hbound", "rewrite hlength at hlengthBound", "exact hlengthBound", "rewrite hlength", "rewrite hlength", "rewrite hlength", "rewrite hlength", "rewrite hlength", "rewrite hlength", "rewrite hlength", "rewrite hlength", "rewrite hlength", "have he : exists qb qc q rb rc R. ((forall fom_index_pfp_gcd_induction_divisioninput. (exists fom_gap_pfp_gcd_induction_divisioninput_index_bound. fom_gap_pfp_gcd_induction_divisioninput_index_bound + S (fom_index_pfp_gcd_induction_divisioninput) = L) -> exists fom_value_pfp_gcd_induction_divisioninput. ((((exists fom_beta_height_pfp_gcd_induction_divisioninput_entry. fom_beta_height_pfp_gcd_induction_divisioninput_entry + S (fom_value_pfp_gcd_induction_divisioninput) = S ((S (fom_index_pfp_gcd_induction_divisioninput)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_induction_divisioninput_entry. ab = fom_beta_quotient_pfp_gcd_induction_divisioninput_entry * S ((S (fom_index_pfp_gcd_induction_divisioninput)) * ac) + (fom_value_pfp_gcd_induction_divisioninput))) /\\ (exists fom_gap_pfp_gcd_induction_divisioninput_value_bound. fom_gap_pfp_gcd_induction_divisioninput_value_bound + S (fom_value_pfp_gcd_induction_divisioninput) = p))) /\\ (((forall fom_index_pfp_gcd_induction_divisiondivisor. (exists fom_gap_pfp_gcd_induction_divisiondivisor_index_bound. fom_gap_pfp_gcd_induction_divisiondivisor_index_bound + S (fom_index_pfp_gcd_induction_divisiondivisor) = S (x3)) -> exists fom_value_pfp_gcd_induction_divisiondivisor. ((((exists fom_beta_height_pfp_gcd_induction_divisiondivisor_entry. fom_beta_height_pfp_gcd_induction_divisiondivisor_entry + S (fom_value_pfp_gcd_induction_divisiondivisor) = S ((S (fom_index_pfp_gcd_induction_divisiondivisor)) * x1)) /\\ exists fom_beta_quotient_pfp_gcd_induction_divisiondivisor_entry. x = fom_beta_quotient_pfp_gcd_induction_divisiondivisor_entry * S ((S (fom_index_pfp_gcd_induction_divisiondivisor)) * x1) + (fom_value_pfp_gcd_induction_divisiondivisor))) /\\ (exists fom_gap_pfp_gcd_induction_divisiondivisor_value_bound. fom_gap_pfp_gcd_induction_divisiondivisor_value_bound + S (fom_value_pfp_gcd_induction_divisiondivisor) = p))) /\\ (((((((q)=0) /\\ ((exists pfc_gap_gcd_induction_divisionlengthshort. pfc_gap_gcd_induction_divisionlengthshort+(L)=(x3))))) \\/ (((~((q)=0)) /\\ (((q)+(x3)=(L)))))) /\\ ((exists pfd_head_gcd_induction_division pfd_inverse_gcd_induction_division pfd_product_code_gcd_induction_division pfd_product_scale_gcd_induction_division pfd_residual_code_gcd_induction_division pfd_residual_scale_gcd_induction_division pfd_cut_gcd_induction_division. ((((exists ff_h_pfp_gcd_induction_divisionhead. ff_h_pfp_gcd_induction_divisionhead + S (pfd_head_gcd_induction_division) = S ((S (0)) * x1)) /\\ exists ff_q_pfp_gcd_induction_divisionhead. x = ff_q_pfp_gcd_induction_divisionhead * S ((S (0)) * x1) + (pfd_head_gcd_induction_division))) /\\ (((((~((pfd_head_gcd_induction_division) = 0)) /\\ ((((exists pfa_gap_gcd_induction_divisioninversemultiplicationleft. pfa_gap_gcd_induction_divisioninversemultiplicationleft + S (pfd_head_gcd_induction_division) = (p)) /\\ (((exists pfa_gap_gcd_induction_divisioninversemultiplicationright. pfa_gap_gcd_induction_divisioninversemultiplicationright + S (pfd_inverse_gcd_induction_division) = (p)) /\\ ((((exists pfa_gap_gcd_induction_divisioninversemultiplicationresultbound. pfa_gap_gcd_induction_divisioninversemultiplicationresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_gcd_induction_divisioninversemultiplicationresultcongruence pfa_offset_right_gcd_induction_divisioninversemultiplicationresultcongruence. ((pfd_head_gcd_induction_division) * (pfd_inverse_gcd_induction_division)) + (p) * pfa_offset_left_gcd_induction_divisioninversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_gcd_induction_divisioninversemultiplicationresultcongruence)))))))))))) /\\ (((forall pfd_index_gcd_induction_divisionquotient. (exists pfa_gap_gcd_induction_divisionquotientbound. pfa_gap_gcd_induction_divisionquotientbound + S (pfd_index_gcd_induction_divisionquotient) = (q)) -> exists pfd_value_gcd_induction_divisionquotient. ((((exists ff_h_pfp_gcd_induction_divisionquotiententry. ff_h_pfp_gcd_induction_divisionquotiententry + S (pfd_value_gcd_induction_divisionquotient) = S ((S (pfd_index_gcd_induction_divisionquotient)) * qc)) /\\ exists ff_q_pfp_gcd_induction_divisionquotiententry. qb = ff_q_pfp_gcd_induction_divisionquotiententry * S ((S (pfd_index_gcd_induction_divisionquotient)) * qc) + (pfd_value_gcd_induction_divisionquotient))) /\\ ((exists pfd_input_gcd_induction_divisionquotientstep pfd_previous_gcd_induction_divisionquotientstep pfd_difference_gcd_induction_divisionquotientstep. ((((exists ff_h_pfp_gcd_induction_divisionquotientstepinput. ff_h_pfp_gcd_induction_divisionquotientstepinput + S (pfd_input_gcd_induction_divisionquotientstep) = S ((S (pfd_index_gcd_induction_divisionquotient)) * ac)) /\\ exists ff_q_pfp_gcd_induction_divisionquotientstepinput. ab = ff_q_pfp_gcd_induction_divisionquotientstepinput * S ((S (pfd_index_gcd_induction_divisionquotient)) * ac) + (pfd_input_gcd_induction_divisionquotientstep))) /\\ (((exists pfc_terms_code_gcd_induction_divisionquotientstepprevious pfc_terms_scale_gcd_induction_divisionquotientstepprevious pfc_natural_sum_gcd_induction_divisionquotientstepprevious. ((forall pfc_index_gcd_induction_divisionquotientsteppreviousdiagonal. (exists pfa_gap_gcd_induction_divisionquotientsteppreviousdiagonalbound. pfa_gap_gcd_induction_divisionquotientsteppreviousdiagonalbound + S (pfc_index_gcd_induction_divisionquotientsteppreviousdiagonal) = (S (pfd_index_gcd_induction_divisionquotient))) -> exists pfc_value_gcd_induction_divisionquotientsteppreviousdiagonal. ((((exists ff_h_pfp_gcd_induction_divisionquotientsteppreviousdiagonalentry. ff_h_pfp_gcd_induction_divisionquotientsteppreviousdiagonalentry + S (pfc_value_gcd_induction_divisionquotientsteppreviousdiagonal) = S ((S (pfc_index_gcd_induction_divisionquotientsteppreviousdiagonal)) * pfc_terms_scale_gcd_induction_divisionquotientstepprevious)) /\\ exists ff_q_pfp_gcd_induction_divisionquotientsteppreviousdiagonalentry. pfc_terms_code_gcd_induction_divisionquotientstepprevious = ff_q_pfp_gcd_induction_divisionquotientsteppreviousdiagonalentry * S ((S (pfc_index_gcd_induction_divisionquotientsteppreviousdiagonal)) * pfc_terms_scale_gcd_induction_divisionquotientstepprevious) + (pfc_value_gcd_induction_divisionquotientsteppreviousdiagonal))) /\\ ((exists pfc_complement_gcd_induction_divisionquotientsteppreviousdiagonalterm pfc_left_gcd_induction_divisionquotientsteppreviousdiagonalterm pfc_right_gcd_induction_divisionquotientsteppreviousdiagonalterm. (((pfc_index_gcd_induction_divisionquotientsteppreviousdiagonal)+pfc_complement_gcd_induction_divisionquotientsteppreviousdiagonalterm=(pfd_index_gcd_induction_divisionquotient)) /\\ ((((((exists pfa_gap_gcd_induction_divisionquotientsteppreviousdiagonaltermleftinside. pfa_gap_gcd_induction_divisionquotientsteppreviousdiagonaltermleftinside + S (pfc_index_gcd_induction_divisionquotientsteppreviousdiagonal) = (pfd_index_gcd_induction_divisionquotient)) /\\ ((((exists ff_h_pfp_gcd_induction_divisionquotientsteppreviousdiagonaltermleftentry. ff_h_pfp_gcd_induction_divisionquotientsteppreviousdiagonaltermleftentry + S (pfc_left_gcd_induction_divisionquotientsteppreviousdiagonalterm) = S ((S (pfc_index_gcd_induction_divisionquotientsteppreviousdiagonal)) * qc)) /\\ exists ff_q_pfp_gcd_induction_divisionquotientsteppreviousdiagonaltermleftentry. qb = ff_q_pfp_gcd_induction_divisionquotientsteppreviousdiagonaltermleftentry * S ((S (pfc_index_gcd_induction_divisionquotientsteppreviousdiagonal)) * qc) + (pfc_left_gcd_induction_divisionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_induction_divisionquotientsteppreviousdiagonaltermleftoutside. pfc_gap_gcd_induction_divisionquotientsteppreviousdiagonaltermleftoutside+(pfd_index_gcd_induction_divisionquotient)=(pfc_index_gcd_induction_divisionquotientsteppreviousdiagonal)) /\\ (((pfc_left_gcd_induction_divisionquotientsteppreviousdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_induction_divisionquotientsteppreviousdiagonaltermrightinside. pfa_gap_gcd_induction_divisionquotientsteppreviousdiagonaltermrightinside + S (pfc_complement_gcd_induction_divisionquotientsteppreviousdiagonalterm) = (S (x3))) /\\ ((((exists ff_h_pfp_gcd_induction_divisionquotientsteppreviousdiagonaltermrightentry. ff_h_pfp_gcd_induction_divisionquotientsteppreviousdiagonaltermrightentry + S (pfc_right_gcd_induction_divisionquotientsteppreviousdiagonalterm) = S ((S (pfc_complement_gcd_induction_divisionquotientsteppreviousdiagonalterm)) * x1)) /\\ exists ff_q_pfp_gcd_induction_divisionquotientsteppreviousdiagonaltermrightentry. x = ff_q_pfp_gcd_induction_divisionquotientsteppreviousdiagonaltermrightentry * S ((S (pfc_complement_gcd_induction_divisionquotientsteppreviousdiagonalterm)) * x1) + (pfc_right_gcd_induction_divisionquotientsteppreviousdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_induction_divisionquotientsteppreviousdiagonaltermrightoutside. pfc_gap_gcd_induction_divisionquotientsteppreviousdiagonaltermrightoutside+(S (x3))=(pfc_complement_gcd_induction_divisionquotientsteppreviousdiagonalterm)) /\\ (((pfc_right_gcd_induction_divisionquotientsteppreviousdiagonalterm)=0))))) /\\ (((pfc_value_gcd_induction_divisionquotientsteppreviousdiagonal)=pfc_left_gcd_induction_divisionquotientsteppreviousdiagonalterm*pfc_right_gcd_induction_divisionquotientsteppreviousdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_induction_divisionquotientstepprevioussum fs_v_pfc_gcd_induction_divisionquotientstepprevioussum. ((((exists fs_h_pfc_gcd_induction_divisionquotientstepprevioussum_body_start. fs_h_pfc_gcd_induction_divisionquotientstepprevioussum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_induction_divisionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_induction_divisionquotientstepprevioussum_body_start. fs_u_pfc_gcd_induction_divisionquotientstepprevioussum = fs_q_pfc_gcd_induction_divisionquotientstepprevioussum_body_start * S ((S (0)) * fs_v_pfc_gcd_induction_divisionquotientstepprevioussum) + (0))) /\\ ((((exists fs_h_pfc_gcd_induction_divisionquotientstepprevioussum_body_terminal. fs_h_pfc_gcd_induction_divisionquotientstepprevioussum_body_terminal + S (pfc_natural_sum_gcd_induction_divisionquotientstepprevious) = S ((S (S (pfd_index_gcd_induction_divisionquotient))) * fs_v_pfc_gcd_induction_divisionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_induction_divisionquotientstepprevioussum_body_terminal. fs_u_pfc_gcd_induction_divisionquotientstepprevioussum = fs_q_pfc_gcd_induction_divisionquotientstepprevioussum_body_terminal * S ((S (S (pfd_index_gcd_induction_divisionquotient))) * fs_v_pfc_gcd_induction_divisionquotientstepprevioussum) + (pfc_natural_sum_gcd_induction_divisionquotientstepprevious))) /\\ forall fs_i_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps. (exists fs_lt_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps_bound. fs_lt_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps_bound + S fs_i_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps = S (pfd_index_gcd_induction_divisionquotient)) -> exists fs_a_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps fs_r_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps fs_s_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps. ((((exists fs_h_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps_summand. fs_h_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps_summand + S (fs_a_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps)) * pfc_terms_scale_gcd_induction_divisionquotientstepprevious)) /\\ exists fs_q_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps_summand. pfc_terms_code_gcd_induction_divisionquotientstepprevious = fs_q_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps_summand * S ((S (fs_i_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps)) * pfc_terms_scale_gcd_induction_divisionquotientstepprevious) + (fs_a_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps_partial. fs_h_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps_partial + S (fs_r_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps) = S ((S (fs_i_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_induction_divisionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps_partial. fs_u_pfc_gcd_induction_divisionquotientstepprevioussum = fs_q_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps_partial * S ((S (fs_i_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_induction_divisionquotientstepprevioussum) + (fs_r_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps_successor. fs_h_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps_successor + S (fs_s_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps) = S ((S (S fs_i_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_induction_divisionquotientstepprevioussum)) /\\ exists fs_q_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps_successor. fs_u_pfc_gcd_induction_divisionquotientstepprevioussum = fs_q_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps_successor * S ((S (S fs_i_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps)) * fs_v_pfc_gcd_induction_divisionquotientstepprevioussum) + (fs_s_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps))) /\\ fs_s_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps = fs_r_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps + fs_a_pfc_gcd_induction_divisionquotientstepprevioussum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_induction_divisionquotientsteppreviousresiduebound. pfa_gap_gcd_induction_divisionquotientsteppreviousresiduebound + S (pfd_previous_gcd_induction_divisionquotientstep) = (p)) /\\ ((exists pfa_offset_left_gcd_induction_divisionquotientsteppreviousresiduecongruence pfa_offset_right_gcd_induction_divisionquotientsteppreviousresiduecongruence. (pfc_natural_sum_gcd_induction_divisionquotientstepprevious) + (p) * pfa_offset_left_gcd_induction_divisionquotientsteppreviousresiduecongruence = (pfd_previous_gcd_induction_divisionquotientstep) + (p) * pfa_offset_right_gcd_induction_divisionquotientsteppreviousresiduecongruence))))))))) /\\ (((((exists pfa_gap_gcd_induction_divisionquotientstepsubtractleft. pfa_gap_gcd_induction_divisionquotientstepsubtractleft + S (pfd_previous_gcd_induction_divisionquotientstep) = (p)) /\\ (((exists pfa_gap_gcd_induction_divisionquotientstepsubtractright. pfa_gap_gcd_induction_divisionquotientstepsubtractright + S (pfd_difference_gcd_induction_divisionquotientstep) = (p)) /\\ ((((exists pfa_gap_gcd_induction_divisionquotientstepsubtractresultbound. pfa_gap_gcd_induction_divisionquotientstepsubtractresultbound + S (pfd_input_gcd_induction_divisionquotientstep) = (p)) /\\ ((exists pfa_offset_left_gcd_induction_divisionquotientstepsubtractresultcongruence pfa_offset_right_gcd_induction_divisionquotientstepsubtractresultcongruence. ((pfd_previous_gcd_induction_divisionquotientstep) + (pfd_difference_gcd_induction_divisionquotientstep)) + (p) * pfa_offset_left_gcd_induction_divisionquotientstepsubtractresultcongruence = (pfd_input_gcd_induction_divisionquotientstep) + (p) * pfa_offset_right_gcd_induction_divisionquotientstepsubtractresultcongruence))))))))) /\\ ((((exists pfa_gap_gcd_induction_divisionquotientstepmultiplyleft. pfa_gap_gcd_induction_divisionquotientstepmultiplyleft + S (pfd_inverse_gcd_induction_division) = (p)) /\\ (((exists pfa_gap_gcd_induction_divisionquotientstepmultiplyright. pfa_gap_gcd_induction_divisionquotientstepmultiplyright + S (pfd_difference_gcd_induction_divisionquotientstep) = (p)) /\\ ((((exists pfa_gap_gcd_induction_divisionquotientstepmultiplyresultbound. pfa_gap_gcd_induction_divisionquotientstepmultiplyresultbound + S (pfd_value_gcd_induction_divisionquotient) = (p)) /\\ ((exists pfa_offset_left_gcd_induction_divisionquotientstepmultiplyresultcongruence pfa_offset_right_gcd_induction_divisionquotientstepmultiplyresultcongruence. ((pfd_inverse_gcd_induction_division) * (pfd_difference_gcd_induction_divisionquotientstep)) + (p) * pfa_offset_left_gcd_induction_divisionquotientstepmultiplyresultcongruence = (pfd_value_gcd_induction_divisionquotient) + (p) * pfa_offset_right_gcd_induction_divisionquotientstepmultiplyresultcongruence))))))))))))))))))) /\\ (((forall pfc_index_gcd_induction_divisionproduct. (exists pfa_gap_gcd_induction_divisionproductbound. pfa_gap_gcd_induction_divisionproductbound + S (pfc_index_gcd_induction_divisionproduct) = (L)) -> exists pfc_value_gcd_induction_divisionproduct. ((((exists ff_h_pfp_gcd_induction_divisionproductentry. ff_h_pfp_gcd_induction_divisionproductentry + S (pfc_value_gcd_induction_divisionproduct) = S ((S (pfc_index_gcd_induction_divisionproduct)) * pfd_product_scale_gcd_induction_division)) /\\ exists ff_q_pfp_gcd_induction_divisionproductentry. pfd_product_code_gcd_induction_division = ff_q_pfp_gcd_induction_divisionproductentry * S ((S (pfc_index_gcd_induction_divisionproduct)) * pfd_product_scale_gcd_induction_division) + (pfc_value_gcd_induction_divisionproduct))) /\\ ((exists pfc_terms_code_gcd_induction_divisionproductcoefficient pfc_terms_scale_gcd_induction_divisionproductcoefficient pfc_natural_sum_gcd_induction_divisionproductcoefficient. ((forall pfc_index_gcd_induction_divisionproductcoefficientdiagonal. (exists pfa_gap_gcd_induction_divisionproductcoefficientdiagonalbound. pfa_gap_gcd_induction_divisionproductcoefficientdiagonalbound + S (pfc_index_gcd_induction_divisionproductcoefficientdiagonal) = (S (pfc_index_gcd_induction_divisionproduct))) -> exists pfc_value_gcd_induction_divisionproductcoefficientdiagonal. ((((exists ff_h_pfp_gcd_induction_divisionproductcoefficientdiagonalentry. ff_h_pfp_gcd_induction_divisionproductcoefficientdiagonalentry + S (pfc_value_gcd_induction_divisionproductcoefficientdiagonal) = S ((S (pfc_index_gcd_induction_divisionproductcoefficientdiagonal)) * pfc_terms_scale_gcd_induction_divisionproductcoefficient)) /\\ exists ff_q_pfp_gcd_induction_divisionproductcoefficientdiagonalentry. pfc_terms_code_gcd_induction_divisionproductcoefficient = ff_q_pfp_gcd_induction_divisionproductcoefficientdiagonalentry * S ((S (pfc_index_gcd_induction_divisionproductcoefficientdiagonal)) * pfc_terms_scale_gcd_induction_divisionproductcoefficient) + (pfc_value_gcd_induction_divisionproductcoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_induction_divisionproductcoefficientdiagonalterm pfc_left_gcd_induction_divisionproductcoefficientdiagonalterm pfc_right_gcd_induction_divisionproductcoefficientdiagonalterm. (((pfc_index_gcd_induction_divisionproductcoefficientdiagonal)+pfc_complement_gcd_induction_divisionproductcoefficientdiagonalterm=(pfc_index_gcd_induction_divisionproduct)) /\\ ((((((exists pfa_gap_gcd_induction_divisionproductcoefficientdiagonaltermleftinside. pfa_gap_gcd_induction_divisionproductcoefficientdiagonaltermleftinside + S (pfc_index_gcd_induction_divisionproductcoefficientdiagonal) = (q)) /\\ ((((exists ff_h_pfp_gcd_induction_divisionproductcoefficientdiagonaltermleftentry. ff_h_pfp_gcd_induction_divisionproductcoefficientdiagonaltermleftentry + S (pfc_left_gcd_induction_divisionproductcoefficientdiagonalterm) = S ((S (pfc_index_gcd_induction_divisionproductcoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_gcd_induction_divisionproductcoefficientdiagonaltermleftentry. qb = ff_q_pfp_gcd_induction_divisionproductcoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_induction_divisionproductcoefficientdiagonal)) * qc) + (pfc_left_gcd_induction_divisionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_induction_divisionproductcoefficientdiagonaltermleftoutside. pfc_gap_gcd_induction_divisionproductcoefficientdiagonaltermleftoutside+(q)=(pfc_index_gcd_induction_divisionproductcoefficientdiagonal)) /\\ (((pfc_left_gcd_induction_divisionproductcoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_induction_divisionproductcoefficientdiagonaltermrightinside. pfa_gap_gcd_induction_divisionproductcoefficientdiagonaltermrightinside + S (pfc_complement_gcd_induction_divisionproductcoefficientdiagonalterm) = (S (x3))) /\\ ((((exists ff_h_pfp_gcd_induction_divisionproductcoefficientdiagonaltermrightentry. ff_h_pfp_gcd_induction_divisionproductcoefficientdiagonaltermrightentry + S (pfc_right_gcd_induction_divisionproductcoefficientdiagonalterm) = S ((S (pfc_complement_gcd_induction_divisionproductcoefficientdiagonalterm)) * x1)) /\\ exists ff_q_pfp_gcd_induction_divisionproductcoefficientdiagonaltermrightentry. x = ff_q_pfp_gcd_induction_divisionproductcoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_induction_divisionproductcoefficientdiagonalterm)) * x1) + (pfc_right_gcd_induction_divisionproductcoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_induction_divisionproductcoefficientdiagonaltermrightoutside. pfc_gap_gcd_induction_divisionproductcoefficientdiagonaltermrightoutside+(S (x3))=(pfc_complement_gcd_induction_divisionproductcoefficientdiagonalterm)) /\\ (((pfc_right_gcd_induction_divisionproductcoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_induction_divisionproductcoefficientdiagonal)=pfc_left_gcd_induction_divisionproductcoefficientdiagonalterm*pfc_right_gcd_induction_divisionproductcoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_induction_divisionproductcoefficientsum fs_v_pfc_gcd_induction_divisionproductcoefficientsum. ((((exists fs_h_pfc_gcd_induction_divisionproductcoefficientsum_body_start. fs_h_pfc_gcd_induction_divisionproductcoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_induction_divisionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_divisionproductcoefficientsum_body_start. fs_u_pfc_gcd_induction_divisionproductcoefficientsum = fs_q_pfc_gcd_induction_divisionproductcoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_induction_divisionproductcoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_induction_divisionproductcoefficientsum_body_terminal. fs_h_pfc_gcd_induction_divisionproductcoefficientsum_body_terminal + S (pfc_natural_sum_gcd_induction_divisionproductcoefficient) = S ((S (S (pfc_index_gcd_induction_divisionproduct))) * fs_v_pfc_gcd_induction_divisionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_divisionproductcoefficientsum_body_terminal. fs_u_pfc_gcd_induction_divisionproductcoefficientsum = fs_q_pfc_gcd_induction_divisionproductcoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_induction_divisionproduct))) * fs_v_pfc_gcd_induction_divisionproductcoefficientsum) + (pfc_natural_sum_gcd_induction_divisionproductcoefficient))) /\\ forall fs_i_pfc_gcd_induction_divisionproductcoefficientsum_body_steps. (exists fs_lt_pfc_gcd_induction_divisionproductcoefficientsum_body_steps_bound. fs_lt_pfc_gcd_induction_divisionproductcoefficientsum_body_steps_bound + S fs_i_pfc_gcd_induction_divisionproductcoefficientsum_body_steps = S (pfc_index_gcd_induction_divisionproduct)) -> exists fs_a_pfc_gcd_induction_divisionproductcoefficientsum_body_steps fs_r_pfc_gcd_induction_divisionproductcoefficientsum_body_steps fs_s_pfc_gcd_induction_divisionproductcoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_induction_divisionproductcoefficientsum_body_steps_summand. fs_h_pfc_gcd_induction_divisionproductcoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_induction_divisionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_induction_divisionproductcoefficientsum_body_steps)) * pfc_terms_scale_gcd_induction_divisionproductcoefficient)) /\\ exists fs_q_pfc_gcd_induction_divisionproductcoefficientsum_body_steps_summand. pfc_terms_code_gcd_induction_divisionproductcoefficient = fs_q_pfc_gcd_induction_divisionproductcoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_induction_divisionproductcoefficientsum_body_steps)) * pfc_terms_scale_gcd_induction_divisionproductcoefficient) + (fs_a_pfc_gcd_induction_divisionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_induction_divisionproductcoefficientsum_body_steps_partial. fs_h_pfc_gcd_induction_divisionproductcoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_induction_divisionproductcoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_induction_divisionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_divisionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_divisionproductcoefficientsum_body_steps_partial. fs_u_pfc_gcd_induction_divisionproductcoefficientsum = fs_q_pfc_gcd_induction_divisionproductcoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_induction_divisionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_divisionproductcoefficientsum) + (fs_r_pfc_gcd_induction_divisionproductcoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_induction_divisionproductcoefficientsum_body_steps_successor. fs_h_pfc_gcd_induction_divisionproductcoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_induction_divisionproductcoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_induction_divisionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_divisionproductcoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_divisionproductcoefficientsum_body_steps_successor. fs_u_pfc_gcd_induction_divisionproductcoefficientsum = fs_q_pfc_gcd_induction_divisionproductcoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_induction_divisionproductcoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_divisionproductcoefficientsum) + (fs_s_pfc_gcd_induction_divisionproductcoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_induction_divisionproductcoefficientsum_body_steps = fs_r_pfc_gcd_induction_divisionproductcoefficientsum_body_steps + fs_a_pfc_gcd_induction_divisionproductcoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_induction_divisionproductcoefficientresiduebound. pfa_gap_gcd_induction_divisionproductcoefficientresiduebound + S (pfc_value_gcd_induction_divisionproduct) = (p)) /\\ ((exists pfa_offset_left_gcd_induction_divisionproductcoefficientresiduecongruence pfa_offset_right_gcd_induction_divisionproductcoefficientresiduecongruence. (pfc_natural_sum_gcd_induction_divisionproductcoefficient) + (p) * pfa_offset_left_gcd_induction_divisionproductcoefficientresiduecongruence = (pfc_value_gcd_induction_divisionproduct) + (p) * pfa_offset_right_gcd_induction_divisionproductcoefficientresiduecongruence)))))))))))) /\\ (((forall pfs_index_gcd_induction_divisiondifference. (exists pfa_gap_gcd_induction_divisiondifferenceindex. pfa_gap_gcd_induction_divisiondifferenceindex + S (pfs_index_gcd_induction_divisiondifference) = (L)) -> exists pfs_left_gcd_induction_divisiondifference pfs_right_gcd_induction_divisiondifference pfs_result_gcd_induction_divisiondifference. ((((exists ff_h_pfp_gcd_induction_divisiondifferenceleft. ff_h_pfp_gcd_induction_divisiondifferenceleft + S (pfs_left_gcd_induction_divisiondifference) = S ((S (pfs_index_gcd_induction_divisiondifference)) * ac)) /\\ exists ff_q_pfp_gcd_induction_divisiondifferenceleft. ab = ff_q_pfp_gcd_induction_divisiondifferenceleft * S ((S (pfs_index_gcd_induction_divisiondifference)) * ac) + (pfs_left_gcd_induction_divisiondifference))) /\\ (((((exists ff_h_pfp_gcd_induction_divisiondifferenceright. ff_h_pfp_gcd_induction_divisiondifferenceright + S (pfs_right_gcd_induction_divisiondifference) = S ((S (pfs_index_gcd_induction_divisiondifference)) * pfd_product_scale_gcd_induction_division)) /\\ exists ff_q_pfp_gcd_induction_divisiondifferenceright. pfd_product_code_gcd_induction_division = ff_q_pfp_gcd_induction_divisiondifferenceright * S ((S (pfs_index_gcd_induction_divisiondifference)) * pfd_product_scale_gcd_induction_division) + (pfs_right_gcd_induction_divisiondifference))) /\\ (((((exists ff_h_pfp_gcd_induction_divisiondifferenceresult. ff_h_pfp_gcd_induction_divisiondifferenceresult + S (pfs_result_gcd_induction_divisiondifference) = S ((S (pfs_index_gcd_induction_divisiondifference)) * pfd_residual_scale_gcd_induction_division)) /\\ exists ff_q_pfp_gcd_induction_divisiondifferenceresult. pfd_residual_code_gcd_induction_division = ff_q_pfp_gcd_induction_divisiondifferenceresult * S ((S (pfs_index_gcd_induction_divisiondifference)) * pfd_residual_scale_gcd_induction_division) + (pfs_result_gcd_induction_divisiondifference))) /\\ ((((exists pfa_gap_gcd_induction_divisiondifferenceoperationleft. pfa_gap_gcd_induction_divisiondifferenceoperationleft + S (pfs_right_gcd_induction_divisiondifference) = (p)) /\\ (((exists pfa_gap_gcd_induction_divisiondifferenceoperationright. pfa_gap_gcd_induction_divisiondifferenceoperationright + S (pfs_result_gcd_induction_divisiondifference) = (p)) /\\ ((((exists pfa_gap_gcd_induction_divisiondifferenceoperationresultbound. pfa_gap_gcd_induction_divisiondifferenceoperationresultbound + S (pfs_left_gcd_induction_divisiondifference) = (p)) /\\ ((exists pfa_offset_left_gcd_induction_divisiondifferenceoperationresultcongruence pfa_offset_right_gcd_induction_divisiondifferenceoperationresultcongruence. ((pfs_right_gcd_induction_divisiondifference) + (pfs_result_gcd_induction_divisiondifference)) + (p) * pfa_offset_left_gcd_induction_divisiondifferenceoperationresultcongruence = (pfs_left_gcd_induction_divisiondifference) + (p) * pfa_offset_right_gcd_induction_divisiondifferenceoperationresultcongruence)))))))))))))))) /\\ (((((L)=(pfd_cut_gcd_induction_division)+(R)) /\\ (((forall fom_index_pfp_gcd_induction_divisiontriminput. (exists fom_gap_pfp_gcd_induction_divisiontriminput_index_bound. fom_gap_pfp_gcd_induction_divisiontriminput_index_bound + S (fom_index_pfp_gcd_induction_divisiontriminput) = L) -> exists fom_value_pfp_gcd_induction_divisiontriminput. ((((exists fom_beta_height_pfp_gcd_induction_divisiontriminput_entry. fom_beta_height_pfp_gcd_induction_divisiontriminput_entry + S (fom_value_pfp_gcd_induction_divisiontriminput) = S ((S (fom_index_pfp_gcd_induction_divisiontriminput)) * pfd_residual_scale_gcd_induction_division)) /\\ exists fom_beta_quotient_pfp_gcd_induction_divisiontriminput_entry. pfd_residual_code_gcd_induction_division = fom_beta_quotient_pfp_gcd_induction_divisiontriminput_entry * S ((S (fom_index_pfp_gcd_induction_divisiontriminput)) * pfd_residual_scale_gcd_induction_division) + (fom_value_pfp_gcd_induction_divisiontriminput))) /\\ (exists fom_gap_pfp_gcd_induction_divisiontriminput_value_bound. fom_gap_pfp_gcd_induction_divisiontriminput_value_bound + S (fom_value_pfp_gcd_induction_divisiontriminput) = p))) /\\ (((forall pfp_repeat_index_gcd_induction_divisiontrimremoved. (exists pfa_gap_gcd_induction_divisiontrimremovedindex. pfa_gap_gcd_induction_divisiontrimremovedindex + S (pfp_repeat_index_gcd_induction_divisiontrimremoved) = (pfd_cut_gcd_induction_division)) -> (((exists ff_h_pfp_gcd_induction_divisiontrimremovedentry. ff_h_pfp_gcd_induction_divisiontrimremovedentry + S (0) = S ((S (pfp_repeat_index_gcd_induction_divisiontrimremoved)) * pfd_residual_scale_gcd_induction_division)) /\\ exists ff_q_pfp_gcd_induction_divisiontrimremovedentry. pfd_residual_code_gcd_induction_division = ff_q_pfp_gcd_induction_divisiontrimremovedentry * S ((S (pfp_repeat_index_gcd_induction_divisiontrimremoved)) * pfd_residual_scale_gcd_induction_division) + (0)))) /\\ (((forall pftrim_index_gcd_induction_divisiontrimsuffix pftrim_value_gcd_induction_divisiontrimsuffix. (exists pfa_gap_gcd_induction_divisiontrimsuffixbound. pfa_gap_gcd_induction_divisiontrimsuffixbound + S (pftrim_index_gcd_induction_divisiontrimsuffix) = (R)) -> (((exists ff_h_pfp_gcd_induction_divisiontrimsuffixsource. ff_h_pfp_gcd_induction_divisiontrimsuffixsource + S (pftrim_value_gcd_induction_divisiontrimsuffix) = S ((S ((pfd_cut_gcd_induction_division)+pftrim_index_gcd_induction_divisiontrimsuffix)) * pfd_residual_scale_gcd_induction_division)) /\\ exists ff_q_pfp_gcd_induction_divisiontrimsuffixsource. pfd_residual_code_gcd_induction_division = ff_q_pfp_gcd_induction_divisiontrimsuffixsource * S ((S ((pfd_cut_gcd_induction_division)+pftrim_index_gcd_induction_divisiontrimsuffix)) * pfd_residual_scale_gcd_induction_division) + (pftrim_value_gcd_induction_divisiontrimsuffix))) -> (((exists ff_h_pfp_gcd_induction_divisiontrimsuffixoutput. ff_h_pfp_gcd_induction_divisiontrimsuffixoutput + S (pftrim_value_gcd_induction_divisiontrimsuffix) = S ((S (pftrim_index_gcd_induction_divisiontrimsuffix)) * rc)) /\\ exists ff_q_pfp_gcd_induction_divisiontrimsuffixoutput. rb = ff_q_pfp_gcd_induction_divisiontrimsuffixoutput * S ((S (pftrim_index_gcd_induction_divisiontrimsuffix)) * rc) + (pftrim_value_gcd_induction_divisiontrimsuffix)))) /\\ (((R)=0 \\/ (exists pftrim_leading_gcd_induction_divisiontrimnormal. ((((exists ff_h_pfp_gcd_induction_divisiontrimnormalentry. ff_h_pfp_gcd_induction_divisiontrimnormalentry + S (pftrim_leading_gcd_induction_divisiontrimnormal) = S ((S (0)) * rc)) /\\ exists ff_q_pfp_gcd_induction_divisiontrimnormalentry. rb = ff_q_pfp_gcd_induction_divisiontrimnormalentry * S ((S (0)) * rc) + (pftrim_leading_gcd_induction_divisiontrimnormal))) /\\ ((~(pftrim_leading_gcd_induction_divisiontrimnormal=0))))))))))))))))))))))))))))))))", "specialize prime_field_polynomial_division_execution_exists (p)", "specialize prime_field_polynomial_division_execution_exists (ab)", "specialize prime_field_polynomial_division_execution_exists (ac)", "specialize prime_field_polynomial_division_execution_exists (L)", "specialize prime_field_polynomial_division_execution_exists (x)", "specialize prime_field_polynomial_division_execution_exists (x1)", "specialize prime_field_polynomial_division_execution_exists (x3)", "apply prime_field_polynomial_division_execution_exists", "exact hp", "exact hA", "rewrite hlength at ht_witness_witness_witness_right_right_right_right_witness", "rewrite hlength at ht_witness_witness_witness_right_right_right_right_witness", "exact ht_witness_witness_witness_right_right_right_right_witness", "cases he", "cases he_witness", "cases he_witness_witness", "cases he_witness_witness_witness", "cases he_witness_witness_witness_witness", "cases he_witness_witness_witness_witness_witness", "specialize prime_field_polynomial_gcd_bezout_division_backward (p)", "specialize prime_field_polynomial_gcd_bezout_division_backward (ab)", "specialize prime_field_polynomial_gcd_bezout_division_backward (ac)", "specialize prime_field_polynomial_gcd_bezout_division_backward (L)", "specialize prime_field_polynomial_gcd_bezout_division_backward (x)", "specialize prime_field_polynomial_gcd_bezout_division_backward (x1)", "specialize prime_field_polynomial_gcd_bezout_division_backward (x3)", "specialize prime_field_polynomial_gcd_bezout_division_backward (x4)", "specialize prime_field_polynomial_gcd_bezout_division_backward (x5)", "specialize prime_field_polynomial_gcd_bezout_division_backward (x6)", "specialize prime_field_polynomial_gcd_bezout_division_backward (x7)", "specialize prime_field_polynomial_gcd_bezout_division_backward (x8)", "specialize prime_field_polynomial_gcd_bezout_division_backward (x9)", "apply prime_field_polynomial_gcd_bezout_division_backward", "exact hp", "exact he_witness_witness_witness_witness_witness_witness", "specialize IH (p)", "specialize IH (x)", "specialize IH (x1)", "specialize IH (S x3)", "specialize IH (x7)", "specialize IH (x8)", "specialize IH (x9)", "apply IH", "exact hp", "rewrite hlength at ht_witness_witness_witness_left", "exact ht_witness_witness_witness_left", "specialize prime_field_polynomial_division_remainder_bounded (p)", "specialize prime_field_polynomial_division_remainder_bounded (ab)", "specialize prime_field_polynomial_division_remainder_bounded (ac)", "specialize prime_field_polynomial_division_remainder_bounded (L)", "specialize prime_field_polynomial_division_remainder_bounded (x)", "specialize prime_field_polynomial_division_remainder_bounded (x1)", "specialize prime_field_polynomial_division_remainder_bounded (x3)", "specialize prime_field_polynomial_division_remainder_bounded (x4)", "specialize prime_field_polynomial_division_remainder_bounded (x5)", "specialize prime_field_polynomial_division_remainder_bounded (x6)", "specialize prime_field_polynomial_division_remainder_bounded (x7)", "specialize prime_field_polynomial_division_remainder_bounded (x8)", "specialize prime_field_polynomial_division_remainder_bounded (x9)", "apply prime_field_polynomial_division_remainder_bounded", "exact he_witness_witness_witness_witness_witness_witness", "specialize le_trans (x9)", "specialize le_trans (x3)", "specialize le_trans (n)", "apply le_trans", "have hr : ((exists pfc_gap_gcd_induction_remainder_bound. pfc_gap_gcd_induction_remainder_bound+(x9)=(x3)) /\\ ((exists pfa_gap_gcd_induction_remainder_strict. pfa_gap_gcd_induction_remainder_strict + S (x9) = (S x3))))", "specialize prime_field_polynomial_division_remainder_length_descent (p)", "specialize prime_field_polynomial_division_remainder_length_descent (ab)", "specialize prime_field_polynomial_division_remainder_length_descent (ac)", "specialize prime_field_polynomial_division_remainder_length_descent (L)", "specialize prime_field_polynomial_division_remainder_length_descent (x)", "specialize prime_field_polynomial_division_remainder_length_descent (x1)", "specialize prime_field_polynomial_division_remainder_length_descent (x3)", "specialize prime_field_polynomial_division_remainder_length_descent (x4)", "specialize prime_field_polynomial_division_remainder_length_descent (x5)", "specialize prime_field_polynomial_division_remainder_length_descent (x6)", "specialize prime_field_polynomial_division_remainder_length_descent (x7)", "specialize prime_field_polynomial_division_remainder_length_descent (x8)", "specialize prime_field_polynomial_division_remainder_length_descent (x9)", "apply prime_field_polynomial_division_remainder_length_descent", "exact hp", "exact he_witness_witness_witness_witness_witness_witness", "cases hr", "exact hr_left", "exact hdBound" ], "script_sha256": "0853305af0b5b660c10e82f46f31542eeb126141b9116b2bb8f7bb49753f6226", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_existence_candidate.py", "sha256": "81f2f48dd2e81894c7a267453646eb6f2b6f9bd3ee320386d8c561f6b9f8b8ca" }, "statement": "forall n. forall p ab ac L bb bc M. (~((p) = 1) /\\ forall pfa_factor_left_gcd_induction_prime pfa_factor_right_gcd_induction_prime. (p) = pfa_factor_left_gcd_induction_prime * pfa_factor_right_gcd_induction_prime -> pfa_factor_left_gcd_induction_prime = 1 \\/ pfa_factor_right_gcd_induction_prime = 1) -> (forall fom_index_pfp_gcd_induction_A. (exists fom_gap_pfp_gcd_induction_A_index_bound. fom_gap_pfp_gcd_induction_A_index_bound + S (fom_index_pfp_gcd_induction_A) = L) -> exists fom_value_pfp_gcd_induction_A. ((((exists fom_beta_height_pfp_gcd_induction_A_entry. fom_beta_height_pfp_gcd_induction_A_entry + S (fom_value_pfp_gcd_induction_A) = S ((S (fom_index_pfp_gcd_induction_A)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_induction_A_entry. ab = fom_beta_quotient_pfp_gcd_induction_A_entry * S ((S (fom_index_pfp_gcd_induction_A)) * ac) + (fom_value_pfp_gcd_induction_A))) /\\ (exists fom_gap_pfp_gcd_induction_A_value_bound. fom_gap_pfp_gcd_induction_A_value_bound + S (fom_value_pfp_gcd_induction_A) = p))) -> (forall fom_index_pfp_gcd_induction_B. (exists fom_gap_pfp_gcd_induction_B_index_bound. fom_gap_pfp_gcd_induction_B_index_bound + S (fom_index_pfp_gcd_induction_B) = M) -> exists fom_value_pfp_gcd_induction_B. ((((exists fom_beta_height_pfp_gcd_induction_B_entry. fom_beta_height_pfp_gcd_induction_B_entry + S (fom_value_pfp_gcd_induction_B) = S ((S (fom_index_pfp_gcd_induction_B)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_induction_B_entry. bb = fom_beta_quotient_pfp_gcd_induction_B_entry * S ((S (fom_index_pfp_gcd_induction_B)) * bc) + (fom_value_pfp_gcd_induction_B))) /\\ (exists fom_gap_pfp_gcd_induction_B_value_bound. fom_gap_pfp_gcd_induction_B_value_bound + S (fom_value_pfp_gcd_induction_B) = p))) -> (exists pfc_gap_gcd_induction_bound. pfc_gap_gcd_induction_bound+(M)=(n)) -> (exists pfgs_gb_gcd_induction_solution pfgs_gc_gcd_induction_solution pfgs_G_gcd_induction_solution pfgs_ub_gcd_induction_solution pfgs_uc_gcd_induction_solution pfgs_U_gcd_induction_solution pfgs_vb_gcd_induction_solution pfgs_vc_gcd_induction_solution pfgs_V_gcd_induction_solution. (((pfgs_G_gcd_induction_solution)=0 \\/ (((~((pfgs_G_gcd_induction_solution) = 0)) /\\ (((forall fom_index_pfp_gcd_induction_solution_witness_normal_moniccoefficients. (exists fom_gap_pfp_gcd_induction_solution_witness_normal_moniccoefficients_index_bound. fom_gap_pfp_gcd_induction_solution_witness_normal_moniccoefficients_index_bound + S (fom_index_pfp_gcd_induction_solution_witness_normal_moniccoefficients) = pfgs_G_gcd_induction_solution) -> exists fom_value_pfp_gcd_induction_solution_witness_normal_moniccoefficients. ((((exists fom_beta_height_pfp_gcd_induction_solution_witness_normal_moniccoefficients_entry. fom_beta_height_pfp_gcd_induction_solution_witness_normal_moniccoefficients_entry + S (fom_value_pfp_gcd_induction_solution_witness_normal_moniccoefficients) = S ((S (fom_index_pfp_gcd_induction_solution_witness_normal_moniccoefficients)) * pfgs_gc_gcd_induction_solution)) /\\ exists fom_beta_quotient_pfp_gcd_induction_solution_witness_normal_moniccoefficients_entry. pfgs_gb_gcd_induction_solution = fom_beta_quotient_pfp_gcd_induction_solution_witness_normal_moniccoefficients_entry * S ((S (fom_index_pfp_gcd_induction_solution_witness_normal_moniccoefficients)) * pfgs_gc_gcd_induction_solution) + (fom_value_pfp_gcd_induction_solution_witness_normal_moniccoefficients))) /\\ (exists fom_gap_pfp_gcd_induction_solution_witness_normal_moniccoefficients_value_bound. fom_gap_pfp_gcd_induction_solution_witness_normal_moniccoefficients_value_bound + S (fom_value_pfp_gcd_induction_solution_witness_normal_moniccoefficients) = p))) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_normal_monicleading. ff_h_pfp_gcd_induction_solution_witness_normal_monicleading + S (1) = S ((S (0)) * pfgs_gc_gcd_induction_solution)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_normal_monicleading. pfgs_gb_gcd_induction_solution = ff_q_pfp_gcd_induction_solution_witness_normal_monicleading * S ((S (0)) * pfgs_gc_gcd_induction_solution) + (1))))))))) /\\ (((((((forall fom_index_pfp_gcd_induction_solution_witness_common_left_bounded. (exists fom_gap_pfp_gcd_induction_solution_witness_common_left_bounded_index_bound. fom_gap_pfp_gcd_induction_solution_witness_common_left_bounded_index_bound + S (fom_index_pfp_gcd_induction_solution_witness_common_left_bounded) = L) -> exists fom_value_pfp_gcd_induction_solution_witness_common_left_bounded. ((((exists fom_beta_height_pfp_gcd_induction_solution_witness_common_left_bounded_entry. fom_beta_height_pfp_gcd_induction_solution_witness_common_left_bounded_entry + S (fom_value_pfp_gcd_induction_solution_witness_common_left_bounded) = S ((S (fom_index_pfp_gcd_induction_solution_witness_common_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_induction_solution_witness_common_left_bounded_entry. ab = fom_beta_quotient_pfp_gcd_induction_solution_witness_common_left_bounded_entry * S ((S (fom_index_pfp_gcd_induction_solution_witness_common_left_bounded)) * ac) + (fom_value_pfp_gcd_induction_solution_witness_common_left_bounded))) /\\ (exists fom_gap_pfp_gcd_induction_solution_witness_common_left_bounded_value_bound. fom_gap_pfp_gcd_induction_solution_witness_common_left_bounded_value_bound + S (fom_value_pfp_gcd_induction_solution_witness_common_left_bounded) = p))) /\\ ((exists pfgd_qb_gcd_induction_solution_witness_common_left pfgd_qc_gcd_induction_solution_witness_common_left pfgd_Q_gcd_induction_solution_witness_common_left pfgd_pb_gcd_induction_solution_witness_common_left pfgd_pc_gcd_induction_solution_witness_common_left pfgd_P_gcd_induction_solution_witness_common_left. ((((forall fom_index_pfp_gcd_induction_solution_witness_common_left_productleft. (exists fom_gap_pfp_gcd_induction_solution_witness_common_left_productleft_index_bound. fom_gap_pfp_gcd_induction_solution_witness_common_left_productleft_index_bound + S (fom_index_pfp_gcd_induction_solution_witness_common_left_productleft) = pfgd_Q_gcd_induction_solution_witness_common_left) -> exists fom_value_pfp_gcd_induction_solution_witness_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_induction_solution_witness_common_left_productleft_entry. fom_beta_height_pfp_gcd_induction_solution_witness_common_left_productleft_entry + S (fom_value_pfp_gcd_induction_solution_witness_common_left_productleft) = S ((S (fom_index_pfp_gcd_induction_solution_witness_common_left_productleft)) * pfgd_qc_gcd_induction_solution_witness_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_induction_solution_witness_common_left_productleft_entry. pfgd_qb_gcd_induction_solution_witness_common_left = fom_beta_quotient_pfp_gcd_induction_solution_witness_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_induction_solution_witness_common_left_productleft)) * pfgd_qc_gcd_induction_solution_witness_common_left) + (fom_value_pfp_gcd_induction_solution_witness_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_induction_solution_witness_common_left_productleft_value_bound. fom_gap_pfp_gcd_induction_solution_witness_common_left_productleft_value_bound + S (fom_value_pfp_gcd_induction_solution_witness_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_induction_solution_witness_common_left_productright. (exists fom_gap_pfp_gcd_induction_solution_witness_common_left_productright_index_bound. fom_gap_pfp_gcd_induction_solution_witness_common_left_productright_index_bound + S (fom_index_pfp_gcd_induction_solution_witness_common_left_productright) = pfgs_G_gcd_induction_solution) -> exists fom_value_pfp_gcd_induction_solution_witness_common_left_productright. ((((exists fom_beta_height_pfp_gcd_induction_solution_witness_common_left_productright_entry. fom_beta_height_pfp_gcd_induction_solution_witness_common_left_productright_entry + S (fom_value_pfp_gcd_induction_solution_witness_common_left_productright) = S ((S (fom_index_pfp_gcd_induction_solution_witness_common_left_productright)) * pfgs_gc_gcd_induction_solution)) /\\ exists fom_beta_quotient_pfp_gcd_induction_solution_witness_common_left_productright_entry. pfgs_gb_gcd_induction_solution = fom_beta_quotient_pfp_gcd_induction_solution_witness_common_left_productright_entry * S ((S (fom_index_pfp_gcd_induction_solution_witness_common_left_productright)) * pfgs_gc_gcd_induction_solution) + (fom_value_pfp_gcd_induction_solution_witness_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_induction_solution_witness_common_left_productright_value_bound. fom_gap_pfp_gcd_induction_solution_witness_common_left_productright_value_bound + S (fom_value_pfp_gcd_induction_solution_witness_common_left_productright) = p))) /\\ (((((((pfgd_Q_gcd_induction_solution_witness_common_left)=0 \\/ (pfgs_G_gcd_induction_solution)=0) /\\ (((pfgd_P_gcd_induction_solution_witness_common_left)=0)))) \\/ (((~((pfgd_Q_gcd_induction_solution_witness_common_left)=0)) /\\ (((~((pfgs_G_gcd_induction_solution)=0)) /\\ (((pfgd_Q_gcd_induction_solution_witness_common_left)+(pfgs_G_gcd_induction_solution)=S (pfgd_P_gcd_induction_solution_witness_common_left)))))))) /\\ ((forall pfc_index_gcd_induction_solution_witness_common_left_productcoefficients. (exists pfa_gap_gcd_induction_solution_witness_common_left_productcoefficientsbound. pfa_gap_gcd_induction_solution_witness_common_left_productcoefficientsbound + S (pfc_index_gcd_induction_solution_witness_common_left_productcoefficients) = (pfgd_P_gcd_induction_solution_witness_common_left)) -> exists pfc_value_gcd_induction_solution_witness_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_induction_solution_witness_common_left_productcoefficientsentry. ff_h_pfp_gcd_induction_solution_witness_common_left_productcoefficientsentry + S (pfc_value_gcd_induction_solution_witness_common_left_productcoefficients) = S ((S (pfc_index_gcd_induction_solution_witness_common_left_productcoefficients)) * pfgd_pc_gcd_induction_solution_witness_common_left)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_common_left_productcoefficientsentry. pfgd_pb_gcd_induction_solution_witness_common_left = ff_q_pfp_gcd_induction_solution_witness_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_induction_solution_witness_common_left_productcoefficients)) * pfgd_pc_gcd_induction_solution_witness_common_left) + (pfc_value_gcd_induction_solution_witness_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_induction_solution_witness_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_induction_solution_witness_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_induction_solution_witness_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_induction_solution_witness_common_left_productcoefficients))) -> exists pfc_value_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_induction_solution_witness_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_induction_solution_witness_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_induction_solution_witness_common_left_productcoefficientscoefficient) + (pfc_value_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_induction_solution_witness_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_induction_solution_witness_common_left)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_induction_solution_witness_common_left)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_induction_solution_witness_common_left = ff_q_pfp_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_induction_solution_witness_common_left) + (pfc_left_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_induction_solution_witness_common_left)=(pfc_index_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm) = (pfgs_G_gcd_induction_solution)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_induction_solution)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_gcd_induction_solution = ff_q_pfp_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_induction_solution) + (pfc_right_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_gcd_induction_solution)=(pfc_complement_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_induction_solution_witness_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_induction_solution_witness_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_induction_solution_witness_common_left_productcoefficients))) * fs_v_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_induction_solution_witness_common_left_productcoefficients))) * fs_v_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_induction_solution_witness_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_induction_solution_witness_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_induction_solution_witness_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_induction_solution_witness_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_induction_solution_witness_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_induction_solution_witness_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_induction_solution_witness_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_induction_solution_witness_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_induction_solution_witness_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_induction_solution_witness_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_induction_solution_witness_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_induction_solution_witness_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_induction_solution_witness_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_induction_solution_witness_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_induction_solution_witness_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_induction_solution_witness_common_left_equivalent pfrep_left_gcd_induction_solution_witness_common_left_equivalent pfrep_right_gcd_induction_solution_witness_common_left_equivalent. ((exists pfrep_position_gcd_induction_solution_witness_common_left_equivalentfirst. ((pfrep_position_gcd_induction_solution_witness_common_left_equivalentfirst+S (pfrep_power_gcd_induction_solution_witness_common_left_equivalent)=(pfgd_P_gcd_induction_solution_witness_common_left)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_common_left_equivalentfirstentry. ff_h_pfp_gcd_induction_solution_witness_common_left_equivalentfirstentry + S (pfrep_left_gcd_induction_solution_witness_common_left_equivalent) = S ((S (pfrep_position_gcd_induction_solution_witness_common_left_equivalentfirst)) * pfgd_pc_gcd_induction_solution_witness_common_left)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_common_left_equivalentfirstentry. pfgd_pb_gcd_induction_solution_witness_common_left = ff_q_pfp_gcd_induction_solution_witness_common_left_equivalentfirstentry * S ((S (pfrep_position_gcd_induction_solution_witness_common_left_equivalentfirst)) * pfgd_pc_gcd_induction_solution_witness_common_left) + (pfrep_left_gcd_induction_solution_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_induction_solution_witness_common_left_equivalentfirstoutside. pfrep_gap_gcd_induction_solution_witness_common_left_equivalentfirstoutside+(pfgd_P_gcd_induction_solution_witness_common_left)=(pfrep_power_gcd_induction_solution_witness_common_left_equivalent)) /\\ (((pfrep_left_gcd_induction_solution_witness_common_left_equivalent)=0))))) -> ((exists pfrep_position_gcd_induction_solution_witness_common_left_equivalentsecond. ((pfrep_position_gcd_induction_solution_witness_common_left_equivalentsecond+S (pfrep_power_gcd_induction_solution_witness_common_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_common_left_equivalentsecondentry. ff_h_pfp_gcd_induction_solution_witness_common_left_equivalentsecondentry + S (pfrep_right_gcd_induction_solution_witness_common_left_equivalent) = S ((S (pfrep_position_gcd_induction_solution_witness_common_left_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_common_left_equivalentsecondentry. ab = ff_q_pfp_gcd_induction_solution_witness_common_left_equivalentsecondentry * S ((S (pfrep_position_gcd_induction_solution_witness_common_left_equivalentsecond)) * ac) + (pfrep_right_gcd_induction_solution_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_induction_solution_witness_common_left_equivalentsecondoutside. pfrep_gap_gcd_induction_solution_witness_common_left_equivalentsecondoutside+(L)=(pfrep_power_gcd_induction_solution_witness_common_left_equivalent)) /\\ (((pfrep_right_gcd_induction_solution_witness_common_left_equivalent)=0))))) -> pfrep_left_gcd_induction_solution_witness_common_left_equivalent=pfrep_right_gcd_induction_solution_witness_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_induction_solution_witness_common_right_bounded. (exists fom_gap_pfp_gcd_induction_solution_witness_common_right_bounded_index_bound. fom_gap_pfp_gcd_induction_solution_witness_common_right_bounded_index_bound + S (fom_index_pfp_gcd_induction_solution_witness_common_right_bounded) = M) -> exists fom_value_pfp_gcd_induction_solution_witness_common_right_bounded. ((((exists fom_beta_height_pfp_gcd_induction_solution_witness_common_right_bounded_entry. fom_beta_height_pfp_gcd_induction_solution_witness_common_right_bounded_entry + S (fom_value_pfp_gcd_induction_solution_witness_common_right_bounded) = S ((S (fom_index_pfp_gcd_induction_solution_witness_common_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_induction_solution_witness_common_right_bounded_entry. bb = fom_beta_quotient_pfp_gcd_induction_solution_witness_common_right_bounded_entry * S ((S (fom_index_pfp_gcd_induction_solution_witness_common_right_bounded)) * bc) + (fom_value_pfp_gcd_induction_solution_witness_common_right_bounded))) /\\ (exists fom_gap_pfp_gcd_induction_solution_witness_common_right_bounded_value_bound. fom_gap_pfp_gcd_induction_solution_witness_common_right_bounded_value_bound + S (fom_value_pfp_gcd_induction_solution_witness_common_right_bounded) = p))) /\\ ((exists pfgd_qb_gcd_induction_solution_witness_common_right pfgd_qc_gcd_induction_solution_witness_common_right pfgd_Q_gcd_induction_solution_witness_common_right pfgd_pb_gcd_induction_solution_witness_common_right pfgd_pc_gcd_induction_solution_witness_common_right pfgd_P_gcd_induction_solution_witness_common_right. ((((forall fom_index_pfp_gcd_induction_solution_witness_common_right_productleft. (exists fom_gap_pfp_gcd_induction_solution_witness_common_right_productleft_index_bound. fom_gap_pfp_gcd_induction_solution_witness_common_right_productleft_index_bound + S (fom_index_pfp_gcd_induction_solution_witness_common_right_productleft) = pfgd_Q_gcd_induction_solution_witness_common_right) -> exists fom_value_pfp_gcd_induction_solution_witness_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_induction_solution_witness_common_right_productleft_entry. fom_beta_height_pfp_gcd_induction_solution_witness_common_right_productleft_entry + S (fom_value_pfp_gcd_induction_solution_witness_common_right_productleft) = S ((S (fom_index_pfp_gcd_induction_solution_witness_common_right_productleft)) * pfgd_qc_gcd_induction_solution_witness_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_induction_solution_witness_common_right_productleft_entry. pfgd_qb_gcd_induction_solution_witness_common_right = fom_beta_quotient_pfp_gcd_induction_solution_witness_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_induction_solution_witness_common_right_productleft)) * pfgd_qc_gcd_induction_solution_witness_common_right) + (fom_value_pfp_gcd_induction_solution_witness_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_induction_solution_witness_common_right_productleft_value_bound. fom_gap_pfp_gcd_induction_solution_witness_common_right_productleft_value_bound + S (fom_value_pfp_gcd_induction_solution_witness_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_induction_solution_witness_common_right_productright. (exists fom_gap_pfp_gcd_induction_solution_witness_common_right_productright_index_bound. fom_gap_pfp_gcd_induction_solution_witness_common_right_productright_index_bound + S (fom_index_pfp_gcd_induction_solution_witness_common_right_productright) = pfgs_G_gcd_induction_solution) -> exists fom_value_pfp_gcd_induction_solution_witness_common_right_productright. ((((exists fom_beta_height_pfp_gcd_induction_solution_witness_common_right_productright_entry. fom_beta_height_pfp_gcd_induction_solution_witness_common_right_productright_entry + S (fom_value_pfp_gcd_induction_solution_witness_common_right_productright) = S ((S (fom_index_pfp_gcd_induction_solution_witness_common_right_productright)) * pfgs_gc_gcd_induction_solution)) /\\ exists fom_beta_quotient_pfp_gcd_induction_solution_witness_common_right_productright_entry. pfgs_gb_gcd_induction_solution = fom_beta_quotient_pfp_gcd_induction_solution_witness_common_right_productright_entry * S ((S (fom_index_pfp_gcd_induction_solution_witness_common_right_productright)) * pfgs_gc_gcd_induction_solution) + (fom_value_pfp_gcd_induction_solution_witness_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_induction_solution_witness_common_right_productright_value_bound. fom_gap_pfp_gcd_induction_solution_witness_common_right_productright_value_bound + S (fom_value_pfp_gcd_induction_solution_witness_common_right_productright) = p))) /\\ (((((((pfgd_Q_gcd_induction_solution_witness_common_right)=0 \\/ (pfgs_G_gcd_induction_solution)=0) /\\ (((pfgd_P_gcd_induction_solution_witness_common_right)=0)))) \\/ (((~((pfgd_Q_gcd_induction_solution_witness_common_right)=0)) /\\ (((~((pfgs_G_gcd_induction_solution)=0)) /\\ (((pfgd_Q_gcd_induction_solution_witness_common_right)+(pfgs_G_gcd_induction_solution)=S (pfgd_P_gcd_induction_solution_witness_common_right)))))))) /\\ ((forall pfc_index_gcd_induction_solution_witness_common_right_productcoefficients. (exists pfa_gap_gcd_induction_solution_witness_common_right_productcoefficientsbound. pfa_gap_gcd_induction_solution_witness_common_right_productcoefficientsbound + S (pfc_index_gcd_induction_solution_witness_common_right_productcoefficients) = (pfgd_P_gcd_induction_solution_witness_common_right)) -> exists pfc_value_gcd_induction_solution_witness_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_induction_solution_witness_common_right_productcoefficientsentry. ff_h_pfp_gcd_induction_solution_witness_common_right_productcoefficientsentry + S (pfc_value_gcd_induction_solution_witness_common_right_productcoefficients) = S ((S (pfc_index_gcd_induction_solution_witness_common_right_productcoefficients)) * pfgd_pc_gcd_induction_solution_witness_common_right)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_common_right_productcoefficientsentry. pfgd_pb_gcd_induction_solution_witness_common_right = ff_q_pfp_gcd_induction_solution_witness_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_induction_solution_witness_common_right_productcoefficients)) * pfgd_pc_gcd_induction_solution_witness_common_right) + (pfc_value_gcd_induction_solution_witness_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_induction_solution_witness_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_induction_solution_witness_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_induction_solution_witness_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_induction_solution_witness_common_right_productcoefficients))) -> exists pfc_value_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_induction_solution_witness_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_induction_solution_witness_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_induction_solution_witness_common_right_productcoefficientscoefficient) + (pfc_value_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_induction_solution_witness_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_induction_solution_witness_common_right)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_induction_solution_witness_common_right)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_induction_solution_witness_common_right = ff_q_pfp_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_induction_solution_witness_common_right) + (pfc_left_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_induction_solution_witness_common_right)=(pfc_index_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm) = (pfgs_G_gcd_induction_solution)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_induction_solution)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_gcd_induction_solution = ff_q_pfp_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_induction_solution) + (pfc_right_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_gcd_induction_solution)=(pfc_complement_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_induction_solution_witness_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_induction_solution_witness_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_induction_solution_witness_common_right_productcoefficients))) * fs_v_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_induction_solution_witness_common_right_productcoefficients))) * fs_v_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_induction_solution_witness_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_induction_solution_witness_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_induction_solution_witness_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_induction_solution_witness_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_induction_solution_witness_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_induction_solution_witness_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_induction_solution_witness_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_induction_solution_witness_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_induction_solution_witness_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_induction_solution_witness_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_induction_solution_witness_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_induction_solution_witness_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_induction_solution_witness_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_induction_solution_witness_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_induction_solution_witness_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_induction_solution_witness_common_right_equivalent pfrep_left_gcd_induction_solution_witness_common_right_equivalent pfrep_right_gcd_induction_solution_witness_common_right_equivalent. ((exists pfrep_position_gcd_induction_solution_witness_common_right_equivalentfirst. ((pfrep_position_gcd_induction_solution_witness_common_right_equivalentfirst+S (pfrep_power_gcd_induction_solution_witness_common_right_equivalent)=(pfgd_P_gcd_induction_solution_witness_common_right)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_common_right_equivalentfirstentry. ff_h_pfp_gcd_induction_solution_witness_common_right_equivalentfirstentry + S (pfrep_left_gcd_induction_solution_witness_common_right_equivalent) = S ((S (pfrep_position_gcd_induction_solution_witness_common_right_equivalentfirst)) * pfgd_pc_gcd_induction_solution_witness_common_right)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_common_right_equivalentfirstentry. pfgd_pb_gcd_induction_solution_witness_common_right = ff_q_pfp_gcd_induction_solution_witness_common_right_equivalentfirstentry * S ((S (pfrep_position_gcd_induction_solution_witness_common_right_equivalentfirst)) * pfgd_pc_gcd_induction_solution_witness_common_right) + (pfrep_left_gcd_induction_solution_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_induction_solution_witness_common_right_equivalentfirstoutside. pfrep_gap_gcd_induction_solution_witness_common_right_equivalentfirstoutside+(pfgd_P_gcd_induction_solution_witness_common_right)=(pfrep_power_gcd_induction_solution_witness_common_right_equivalent)) /\\ (((pfrep_left_gcd_induction_solution_witness_common_right_equivalent)=0))))) -> ((exists pfrep_position_gcd_induction_solution_witness_common_right_equivalentsecond. ((pfrep_position_gcd_induction_solution_witness_common_right_equivalentsecond+S (pfrep_power_gcd_induction_solution_witness_common_right_equivalent)=(M)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_common_right_equivalentsecondentry. ff_h_pfp_gcd_induction_solution_witness_common_right_equivalentsecondentry + S (pfrep_right_gcd_induction_solution_witness_common_right_equivalent) = S ((S (pfrep_position_gcd_induction_solution_witness_common_right_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_common_right_equivalentsecondentry. bb = ff_q_pfp_gcd_induction_solution_witness_common_right_equivalentsecondentry * S ((S (pfrep_position_gcd_induction_solution_witness_common_right_equivalentsecond)) * bc) + (pfrep_right_gcd_induction_solution_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_induction_solution_witness_common_right_equivalentsecondoutside. pfrep_gap_gcd_induction_solution_witness_common_right_equivalentsecondoutside+(M)=(pfrep_power_gcd_induction_solution_witness_common_right_equivalent)) /\\ (((pfrep_right_gcd_induction_solution_witness_common_right_equivalent)=0))))) -> pfrep_left_gcd_induction_solution_witness_common_right_equivalent=pfrep_right_gcd_induction_solution_witness_common_right_equivalent)))))))))) /\\ ((exists pfgb_pb_gcd_induction_solution_witness_bezout pfgb_pc_gcd_induction_solution_witness_bezout pfgb_P_gcd_induction_solution_witness_bezout pfgb_qb_gcd_induction_solution_witness_bezout pfgb_qc_gcd_induction_solution_witness_bezout pfgb_Q_gcd_induction_solution_witness_bezout. ((((forall fom_index_pfp_gcd_induction_solution_witness_bezout_leftleft. (exists fom_gap_pfp_gcd_induction_solution_witness_bezout_leftleft_index_bound. fom_gap_pfp_gcd_induction_solution_witness_bezout_leftleft_index_bound + S (fom_index_pfp_gcd_induction_solution_witness_bezout_leftleft) = pfgs_U_gcd_induction_solution) -> exists fom_value_pfp_gcd_induction_solution_witness_bezout_leftleft. ((((exists fom_beta_height_pfp_gcd_induction_solution_witness_bezout_leftleft_entry. fom_beta_height_pfp_gcd_induction_solution_witness_bezout_leftleft_entry + S (fom_value_pfp_gcd_induction_solution_witness_bezout_leftleft) = S ((S (fom_index_pfp_gcd_induction_solution_witness_bezout_leftleft)) * pfgs_uc_gcd_induction_solution)) /\\ exists fom_beta_quotient_pfp_gcd_induction_solution_witness_bezout_leftleft_entry. pfgs_ub_gcd_induction_solution = fom_beta_quotient_pfp_gcd_induction_solution_witness_bezout_leftleft_entry * S ((S (fom_index_pfp_gcd_induction_solution_witness_bezout_leftleft)) * pfgs_uc_gcd_induction_solution) + (fom_value_pfp_gcd_induction_solution_witness_bezout_leftleft))) /\\ (exists fom_gap_pfp_gcd_induction_solution_witness_bezout_leftleft_value_bound. fom_gap_pfp_gcd_induction_solution_witness_bezout_leftleft_value_bound + S (fom_value_pfp_gcd_induction_solution_witness_bezout_leftleft) = p))) /\\ (((forall fom_index_pfp_gcd_induction_solution_witness_bezout_leftright. (exists fom_gap_pfp_gcd_induction_solution_witness_bezout_leftright_index_bound. fom_gap_pfp_gcd_induction_solution_witness_bezout_leftright_index_bound + S (fom_index_pfp_gcd_induction_solution_witness_bezout_leftright) = L) -> exists fom_value_pfp_gcd_induction_solution_witness_bezout_leftright. ((((exists fom_beta_height_pfp_gcd_induction_solution_witness_bezout_leftright_entry. fom_beta_height_pfp_gcd_induction_solution_witness_bezout_leftright_entry + S (fom_value_pfp_gcd_induction_solution_witness_bezout_leftright) = S ((S (fom_index_pfp_gcd_induction_solution_witness_bezout_leftright)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_induction_solution_witness_bezout_leftright_entry. ab = fom_beta_quotient_pfp_gcd_induction_solution_witness_bezout_leftright_entry * S ((S (fom_index_pfp_gcd_induction_solution_witness_bezout_leftright)) * ac) + (fom_value_pfp_gcd_induction_solution_witness_bezout_leftright))) /\\ (exists fom_gap_pfp_gcd_induction_solution_witness_bezout_leftright_value_bound. fom_gap_pfp_gcd_induction_solution_witness_bezout_leftright_value_bound + S (fom_value_pfp_gcd_induction_solution_witness_bezout_leftright) = p))) /\\ (((((((pfgs_U_gcd_induction_solution)=0 \\/ (L)=0) /\\ (((pfgb_P_gcd_induction_solution_witness_bezout)=0)))) \\/ (((~((pfgs_U_gcd_induction_solution)=0)) /\\ (((~((L)=0)) /\\ (((pfgs_U_gcd_induction_solution)+(L)=S (pfgb_P_gcd_induction_solution_witness_bezout)))))))) /\\ ((forall pfc_index_gcd_induction_solution_witness_bezout_leftcoefficients. (exists pfa_gap_gcd_induction_solution_witness_bezout_leftcoefficientsbound. pfa_gap_gcd_induction_solution_witness_bezout_leftcoefficientsbound + S (pfc_index_gcd_induction_solution_witness_bezout_leftcoefficients) = (pfgb_P_gcd_induction_solution_witness_bezout)) -> exists pfc_value_gcd_induction_solution_witness_bezout_leftcoefficients. ((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_leftcoefficientsentry. ff_h_pfp_gcd_induction_solution_witness_bezout_leftcoefficientsentry + S (pfc_value_gcd_induction_solution_witness_bezout_leftcoefficients) = S ((S (pfc_index_gcd_induction_solution_witness_bezout_leftcoefficients)) * pfgb_pc_gcd_induction_solution_witness_bezout)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_leftcoefficientsentry. pfgb_pb_gcd_induction_solution_witness_bezout = ff_q_pfp_gcd_induction_solution_witness_bezout_leftcoefficientsentry * S ((S (pfc_index_gcd_induction_solution_witness_bezout_leftcoefficients)) * pfgb_pc_gcd_induction_solution_witness_bezout) + (pfc_value_gcd_induction_solution_witness_bezout_leftcoefficients))) /\\ ((exists pfc_terms_code_gcd_induction_solution_witness_bezout_leftcoefficientscoefficient pfc_terms_scale_gcd_induction_solution_witness_bezout_leftcoefficientscoefficient pfc_natural_sum_gcd_induction_solution_witness_bezout_leftcoefficientscoefficient. ((forall pfc_index_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalbound. pfa_gap_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_induction_solution_witness_bezout_leftcoefficients))) -> exists pfc_value_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_induction_solution_witness_bezout_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_induction_solution_witness_bezout_leftcoefficientscoefficient = ff_q_pfp_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_induction_solution_witness_bezout_leftcoefficientscoefficient) + (pfc_value_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_left_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_right_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonal)+pfc_complement_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm=(pfc_index_gcd_induction_solution_witness_bezout_leftcoefficients)) /\\ ((((((exists pfa_gap_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonal) = (pfgs_U_gcd_induction_solution)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_gcd_induction_solution)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. pfgs_ub_gcd_induction_solution = ff_q_pfp_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_gcd_induction_solution) + (pfc_left_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside+(pfgs_U_gcd_induction_solution)=(pfc_index_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonal)=pfc_left_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm*pfc_right_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum fs_v_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_induction_solution_witness_bezout_leftcoefficientscoefficient) = S ((S (S (pfc_index_gcd_induction_solution_witness_bezout_leftcoefficients))) * fs_v_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_induction_solution_witness_bezout_leftcoefficients))) * fs_v_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum) + (pfc_natural_sum_gcd_induction_solution_witness_bezout_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_induction_solution_witness_bezout_leftcoefficients)) -> exists fs_a_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_induction_solution_witness_bezout_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_induction_solution_witness_bezout_leftcoefficientscoefficient = fs_q_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_induction_solution_witness_bezout_leftcoefficientscoefficient) + (fs_a_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum) + (fs_r_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum) + (fs_s_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientresiduebound. pfa_gap_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientresiduebound + S (pfc_value_gcd_induction_solution_witness_bezout_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_induction_solution_witness_bezout_leftcoefficientscoefficient) + (p) * pfa_offset_left_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientresiduecongruence = (pfc_value_gcd_induction_solution_witness_bezout_leftcoefficients) + (p) * pfa_offset_right_gcd_induction_solution_witness_bezout_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_gcd_induction_solution_witness_bezout_rightleft. (exists fom_gap_pfp_gcd_induction_solution_witness_bezout_rightleft_index_bound. fom_gap_pfp_gcd_induction_solution_witness_bezout_rightleft_index_bound + S (fom_index_pfp_gcd_induction_solution_witness_bezout_rightleft) = pfgs_V_gcd_induction_solution) -> exists fom_value_pfp_gcd_induction_solution_witness_bezout_rightleft. ((((exists fom_beta_height_pfp_gcd_induction_solution_witness_bezout_rightleft_entry. fom_beta_height_pfp_gcd_induction_solution_witness_bezout_rightleft_entry + S (fom_value_pfp_gcd_induction_solution_witness_bezout_rightleft) = S ((S (fom_index_pfp_gcd_induction_solution_witness_bezout_rightleft)) * pfgs_vc_gcd_induction_solution)) /\\ exists fom_beta_quotient_pfp_gcd_induction_solution_witness_bezout_rightleft_entry. pfgs_vb_gcd_induction_solution = fom_beta_quotient_pfp_gcd_induction_solution_witness_bezout_rightleft_entry * S ((S (fom_index_pfp_gcd_induction_solution_witness_bezout_rightleft)) * pfgs_vc_gcd_induction_solution) + (fom_value_pfp_gcd_induction_solution_witness_bezout_rightleft))) /\\ (exists fom_gap_pfp_gcd_induction_solution_witness_bezout_rightleft_value_bound. fom_gap_pfp_gcd_induction_solution_witness_bezout_rightleft_value_bound + S (fom_value_pfp_gcd_induction_solution_witness_bezout_rightleft) = p))) /\\ (((forall fom_index_pfp_gcd_induction_solution_witness_bezout_rightright. (exists fom_gap_pfp_gcd_induction_solution_witness_bezout_rightright_index_bound. fom_gap_pfp_gcd_induction_solution_witness_bezout_rightright_index_bound + S (fom_index_pfp_gcd_induction_solution_witness_bezout_rightright) = M) -> exists fom_value_pfp_gcd_induction_solution_witness_bezout_rightright. ((((exists fom_beta_height_pfp_gcd_induction_solution_witness_bezout_rightright_entry. fom_beta_height_pfp_gcd_induction_solution_witness_bezout_rightright_entry + S (fom_value_pfp_gcd_induction_solution_witness_bezout_rightright) = S ((S (fom_index_pfp_gcd_induction_solution_witness_bezout_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_induction_solution_witness_bezout_rightright_entry. bb = fom_beta_quotient_pfp_gcd_induction_solution_witness_bezout_rightright_entry * S ((S (fom_index_pfp_gcd_induction_solution_witness_bezout_rightright)) * bc) + (fom_value_pfp_gcd_induction_solution_witness_bezout_rightright))) /\\ (exists fom_gap_pfp_gcd_induction_solution_witness_bezout_rightright_value_bound. fom_gap_pfp_gcd_induction_solution_witness_bezout_rightright_value_bound + S (fom_value_pfp_gcd_induction_solution_witness_bezout_rightright) = p))) /\\ (((((((pfgs_V_gcd_induction_solution)=0 \\/ (M)=0) /\\ (((pfgb_Q_gcd_induction_solution_witness_bezout)=0)))) \\/ (((~((pfgs_V_gcd_induction_solution)=0)) /\\ (((~((M)=0)) /\\ (((pfgs_V_gcd_induction_solution)+(M)=S (pfgb_Q_gcd_induction_solution_witness_bezout)))))))) /\\ ((forall pfc_index_gcd_induction_solution_witness_bezout_rightcoefficients. (exists pfa_gap_gcd_induction_solution_witness_bezout_rightcoefficientsbound. pfa_gap_gcd_induction_solution_witness_bezout_rightcoefficientsbound + S (pfc_index_gcd_induction_solution_witness_bezout_rightcoefficients) = (pfgb_Q_gcd_induction_solution_witness_bezout)) -> exists pfc_value_gcd_induction_solution_witness_bezout_rightcoefficients. ((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_rightcoefficientsentry. ff_h_pfp_gcd_induction_solution_witness_bezout_rightcoefficientsentry + S (pfc_value_gcd_induction_solution_witness_bezout_rightcoefficients) = S ((S (pfc_index_gcd_induction_solution_witness_bezout_rightcoefficients)) * pfgb_qc_gcd_induction_solution_witness_bezout)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_rightcoefficientsentry. pfgb_qb_gcd_induction_solution_witness_bezout = ff_q_pfp_gcd_induction_solution_witness_bezout_rightcoefficientsentry * S ((S (pfc_index_gcd_induction_solution_witness_bezout_rightcoefficients)) * pfgb_qc_gcd_induction_solution_witness_bezout) + (pfc_value_gcd_induction_solution_witness_bezout_rightcoefficients))) /\\ ((exists pfc_terms_code_gcd_induction_solution_witness_bezout_rightcoefficientscoefficient pfc_terms_scale_gcd_induction_solution_witness_bezout_rightcoefficientscoefficient pfc_natural_sum_gcd_induction_solution_witness_bezout_rightcoefficientscoefficient. ((forall pfc_index_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalbound. pfa_gap_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_induction_solution_witness_bezout_rightcoefficients))) -> exists pfc_value_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_induction_solution_witness_bezout_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_induction_solution_witness_bezout_rightcoefficientscoefficient = ff_q_pfp_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_induction_solution_witness_bezout_rightcoefficientscoefficient) + (pfc_value_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_left_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_right_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonal)+pfc_complement_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm=(pfc_index_gcd_induction_solution_witness_bezout_rightcoefficients)) /\\ ((((((exists pfa_gap_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonal) = (pfgs_V_gcd_induction_solution)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_gcd_induction_solution)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. pfgs_vb_gcd_induction_solution = ff_q_pfp_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_gcd_induction_solution) + (pfc_left_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside+(pfgs_V_gcd_induction_solution)=(pfc_index_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonal)=pfc_left_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm*pfc_right_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum fs_v_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_induction_solution_witness_bezout_rightcoefficientscoefficient) = S ((S (S (pfc_index_gcd_induction_solution_witness_bezout_rightcoefficients))) * fs_v_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_induction_solution_witness_bezout_rightcoefficients))) * fs_v_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum) + (pfc_natural_sum_gcd_induction_solution_witness_bezout_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_induction_solution_witness_bezout_rightcoefficients)) -> exists fs_a_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_induction_solution_witness_bezout_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_induction_solution_witness_bezout_rightcoefficientscoefficient = fs_q_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_induction_solution_witness_bezout_rightcoefficientscoefficient) + (fs_a_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum) + (fs_r_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum) + (fs_s_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientresiduebound. pfa_gap_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientresiduebound + S (pfc_value_gcd_induction_solution_witness_bezout_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_induction_solution_witness_bezout_rightcoefficientscoefficient) + (p) * pfa_offset_left_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientresiduecongruence = (pfc_value_gcd_induction_solution_witness_bezout_rightcoefficients) + (p) * pfa_offset_right_gcd_induction_solution_witness_bezout_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded. (exists fom_gap_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded_index_bound. fom_gap_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded_index_bound + S (fom_index_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded) = pfgb_P_gcd_induction_solution_witness_bezout) -> exists fom_value_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded. ((((exists fom_beta_height_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded_entry. fom_beta_height_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded_entry + S (fom_value_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded) = S ((S (fom_index_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded)) * pfgb_pc_gcd_induction_solution_witness_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded_entry. pfgb_pb_gcd_induction_solution_witness_bezout = fom_beta_quotient_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded_entry * S ((S (fom_index_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded)) * pfgb_pc_gcd_induction_solution_witness_bezout) + (fom_value_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded))) /\\ (exists fom_gap_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded_value_bound. fom_gap_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded_value_bound + S (fom_value_pfp_gcd_induction_solution_witness_bezout_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded. (exists fom_gap_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded_index_bound. fom_gap_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded_index_bound + S (fom_index_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded) = pfgb_Q_gcd_induction_solution_witness_bezout) -> exists fom_value_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded. ((((exists fom_beta_height_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded_entry. fom_beta_height_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded_entry + S (fom_value_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded) = S ((S (fom_index_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded)) * pfgb_qc_gcd_induction_solution_witness_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded_entry. pfgb_qb_gcd_induction_solution_witness_bezout = fom_beta_quotient_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded_entry * S ((S (fom_index_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded)) * pfgb_qc_gcd_induction_solution_witness_bezout) + (fom_value_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded))) /\\ (exists fom_gap_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded_value_bound. fom_gap_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded_value_bound + S (fom_value_pfp_gcd_induction_solution_witness_bezout_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded. (exists fom_gap_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded_index_bound. fom_gap_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded_index_bound + S (fom_index_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded) = pfgs_G_gcd_induction_solution) -> exists fom_value_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded. ((((exists fom_beta_height_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded_entry. fom_beta_height_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded_entry + S (fom_value_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded) = S ((S (fom_index_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded)) * pfgs_gc_gcd_induction_solution)) /\\ exists fom_beta_quotient_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded_entry. pfgs_gb_gcd_induction_solution = fom_beta_quotient_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded_entry * S ((S (fom_index_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded)) * pfgs_gc_gcd_induction_solution) + (fom_value_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded))) /\\ (exists fom_gap_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded_value_bound. fom_gap_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded_value_bound + S (fom_value_pfp_gcd_induction_solution_witness_bezout_sum_result_bounded) = p))) /\\ ((exists pfga_ub_gcd_induction_solution_witness_bezout_sum pfga_uc_gcd_induction_solution_witness_bezout_sum pfga_vb_gcd_induction_solution_witness_bezout_sum pfga_vc_gcd_induction_solution_witness_bezout_sum pfga_tb_gcd_induction_solution_witness_bezout_sum pfga_tc_gcd_induction_solution_witness_bezout_sum pfga_K_gcd_induction_solution_witness_bezout_sum. ((((forall pfrep_power_gcd_induction_solution_witness_bezout_sum_left pfrep_left_gcd_induction_solution_witness_bezout_sum_left pfrep_right_gcd_induction_solution_witness_bezout_sum_left. ((exists pfrep_position_gcd_induction_solution_witness_bezout_sum_leftfirst. ((pfrep_position_gcd_induction_solution_witness_bezout_sum_leftfirst+S (pfrep_power_gcd_induction_solution_witness_bezout_sum_left)=(pfgb_P_gcd_induction_solution_witness_bezout)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_sum_leftfirstentry. ff_h_pfp_gcd_induction_solution_witness_bezout_sum_leftfirstentry + S (pfrep_left_gcd_induction_solution_witness_bezout_sum_left) = S ((S (pfrep_position_gcd_induction_solution_witness_bezout_sum_leftfirst)) * pfgb_pc_gcd_induction_solution_witness_bezout)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_sum_leftfirstentry. pfgb_pb_gcd_induction_solution_witness_bezout = ff_q_pfp_gcd_induction_solution_witness_bezout_sum_leftfirstentry * S ((S (pfrep_position_gcd_induction_solution_witness_bezout_sum_leftfirst)) * pfgb_pc_gcd_induction_solution_witness_bezout) + (pfrep_left_gcd_induction_solution_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_induction_solution_witness_bezout_sum_leftfirstoutside. pfrep_gap_gcd_induction_solution_witness_bezout_sum_leftfirstoutside+(pfgb_P_gcd_induction_solution_witness_bezout)=(pfrep_power_gcd_induction_solution_witness_bezout_sum_left)) /\\ (((pfrep_left_gcd_induction_solution_witness_bezout_sum_left)=0))))) -> ((exists pfrep_position_gcd_induction_solution_witness_bezout_sum_leftsecond. ((pfrep_position_gcd_induction_solution_witness_bezout_sum_leftsecond+S (pfrep_power_gcd_induction_solution_witness_bezout_sum_left)=(pfga_K_gcd_induction_solution_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_sum_leftsecondentry. ff_h_pfp_gcd_induction_solution_witness_bezout_sum_leftsecondentry + S (pfrep_right_gcd_induction_solution_witness_bezout_sum_left) = S ((S (pfrep_position_gcd_induction_solution_witness_bezout_sum_leftsecond)) * pfga_uc_gcd_induction_solution_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_sum_leftsecondentry. pfga_ub_gcd_induction_solution_witness_bezout_sum = ff_q_pfp_gcd_induction_solution_witness_bezout_sum_leftsecondentry * S ((S (pfrep_position_gcd_induction_solution_witness_bezout_sum_leftsecond)) * pfga_uc_gcd_induction_solution_witness_bezout_sum) + (pfrep_right_gcd_induction_solution_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_induction_solution_witness_bezout_sum_leftsecondoutside. pfrep_gap_gcd_induction_solution_witness_bezout_sum_leftsecondoutside+(pfga_K_gcd_induction_solution_witness_bezout_sum)=(pfrep_power_gcd_induction_solution_witness_bezout_sum_left)) /\\ (((pfrep_right_gcd_induction_solution_witness_bezout_sum_left)=0))))) -> pfrep_left_gcd_induction_solution_witness_bezout_sum_left=pfrep_right_gcd_induction_solution_witness_bezout_sum_left) /\\ ((forall pfrep_power_gcd_induction_solution_witness_bezout_sum_right pfrep_left_gcd_induction_solution_witness_bezout_sum_right pfrep_right_gcd_induction_solution_witness_bezout_sum_right. ((exists pfrep_position_gcd_induction_solution_witness_bezout_sum_rightfirst. ((pfrep_position_gcd_induction_solution_witness_bezout_sum_rightfirst+S (pfrep_power_gcd_induction_solution_witness_bezout_sum_right)=(pfgb_Q_gcd_induction_solution_witness_bezout)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_sum_rightfirstentry. ff_h_pfp_gcd_induction_solution_witness_bezout_sum_rightfirstentry + S (pfrep_left_gcd_induction_solution_witness_bezout_sum_right) = S ((S (pfrep_position_gcd_induction_solution_witness_bezout_sum_rightfirst)) * pfgb_qc_gcd_induction_solution_witness_bezout)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_sum_rightfirstentry. pfgb_qb_gcd_induction_solution_witness_bezout = ff_q_pfp_gcd_induction_solution_witness_bezout_sum_rightfirstentry * S ((S (pfrep_position_gcd_induction_solution_witness_bezout_sum_rightfirst)) * pfgb_qc_gcd_induction_solution_witness_bezout) + (pfrep_left_gcd_induction_solution_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_induction_solution_witness_bezout_sum_rightfirstoutside. pfrep_gap_gcd_induction_solution_witness_bezout_sum_rightfirstoutside+(pfgb_Q_gcd_induction_solution_witness_bezout)=(pfrep_power_gcd_induction_solution_witness_bezout_sum_right)) /\\ (((pfrep_left_gcd_induction_solution_witness_bezout_sum_right)=0))))) -> ((exists pfrep_position_gcd_induction_solution_witness_bezout_sum_rightsecond. ((pfrep_position_gcd_induction_solution_witness_bezout_sum_rightsecond+S (pfrep_power_gcd_induction_solution_witness_bezout_sum_right)=(pfga_K_gcd_induction_solution_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_sum_rightsecondentry. ff_h_pfp_gcd_induction_solution_witness_bezout_sum_rightsecondentry + S (pfrep_right_gcd_induction_solution_witness_bezout_sum_right) = S ((S (pfrep_position_gcd_induction_solution_witness_bezout_sum_rightsecond)) * pfga_vc_gcd_induction_solution_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_sum_rightsecondentry. pfga_vb_gcd_induction_solution_witness_bezout_sum = ff_q_pfp_gcd_induction_solution_witness_bezout_sum_rightsecondentry * S ((S (pfrep_position_gcd_induction_solution_witness_bezout_sum_rightsecond)) * pfga_vc_gcd_induction_solution_witness_bezout_sum) + (pfrep_right_gcd_induction_solution_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_induction_solution_witness_bezout_sum_rightsecondoutside. pfrep_gap_gcd_induction_solution_witness_bezout_sum_rightsecondoutside+(pfga_K_gcd_induction_solution_witness_bezout_sum)=(pfrep_power_gcd_induction_solution_witness_bezout_sum_right)) /\\ (((pfrep_right_gcd_induction_solution_witness_bezout_sum_right)=0))))) -> pfrep_left_gcd_induction_solution_witness_bezout_sum_right=pfrep_right_gcd_induction_solution_witness_bezout_sum_right)))) /\\ (((forall pfp_index_gcd_induction_solution_witness_bezout_sum_add. (exists pfa_gap_gcd_induction_solution_witness_bezout_sum_addindex. pfa_gap_gcd_induction_solution_witness_bezout_sum_addindex + S (pfp_index_gcd_induction_solution_witness_bezout_sum_add) = (pfga_K_gcd_induction_solution_witness_bezout_sum)) -> exists pfp_left_gcd_induction_solution_witness_bezout_sum_add pfp_right_gcd_induction_solution_witness_bezout_sum_add pfp_value_gcd_induction_solution_witness_bezout_sum_add. ((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_sum_addleft. ff_h_pfp_gcd_induction_solution_witness_bezout_sum_addleft + S (pfp_left_gcd_induction_solution_witness_bezout_sum_add) = S ((S (pfp_index_gcd_induction_solution_witness_bezout_sum_add)) * pfga_uc_gcd_induction_solution_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_sum_addleft. pfga_ub_gcd_induction_solution_witness_bezout_sum = ff_q_pfp_gcd_induction_solution_witness_bezout_sum_addleft * S ((S (pfp_index_gcd_induction_solution_witness_bezout_sum_add)) * pfga_uc_gcd_induction_solution_witness_bezout_sum) + (pfp_left_gcd_induction_solution_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_sum_addright. ff_h_pfp_gcd_induction_solution_witness_bezout_sum_addright + S (pfp_right_gcd_induction_solution_witness_bezout_sum_add) = S ((S (pfp_index_gcd_induction_solution_witness_bezout_sum_add)) * pfga_vc_gcd_induction_solution_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_sum_addright. pfga_vb_gcd_induction_solution_witness_bezout_sum = ff_q_pfp_gcd_induction_solution_witness_bezout_sum_addright * S ((S (pfp_index_gcd_induction_solution_witness_bezout_sum_add)) * pfga_vc_gcd_induction_solution_witness_bezout_sum) + (pfp_right_gcd_induction_solution_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_sum_addtarget. ff_h_pfp_gcd_induction_solution_witness_bezout_sum_addtarget + S (pfp_value_gcd_induction_solution_witness_bezout_sum_add) = S ((S (pfp_index_gcd_induction_solution_witness_bezout_sum_add)) * pfga_tc_gcd_induction_solution_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_sum_addtarget. pfga_tb_gcd_induction_solution_witness_bezout_sum = ff_q_pfp_gcd_induction_solution_witness_bezout_sum_addtarget * S ((S (pfp_index_gcd_induction_solution_witness_bezout_sum_add)) * pfga_tc_gcd_induction_solution_witness_bezout_sum) + (pfp_value_gcd_induction_solution_witness_bezout_sum_add))) /\\ ((((exists pfa_gap_gcd_induction_solution_witness_bezout_sum_addoperationleft. pfa_gap_gcd_induction_solution_witness_bezout_sum_addoperationleft + S (pfp_left_gcd_induction_solution_witness_bezout_sum_add) = (p)) /\\ (((exists pfa_gap_gcd_induction_solution_witness_bezout_sum_addoperationright. pfa_gap_gcd_induction_solution_witness_bezout_sum_addoperationright + S (pfp_right_gcd_induction_solution_witness_bezout_sum_add) = (p)) /\\ ((((exists pfa_gap_gcd_induction_solution_witness_bezout_sum_addoperationresultbound. pfa_gap_gcd_induction_solution_witness_bezout_sum_addoperationresultbound + S (pfp_value_gcd_induction_solution_witness_bezout_sum_add) = (p)) /\\ ((exists pfa_offset_left_gcd_induction_solution_witness_bezout_sum_addoperationresultcongruence pfa_offset_right_gcd_induction_solution_witness_bezout_sum_addoperationresultcongruence. ((pfp_left_gcd_induction_solution_witness_bezout_sum_add) + (pfp_right_gcd_induction_solution_witness_bezout_sum_add)) + (p) * pfa_offset_left_gcd_induction_solution_witness_bezout_sum_addoperationresultcongruence = (pfp_value_gcd_induction_solution_witness_bezout_sum_add) + (p) * pfa_offset_right_gcd_induction_solution_witness_bezout_sum_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_gcd_induction_solution_witness_bezout_sum_result pfrep_left_gcd_induction_solution_witness_bezout_sum_result pfrep_right_gcd_induction_solution_witness_bezout_sum_result. ((exists pfrep_position_gcd_induction_solution_witness_bezout_sum_resultfirst. ((pfrep_position_gcd_induction_solution_witness_bezout_sum_resultfirst+S (pfrep_power_gcd_induction_solution_witness_bezout_sum_result)=(pfga_K_gcd_induction_solution_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_sum_resultfirstentry. ff_h_pfp_gcd_induction_solution_witness_bezout_sum_resultfirstentry + S (pfrep_left_gcd_induction_solution_witness_bezout_sum_result) = S ((S (pfrep_position_gcd_induction_solution_witness_bezout_sum_resultfirst)) * pfga_tc_gcd_induction_solution_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_sum_resultfirstentry. pfga_tb_gcd_induction_solution_witness_bezout_sum = ff_q_pfp_gcd_induction_solution_witness_bezout_sum_resultfirstentry * S ((S (pfrep_position_gcd_induction_solution_witness_bezout_sum_resultfirst)) * pfga_tc_gcd_induction_solution_witness_bezout_sum) + (pfrep_left_gcd_induction_solution_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_induction_solution_witness_bezout_sum_resultfirstoutside. pfrep_gap_gcd_induction_solution_witness_bezout_sum_resultfirstoutside+(pfga_K_gcd_induction_solution_witness_bezout_sum)=(pfrep_power_gcd_induction_solution_witness_bezout_sum_result)) /\\ (((pfrep_left_gcd_induction_solution_witness_bezout_sum_result)=0))))) -> ((exists pfrep_position_gcd_induction_solution_witness_bezout_sum_resultsecond. ((pfrep_position_gcd_induction_solution_witness_bezout_sum_resultsecond+S (pfrep_power_gcd_induction_solution_witness_bezout_sum_result)=(pfgs_G_gcd_induction_solution)) /\\ ((((exists ff_h_pfp_gcd_induction_solution_witness_bezout_sum_resultsecondentry. ff_h_pfp_gcd_induction_solution_witness_bezout_sum_resultsecondentry + S (pfrep_right_gcd_induction_solution_witness_bezout_sum_result) = S ((S (pfrep_position_gcd_induction_solution_witness_bezout_sum_resultsecond)) * pfgs_gc_gcd_induction_solution)) /\\ exists ff_q_pfp_gcd_induction_solution_witness_bezout_sum_resultsecondentry. pfgs_gb_gcd_induction_solution = ff_q_pfp_gcd_induction_solution_witness_bezout_sum_resultsecondentry * S ((S (pfrep_position_gcd_induction_solution_witness_bezout_sum_resultsecond)) * pfgs_gc_gcd_induction_solution) + (pfrep_right_gcd_induction_solution_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_induction_solution_witness_bezout_sum_resultsecondoutside. pfrep_gap_gcd_induction_solution_witness_bezout_sum_resultsecondoutside+(pfgs_G_gcd_induction_solution)=(pfrep_power_gcd_induction_solution_witness_bezout_sum_result)) /\\ (((pfrep_right_gcd_induction_solution_witness_bezout_sum_result)=0))))) -> pfrep_left_gcd_induction_solution_witness_bezout_sum_result=pfrep_right_gcd_induction_solution_witness_bezout_sum_result)))))))))))))))))))))))", "statement_sha256": "8e2c4b648890d1b26f2782de3c3597a79d0d7800cb6cc406c2b0a9d1d025c1e2", "summary": "Ordinary natural induction constructs actual normalized gcd and Bezout witnesses for every pair with second retained length at most n. 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(bc)", "specialize prime_field_polynomial_gcd_bezout_exists_up_to (M)", "apply prime_field_polynomial_gcd_bezout_exists_up_to", "exact hp", "exact hA", "exact hB", "specialize le_refl (M)", "apply le_refl" ], "script_sha256": "3967d559048508e2a38ebe36ec43255c355372d4564180c48d2cca47d8cc04b9", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_existence_candidate.py", "sha256": "81f2f48dd2e81894c7a267453646eb6f2b6f9bd3ee320386d8c561f6b9f8b8ca" }, "statement": "forall p ab ac L bb bc M. (~((p) = 1) /\\ forall pfa_factor_left_gcd_exists_prime pfa_factor_right_gcd_exists_prime. (p) = pfa_factor_left_gcd_exists_prime * pfa_factor_right_gcd_exists_prime -> pfa_factor_left_gcd_exists_prime = 1 \\/ pfa_factor_right_gcd_exists_prime = 1) -> (forall fom_index_pfp_gcd_exists_A. (exists fom_gap_pfp_gcd_exists_A_index_bound. fom_gap_pfp_gcd_exists_A_index_bound + S (fom_index_pfp_gcd_exists_A) = L) -> exists fom_value_pfp_gcd_exists_A. ((((exists fom_beta_height_pfp_gcd_exists_A_entry. fom_beta_height_pfp_gcd_exists_A_entry + S (fom_value_pfp_gcd_exists_A) = S ((S (fom_index_pfp_gcd_exists_A)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_exists_A_entry. ab = fom_beta_quotient_pfp_gcd_exists_A_entry * S ((S (fom_index_pfp_gcd_exists_A)) * ac) + (fom_value_pfp_gcd_exists_A))) /\\ (exists fom_gap_pfp_gcd_exists_A_value_bound. fom_gap_pfp_gcd_exists_A_value_bound + S (fom_value_pfp_gcd_exists_A) = p))) -> (forall fom_index_pfp_gcd_exists_B. (exists fom_gap_pfp_gcd_exists_B_index_bound. fom_gap_pfp_gcd_exists_B_index_bound + S (fom_index_pfp_gcd_exists_B) = M) -> exists fom_value_pfp_gcd_exists_B. ((((exists fom_beta_height_pfp_gcd_exists_B_entry. fom_beta_height_pfp_gcd_exists_B_entry + S (fom_value_pfp_gcd_exists_B) = S ((S (fom_index_pfp_gcd_exists_B)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_exists_B_entry. bb = fom_beta_quotient_pfp_gcd_exists_B_entry * S ((S (fom_index_pfp_gcd_exists_B)) * bc) + (fom_value_pfp_gcd_exists_B))) /\\ (exists fom_gap_pfp_gcd_exists_B_value_bound. fom_gap_pfp_gcd_exists_B_value_bound + S (fom_value_pfp_gcd_exists_B) = p))) -> (exists pfgs_gb_gcd_exists_result pfgs_gc_gcd_exists_result pfgs_G_gcd_exists_result pfgs_ub_gcd_exists_result pfgs_uc_gcd_exists_result pfgs_U_gcd_exists_result pfgs_vb_gcd_exists_result pfgs_vc_gcd_exists_result pfgs_V_gcd_exists_result. (((pfgs_G_gcd_exists_result)=0 \\/ (((~((pfgs_G_gcd_exists_result) = 0)) /\\ (((forall fom_index_pfp_gcd_exists_result_witness_normal_moniccoefficients. (exists fom_gap_pfp_gcd_exists_result_witness_normal_moniccoefficients_index_bound. fom_gap_pfp_gcd_exists_result_witness_normal_moniccoefficients_index_bound + S (fom_index_pfp_gcd_exists_result_witness_normal_moniccoefficients) = pfgs_G_gcd_exists_result) -> exists fom_value_pfp_gcd_exists_result_witness_normal_moniccoefficients. ((((exists fom_beta_height_pfp_gcd_exists_result_witness_normal_moniccoefficients_entry. fom_beta_height_pfp_gcd_exists_result_witness_normal_moniccoefficients_entry + S (fom_value_pfp_gcd_exists_result_witness_normal_moniccoefficients) = S ((S (fom_index_pfp_gcd_exists_result_witness_normal_moniccoefficients)) * pfgs_gc_gcd_exists_result)) /\\ exists fom_beta_quotient_pfp_gcd_exists_result_witness_normal_moniccoefficients_entry. pfgs_gb_gcd_exists_result = fom_beta_quotient_pfp_gcd_exists_result_witness_normal_moniccoefficients_entry * S ((S (fom_index_pfp_gcd_exists_result_witness_normal_moniccoefficients)) * pfgs_gc_gcd_exists_result) + (fom_value_pfp_gcd_exists_result_witness_normal_moniccoefficients))) /\\ (exists fom_gap_pfp_gcd_exists_result_witness_normal_moniccoefficients_value_bound. fom_gap_pfp_gcd_exists_result_witness_normal_moniccoefficients_value_bound + S (fom_value_pfp_gcd_exists_result_witness_normal_moniccoefficients) = p))) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_normal_monicleading. ff_h_pfp_gcd_exists_result_witness_normal_monicleading + S (1) = S ((S (0)) * pfgs_gc_gcd_exists_result)) /\\ exists ff_q_pfp_gcd_exists_result_witness_normal_monicleading. pfgs_gb_gcd_exists_result = ff_q_pfp_gcd_exists_result_witness_normal_monicleading * S ((S (0)) * pfgs_gc_gcd_exists_result) + (1))))))))) /\\ (((((((forall fom_index_pfp_gcd_exists_result_witness_common_left_bounded. (exists fom_gap_pfp_gcd_exists_result_witness_common_left_bounded_index_bound. fom_gap_pfp_gcd_exists_result_witness_common_left_bounded_index_bound + S (fom_index_pfp_gcd_exists_result_witness_common_left_bounded) = L) -> exists fom_value_pfp_gcd_exists_result_witness_common_left_bounded. ((((exists fom_beta_height_pfp_gcd_exists_result_witness_common_left_bounded_entry. fom_beta_height_pfp_gcd_exists_result_witness_common_left_bounded_entry + S (fom_value_pfp_gcd_exists_result_witness_common_left_bounded) = S ((S (fom_index_pfp_gcd_exists_result_witness_common_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_exists_result_witness_common_left_bounded_entry. ab = fom_beta_quotient_pfp_gcd_exists_result_witness_common_left_bounded_entry * S ((S (fom_index_pfp_gcd_exists_result_witness_common_left_bounded)) * ac) + (fom_value_pfp_gcd_exists_result_witness_common_left_bounded))) /\\ (exists fom_gap_pfp_gcd_exists_result_witness_common_left_bounded_value_bound. fom_gap_pfp_gcd_exists_result_witness_common_left_bounded_value_bound + S (fom_value_pfp_gcd_exists_result_witness_common_left_bounded) = p))) /\\ ((exists pfgd_qb_gcd_exists_result_witness_common_left pfgd_qc_gcd_exists_result_witness_common_left pfgd_Q_gcd_exists_result_witness_common_left pfgd_pb_gcd_exists_result_witness_common_left pfgd_pc_gcd_exists_result_witness_common_left pfgd_P_gcd_exists_result_witness_common_left. ((((forall fom_index_pfp_gcd_exists_result_witness_common_left_productleft. (exists fom_gap_pfp_gcd_exists_result_witness_common_left_productleft_index_bound. fom_gap_pfp_gcd_exists_result_witness_common_left_productleft_index_bound + S (fom_index_pfp_gcd_exists_result_witness_common_left_productleft) = pfgd_Q_gcd_exists_result_witness_common_left) -> exists fom_value_pfp_gcd_exists_result_witness_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_exists_result_witness_common_left_productleft_entry. fom_beta_height_pfp_gcd_exists_result_witness_common_left_productleft_entry + S (fom_value_pfp_gcd_exists_result_witness_common_left_productleft) = S ((S (fom_index_pfp_gcd_exists_result_witness_common_left_productleft)) * pfgd_qc_gcd_exists_result_witness_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_exists_result_witness_common_left_productleft_entry. pfgd_qb_gcd_exists_result_witness_common_left = fom_beta_quotient_pfp_gcd_exists_result_witness_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_exists_result_witness_common_left_productleft)) * pfgd_qc_gcd_exists_result_witness_common_left) + (fom_value_pfp_gcd_exists_result_witness_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_exists_result_witness_common_left_productleft_value_bound. fom_gap_pfp_gcd_exists_result_witness_common_left_productleft_value_bound + S (fom_value_pfp_gcd_exists_result_witness_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_exists_result_witness_common_left_productright. (exists fom_gap_pfp_gcd_exists_result_witness_common_left_productright_index_bound. fom_gap_pfp_gcd_exists_result_witness_common_left_productright_index_bound + S (fom_index_pfp_gcd_exists_result_witness_common_left_productright) = pfgs_G_gcd_exists_result) -> exists fom_value_pfp_gcd_exists_result_witness_common_left_productright. ((((exists fom_beta_height_pfp_gcd_exists_result_witness_common_left_productright_entry. fom_beta_height_pfp_gcd_exists_result_witness_common_left_productright_entry + S (fom_value_pfp_gcd_exists_result_witness_common_left_productright) = S ((S (fom_index_pfp_gcd_exists_result_witness_common_left_productright)) * pfgs_gc_gcd_exists_result)) /\\ exists fom_beta_quotient_pfp_gcd_exists_result_witness_common_left_productright_entry. pfgs_gb_gcd_exists_result = fom_beta_quotient_pfp_gcd_exists_result_witness_common_left_productright_entry * S ((S (fom_index_pfp_gcd_exists_result_witness_common_left_productright)) * pfgs_gc_gcd_exists_result) + (fom_value_pfp_gcd_exists_result_witness_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_exists_result_witness_common_left_productright_value_bound. fom_gap_pfp_gcd_exists_result_witness_common_left_productright_value_bound + S (fom_value_pfp_gcd_exists_result_witness_common_left_productright) = p))) /\\ (((((((pfgd_Q_gcd_exists_result_witness_common_left)=0 \\/ (pfgs_G_gcd_exists_result)=0) /\\ (((pfgd_P_gcd_exists_result_witness_common_left)=0)))) \\/ (((~((pfgd_Q_gcd_exists_result_witness_common_left)=0)) /\\ (((~((pfgs_G_gcd_exists_result)=0)) /\\ (((pfgd_Q_gcd_exists_result_witness_common_left)+(pfgs_G_gcd_exists_result)=S (pfgd_P_gcd_exists_result_witness_common_left)))))))) /\\ ((forall pfc_index_gcd_exists_result_witness_common_left_productcoefficients. (exists pfa_gap_gcd_exists_result_witness_common_left_productcoefficientsbound. pfa_gap_gcd_exists_result_witness_common_left_productcoefficientsbound + S (pfc_index_gcd_exists_result_witness_common_left_productcoefficients) = (pfgd_P_gcd_exists_result_witness_common_left)) -> exists pfc_value_gcd_exists_result_witness_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_exists_result_witness_common_left_productcoefficientsentry. ff_h_pfp_gcd_exists_result_witness_common_left_productcoefficientsentry + S (pfc_value_gcd_exists_result_witness_common_left_productcoefficients) = S ((S (pfc_index_gcd_exists_result_witness_common_left_productcoefficients)) * pfgd_pc_gcd_exists_result_witness_common_left)) /\\ exists ff_q_pfp_gcd_exists_result_witness_common_left_productcoefficientsentry. pfgd_pb_gcd_exists_result_witness_common_left = ff_q_pfp_gcd_exists_result_witness_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_exists_result_witness_common_left_productcoefficients)) * pfgd_pc_gcd_exists_result_witness_common_left) + (pfc_value_gcd_exists_result_witness_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_exists_result_witness_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_exists_result_witness_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_exists_result_witness_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_exists_result_witness_common_left_productcoefficients))) -> exists pfc_value_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_exists_result_witness_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_exists_result_witness_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_exists_result_witness_common_left_productcoefficientscoefficient) + (pfc_value_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_exists_result_witness_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_exists_result_witness_common_left)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_exists_result_witness_common_left)) /\\ exists ff_q_pfp_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_exists_result_witness_common_left = ff_q_pfp_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_exists_result_witness_common_left) + (pfc_left_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_exists_result_witness_common_left)=(pfc_index_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm) = (pfgs_G_gcd_exists_result)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_exists_result)) /\\ exists ff_q_pfp_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_gcd_exists_result = ff_q_pfp_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_exists_result) + (pfc_right_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_gcd_exists_result)=(pfc_complement_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_exists_result_witness_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_exists_result_witness_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_exists_result_witness_common_left_productcoefficients))) * fs_v_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_exists_result_witness_common_left_productcoefficients))) * fs_v_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_exists_result_witness_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_exists_result_witness_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_exists_result_witness_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_exists_result_witness_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_exists_result_witness_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_exists_result_witness_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_exists_result_witness_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_exists_result_witness_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_exists_result_witness_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_exists_result_witness_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_exists_result_witness_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_exists_result_witness_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_exists_result_witness_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_exists_result_witness_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_exists_result_witness_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_exists_result_witness_common_left_equivalent pfrep_left_gcd_exists_result_witness_common_left_equivalent pfrep_right_gcd_exists_result_witness_common_left_equivalent. ((exists pfrep_position_gcd_exists_result_witness_common_left_equivalentfirst. ((pfrep_position_gcd_exists_result_witness_common_left_equivalentfirst+S (pfrep_power_gcd_exists_result_witness_common_left_equivalent)=(pfgd_P_gcd_exists_result_witness_common_left)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_common_left_equivalentfirstentry. ff_h_pfp_gcd_exists_result_witness_common_left_equivalentfirstentry + S (pfrep_left_gcd_exists_result_witness_common_left_equivalent) = S ((S (pfrep_position_gcd_exists_result_witness_common_left_equivalentfirst)) * pfgd_pc_gcd_exists_result_witness_common_left)) /\\ exists ff_q_pfp_gcd_exists_result_witness_common_left_equivalentfirstentry. pfgd_pb_gcd_exists_result_witness_common_left = ff_q_pfp_gcd_exists_result_witness_common_left_equivalentfirstentry * S ((S (pfrep_position_gcd_exists_result_witness_common_left_equivalentfirst)) * pfgd_pc_gcd_exists_result_witness_common_left) + (pfrep_left_gcd_exists_result_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_exists_result_witness_common_left_equivalentfirstoutside. pfrep_gap_gcd_exists_result_witness_common_left_equivalentfirstoutside+(pfgd_P_gcd_exists_result_witness_common_left)=(pfrep_power_gcd_exists_result_witness_common_left_equivalent)) /\\ (((pfrep_left_gcd_exists_result_witness_common_left_equivalent)=0))))) -> ((exists pfrep_position_gcd_exists_result_witness_common_left_equivalentsecond. ((pfrep_position_gcd_exists_result_witness_common_left_equivalentsecond+S (pfrep_power_gcd_exists_result_witness_common_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_common_left_equivalentsecondentry. ff_h_pfp_gcd_exists_result_witness_common_left_equivalentsecondentry + S (pfrep_right_gcd_exists_result_witness_common_left_equivalent) = S ((S (pfrep_position_gcd_exists_result_witness_common_left_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_gcd_exists_result_witness_common_left_equivalentsecondentry. ab = ff_q_pfp_gcd_exists_result_witness_common_left_equivalentsecondentry * S ((S (pfrep_position_gcd_exists_result_witness_common_left_equivalentsecond)) * ac) + (pfrep_right_gcd_exists_result_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_exists_result_witness_common_left_equivalentsecondoutside. pfrep_gap_gcd_exists_result_witness_common_left_equivalentsecondoutside+(L)=(pfrep_power_gcd_exists_result_witness_common_left_equivalent)) /\\ (((pfrep_right_gcd_exists_result_witness_common_left_equivalent)=0))))) -> pfrep_left_gcd_exists_result_witness_common_left_equivalent=pfrep_right_gcd_exists_result_witness_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_exists_result_witness_common_right_bounded. (exists fom_gap_pfp_gcd_exists_result_witness_common_right_bounded_index_bound. fom_gap_pfp_gcd_exists_result_witness_common_right_bounded_index_bound + S (fom_index_pfp_gcd_exists_result_witness_common_right_bounded) = M) -> exists fom_value_pfp_gcd_exists_result_witness_common_right_bounded. ((((exists fom_beta_height_pfp_gcd_exists_result_witness_common_right_bounded_entry. fom_beta_height_pfp_gcd_exists_result_witness_common_right_bounded_entry + S (fom_value_pfp_gcd_exists_result_witness_common_right_bounded) = S ((S (fom_index_pfp_gcd_exists_result_witness_common_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_exists_result_witness_common_right_bounded_entry. bb = fom_beta_quotient_pfp_gcd_exists_result_witness_common_right_bounded_entry * S ((S (fom_index_pfp_gcd_exists_result_witness_common_right_bounded)) * bc) + (fom_value_pfp_gcd_exists_result_witness_common_right_bounded))) /\\ (exists fom_gap_pfp_gcd_exists_result_witness_common_right_bounded_value_bound. fom_gap_pfp_gcd_exists_result_witness_common_right_bounded_value_bound + S (fom_value_pfp_gcd_exists_result_witness_common_right_bounded) = p))) /\\ ((exists pfgd_qb_gcd_exists_result_witness_common_right pfgd_qc_gcd_exists_result_witness_common_right pfgd_Q_gcd_exists_result_witness_common_right pfgd_pb_gcd_exists_result_witness_common_right pfgd_pc_gcd_exists_result_witness_common_right pfgd_P_gcd_exists_result_witness_common_right. ((((forall fom_index_pfp_gcd_exists_result_witness_common_right_productleft. (exists fom_gap_pfp_gcd_exists_result_witness_common_right_productleft_index_bound. fom_gap_pfp_gcd_exists_result_witness_common_right_productleft_index_bound + S (fom_index_pfp_gcd_exists_result_witness_common_right_productleft) = pfgd_Q_gcd_exists_result_witness_common_right) -> exists fom_value_pfp_gcd_exists_result_witness_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_exists_result_witness_common_right_productleft_entry. fom_beta_height_pfp_gcd_exists_result_witness_common_right_productleft_entry + S (fom_value_pfp_gcd_exists_result_witness_common_right_productleft) = S ((S (fom_index_pfp_gcd_exists_result_witness_common_right_productleft)) * pfgd_qc_gcd_exists_result_witness_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_exists_result_witness_common_right_productleft_entry. pfgd_qb_gcd_exists_result_witness_common_right = fom_beta_quotient_pfp_gcd_exists_result_witness_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_exists_result_witness_common_right_productleft)) * pfgd_qc_gcd_exists_result_witness_common_right) + (fom_value_pfp_gcd_exists_result_witness_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_exists_result_witness_common_right_productleft_value_bound. fom_gap_pfp_gcd_exists_result_witness_common_right_productleft_value_bound + S (fom_value_pfp_gcd_exists_result_witness_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_exists_result_witness_common_right_productright. (exists fom_gap_pfp_gcd_exists_result_witness_common_right_productright_index_bound. fom_gap_pfp_gcd_exists_result_witness_common_right_productright_index_bound + S (fom_index_pfp_gcd_exists_result_witness_common_right_productright) = pfgs_G_gcd_exists_result) -> exists fom_value_pfp_gcd_exists_result_witness_common_right_productright. ((((exists fom_beta_height_pfp_gcd_exists_result_witness_common_right_productright_entry. fom_beta_height_pfp_gcd_exists_result_witness_common_right_productright_entry + S (fom_value_pfp_gcd_exists_result_witness_common_right_productright) = S ((S (fom_index_pfp_gcd_exists_result_witness_common_right_productright)) * pfgs_gc_gcd_exists_result)) /\\ exists fom_beta_quotient_pfp_gcd_exists_result_witness_common_right_productright_entry. pfgs_gb_gcd_exists_result = fom_beta_quotient_pfp_gcd_exists_result_witness_common_right_productright_entry * S ((S (fom_index_pfp_gcd_exists_result_witness_common_right_productright)) * pfgs_gc_gcd_exists_result) + (fom_value_pfp_gcd_exists_result_witness_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_exists_result_witness_common_right_productright_value_bound. fom_gap_pfp_gcd_exists_result_witness_common_right_productright_value_bound + S (fom_value_pfp_gcd_exists_result_witness_common_right_productright) = p))) /\\ (((((((pfgd_Q_gcd_exists_result_witness_common_right)=0 \\/ (pfgs_G_gcd_exists_result)=0) /\\ (((pfgd_P_gcd_exists_result_witness_common_right)=0)))) \\/ (((~((pfgd_Q_gcd_exists_result_witness_common_right)=0)) /\\ (((~((pfgs_G_gcd_exists_result)=0)) /\\ (((pfgd_Q_gcd_exists_result_witness_common_right)+(pfgs_G_gcd_exists_result)=S (pfgd_P_gcd_exists_result_witness_common_right)))))))) /\\ ((forall pfc_index_gcd_exists_result_witness_common_right_productcoefficients. (exists pfa_gap_gcd_exists_result_witness_common_right_productcoefficientsbound. pfa_gap_gcd_exists_result_witness_common_right_productcoefficientsbound + S (pfc_index_gcd_exists_result_witness_common_right_productcoefficients) = (pfgd_P_gcd_exists_result_witness_common_right)) -> exists pfc_value_gcd_exists_result_witness_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_exists_result_witness_common_right_productcoefficientsentry. ff_h_pfp_gcd_exists_result_witness_common_right_productcoefficientsentry + S (pfc_value_gcd_exists_result_witness_common_right_productcoefficients) = S ((S (pfc_index_gcd_exists_result_witness_common_right_productcoefficients)) * pfgd_pc_gcd_exists_result_witness_common_right)) /\\ exists ff_q_pfp_gcd_exists_result_witness_common_right_productcoefficientsentry. pfgd_pb_gcd_exists_result_witness_common_right = ff_q_pfp_gcd_exists_result_witness_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_exists_result_witness_common_right_productcoefficients)) * pfgd_pc_gcd_exists_result_witness_common_right) + (pfc_value_gcd_exists_result_witness_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_exists_result_witness_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_exists_result_witness_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_exists_result_witness_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_exists_result_witness_common_right_productcoefficients))) -> exists pfc_value_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_exists_result_witness_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_exists_result_witness_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_exists_result_witness_common_right_productcoefficientscoefficient) + (pfc_value_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_exists_result_witness_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_exists_result_witness_common_right)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_exists_result_witness_common_right)) /\\ exists ff_q_pfp_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_exists_result_witness_common_right = ff_q_pfp_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_exists_result_witness_common_right) + (pfc_left_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_exists_result_witness_common_right)=(pfc_index_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm) = (pfgs_G_gcd_exists_result)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_exists_result)) /\\ exists ff_q_pfp_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_gcd_exists_result = ff_q_pfp_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_gcd_exists_result) + (pfc_right_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_gcd_exists_result)=(pfc_complement_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_exists_result_witness_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_exists_result_witness_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_exists_result_witness_common_right_productcoefficients))) * fs_v_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_exists_result_witness_common_right_productcoefficients))) * fs_v_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_exists_result_witness_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_exists_result_witness_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_exists_result_witness_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_exists_result_witness_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_exists_result_witness_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_exists_result_witness_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_exists_result_witness_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_exists_result_witness_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_exists_result_witness_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_exists_result_witness_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_exists_result_witness_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_exists_result_witness_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_exists_result_witness_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_exists_result_witness_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_exists_result_witness_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_exists_result_witness_common_right_equivalent pfrep_left_gcd_exists_result_witness_common_right_equivalent pfrep_right_gcd_exists_result_witness_common_right_equivalent. ((exists pfrep_position_gcd_exists_result_witness_common_right_equivalentfirst. ((pfrep_position_gcd_exists_result_witness_common_right_equivalentfirst+S (pfrep_power_gcd_exists_result_witness_common_right_equivalent)=(pfgd_P_gcd_exists_result_witness_common_right)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_common_right_equivalentfirstentry. ff_h_pfp_gcd_exists_result_witness_common_right_equivalentfirstentry + S (pfrep_left_gcd_exists_result_witness_common_right_equivalent) = S ((S (pfrep_position_gcd_exists_result_witness_common_right_equivalentfirst)) * pfgd_pc_gcd_exists_result_witness_common_right)) /\\ exists ff_q_pfp_gcd_exists_result_witness_common_right_equivalentfirstentry. pfgd_pb_gcd_exists_result_witness_common_right = ff_q_pfp_gcd_exists_result_witness_common_right_equivalentfirstentry * S ((S (pfrep_position_gcd_exists_result_witness_common_right_equivalentfirst)) * pfgd_pc_gcd_exists_result_witness_common_right) + (pfrep_left_gcd_exists_result_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_exists_result_witness_common_right_equivalentfirstoutside. pfrep_gap_gcd_exists_result_witness_common_right_equivalentfirstoutside+(pfgd_P_gcd_exists_result_witness_common_right)=(pfrep_power_gcd_exists_result_witness_common_right_equivalent)) /\\ (((pfrep_left_gcd_exists_result_witness_common_right_equivalent)=0))))) -> ((exists pfrep_position_gcd_exists_result_witness_common_right_equivalentsecond. ((pfrep_position_gcd_exists_result_witness_common_right_equivalentsecond+S (pfrep_power_gcd_exists_result_witness_common_right_equivalent)=(M)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_common_right_equivalentsecondentry. ff_h_pfp_gcd_exists_result_witness_common_right_equivalentsecondentry + S (pfrep_right_gcd_exists_result_witness_common_right_equivalent) = S ((S (pfrep_position_gcd_exists_result_witness_common_right_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_gcd_exists_result_witness_common_right_equivalentsecondentry. bb = ff_q_pfp_gcd_exists_result_witness_common_right_equivalentsecondentry * S ((S (pfrep_position_gcd_exists_result_witness_common_right_equivalentsecond)) * bc) + (pfrep_right_gcd_exists_result_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_exists_result_witness_common_right_equivalentsecondoutside. pfrep_gap_gcd_exists_result_witness_common_right_equivalentsecondoutside+(M)=(pfrep_power_gcd_exists_result_witness_common_right_equivalent)) /\\ (((pfrep_right_gcd_exists_result_witness_common_right_equivalent)=0))))) -> pfrep_left_gcd_exists_result_witness_common_right_equivalent=pfrep_right_gcd_exists_result_witness_common_right_equivalent)))))))))) /\\ ((exists pfgb_pb_gcd_exists_result_witness_bezout pfgb_pc_gcd_exists_result_witness_bezout pfgb_P_gcd_exists_result_witness_bezout pfgb_qb_gcd_exists_result_witness_bezout pfgb_qc_gcd_exists_result_witness_bezout pfgb_Q_gcd_exists_result_witness_bezout. ((((forall fom_index_pfp_gcd_exists_result_witness_bezout_leftleft. (exists fom_gap_pfp_gcd_exists_result_witness_bezout_leftleft_index_bound. fom_gap_pfp_gcd_exists_result_witness_bezout_leftleft_index_bound + S (fom_index_pfp_gcd_exists_result_witness_bezout_leftleft) = pfgs_U_gcd_exists_result) -> exists fom_value_pfp_gcd_exists_result_witness_bezout_leftleft. ((((exists fom_beta_height_pfp_gcd_exists_result_witness_bezout_leftleft_entry. fom_beta_height_pfp_gcd_exists_result_witness_bezout_leftleft_entry + S (fom_value_pfp_gcd_exists_result_witness_bezout_leftleft) = S ((S (fom_index_pfp_gcd_exists_result_witness_bezout_leftleft)) * pfgs_uc_gcd_exists_result)) /\\ exists fom_beta_quotient_pfp_gcd_exists_result_witness_bezout_leftleft_entry. pfgs_ub_gcd_exists_result = fom_beta_quotient_pfp_gcd_exists_result_witness_bezout_leftleft_entry * S ((S (fom_index_pfp_gcd_exists_result_witness_bezout_leftleft)) * pfgs_uc_gcd_exists_result) + (fom_value_pfp_gcd_exists_result_witness_bezout_leftleft))) /\\ (exists fom_gap_pfp_gcd_exists_result_witness_bezout_leftleft_value_bound. fom_gap_pfp_gcd_exists_result_witness_bezout_leftleft_value_bound + S (fom_value_pfp_gcd_exists_result_witness_bezout_leftleft) = p))) /\\ (((forall fom_index_pfp_gcd_exists_result_witness_bezout_leftright. (exists fom_gap_pfp_gcd_exists_result_witness_bezout_leftright_index_bound. fom_gap_pfp_gcd_exists_result_witness_bezout_leftright_index_bound + S (fom_index_pfp_gcd_exists_result_witness_bezout_leftright) = L) -> exists fom_value_pfp_gcd_exists_result_witness_bezout_leftright. ((((exists fom_beta_height_pfp_gcd_exists_result_witness_bezout_leftright_entry. fom_beta_height_pfp_gcd_exists_result_witness_bezout_leftright_entry + S (fom_value_pfp_gcd_exists_result_witness_bezout_leftright) = S ((S (fom_index_pfp_gcd_exists_result_witness_bezout_leftright)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_exists_result_witness_bezout_leftright_entry. ab = fom_beta_quotient_pfp_gcd_exists_result_witness_bezout_leftright_entry * S ((S (fom_index_pfp_gcd_exists_result_witness_bezout_leftright)) * ac) + (fom_value_pfp_gcd_exists_result_witness_bezout_leftright))) /\\ (exists fom_gap_pfp_gcd_exists_result_witness_bezout_leftright_value_bound. fom_gap_pfp_gcd_exists_result_witness_bezout_leftright_value_bound + S (fom_value_pfp_gcd_exists_result_witness_bezout_leftright) = p))) /\\ (((((((pfgs_U_gcd_exists_result)=0 \\/ (L)=0) /\\ (((pfgb_P_gcd_exists_result_witness_bezout)=0)))) \\/ (((~((pfgs_U_gcd_exists_result)=0)) /\\ (((~((L)=0)) /\\ (((pfgs_U_gcd_exists_result)+(L)=S (pfgb_P_gcd_exists_result_witness_bezout)))))))) /\\ ((forall pfc_index_gcd_exists_result_witness_bezout_leftcoefficients. (exists pfa_gap_gcd_exists_result_witness_bezout_leftcoefficientsbound. pfa_gap_gcd_exists_result_witness_bezout_leftcoefficientsbound + S (pfc_index_gcd_exists_result_witness_bezout_leftcoefficients) = (pfgb_P_gcd_exists_result_witness_bezout)) -> exists pfc_value_gcd_exists_result_witness_bezout_leftcoefficients. ((((exists ff_h_pfp_gcd_exists_result_witness_bezout_leftcoefficientsentry. ff_h_pfp_gcd_exists_result_witness_bezout_leftcoefficientsentry + S (pfc_value_gcd_exists_result_witness_bezout_leftcoefficients) = S ((S (pfc_index_gcd_exists_result_witness_bezout_leftcoefficients)) * pfgb_pc_gcd_exists_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_leftcoefficientsentry. pfgb_pb_gcd_exists_result_witness_bezout = ff_q_pfp_gcd_exists_result_witness_bezout_leftcoefficientsentry * S ((S (pfc_index_gcd_exists_result_witness_bezout_leftcoefficients)) * pfgb_pc_gcd_exists_result_witness_bezout) + (pfc_value_gcd_exists_result_witness_bezout_leftcoefficients))) /\\ ((exists pfc_terms_code_gcd_exists_result_witness_bezout_leftcoefficientscoefficient pfc_terms_scale_gcd_exists_result_witness_bezout_leftcoefficientscoefficient pfc_natural_sum_gcd_exists_result_witness_bezout_leftcoefficientscoefficient. ((forall pfc_index_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalbound. pfa_gap_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_exists_result_witness_bezout_leftcoefficients))) -> exists pfc_value_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_exists_result_witness_bezout_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_exists_result_witness_bezout_leftcoefficientscoefficient = ff_q_pfp_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_exists_result_witness_bezout_leftcoefficientscoefficient) + (pfc_value_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_left_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_right_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonal)+pfc_complement_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm=(pfc_index_gcd_exists_result_witness_bezout_leftcoefficients)) /\\ ((((((exists pfa_gap_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonal) = (pfgs_U_gcd_exists_result)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_gcd_exists_result)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. pfgs_ub_gcd_exists_result = ff_q_pfp_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_gcd_exists_result) + (pfc_left_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside+(pfgs_U_gcd_exists_result)=(pfc_index_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonal)=pfc_left_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm*pfc_right_gcd_exists_result_witness_bezout_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum fs_v_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_exists_result_witness_bezout_leftcoefficientscoefficient) = S ((S (S (pfc_index_gcd_exists_result_witness_bezout_leftcoefficients))) * fs_v_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_exists_result_witness_bezout_leftcoefficients))) * fs_v_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum) + (pfc_natural_sum_gcd_exists_result_witness_bezout_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_exists_result_witness_bezout_leftcoefficients)) -> exists fs_a_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_exists_result_witness_bezout_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_exists_result_witness_bezout_leftcoefficientscoefficient = fs_q_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_exists_result_witness_bezout_leftcoefficientscoefficient) + (fs_a_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum) + (fs_r_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum) + (fs_s_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_exists_result_witness_bezout_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_exists_result_witness_bezout_leftcoefficientscoefficientresiduebound. pfa_gap_gcd_exists_result_witness_bezout_leftcoefficientscoefficientresiduebound + S (pfc_value_gcd_exists_result_witness_bezout_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_exists_result_witness_bezout_leftcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_exists_result_witness_bezout_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_exists_result_witness_bezout_leftcoefficientscoefficient) + (p) * pfa_offset_left_gcd_exists_result_witness_bezout_leftcoefficientscoefficientresiduecongruence = (pfc_value_gcd_exists_result_witness_bezout_leftcoefficients) + (p) * pfa_offset_right_gcd_exists_result_witness_bezout_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_gcd_exists_result_witness_bezout_rightleft. (exists fom_gap_pfp_gcd_exists_result_witness_bezout_rightleft_index_bound. fom_gap_pfp_gcd_exists_result_witness_bezout_rightleft_index_bound + S (fom_index_pfp_gcd_exists_result_witness_bezout_rightleft) = pfgs_V_gcd_exists_result) -> exists fom_value_pfp_gcd_exists_result_witness_bezout_rightleft. ((((exists fom_beta_height_pfp_gcd_exists_result_witness_bezout_rightleft_entry. fom_beta_height_pfp_gcd_exists_result_witness_bezout_rightleft_entry + S (fom_value_pfp_gcd_exists_result_witness_bezout_rightleft) = S ((S (fom_index_pfp_gcd_exists_result_witness_bezout_rightleft)) * pfgs_vc_gcd_exists_result)) /\\ exists fom_beta_quotient_pfp_gcd_exists_result_witness_bezout_rightleft_entry. pfgs_vb_gcd_exists_result = fom_beta_quotient_pfp_gcd_exists_result_witness_bezout_rightleft_entry * S ((S (fom_index_pfp_gcd_exists_result_witness_bezout_rightleft)) * pfgs_vc_gcd_exists_result) + (fom_value_pfp_gcd_exists_result_witness_bezout_rightleft))) /\\ (exists fom_gap_pfp_gcd_exists_result_witness_bezout_rightleft_value_bound. fom_gap_pfp_gcd_exists_result_witness_bezout_rightleft_value_bound + S (fom_value_pfp_gcd_exists_result_witness_bezout_rightleft) = p))) /\\ (((forall fom_index_pfp_gcd_exists_result_witness_bezout_rightright. (exists fom_gap_pfp_gcd_exists_result_witness_bezout_rightright_index_bound. fom_gap_pfp_gcd_exists_result_witness_bezout_rightright_index_bound + S (fom_index_pfp_gcd_exists_result_witness_bezout_rightright) = M) -> exists fom_value_pfp_gcd_exists_result_witness_bezout_rightright. ((((exists fom_beta_height_pfp_gcd_exists_result_witness_bezout_rightright_entry. fom_beta_height_pfp_gcd_exists_result_witness_bezout_rightright_entry + S (fom_value_pfp_gcd_exists_result_witness_bezout_rightright) = S ((S (fom_index_pfp_gcd_exists_result_witness_bezout_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_exists_result_witness_bezout_rightright_entry. bb = fom_beta_quotient_pfp_gcd_exists_result_witness_bezout_rightright_entry * S ((S (fom_index_pfp_gcd_exists_result_witness_bezout_rightright)) * bc) + (fom_value_pfp_gcd_exists_result_witness_bezout_rightright))) /\\ (exists fom_gap_pfp_gcd_exists_result_witness_bezout_rightright_value_bound. fom_gap_pfp_gcd_exists_result_witness_bezout_rightright_value_bound + S (fom_value_pfp_gcd_exists_result_witness_bezout_rightright) = p))) /\\ (((((((pfgs_V_gcd_exists_result)=0 \\/ (M)=0) /\\ (((pfgb_Q_gcd_exists_result_witness_bezout)=0)))) \\/ (((~((pfgs_V_gcd_exists_result)=0)) /\\ (((~((M)=0)) /\\ (((pfgs_V_gcd_exists_result)+(M)=S (pfgb_Q_gcd_exists_result_witness_bezout)))))))) /\\ ((forall pfc_index_gcd_exists_result_witness_bezout_rightcoefficients. (exists pfa_gap_gcd_exists_result_witness_bezout_rightcoefficientsbound. pfa_gap_gcd_exists_result_witness_bezout_rightcoefficientsbound + S (pfc_index_gcd_exists_result_witness_bezout_rightcoefficients) = (pfgb_Q_gcd_exists_result_witness_bezout)) -> exists pfc_value_gcd_exists_result_witness_bezout_rightcoefficients. ((((exists ff_h_pfp_gcd_exists_result_witness_bezout_rightcoefficientsentry. ff_h_pfp_gcd_exists_result_witness_bezout_rightcoefficientsentry + S (pfc_value_gcd_exists_result_witness_bezout_rightcoefficients) = S ((S (pfc_index_gcd_exists_result_witness_bezout_rightcoefficients)) * pfgb_qc_gcd_exists_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_rightcoefficientsentry. pfgb_qb_gcd_exists_result_witness_bezout = ff_q_pfp_gcd_exists_result_witness_bezout_rightcoefficientsentry * S ((S (pfc_index_gcd_exists_result_witness_bezout_rightcoefficients)) * pfgb_qc_gcd_exists_result_witness_bezout) + (pfc_value_gcd_exists_result_witness_bezout_rightcoefficients))) /\\ ((exists pfc_terms_code_gcd_exists_result_witness_bezout_rightcoefficientscoefficient pfc_terms_scale_gcd_exists_result_witness_bezout_rightcoefficientscoefficient pfc_natural_sum_gcd_exists_result_witness_bezout_rightcoefficientscoefficient. ((forall pfc_index_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalbound. pfa_gap_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_exists_result_witness_bezout_rightcoefficients))) -> exists pfc_value_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_exists_result_witness_bezout_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_exists_result_witness_bezout_rightcoefficientscoefficient = ff_q_pfp_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_exists_result_witness_bezout_rightcoefficientscoefficient) + (pfc_value_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_left_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_right_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonal)+pfc_complement_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm=(pfc_index_gcd_exists_result_witness_bezout_rightcoefficients)) /\\ ((((((exists pfa_gap_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonal) = (pfgs_V_gcd_exists_result)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_gcd_exists_result)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. pfgs_vb_gcd_exists_result = ff_q_pfp_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_gcd_exists_result) + (pfc_left_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside+(pfgs_V_gcd_exists_result)=(pfc_index_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonal)=pfc_left_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm*pfc_right_gcd_exists_result_witness_bezout_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum fs_v_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_exists_result_witness_bezout_rightcoefficientscoefficient) = S ((S (S (pfc_index_gcd_exists_result_witness_bezout_rightcoefficients))) * fs_v_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_exists_result_witness_bezout_rightcoefficients))) * fs_v_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum) + (pfc_natural_sum_gcd_exists_result_witness_bezout_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_exists_result_witness_bezout_rightcoefficients)) -> exists fs_a_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_exists_result_witness_bezout_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_exists_result_witness_bezout_rightcoefficientscoefficient = fs_q_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_exists_result_witness_bezout_rightcoefficientscoefficient) + (fs_a_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum) + (fs_r_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum) + (fs_s_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_exists_result_witness_bezout_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_exists_result_witness_bezout_rightcoefficientscoefficientresiduebound. pfa_gap_gcd_exists_result_witness_bezout_rightcoefficientscoefficientresiduebound + S (pfc_value_gcd_exists_result_witness_bezout_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_exists_result_witness_bezout_rightcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_exists_result_witness_bezout_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_exists_result_witness_bezout_rightcoefficientscoefficient) + (p) * pfa_offset_left_gcd_exists_result_witness_bezout_rightcoefficientscoefficientresiduecongruence = (pfc_value_gcd_exists_result_witness_bezout_rightcoefficients) + (p) * pfa_offset_right_gcd_exists_result_witness_bezout_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_gcd_exists_result_witness_bezout_sum_left_bounded. (exists fom_gap_pfp_gcd_exists_result_witness_bezout_sum_left_bounded_index_bound. fom_gap_pfp_gcd_exists_result_witness_bezout_sum_left_bounded_index_bound + S (fom_index_pfp_gcd_exists_result_witness_bezout_sum_left_bounded) = pfgb_P_gcd_exists_result_witness_bezout) -> exists fom_value_pfp_gcd_exists_result_witness_bezout_sum_left_bounded. ((((exists fom_beta_height_pfp_gcd_exists_result_witness_bezout_sum_left_bounded_entry. fom_beta_height_pfp_gcd_exists_result_witness_bezout_sum_left_bounded_entry + S (fom_value_pfp_gcd_exists_result_witness_bezout_sum_left_bounded) = S ((S (fom_index_pfp_gcd_exists_result_witness_bezout_sum_left_bounded)) * pfgb_pc_gcd_exists_result_witness_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_exists_result_witness_bezout_sum_left_bounded_entry. pfgb_pb_gcd_exists_result_witness_bezout = fom_beta_quotient_pfp_gcd_exists_result_witness_bezout_sum_left_bounded_entry * S ((S (fom_index_pfp_gcd_exists_result_witness_bezout_sum_left_bounded)) * pfgb_pc_gcd_exists_result_witness_bezout) + (fom_value_pfp_gcd_exists_result_witness_bezout_sum_left_bounded))) /\\ (exists fom_gap_pfp_gcd_exists_result_witness_bezout_sum_left_bounded_value_bound. fom_gap_pfp_gcd_exists_result_witness_bezout_sum_left_bounded_value_bound + S (fom_value_pfp_gcd_exists_result_witness_bezout_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_exists_result_witness_bezout_sum_right_bounded. (exists fom_gap_pfp_gcd_exists_result_witness_bezout_sum_right_bounded_index_bound. fom_gap_pfp_gcd_exists_result_witness_bezout_sum_right_bounded_index_bound + S (fom_index_pfp_gcd_exists_result_witness_bezout_sum_right_bounded) = pfgb_Q_gcd_exists_result_witness_bezout) -> exists fom_value_pfp_gcd_exists_result_witness_bezout_sum_right_bounded. ((((exists fom_beta_height_pfp_gcd_exists_result_witness_bezout_sum_right_bounded_entry. fom_beta_height_pfp_gcd_exists_result_witness_bezout_sum_right_bounded_entry + S (fom_value_pfp_gcd_exists_result_witness_bezout_sum_right_bounded) = S ((S (fom_index_pfp_gcd_exists_result_witness_bezout_sum_right_bounded)) * pfgb_qc_gcd_exists_result_witness_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_exists_result_witness_bezout_sum_right_bounded_entry. pfgb_qb_gcd_exists_result_witness_bezout = fom_beta_quotient_pfp_gcd_exists_result_witness_bezout_sum_right_bounded_entry * S ((S (fom_index_pfp_gcd_exists_result_witness_bezout_sum_right_bounded)) * pfgb_qc_gcd_exists_result_witness_bezout) + (fom_value_pfp_gcd_exists_result_witness_bezout_sum_right_bounded))) /\\ (exists fom_gap_pfp_gcd_exists_result_witness_bezout_sum_right_bounded_value_bound. fom_gap_pfp_gcd_exists_result_witness_bezout_sum_right_bounded_value_bound + S (fom_value_pfp_gcd_exists_result_witness_bezout_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_exists_result_witness_bezout_sum_result_bounded. (exists fom_gap_pfp_gcd_exists_result_witness_bezout_sum_result_bounded_index_bound. fom_gap_pfp_gcd_exists_result_witness_bezout_sum_result_bounded_index_bound + S (fom_index_pfp_gcd_exists_result_witness_bezout_sum_result_bounded) = pfgs_G_gcd_exists_result) -> exists fom_value_pfp_gcd_exists_result_witness_bezout_sum_result_bounded. ((((exists fom_beta_height_pfp_gcd_exists_result_witness_bezout_sum_result_bounded_entry. fom_beta_height_pfp_gcd_exists_result_witness_bezout_sum_result_bounded_entry + S (fom_value_pfp_gcd_exists_result_witness_bezout_sum_result_bounded) = S ((S (fom_index_pfp_gcd_exists_result_witness_bezout_sum_result_bounded)) * pfgs_gc_gcd_exists_result)) /\\ exists fom_beta_quotient_pfp_gcd_exists_result_witness_bezout_sum_result_bounded_entry. pfgs_gb_gcd_exists_result = fom_beta_quotient_pfp_gcd_exists_result_witness_bezout_sum_result_bounded_entry * S ((S (fom_index_pfp_gcd_exists_result_witness_bezout_sum_result_bounded)) * pfgs_gc_gcd_exists_result) + (fom_value_pfp_gcd_exists_result_witness_bezout_sum_result_bounded))) /\\ (exists fom_gap_pfp_gcd_exists_result_witness_bezout_sum_result_bounded_value_bound. fom_gap_pfp_gcd_exists_result_witness_bezout_sum_result_bounded_value_bound + S (fom_value_pfp_gcd_exists_result_witness_bezout_sum_result_bounded) = p))) /\\ ((exists pfga_ub_gcd_exists_result_witness_bezout_sum pfga_uc_gcd_exists_result_witness_bezout_sum pfga_vb_gcd_exists_result_witness_bezout_sum pfga_vc_gcd_exists_result_witness_bezout_sum pfga_tb_gcd_exists_result_witness_bezout_sum pfga_tc_gcd_exists_result_witness_bezout_sum pfga_K_gcd_exists_result_witness_bezout_sum. ((((forall pfrep_power_gcd_exists_result_witness_bezout_sum_left pfrep_left_gcd_exists_result_witness_bezout_sum_left pfrep_right_gcd_exists_result_witness_bezout_sum_left. ((exists pfrep_position_gcd_exists_result_witness_bezout_sum_leftfirst. ((pfrep_position_gcd_exists_result_witness_bezout_sum_leftfirst+S (pfrep_power_gcd_exists_result_witness_bezout_sum_left)=(pfgb_P_gcd_exists_result_witness_bezout)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_bezout_sum_leftfirstentry. ff_h_pfp_gcd_exists_result_witness_bezout_sum_leftfirstentry + S (pfrep_left_gcd_exists_result_witness_bezout_sum_left) = S ((S (pfrep_position_gcd_exists_result_witness_bezout_sum_leftfirst)) * pfgb_pc_gcd_exists_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_sum_leftfirstentry. pfgb_pb_gcd_exists_result_witness_bezout = ff_q_pfp_gcd_exists_result_witness_bezout_sum_leftfirstentry * S ((S (pfrep_position_gcd_exists_result_witness_bezout_sum_leftfirst)) * pfgb_pc_gcd_exists_result_witness_bezout) + (pfrep_left_gcd_exists_result_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_exists_result_witness_bezout_sum_leftfirstoutside. pfrep_gap_gcd_exists_result_witness_bezout_sum_leftfirstoutside+(pfgb_P_gcd_exists_result_witness_bezout)=(pfrep_power_gcd_exists_result_witness_bezout_sum_left)) /\\ (((pfrep_left_gcd_exists_result_witness_bezout_sum_left)=0))))) -> ((exists pfrep_position_gcd_exists_result_witness_bezout_sum_leftsecond. ((pfrep_position_gcd_exists_result_witness_bezout_sum_leftsecond+S (pfrep_power_gcd_exists_result_witness_bezout_sum_left)=(pfga_K_gcd_exists_result_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_bezout_sum_leftsecondentry. ff_h_pfp_gcd_exists_result_witness_bezout_sum_leftsecondentry + S (pfrep_right_gcd_exists_result_witness_bezout_sum_left) = S ((S (pfrep_position_gcd_exists_result_witness_bezout_sum_leftsecond)) * pfga_uc_gcd_exists_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_sum_leftsecondentry. pfga_ub_gcd_exists_result_witness_bezout_sum = ff_q_pfp_gcd_exists_result_witness_bezout_sum_leftsecondentry * S ((S (pfrep_position_gcd_exists_result_witness_bezout_sum_leftsecond)) * pfga_uc_gcd_exists_result_witness_bezout_sum) + (pfrep_right_gcd_exists_result_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_exists_result_witness_bezout_sum_leftsecondoutside. pfrep_gap_gcd_exists_result_witness_bezout_sum_leftsecondoutside+(pfga_K_gcd_exists_result_witness_bezout_sum)=(pfrep_power_gcd_exists_result_witness_bezout_sum_left)) /\\ (((pfrep_right_gcd_exists_result_witness_bezout_sum_left)=0))))) -> pfrep_left_gcd_exists_result_witness_bezout_sum_left=pfrep_right_gcd_exists_result_witness_bezout_sum_left) /\\ ((forall pfrep_power_gcd_exists_result_witness_bezout_sum_right pfrep_left_gcd_exists_result_witness_bezout_sum_right pfrep_right_gcd_exists_result_witness_bezout_sum_right. ((exists pfrep_position_gcd_exists_result_witness_bezout_sum_rightfirst. ((pfrep_position_gcd_exists_result_witness_bezout_sum_rightfirst+S (pfrep_power_gcd_exists_result_witness_bezout_sum_right)=(pfgb_Q_gcd_exists_result_witness_bezout)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_bezout_sum_rightfirstentry. ff_h_pfp_gcd_exists_result_witness_bezout_sum_rightfirstentry + S (pfrep_left_gcd_exists_result_witness_bezout_sum_right) = S ((S (pfrep_position_gcd_exists_result_witness_bezout_sum_rightfirst)) * pfgb_qc_gcd_exists_result_witness_bezout)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_sum_rightfirstentry. pfgb_qb_gcd_exists_result_witness_bezout = ff_q_pfp_gcd_exists_result_witness_bezout_sum_rightfirstentry * S ((S (pfrep_position_gcd_exists_result_witness_bezout_sum_rightfirst)) * pfgb_qc_gcd_exists_result_witness_bezout) + (pfrep_left_gcd_exists_result_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_exists_result_witness_bezout_sum_rightfirstoutside. pfrep_gap_gcd_exists_result_witness_bezout_sum_rightfirstoutside+(pfgb_Q_gcd_exists_result_witness_bezout)=(pfrep_power_gcd_exists_result_witness_bezout_sum_right)) /\\ (((pfrep_left_gcd_exists_result_witness_bezout_sum_right)=0))))) -> ((exists pfrep_position_gcd_exists_result_witness_bezout_sum_rightsecond. ((pfrep_position_gcd_exists_result_witness_bezout_sum_rightsecond+S (pfrep_power_gcd_exists_result_witness_bezout_sum_right)=(pfga_K_gcd_exists_result_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_bezout_sum_rightsecondentry. ff_h_pfp_gcd_exists_result_witness_bezout_sum_rightsecondentry + S (pfrep_right_gcd_exists_result_witness_bezout_sum_right) = S ((S (pfrep_position_gcd_exists_result_witness_bezout_sum_rightsecond)) * pfga_vc_gcd_exists_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_sum_rightsecondentry. pfga_vb_gcd_exists_result_witness_bezout_sum = ff_q_pfp_gcd_exists_result_witness_bezout_sum_rightsecondentry * S ((S (pfrep_position_gcd_exists_result_witness_bezout_sum_rightsecond)) * pfga_vc_gcd_exists_result_witness_bezout_sum) + (pfrep_right_gcd_exists_result_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_exists_result_witness_bezout_sum_rightsecondoutside. pfrep_gap_gcd_exists_result_witness_bezout_sum_rightsecondoutside+(pfga_K_gcd_exists_result_witness_bezout_sum)=(pfrep_power_gcd_exists_result_witness_bezout_sum_right)) /\\ (((pfrep_right_gcd_exists_result_witness_bezout_sum_right)=0))))) -> pfrep_left_gcd_exists_result_witness_bezout_sum_right=pfrep_right_gcd_exists_result_witness_bezout_sum_right)))) /\\ (((forall pfp_index_gcd_exists_result_witness_bezout_sum_add. (exists pfa_gap_gcd_exists_result_witness_bezout_sum_addindex. pfa_gap_gcd_exists_result_witness_bezout_sum_addindex + S (pfp_index_gcd_exists_result_witness_bezout_sum_add) = (pfga_K_gcd_exists_result_witness_bezout_sum)) -> exists pfp_left_gcd_exists_result_witness_bezout_sum_add pfp_right_gcd_exists_result_witness_bezout_sum_add pfp_value_gcd_exists_result_witness_bezout_sum_add. ((((exists ff_h_pfp_gcd_exists_result_witness_bezout_sum_addleft. ff_h_pfp_gcd_exists_result_witness_bezout_sum_addleft + S (pfp_left_gcd_exists_result_witness_bezout_sum_add) = S ((S (pfp_index_gcd_exists_result_witness_bezout_sum_add)) * pfga_uc_gcd_exists_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_sum_addleft. pfga_ub_gcd_exists_result_witness_bezout_sum = ff_q_pfp_gcd_exists_result_witness_bezout_sum_addleft * S ((S (pfp_index_gcd_exists_result_witness_bezout_sum_add)) * pfga_uc_gcd_exists_result_witness_bezout_sum) + (pfp_left_gcd_exists_result_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_exists_result_witness_bezout_sum_addright. ff_h_pfp_gcd_exists_result_witness_bezout_sum_addright + S (pfp_right_gcd_exists_result_witness_bezout_sum_add) = S ((S (pfp_index_gcd_exists_result_witness_bezout_sum_add)) * pfga_vc_gcd_exists_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_sum_addright. pfga_vb_gcd_exists_result_witness_bezout_sum = ff_q_pfp_gcd_exists_result_witness_bezout_sum_addright * S ((S (pfp_index_gcd_exists_result_witness_bezout_sum_add)) * pfga_vc_gcd_exists_result_witness_bezout_sum) + (pfp_right_gcd_exists_result_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_exists_result_witness_bezout_sum_addtarget. ff_h_pfp_gcd_exists_result_witness_bezout_sum_addtarget + S (pfp_value_gcd_exists_result_witness_bezout_sum_add) = S ((S (pfp_index_gcd_exists_result_witness_bezout_sum_add)) * pfga_tc_gcd_exists_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_sum_addtarget. pfga_tb_gcd_exists_result_witness_bezout_sum = ff_q_pfp_gcd_exists_result_witness_bezout_sum_addtarget * S ((S (pfp_index_gcd_exists_result_witness_bezout_sum_add)) * pfga_tc_gcd_exists_result_witness_bezout_sum) + (pfp_value_gcd_exists_result_witness_bezout_sum_add))) /\\ ((((exists pfa_gap_gcd_exists_result_witness_bezout_sum_addoperationleft. pfa_gap_gcd_exists_result_witness_bezout_sum_addoperationleft + S (pfp_left_gcd_exists_result_witness_bezout_sum_add) = (p)) /\\ (((exists pfa_gap_gcd_exists_result_witness_bezout_sum_addoperationright. pfa_gap_gcd_exists_result_witness_bezout_sum_addoperationright + S (pfp_right_gcd_exists_result_witness_bezout_sum_add) = (p)) /\\ ((((exists pfa_gap_gcd_exists_result_witness_bezout_sum_addoperationresultbound. pfa_gap_gcd_exists_result_witness_bezout_sum_addoperationresultbound + S (pfp_value_gcd_exists_result_witness_bezout_sum_add) = (p)) /\\ ((exists pfa_offset_left_gcd_exists_result_witness_bezout_sum_addoperationresultcongruence pfa_offset_right_gcd_exists_result_witness_bezout_sum_addoperationresultcongruence. ((pfp_left_gcd_exists_result_witness_bezout_sum_add) + (pfp_right_gcd_exists_result_witness_bezout_sum_add)) + (p) * pfa_offset_left_gcd_exists_result_witness_bezout_sum_addoperationresultcongruence = (pfp_value_gcd_exists_result_witness_bezout_sum_add) + (p) * pfa_offset_right_gcd_exists_result_witness_bezout_sum_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_gcd_exists_result_witness_bezout_sum_result pfrep_left_gcd_exists_result_witness_bezout_sum_result pfrep_right_gcd_exists_result_witness_bezout_sum_result. ((exists pfrep_position_gcd_exists_result_witness_bezout_sum_resultfirst. ((pfrep_position_gcd_exists_result_witness_bezout_sum_resultfirst+S (pfrep_power_gcd_exists_result_witness_bezout_sum_result)=(pfga_K_gcd_exists_result_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_bezout_sum_resultfirstentry. ff_h_pfp_gcd_exists_result_witness_bezout_sum_resultfirstentry + S (pfrep_left_gcd_exists_result_witness_bezout_sum_result) = S ((S (pfrep_position_gcd_exists_result_witness_bezout_sum_resultfirst)) * pfga_tc_gcd_exists_result_witness_bezout_sum)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_sum_resultfirstentry. pfga_tb_gcd_exists_result_witness_bezout_sum = ff_q_pfp_gcd_exists_result_witness_bezout_sum_resultfirstentry * S ((S (pfrep_position_gcd_exists_result_witness_bezout_sum_resultfirst)) * pfga_tc_gcd_exists_result_witness_bezout_sum) + (pfrep_left_gcd_exists_result_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_exists_result_witness_bezout_sum_resultfirstoutside. pfrep_gap_gcd_exists_result_witness_bezout_sum_resultfirstoutside+(pfga_K_gcd_exists_result_witness_bezout_sum)=(pfrep_power_gcd_exists_result_witness_bezout_sum_result)) /\\ (((pfrep_left_gcd_exists_result_witness_bezout_sum_result)=0))))) -> ((exists pfrep_position_gcd_exists_result_witness_bezout_sum_resultsecond. ((pfrep_position_gcd_exists_result_witness_bezout_sum_resultsecond+S (pfrep_power_gcd_exists_result_witness_bezout_sum_result)=(pfgs_G_gcd_exists_result)) /\\ ((((exists ff_h_pfp_gcd_exists_result_witness_bezout_sum_resultsecondentry. ff_h_pfp_gcd_exists_result_witness_bezout_sum_resultsecondentry + S (pfrep_right_gcd_exists_result_witness_bezout_sum_result) = S ((S (pfrep_position_gcd_exists_result_witness_bezout_sum_resultsecond)) * pfgs_gc_gcd_exists_result)) /\\ exists ff_q_pfp_gcd_exists_result_witness_bezout_sum_resultsecondentry. pfgs_gb_gcd_exists_result = ff_q_pfp_gcd_exists_result_witness_bezout_sum_resultsecondentry * S ((S (pfrep_position_gcd_exists_result_witness_bezout_sum_resultsecond)) * pfgs_gc_gcd_exists_result) + (pfrep_right_gcd_exists_result_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_exists_result_witness_bezout_sum_resultsecondoutside. pfrep_gap_gcd_exists_result_witness_bezout_sum_resultsecondoutside+(pfgs_G_gcd_exists_result)=(pfrep_power_gcd_exists_result_witness_bezout_sum_result)) /\\ (((pfrep_right_gcd_exists_result_witness_bezout_sum_result)=0))))) -> pfrep_left_gcd_exists_result_witness_bezout_sum_result=pfrep_right_gcd_exists_result_witness_bezout_sum_result)))))))))))))))))))))))", "statement_sha256": "309f81ff6499fef4034c503f33c5cba15c358b9bd849af1800825d638ec8e787", "summary": "Take the actual second representation length as induction bound. 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prime_field_polynomial_bezout_common_right_divisor (V)", "apply prime_field_polynomial_bezout_common_right_divisor", "exact hp", "exact hd", "exact hb" ], "script_sha256": "a10d8a23524bb3ea1ac97d7b98d6736e41dd4f466c2c21988056d01749570731", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_existence_candidate.py", "sha256": "81f2f48dd2e81894c7a267453646eb6f2b6f9bd3ee320386d8c561f6b9f8b8ca" }, "statement": "forall p gb gc G ab ac L bb bc M ub uc U vb vc V. (~((p) = 1) /\\ forall pfa_factor_left_gcd_greatest_prime pfa_factor_right_gcd_greatest_prime. (p) = pfa_factor_left_gcd_greatest_prime * pfa_factor_right_gcd_greatest_prime -> pfa_factor_left_gcd_greatest_prime = 1 \\/ pfa_factor_right_gcd_greatest_prime = 1) -> (((((forall fom_index_pfp_gcd_greatest_common_left_bounded. (exists fom_gap_pfp_gcd_greatest_common_left_bounded_index_bound. fom_gap_pfp_gcd_greatest_common_left_bounded_index_bound + S (fom_index_pfp_gcd_greatest_common_left_bounded) = L) -> exists fom_value_pfp_gcd_greatest_common_left_bounded. ((((exists fom_beta_height_pfp_gcd_greatest_common_left_bounded_entry. fom_beta_height_pfp_gcd_greatest_common_left_bounded_entry + S (fom_value_pfp_gcd_greatest_common_left_bounded) = S ((S (fom_index_pfp_gcd_greatest_common_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_common_left_bounded_entry. ab = fom_beta_quotient_pfp_gcd_greatest_common_left_bounded_entry * S ((S (fom_index_pfp_gcd_greatest_common_left_bounded)) * ac) + (fom_value_pfp_gcd_greatest_common_left_bounded))) /\\ (exists fom_gap_pfp_gcd_greatest_common_left_bounded_value_bound. fom_gap_pfp_gcd_greatest_common_left_bounded_value_bound + S (fom_value_pfp_gcd_greatest_common_left_bounded) = p))) /\\ ((exists pfgd_qb_gcd_greatest_common_left pfgd_qc_gcd_greatest_common_left pfgd_Q_gcd_greatest_common_left pfgd_pb_gcd_greatest_common_left pfgd_pc_gcd_greatest_common_left pfgd_P_gcd_greatest_common_left. ((((forall fom_index_pfp_gcd_greatest_common_left_productleft. (exists fom_gap_pfp_gcd_greatest_common_left_productleft_index_bound. fom_gap_pfp_gcd_greatest_common_left_productleft_index_bound + S (fom_index_pfp_gcd_greatest_common_left_productleft) = pfgd_Q_gcd_greatest_common_left) -> exists fom_value_pfp_gcd_greatest_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_greatest_common_left_productleft_entry. fom_beta_height_pfp_gcd_greatest_common_left_productleft_entry + S (fom_value_pfp_gcd_greatest_common_left_productleft) = S ((S (fom_index_pfp_gcd_greatest_common_left_productleft)) * pfgd_qc_gcd_greatest_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_common_left_productleft_entry. pfgd_qb_gcd_greatest_common_left = fom_beta_quotient_pfp_gcd_greatest_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_greatest_common_left_productleft)) * pfgd_qc_gcd_greatest_common_left) + (fom_value_pfp_gcd_greatest_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_greatest_common_left_productleft_value_bound. fom_gap_pfp_gcd_greatest_common_left_productleft_value_bound + S (fom_value_pfp_gcd_greatest_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_greatest_common_left_productright. (exists fom_gap_pfp_gcd_greatest_common_left_productright_index_bound. fom_gap_pfp_gcd_greatest_common_left_productright_index_bound + S (fom_index_pfp_gcd_greatest_common_left_productright) = G) -> exists fom_value_pfp_gcd_greatest_common_left_productright. ((((exists fom_beta_height_pfp_gcd_greatest_common_left_productright_entry. fom_beta_height_pfp_gcd_greatest_common_left_productright_entry + S (fom_value_pfp_gcd_greatest_common_left_productright) = S ((S (fom_index_pfp_gcd_greatest_common_left_productright)) * gc)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_common_left_productright_entry. gb = fom_beta_quotient_pfp_gcd_greatest_common_left_productright_entry * S ((S (fom_index_pfp_gcd_greatest_common_left_productright)) * gc) + (fom_value_pfp_gcd_greatest_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_greatest_common_left_productright_value_bound. fom_gap_pfp_gcd_greatest_common_left_productright_value_bound + S (fom_value_pfp_gcd_greatest_common_left_productright) = p))) /\\ (((((((pfgd_Q_gcd_greatest_common_left)=0 \\/ (G)=0) /\\ (((pfgd_P_gcd_greatest_common_left)=0)))) \\/ (((~((pfgd_Q_gcd_greatest_common_left)=0)) /\\ (((~((G)=0)) /\\ (((pfgd_Q_gcd_greatest_common_left)+(G)=S (pfgd_P_gcd_greatest_common_left)))))))) /\\ ((forall pfc_index_gcd_greatest_common_left_productcoefficients. (exists pfa_gap_gcd_greatest_common_left_productcoefficientsbound. pfa_gap_gcd_greatest_common_left_productcoefficientsbound + S (pfc_index_gcd_greatest_common_left_productcoefficients) = (pfgd_P_gcd_greatest_common_left)) -> exists pfc_value_gcd_greatest_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_greatest_common_left_productcoefficientsentry. ff_h_pfp_gcd_greatest_common_left_productcoefficientsentry + S (pfc_value_gcd_greatest_common_left_productcoefficients) = S ((S (pfc_index_gcd_greatest_common_left_productcoefficients)) * pfgd_pc_gcd_greatest_common_left)) /\\ exists ff_q_pfp_gcd_greatest_common_left_productcoefficientsentry. pfgd_pb_gcd_greatest_common_left = ff_q_pfp_gcd_greatest_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_greatest_common_left_productcoefficients)) * pfgd_pc_gcd_greatest_common_left) + (pfc_value_gcd_greatest_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_greatest_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_greatest_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_greatest_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_greatest_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_greatest_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_greatest_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_greatest_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_greatest_common_left_productcoefficients))) -> exists pfc_value_gcd_greatest_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_greatest_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_greatest_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_greatest_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_greatest_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_greatest_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_greatest_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_greatest_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_greatest_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_common_left_productcoefficientscoefficient) + (pfc_value_gcd_greatest_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_greatest_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_greatest_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_greatest_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_greatest_common_left)) /\\ ((((exists ff_h_pfp_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_greatest_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_greatest_common_left)) /\\ exists ff_q_pfp_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_greatest_common_left = ff_q_pfp_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_greatest_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_greatest_common_left) + (pfc_left_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_greatest_common_left)=(pfc_index_gcd_greatest_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm) = (G)) /\\ ((((exists ff_h_pfp_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm)) * gc)) /\\ exists ff_q_pfp_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermrightentry. gb = ff_q_pfp_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm)) * gc) + (pfc_right_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_greatest_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(G)=(pfc_complement_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_greatest_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_greatest_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_greatest_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_greatest_common_left_productcoefficients))) * fs_v_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_greatest_common_left_productcoefficients))) * fs_v_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_greatest_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_greatest_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_greatest_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_greatest_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_greatest_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_greatest_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_greatest_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_greatest_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_greatest_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_greatest_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_greatest_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_greatest_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_greatest_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_greatest_common_left_equivalent pfrep_left_gcd_greatest_common_left_equivalent pfrep_right_gcd_greatest_common_left_equivalent. ((exists pfrep_position_gcd_greatest_common_left_equivalentfirst. ((pfrep_position_gcd_greatest_common_left_equivalentfirst+S (pfrep_power_gcd_greatest_common_left_equivalent)=(pfgd_P_gcd_greatest_common_left)) /\\ ((((exists ff_h_pfp_gcd_greatest_common_left_equivalentfirstentry. ff_h_pfp_gcd_greatest_common_left_equivalentfirstentry + S (pfrep_left_gcd_greatest_common_left_equivalent) = S ((S (pfrep_position_gcd_greatest_common_left_equivalentfirst)) * pfgd_pc_gcd_greatest_common_left)) /\\ exists ff_q_pfp_gcd_greatest_common_left_equivalentfirstentry. pfgd_pb_gcd_greatest_common_left = ff_q_pfp_gcd_greatest_common_left_equivalentfirstentry * S ((S (pfrep_position_gcd_greatest_common_left_equivalentfirst)) * pfgd_pc_gcd_greatest_common_left) + (pfrep_left_gcd_greatest_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_greatest_common_left_equivalentfirstoutside. pfrep_gap_gcd_greatest_common_left_equivalentfirstoutside+(pfgd_P_gcd_greatest_common_left)=(pfrep_power_gcd_greatest_common_left_equivalent)) /\\ (((pfrep_left_gcd_greatest_common_left_equivalent)=0))))) -> ((exists pfrep_position_gcd_greatest_common_left_equivalentsecond. ((pfrep_position_gcd_greatest_common_left_equivalentsecond+S (pfrep_power_gcd_greatest_common_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_gcd_greatest_common_left_equivalentsecondentry. ff_h_pfp_gcd_greatest_common_left_equivalentsecondentry + S (pfrep_right_gcd_greatest_common_left_equivalent) = S ((S (pfrep_position_gcd_greatest_common_left_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_gcd_greatest_common_left_equivalentsecondentry. ab = ff_q_pfp_gcd_greatest_common_left_equivalentsecondentry * S ((S (pfrep_position_gcd_greatest_common_left_equivalentsecond)) * ac) + (pfrep_right_gcd_greatest_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_greatest_common_left_equivalentsecondoutside. pfrep_gap_gcd_greatest_common_left_equivalentsecondoutside+(L)=(pfrep_power_gcd_greatest_common_left_equivalent)) /\\ (((pfrep_right_gcd_greatest_common_left_equivalent)=0))))) -> pfrep_left_gcd_greatest_common_left_equivalent=pfrep_right_gcd_greatest_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_greatest_common_right_bounded. (exists fom_gap_pfp_gcd_greatest_common_right_bounded_index_bound. fom_gap_pfp_gcd_greatest_common_right_bounded_index_bound + S (fom_index_pfp_gcd_greatest_common_right_bounded) = M) -> exists fom_value_pfp_gcd_greatest_common_right_bounded. ((((exists fom_beta_height_pfp_gcd_greatest_common_right_bounded_entry. fom_beta_height_pfp_gcd_greatest_common_right_bounded_entry + S (fom_value_pfp_gcd_greatest_common_right_bounded) = S ((S (fom_index_pfp_gcd_greatest_common_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_common_right_bounded_entry. bb = fom_beta_quotient_pfp_gcd_greatest_common_right_bounded_entry * S ((S (fom_index_pfp_gcd_greatest_common_right_bounded)) * bc) + (fom_value_pfp_gcd_greatest_common_right_bounded))) /\\ (exists fom_gap_pfp_gcd_greatest_common_right_bounded_value_bound. fom_gap_pfp_gcd_greatest_common_right_bounded_value_bound + S (fom_value_pfp_gcd_greatest_common_right_bounded) = p))) /\\ ((exists pfgd_qb_gcd_greatest_common_right pfgd_qc_gcd_greatest_common_right pfgd_Q_gcd_greatest_common_right pfgd_pb_gcd_greatest_common_right pfgd_pc_gcd_greatest_common_right pfgd_P_gcd_greatest_common_right. ((((forall fom_index_pfp_gcd_greatest_common_right_productleft. (exists fom_gap_pfp_gcd_greatest_common_right_productleft_index_bound. fom_gap_pfp_gcd_greatest_common_right_productleft_index_bound + S (fom_index_pfp_gcd_greatest_common_right_productleft) = pfgd_Q_gcd_greatest_common_right) -> exists fom_value_pfp_gcd_greatest_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_greatest_common_right_productleft_entry. fom_beta_height_pfp_gcd_greatest_common_right_productleft_entry + S (fom_value_pfp_gcd_greatest_common_right_productleft) = S ((S (fom_index_pfp_gcd_greatest_common_right_productleft)) * pfgd_qc_gcd_greatest_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_common_right_productleft_entry. pfgd_qb_gcd_greatest_common_right = fom_beta_quotient_pfp_gcd_greatest_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_greatest_common_right_productleft)) * pfgd_qc_gcd_greatest_common_right) + (fom_value_pfp_gcd_greatest_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_greatest_common_right_productleft_value_bound. fom_gap_pfp_gcd_greatest_common_right_productleft_value_bound + S (fom_value_pfp_gcd_greatest_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_greatest_common_right_productright. (exists fom_gap_pfp_gcd_greatest_common_right_productright_index_bound. fom_gap_pfp_gcd_greatest_common_right_productright_index_bound + S (fom_index_pfp_gcd_greatest_common_right_productright) = G) -> exists fom_value_pfp_gcd_greatest_common_right_productright. ((((exists fom_beta_height_pfp_gcd_greatest_common_right_productright_entry. fom_beta_height_pfp_gcd_greatest_common_right_productright_entry + S (fom_value_pfp_gcd_greatest_common_right_productright) = S ((S (fom_index_pfp_gcd_greatest_common_right_productright)) * gc)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_common_right_productright_entry. gb = fom_beta_quotient_pfp_gcd_greatest_common_right_productright_entry * S ((S (fom_index_pfp_gcd_greatest_common_right_productright)) * gc) + (fom_value_pfp_gcd_greatest_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_greatest_common_right_productright_value_bound. fom_gap_pfp_gcd_greatest_common_right_productright_value_bound + S (fom_value_pfp_gcd_greatest_common_right_productright) = p))) /\\ (((((((pfgd_Q_gcd_greatest_common_right)=0 \\/ (G)=0) /\\ (((pfgd_P_gcd_greatest_common_right)=0)))) \\/ (((~((pfgd_Q_gcd_greatest_common_right)=0)) /\\ (((~((G)=0)) /\\ (((pfgd_Q_gcd_greatest_common_right)+(G)=S (pfgd_P_gcd_greatest_common_right)))))))) /\\ ((forall pfc_index_gcd_greatest_common_right_productcoefficients. (exists pfa_gap_gcd_greatest_common_right_productcoefficientsbound. pfa_gap_gcd_greatest_common_right_productcoefficientsbound + S (pfc_index_gcd_greatest_common_right_productcoefficients) = (pfgd_P_gcd_greatest_common_right)) -> exists pfc_value_gcd_greatest_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_greatest_common_right_productcoefficientsentry. ff_h_pfp_gcd_greatest_common_right_productcoefficientsentry + S (pfc_value_gcd_greatest_common_right_productcoefficients) = S ((S (pfc_index_gcd_greatest_common_right_productcoefficients)) * pfgd_pc_gcd_greatest_common_right)) /\\ exists ff_q_pfp_gcd_greatest_common_right_productcoefficientsentry. pfgd_pb_gcd_greatest_common_right = ff_q_pfp_gcd_greatest_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_greatest_common_right_productcoefficients)) * pfgd_pc_gcd_greatest_common_right) + (pfc_value_gcd_greatest_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_greatest_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_greatest_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_greatest_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_greatest_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_greatest_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_greatest_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_greatest_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_greatest_common_right_productcoefficients))) -> exists pfc_value_gcd_greatest_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_greatest_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_greatest_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_greatest_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_greatest_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_greatest_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_greatest_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_greatest_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_greatest_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_common_right_productcoefficientscoefficient) + (pfc_value_gcd_greatest_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_greatest_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_greatest_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_greatest_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_greatest_common_right)) /\\ ((((exists ff_h_pfp_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_greatest_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_greatest_common_right)) /\\ exists ff_q_pfp_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_greatest_common_right = ff_q_pfp_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_greatest_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_greatest_common_right) + (pfc_left_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_greatest_common_right)=(pfc_index_gcd_greatest_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm) = (G)) /\\ ((((exists ff_h_pfp_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm)) * gc)) /\\ exists ff_q_pfp_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermrightentry. gb = ff_q_pfp_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm)) * gc) + (pfc_right_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_greatest_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(G)=(pfc_complement_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_greatest_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_greatest_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_greatest_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_greatest_common_right_productcoefficients))) * fs_v_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_greatest_common_right_productcoefficients))) * fs_v_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_greatest_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_greatest_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_greatest_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_greatest_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_greatest_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_greatest_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_greatest_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_greatest_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_greatest_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_greatest_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_greatest_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_greatest_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_greatest_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_greatest_common_right_equivalent pfrep_left_gcd_greatest_common_right_equivalent pfrep_right_gcd_greatest_common_right_equivalent. ((exists pfrep_position_gcd_greatest_common_right_equivalentfirst. ((pfrep_position_gcd_greatest_common_right_equivalentfirst+S (pfrep_power_gcd_greatest_common_right_equivalent)=(pfgd_P_gcd_greatest_common_right)) /\\ ((((exists ff_h_pfp_gcd_greatest_common_right_equivalentfirstentry. ff_h_pfp_gcd_greatest_common_right_equivalentfirstentry + S (pfrep_left_gcd_greatest_common_right_equivalent) = S ((S (pfrep_position_gcd_greatest_common_right_equivalentfirst)) * pfgd_pc_gcd_greatest_common_right)) /\\ exists ff_q_pfp_gcd_greatest_common_right_equivalentfirstentry. pfgd_pb_gcd_greatest_common_right = ff_q_pfp_gcd_greatest_common_right_equivalentfirstentry * S ((S (pfrep_position_gcd_greatest_common_right_equivalentfirst)) * pfgd_pc_gcd_greatest_common_right) + (pfrep_left_gcd_greatest_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_greatest_common_right_equivalentfirstoutside. pfrep_gap_gcd_greatest_common_right_equivalentfirstoutside+(pfgd_P_gcd_greatest_common_right)=(pfrep_power_gcd_greatest_common_right_equivalent)) /\\ (((pfrep_left_gcd_greatest_common_right_equivalent)=0))))) -> ((exists pfrep_position_gcd_greatest_common_right_equivalentsecond. ((pfrep_position_gcd_greatest_common_right_equivalentsecond+S (pfrep_power_gcd_greatest_common_right_equivalent)=(M)) /\\ ((((exists ff_h_pfp_gcd_greatest_common_right_equivalentsecondentry. ff_h_pfp_gcd_greatest_common_right_equivalentsecondentry + S (pfrep_right_gcd_greatest_common_right_equivalent) = S ((S (pfrep_position_gcd_greatest_common_right_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_gcd_greatest_common_right_equivalentsecondentry. bb = ff_q_pfp_gcd_greatest_common_right_equivalentsecondentry * S ((S (pfrep_position_gcd_greatest_common_right_equivalentsecond)) * bc) + (pfrep_right_gcd_greatest_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_greatest_common_right_equivalentsecondoutside. pfrep_gap_gcd_greatest_common_right_equivalentsecondoutside+(M)=(pfrep_power_gcd_greatest_common_right_equivalent)) /\\ (((pfrep_right_gcd_greatest_common_right_equivalent)=0))))) -> pfrep_left_gcd_greatest_common_right_equivalent=pfrep_right_gcd_greatest_common_right_equivalent)))))))))) -> (exists pfgb_pb_gcd_greatest_bezout pfgb_pc_gcd_greatest_bezout pfgb_P_gcd_greatest_bezout pfgb_qb_gcd_greatest_bezout pfgb_qc_gcd_greatest_bezout pfgb_Q_gcd_greatest_bezout. ((((forall fom_index_pfp_gcd_greatest_bezout_leftleft. (exists fom_gap_pfp_gcd_greatest_bezout_leftleft_index_bound. fom_gap_pfp_gcd_greatest_bezout_leftleft_index_bound + S (fom_index_pfp_gcd_greatest_bezout_leftleft) = U) -> exists fom_value_pfp_gcd_greatest_bezout_leftleft. ((((exists fom_beta_height_pfp_gcd_greatest_bezout_leftleft_entry. fom_beta_height_pfp_gcd_greatest_bezout_leftleft_entry + S (fom_value_pfp_gcd_greatest_bezout_leftleft) = S ((S (fom_index_pfp_gcd_greatest_bezout_leftleft)) * uc)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_bezout_leftleft_entry. ub = fom_beta_quotient_pfp_gcd_greatest_bezout_leftleft_entry * S ((S (fom_index_pfp_gcd_greatest_bezout_leftleft)) * uc) + (fom_value_pfp_gcd_greatest_bezout_leftleft))) /\\ (exists fom_gap_pfp_gcd_greatest_bezout_leftleft_value_bound. fom_gap_pfp_gcd_greatest_bezout_leftleft_value_bound + S (fom_value_pfp_gcd_greatest_bezout_leftleft) = p))) /\\ (((forall fom_index_pfp_gcd_greatest_bezout_leftright. (exists fom_gap_pfp_gcd_greatest_bezout_leftright_index_bound. fom_gap_pfp_gcd_greatest_bezout_leftright_index_bound + S (fom_index_pfp_gcd_greatest_bezout_leftright) = L) -> exists fom_value_pfp_gcd_greatest_bezout_leftright. ((((exists fom_beta_height_pfp_gcd_greatest_bezout_leftright_entry. fom_beta_height_pfp_gcd_greatest_bezout_leftright_entry + S (fom_value_pfp_gcd_greatest_bezout_leftright) = S ((S (fom_index_pfp_gcd_greatest_bezout_leftright)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_bezout_leftright_entry. ab = fom_beta_quotient_pfp_gcd_greatest_bezout_leftright_entry * S ((S (fom_index_pfp_gcd_greatest_bezout_leftright)) * ac) + (fom_value_pfp_gcd_greatest_bezout_leftright))) /\\ (exists fom_gap_pfp_gcd_greatest_bezout_leftright_value_bound. fom_gap_pfp_gcd_greatest_bezout_leftright_value_bound + S (fom_value_pfp_gcd_greatest_bezout_leftright) = p))) /\\ (((((((U)=0 \\/ (L)=0) /\\ (((pfgb_P_gcd_greatest_bezout)=0)))) \\/ (((~((U)=0)) /\\ (((~((L)=0)) /\\ (((U)+(L)=S (pfgb_P_gcd_greatest_bezout)))))))) /\\ ((forall pfc_index_gcd_greatest_bezout_leftcoefficients. (exists pfa_gap_gcd_greatest_bezout_leftcoefficientsbound. pfa_gap_gcd_greatest_bezout_leftcoefficientsbound + S (pfc_index_gcd_greatest_bezout_leftcoefficients) = (pfgb_P_gcd_greatest_bezout)) -> exists pfc_value_gcd_greatest_bezout_leftcoefficients. ((((exists ff_h_pfp_gcd_greatest_bezout_leftcoefficientsentry. ff_h_pfp_gcd_greatest_bezout_leftcoefficientsentry + S (pfc_value_gcd_greatest_bezout_leftcoefficients) = S ((S (pfc_index_gcd_greatest_bezout_leftcoefficients)) * pfgb_pc_gcd_greatest_bezout)) /\\ exists ff_q_pfp_gcd_greatest_bezout_leftcoefficientsentry. pfgb_pb_gcd_greatest_bezout = ff_q_pfp_gcd_greatest_bezout_leftcoefficientsentry * S ((S (pfc_index_gcd_greatest_bezout_leftcoefficients)) * pfgb_pc_gcd_greatest_bezout) + (pfc_value_gcd_greatest_bezout_leftcoefficients))) /\\ ((exists pfc_terms_code_gcd_greatest_bezout_leftcoefficientscoefficient pfc_terms_scale_gcd_greatest_bezout_leftcoefficientscoefficient pfc_natural_sum_gcd_greatest_bezout_leftcoefficientscoefficient. ((forall pfc_index_gcd_greatest_bezout_leftcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalbound. pfa_gap_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_greatest_bezout_leftcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_greatest_bezout_leftcoefficients))) -> exists pfc_value_gcd_greatest_bezout_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_greatest_bezout_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_greatest_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_bezout_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_greatest_bezout_leftcoefficientscoefficient = ff_q_pfp_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_greatest_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_bezout_leftcoefficientscoefficient) + (pfc_value_gcd_greatest_bezout_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm pfc_left_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm pfc_right_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_greatest_bezout_leftcoefficientscoefficientdiagonal)+pfc_complement_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm=(pfc_index_gcd_greatest_bezout_leftcoefficients)) /\\ ((((((exists pfa_gap_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_greatest_bezout_leftcoefficientscoefficientdiagonal) = (U)) /\\ ((((exists ff_h_pfp_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_greatest_bezout_leftcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_greatest_bezout_leftcoefficientscoefficientdiagonal)) * uc) + (pfc_left_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermleftoutside+(U)=(pfc_index_gcd_greatest_bezout_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_greatest_bezout_leftcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_greatest_bezout_leftcoefficientscoefficientdiagonal)=pfc_left_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm*pfc_right_gcd_greatest_bezout_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum fs_v_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_greatest_bezout_leftcoefficientscoefficient) = S ((S (S (pfc_index_gcd_greatest_bezout_leftcoefficients))) * fs_v_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_greatest_bezout_leftcoefficients))) * fs_v_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum) + (pfc_natural_sum_gcd_greatest_bezout_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_greatest_bezout_leftcoefficients)) -> exists fs_a_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_bezout_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_greatest_bezout_leftcoefficientscoefficient = fs_q_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_bezout_leftcoefficientscoefficient) + (fs_a_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum) + (fs_r_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum) + (fs_s_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_greatest_bezout_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_greatest_bezout_leftcoefficientscoefficientresiduebound. pfa_gap_gcd_greatest_bezout_leftcoefficientscoefficientresiduebound + S (pfc_value_gcd_greatest_bezout_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_greatest_bezout_leftcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_greatest_bezout_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_greatest_bezout_leftcoefficientscoefficient) + (p) * pfa_offset_left_gcd_greatest_bezout_leftcoefficientscoefficientresiduecongruence = (pfc_value_gcd_greatest_bezout_leftcoefficients) + (p) * pfa_offset_right_gcd_greatest_bezout_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_gcd_greatest_bezout_rightleft. (exists fom_gap_pfp_gcd_greatest_bezout_rightleft_index_bound. fom_gap_pfp_gcd_greatest_bezout_rightleft_index_bound + S (fom_index_pfp_gcd_greatest_bezout_rightleft) = V) -> exists fom_value_pfp_gcd_greatest_bezout_rightleft. ((((exists fom_beta_height_pfp_gcd_greatest_bezout_rightleft_entry. fom_beta_height_pfp_gcd_greatest_bezout_rightleft_entry + S (fom_value_pfp_gcd_greatest_bezout_rightleft) = S ((S (fom_index_pfp_gcd_greatest_bezout_rightleft)) * vc)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_bezout_rightleft_entry. vb = fom_beta_quotient_pfp_gcd_greatest_bezout_rightleft_entry * S ((S (fom_index_pfp_gcd_greatest_bezout_rightleft)) * vc) + (fom_value_pfp_gcd_greatest_bezout_rightleft))) /\\ (exists fom_gap_pfp_gcd_greatest_bezout_rightleft_value_bound. fom_gap_pfp_gcd_greatest_bezout_rightleft_value_bound + S (fom_value_pfp_gcd_greatest_bezout_rightleft) = p))) /\\ (((forall fom_index_pfp_gcd_greatest_bezout_rightright. (exists fom_gap_pfp_gcd_greatest_bezout_rightright_index_bound. fom_gap_pfp_gcd_greatest_bezout_rightright_index_bound + S (fom_index_pfp_gcd_greatest_bezout_rightright) = M) -> exists fom_value_pfp_gcd_greatest_bezout_rightright. ((((exists fom_beta_height_pfp_gcd_greatest_bezout_rightright_entry. fom_beta_height_pfp_gcd_greatest_bezout_rightright_entry + S (fom_value_pfp_gcd_greatest_bezout_rightright) = S ((S (fom_index_pfp_gcd_greatest_bezout_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_bezout_rightright_entry. bb = fom_beta_quotient_pfp_gcd_greatest_bezout_rightright_entry * S ((S (fom_index_pfp_gcd_greatest_bezout_rightright)) * bc) + (fom_value_pfp_gcd_greatest_bezout_rightright))) /\\ (exists fom_gap_pfp_gcd_greatest_bezout_rightright_value_bound. fom_gap_pfp_gcd_greatest_bezout_rightright_value_bound + S (fom_value_pfp_gcd_greatest_bezout_rightright) = p))) /\\ (((((((V)=0 \\/ (M)=0) /\\ (((pfgb_Q_gcd_greatest_bezout)=0)))) \\/ (((~((V)=0)) /\\ (((~((M)=0)) /\\ (((V)+(M)=S (pfgb_Q_gcd_greatest_bezout)))))))) /\\ ((forall pfc_index_gcd_greatest_bezout_rightcoefficients. (exists pfa_gap_gcd_greatest_bezout_rightcoefficientsbound. pfa_gap_gcd_greatest_bezout_rightcoefficientsbound + S (pfc_index_gcd_greatest_bezout_rightcoefficients) = (pfgb_Q_gcd_greatest_bezout)) -> exists pfc_value_gcd_greatest_bezout_rightcoefficients. ((((exists ff_h_pfp_gcd_greatest_bezout_rightcoefficientsentry. ff_h_pfp_gcd_greatest_bezout_rightcoefficientsentry + S (pfc_value_gcd_greatest_bezout_rightcoefficients) = S ((S (pfc_index_gcd_greatest_bezout_rightcoefficients)) * pfgb_qc_gcd_greatest_bezout)) /\\ exists ff_q_pfp_gcd_greatest_bezout_rightcoefficientsentry. pfgb_qb_gcd_greatest_bezout = ff_q_pfp_gcd_greatest_bezout_rightcoefficientsentry * S ((S (pfc_index_gcd_greatest_bezout_rightcoefficients)) * pfgb_qc_gcd_greatest_bezout) + (pfc_value_gcd_greatest_bezout_rightcoefficients))) /\\ ((exists pfc_terms_code_gcd_greatest_bezout_rightcoefficientscoefficient pfc_terms_scale_gcd_greatest_bezout_rightcoefficientscoefficient pfc_natural_sum_gcd_greatest_bezout_rightcoefficientscoefficient. ((forall pfc_index_gcd_greatest_bezout_rightcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalbound. pfa_gap_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_greatest_bezout_rightcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_greatest_bezout_rightcoefficients))) -> exists pfc_value_gcd_greatest_bezout_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_greatest_bezout_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_greatest_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_bezout_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_greatest_bezout_rightcoefficientscoefficient = ff_q_pfp_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_greatest_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_bezout_rightcoefficientscoefficient) + (pfc_value_gcd_greatest_bezout_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm pfc_left_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm pfc_right_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_greatest_bezout_rightcoefficientscoefficientdiagonal)+pfc_complement_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm=(pfc_index_gcd_greatest_bezout_rightcoefficients)) /\\ ((((((exists pfa_gap_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_greatest_bezout_rightcoefficientscoefficientdiagonal) = (V)) /\\ ((((exists ff_h_pfp_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_greatest_bezout_rightcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_greatest_bezout_rightcoefficientscoefficientdiagonal)) * vc) + (pfc_left_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermleftoutside+(V)=(pfc_index_gcd_greatest_bezout_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_greatest_bezout_rightcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_greatest_bezout_rightcoefficientscoefficientdiagonal)=pfc_left_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm*pfc_right_gcd_greatest_bezout_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum fs_v_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_greatest_bezout_rightcoefficientscoefficient) = S ((S (S (pfc_index_gcd_greatest_bezout_rightcoefficients))) * fs_v_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_greatest_bezout_rightcoefficients))) * fs_v_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum) + (pfc_natural_sum_gcd_greatest_bezout_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_greatest_bezout_rightcoefficients)) -> exists fs_a_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_bezout_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_greatest_bezout_rightcoefficientscoefficient = fs_q_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_bezout_rightcoefficientscoefficient) + (fs_a_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum) + (fs_r_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum) + (fs_s_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_greatest_bezout_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_greatest_bezout_rightcoefficientscoefficientresiduebound. pfa_gap_gcd_greatest_bezout_rightcoefficientscoefficientresiduebound + S (pfc_value_gcd_greatest_bezout_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_greatest_bezout_rightcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_greatest_bezout_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_greatest_bezout_rightcoefficientscoefficient) + (p) * pfa_offset_left_gcd_greatest_bezout_rightcoefficientscoefficientresiduecongruence = (pfc_value_gcd_greatest_bezout_rightcoefficients) + (p) * pfa_offset_right_gcd_greatest_bezout_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_gcd_greatest_bezout_sum_left_bounded. (exists fom_gap_pfp_gcd_greatest_bezout_sum_left_bounded_index_bound. fom_gap_pfp_gcd_greatest_bezout_sum_left_bounded_index_bound + S (fom_index_pfp_gcd_greatest_bezout_sum_left_bounded) = pfgb_P_gcd_greatest_bezout) -> exists fom_value_pfp_gcd_greatest_bezout_sum_left_bounded. ((((exists fom_beta_height_pfp_gcd_greatest_bezout_sum_left_bounded_entry. fom_beta_height_pfp_gcd_greatest_bezout_sum_left_bounded_entry + S (fom_value_pfp_gcd_greatest_bezout_sum_left_bounded) = S ((S (fom_index_pfp_gcd_greatest_bezout_sum_left_bounded)) * pfgb_pc_gcd_greatest_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_bezout_sum_left_bounded_entry. pfgb_pb_gcd_greatest_bezout = fom_beta_quotient_pfp_gcd_greatest_bezout_sum_left_bounded_entry * S ((S (fom_index_pfp_gcd_greatest_bezout_sum_left_bounded)) * pfgb_pc_gcd_greatest_bezout) + (fom_value_pfp_gcd_greatest_bezout_sum_left_bounded))) /\\ (exists fom_gap_pfp_gcd_greatest_bezout_sum_left_bounded_value_bound. fom_gap_pfp_gcd_greatest_bezout_sum_left_bounded_value_bound + S (fom_value_pfp_gcd_greatest_bezout_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_greatest_bezout_sum_right_bounded. (exists fom_gap_pfp_gcd_greatest_bezout_sum_right_bounded_index_bound. fom_gap_pfp_gcd_greatest_bezout_sum_right_bounded_index_bound + S (fom_index_pfp_gcd_greatest_bezout_sum_right_bounded) = pfgb_Q_gcd_greatest_bezout) -> exists fom_value_pfp_gcd_greatest_bezout_sum_right_bounded. ((((exists fom_beta_height_pfp_gcd_greatest_bezout_sum_right_bounded_entry. fom_beta_height_pfp_gcd_greatest_bezout_sum_right_bounded_entry + S (fom_value_pfp_gcd_greatest_bezout_sum_right_bounded) = S ((S (fom_index_pfp_gcd_greatest_bezout_sum_right_bounded)) * pfgb_qc_gcd_greatest_bezout)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_bezout_sum_right_bounded_entry. pfgb_qb_gcd_greatest_bezout = fom_beta_quotient_pfp_gcd_greatest_bezout_sum_right_bounded_entry * S ((S (fom_index_pfp_gcd_greatest_bezout_sum_right_bounded)) * pfgb_qc_gcd_greatest_bezout) + (fom_value_pfp_gcd_greatest_bezout_sum_right_bounded))) /\\ (exists fom_gap_pfp_gcd_greatest_bezout_sum_right_bounded_value_bound. fom_gap_pfp_gcd_greatest_bezout_sum_right_bounded_value_bound + S (fom_value_pfp_gcd_greatest_bezout_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_gcd_greatest_bezout_sum_result_bounded. (exists fom_gap_pfp_gcd_greatest_bezout_sum_result_bounded_index_bound. fom_gap_pfp_gcd_greatest_bezout_sum_result_bounded_index_bound + S (fom_index_pfp_gcd_greatest_bezout_sum_result_bounded) = G) -> exists fom_value_pfp_gcd_greatest_bezout_sum_result_bounded. ((((exists fom_beta_height_pfp_gcd_greatest_bezout_sum_result_bounded_entry. fom_beta_height_pfp_gcd_greatest_bezout_sum_result_bounded_entry + S (fom_value_pfp_gcd_greatest_bezout_sum_result_bounded) = S ((S (fom_index_pfp_gcd_greatest_bezout_sum_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_bezout_sum_result_bounded_entry. gb = fom_beta_quotient_pfp_gcd_greatest_bezout_sum_result_bounded_entry * S ((S (fom_index_pfp_gcd_greatest_bezout_sum_result_bounded)) * gc) + (fom_value_pfp_gcd_greatest_bezout_sum_result_bounded))) /\\ (exists fom_gap_pfp_gcd_greatest_bezout_sum_result_bounded_value_bound. fom_gap_pfp_gcd_greatest_bezout_sum_result_bounded_value_bound + S (fom_value_pfp_gcd_greatest_bezout_sum_result_bounded) = p))) /\\ ((exists pfga_ub_gcd_greatest_bezout_sum pfga_uc_gcd_greatest_bezout_sum pfga_vb_gcd_greatest_bezout_sum pfga_vc_gcd_greatest_bezout_sum pfga_tb_gcd_greatest_bezout_sum pfga_tc_gcd_greatest_bezout_sum pfga_K_gcd_greatest_bezout_sum. ((((forall pfrep_power_gcd_greatest_bezout_sum_left pfrep_left_gcd_greatest_bezout_sum_left pfrep_right_gcd_greatest_bezout_sum_left. ((exists pfrep_position_gcd_greatest_bezout_sum_leftfirst. ((pfrep_position_gcd_greatest_bezout_sum_leftfirst+S (pfrep_power_gcd_greatest_bezout_sum_left)=(pfgb_P_gcd_greatest_bezout)) /\\ ((((exists ff_h_pfp_gcd_greatest_bezout_sum_leftfirstentry. ff_h_pfp_gcd_greatest_bezout_sum_leftfirstentry + S (pfrep_left_gcd_greatest_bezout_sum_left) = S ((S (pfrep_position_gcd_greatest_bezout_sum_leftfirst)) * pfgb_pc_gcd_greatest_bezout)) /\\ exists ff_q_pfp_gcd_greatest_bezout_sum_leftfirstentry. pfgb_pb_gcd_greatest_bezout = ff_q_pfp_gcd_greatest_bezout_sum_leftfirstentry * S ((S (pfrep_position_gcd_greatest_bezout_sum_leftfirst)) * pfgb_pc_gcd_greatest_bezout) + (pfrep_left_gcd_greatest_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_greatest_bezout_sum_leftfirstoutside. pfrep_gap_gcd_greatest_bezout_sum_leftfirstoutside+(pfgb_P_gcd_greatest_bezout)=(pfrep_power_gcd_greatest_bezout_sum_left)) /\\ (((pfrep_left_gcd_greatest_bezout_sum_left)=0))))) -> ((exists pfrep_position_gcd_greatest_bezout_sum_leftsecond. ((pfrep_position_gcd_greatest_bezout_sum_leftsecond+S (pfrep_power_gcd_greatest_bezout_sum_left)=(pfga_K_gcd_greatest_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_greatest_bezout_sum_leftsecondentry. ff_h_pfp_gcd_greatest_bezout_sum_leftsecondentry + S (pfrep_right_gcd_greatest_bezout_sum_left) = S ((S (pfrep_position_gcd_greatest_bezout_sum_leftsecond)) * pfga_uc_gcd_greatest_bezout_sum)) /\\ exists ff_q_pfp_gcd_greatest_bezout_sum_leftsecondentry. pfga_ub_gcd_greatest_bezout_sum = ff_q_pfp_gcd_greatest_bezout_sum_leftsecondentry * S ((S (pfrep_position_gcd_greatest_bezout_sum_leftsecond)) * pfga_uc_gcd_greatest_bezout_sum) + (pfrep_right_gcd_greatest_bezout_sum_left)))))) \\/ (((exists pfrep_gap_gcd_greatest_bezout_sum_leftsecondoutside. pfrep_gap_gcd_greatest_bezout_sum_leftsecondoutside+(pfga_K_gcd_greatest_bezout_sum)=(pfrep_power_gcd_greatest_bezout_sum_left)) /\\ (((pfrep_right_gcd_greatest_bezout_sum_left)=0))))) -> pfrep_left_gcd_greatest_bezout_sum_left=pfrep_right_gcd_greatest_bezout_sum_left) /\\ ((forall pfrep_power_gcd_greatest_bezout_sum_right pfrep_left_gcd_greatest_bezout_sum_right pfrep_right_gcd_greatest_bezout_sum_right. ((exists pfrep_position_gcd_greatest_bezout_sum_rightfirst. ((pfrep_position_gcd_greatest_bezout_sum_rightfirst+S (pfrep_power_gcd_greatest_bezout_sum_right)=(pfgb_Q_gcd_greatest_bezout)) /\\ ((((exists ff_h_pfp_gcd_greatest_bezout_sum_rightfirstentry. ff_h_pfp_gcd_greatest_bezout_sum_rightfirstentry + S (pfrep_left_gcd_greatest_bezout_sum_right) = S ((S (pfrep_position_gcd_greatest_bezout_sum_rightfirst)) * pfgb_qc_gcd_greatest_bezout)) /\\ exists ff_q_pfp_gcd_greatest_bezout_sum_rightfirstentry. pfgb_qb_gcd_greatest_bezout = ff_q_pfp_gcd_greatest_bezout_sum_rightfirstentry * S ((S (pfrep_position_gcd_greatest_bezout_sum_rightfirst)) * pfgb_qc_gcd_greatest_bezout) + (pfrep_left_gcd_greatest_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_greatest_bezout_sum_rightfirstoutside. pfrep_gap_gcd_greatest_bezout_sum_rightfirstoutside+(pfgb_Q_gcd_greatest_bezout)=(pfrep_power_gcd_greatest_bezout_sum_right)) /\\ (((pfrep_left_gcd_greatest_bezout_sum_right)=0))))) -> ((exists pfrep_position_gcd_greatest_bezout_sum_rightsecond. ((pfrep_position_gcd_greatest_bezout_sum_rightsecond+S (pfrep_power_gcd_greatest_bezout_sum_right)=(pfga_K_gcd_greatest_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_greatest_bezout_sum_rightsecondentry. ff_h_pfp_gcd_greatest_bezout_sum_rightsecondentry + S (pfrep_right_gcd_greatest_bezout_sum_right) = S ((S (pfrep_position_gcd_greatest_bezout_sum_rightsecond)) * pfga_vc_gcd_greatest_bezout_sum)) /\\ exists ff_q_pfp_gcd_greatest_bezout_sum_rightsecondentry. pfga_vb_gcd_greatest_bezout_sum = ff_q_pfp_gcd_greatest_bezout_sum_rightsecondentry * S ((S (pfrep_position_gcd_greatest_bezout_sum_rightsecond)) * pfga_vc_gcd_greatest_bezout_sum) + (pfrep_right_gcd_greatest_bezout_sum_right)))))) \\/ (((exists pfrep_gap_gcd_greatest_bezout_sum_rightsecondoutside. pfrep_gap_gcd_greatest_bezout_sum_rightsecondoutside+(pfga_K_gcd_greatest_bezout_sum)=(pfrep_power_gcd_greatest_bezout_sum_right)) /\\ (((pfrep_right_gcd_greatest_bezout_sum_right)=0))))) -> pfrep_left_gcd_greatest_bezout_sum_right=pfrep_right_gcd_greatest_bezout_sum_right)))) /\\ (((forall pfp_index_gcd_greatest_bezout_sum_add. (exists pfa_gap_gcd_greatest_bezout_sum_addindex. pfa_gap_gcd_greatest_bezout_sum_addindex + S (pfp_index_gcd_greatest_bezout_sum_add) = (pfga_K_gcd_greatest_bezout_sum)) -> exists pfp_left_gcd_greatest_bezout_sum_add pfp_right_gcd_greatest_bezout_sum_add pfp_value_gcd_greatest_bezout_sum_add. ((((exists ff_h_pfp_gcd_greatest_bezout_sum_addleft. ff_h_pfp_gcd_greatest_bezout_sum_addleft + S (pfp_left_gcd_greatest_bezout_sum_add) = S ((S (pfp_index_gcd_greatest_bezout_sum_add)) * pfga_uc_gcd_greatest_bezout_sum)) /\\ exists ff_q_pfp_gcd_greatest_bezout_sum_addleft. pfga_ub_gcd_greatest_bezout_sum = ff_q_pfp_gcd_greatest_bezout_sum_addleft * S ((S (pfp_index_gcd_greatest_bezout_sum_add)) * pfga_uc_gcd_greatest_bezout_sum) + (pfp_left_gcd_greatest_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_greatest_bezout_sum_addright. ff_h_pfp_gcd_greatest_bezout_sum_addright + S (pfp_right_gcd_greatest_bezout_sum_add) = S ((S (pfp_index_gcd_greatest_bezout_sum_add)) * pfga_vc_gcd_greatest_bezout_sum)) /\\ exists ff_q_pfp_gcd_greatest_bezout_sum_addright. pfga_vb_gcd_greatest_bezout_sum = ff_q_pfp_gcd_greatest_bezout_sum_addright * S ((S (pfp_index_gcd_greatest_bezout_sum_add)) * pfga_vc_gcd_greatest_bezout_sum) + (pfp_right_gcd_greatest_bezout_sum_add))) /\\ (((((exists ff_h_pfp_gcd_greatest_bezout_sum_addtarget. ff_h_pfp_gcd_greatest_bezout_sum_addtarget + S (pfp_value_gcd_greatest_bezout_sum_add) = S ((S (pfp_index_gcd_greatest_bezout_sum_add)) * pfga_tc_gcd_greatest_bezout_sum)) /\\ exists ff_q_pfp_gcd_greatest_bezout_sum_addtarget. pfga_tb_gcd_greatest_bezout_sum = ff_q_pfp_gcd_greatest_bezout_sum_addtarget * S ((S (pfp_index_gcd_greatest_bezout_sum_add)) * pfga_tc_gcd_greatest_bezout_sum) + (pfp_value_gcd_greatest_bezout_sum_add))) /\\ ((((exists pfa_gap_gcd_greatest_bezout_sum_addoperationleft. pfa_gap_gcd_greatest_bezout_sum_addoperationleft + S (pfp_left_gcd_greatest_bezout_sum_add) = (p)) /\\ (((exists pfa_gap_gcd_greatest_bezout_sum_addoperationright. pfa_gap_gcd_greatest_bezout_sum_addoperationright + S (pfp_right_gcd_greatest_bezout_sum_add) = (p)) /\\ ((((exists pfa_gap_gcd_greatest_bezout_sum_addoperationresultbound. pfa_gap_gcd_greatest_bezout_sum_addoperationresultbound + S (pfp_value_gcd_greatest_bezout_sum_add) = (p)) /\\ ((exists pfa_offset_left_gcd_greatest_bezout_sum_addoperationresultcongruence pfa_offset_right_gcd_greatest_bezout_sum_addoperationresultcongruence. ((pfp_left_gcd_greatest_bezout_sum_add) + (pfp_right_gcd_greatest_bezout_sum_add)) + (p) * pfa_offset_left_gcd_greatest_bezout_sum_addoperationresultcongruence = (pfp_value_gcd_greatest_bezout_sum_add) + (p) * pfa_offset_right_gcd_greatest_bezout_sum_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_gcd_greatest_bezout_sum_result pfrep_left_gcd_greatest_bezout_sum_result pfrep_right_gcd_greatest_bezout_sum_result. ((exists pfrep_position_gcd_greatest_bezout_sum_resultfirst. ((pfrep_position_gcd_greatest_bezout_sum_resultfirst+S (pfrep_power_gcd_greatest_bezout_sum_result)=(pfga_K_gcd_greatest_bezout_sum)) /\\ ((((exists ff_h_pfp_gcd_greatest_bezout_sum_resultfirstentry. ff_h_pfp_gcd_greatest_bezout_sum_resultfirstentry + S (pfrep_left_gcd_greatest_bezout_sum_result) = S ((S (pfrep_position_gcd_greatest_bezout_sum_resultfirst)) * pfga_tc_gcd_greatest_bezout_sum)) /\\ exists ff_q_pfp_gcd_greatest_bezout_sum_resultfirstentry. pfga_tb_gcd_greatest_bezout_sum = ff_q_pfp_gcd_greatest_bezout_sum_resultfirstentry * S ((S (pfrep_position_gcd_greatest_bezout_sum_resultfirst)) * pfga_tc_gcd_greatest_bezout_sum) + (pfrep_left_gcd_greatest_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_greatest_bezout_sum_resultfirstoutside. pfrep_gap_gcd_greatest_bezout_sum_resultfirstoutside+(pfga_K_gcd_greatest_bezout_sum)=(pfrep_power_gcd_greatest_bezout_sum_result)) /\\ (((pfrep_left_gcd_greatest_bezout_sum_result)=0))))) -> ((exists pfrep_position_gcd_greatest_bezout_sum_resultsecond. ((pfrep_position_gcd_greatest_bezout_sum_resultsecond+S (pfrep_power_gcd_greatest_bezout_sum_result)=(G)) /\\ ((((exists ff_h_pfp_gcd_greatest_bezout_sum_resultsecondentry. ff_h_pfp_gcd_greatest_bezout_sum_resultsecondentry + S (pfrep_right_gcd_greatest_bezout_sum_result) = S ((S (pfrep_position_gcd_greatest_bezout_sum_resultsecond)) * gc)) /\\ exists ff_q_pfp_gcd_greatest_bezout_sum_resultsecondentry. gb = ff_q_pfp_gcd_greatest_bezout_sum_resultsecondentry * S ((S (pfrep_position_gcd_greatest_bezout_sum_resultsecond)) * gc) + (pfrep_right_gcd_greatest_bezout_sum_result)))))) \\/ (((exists pfrep_gap_gcd_greatest_bezout_sum_resultsecondoutside. pfrep_gap_gcd_greatest_bezout_sum_resultsecondoutside+(G)=(pfrep_power_gcd_greatest_bezout_sum_result)) /\\ (((pfrep_right_gcd_greatest_bezout_sum_result)=0))))) -> pfrep_left_gcd_greatest_bezout_sum_result=pfrep_right_gcd_greatest_bezout_sum_result)))))))))))))))))) -> (((((((forall fom_index_pfp_gcd_greatest_result_common_left_bounded. (exists fom_gap_pfp_gcd_greatest_result_common_left_bounded_index_bound. fom_gap_pfp_gcd_greatest_result_common_left_bounded_index_bound + S (fom_index_pfp_gcd_greatest_result_common_left_bounded) = L) -> exists fom_value_pfp_gcd_greatest_result_common_left_bounded. ((((exists fom_beta_height_pfp_gcd_greatest_result_common_left_bounded_entry. fom_beta_height_pfp_gcd_greatest_result_common_left_bounded_entry + S (fom_value_pfp_gcd_greatest_result_common_left_bounded) = S ((S (fom_index_pfp_gcd_greatest_result_common_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_result_common_left_bounded_entry. ab = fom_beta_quotient_pfp_gcd_greatest_result_common_left_bounded_entry * S ((S (fom_index_pfp_gcd_greatest_result_common_left_bounded)) * ac) + (fom_value_pfp_gcd_greatest_result_common_left_bounded))) /\\ (exists fom_gap_pfp_gcd_greatest_result_common_left_bounded_value_bound. fom_gap_pfp_gcd_greatest_result_common_left_bounded_value_bound + S (fom_value_pfp_gcd_greatest_result_common_left_bounded) = p))) /\\ ((exists pfgd_qb_gcd_greatest_result_common_left pfgd_qc_gcd_greatest_result_common_left pfgd_Q_gcd_greatest_result_common_left pfgd_pb_gcd_greatest_result_common_left pfgd_pc_gcd_greatest_result_common_left pfgd_P_gcd_greatest_result_common_left. ((((forall fom_index_pfp_gcd_greatest_result_common_left_productleft. (exists fom_gap_pfp_gcd_greatest_result_common_left_productleft_index_bound. fom_gap_pfp_gcd_greatest_result_common_left_productleft_index_bound + S (fom_index_pfp_gcd_greatest_result_common_left_productleft) = pfgd_Q_gcd_greatest_result_common_left) -> exists fom_value_pfp_gcd_greatest_result_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_greatest_result_common_left_productleft_entry. fom_beta_height_pfp_gcd_greatest_result_common_left_productleft_entry + S (fom_value_pfp_gcd_greatest_result_common_left_productleft) = S ((S (fom_index_pfp_gcd_greatest_result_common_left_productleft)) * pfgd_qc_gcd_greatest_result_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_result_common_left_productleft_entry. pfgd_qb_gcd_greatest_result_common_left = fom_beta_quotient_pfp_gcd_greatest_result_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_greatest_result_common_left_productleft)) * pfgd_qc_gcd_greatest_result_common_left) + (fom_value_pfp_gcd_greatest_result_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_greatest_result_common_left_productleft_value_bound. fom_gap_pfp_gcd_greatest_result_common_left_productleft_value_bound + S (fom_value_pfp_gcd_greatest_result_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_greatest_result_common_left_productright. (exists fom_gap_pfp_gcd_greatest_result_common_left_productright_index_bound. fom_gap_pfp_gcd_greatest_result_common_left_productright_index_bound + S (fom_index_pfp_gcd_greatest_result_common_left_productright) = G) -> exists fom_value_pfp_gcd_greatest_result_common_left_productright. ((((exists fom_beta_height_pfp_gcd_greatest_result_common_left_productright_entry. fom_beta_height_pfp_gcd_greatest_result_common_left_productright_entry + S (fom_value_pfp_gcd_greatest_result_common_left_productright) = S ((S (fom_index_pfp_gcd_greatest_result_common_left_productright)) * gc)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_result_common_left_productright_entry. gb = fom_beta_quotient_pfp_gcd_greatest_result_common_left_productright_entry * S ((S (fom_index_pfp_gcd_greatest_result_common_left_productright)) * gc) + (fom_value_pfp_gcd_greatest_result_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_greatest_result_common_left_productright_value_bound. fom_gap_pfp_gcd_greatest_result_common_left_productright_value_bound + S (fom_value_pfp_gcd_greatest_result_common_left_productright) = p))) /\\ (((((((pfgd_Q_gcd_greatest_result_common_left)=0 \\/ (G)=0) /\\ (((pfgd_P_gcd_greatest_result_common_left)=0)))) \\/ (((~((pfgd_Q_gcd_greatest_result_common_left)=0)) /\\ (((~((G)=0)) /\\ (((pfgd_Q_gcd_greatest_result_common_left)+(G)=S (pfgd_P_gcd_greatest_result_common_left)))))))) /\\ ((forall pfc_index_gcd_greatest_result_common_left_productcoefficients. (exists pfa_gap_gcd_greatest_result_common_left_productcoefficientsbound. pfa_gap_gcd_greatest_result_common_left_productcoefficientsbound + S (pfc_index_gcd_greatest_result_common_left_productcoefficients) = (pfgd_P_gcd_greatest_result_common_left)) -> exists pfc_value_gcd_greatest_result_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_greatest_result_common_left_productcoefficientsentry. ff_h_pfp_gcd_greatest_result_common_left_productcoefficientsentry + S (pfc_value_gcd_greatest_result_common_left_productcoefficients) = S ((S (pfc_index_gcd_greatest_result_common_left_productcoefficients)) * pfgd_pc_gcd_greatest_result_common_left)) /\\ exists ff_q_pfp_gcd_greatest_result_common_left_productcoefficientsentry. pfgd_pb_gcd_greatest_result_common_left = ff_q_pfp_gcd_greatest_result_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_greatest_result_common_left_productcoefficients)) * pfgd_pc_gcd_greatest_result_common_left) + (pfc_value_gcd_greatest_result_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_greatest_result_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_greatest_result_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_greatest_result_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_greatest_result_common_left_productcoefficients))) -> exists pfc_value_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_result_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_greatest_result_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_result_common_left_productcoefficientscoefficient) + (pfc_value_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_greatest_result_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_greatest_result_common_left)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_greatest_result_common_left)) /\\ exists ff_q_pfp_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_greatest_result_common_left = ff_q_pfp_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_greatest_result_common_left) + (pfc_left_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_greatest_result_common_left)=(pfc_index_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm) = (G)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm)) * gc)) /\\ exists ff_q_pfp_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermrightentry. gb = ff_q_pfp_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm)) * gc) + (pfc_right_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(G)=(pfc_complement_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_greatest_result_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_greatest_result_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_greatest_result_common_left_productcoefficients))) * fs_v_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_greatest_result_common_left_productcoefficients))) * fs_v_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_greatest_result_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_greatest_result_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_result_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_greatest_result_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_result_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_greatest_result_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_greatest_result_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_greatest_result_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_greatest_result_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_greatest_result_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_greatest_result_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_greatest_result_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_greatest_result_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_greatest_result_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_greatest_result_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_greatest_result_common_left_equivalent pfrep_left_gcd_greatest_result_common_left_equivalent pfrep_right_gcd_greatest_result_common_left_equivalent. ((exists pfrep_position_gcd_greatest_result_common_left_equivalentfirst. ((pfrep_position_gcd_greatest_result_common_left_equivalentfirst+S (pfrep_power_gcd_greatest_result_common_left_equivalent)=(pfgd_P_gcd_greatest_result_common_left)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_common_left_equivalentfirstentry. ff_h_pfp_gcd_greatest_result_common_left_equivalentfirstentry + S (pfrep_left_gcd_greatest_result_common_left_equivalent) = S ((S (pfrep_position_gcd_greatest_result_common_left_equivalentfirst)) * pfgd_pc_gcd_greatest_result_common_left)) /\\ exists ff_q_pfp_gcd_greatest_result_common_left_equivalentfirstentry. pfgd_pb_gcd_greatest_result_common_left = ff_q_pfp_gcd_greatest_result_common_left_equivalentfirstentry * S ((S (pfrep_position_gcd_greatest_result_common_left_equivalentfirst)) * pfgd_pc_gcd_greatest_result_common_left) + (pfrep_left_gcd_greatest_result_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_greatest_result_common_left_equivalentfirstoutside. pfrep_gap_gcd_greatest_result_common_left_equivalentfirstoutside+(pfgd_P_gcd_greatest_result_common_left)=(pfrep_power_gcd_greatest_result_common_left_equivalent)) /\\ (((pfrep_left_gcd_greatest_result_common_left_equivalent)=0))))) -> ((exists pfrep_position_gcd_greatest_result_common_left_equivalentsecond. ((pfrep_position_gcd_greatest_result_common_left_equivalentsecond+S (pfrep_power_gcd_greatest_result_common_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_common_left_equivalentsecondentry. ff_h_pfp_gcd_greatest_result_common_left_equivalentsecondentry + S (pfrep_right_gcd_greatest_result_common_left_equivalent) = S ((S (pfrep_position_gcd_greatest_result_common_left_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_gcd_greatest_result_common_left_equivalentsecondentry. ab = ff_q_pfp_gcd_greatest_result_common_left_equivalentsecondentry * S ((S (pfrep_position_gcd_greatest_result_common_left_equivalentsecond)) * ac) + (pfrep_right_gcd_greatest_result_common_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_greatest_result_common_left_equivalentsecondoutside. pfrep_gap_gcd_greatest_result_common_left_equivalentsecondoutside+(L)=(pfrep_power_gcd_greatest_result_common_left_equivalent)) /\\ (((pfrep_right_gcd_greatest_result_common_left_equivalent)=0))))) -> pfrep_left_gcd_greatest_result_common_left_equivalent=pfrep_right_gcd_greatest_result_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_greatest_result_common_right_bounded. (exists fom_gap_pfp_gcd_greatest_result_common_right_bounded_index_bound. fom_gap_pfp_gcd_greatest_result_common_right_bounded_index_bound + S (fom_index_pfp_gcd_greatest_result_common_right_bounded) = M) -> exists fom_value_pfp_gcd_greatest_result_common_right_bounded. ((((exists fom_beta_height_pfp_gcd_greatest_result_common_right_bounded_entry. fom_beta_height_pfp_gcd_greatest_result_common_right_bounded_entry + S (fom_value_pfp_gcd_greatest_result_common_right_bounded) = S ((S (fom_index_pfp_gcd_greatest_result_common_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_result_common_right_bounded_entry. bb = fom_beta_quotient_pfp_gcd_greatest_result_common_right_bounded_entry * S ((S (fom_index_pfp_gcd_greatest_result_common_right_bounded)) * bc) + (fom_value_pfp_gcd_greatest_result_common_right_bounded))) /\\ (exists fom_gap_pfp_gcd_greatest_result_common_right_bounded_value_bound. fom_gap_pfp_gcd_greatest_result_common_right_bounded_value_bound + S (fom_value_pfp_gcd_greatest_result_common_right_bounded) = p))) /\\ ((exists pfgd_qb_gcd_greatest_result_common_right pfgd_qc_gcd_greatest_result_common_right pfgd_Q_gcd_greatest_result_common_right pfgd_pb_gcd_greatest_result_common_right pfgd_pc_gcd_greatest_result_common_right pfgd_P_gcd_greatest_result_common_right. ((((forall fom_index_pfp_gcd_greatest_result_common_right_productleft. (exists fom_gap_pfp_gcd_greatest_result_common_right_productleft_index_bound. fom_gap_pfp_gcd_greatest_result_common_right_productleft_index_bound + S (fom_index_pfp_gcd_greatest_result_common_right_productleft) = pfgd_Q_gcd_greatest_result_common_right) -> exists fom_value_pfp_gcd_greatest_result_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_greatest_result_common_right_productleft_entry. fom_beta_height_pfp_gcd_greatest_result_common_right_productleft_entry + S (fom_value_pfp_gcd_greatest_result_common_right_productleft) = S ((S (fom_index_pfp_gcd_greatest_result_common_right_productleft)) * pfgd_qc_gcd_greatest_result_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_result_common_right_productleft_entry. pfgd_qb_gcd_greatest_result_common_right = fom_beta_quotient_pfp_gcd_greatest_result_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_greatest_result_common_right_productleft)) * pfgd_qc_gcd_greatest_result_common_right) + (fom_value_pfp_gcd_greatest_result_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_greatest_result_common_right_productleft_value_bound. fom_gap_pfp_gcd_greatest_result_common_right_productleft_value_bound + S (fom_value_pfp_gcd_greatest_result_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_greatest_result_common_right_productright. (exists fom_gap_pfp_gcd_greatest_result_common_right_productright_index_bound. fom_gap_pfp_gcd_greatest_result_common_right_productright_index_bound + S (fom_index_pfp_gcd_greatest_result_common_right_productright) = G) -> exists fom_value_pfp_gcd_greatest_result_common_right_productright. ((((exists fom_beta_height_pfp_gcd_greatest_result_common_right_productright_entry. fom_beta_height_pfp_gcd_greatest_result_common_right_productright_entry + S (fom_value_pfp_gcd_greatest_result_common_right_productright) = S ((S (fom_index_pfp_gcd_greatest_result_common_right_productright)) * gc)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_result_common_right_productright_entry. gb = fom_beta_quotient_pfp_gcd_greatest_result_common_right_productright_entry * S ((S (fom_index_pfp_gcd_greatest_result_common_right_productright)) * gc) + (fom_value_pfp_gcd_greatest_result_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_greatest_result_common_right_productright_value_bound. fom_gap_pfp_gcd_greatest_result_common_right_productright_value_bound + S (fom_value_pfp_gcd_greatest_result_common_right_productright) = p))) /\\ (((((((pfgd_Q_gcd_greatest_result_common_right)=0 \\/ (G)=0) /\\ (((pfgd_P_gcd_greatest_result_common_right)=0)))) \\/ (((~((pfgd_Q_gcd_greatest_result_common_right)=0)) /\\ (((~((G)=0)) /\\ (((pfgd_Q_gcd_greatest_result_common_right)+(G)=S (pfgd_P_gcd_greatest_result_common_right)))))))) /\\ ((forall pfc_index_gcd_greatest_result_common_right_productcoefficients. (exists pfa_gap_gcd_greatest_result_common_right_productcoefficientsbound. pfa_gap_gcd_greatest_result_common_right_productcoefficientsbound + S (pfc_index_gcd_greatest_result_common_right_productcoefficients) = (pfgd_P_gcd_greatest_result_common_right)) -> exists pfc_value_gcd_greatest_result_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_greatest_result_common_right_productcoefficientsentry. ff_h_pfp_gcd_greatest_result_common_right_productcoefficientsentry + S (pfc_value_gcd_greatest_result_common_right_productcoefficients) = S ((S (pfc_index_gcd_greatest_result_common_right_productcoefficients)) * pfgd_pc_gcd_greatest_result_common_right)) /\\ exists ff_q_pfp_gcd_greatest_result_common_right_productcoefficientsentry. pfgd_pb_gcd_greatest_result_common_right = ff_q_pfp_gcd_greatest_result_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_greatest_result_common_right_productcoefficients)) * pfgd_pc_gcd_greatest_result_common_right) + (pfc_value_gcd_greatest_result_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_greatest_result_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_greatest_result_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_greatest_result_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_greatest_result_common_right_productcoefficients))) -> exists pfc_value_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_result_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_greatest_result_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_result_common_right_productcoefficientscoefficient) + (pfc_value_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_greatest_result_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_greatest_result_common_right)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_greatest_result_common_right)) /\\ exists ff_q_pfp_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_greatest_result_common_right = ff_q_pfp_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_greatest_result_common_right) + (pfc_left_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_greatest_result_common_right)=(pfc_index_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm) = (G)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm)) * gc)) /\\ exists ff_q_pfp_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermrightentry. gb = ff_q_pfp_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm)) * gc) + (pfc_right_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(G)=(pfc_complement_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_greatest_result_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_greatest_result_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_greatest_result_common_right_productcoefficients))) * fs_v_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_greatest_result_common_right_productcoefficients))) * fs_v_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_greatest_result_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_greatest_result_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_result_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_greatest_result_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_result_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_greatest_result_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_greatest_result_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_greatest_result_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_greatest_result_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_greatest_result_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_greatest_result_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_greatest_result_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_greatest_result_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_greatest_result_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_greatest_result_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_greatest_result_common_right_equivalent pfrep_left_gcd_greatest_result_common_right_equivalent pfrep_right_gcd_greatest_result_common_right_equivalent. ((exists pfrep_position_gcd_greatest_result_common_right_equivalentfirst. ((pfrep_position_gcd_greatest_result_common_right_equivalentfirst+S (pfrep_power_gcd_greatest_result_common_right_equivalent)=(pfgd_P_gcd_greatest_result_common_right)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_common_right_equivalentfirstentry. ff_h_pfp_gcd_greatest_result_common_right_equivalentfirstentry + S (pfrep_left_gcd_greatest_result_common_right_equivalent) = S ((S (pfrep_position_gcd_greatest_result_common_right_equivalentfirst)) * pfgd_pc_gcd_greatest_result_common_right)) /\\ exists ff_q_pfp_gcd_greatest_result_common_right_equivalentfirstentry. pfgd_pb_gcd_greatest_result_common_right = ff_q_pfp_gcd_greatest_result_common_right_equivalentfirstentry * S ((S (pfrep_position_gcd_greatest_result_common_right_equivalentfirst)) * pfgd_pc_gcd_greatest_result_common_right) + (pfrep_left_gcd_greatest_result_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_greatest_result_common_right_equivalentfirstoutside. pfrep_gap_gcd_greatest_result_common_right_equivalentfirstoutside+(pfgd_P_gcd_greatest_result_common_right)=(pfrep_power_gcd_greatest_result_common_right_equivalent)) /\\ (((pfrep_left_gcd_greatest_result_common_right_equivalent)=0))))) -> ((exists pfrep_position_gcd_greatest_result_common_right_equivalentsecond. ((pfrep_position_gcd_greatest_result_common_right_equivalentsecond+S (pfrep_power_gcd_greatest_result_common_right_equivalent)=(M)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_common_right_equivalentsecondentry. ff_h_pfp_gcd_greatest_result_common_right_equivalentsecondentry + S (pfrep_right_gcd_greatest_result_common_right_equivalent) = S ((S (pfrep_position_gcd_greatest_result_common_right_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_gcd_greatest_result_common_right_equivalentsecondentry. bb = ff_q_pfp_gcd_greatest_result_common_right_equivalentsecondentry * S ((S (pfrep_position_gcd_greatest_result_common_right_equivalentsecond)) * bc) + (pfrep_right_gcd_greatest_result_common_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_greatest_result_common_right_equivalentsecondoutside. pfrep_gap_gcd_greatest_result_common_right_equivalentsecondoutside+(M)=(pfrep_power_gcd_greatest_result_common_right_equivalent)) /\\ (((pfrep_right_gcd_greatest_result_common_right_equivalent)=0))))) -> pfrep_left_gcd_greatest_result_common_right_equivalent=pfrep_right_gcd_greatest_result_common_right_equivalent)))))))))) /\\ ((forall pfgg_db_gcd_greatest_result pfgg_dc_gcd_greatest_result pfgg_D_gcd_greatest_result. (((((forall fom_index_pfp_gcd_greatest_result_divisor_left_bounded. (exists fom_gap_pfp_gcd_greatest_result_divisor_left_bounded_index_bound. fom_gap_pfp_gcd_greatest_result_divisor_left_bounded_index_bound + S (fom_index_pfp_gcd_greatest_result_divisor_left_bounded) = L) -> exists fom_value_pfp_gcd_greatest_result_divisor_left_bounded. ((((exists fom_beta_height_pfp_gcd_greatest_result_divisor_left_bounded_entry. fom_beta_height_pfp_gcd_greatest_result_divisor_left_bounded_entry + S (fom_value_pfp_gcd_greatest_result_divisor_left_bounded) = S ((S (fom_index_pfp_gcd_greatest_result_divisor_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_result_divisor_left_bounded_entry. ab = fom_beta_quotient_pfp_gcd_greatest_result_divisor_left_bounded_entry * S ((S (fom_index_pfp_gcd_greatest_result_divisor_left_bounded)) * ac) + (fom_value_pfp_gcd_greatest_result_divisor_left_bounded))) /\\ (exists fom_gap_pfp_gcd_greatest_result_divisor_left_bounded_value_bound. fom_gap_pfp_gcd_greatest_result_divisor_left_bounded_value_bound + S (fom_value_pfp_gcd_greatest_result_divisor_left_bounded) = p))) /\\ ((exists pfgd_qb_gcd_greatest_result_divisor_left pfgd_qc_gcd_greatest_result_divisor_left pfgd_Q_gcd_greatest_result_divisor_left pfgd_pb_gcd_greatest_result_divisor_left pfgd_pc_gcd_greatest_result_divisor_left pfgd_P_gcd_greatest_result_divisor_left. ((((forall fom_index_pfp_gcd_greatest_result_divisor_left_productleft. (exists fom_gap_pfp_gcd_greatest_result_divisor_left_productleft_index_bound. fom_gap_pfp_gcd_greatest_result_divisor_left_productleft_index_bound + S (fom_index_pfp_gcd_greatest_result_divisor_left_productleft) = pfgd_Q_gcd_greatest_result_divisor_left) -> exists fom_value_pfp_gcd_greatest_result_divisor_left_productleft. ((((exists fom_beta_height_pfp_gcd_greatest_result_divisor_left_productleft_entry. fom_beta_height_pfp_gcd_greatest_result_divisor_left_productleft_entry + S (fom_value_pfp_gcd_greatest_result_divisor_left_productleft) = S ((S (fom_index_pfp_gcd_greatest_result_divisor_left_productleft)) * pfgd_qc_gcd_greatest_result_divisor_left)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_result_divisor_left_productleft_entry. pfgd_qb_gcd_greatest_result_divisor_left = fom_beta_quotient_pfp_gcd_greatest_result_divisor_left_productleft_entry * S ((S (fom_index_pfp_gcd_greatest_result_divisor_left_productleft)) * pfgd_qc_gcd_greatest_result_divisor_left) + (fom_value_pfp_gcd_greatest_result_divisor_left_productleft))) /\\ (exists fom_gap_pfp_gcd_greatest_result_divisor_left_productleft_value_bound. fom_gap_pfp_gcd_greatest_result_divisor_left_productleft_value_bound + S (fom_value_pfp_gcd_greatest_result_divisor_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_greatest_result_divisor_left_productright. (exists fom_gap_pfp_gcd_greatest_result_divisor_left_productright_index_bound. fom_gap_pfp_gcd_greatest_result_divisor_left_productright_index_bound + S (fom_index_pfp_gcd_greatest_result_divisor_left_productright) = pfgg_D_gcd_greatest_result) -> exists fom_value_pfp_gcd_greatest_result_divisor_left_productright. ((((exists fom_beta_height_pfp_gcd_greatest_result_divisor_left_productright_entry. fom_beta_height_pfp_gcd_greatest_result_divisor_left_productright_entry + S (fom_value_pfp_gcd_greatest_result_divisor_left_productright) = S ((S (fom_index_pfp_gcd_greatest_result_divisor_left_productright)) * pfgg_dc_gcd_greatest_result)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_result_divisor_left_productright_entry. pfgg_db_gcd_greatest_result = fom_beta_quotient_pfp_gcd_greatest_result_divisor_left_productright_entry * S ((S (fom_index_pfp_gcd_greatest_result_divisor_left_productright)) * pfgg_dc_gcd_greatest_result) + (fom_value_pfp_gcd_greatest_result_divisor_left_productright))) /\\ (exists fom_gap_pfp_gcd_greatest_result_divisor_left_productright_value_bound. fom_gap_pfp_gcd_greatest_result_divisor_left_productright_value_bound + S (fom_value_pfp_gcd_greatest_result_divisor_left_productright) = p))) /\\ (((((((pfgd_Q_gcd_greatest_result_divisor_left)=0 \\/ (pfgg_D_gcd_greatest_result)=0) /\\ (((pfgd_P_gcd_greatest_result_divisor_left)=0)))) \\/ (((~((pfgd_Q_gcd_greatest_result_divisor_left)=0)) /\\ (((~((pfgg_D_gcd_greatest_result)=0)) /\\ (((pfgd_Q_gcd_greatest_result_divisor_left)+(pfgg_D_gcd_greatest_result)=S (pfgd_P_gcd_greatest_result_divisor_left)))))))) /\\ ((forall pfc_index_gcd_greatest_result_divisor_left_productcoefficients. (exists pfa_gap_gcd_greatest_result_divisor_left_productcoefficientsbound. pfa_gap_gcd_greatest_result_divisor_left_productcoefficientsbound + S (pfc_index_gcd_greatest_result_divisor_left_productcoefficients) = (pfgd_P_gcd_greatest_result_divisor_left)) -> exists pfc_value_gcd_greatest_result_divisor_left_productcoefficients. ((((exists ff_h_pfp_gcd_greatest_result_divisor_left_productcoefficientsentry. ff_h_pfp_gcd_greatest_result_divisor_left_productcoefficientsentry + S (pfc_value_gcd_greatest_result_divisor_left_productcoefficients) = S ((S (pfc_index_gcd_greatest_result_divisor_left_productcoefficients)) * pfgd_pc_gcd_greatest_result_divisor_left)) /\\ exists ff_q_pfp_gcd_greatest_result_divisor_left_productcoefficientsentry. pfgd_pb_gcd_greatest_result_divisor_left = ff_q_pfp_gcd_greatest_result_divisor_left_productcoefficientsentry * S ((S (pfc_index_gcd_greatest_result_divisor_left_productcoefficients)) * pfgd_pc_gcd_greatest_result_divisor_left) + (pfc_value_gcd_greatest_result_divisor_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_greatest_result_divisor_left_productcoefficientscoefficient pfc_terms_scale_gcd_greatest_result_divisor_left_productcoefficientscoefficient pfc_natural_sum_gcd_greatest_result_divisor_left_productcoefficientscoefficient. ((forall pfc_index_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_greatest_result_divisor_left_productcoefficients))) -> exists pfc_value_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_result_divisor_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_greatest_result_divisor_left_productcoefficientscoefficient = ff_q_pfp_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_result_divisor_left_productcoefficientscoefficient) + (pfc_value_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_greatest_result_divisor_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_greatest_result_divisor_left)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_greatest_result_divisor_left)) /\\ exists ff_q_pfp_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_greatest_result_divisor_left = ff_q_pfp_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_greatest_result_divisor_left) + (pfc_left_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_greatest_result_divisor_left)=(pfc_index_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm) = (pfgg_D_gcd_greatest_result)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm)) * pfgg_dc_gcd_greatest_result)) /\\ exists ff_q_pfp_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermrightentry. pfgg_db_gcd_greatest_result = ff_q_pfp_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm)) * pfgg_dc_gcd_greatest_result) + (pfc_right_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonaltermrightoutside+(pfgg_D_gcd_greatest_result)=(pfc_complement_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_greatest_result_divisor_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum fs_v_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_greatest_result_divisor_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_greatest_result_divisor_left_productcoefficients))) * fs_v_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_greatest_result_divisor_left_productcoefficients))) * fs_v_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_greatest_result_divisor_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_greatest_result_divisor_left_productcoefficients)) -> exists fs_a_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_result_divisor_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_greatest_result_divisor_left_productcoefficientscoefficient = fs_q_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_result_divisor_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_greatest_result_divisor_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_greatest_result_divisor_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_greatest_result_divisor_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_greatest_result_divisor_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_greatest_result_divisor_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_greatest_result_divisor_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_greatest_result_divisor_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_greatest_result_divisor_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_greatest_result_divisor_left_productcoefficients) + (p) * pfa_offset_right_gcd_greatest_result_divisor_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_greatest_result_divisor_left_equivalent pfrep_left_gcd_greatest_result_divisor_left_equivalent pfrep_right_gcd_greatest_result_divisor_left_equivalent. ((exists pfrep_position_gcd_greatest_result_divisor_left_equivalentfirst. ((pfrep_position_gcd_greatest_result_divisor_left_equivalentfirst+S (pfrep_power_gcd_greatest_result_divisor_left_equivalent)=(pfgd_P_gcd_greatest_result_divisor_left)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_divisor_left_equivalentfirstentry. ff_h_pfp_gcd_greatest_result_divisor_left_equivalentfirstentry + S (pfrep_left_gcd_greatest_result_divisor_left_equivalent) = S ((S (pfrep_position_gcd_greatest_result_divisor_left_equivalentfirst)) * pfgd_pc_gcd_greatest_result_divisor_left)) /\\ exists ff_q_pfp_gcd_greatest_result_divisor_left_equivalentfirstentry. pfgd_pb_gcd_greatest_result_divisor_left = ff_q_pfp_gcd_greatest_result_divisor_left_equivalentfirstentry * S ((S (pfrep_position_gcd_greatest_result_divisor_left_equivalentfirst)) * pfgd_pc_gcd_greatest_result_divisor_left) + (pfrep_left_gcd_greatest_result_divisor_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_greatest_result_divisor_left_equivalentfirstoutside. pfrep_gap_gcd_greatest_result_divisor_left_equivalentfirstoutside+(pfgd_P_gcd_greatest_result_divisor_left)=(pfrep_power_gcd_greatest_result_divisor_left_equivalent)) /\\ (((pfrep_left_gcd_greatest_result_divisor_left_equivalent)=0))))) -> ((exists pfrep_position_gcd_greatest_result_divisor_left_equivalentsecond. ((pfrep_position_gcd_greatest_result_divisor_left_equivalentsecond+S (pfrep_power_gcd_greatest_result_divisor_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_divisor_left_equivalentsecondentry. ff_h_pfp_gcd_greatest_result_divisor_left_equivalentsecondentry + S (pfrep_right_gcd_greatest_result_divisor_left_equivalent) = S ((S (pfrep_position_gcd_greatest_result_divisor_left_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_gcd_greatest_result_divisor_left_equivalentsecondentry. ab = ff_q_pfp_gcd_greatest_result_divisor_left_equivalentsecondentry * S ((S (pfrep_position_gcd_greatest_result_divisor_left_equivalentsecond)) * ac) + (pfrep_right_gcd_greatest_result_divisor_left_equivalent)))))) \\/ (((exists pfrep_gap_gcd_greatest_result_divisor_left_equivalentsecondoutside. pfrep_gap_gcd_greatest_result_divisor_left_equivalentsecondoutside+(L)=(pfrep_power_gcd_greatest_result_divisor_left_equivalent)) /\\ (((pfrep_right_gcd_greatest_result_divisor_left_equivalent)=0))))) -> pfrep_left_gcd_greatest_result_divisor_left_equivalent=pfrep_right_gcd_greatest_result_divisor_left_equivalent))))))) /\\ ((((forall fom_index_pfp_gcd_greatest_result_divisor_right_bounded. (exists fom_gap_pfp_gcd_greatest_result_divisor_right_bounded_index_bound. fom_gap_pfp_gcd_greatest_result_divisor_right_bounded_index_bound + S (fom_index_pfp_gcd_greatest_result_divisor_right_bounded) = M) -> exists fom_value_pfp_gcd_greatest_result_divisor_right_bounded. ((((exists fom_beta_height_pfp_gcd_greatest_result_divisor_right_bounded_entry. fom_beta_height_pfp_gcd_greatest_result_divisor_right_bounded_entry + S (fom_value_pfp_gcd_greatest_result_divisor_right_bounded) = S ((S (fom_index_pfp_gcd_greatest_result_divisor_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_result_divisor_right_bounded_entry. bb = fom_beta_quotient_pfp_gcd_greatest_result_divisor_right_bounded_entry * S ((S (fom_index_pfp_gcd_greatest_result_divisor_right_bounded)) * bc) + (fom_value_pfp_gcd_greatest_result_divisor_right_bounded))) /\\ (exists fom_gap_pfp_gcd_greatest_result_divisor_right_bounded_value_bound. fom_gap_pfp_gcd_greatest_result_divisor_right_bounded_value_bound + S (fom_value_pfp_gcd_greatest_result_divisor_right_bounded) = p))) /\\ ((exists pfgd_qb_gcd_greatest_result_divisor_right pfgd_qc_gcd_greatest_result_divisor_right pfgd_Q_gcd_greatest_result_divisor_right pfgd_pb_gcd_greatest_result_divisor_right pfgd_pc_gcd_greatest_result_divisor_right pfgd_P_gcd_greatest_result_divisor_right. ((((forall fom_index_pfp_gcd_greatest_result_divisor_right_productleft. (exists fom_gap_pfp_gcd_greatest_result_divisor_right_productleft_index_bound. fom_gap_pfp_gcd_greatest_result_divisor_right_productleft_index_bound + S (fom_index_pfp_gcd_greatest_result_divisor_right_productleft) = pfgd_Q_gcd_greatest_result_divisor_right) -> exists fom_value_pfp_gcd_greatest_result_divisor_right_productleft. ((((exists fom_beta_height_pfp_gcd_greatest_result_divisor_right_productleft_entry. fom_beta_height_pfp_gcd_greatest_result_divisor_right_productleft_entry + S (fom_value_pfp_gcd_greatest_result_divisor_right_productleft) = S ((S (fom_index_pfp_gcd_greatest_result_divisor_right_productleft)) * pfgd_qc_gcd_greatest_result_divisor_right)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_result_divisor_right_productleft_entry. pfgd_qb_gcd_greatest_result_divisor_right = fom_beta_quotient_pfp_gcd_greatest_result_divisor_right_productleft_entry * S ((S (fom_index_pfp_gcd_greatest_result_divisor_right_productleft)) * pfgd_qc_gcd_greatest_result_divisor_right) + (fom_value_pfp_gcd_greatest_result_divisor_right_productleft))) /\\ (exists fom_gap_pfp_gcd_greatest_result_divisor_right_productleft_value_bound. fom_gap_pfp_gcd_greatest_result_divisor_right_productleft_value_bound + S (fom_value_pfp_gcd_greatest_result_divisor_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_greatest_result_divisor_right_productright. (exists fom_gap_pfp_gcd_greatest_result_divisor_right_productright_index_bound. fom_gap_pfp_gcd_greatest_result_divisor_right_productright_index_bound + S (fom_index_pfp_gcd_greatest_result_divisor_right_productright) = pfgg_D_gcd_greatest_result) -> exists fom_value_pfp_gcd_greatest_result_divisor_right_productright. ((((exists fom_beta_height_pfp_gcd_greatest_result_divisor_right_productright_entry. fom_beta_height_pfp_gcd_greatest_result_divisor_right_productright_entry + S (fom_value_pfp_gcd_greatest_result_divisor_right_productright) = S ((S (fom_index_pfp_gcd_greatest_result_divisor_right_productright)) * pfgg_dc_gcd_greatest_result)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_result_divisor_right_productright_entry. pfgg_db_gcd_greatest_result = fom_beta_quotient_pfp_gcd_greatest_result_divisor_right_productright_entry * S ((S (fom_index_pfp_gcd_greatest_result_divisor_right_productright)) * pfgg_dc_gcd_greatest_result) + (fom_value_pfp_gcd_greatest_result_divisor_right_productright))) /\\ (exists fom_gap_pfp_gcd_greatest_result_divisor_right_productright_value_bound. fom_gap_pfp_gcd_greatest_result_divisor_right_productright_value_bound + S (fom_value_pfp_gcd_greatest_result_divisor_right_productright) = p))) /\\ (((((((pfgd_Q_gcd_greatest_result_divisor_right)=0 \\/ (pfgg_D_gcd_greatest_result)=0) /\\ (((pfgd_P_gcd_greatest_result_divisor_right)=0)))) \\/ (((~((pfgd_Q_gcd_greatest_result_divisor_right)=0)) /\\ (((~((pfgg_D_gcd_greatest_result)=0)) /\\ (((pfgd_Q_gcd_greatest_result_divisor_right)+(pfgg_D_gcd_greatest_result)=S (pfgd_P_gcd_greatest_result_divisor_right)))))))) /\\ ((forall pfc_index_gcd_greatest_result_divisor_right_productcoefficients. (exists pfa_gap_gcd_greatest_result_divisor_right_productcoefficientsbound. pfa_gap_gcd_greatest_result_divisor_right_productcoefficientsbound + S (pfc_index_gcd_greatest_result_divisor_right_productcoefficients) = (pfgd_P_gcd_greatest_result_divisor_right)) -> exists pfc_value_gcd_greatest_result_divisor_right_productcoefficients. ((((exists ff_h_pfp_gcd_greatest_result_divisor_right_productcoefficientsentry. ff_h_pfp_gcd_greatest_result_divisor_right_productcoefficientsentry + S (pfc_value_gcd_greatest_result_divisor_right_productcoefficients) = S ((S (pfc_index_gcd_greatest_result_divisor_right_productcoefficients)) * pfgd_pc_gcd_greatest_result_divisor_right)) /\\ exists ff_q_pfp_gcd_greatest_result_divisor_right_productcoefficientsentry. pfgd_pb_gcd_greatest_result_divisor_right = ff_q_pfp_gcd_greatest_result_divisor_right_productcoefficientsentry * S ((S (pfc_index_gcd_greatest_result_divisor_right_productcoefficients)) * pfgd_pc_gcd_greatest_result_divisor_right) + (pfc_value_gcd_greatest_result_divisor_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_greatest_result_divisor_right_productcoefficientscoefficient pfc_terms_scale_gcd_greatest_result_divisor_right_productcoefficientscoefficient pfc_natural_sum_gcd_greatest_result_divisor_right_productcoefficientscoefficient. ((forall pfc_index_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_greatest_result_divisor_right_productcoefficients))) -> exists pfc_value_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_result_divisor_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_greatest_result_divisor_right_productcoefficientscoefficient = ff_q_pfp_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_result_divisor_right_productcoefficientscoefficient) + (pfc_value_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_greatest_result_divisor_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_greatest_result_divisor_right)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_greatest_result_divisor_right)) /\\ exists ff_q_pfp_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_greatest_result_divisor_right = ff_q_pfp_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_greatest_result_divisor_right) + (pfc_left_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_greatest_result_divisor_right)=(pfc_index_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm) = (pfgg_D_gcd_greatest_result)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm)) * pfgg_dc_gcd_greatest_result)) /\\ exists ff_q_pfp_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermrightentry. pfgg_db_gcd_greatest_result = ff_q_pfp_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm)) * pfgg_dc_gcd_greatest_result) + (pfc_right_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonaltermrightoutside+(pfgg_D_gcd_greatest_result)=(pfc_complement_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_greatest_result_divisor_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum fs_v_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_greatest_result_divisor_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_greatest_result_divisor_right_productcoefficients))) * fs_v_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_greatest_result_divisor_right_productcoefficients))) * fs_v_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_greatest_result_divisor_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_greatest_result_divisor_right_productcoefficients)) -> exists fs_a_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_result_divisor_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_greatest_result_divisor_right_productcoefficientscoefficient = fs_q_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_result_divisor_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_greatest_result_divisor_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_greatest_result_divisor_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_greatest_result_divisor_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_greatest_result_divisor_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_greatest_result_divisor_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_greatest_result_divisor_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_greatest_result_divisor_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_greatest_result_divisor_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_greatest_result_divisor_right_productcoefficients) + (p) * pfa_offset_right_gcd_greatest_result_divisor_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_greatest_result_divisor_right_equivalent pfrep_left_gcd_greatest_result_divisor_right_equivalent pfrep_right_gcd_greatest_result_divisor_right_equivalent. ((exists pfrep_position_gcd_greatest_result_divisor_right_equivalentfirst. ((pfrep_position_gcd_greatest_result_divisor_right_equivalentfirst+S (pfrep_power_gcd_greatest_result_divisor_right_equivalent)=(pfgd_P_gcd_greatest_result_divisor_right)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_divisor_right_equivalentfirstentry. ff_h_pfp_gcd_greatest_result_divisor_right_equivalentfirstentry + S (pfrep_left_gcd_greatest_result_divisor_right_equivalent) = S ((S (pfrep_position_gcd_greatest_result_divisor_right_equivalentfirst)) * pfgd_pc_gcd_greatest_result_divisor_right)) /\\ exists ff_q_pfp_gcd_greatest_result_divisor_right_equivalentfirstentry. pfgd_pb_gcd_greatest_result_divisor_right = ff_q_pfp_gcd_greatest_result_divisor_right_equivalentfirstentry * S ((S (pfrep_position_gcd_greatest_result_divisor_right_equivalentfirst)) * pfgd_pc_gcd_greatest_result_divisor_right) + (pfrep_left_gcd_greatest_result_divisor_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_greatest_result_divisor_right_equivalentfirstoutside. pfrep_gap_gcd_greatest_result_divisor_right_equivalentfirstoutside+(pfgd_P_gcd_greatest_result_divisor_right)=(pfrep_power_gcd_greatest_result_divisor_right_equivalent)) /\\ (((pfrep_left_gcd_greatest_result_divisor_right_equivalent)=0))))) -> ((exists pfrep_position_gcd_greatest_result_divisor_right_equivalentsecond. ((pfrep_position_gcd_greatest_result_divisor_right_equivalentsecond+S (pfrep_power_gcd_greatest_result_divisor_right_equivalent)=(M)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_divisor_right_equivalentsecondentry. ff_h_pfp_gcd_greatest_result_divisor_right_equivalentsecondentry + S (pfrep_right_gcd_greatest_result_divisor_right_equivalent) = S ((S (pfrep_position_gcd_greatest_result_divisor_right_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_gcd_greatest_result_divisor_right_equivalentsecondentry. bb = ff_q_pfp_gcd_greatest_result_divisor_right_equivalentsecondentry * S ((S (pfrep_position_gcd_greatest_result_divisor_right_equivalentsecond)) * bc) + (pfrep_right_gcd_greatest_result_divisor_right_equivalent)))))) \\/ (((exists pfrep_gap_gcd_greatest_result_divisor_right_equivalentsecondoutside. pfrep_gap_gcd_greatest_result_divisor_right_equivalentsecondoutside+(M)=(pfrep_power_gcd_greatest_result_divisor_right_equivalent)) /\\ (((pfrep_right_gcd_greatest_result_divisor_right_equivalent)=0))))) -> pfrep_left_gcd_greatest_result_divisor_right_equivalent=pfrep_right_gcd_greatest_result_divisor_right_equivalent)))))))))) -> (((forall fom_index_pfp_gcd_greatest_result_greatest_bounded. (exists fom_gap_pfp_gcd_greatest_result_greatest_bounded_index_bound. fom_gap_pfp_gcd_greatest_result_greatest_bounded_index_bound + S (fom_index_pfp_gcd_greatest_result_greatest_bounded) = G) -> exists fom_value_pfp_gcd_greatest_result_greatest_bounded. ((((exists fom_beta_height_pfp_gcd_greatest_result_greatest_bounded_entry. fom_beta_height_pfp_gcd_greatest_result_greatest_bounded_entry + S (fom_value_pfp_gcd_greatest_result_greatest_bounded) = S ((S (fom_index_pfp_gcd_greatest_result_greatest_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_result_greatest_bounded_entry. gb = fom_beta_quotient_pfp_gcd_greatest_result_greatest_bounded_entry * S ((S (fom_index_pfp_gcd_greatest_result_greatest_bounded)) * gc) + (fom_value_pfp_gcd_greatest_result_greatest_bounded))) /\\ (exists fom_gap_pfp_gcd_greatest_result_greatest_bounded_value_bound. fom_gap_pfp_gcd_greatest_result_greatest_bounded_value_bound + S (fom_value_pfp_gcd_greatest_result_greatest_bounded) = p))) /\\ ((exists pfgd_qb_gcd_greatest_result_greatest pfgd_qc_gcd_greatest_result_greatest pfgd_Q_gcd_greatest_result_greatest pfgd_pb_gcd_greatest_result_greatest pfgd_pc_gcd_greatest_result_greatest pfgd_P_gcd_greatest_result_greatest. ((((forall fom_index_pfp_gcd_greatest_result_greatest_productleft. (exists fom_gap_pfp_gcd_greatest_result_greatest_productleft_index_bound. fom_gap_pfp_gcd_greatest_result_greatest_productleft_index_bound + S (fom_index_pfp_gcd_greatest_result_greatest_productleft) = pfgd_Q_gcd_greatest_result_greatest) -> exists fom_value_pfp_gcd_greatest_result_greatest_productleft. ((((exists fom_beta_height_pfp_gcd_greatest_result_greatest_productleft_entry. fom_beta_height_pfp_gcd_greatest_result_greatest_productleft_entry + S (fom_value_pfp_gcd_greatest_result_greatest_productleft) = S ((S (fom_index_pfp_gcd_greatest_result_greatest_productleft)) * pfgd_qc_gcd_greatest_result_greatest)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_result_greatest_productleft_entry. pfgd_qb_gcd_greatest_result_greatest = fom_beta_quotient_pfp_gcd_greatest_result_greatest_productleft_entry * S ((S (fom_index_pfp_gcd_greatest_result_greatest_productleft)) * pfgd_qc_gcd_greatest_result_greatest) + (fom_value_pfp_gcd_greatest_result_greatest_productleft))) /\\ (exists fom_gap_pfp_gcd_greatest_result_greatest_productleft_value_bound. fom_gap_pfp_gcd_greatest_result_greatest_productleft_value_bound + S (fom_value_pfp_gcd_greatest_result_greatest_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_greatest_result_greatest_productright. (exists fom_gap_pfp_gcd_greatest_result_greatest_productright_index_bound. fom_gap_pfp_gcd_greatest_result_greatest_productright_index_bound + S (fom_index_pfp_gcd_greatest_result_greatest_productright) = pfgg_D_gcd_greatest_result) -> exists fom_value_pfp_gcd_greatest_result_greatest_productright. ((((exists fom_beta_height_pfp_gcd_greatest_result_greatest_productright_entry. fom_beta_height_pfp_gcd_greatest_result_greatest_productright_entry + S (fom_value_pfp_gcd_greatest_result_greatest_productright) = S ((S (fom_index_pfp_gcd_greatest_result_greatest_productright)) * pfgg_dc_gcd_greatest_result)) /\\ exists fom_beta_quotient_pfp_gcd_greatest_result_greatest_productright_entry. pfgg_db_gcd_greatest_result = fom_beta_quotient_pfp_gcd_greatest_result_greatest_productright_entry * S ((S (fom_index_pfp_gcd_greatest_result_greatest_productright)) * pfgg_dc_gcd_greatest_result) + (fom_value_pfp_gcd_greatest_result_greatest_productright))) /\\ (exists fom_gap_pfp_gcd_greatest_result_greatest_productright_value_bound. fom_gap_pfp_gcd_greatest_result_greatest_productright_value_bound + S (fom_value_pfp_gcd_greatest_result_greatest_productright) = p))) /\\ (((((((pfgd_Q_gcd_greatest_result_greatest)=0 \\/ (pfgg_D_gcd_greatest_result)=0) /\\ (((pfgd_P_gcd_greatest_result_greatest)=0)))) \\/ (((~((pfgd_Q_gcd_greatest_result_greatest)=0)) /\\ (((~((pfgg_D_gcd_greatest_result)=0)) /\\ (((pfgd_Q_gcd_greatest_result_greatest)+(pfgg_D_gcd_greatest_result)=S (pfgd_P_gcd_greatest_result_greatest)))))))) /\\ ((forall pfc_index_gcd_greatest_result_greatest_productcoefficients. (exists pfa_gap_gcd_greatest_result_greatest_productcoefficientsbound. pfa_gap_gcd_greatest_result_greatest_productcoefficientsbound + S (pfc_index_gcd_greatest_result_greatest_productcoefficients) = (pfgd_P_gcd_greatest_result_greatest)) -> exists pfc_value_gcd_greatest_result_greatest_productcoefficients. ((((exists ff_h_pfp_gcd_greatest_result_greatest_productcoefficientsentry. ff_h_pfp_gcd_greatest_result_greatest_productcoefficientsentry + S (pfc_value_gcd_greatest_result_greatest_productcoefficients) = S ((S (pfc_index_gcd_greatest_result_greatest_productcoefficients)) * pfgd_pc_gcd_greatest_result_greatest)) /\\ exists ff_q_pfp_gcd_greatest_result_greatest_productcoefficientsentry. pfgd_pb_gcd_greatest_result_greatest = ff_q_pfp_gcd_greatest_result_greatest_productcoefficientsentry * S ((S (pfc_index_gcd_greatest_result_greatest_productcoefficients)) * pfgd_pc_gcd_greatest_result_greatest) + (pfc_value_gcd_greatest_result_greatest_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_greatest_result_greatest_productcoefficientscoefficient pfc_terms_scale_gcd_greatest_result_greatest_productcoefficientscoefficient pfc_natural_sum_gcd_greatest_result_greatest_productcoefficientscoefficient. ((forall pfc_index_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_greatest_result_greatest_productcoefficients))) -> exists pfc_value_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_result_greatest_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_greatest_result_greatest_productcoefficientscoefficient = ff_q_pfp_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_greatest_result_greatest_productcoefficientscoefficient) + (pfc_value_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm pfc_left_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm pfc_right_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_greatest_result_greatest_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonal) = (pfgd_Q_gcd_greatest_result_greatest)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_greatest_result_greatest)) /\\ exists ff_q_pfp_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_gcd_greatest_result_greatest = ff_q_pfp_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonal)) * pfgd_qc_gcd_greatest_result_greatest) + (pfc_left_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_gcd_greatest_result_greatest)=(pfc_index_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm) = (pfgg_D_gcd_greatest_result)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm)) * pfgg_dc_gcd_greatest_result)) /\\ exists ff_q_pfp_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermrightentry. pfgg_db_gcd_greatest_result = ff_q_pfp_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm)) * pfgg_dc_gcd_greatest_result) + (pfc_right_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonaltermrightoutside+(pfgg_D_gcd_greatest_result)=(pfc_complement_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonal)=pfc_left_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_greatest_result_greatest_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum fs_v_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_greatest_result_greatest_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_greatest_result_greatest_productcoefficients))) * fs_v_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_greatest_result_greatest_productcoefficients))) * fs_v_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_greatest_result_greatest_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_greatest_result_greatest_productcoefficients)) -> exists fs_a_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_result_greatest_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_greatest_result_greatest_productcoefficientscoefficient = fs_q_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_greatest_result_greatest_productcoefficientscoefficient) + (fs_a_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum = fs_q_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_greatest_result_greatest_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_greatest_result_greatest_productcoefficientscoefficientresiduebound. pfa_gap_gcd_greatest_result_greatest_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_greatest_result_greatest_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_greatest_result_greatest_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_greatest_result_greatest_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_greatest_result_greatest_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_greatest_result_greatest_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_greatest_result_greatest_productcoefficients) + (p) * pfa_offset_right_gcd_greatest_result_greatest_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_greatest_result_greatest_equivalent pfrep_left_gcd_greatest_result_greatest_equivalent pfrep_right_gcd_greatest_result_greatest_equivalent. ((exists pfrep_position_gcd_greatest_result_greatest_equivalentfirst. ((pfrep_position_gcd_greatest_result_greatest_equivalentfirst+S (pfrep_power_gcd_greatest_result_greatest_equivalent)=(pfgd_P_gcd_greatest_result_greatest)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_greatest_equivalentfirstentry. ff_h_pfp_gcd_greatest_result_greatest_equivalentfirstentry + S (pfrep_left_gcd_greatest_result_greatest_equivalent) = S ((S (pfrep_position_gcd_greatest_result_greatest_equivalentfirst)) * pfgd_pc_gcd_greatest_result_greatest)) /\\ exists ff_q_pfp_gcd_greatest_result_greatest_equivalentfirstentry. pfgd_pb_gcd_greatest_result_greatest = ff_q_pfp_gcd_greatest_result_greatest_equivalentfirstentry * S ((S (pfrep_position_gcd_greatest_result_greatest_equivalentfirst)) * pfgd_pc_gcd_greatest_result_greatest) + (pfrep_left_gcd_greatest_result_greatest_equivalent)))))) \\/ (((exists pfrep_gap_gcd_greatest_result_greatest_equivalentfirstoutside. pfrep_gap_gcd_greatest_result_greatest_equivalentfirstoutside+(pfgd_P_gcd_greatest_result_greatest)=(pfrep_power_gcd_greatest_result_greatest_equivalent)) /\\ (((pfrep_left_gcd_greatest_result_greatest_equivalent)=0))))) -> ((exists pfrep_position_gcd_greatest_result_greatest_equivalentsecond. ((pfrep_position_gcd_greatest_result_greatest_equivalentsecond+S (pfrep_power_gcd_greatest_result_greatest_equivalent)=(G)) /\\ ((((exists ff_h_pfp_gcd_greatest_result_greatest_equivalentsecondentry. ff_h_pfp_gcd_greatest_result_greatest_equivalentsecondentry + S (pfrep_right_gcd_greatest_result_greatest_equivalent) = S ((S (pfrep_position_gcd_greatest_result_greatest_equivalentsecond)) * gc)) /\\ exists ff_q_pfp_gcd_greatest_result_greatest_equivalentsecondentry. gb = ff_q_pfp_gcd_greatest_result_greatest_equivalentsecondentry * S ((S (pfrep_position_gcd_greatest_result_greatest_equivalentsecond)) * gc) + (pfrep_right_gcd_greatest_result_greatest_equivalent)))))) \\/ (((exists pfrep_gap_gcd_greatest_result_greatest_equivalentsecondoutside. pfrep_gap_gcd_greatest_result_greatest_equivalentsecondoutside+(G)=(pfrep_power_gcd_greatest_result_greatest_equivalent)) /\\ (((pfrep_right_gcd_greatest_result_greatest_equivalent)=0))))) -> pfrep_left_gcd_greatest_result_greatest_equivalent=pfrep_right_gcd_greatest_result_greatest_equivalent)))))))))))", "statement_sha256": "91a89630be8631cd892a7e0dd57bc4a36c2f3a3b734b16f12390124493a0ab43", "summary": "A common right divisor with an actual Bezout representation satisfies the full universally quantified greatestness property, including a zero gcd; 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(((pfgs_G_normalized_exists_solution)=0 \\/ (((~((pfgs_G_normalized_exists_solution) = 0)) /\\ (((forall fom_index_pfp_normalized_exists_solution_witness_normal_moniccoefficients. (exists fom_gap_pfp_normalized_exists_solution_witness_normal_moniccoefficients_index_bound. fom_gap_pfp_normalized_exists_solution_witness_normal_moniccoefficients_index_bound + S (fom_index_pfp_normalized_exists_solution_witness_normal_moniccoefficients) = pfgs_G_normalized_exists_solution) -> exists fom_value_pfp_normalized_exists_solution_witness_normal_moniccoefficients. ((((exists fom_beta_height_pfp_normalized_exists_solution_witness_normal_moniccoefficients_entry. fom_beta_height_pfp_normalized_exists_solution_witness_normal_moniccoefficients_entry + S (fom_value_pfp_normalized_exists_solution_witness_normal_moniccoefficients) = S ((S (fom_index_pfp_normalized_exists_solution_witness_normal_moniccoefficients)) * pfgs_gc_normalized_exists_solution)) /\\ exists fom_beta_quotient_pfp_normalized_exists_solution_witness_normal_moniccoefficients_entry. pfgs_gb_normalized_exists_solution = fom_beta_quotient_pfp_normalized_exists_solution_witness_normal_moniccoefficients_entry * S ((S (fom_index_pfp_normalized_exists_solution_witness_normal_moniccoefficients)) * pfgs_gc_normalized_exists_solution) + (fom_value_pfp_normalized_exists_solution_witness_normal_moniccoefficients))) /\\ (exists fom_gap_pfp_normalized_exists_solution_witness_normal_moniccoefficients_value_bound. fom_gap_pfp_normalized_exists_solution_witness_normal_moniccoefficients_value_bound + S (fom_value_pfp_normalized_exists_solution_witness_normal_moniccoefficients) = p))) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_normal_monicleading. ff_h_pfp_normalized_exists_solution_witness_normal_monicleading + S (1) = S ((S (0)) * pfgs_gc_normalized_exists_solution)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_normal_monicleading. pfgs_gb_normalized_exists_solution = ff_q_pfp_normalized_exists_solution_witness_normal_monicleading * S ((S (0)) * pfgs_gc_normalized_exists_solution) + (1))))))))) /\\ (((((((forall fom_index_pfp_normalized_exists_solution_witness_common_left_bounded. (exists fom_gap_pfp_normalized_exists_solution_witness_common_left_bounded_index_bound. fom_gap_pfp_normalized_exists_solution_witness_common_left_bounded_index_bound + S (fom_index_pfp_normalized_exists_solution_witness_common_left_bounded) = L) -> exists fom_value_pfp_normalized_exists_solution_witness_common_left_bounded. ((((exists fom_beta_height_pfp_normalized_exists_solution_witness_common_left_bounded_entry. fom_beta_height_pfp_normalized_exists_solution_witness_common_left_bounded_entry + S (fom_value_pfp_normalized_exists_solution_witness_common_left_bounded) = S ((S (fom_index_pfp_normalized_exists_solution_witness_common_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_normalized_exists_solution_witness_common_left_bounded_entry. ab = fom_beta_quotient_pfp_normalized_exists_solution_witness_common_left_bounded_entry * S ((S (fom_index_pfp_normalized_exists_solution_witness_common_left_bounded)) * ac) + (fom_value_pfp_normalized_exists_solution_witness_common_left_bounded))) /\\ (exists fom_gap_pfp_normalized_exists_solution_witness_common_left_bounded_value_bound. fom_gap_pfp_normalized_exists_solution_witness_common_left_bounded_value_bound + S (fom_value_pfp_normalized_exists_solution_witness_common_left_bounded) = p))) /\\ ((exists pfgd_qb_normalized_exists_solution_witness_common_left pfgd_qc_normalized_exists_solution_witness_common_left pfgd_Q_normalized_exists_solution_witness_common_left pfgd_pb_normalized_exists_solution_witness_common_left pfgd_pc_normalized_exists_solution_witness_common_left pfgd_P_normalized_exists_solution_witness_common_left. ((((forall fom_index_pfp_normalized_exists_solution_witness_common_left_productleft. (exists fom_gap_pfp_normalized_exists_solution_witness_common_left_productleft_index_bound. fom_gap_pfp_normalized_exists_solution_witness_common_left_productleft_index_bound + S (fom_index_pfp_normalized_exists_solution_witness_common_left_productleft) = pfgd_Q_normalized_exists_solution_witness_common_left) -> exists fom_value_pfp_normalized_exists_solution_witness_common_left_productleft. ((((exists fom_beta_height_pfp_normalized_exists_solution_witness_common_left_productleft_entry. fom_beta_height_pfp_normalized_exists_solution_witness_common_left_productleft_entry + S (fom_value_pfp_normalized_exists_solution_witness_common_left_productleft) = S ((S (fom_index_pfp_normalized_exists_solution_witness_common_left_productleft)) * pfgd_qc_normalized_exists_solution_witness_common_left)) /\\ exists fom_beta_quotient_pfp_normalized_exists_solution_witness_common_left_productleft_entry. pfgd_qb_normalized_exists_solution_witness_common_left = fom_beta_quotient_pfp_normalized_exists_solution_witness_common_left_productleft_entry * S ((S (fom_index_pfp_normalized_exists_solution_witness_common_left_productleft)) * pfgd_qc_normalized_exists_solution_witness_common_left) + (fom_value_pfp_normalized_exists_solution_witness_common_left_productleft))) /\\ (exists fom_gap_pfp_normalized_exists_solution_witness_common_left_productleft_value_bound. fom_gap_pfp_normalized_exists_solution_witness_common_left_productleft_value_bound + S (fom_value_pfp_normalized_exists_solution_witness_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_normalized_exists_solution_witness_common_left_productright. (exists fom_gap_pfp_normalized_exists_solution_witness_common_left_productright_index_bound. fom_gap_pfp_normalized_exists_solution_witness_common_left_productright_index_bound + S (fom_index_pfp_normalized_exists_solution_witness_common_left_productright) = pfgs_G_normalized_exists_solution) -> exists fom_value_pfp_normalized_exists_solution_witness_common_left_productright. ((((exists fom_beta_height_pfp_normalized_exists_solution_witness_common_left_productright_entry. fom_beta_height_pfp_normalized_exists_solution_witness_common_left_productright_entry + S (fom_value_pfp_normalized_exists_solution_witness_common_left_productright) = S ((S (fom_index_pfp_normalized_exists_solution_witness_common_left_productright)) * pfgs_gc_normalized_exists_solution)) /\\ exists fom_beta_quotient_pfp_normalized_exists_solution_witness_common_left_productright_entry. pfgs_gb_normalized_exists_solution = fom_beta_quotient_pfp_normalized_exists_solution_witness_common_left_productright_entry * S ((S (fom_index_pfp_normalized_exists_solution_witness_common_left_productright)) * pfgs_gc_normalized_exists_solution) + (fom_value_pfp_normalized_exists_solution_witness_common_left_productright))) /\\ (exists fom_gap_pfp_normalized_exists_solution_witness_common_left_productright_value_bound. fom_gap_pfp_normalized_exists_solution_witness_common_left_productright_value_bound + S (fom_value_pfp_normalized_exists_solution_witness_common_left_productright) = p))) /\\ (((((((pfgd_Q_normalized_exists_solution_witness_common_left)=0 \\/ (pfgs_G_normalized_exists_solution)=0) /\\ (((pfgd_P_normalized_exists_solution_witness_common_left)=0)))) \\/ (((~((pfgd_Q_normalized_exists_solution_witness_common_left)=0)) /\\ (((~((pfgs_G_normalized_exists_solution)=0)) /\\ (((pfgd_Q_normalized_exists_solution_witness_common_left)+(pfgs_G_normalized_exists_solution)=S (pfgd_P_normalized_exists_solution_witness_common_left)))))))) /\\ ((forall pfc_index_normalized_exists_solution_witness_common_left_productcoefficients. (exists pfa_gap_normalized_exists_solution_witness_common_left_productcoefficientsbound. pfa_gap_normalized_exists_solution_witness_common_left_productcoefficientsbound + S (pfc_index_normalized_exists_solution_witness_common_left_productcoefficients) = (pfgd_P_normalized_exists_solution_witness_common_left)) -> exists pfc_value_normalized_exists_solution_witness_common_left_productcoefficients. ((((exists ff_h_pfp_normalized_exists_solution_witness_common_left_productcoefficientsentry. ff_h_pfp_normalized_exists_solution_witness_common_left_productcoefficientsentry + S (pfc_value_normalized_exists_solution_witness_common_left_productcoefficients) = S ((S (pfc_index_normalized_exists_solution_witness_common_left_productcoefficients)) * pfgd_pc_normalized_exists_solution_witness_common_left)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_common_left_productcoefficientsentry. pfgd_pb_normalized_exists_solution_witness_common_left = ff_q_pfp_normalized_exists_solution_witness_common_left_productcoefficientsentry * S ((S (pfc_index_normalized_exists_solution_witness_common_left_productcoefficients)) * pfgd_pc_normalized_exists_solution_witness_common_left) + (pfc_value_normalized_exists_solution_witness_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_normalized_exists_solution_witness_common_left_productcoefficientscoefficient pfc_terms_scale_normalized_exists_solution_witness_common_left_productcoefficientscoefficient pfc_natural_sum_normalized_exists_solution_witness_common_left_productcoefficientscoefficient. ((forall pfc_index_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_normalized_exists_solution_witness_common_left_productcoefficients))) -> exists pfc_value_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_solution_witness_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normalized_exists_solution_witness_common_left_productcoefficientscoefficient = ff_q_pfp_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_solution_witness_common_left_productcoefficientscoefficient) + (pfc_value_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_left_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm pfc_right_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_normalized_exists_solution_witness_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_normalized_exists_solution_witness_common_left)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_normalized_exists_solution_witness_common_left)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_normalized_exists_solution_witness_common_left = ff_q_pfp_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_normalized_exists_solution_witness_common_left) + (pfc_left_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_normalized_exists_solution_witness_common_left)=(pfc_index_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm) = (pfgs_G_normalized_exists_solution)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_normalized_exists_solution)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_normalized_exists_solution = ff_q_pfp_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_normalized_exists_solution) + (pfc_right_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_normalized_exists_solution)=(pfc_complement_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonal)=pfc_left_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_normalized_exists_solution_witness_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum fs_v_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalized_exists_solution_witness_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_normalized_exists_solution_witness_common_left_productcoefficients))) * fs_v_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalized_exists_solution_witness_common_left_productcoefficients))) * fs_v_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_normalized_exists_solution_witness_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_normalized_exists_solution_witness_common_left_productcoefficients)) -> exists fs_a_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_solution_witness_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalized_exists_solution_witness_common_left_productcoefficientscoefficient = fs_q_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_solution_witness_common_left_productcoefficientscoefficient) + (fs_a_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normalized_exists_solution_witness_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalized_exists_solution_witness_common_left_productcoefficientscoefficientresiduebound. pfa_gap_normalized_exists_solution_witness_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_normalized_exists_solution_witness_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalized_exists_solution_witness_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_normalized_exists_solution_witness_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalized_exists_solution_witness_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_normalized_exists_solution_witness_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_normalized_exists_solution_witness_common_left_productcoefficients) + (p) * pfa_offset_right_normalized_exists_solution_witness_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normalized_exists_solution_witness_common_left_equivalent pfrep_left_normalized_exists_solution_witness_common_left_equivalent pfrep_right_normalized_exists_solution_witness_common_left_equivalent. ((exists pfrep_position_normalized_exists_solution_witness_common_left_equivalentfirst. ((pfrep_position_normalized_exists_solution_witness_common_left_equivalentfirst+S (pfrep_power_normalized_exists_solution_witness_common_left_equivalent)=(pfgd_P_normalized_exists_solution_witness_common_left)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_common_left_equivalentfirstentry. ff_h_pfp_normalized_exists_solution_witness_common_left_equivalentfirstentry + S (pfrep_left_normalized_exists_solution_witness_common_left_equivalent) = S ((S (pfrep_position_normalized_exists_solution_witness_common_left_equivalentfirst)) * pfgd_pc_normalized_exists_solution_witness_common_left)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_common_left_equivalentfirstentry. pfgd_pb_normalized_exists_solution_witness_common_left = ff_q_pfp_normalized_exists_solution_witness_common_left_equivalentfirstentry * S ((S (pfrep_position_normalized_exists_solution_witness_common_left_equivalentfirst)) * pfgd_pc_normalized_exists_solution_witness_common_left) + (pfrep_left_normalized_exists_solution_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_normalized_exists_solution_witness_common_left_equivalentfirstoutside. pfrep_gap_normalized_exists_solution_witness_common_left_equivalentfirstoutside+(pfgd_P_normalized_exists_solution_witness_common_left)=(pfrep_power_normalized_exists_solution_witness_common_left_equivalent)) /\\ (((pfrep_left_normalized_exists_solution_witness_common_left_equivalent)=0))))) -> ((exists pfrep_position_normalized_exists_solution_witness_common_left_equivalentsecond. ((pfrep_position_normalized_exists_solution_witness_common_left_equivalentsecond+S (pfrep_power_normalized_exists_solution_witness_common_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_common_left_equivalentsecondentry. ff_h_pfp_normalized_exists_solution_witness_common_left_equivalentsecondentry + S (pfrep_right_normalized_exists_solution_witness_common_left_equivalent) = S ((S (pfrep_position_normalized_exists_solution_witness_common_left_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_common_left_equivalentsecondentry. ab = ff_q_pfp_normalized_exists_solution_witness_common_left_equivalentsecondentry * S ((S (pfrep_position_normalized_exists_solution_witness_common_left_equivalentsecond)) * ac) + (pfrep_right_normalized_exists_solution_witness_common_left_equivalent)))))) \\/ (((exists pfrep_gap_normalized_exists_solution_witness_common_left_equivalentsecondoutside. pfrep_gap_normalized_exists_solution_witness_common_left_equivalentsecondoutside+(L)=(pfrep_power_normalized_exists_solution_witness_common_left_equivalent)) /\\ (((pfrep_right_normalized_exists_solution_witness_common_left_equivalent)=0))))) -> pfrep_left_normalized_exists_solution_witness_common_left_equivalent=pfrep_right_normalized_exists_solution_witness_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_normalized_exists_solution_witness_common_right_bounded. (exists fom_gap_pfp_normalized_exists_solution_witness_common_right_bounded_index_bound. fom_gap_pfp_normalized_exists_solution_witness_common_right_bounded_index_bound + S (fom_index_pfp_normalized_exists_solution_witness_common_right_bounded) = M) -> exists fom_value_pfp_normalized_exists_solution_witness_common_right_bounded. ((((exists fom_beta_height_pfp_normalized_exists_solution_witness_common_right_bounded_entry. fom_beta_height_pfp_normalized_exists_solution_witness_common_right_bounded_entry + S (fom_value_pfp_normalized_exists_solution_witness_common_right_bounded) = S ((S (fom_index_pfp_normalized_exists_solution_witness_common_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_normalized_exists_solution_witness_common_right_bounded_entry. bb = fom_beta_quotient_pfp_normalized_exists_solution_witness_common_right_bounded_entry * S ((S (fom_index_pfp_normalized_exists_solution_witness_common_right_bounded)) * bc) + (fom_value_pfp_normalized_exists_solution_witness_common_right_bounded))) /\\ (exists fom_gap_pfp_normalized_exists_solution_witness_common_right_bounded_value_bound. fom_gap_pfp_normalized_exists_solution_witness_common_right_bounded_value_bound + S (fom_value_pfp_normalized_exists_solution_witness_common_right_bounded) = p))) /\\ ((exists pfgd_qb_normalized_exists_solution_witness_common_right pfgd_qc_normalized_exists_solution_witness_common_right pfgd_Q_normalized_exists_solution_witness_common_right pfgd_pb_normalized_exists_solution_witness_common_right pfgd_pc_normalized_exists_solution_witness_common_right pfgd_P_normalized_exists_solution_witness_common_right. ((((forall fom_index_pfp_normalized_exists_solution_witness_common_right_productleft. (exists fom_gap_pfp_normalized_exists_solution_witness_common_right_productleft_index_bound. fom_gap_pfp_normalized_exists_solution_witness_common_right_productleft_index_bound + S (fom_index_pfp_normalized_exists_solution_witness_common_right_productleft) = pfgd_Q_normalized_exists_solution_witness_common_right) -> exists fom_value_pfp_normalized_exists_solution_witness_common_right_productleft. ((((exists fom_beta_height_pfp_normalized_exists_solution_witness_common_right_productleft_entry. fom_beta_height_pfp_normalized_exists_solution_witness_common_right_productleft_entry + S (fom_value_pfp_normalized_exists_solution_witness_common_right_productleft) = S ((S (fom_index_pfp_normalized_exists_solution_witness_common_right_productleft)) * pfgd_qc_normalized_exists_solution_witness_common_right)) /\\ exists fom_beta_quotient_pfp_normalized_exists_solution_witness_common_right_productleft_entry. pfgd_qb_normalized_exists_solution_witness_common_right = fom_beta_quotient_pfp_normalized_exists_solution_witness_common_right_productleft_entry * S ((S (fom_index_pfp_normalized_exists_solution_witness_common_right_productleft)) * pfgd_qc_normalized_exists_solution_witness_common_right) + (fom_value_pfp_normalized_exists_solution_witness_common_right_productleft))) /\\ (exists fom_gap_pfp_normalized_exists_solution_witness_common_right_productleft_value_bound. fom_gap_pfp_normalized_exists_solution_witness_common_right_productleft_value_bound + S (fom_value_pfp_normalized_exists_solution_witness_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_normalized_exists_solution_witness_common_right_productright. (exists fom_gap_pfp_normalized_exists_solution_witness_common_right_productright_index_bound. fom_gap_pfp_normalized_exists_solution_witness_common_right_productright_index_bound + S (fom_index_pfp_normalized_exists_solution_witness_common_right_productright) = pfgs_G_normalized_exists_solution) -> exists fom_value_pfp_normalized_exists_solution_witness_common_right_productright. ((((exists fom_beta_height_pfp_normalized_exists_solution_witness_common_right_productright_entry. fom_beta_height_pfp_normalized_exists_solution_witness_common_right_productright_entry + S (fom_value_pfp_normalized_exists_solution_witness_common_right_productright) = S ((S (fom_index_pfp_normalized_exists_solution_witness_common_right_productright)) * pfgs_gc_normalized_exists_solution)) /\\ exists fom_beta_quotient_pfp_normalized_exists_solution_witness_common_right_productright_entry. pfgs_gb_normalized_exists_solution = fom_beta_quotient_pfp_normalized_exists_solution_witness_common_right_productright_entry * S ((S (fom_index_pfp_normalized_exists_solution_witness_common_right_productright)) * pfgs_gc_normalized_exists_solution) + (fom_value_pfp_normalized_exists_solution_witness_common_right_productright))) /\\ (exists fom_gap_pfp_normalized_exists_solution_witness_common_right_productright_value_bound. fom_gap_pfp_normalized_exists_solution_witness_common_right_productright_value_bound + S (fom_value_pfp_normalized_exists_solution_witness_common_right_productright) = p))) /\\ (((((((pfgd_Q_normalized_exists_solution_witness_common_right)=0 \\/ (pfgs_G_normalized_exists_solution)=0) /\\ (((pfgd_P_normalized_exists_solution_witness_common_right)=0)))) \\/ (((~((pfgd_Q_normalized_exists_solution_witness_common_right)=0)) /\\ (((~((pfgs_G_normalized_exists_solution)=0)) /\\ (((pfgd_Q_normalized_exists_solution_witness_common_right)+(pfgs_G_normalized_exists_solution)=S (pfgd_P_normalized_exists_solution_witness_common_right)))))))) /\\ ((forall pfc_index_normalized_exists_solution_witness_common_right_productcoefficients. (exists pfa_gap_normalized_exists_solution_witness_common_right_productcoefficientsbound. pfa_gap_normalized_exists_solution_witness_common_right_productcoefficientsbound + S (pfc_index_normalized_exists_solution_witness_common_right_productcoefficients) = (pfgd_P_normalized_exists_solution_witness_common_right)) -> exists pfc_value_normalized_exists_solution_witness_common_right_productcoefficients. ((((exists ff_h_pfp_normalized_exists_solution_witness_common_right_productcoefficientsentry. ff_h_pfp_normalized_exists_solution_witness_common_right_productcoefficientsentry + S (pfc_value_normalized_exists_solution_witness_common_right_productcoefficients) = S ((S (pfc_index_normalized_exists_solution_witness_common_right_productcoefficients)) * pfgd_pc_normalized_exists_solution_witness_common_right)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_common_right_productcoefficientsentry. pfgd_pb_normalized_exists_solution_witness_common_right = ff_q_pfp_normalized_exists_solution_witness_common_right_productcoefficientsentry * S ((S (pfc_index_normalized_exists_solution_witness_common_right_productcoefficients)) * pfgd_pc_normalized_exists_solution_witness_common_right) + (pfc_value_normalized_exists_solution_witness_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_normalized_exists_solution_witness_common_right_productcoefficientscoefficient pfc_terms_scale_normalized_exists_solution_witness_common_right_productcoefficientscoefficient pfc_natural_sum_normalized_exists_solution_witness_common_right_productcoefficientscoefficient. ((forall pfc_index_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_normalized_exists_solution_witness_common_right_productcoefficients))) -> exists pfc_value_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_solution_witness_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normalized_exists_solution_witness_common_right_productcoefficientscoefficient = ff_q_pfp_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_solution_witness_common_right_productcoefficientscoefficient) + (pfc_value_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_left_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm pfc_right_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_normalized_exists_solution_witness_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_normalized_exists_solution_witness_common_right)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_normalized_exists_solution_witness_common_right)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_normalized_exists_solution_witness_common_right = ff_q_pfp_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_normalized_exists_solution_witness_common_right) + (pfc_left_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_normalized_exists_solution_witness_common_right)=(pfc_index_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm) = (pfgs_G_normalized_exists_solution)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_normalized_exists_solution)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry. pfgs_gb_normalized_exists_solution = ff_q_pfp_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm)) * pfgs_gc_normalized_exists_solution) + (pfc_right_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(pfgs_G_normalized_exists_solution)=(pfc_complement_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonal)=pfc_left_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_normalized_exists_solution_witness_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum fs_v_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalized_exists_solution_witness_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_normalized_exists_solution_witness_common_right_productcoefficients))) * fs_v_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalized_exists_solution_witness_common_right_productcoefficients))) * fs_v_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_normalized_exists_solution_witness_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_normalized_exists_solution_witness_common_right_productcoefficients)) -> exists fs_a_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_solution_witness_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalized_exists_solution_witness_common_right_productcoefficientscoefficient = fs_q_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_solution_witness_common_right_productcoefficientscoefficient) + (fs_a_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normalized_exists_solution_witness_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalized_exists_solution_witness_common_right_productcoefficientscoefficientresiduebound. pfa_gap_normalized_exists_solution_witness_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_normalized_exists_solution_witness_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalized_exists_solution_witness_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_normalized_exists_solution_witness_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalized_exists_solution_witness_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_normalized_exists_solution_witness_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_normalized_exists_solution_witness_common_right_productcoefficients) + (p) * pfa_offset_right_normalized_exists_solution_witness_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normalized_exists_solution_witness_common_right_equivalent pfrep_left_normalized_exists_solution_witness_common_right_equivalent pfrep_right_normalized_exists_solution_witness_common_right_equivalent. ((exists pfrep_position_normalized_exists_solution_witness_common_right_equivalentfirst. ((pfrep_position_normalized_exists_solution_witness_common_right_equivalentfirst+S (pfrep_power_normalized_exists_solution_witness_common_right_equivalent)=(pfgd_P_normalized_exists_solution_witness_common_right)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_common_right_equivalentfirstentry. ff_h_pfp_normalized_exists_solution_witness_common_right_equivalentfirstentry + S (pfrep_left_normalized_exists_solution_witness_common_right_equivalent) = S ((S (pfrep_position_normalized_exists_solution_witness_common_right_equivalentfirst)) * pfgd_pc_normalized_exists_solution_witness_common_right)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_common_right_equivalentfirstentry. pfgd_pb_normalized_exists_solution_witness_common_right = ff_q_pfp_normalized_exists_solution_witness_common_right_equivalentfirstentry * S ((S (pfrep_position_normalized_exists_solution_witness_common_right_equivalentfirst)) * pfgd_pc_normalized_exists_solution_witness_common_right) + (pfrep_left_normalized_exists_solution_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_normalized_exists_solution_witness_common_right_equivalentfirstoutside. pfrep_gap_normalized_exists_solution_witness_common_right_equivalentfirstoutside+(pfgd_P_normalized_exists_solution_witness_common_right)=(pfrep_power_normalized_exists_solution_witness_common_right_equivalent)) /\\ (((pfrep_left_normalized_exists_solution_witness_common_right_equivalent)=0))))) -> ((exists pfrep_position_normalized_exists_solution_witness_common_right_equivalentsecond. ((pfrep_position_normalized_exists_solution_witness_common_right_equivalentsecond+S (pfrep_power_normalized_exists_solution_witness_common_right_equivalent)=(M)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_common_right_equivalentsecondentry. ff_h_pfp_normalized_exists_solution_witness_common_right_equivalentsecondentry + S (pfrep_right_normalized_exists_solution_witness_common_right_equivalent) = S ((S (pfrep_position_normalized_exists_solution_witness_common_right_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_common_right_equivalentsecondentry. bb = ff_q_pfp_normalized_exists_solution_witness_common_right_equivalentsecondentry * S ((S (pfrep_position_normalized_exists_solution_witness_common_right_equivalentsecond)) * bc) + (pfrep_right_normalized_exists_solution_witness_common_right_equivalent)))))) \\/ (((exists pfrep_gap_normalized_exists_solution_witness_common_right_equivalentsecondoutside. pfrep_gap_normalized_exists_solution_witness_common_right_equivalentsecondoutside+(M)=(pfrep_power_normalized_exists_solution_witness_common_right_equivalent)) /\\ (((pfrep_right_normalized_exists_solution_witness_common_right_equivalent)=0))))) -> pfrep_left_normalized_exists_solution_witness_common_right_equivalent=pfrep_right_normalized_exists_solution_witness_common_right_equivalent)))))))))) /\\ ((exists pfgb_pb_normalized_exists_solution_witness_bezout pfgb_pc_normalized_exists_solution_witness_bezout pfgb_P_normalized_exists_solution_witness_bezout pfgb_qb_normalized_exists_solution_witness_bezout pfgb_qc_normalized_exists_solution_witness_bezout pfgb_Q_normalized_exists_solution_witness_bezout. ((((forall fom_index_pfp_normalized_exists_solution_witness_bezout_leftleft. (exists fom_gap_pfp_normalized_exists_solution_witness_bezout_leftleft_index_bound. fom_gap_pfp_normalized_exists_solution_witness_bezout_leftleft_index_bound + S (fom_index_pfp_normalized_exists_solution_witness_bezout_leftleft) = pfgs_U_normalized_exists_solution) -> exists fom_value_pfp_normalized_exists_solution_witness_bezout_leftleft. ((((exists fom_beta_height_pfp_normalized_exists_solution_witness_bezout_leftleft_entry. fom_beta_height_pfp_normalized_exists_solution_witness_bezout_leftleft_entry + S (fom_value_pfp_normalized_exists_solution_witness_bezout_leftleft) = S ((S (fom_index_pfp_normalized_exists_solution_witness_bezout_leftleft)) * pfgs_uc_normalized_exists_solution)) /\\ exists fom_beta_quotient_pfp_normalized_exists_solution_witness_bezout_leftleft_entry. pfgs_ub_normalized_exists_solution = fom_beta_quotient_pfp_normalized_exists_solution_witness_bezout_leftleft_entry * S ((S (fom_index_pfp_normalized_exists_solution_witness_bezout_leftleft)) * pfgs_uc_normalized_exists_solution) + (fom_value_pfp_normalized_exists_solution_witness_bezout_leftleft))) /\\ (exists fom_gap_pfp_normalized_exists_solution_witness_bezout_leftleft_value_bound. fom_gap_pfp_normalized_exists_solution_witness_bezout_leftleft_value_bound + S (fom_value_pfp_normalized_exists_solution_witness_bezout_leftleft) = p))) /\\ (((forall fom_index_pfp_normalized_exists_solution_witness_bezout_leftright. (exists fom_gap_pfp_normalized_exists_solution_witness_bezout_leftright_index_bound. fom_gap_pfp_normalized_exists_solution_witness_bezout_leftright_index_bound + S (fom_index_pfp_normalized_exists_solution_witness_bezout_leftright) = L) -> exists fom_value_pfp_normalized_exists_solution_witness_bezout_leftright. ((((exists fom_beta_height_pfp_normalized_exists_solution_witness_bezout_leftright_entry. fom_beta_height_pfp_normalized_exists_solution_witness_bezout_leftright_entry + S (fom_value_pfp_normalized_exists_solution_witness_bezout_leftright) = S ((S (fom_index_pfp_normalized_exists_solution_witness_bezout_leftright)) * ac)) /\\ exists fom_beta_quotient_pfp_normalized_exists_solution_witness_bezout_leftright_entry. ab = fom_beta_quotient_pfp_normalized_exists_solution_witness_bezout_leftright_entry * S ((S (fom_index_pfp_normalized_exists_solution_witness_bezout_leftright)) * ac) + (fom_value_pfp_normalized_exists_solution_witness_bezout_leftright))) /\\ (exists fom_gap_pfp_normalized_exists_solution_witness_bezout_leftright_value_bound. fom_gap_pfp_normalized_exists_solution_witness_bezout_leftright_value_bound + S (fom_value_pfp_normalized_exists_solution_witness_bezout_leftright) = p))) /\\ (((((((pfgs_U_normalized_exists_solution)=0 \\/ (L)=0) /\\ (((pfgb_P_normalized_exists_solution_witness_bezout)=0)))) \\/ (((~((pfgs_U_normalized_exists_solution)=0)) /\\ (((~((L)=0)) /\\ (((pfgs_U_normalized_exists_solution)+(L)=S (pfgb_P_normalized_exists_solution_witness_bezout)))))))) /\\ ((forall pfc_index_normalized_exists_solution_witness_bezout_leftcoefficients. (exists pfa_gap_normalized_exists_solution_witness_bezout_leftcoefficientsbound. pfa_gap_normalized_exists_solution_witness_bezout_leftcoefficientsbound + S (pfc_index_normalized_exists_solution_witness_bezout_leftcoefficients) = (pfgb_P_normalized_exists_solution_witness_bezout)) -> exists pfc_value_normalized_exists_solution_witness_bezout_leftcoefficients. ((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_leftcoefficientsentry. ff_h_pfp_normalized_exists_solution_witness_bezout_leftcoefficientsentry + S (pfc_value_normalized_exists_solution_witness_bezout_leftcoefficients) = S ((S (pfc_index_normalized_exists_solution_witness_bezout_leftcoefficients)) * pfgb_pc_normalized_exists_solution_witness_bezout)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_leftcoefficientsentry. pfgb_pb_normalized_exists_solution_witness_bezout = ff_q_pfp_normalized_exists_solution_witness_bezout_leftcoefficientsentry * S ((S (pfc_index_normalized_exists_solution_witness_bezout_leftcoefficients)) * pfgb_pc_normalized_exists_solution_witness_bezout) + (pfc_value_normalized_exists_solution_witness_bezout_leftcoefficients))) /\\ ((exists pfc_terms_code_normalized_exists_solution_witness_bezout_leftcoefficientscoefficient pfc_terms_scale_normalized_exists_solution_witness_bezout_leftcoefficientscoefficient pfc_natural_sum_normalized_exists_solution_witness_bezout_leftcoefficientscoefficient. ((forall pfc_index_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonal. (exists pfa_gap_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalbound. pfa_gap_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalbound + S (pfc_index_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonal) = (S (pfc_index_normalized_exists_solution_witness_bezout_leftcoefficients))) -> exists pfc_value_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalentry + S (pfc_value_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_solution_witness_bezout_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_normalized_exists_solution_witness_bezout_leftcoefficientscoefficient = ff_q_pfp_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_solution_witness_bezout_leftcoefficientscoefficient) + (pfc_value_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_left_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm pfc_right_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm. (((pfc_index_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonal)+pfc_complement_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm=(pfc_index_normalized_exists_solution_witness_bezout_leftcoefficients)) /\\ ((((((exists pfa_gap_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonal) = (pfgs_U_normalized_exists_solution)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_normalized_exists_solution)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry. pfgs_ub_normalized_exists_solution = ff_q_pfp_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonal)) * pfgs_uc_normalized_exists_solution) + (pfc_left_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermleftoutside+(pfgs_U_normalized_exists_solution)=(pfc_index_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonal)=pfc_left_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm*pfc_right_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum fs_v_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_h_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_start. fs_u_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalized_exists_solution_witness_bezout_leftcoefficientscoefficient) = S ((S (S (pfc_index_normalized_exists_solution_witness_bezout_leftcoefficients))) * fs_v_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalized_exists_solution_witness_bezout_leftcoefficients))) * fs_v_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum) + (pfc_natural_sum_normalized_exists_solution_witness_bezout_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps = S (pfc_index_normalized_exists_solution_witness_bezout_leftcoefficients)) -> exists fs_a_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_r_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps fs_s_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_solution_witness_bezout_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalized_exists_solution_witness_bezout_leftcoefficientscoefficient = fs_q_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_solution_witness_bezout_leftcoefficientscoefficient) + (fs_a_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum) + (fs_r_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum) + (fs_s_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientresiduebound. pfa_gap_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientresiduebound + S (pfc_value_normalized_exists_solution_witness_bezout_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientresiduecongruence pfa_offset_right_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalized_exists_solution_witness_bezout_leftcoefficientscoefficient) + (p) * pfa_offset_left_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientresiduecongruence = (pfc_value_normalized_exists_solution_witness_bezout_leftcoefficients) + (p) * pfa_offset_right_normalized_exists_solution_witness_bezout_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_normalized_exists_solution_witness_bezout_rightleft. (exists fom_gap_pfp_normalized_exists_solution_witness_bezout_rightleft_index_bound. fom_gap_pfp_normalized_exists_solution_witness_bezout_rightleft_index_bound + S (fom_index_pfp_normalized_exists_solution_witness_bezout_rightleft) = pfgs_V_normalized_exists_solution) -> exists fom_value_pfp_normalized_exists_solution_witness_bezout_rightleft. ((((exists fom_beta_height_pfp_normalized_exists_solution_witness_bezout_rightleft_entry. fom_beta_height_pfp_normalized_exists_solution_witness_bezout_rightleft_entry + S (fom_value_pfp_normalized_exists_solution_witness_bezout_rightleft) = S ((S (fom_index_pfp_normalized_exists_solution_witness_bezout_rightleft)) * pfgs_vc_normalized_exists_solution)) /\\ exists fom_beta_quotient_pfp_normalized_exists_solution_witness_bezout_rightleft_entry. pfgs_vb_normalized_exists_solution = fom_beta_quotient_pfp_normalized_exists_solution_witness_bezout_rightleft_entry * S ((S (fom_index_pfp_normalized_exists_solution_witness_bezout_rightleft)) * pfgs_vc_normalized_exists_solution) + (fom_value_pfp_normalized_exists_solution_witness_bezout_rightleft))) /\\ (exists fom_gap_pfp_normalized_exists_solution_witness_bezout_rightleft_value_bound. fom_gap_pfp_normalized_exists_solution_witness_bezout_rightleft_value_bound + S (fom_value_pfp_normalized_exists_solution_witness_bezout_rightleft) = p))) /\\ (((forall fom_index_pfp_normalized_exists_solution_witness_bezout_rightright. (exists fom_gap_pfp_normalized_exists_solution_witness_bezout_rightright_index_bound. fom_gap_pfp_normalized_exists_solution_witness_bezout_rightright_index_bound + S (fom_index_pfp_normalized_exists_solution_witness_bezout_rightright) = M) -> exists fom_value_pfp_normalized_exists_solution_witness_bezout_rightright. ((((exists fom_beta_height_pfp_normalized_exists_solution_witness_bezout_rightright_entry. fom_beta_height_pfp_normalized_exists_solution_witness_bezout_rightright_entry + S (fom_value_pfp_normalized_exists_solution_witness_bezout_rightright) = S ((S (fom_index_pfp_normalized_exists_solution_witness_bezout_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_normalized_exists_solution_witness_bezout_rightright_entry. bb = fom_beta_quotient_pfp_normalized_exists_solution_witness_bezout_rightright_entry * S ((S (fom_index_pfp_normalized_exists_solution_witness_bezout_rightright)) * bc) + (fom_value_pfp_normalized_exists_solution_witness_bezout_rightright))) /\\ (exists fom_gap_pfp_normalized_exists_solution_witness_bezout_rightright_value_bound. fom_gap_pfp_normalized_exists_solution_witness_bezout_rightright_value_bound + S (fom_value_pfp_normalized_exists_solution_witness_bezout_rightright) = p))) /\\ (((((((pfgs_V_normalized_exists_solution)=0 \\/ (M)=0) /\\ (((pfgb_Q_normalized_exists_solution_witness_bezout)=0)))) \\/ (((~((pfgs_V_normalized_exists_solution)=0)) /\\ (((~((M)=0)) /\\ (((pfgs_V_normalized_exists_solution)+(M)=S (pfgb_Q_normalized_exists_solution_witness_bezout)))))))) /\\ ((forall pfc_index_normalized_exists_solution_witness_bezout_rightcoefficients. (exists pfa_gap_normalized_exists_solution_witness_bezout_rightcoefficientsbound. pfa_gap_normalized_exists_solution_witness_bezout_rightcoefficientsbound + S (pfc_index_normalized_exists_solution_witness_bezout_rightcoefficients) = (pfgb_Q_normalized_exists_solution_witness_bezout)) -> exists pfc_value_normalized_exists_solution_witness_bezout_rightcoefficients. ((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_rightcoefficientsentry. ff_h_pfp_normalized_exists_solution_witness_bezout_rightcoefficientsentry + S (pfc_value_normalized_exists_solution_witness_bezout_rightcoefficients) = S ((S (pfc_index_normalized_exists_solution_witness_bezout_rightcoefficients)) * pfgb_qc_normalized_exists_solution_witness_bezout)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_rightcoefficientsentry. pfgb_qb_normalized_exists_solution_witness_bezout = ff_q_pfp_normalized_exists_solution_witness_bezout_rightcoefficientsentry * S ((S (pfc_index_normalized_exists_solution_witness_bezout_rightcoefficients)) * pfgb_qc_normalized_exists_solution_witness_bezout) + (pfc_value_normalized_exists_solution_witness_bezout_rightcoefficients))) /\\ ((exists pfc_terms_code_normalized_exists_solution_witness_bezout_rightcoefficientscoefficient pfc_terms_scale_normalized_exists_solution_witness_bezout_rightcoefficientscoefficient pfc_natural_sum_normalized_exists_solution_witness_bezout_rightcoefficientscoefficient. ((forall pfc_index_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonal. (exists pfa_gap_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalbound. pfa_gap_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalbound + S (pfc_index_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonal) = (S (pfc_index_normalized_exists_solution_witness_bezout_rightcoefficients))) -> exists pfc_value_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalentry + S (pfc_value_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_solution_witness_bezout_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_normalized_exists_solution_witness_bezout_rightcoefficientscoefficient = ff_q_pfp_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_solution_witness_bezout_rightcoefficientscoefficient) + (pfc_value_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_left_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm pfc_right_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm. (((pfc_index_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonal)+pfc_complement_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm=(pfc_index_normalized_exists_solution_witness_bezout_rightcoefficients)) /\\ ((((((exists pfa_gap_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonal) = (pfgs_V_normalized_exists_solution)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_normalized_exists_solution)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry. pfgs_vb_normalized_exists_solution = ff_q_pfp_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonal)) * pfgs_vc_normalized_exists_solution) + (pfc_left_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermleftoutside+(pfgs_V_normalized_exists_solution)=(pfc_index_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonal)=pfc_left_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm*pfc_right_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum fs_v_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_h_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_start. fs_u_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalized_exists_solution_witness_bezout_rightcoefficientscoefficient) = S ((S (S (pfc_index_normalized_exists_solution_witness_bezout_rightcoefficients))) * fs_v_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalized_exists_solution_witness_bezout_rightcoefficients))) * fs_v_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum) + (pfc_natural_sum_normalized_exists_solution_witness_bezout_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps = S (pfc_index_normalized_exists_solution_witness_bezout_rightcoefficients)) -> exists fs_a_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_r_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps fs_s_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_solution_witness_bezout_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalized_exists_solution_witness_bezout_rightcoefficientscoefficient = fs_q_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_solution_witness_bezout_rightcoefficientscoefficient) + (fs_a_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum) + (fs_r_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum = fs_q_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum) + (fs_s_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientresiduebound. pfa_gap_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientresiduebound + S (pfc_value_normalized_exists_solution_witness_bezout_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientresiduecongruence pfa_offset_right_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalized_exists_solution_witness_bezout_rightcoefficientscoefficient) + (p) * pfa_offset_left_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientresiduecongruence = (pfc_value_normalized_exists_solution_witness_bezout_rightcoefficients) + (p) * pfa_offset_right_normalized_exists_solution_witness_bezout_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded. (exists fom_gap_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded_index_bound. fom_gap_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded_index_bound + S (fom_index_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded) = pfgb_P_normalized_exists_solution_witness_bezout) -> exists fom_value_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded. ((((exists fom_beta_height_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded_entry. fom_beta_height_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded_entry + S (fom_value_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded) = S ((S (fom_index_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded)) * pfgb_pc_normalized_exists_solution_witness_bezout)) /\\ exists fom_beta_quotient_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded_entry. pfgb_pb_normalized_exists_solution_witness_bezout = fom_beta_quotient_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded_entry * S ((S (fom_index_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded)) * pfgb_pc_normalized_exists_solution_witness_bezout) + (fom_value_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded))) /\\ (exists fom_gap_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded_value_bound. fom_gap_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded_value_bound + S (fom_value_pfp_normalized_exists_solution_witness_bezout_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded. (exists fom_gap_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded_index_bound. fom_gap_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded_index_bound + S (fom_index_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded) = pfgb_Q_normalized_exists_solution_witness_bezout) -> exists fom_value_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded. ((((exists fom_beta_height_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded_entry. fom_beta_height_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded_entry + S (fom_value_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded) = S ((S (fom_index_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded)) * pfgb_qc_normalized_exists_solution_witness_bezout)) /\\ exists fom_beta_quotient_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded_entry. pfgb_qb_normalized_exists_solution_witness_bezout = fom_beta_quotient_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded_entry * S ((S (fom_index_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded)) * pfgb_qc_normalized_exists_solution_witness_bezout) + (fom_value_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded))) /\\ (exists fom_gap_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded_value_bound. fom_gap_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded_value_bound + S (fom_value_pfp_normalized_exists_solution_witness_bezout_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded. (exists fom_gap_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded_index_bound. fom_gap_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded_index_bound + S (fom_index_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded) = pfgs_G_normalized_exists_solution) -> exists fom_value_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded. ((((exists fom_beta_height_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded_entry. fom_beta_height_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded_entry + S (fom_value_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded) = S ((S (fom_index_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded)) * pfgs_gc_normalized_exists_solution)) /\\ exists fom_beta_quotient_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded_entry. pfgs_gb_normalized_exists_solution = fom_beta_quotient_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded_entry * S ((S (fom_index_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded)) * pfgs_gc_normalized_exists_solution) + (fom_value_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded))) /\\ (exists fom_gap_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded_value_bound. fom_gap_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded_value_bound + S (fom_value_pfp_normalized_exists_solution_witness_bezout_sum_result_bounded) = p))) /\\ ((exists pfga_ub_normalized_exists_solution_witness_bezout_sum pfga_uc_normalized_exists_solution_witness_bezout_sum pfga_vb_normalized_exists_solution_witness_bezout_sum pfga_vc_normalized_exists_solution_witness_bezout_sum pfga_tb_normalized_exists_solution_witness_bezout_sum pfga_tc_normalized_exists_solution_witness_bezout_sum pfga_K_normalized_exists_solution_witness_bezout_sum. ((((forall pfrep_power_normalized_exists_solution_witness_bezout_sum_left pfrep_left_normalized_exists_solution_witness_bezout_sum_left pfrep_right_normalized_exists_solution_witness_bezout_sum_left. ((exists pfrep_position_normalized_exists_solution_witness_bezout_sum_leftfirst. ((pfrep_position_normalized_exists_solution_witness_bezout_sum_leftfirst+S (pfrep_power_normalized_exists_solution_witness_bezout_sum_left)=(pfgb_P_normalized_exists_solution_witness_bezout)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_sum_leftfirstentry. ff_h_pfp_normalized_exists_solution_witness_bezout_sum_leftfirstentry + S (pfrep_left_normalized_exists_solution_witness_bezout_sum_left) = S ((S (pfrep_position_normalized_exists_solution_witness_bezout_sum_leftfirst)) * pfgb_pc_normalized_exists_solution_witness_bezout)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_sum_leftfirstentry. pfgb_pb_normalized_exists_solution_witness_bezout = ff_q_pfp_normalized_exists_solution_witness_bezout_sum_leftfirstentry * S ((S (pfrep_position_normalized_exists_solution_witness_bezout_sum_leftfirst)) * pfgb_pc_normalized_exists_solution_witness_bezout) + (pfrep_left_normalized_exists_solution_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_normalized_exists_solution_witness_bezout_sum_leftfirstoutside. pfrep_gap_normalized_exists_solution_witness_bezout_sum_leftfirstoutside+(pfgb_P_normalized_exists_solution_witness_bezout)=(pfrep_power_normalized_exists_solution_witness_bezout_sum_left)) /\\ (((pfrep_left_normalized_exists_solution_witness_bezout_sum_left)=0))))) -> ((exists pfrep_position_normalized_exists_solution_witness_bezout_sum_leftsecond. ((pfrep_position_normalized_exists_solution_witness_bezout_sum_leftsecond+S (pfrep_power_normalized_exists_solution_witness_bezout_sum_left)=(pfga_K_normalized_exists_solution_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_sum_leftsecondentry. ff_h_pfp_normalized_exists_solution_witness_bezout_sum_leftsecondentry + S (pfrep_right_normalized_exists_solution_witness_bezout_sum_left) = S ((S (pfrep_position_normalized_exists_solution_witness_bezout_sum_leftsecond)) * pfga_uc_normalized_exists_solution_witness_bezout_sum)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_sum_leftsecondentry. pfga_ub_normalized_exists_solution_witness_bezout_sum = ff_q_pfp_normalized_exists_solution_witness_bezout_sum_leftsecondentry * S ((S (pfrep_position_normalized_exists_solution_witness_bezout_sum_leftsecond)) * pfga_uc_normalized_exists_solution_witness_bezout_sum) + (pfrep_right_normalized_exists_solution_witness_bezout_sum_left)))))) \\/ (((exists pfrep_gap_normalized_exists_solution_witness_bezout_sum_leftsecondoutside. pfrep_gap_normalized_exists_solution_witness_bezout_sum_leftsecondoutside+(pfga_K_normalized_exists_solution_witness_bezout_sum)=(pfrep_power_normalized_exists_solution_witness_bezout_sum_left)) /\\ (((pfrep_right_normalized_exists_solution_witness_bezout_sum_left)=0))))) -> pfrep_left_normalized_exists_solution_witness_bezout_sum_left=pfrep_right_normalized_exists_solution_witness_bezout_sum_left) /\\ ((forall pfrep_power_normalized_exists_solution_witness_bezout_sum_right pfrep_left_normalized_exists_solution_witness_bezout_sum_right pfrep_right_normalized_exists_solution_witness_bezout_sum_right. ((exists pfrep_position_normalized_exists_solution_witness_bezout_sum_rightfirst. ((pfrep_position_normalized_exists_solution_witness_bezout_sum_rightfirst+S (pfrep_power_normalized_exists_solution_witness_bezout_sum_right)=(pfgb_Q_normalized_exists_solution_witness_bezout)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_sum_rightfirstentry. ff_h_pfp_normalized_exists_solution_witness_bezout_sum_rightfirstentry + S (pfrep_left_normalized_exists_solution_witness_bezout_sum_right) = S ((S (pfrep_position_normalized_exists_solution_witness_bezout_sum_rightfirst)) * pfgb_qc_normalized_exists_solution_witness_bezout)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_sum_rightfirstentry. pfgb_qb_normalized_exists_solution_witness_bezout = ff_q_pfp_normalized_exists_solution_witness_bezout_sum_rightfirstentry * S ((S (pfrep_position_normalized_exists_solution_witness_bezout_sum_rightfirst)) * pfgb_qc_normalized_exists_solution_witness_bezout) + (pfrep_left_normalized_exists_solution_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_normalized_exists_solution_witness_bezout_sum_rightfirstoutside. pfrep_gap_normalized_exists_solution_witness_bezout_sum_rightfirstoutside+(pfgb_Q_normalized_exists_solution_witness_bezout)=(pfrep_power_normalized_exists_solution_witness_bezout_sum_right)) /\\ (((pfrep_left_normalized_exists_solution_witness_bezout_sum_right)=0))))) -> ((exists pfrep_position_normalized_exists_solution_witness_bezout_sum_rightsecond. ((pfrep_position_normalized_exists_solution_witness_bezout_sum_rightsecond+S (pfrep_power_normalized_exists_solution_witness_bezout_sum_right)=(pfga_K_normalized_exists_solution_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_sum_rightsecondentry. ff_h_pfp_normalized_exists_solution_witness_bezout_sum_rightsecondentry + S (pfrep_right_normalized_exists_solution_witness_bezout_sum_right) = S ((S (pfrep_position_normalized_exists_solution_witness_bezout_sum_rightsecond)) * pfga_vc_normalized_exists_solution_witness_bezout_sum)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_sum_rightsecondentry. pfga_vb_normalized_exists_solution_witness_bezout_sum = ff_q_pfp_normalized_exists_solution_witness_bezout_sum_rightsecondentry * S ((S (pfrep_position_normalized_exists_solution_witness_bezout_sum_rightsecond)) * pfga_vc_normalized_exists_solution_witness_bezout_sum) + (pfrep_right_normalized_exists_solution_witness_bezout_sum_right)))))) \\/ (((exists pfrep_gap_normalized_exists_solution_witness_bezout_sum_rightsecondoutside. pfrep_gap_normalized_exists_solution_witness_bezout_sum_rightsecondoutside+(pfga_K_normalized_exists_solution_witness_bezout_sum)=(pfrep_power_normalized_exists_solution_witness_bezout_sum_right)) /\\ (((pfrep_right_normalized_exists_solution_witness_bezout_sum_right)=0))))) -> pfrep_left_normalized_exists_solution_witness_bezout_sum_right=pfrep_right_normalized_exists_solution_witness_bezout_sum_right)))) /\\ (((forall pfp_index_normalized_exists_solution_witness_bezout_sum_add. (exists pfa_gap_normalized_exists_solution_witness_bezout_sum_addindex. pfa_gap_normalized_exists_solution_witness_bezout_sum_addindex + S (pfp_index_normalized_exists_solution_witness_bezout_sum_add) = (pfga_K_normalized_exists_solution_witness_bezout_sum)) -> exists pfp_left_normalized_exists_solution_witness_bezout_sum_add pfp_right_normalized_exists_solution_witness_bezout_sum_add pfp_value_normalized_exists_solution_witness_bezout_sum_add. ((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_sum_addleft. ff_h_pfp_normalized_exists_solution_witness_bezout_sum_addleft + S (pfp_left_normalized_exists_solution_witness_bezout_sum_add) = S ((S (pfp_index_normalized_exists_solution_witness_bezout_sum_add)) * pfga_uc_normalized_exists_solution_witness_bezout_sum)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_sum_addleft. pfga_ub_normalized_exists_solution_witness_bezout_sum = ff_q_pfp_normalized_exists_solution_witness_bezout_sum_addleft * S ((S (pfp_index_normalized_exists_solution_witness_bezout_sum_add)) * pfga_uc_normalized_exists_solution_witness_bezout_sum) + (pfp_left_normalized_exists_solution_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_sum_addright. ff_h_pfp_normalized_exists_solution_witness_bezout_sum_addright + S (pfp_right_normalized_exists_solution_witness_bezout_sum_add) = S ((S (pfp_index_normalized_exists_solution_witness_bezout_sum_add)) * pfga_vc_normalized_exists_solution_witness_bezout_sum)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_sum_addright. pfga_vb_normalized_exists_solution_witness_bezout_sum = ff_q_pfp_normalized_exists_solution_witness_bezout_sum_addright * S ((S (pfp_index_normalized_exists_solution_witness_bezout_sum_add)) * pfga_vc_normalized_exists_solution_witness_bezout_sum) + (pfp_right_normalized_exists_solution_witness_bezout_sum_add))) /\\ (((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_sum_addtarget. ff_h_pfp_normalized_exists_solution_witness_bezout_sum_addtarget + S (pfp_value_normalized_exists_solution_witness_bezout_sum_add) = S ((S (pfp_index_normalized_exists_solution_witness_bezout_sum_add)) * pfga_tc_normalized_exists_solution_witness_bezout_sum)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_sum_addtarget. pfga_tb_normalized_exists_solution_witness_bezout_sum = ff_q_pfp_normalized_exists_solution_witness_bezout_sum_addtarget * S ((S (pfp_index_normalized_exists_solution_witness_bezout_sum_add)) * pfga_tc_normalized_exists_solution_witness_bezout_sum) + (pfp_value_normalized_exists_solution_witness_bezout_sum_add))) /\\ ((((exists pfa_gap_normalized_exists_solution_witness_bezout_sum_addoperationleft. pfa_gap_normalized_exists_solution_witness_bezout_sum_addoperationleft + S (pfp_left_normalized_exists_solution_witness_bezout_sum_add) = (p)) /\\ (((exists pfa_gap_normalized_exists_solution_witness_bezout_sum_addoperationright. pfa_gap_normalized_exists_solution_witness_bezout_sum_addoperationright + S (pfp_right_normalized_exists_solution_witness_bezout_sum_add) = (p)) /\\ ((((exists pfa_gap_normalized_exists_solution_witness_bezout_sum_addoperationresultbound. pfa_gap_normalized_exists_solution_witness_bezout_sum_addoperationresultbound + S (pfp_value_normalized_exists_solution_witness_bezout_sum_add) = (p)) /\\ ((exists pfa_offset_left_normalized_exists_solution_witness_bezout_sum_addoperationresultcongruence pfa_offset_right_normalized_exists_solution_witness_bezout_sum_addoperationresultcongruence. ((pfp_left_normalized_exists_solution_witness_bezout_sum_add) + (pfp_right_normalized_exists_solution_witness_bezout_sum_add)) + (p) * pfa_offset_left_normalized_exists_solution_witness_bezout_sum_addoperationresultcongruence = (pfp_value_normalized_exists_solution_witness_bezout_sum_add) + (p) * pfa_offset_right_normalized_exists_solution_witness_bezout_sum_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_normalized_exists_solution_witness_bezout_sum_result pfrep_left_normalized_exists_solution_witness_bezout_sum_result pfrep_right_normalized_exists_solution_witness_bezout_sum_result. ((exists pfrep_position_normalized_exists_solution_witness_bezout_sum_resultfirst. ((pfrep_position_normalized_exists_solution_witness_bezout_sum_resultfirst+S (pfrep_power_normalized_exists_solution_witness_bezout_sum_result)=(pfga_K_normalized_exists_solution_witness_bezout_sum)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_sum_resultfirstentry. ff_h_pfp_normalized_exists_solution_witness_bezout_sum_resultfirstentry + S (pfrep_left_normalized_exists_solution_witness_bezout_sum_result) = S ((S (pfrep_position_normalized_exists_solution_witness_bezout_sum_resultfirst)) * pfga_tc_normalized_exists_solution_witness_bezout_sum)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_sum_resultfirstentry. pfga_tb_normalized_exists_solution_witness_bezout_sum = ff_q_pfp_normalized_exists_solution_witness_bezout_sum_resultfirstentry * S ((S (pfrep_position_normalized_exists_solution_witness_bezout_sum_resultfirst)) * pfga_tc_normalized_exists_solution_witness_bezout_sum) + (pfrep_left_normalized_exists_solution_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_normalized_exists_solution_witness_bezout_sum_resultfirstoutside. pfrep_gap_normalized_exists_solution_witness_bezout_sum_resultfirstoutside+(pfga_K_normalized_exists_solution_witness_bezout_sum)=(pfrep_power_normalized_exists_solution_witness_bezout_sum_result)) /\\ (((pfrep_left_normalized_exists_solution_witness_bezout_sum_result)=0))))) -> ((exists pfrep_position_normalized_exists_solution_witness_bezout_sum_resultsecond. ((pfrep_position_normalized_exists_solution_witness_bezout_sum_resultsecond+S (pfrep_power_normalized_exists_solution_witness_bezout_sum_result)=(pfgs_G_normalized_exists_solution)) /\\ ((((exists ff_h_pfp_normalized_exists_solution_witness_bezout_sum_resultsecondentry. ff_h_pfp_normalized_exists_solution_witness_bezout_sum_resultsecondentry + S (pfrep_right_normalized_exists_solution_witness_bezout_sum_result) = S ((S (pfrep_position_normalized_exists_solution_witness_bezout_sum_resultsecond)) * pfgs_gc_normalized_exists_solution)) /\\ exists ff_q_pfp_normalized_exists_solution_witness_bezout_sum_resultsecondentry. pfgs_gb_normalized_exists_solution = ff_q_pfp_normalized_exists_solution_witness_bezout_sum_resultsecondentry * S ((S (pfrep_position_normalized_exists_solution_witness_bezout_sum_resultsecond)) * pfgs_gc_normalized_exists_solution) + (pfrep_right_normalized_exists_solution_witness_bezout_sum_result)))))) \\/ (((exists pfrep_gap_normalized_exists_solution_witness_bezout_sum_resultsecondoutside. pfrep_gap_normalized_exists_solution_witness_bezout_sum_resultsecondoutside+(pfgs_G_normalized_exists_solution)=(pfrep_power_normalized_exists_solution_witness_bezout_sum_result)) /\\ (((pfrep_right_normalized_exists_solution_witness_bezout_sum_result)=0))))) -> pfrep_left_normalized_exists_solution_witness_bezout_sum_result=pfrep_right_normalized_exists_solution_witness_bezout_sum_result))))))))))))))))))))))", "specialize prime_field_polynomial_gcd_bezout_exists (p)", "specialize prime_field_polynomial_gcd_bezout_exists (ab)", "specialize prime_field_polynomial_gcd_bezout_exists (ac)", "specialize prime_field_polynomial_gcd_bezout_exists (L)", "specialize prime_field_polynomial_gcd_bezout_exists (bb)", "specialize prime_field_polynomial_gcd_bezout_exists (bc)", "specialize prime_field_polynomial_gcd_bezout_exists (M)", "apply prime_field_polynomial_gcd_bezout_exists", "exact hp", "exact hA", "exact hB", "cases hs", "cases hs_witness", "cases hs_witness_witness", "cases hs_witness_witness_witness", "cases hs_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness_witness_witness_witness_witness", "cases hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_right", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "exists x6", "exists x7", "exists x8", "split", "split", "exact hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_left", "specialize prime_field_polynomial_bezout_is_right_gcd (p)", "specialize prime_field_polynomial_bezout_is_right_gcd (x)", "specialize prime_field_polynomial_bezout_is_right_gcd (x1)", "specialize prime_field_polynomial_bezout_is_right_gcd (x2)", "specialize prime_field_polynomial_bezout_is_right_gcd (ab)", "specialize prime_field_polynomial_bezout_is_right_gcd (ac)", "specialize prime_field_polynomial_bezout_is_right_gcd (L)", "specialize prime_field_polynomial_bezout_is_right_gcd (bb)", "specialize prime_field_polynomial_bezout_is_right_gcd (bc)", "specialize prime_field_polynomial_bezout_is_right_gcd (M)", "specialize prime_field_polynomial_bezout_is_right_gcd (x3)", "specialize prime_field_polynomial_bezout_is_right_gcd (x4)", "specialize prime_field_polynomial_bezout_is_right_gcd (x5)", "specialize prime_field_polynomial_bezout_is_right_gcd (x6)", "specialize prime_field_polynomial_bezout_is_right_gcd (x7)", "specialize prime_field_polynomial_bezout_is_right_gcd (x8)", "apply prime_field_polynomial_bezout_is_right_gcd", "exact hp", "exact hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "exact hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right", "exact hs_witness_witness_witness_witness_witness_witness_witness_witness_witness_right_right" ], "script_sha256": "c91e5394c617eaf2d79647dc9db55548f780f4da363d78ca38ebe3455b96f692", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_existence_candidate.py", "sha256": "81f2f48dd2e81894c7a267453646eb6f2b6f9bd3ee320386d8c561f6b9f8b8ca" }, "statement": "forall p ab ac L bb bc M. (~((p) = 1) /\\ forall pfa_factor_left_normalized_exists_prime pfa_factor_right_normalized_exists_prime. (p) = pfa_factor_left_normalized_exists_prime * pfa_factor_right_normalized_exists_prime -> pfa_factor_left_normalized_exists_prime = 1 \\/ pfa_factor_right_normalized_exists_prime = 1) -> (forall fom_index_pfp_normalized_exists_A. (exists fom_gap_pfp_normalized_exists_A_index_bound. fom_gap_pfp_normalized_exists_A_index_bound + S (fom_index_pfp_normalized_exists_A) = L) -> exists fom_value_pfp_normalized_exists_A. ((((exists fom_beta_height_pfp_normalized_exists_A_entry. fom_beta_height_pfp_normalized_exists_A_entry + S (fom_value_pfp_normalized_exists_A) = S ((S (fom_index_pfp_normalized_exists_A)) * ac)) /\\ exists fom_beta_quotient_pfp_normalized_exists_A_entry. ab = fom_beta_quotient_pfp_normalized_exists_A_entry * S ((S (fom_index_pfp_normalized_exists_A)) * ac) + (fom_value_pfp_normalized_exists_A))) /\\ (exists fom_gap_pfp_normalized_exists_A_value_bound. fom_gap_pfp_normalized_exists_A_value_bound + S (fom_value_pfp_normalized_exists_A) = p))) -> (forall fom_index_pfp_normalized_exists_B. (exists fom_gap_pfp_normalized_exists_B_index_bound. fom_gap_pfp_normalized_exists_B_index_bound + S (fom_index_pfp_normalized_exists_B) = M) -> exists fom_value_pfp_normalized_exists_B. ((((exists fom_beta_height_pfp_normalized_exists_B_entry. fom_beta_height_pfp_normalized_exists_B_entry + S (fom_value_pfp_normalized_exists_B) = S ((S (fom_index_pfp_normalized_exists_B)) * bc)) /\\ exists fom_beta_quotient_pfp_normalized_exists_B_entry. bb = fom_beta_quotient_pfp_normalized_exists_B_entry * S ((S (fom_index_pfp_normalized_exists_B)) * bc) + (fom_value_pfp_normalized_exists_B))) /\\ (exists fom_gap_pfp_normalized_exists_B_value_bound. fom_gap_pfp_normalized_exists_B_value_bound + S (fom_value_pfp_normalized_exists_B) = p))) -> (exists gb gc G ub uc U vb vc V. (((((G)=0 \\/ (((~((G) = 0)) /\\ (((forall fom_index_pfp_normalized_exists_gcd_normal_moniccoefficients. (exists fom_gap_pfp_normalized_exists_gcd_normal_moniccoefficients_index_bound. fom_gap_pfp_normalized_exists_gcd_normal_moniccoefficients_index_bound + S (fom_index_pfp_normalized_exists_gcd_normal_moniccoefficients) = G) -> exists fom_value_pfp_normalized_exists_gcd_normal_moniccoefficients. ((((exists fom_beta_height_pfp_normalized_exists_gcd_normal_moniccoefficients_entry. fom_beta_height_pfp_normalized_exists_gcd_normal_moniccoefficients_entry + S (fom_value_pfp_normalized_exists_gcd_normal_moniccoefficients) = S ((S (fom_index_pfp_normalized_exists_gcd_normal_moniccoefficients)) * gc)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_normal_moniccoefficients_entry. gb = fom_beta_quotient_pfp_normalized_exists_gcd_normal_moniccoefficients_entry * S ((S (fom_index_pfp_normalized_exists_gcd_normal_moniccoefficients)) * gc) + (fom_value_pfp_normalized_exists_gcd_normal_moniccoefficients))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_normal_moniccoefficients_value_bound. fom_gap_pfp_normalized_exists_gcd_normal_moniccoefficients_value_bound + S (fom_value_pfp_normalized_exists_gcd_normal_moniccoefficients) = p))) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_normal_monicleading. ff_h_pfp_normalized_exists_gcd_normal_monicleading + S (1) = S ((S (0)) * gc)) /\\ exists ff_q_pfp_normalized_exists_gcd_normal_monicleading. gb = ff_q_pfp_normalized_exists_gcd_normal_monicleading * S ((S (0)) * gc) + (1))))))))) /\\ ((((((((forall fom_index_pfp_normalized_exists_gcd_gcd_common_left_bounded. (exists fom_gap_pfp_normalized_exists_gcd_gcd_common_left_bounded_index_bound. fom_gap_pfp_normalized_exists_gcd_gcd_common_left_bounded_index_bound + S (fom_index_pfp_normalized_exists_gcd_gcd_common_left_bounded) = L) -> exists fom_value_pfp_normalized_exists_gcd_gcd_common_left_bounded. ((((exists fom_beta_height_pfp_normalized_exists_gcd_gcd_common_left_bounded_entry. fom_beta_height_pfp_normalized_exists_gcd_gcd_common_left_bounded_entry + S (fom_value_pfp_normalized_exists_gcd_gcd_common_left_bounded) = S ((S (fom_index_pfp_normalized_exists_gcd_gcd_common_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_gcd_common_left_bounded_entry. ab = fom_beta_quotient_pfp_normalized_exists_gcd_gcd_common_left_bounded_entry * S ((S (fom_index_pfp_normalized_exists_gcd_gcd_common_left_bounded)) * ac) + (fom_value_pfp_normalized_exists_gcd_gcd_common_left_bounded))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_gcd_common_left_bounded_value_bound. fom_gap_pfp_normalized_exists_gcd_gcd_common_left_bounded_value_bound + S (fom_value_pfp_normalized_exists_gcd_gcd_common_left_bounded) = p))) /\\ ((exists pfgd_qb_normalized_exists_gcd_gcd_common_left pfgd_qc_normalized_exists_gcd_gcd_common_left pfgd_Q_normalized_exists_gcd_gcd_common_left pfgd_pb_normalized_exists_gcd_gcd_common_left pfgd_pc_normalized_exists_gcd_gcd_common_left pfgd_P_normalized_exists_gcd_gcd_common_left. ((((forall fom_index_pfp_normalized_exists_gcd_gcd_common_left_productleft. (exists fom_gap_pfp_normalized_exists_gcd_gcd_common_left_productleft_index_bound. fom_gap_pfp_normalized_exists_gcd_gcd_common_left_productleft_index_bound + S (fom_index_pfp_normalized_exists_gcd_gcd_common_left_productleft) = pfgd_Q_normalized_exists_gcd_gcd_common_left) -> exists fom_value_pfp_normalized_exists_gcd_gcd_common_left_productleft. ((((exists fom_beta_height_pfp_normalized_exists_gcd_gcd_common_left_productleft_entry. fom_beta_height_pfp_normalized_exists_gcd_gcd_common_left_productleft_entry + S (fom_value_pfp_normalized_exists_gcd_gcd_common_left_productleft) = S ((S (fom_index_pfp_normalized_exists_gcd_gcd_common_left_productleft)) * pfgd_qc_normalized_exists_gcd_gcd_common_left)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_gcd_common_left_productleft_entry. pfgd_qb_normalized_exists_gcd_gcd_common_left = fom_beta_quotient_pfp_normalized_exists_gcd_gcd_common_left_productleft_entry * S ((S (fom_index_pfp_normalized_exists_gcd_gcd_common_left_productleft)) * pfgd_qc_normalized_exists_gcd_gcd_common_left) + (fom_value_pfp_normalized_exists_gcd_gcd_common_left_productleft))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_gcd_common_left_productleft_value_bound. fom_gap_pfp_normalized_exists_gcd_gcd_common_left_productleft_value_bound + S (fom_value_pfp_normalized_exists_gcd_gcd_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_normalized_exists_gcd_gcd_common_left_productright. (exists fom_gap_pfp_normalized_exists_gcd_gcd_common_left_productright_index_bound. fom_gap_pfp_normalized_exists_gcd_gcd_common_left_productright_index_bound + S (fom_index_pfp_normalized_exists_gcd_gcd_common_left_productright) = G) -> exists fom_value_pfp_normalized_exists_gcd_gcd_common_left_productright. ((((exists fom_beta_height_pfp_normalized_exists_gcd_gcd_common_left_productright_entry. fom_beta_height_pfp_normalized_exists_gcd_gcd_common_left_productright_entry + S (fom_value_pfp_normalized_exists_gcd_gcd_common_left_productright) = S ((S (fom_index_pfp_normalized_exists_gcd_gcd_common_left_productright)) * gc)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_gcd_common_left_productright_entry. gb = fom_beta_quotient_pfp_normalized_exists_gcd_gcd_common_left_productright_entry * S ((S (fom_index_pfp_normalized_exists_gcd_gcd_common_left_productright)) * gc) + (fom_value_pfp_normalized_exists_gcd_gcd_common_left_productright))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_gcd_common_left_productright_value_bound. fom_gap_pfp_normalized_exists_gcd_gcd_common_left_productright_value_bound + S (fom_value_pfp_normalized_exists_gcd_gcd_common_left_productright) = p))) /\\ (((((((pfgd_Q_normalized_exists_gcd_gcd_common_left)=0 \\/ (G)=0) /\\ (((pfgd_P_normalized_exists_gcd_gcd_common_left)=0)))) \\/ (((~((pfgd_Q_normalized_exists_gcd_gcd_common_left)=0)) /\\ (((~((G)=0)) /\\ (((pfgd_Q_normalized_exists_gcd_gcd_common_left)+(G)=S (pfgd_P_normalized_exists_gcd_gcd_common_left)))))))) /\\ ((forall pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficients. (exists pfa_gap_normalized_exists_gcd_gcd_common_left_productcoefficientsbound. pfa_gap_normalized_exists_gcd_gcd_common_left_productcoefficientsbound + S (pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficients) = (pfgd_P_normalized_exists_gcd_gcd_common_left)) -> exists pfc_value_normalized_exists_gcd_gcd_common_left_productcoefficients. ((((exists ff_h_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientsentry. ff_h_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientsentry + S (pfc_value_normalized_exists_gcd_gcd_common_left_productcoefficients) = S ((S (pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficients)) * pfgd_pc_normalized_exists_gcd_gcd_common_left)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientsentry. pfgd_pb_normalized_exists_gcd_gcd_common_left = ff_q_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientsentry * S ((S (pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficients)) * pfgd_pc_normalized_exists_gcd_gcd_common_left) + (pfc_value_normalized_exists_gcd_gcd_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficient pfc_terms_scale_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficient pfc_natural_sum_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficient. ((forall pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficients))) -> exists pfc_value_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficient = ff_q_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficient) + (pfc_value_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm pfc_left_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm pfc_right_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_normalized_exists_gcd_gcd_common_left)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_normalized_exists_gcd_gcd_common_left)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_normalized_exists_gcd_gcd_common_left = ff_q_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_normalized_exists_gcd_gcd_common_left) + (pfc_left_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_normalized_exists_gcd_gcd_common_left)=(pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm) = (G)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm)) * gc)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermrightentry. gb = ff_q_pfp_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm)) * gc) + (pfc_right_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(G)=(pfc_complement_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonal)=pfc_left_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum fs_v_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficients))) * fs_v_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficients))) * fs_v_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_normalized_exists_gcd_gcd_common_left_productcoefficients)) -> exists fs_a_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficient = fs_q_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficient) + (fs_a_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientresiduebound. pfa_gap_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_normalized_exists_gcd_gcd_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_normalized_exists_gcd_gcd_common_left_productcoefficients) + (p) * pfa_offset_right_normalized_exists_gcd_gcd_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normalized_exists_gcd_gcd_common_left_equivalent pfrep_left_normalized_exists_gcd_gcd_common_left_equivalent pfrep_right_normalized_exists_gcd_gcd_common_left_equivalent. ((exists pfrep_position_normalized_exists_gcd_gcd_common_left_equivalentfirst. ((pfrep_position_normalized_exists_gcd_gcd_common_left_equivalentfirst+S (pfrep_power_normalized_exists_gcd_gcd_common_left_equivalent)=(pfgd_P_normalized_exists_gcd_gcd_common_left)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_common_left_equivalentfirstentry. ff_h_pfp_normalized_exists_gcd_gcd_common_left_equivalentfirstentry + S (pfrep_left_normalized_exists_gcd_gcd_common_left_equivalent) = S ((S (pfrep_position_normalized_exists_gcd_gcd_common_left_equivalentfirst)) * pfgd_pc_normalized_exists_gcd_gcd_common_left)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_common_left_equivalentfirstentry. pfgd_pb_normalized_exists_gcd_gcd_common_left = ff_q_pfp_normalized_exists_gcd_gcd_common_left_equivalentfirstentry * S ((S (pfrep_position_normalized_exists_gcd_gcd_common_left_equivalentfirst)) * pfgd_pc_normalized_exists_gcd_gcd_common_left) + (pfrep_left_normalized_exists_gcd_gcd_common_left_equivalent)))))) \\/ (((exists pfrep_gap_normalized_exists_gcd_gcd_common_left_equivalentfirstoutside. pfrep_gap_normalized_exists_gcd_gcd_common_left_equivalentfirstoutside+(pfgd_P_normalized_exists_gcd_gcd_common_left)=(pfrep_power_normalized_exists_gcd_gcd_common_left_equivalent)) /\\ (((pfrep_left_normalized_exists_gcd_gcd_common_left_equivalent)=0))))) -> ((exists pfrep_position_normalized_exists_gcd_gcd_common_left_equivalentsecond. ((pfrep_position_normalized_exists_gcd_gcd_common_left_equivalentsecond+S (pfrep_power_normalized_exists_gcd_gcd_common_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_common_left_equivalentsecondentry. ff_h_pfp_normalized_exists_gcd_gcd_common_left_equivalentsecondentry + S (pfrep_right_normalized_exists_gcd_gcd_common_left_equivalent) = S ((S (pfrep_position_normalized_exists_gcd_gcd_common_left_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_common_left_equivalentsecondentry. ab = ff_q_pfp_normalized_exists_gcd_gcd_common_left_equivalentsecondentry * S ((S (pfrep_position_normalized_exists_gcd_gcd_common_left_equivalentsecond)) * ac) + (pfrep_right_normalized_exists_gcd_gcd_common_left_equivalent)))))) \\/ (((exists pfrep_gap_normalized_exists_gcd_gcd_common_left_equivalentsecondoutside. pfrep_gap_normalized_exists_gcd_gcd_common_left_equivalentsecondoutside+(L)=(pfrep_power_normalized_exists_gcd_gcd_common_left_equivalent)) /\\ (((pfrep_right_normalized_exists_gcd_gcd_common_left_equivalent)=0))))) -> pfrep_left_normalized_exists_gcd_gcd_common_left_equivalent=pfrep_right_normalized_exists_gcd_gcd_common_left_equivalent))))))) /\\ ((((forall fom_index_pfp_normalized_exists_gcd_gcd_common_right_bounded. (exists fom_gap_pfp_normalized_exists_gcd_gcd_common_right_bounded_index_bound. fom_gap_pfp_normalized_exists_gcd_gcd_common_right_bounded_index_bound + S (fom_index_pfp_normalized_exists_gcd_gcd_common_right_bounded) = M) -> exists fom_value_pfp_normalized_exists_gcd_gcd_common_right_bounded. ((((exists fom_beta_height_pfp_normalized_exists_gcd_gcd_common_right_bounded_entry. fom_beta_height_pfp_normalized_exists_gcd_gcd_common_right_bounded_entry + S (fom_value_pfp_normalized_exists_gcd_gcd_common_right_bounded) = S ((S (fom_index_pfp_normalized_exists_gcd_gcd_common_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_gcd_common_right_bounded_entry. bb = fom_beta_quotient_pfp_normalized_exists_gcd_gcd_common_right_bounded_entry * S ((S (fom_index_pfp_normalized_exists_gcd_gcd_common_right_bounded)) * bc) + (fom_value_pfp_normalized_exists_gcd_gcd_common_right_bounded))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_gcd_common_right_bounded_value_bound. fom_gap_pfp_normalized_exists_gcd_gcd_common_right_bounded_value_bound + S (fom_value_pfp_normalized_exists_gcd_gcd_common_right_bounded) = p))) /\\ ((exists pfgd_qb_normalized_exists_gcd_gcd_common_right pfgd_qc_normalized_exists_gcd_gcd_common_right pfgd_Q_normalized_exists_gcd_gcd_common_right pfgd_pb_normalized_exists_gcd_gcd_common_right pfgd_pc_normalized_exists_gcd_gcd_common_right pfgd_P_normalized_exists_gcd_gcd_common_right. ((((forall fom_index_pfp_normalized_exists_gcd_gcd_common_right_productleft. (exists fom_gap_pfp_normalized_exists_gcd_gcd_common_right_productleft_index_bound. fom_gap_pfp_normalized_exists_gcd_gcd_common_right_productleft_index_bound + S (fom_index_pfp_normalized_exists_gcd_gcd_common_right_productleft) = pfgd_Q_normalized_exists_gcd_gcd_common_right) -> exists fom_value_pfp_normalized_exists_gcd_gcd_common_right_productleft. ((((exists fom_beta_height_pfp_normalized_exists_gcd_gcd_common_right_productleft_entry. fom_beta_height_pfp_normalized_exists_gcd_gcd_common_right_productleft_entry + S (fom_value_pfp_normalized_exists_gcd_gcd_common_right_productleft) = S ((S (fom_index_pfp_normalized_exists_gcd_gcd_common_right_productleft)) * pfgd_qc_normalized_exists_gcd_gcd_common_right)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_gcd_common_right_productleft_entry. pfgd_qb_normalized_exists_gcd_gcd_common_right = fom_beta_quotient_pfp_normalized_exists_gcd_gcd_common_right_productleft_entry * S ((S (fom_index_pfp_normalized_exists_gcd_gcd_common_right_productleft)) * pfgd_qc_normalized_exists_gcd_gcd_common_right) + (fom_value_pfp_normalized_exists_gcd_gcd_common_right_productleft))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_gcd_common_right_productleft_value_bound. fom_gap_pfp_normalized_exists_gcd_gcd_common_right_productleft_value_bound + S (fom_value_pfp_normalized_exists_gcd_gcd_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_normalized_exists_gcd_gcd_common_right_productright. (exists fom_gap_pfp_normalized_exists_gcd_gcd_common_right_productright_index_bound. fom_gap_pfp_normalized_exists_gcd_gcd_common_right_productright_index_bound + S (fom_index_pfp_normalized_exists_gcd_gcd_common_right_productright) = G) -> exists fom_value_pfp_normalized_exists_gcd_gcd_common_right_productright. ((((exists fom_beta_height_pfp_normalized_exists_gcd_gcd_common_right_productright_entry. fom_beta_height_pfp_normalized_exists_gcd_gcd_common_right_productright_entry + S (fom_value_pfp_normalized_exists_gcd_gcd_common_right_productright) = S ((S (fom_index_pfp_normalized_exists_gcd_gcd_common_right_productright)) * gc)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_gcd_common_right_productright_entry. gb = fom_beta_quotient_pfp_normalized_exists_gcd_gcd_common_right_productright_entry * S ((S (fom_index_pfp_normalized_exists_gcd_gcd_common_right_productright)) * gc) + (fom_value_pfp_normalized_exists_gcd_gcd_common_right_productright))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_gcd_common_right_productright_value_bound. fom_gap_pfp_normalized_exists_gcd_gcd_common_right_productright_value_bound + S (fom_value_pfp_normalized_exists_gcd_gcd_common_right_productright) = p))) /\\ (((((((pfgd_Q_normalized_exists_gcd_gcd_common_right)=0 \\/ (G)=0) /\\ (((pfgd_P_normalized_exists_gcd_gcd_common_right)=0)))) \\/ (((~((pfgd_Q_normalized_exists_gcd_gcd_common_right)=0)) /\\ (((~((G)=0)) /\\ (((pfgd_Q_normalized_exists_gcd_gcd_common_right)+(G)=S (pfgd_P_normalized_exists_gcd_gcd_common_right)))))))) /\\ ((forall pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficients. (exists pfa_gap_normalized_exists_gcd_gcd_common_right_productcoefficientsbound. pfa_gap_normalized_exists_gcd_gcd_common_right_productcoefficientsbound + S (pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficients) = (pfgd_P_normalized_exists_gcd_gcd_common_right)) -> exists pfc_value_normalized_exists_gcd_gcd_common_right_productcoefficients. ((((exists ff_h_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientsentry. ff_h_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientsentry + S (pfc_value_normalized_exists_gcd_gcd_common_right_productcoefficients) = S ((S (pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficients)) * pfgd_pc_normalized_exists_gcd_gcd_common_right)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientsentry. pfgd_pb_normalized_exists_gcd_gcd_common_right = ff_q_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientsentry * S ((S (pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficients)) * pfgd_pc_normalized_exists_gcd_gcd_common_right) + (pfc_value_normalized_exists_gcd_gcd_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficient pfc_terms_scale_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficient pfc_natural_sum_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficient. ((forall pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficients))) -> exists pfc_value_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficient = ff_q_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficient) + (pfc_value_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm pfc_left_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm pfc_right_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_normalized_exists_gcd_gcd_common_right)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_normalized_exists_gcd_gcd_common_right)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_normalized_exists_gcd_gcd_common_right = ff_q_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_normalized_exists_gcd_gcd_common_right) + (pfc_left_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_normalized_exists_gcd_gcd_common_right)=(pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm) = (G)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm)) * gc)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermrightentry. gb = ff_q_pfp_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm)) * gc) + (pfc_right_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(G)=(pfc_complement_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonal)=pfc_left_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum fs_v_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficients))) * fs_v_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficients))) * fs_v_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_normalized_exists_gcd_gcd_common_right_productcoefficients)) -> exists fs_a_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficient = fs_q_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficient) + (fs_a_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientresiduebound. pfa_gap_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_normalized_exists_gcd_gcd_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_normalized_exists_gcd_gcd_common_right_productcoefficients) + (p) * pfa_offset_right_normalized_exists_gcd_gcd_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normalized_exists_gcd_gcd_common_right_equivalent pfrep_left_normalized_exists_gcd_gcd_common_right_equivalent pfrep_right_normalized_exists_gcd_gcd_common_right_equivalent. ((exists pfrep_position_normalized_exists_gcd_gcd_common_right_equivalentfirst. ((pfrep_position_normalized_exists_gcd_gcd_common_right_equivalentfirst+S (pfrep_power_normalized_exists_gcd_gcd_common_right_equivalent)=(pfgd_P_normalized_exists_gcd_gcd_common_right)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_common_right_equivalentfirstentry. ff_h_pfp_normalized_exists_gcd_gcd_common_right_equivalentfirstentry + S (pfrep_left_normalized_exists_gcd_gcd_common_right_equivalent) = S ((S (pfrep_position_normalized_exists_gcd_gcd_common_right_equivalentfirst)) * pfgd_pc_normalized_exists_gcd_gcd_common_right)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_common_right_equivalentfirstentry. pfgd_pb_normalized_exists_gcd_gcd_common_right = ff_q_pfp_normalized_exists_gcd_gcd_common_right_equivalentfirstentry * S ((S (pfrep_position_normalized_exists_gcd_gcd_common_right_equivalentfirst)) * pfgd_pc_normalized_exists_gcd_gcd_common_right) + (pfrep_left_normalized_exists_gcd_gcd_common_right_equivalent)))))) \\/ (((exists pfrep_gap_normalized_exists_gcd_gcd_common_right_equivalentfirstoutside. pfrep_gap_normalized_exists_gcd_gcd_common_right_equivalentfirstoutside+(pfgd_P_normalized_exists_gcd_gcd_common_right)=(pfrep_power_normalized_exists_gcd_gcd_common_right_equivalent)) /\\ (((pfrep_left_normalized_exists_gcd_gcd_common_right_equivalent)=0))))) -> ((exists pfrep_position_normalized_exists_gcd_gcd_common_right_equivalentsecond. ((pfrep_position_normalized_exists_gcd_gcd_common_right_equivalentsecond+S (pfrep_power_normalized_exists_gcd_gcd_common_right_equivalent)=(M)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_common_right_equivalentsecondentry. ff_h_pfp_normalized_exists_gcd_gcd_common_right_equivalentsecondentry + S (pfrep_right_normalized_exists_gcd_gcd_common_right_equivalent) = S ((S (pfrep_position_normalized_exists_gcd_gcd_common_right_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_common_right_equivalentsecondentry. bb = ff_q_pfp_normalized_exists_gcd_gcd_common_right_equivalentsecondentry * S ((S (pfrep_position_normalized_exists_gcd_gcd_common_right_equivalentsecond)) * bc) + (pfrep_right_normalized_exists_gcd_gcd_common_right_equivalent)))))) \\/ (((exists pfrep_gap_normalized_exists_gcd_gcd_common_right_equivalentsecondoutside. pfrep_gap_normalized_exists_gcd_gcd_common_right_equivalentsecondoutside+(M)=(pfrep_power_normalized_exists_gcd_gcd_common_right_equivalent)) /\\ (((pfrep_right_normalized_exists_gcd_gcd_common_right_equivalent)=0))))) -> pfrep_left_normalized_exists_gcd_gcd_common_right_equivalent=pfrep_right_normalized_exists_gcd_gcd_common_right_equivalent)))))))))) /\\ ((forall pfgg_db_normalized_exists_gcd_gcd pfgg_dc_normalized_exists_gcd_gcd pfgg_D_normalized_exists_gcd_gcd. (((((forall fom_index_pfp_normalized_exists_gcd_gcd_divisor_left_bounded. (exists fom_gap_pfp_normalized_exists_gcd_gcd_divisor_left_bounded_index_bound. fom_gap_pfp_normalized_exists_gcd_gcd_divisor_left_bounded_index_bound + S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_left_bounded) = L) -> exists fom_value_pfp_normalized_exists_gcd_gcd_divisor_left_bounded. ((((exists fom_beta_height_pfp_normalized_exists_gcd_gcd_divisor_left_bounded_entry. fom_beta_height_pfp_normalized_exists_gcd_gcd_divisor_left_bounded_entry + S (fom_value_pfp_normalized_exists_gcd_gcd_divisor_left_bounded) = S ((S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_left_bounded)) * ac)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_gcd_divisor_left_bounded_entry. ab = fom_beta_quotient_pfp_normalized_exists_gcd_gcd_divisor_left_bounded_entry * S ((S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_left_bounded)) * ac) + (fom_value_pfp_normalized_exists_gcd_gcd_divisor_left_bounded))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_gcd_divisor_left_bounded_value_bound. fom_gap_pfp_normalized_exists_gcd_gcd_divisor_left_bounded_value_bound + S (fom_value_pfp_normalized_exists_gcd_gcd_divisor_left_bounded) = p))) /\\ ((exists pfgd_qb_normalized_exists_gcd_gcd_divisor_left pfgd_qc_normalized_exists_gcd_gcd_divisor_left pfgd_Q_normalized_exists_gcd_gcd_divisor_left pfgd_pb_normalized_exists_gcd_gcd_divisor_left pfgd_pc_normalized_exists_gcd_gcd_divisor_left pfgd_P_normalized_exists_gcd_gcd_divisor_left. ((((forall fom_index_pfp_normalized_exists_gcd_gcd_divisor_left_productleft. (exists fom_gap_pfp_normalized_exists_gcd_gcd_divisor_left_productleft_index_bound. fom_gap_pfp_normalized_exists_gcd_gcd_divisor_left_productleft_index_bound + S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_left_productleft) = pfgd_Q_normalized_exists_gcd_gcd_divisor_left) -> exists fom_value_pfp_normalized_exists_gcd_gcd_divisor_left_productleft. ((((exists fom_beta_height_pfp_normalized_exists_gcd_gcd_divisor_left_productleft_entry. fom_beta_height_pfp_normalized_exists_gcd_gcd_divisor_left_productleft_entry + S (fom_value_pfp_normalized_exists_gcd_gcd_divisor_left_productleft) = S ((S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_left_productleft)) * pfgd_qc_normalized_exists_gcd_gcd_divisor_left)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_gcd_divisor_left_productleft_entry. pfgd_qb_normalized_exists_gcd_gcd_divisor_left = fom_beta_quotient_pfp_normalized_exists_gcd_gcd_divisor_left_productleft_entry * S ((S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_left_productleft)) * pfgd_qc_normalized_exists_gcd_gcd_divisor_left) + (fom_value_pfp_normalized_exists_gcd_gcd_divisor_left_productleft))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_gcd_divisor_left_productleft_value_bound. fom_gap_pfp_normalized_exists_gcd_gcd_divisor_left_productleft_value_bound + S (fom_value_pfp_normalized_exists_gcd_gcd_divisor_left_productleft) = p))) /\\ (((forall fom_index_pfp_normalized_exists_gcd_gcd_divisor_left_productright. (exists fom_gap_pfp_normalized_exists_gcd_gcd_divisor_left_productright_index_bound. fom_gap_pfp_normalized_exists_gcd_gcd_divisor_left_productright_index_bound + S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_left_productright) = pfgg_D_normalized_exists_gcd_gcd) -> exists fom_value_pfp_normalized_exists_gcd_gcd_divisor_left_productright. ((((exists fom_beta_height_pfp_normalized_exists_gcd_gcd_divisor_left_productright_entry. fom_beta_height_pfp_normalized_exists_gcd_gcd_divisor_left_productright_entry + S (fom_value_pfp_normalized_exists_gcd_gcd_divisor_left_productright) = S ((S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_left_productright)) * pfgg_dc_normalized_exists_gcd_gcd)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_gcd_divisor_left_productright_entry. pfgg_db_normalized_exists_gcd_gcd = fom_beta_quotient_pfp_normalized_exists_gcd_gcd_divisor_left_productright_entry * S ((S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_left_productright)) * pfgg_dc_normalized_exists_gcd_gcd) + (fom_value_pfp_normalized_exists_gcd_gcd_divisor_left_productright))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_gcd_divisor_left_productright_value_bound. fom_gap_pfp_normalized_exists_gcd_gcd_divisor_left_productright_value_bound + S (fom_value_pfp_normalized_exists_gcd_gcd_divisor_left_productright) = p))) /\\ (((((((pfgd_Q_normalized_exists_gcd_gcd_divisor_left)=0 \\/ (pfgg_D_normalized_exists_gcd_gcd)=0) /\\ (((pfgd_P_normalized_exists_gcd_gcd_divisor_left)=0)))) \\/ (((~((pfgd_Q_normalized_exists_gcd_gcd_divisor_left)=0)) /\\ (((~((pfgg_D_normalized_exists_gcd_gcd)=0)) /\\ (((pfgd_Q_normalized_exists_gcd_gcd_divisor_left)+(pfgg_D_normalized_exists_gcd_gcd)=S (pfgd_P_normalized_exists_gcd_gcd_divisor_left)))))))) /\\ ((forall pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficients. (exists pfa_gap_normalized_exists_gcd_gcd_divisor_left_productcoefficientsbound. pfa_gap_normalized_exists_gcd_gcd_divisor_left_productcoefficientsbound + S (pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficients) = (pfgd_P_normalized_exists_gcd_gcd_divisor_left)) -> exists pfc_value_normalized_exists_gcd_gcd_divisor_left_productcoefficients. ((((exists ff_h_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientsentry. ff_h_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientsentry + S (pfc_value_normalized_exists_gcd_gcd_divisor_left_productcoefficients) = S ((S (pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficients)) * pfgd_pc_normalized_exists_gcd_gcd_divisor_left)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientsentry. pfgd_pb_normalized_exists_gcd_gcd_divisor_left = ff_q_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientsentry * S ((S (pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficients)) * pfgd_pc_normalized_exists_gcd_gcd_divisor_left) + (pfc_value_normalized_exists_gcd_gcd_divisor_left_productcoefficients))) /\\ ((exists pfc_terms_code_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficient pfc_terms_scale_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficient pfc_natural_sum_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficient. ((forall pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalbound. pfa_gap_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficients))) -> exists pfc_value_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficient = ff_q_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficient) + (pfc_value_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm pfc_left_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm pfc_right_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonal)+pfc_complement_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm=(pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficients)) /\\ ((((((exists pfa_gap_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonal) = (pfgd_Q_normalized_exists_gcd_gcd_divisor_left)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_normalized_exists_gcd_gcd_divisor_left)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_normalized_exists_gcd_gcd_divisor_left = ff_q_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonal)) * pfgd_qc_normalized_exists_gcd_gcd_divisor_left) + (pfc_left_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_normalized_exists_gcd_gcd_divisor_left)=(pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm) = (pfgg_D_normalized_exists_gcd_gcd)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)) * pfgg_dc_normalized_exists_gcd_gcd)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightentry. pfgg_db_normalized_exists_gcd_gcd = ff_q_pfp_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)) * pfgg_dc_normalized_exists_gcd_gcd) + (pfc_right_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightoutside+(pfgg_D_normalized_exists_gcd_gcd)=(pfc_complement_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonal)=pfc_left_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm*pfc_right_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum fs_v_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficient) = S ((S (S (pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficients))) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficients))) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum) + (pfc_natural_sum_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_normalized_exists_gcd_gcd_divisor_left_productcoefficients)) -> exists fs_a_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficient = fs_q_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficient) + (fs_a_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum) + (fs_r_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum) + (fs_s_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientresiduebound. pfa_gap_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientresiduebound + S (pfc_value_normalized_exists_gcd_gcd_divisor_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficient) + (p) * pfa_offset_left_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientresiduecongruence = (pfc_value_normalized_exists_gcd_gcd_divisor_left_productcoefficients) + (p) * pfa_offset_right_normalized_exists_gcd_gcd_divisor_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normalized_exists_gcd_gcd_divisor_left_equivalent pfrep_left_normalized_exists_gcd_gcd_divisor_left_equivalent pfrep_right_normalized_exists_gcd_gcd_divisor_left_equivalent. ((exists pfrep_position_normalized_exists_gcd_gcd_divisor_left_equivalentfirst. ((pfrep_position_normalized_exists_gcd_gcd_divisor_left_equivalentfirst+S (pfrep_power_normalized_exists_gcd_gcd_divisor_left_equivalent)=(pfgd_P_normalized_exists_gcd_gcd_divisor_left)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_divisor_left_equivalentfirstentry. ff_h_pfp_normalized_exists_gcd_gcd_divisor_left_equivalentfirstentry + S (pfrep_left_normalized_exists_gcd_gcd_divisor_left_equivalent) = S ((S (pfrep_position_normalized_exists_gcd_gcd_divisor_left_equivalentfirst)) * pfgd_pc_normalized_exists_gcd_gcd_divisor_left)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_divisor_left_equivalentfirstentry. pfgd_pb_normalized_exists_gcd_gcd_divisor_left = ff_q_pfp_normalized_exists_gcd_gcd_divisor_left_equivalentfirstentry * S ((S (pfrep_position_normalized_exists_gcd_gcd_divisor_left_equivalentfirst)) * pfgd_pc_normalized_exists_gcd_gcd_divisor_left) + (pfrep_left_normalized_exists_gcd_gcd_divisor_left_equivalent)))))) \\/ (((exists pfrep_gap_normalized_exists_gcd_gcd_divisor_left_equivalentfirstoutside. pfrep_gap_normalized_exists_gcd_gcd_divisor_left_equivalentfirstoutside+(pfgd_P_normalized_exists_gcd_gcd_divisor_left)=(pfrep_power_normalized_exists_gcd_gcd_divisor_left_equivalent)) /\\ (((pfrep_left_normalized_exists_gcd_gcd_divisor_left_equivalent)=0))))) -> ((exists pfrep_position_normalized_exists_gcd_gcd_divisor_left_equivalentsecond. ((pfrep_position_normalized_exists_gcd_gcd_divisor_left_equivalentsecond+S (pfrep_power_normalized_exists_gcd_gcd_divisor_left_equivalent)=(L)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_divisor_left_equivalentsecondentry. ff_h_pfp_normalized_exists_gcd_gcd_divisor_left_equivalentsecondentry + S (pfrep_right_normalized_exists_gcd_gcd_divisor_left_equivalent) = S ((S (pfrep_position_normalized_exists_gcd_gcd_divisor_left_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_divisor_left_equivalentsecondentry. ab = ff_q_pfp_normalized_exists_gcd_gcd_divisor_left_equivalentsecondentry * S ((S (pfrep_position_normalized_exists_gcd_gcd_divisor_left_equivalentsecond)) * ac) + (pfrep_right_normalized_exists_gcd_gcd_divisor_left_equivalent)))))) \\/ (((exists pfrep_gap_normalized_exists_gcd_gcd_divisor_left_equivalentsecondoutside. pfrep_gap_normalized_exists_gcd_gcd_divisor_left_equivalentsecondoutside+(L)=(pfrep_power_normalized_exists_gcd_gcd_divisor_left_equivalent)) /\\ (((pfrep_right_normalized_exists_gcd_gcd_divisor_left_equivalent)=0))))) -> pfrep_left_normalized_exists_gcd_gcd_divisor_left_equivalent=pfrep_right_normalized_exists_gcd_gcd_divisor_left_equivalent))))))) /\\ ((((forall fom_index_pfp_normalized_exists_gcd_gcd_divisor_right_bounded. (exists fom_gap_pfp_normalized_exists_gcd_gcd_divisor_right_bounded_index_bound. fom_gap_pfp_normalized_exists_gcd_gcd_divisor_right_bounded_index_bound + S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_right_bounded) = M) -> exists fom_value_pfp_normalized_exists_gcd_gcd_divisor_right_bounded. ((((exists fom_beta_height_pfp_normalized_exists_gcd_gcd_divisor_right_bounded_entry. fom_beta_height_pfp_normalized_exists_gcd_gcd_divisor_right_bounded_entry + S (fom_value_pfp_normalized_exists_gcd_gcd_divisor_right_bounded) = S ((S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_right_bounded)) * bc)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_gcd_divisor_right_bounded_entry. bb = fom_beta_quotient_pfp_normalized_exists_gcd_gcd_divisor_right_bounded_entry * S ((S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_right_bounded)) * bc) + (fom_value_pfp_normalized_exists_gcd_gcd_divisor_right_bounded))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_gcd_divisor_right_bounded_value_bound. fom_gap_pfp_normalized_exists_gcd_gcd_divisor_right_bounded_value_bound + S (fom_value_pfp_normalized_exists_gcd_gcd_divisor_right_bounded) = p))) /\\ ((exists pfgd_qb_normalized_exists_gcd_gcd_divisor_right pfgd_qc_normalized_exists_gcd_gcd_divisor_right pfgd_Q_normalized_exists_gcd_gcd_divisor_right pfgd_pb_normalized_exists_gcd_gcd_divisor_right pfgd_pc_normalized_exists_gcd_gcd_divisor_right pfgd_P_normalized_exists_gcd_gcd_divisor_right. ((((forall fom_index_pfp_normalized_exists_gcd_gcd_divisor_right_productleft. (exists fom_gap_pfp_normalized_exists_gcd_gcd_divisor_right_productleft_index_bound. fom_gap_pfp_normalized_exists_gcd_gcd_divisor_right_productleft_index_bound + S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_right_productleft) = pfgd_Q_normalized_exists_gcd_gcd_divisor_right) -> exists fom_value_pfp_normalized_exists_gcd_gcd_divisor_right_productleft. ((((exists fom_beta_height_pfp_normalized_exists_gcd_gcd_divisor_right_productleft_entry. fom_beta_height_pfp_normalized_exists_gcd_gcd_divisor_right_productleft_entry + S (fom_value_pfp_normalized_exists_gcd_gcd_divisor_right_productleft) = S ((S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_right_productleft)) * pfgd_qc_normalized_exists_gcd_gcd_divisor_right)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_gcd_divisor_right_productleft_entry. pfgd_qb_normalized_exists_gcd_gcd_divisor_right = fom_beta_quotient_pfp_normalized_exists_gcd_gcd_divisor_right_productleft_entry * S ((S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_right_productleft)) * pfgd_qc_normalized_exists_gcd_gcd_divisor_right) + (fom_value_pfp_normalized_exists_gcd_gcd_divisor_right_productleft))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_gcd_divisor_right_productleft_value_bound. fom_gap_pfp_normalized_exists_gcd_gcd_divisor_right_productleft_value_bound + S (fom_value_pfp_normalized_exists_gcd_gcd_divisor_right_productleft) = p))) /\\ (((forall fom_index_pfp_normalized_exists_gcd_gcd_divisor_right_productright. (exists fom_gap_pfp_normalized_exists_gcd_gcd_divisor_right_productright_index_bound. fom_gap_pfp_normalized_exists_gcd_gcd_divisor_right_productright_index_bound + S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_right_productright) = pfgg_D_normalized_exists_gcd_gcd) -> exists fom_value_pfp_normalized_exists_gcd_gcd_divisor_right_productright. ((((exists fom_beta_height_pfp_normalized_exists_gcd_gcd_divisor_right_productright_entry. fom_beta_height_pfp_normalized_exists_gcd_gcd_divisor_right_productright_entry + S (fom_value_pfp_normalized_exists_gcd_gcd_divisor_right_productright) = S ((S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_right_productright)) * pfgg_dc_normalized_exists_gcd_gcd)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_gcd_divisor_right_productright_entry. pfgg_db_normalized_exists_gcd_gcd = fom_beta_quotient_pfp_normalized_exists_gcd_gcd_divisor_right_productright_entry * S ((S (fom_index_pfp_normalized_exists_gcd_gcd_divisor_right_productright)) * pfgg_dc_normalized_exists_gcd_gcd) + (fom_value_pfp_normalized_exists_gcd_gcd_divisor_right_productright))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_gcd_divisor_right_productright_value_bound. fom_gap_pfp_normalized_exists_gcd_gcd_divisor_right_productright_value_bound + S (fom_value_pfp_normalized_exists_gcd_gcd_divisor_right_productright) = p))) /\\ (((((((pfgd_Q_normalized_exists_gcd_gcd_divisor_right)=0 \\/ (pfgg_D_normalized_exists_gcd_gcd)=0) /\\ (((pfgd_P_normalized_exists_gcd_gcd_divisor_right)=0)))) \\/ (((~((pfgd_Q_normalized_exists_gcd_gcd_divisor_right)=0)) /\\ (((~((pfgg_D_normalized_exists_gcd_gcd)=0)) /\\ (((pfgd_Q_normalized_exists_gcd_gcd_divisor_right)+(pfgg_D_normalized_exists_gcd_gcd)=S (pfgd_P_normalized_exists_gcd_gcd_divisor_right)))))))) /\\ ((forall pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficients. (exists pfa_gap_normalized_exists_gcd_gcd_divisor_right_productcoefficientsbound. pfa_gap_normalized_exists_gcd_gcd_divisor_right_productcoefficientsbound + S (pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficients) = (pfgd_P_normalized_exists_gcd_gcd_divisor_right)) -> exists pfc_value_normalized_exists_gcd_gcd_divisor_right_productcoefficients. ((((exists ff_h_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientsentry. ff_h_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientsentry + S (pfc_value_normalized_exists_gcd_gcd_divisor_right_productcoefficients) = S ((S (pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficients)) * pfgd_pc_normalized_exists_gcd_gcd_divisor_right)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientsentry. pfgd_pb_normalized_exists_gcd_gcd_divisor_right = ff_q_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientsentry * S ((S (pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficients)) * pfgd_pc_normalized_exists_gcd_gcd_divisor_right) + (pfc_value_normalized_exists_gcd_gcd_divisor_right_productcoefficients))) /\\ ((exists pfc_terms_code_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficient pfc_terms_scale_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficient pfc_natural_sum_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficient. ((forall pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalbound. pfa_gap_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficients))) -> exists pfc_value_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficient = ff_q_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficient) + (pfc_value_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm pfc_left_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm pfc_right_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonal)+pfc_complement_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm=(pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficients)) /\\ ((((((exists pfa_gap_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonal) = (pfgd_Q_normalized_exists_gcd_gcd_divisor_right)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_normalized_exists_gcd_gcd_divisor_right)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_normalized_exists_gcd_gcd_divisor_right = ff_q_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonal)) * pfgd_qc_normalized_exists_gcd_gcd_divisor_right) + (pfc_left_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_normalized_exists_gcd_gcd_divisor_right)=(pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm) = (pfgg_D_normalized_exists_gcd_gcd)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)) * pfgg_dc_normalized_exists_gcd_gcd)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightentry. pfgg_db_normalized_exists_gcd_gcd = ff_q_pfp_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)) * pfgg_dc_normalized_exists_gcd_gcd) + (pfc_right_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightoutside+(pfgg_D_normalized_exists_gcd_gcd)=(pfc_complement_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonal)=pfc_left_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm*pfc_right_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum fs_v_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficient) = S ((S (S (pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficients))) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficients))) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum) + (pfc_natural_sum_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_normalized_exists_gcd_gcd_divisor_right_productcoefficients)) -> exists fs_a_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficient = fs_q_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficient) + (fs_a_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum) + (fs_r_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum) + (fs_s_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientresiduebound. pfa_gap_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientresiduebound + S (pfc_value_normalized_exists_gcd_gcd_divisor_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficient) + (p) * pfa_offset_left_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientresiduecongruence = (pfc_value_normalized_exists_gcd_gcd_divisor_right_productcoefficients) + (p) * pfa_offset_right_normalized_exists_gcd_gcd_divisor_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normalized_exists_gcd_gcd_divisor_right_equivalent pfrep_left_normalized_exists_gcd_gcd_divisor_right_equivalent pfrep_right_normalized_exists_gcd_gcd_divisor_right_equivalent. ((exists pfrep_position_normalized_exists_gcd_gcd_divisor_right_equivalentfirst. ((pfrep_position_normalized_exists_gcd_gcd_divisor_right_equivalentfirst+S (pfrep_power_normalized_exists_gcd_gcd_divisor_right_equivalent)=(pfgd_P_normalized_exists_gcd_gcd_divisor_right)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_divisor_right_equivalentfirstentry. ff_h_pfp_normalized_exists_gcd_gcd_divisor_right_equivalentfirstentry + S (pfrep_left_normalized_exists_gcd_gcd_divisor_right_equivalent) = S ((S (pfrep_position_normalized_exists_gcd_gcd_divisor_right_equivalentfirst)) * pfgd_pc_normalized_exists_gcd_gcd_divisor_right)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_divisor_right_equivalentfirstentry. pfgd_pb_normalized_exists_gcd_gcd_divisor_right = ff_q_pfp_normalized_exists_gcd_gcd_divisor_right_equivalentfirstentry * S ((S (pfrep_position_normalized_exists_gcd_gcd_divisor_right_equivalentfirst)) * pfgd_pc_normalized_exists_gcd_gcd_divisor_right) + (pfrep_left_normalized_exists_gcd_gcd_divisor_right_equivalent)))))) \\/ (((exists pfrep_gap_normalized_exists_gcd_gcd_divisor_right_equivalentfirstoutside. pfrep_gap_normalized_exists_gcd_gcd_divisor_right_equivalentfirstoutside+(pfgd_P_normalized_exists_gcd_gcd_divisor_right)=(pfrep_power_normalized_exists_gcd_gcd_divisor_right_equivalent)) /\\ (((pfrep_left_normalized_exists_gcd_gcd_divisor_right_equivalent)=0))))) -> ((exists pfrep_position_normalized_exists_gcd_gcd_divisor_right_equivalentsecond. ((pfrep_position_normalized_exists_gcd_gcd_divisor_right_equivalentsecond+S (pfrep_power_normalized_exists_gcd_gcd_divisor_right_equivalent)=(M)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_divisor_right_equivalentsecondentry. ff_h_pfp_normalized_exists_gcd_gcd_divisor_right_equivalentsecondentry + S (pfrep_right_normalized_exists_gcd_gcd_divisor_right_equivalent) = S ((S (pfrep_position_normalized_exists_gcd_gcd_divisor_right_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_divisor_right_equivalentsecondentry. bb = ff_q_pfp_normalized_exists_gcd_gcd_divisor_right_equivalentsecondentry * S ((S (pfrep_position_normalized_exists_gcd_gcd_divisor_right_equivalentsecond)) * bc) + (pfrep_right_normalized_exists_gcd_gcd_divisor_right_equivalent)))))) \\/ (((exists pfrep_gap_normalized_exists_gcd_gcd_divisor_right_equivalentsecondoutside. pfrep_gap_normalized_exists_gcd_gcd_divisor_right_equivalentsecondoutside+(M)=(pfrep_power_normalized_exists_gcd_gcd_divisor_right_equivalent)) /\\ (((pfrep_right_normalized_exists_gcd_gcd_divisor_right_equivalent)=0))))) -> pfrep_left_normalized_exists_gcd_gcd_divisor_right_equivalent=pfrep_right_normalized_exists_gcd_gcd_divisor_right_equivalent)))))))))) -> (((forall fom_index_pfp_normalized_exists_gcd_gcd_greatest_bounded. (exists fom_gap_pfp_normalized_exists_gcd_gcd_greatest_bounded_index_bound. fom_gap_pfp_normalized_exists_gcd_gcd_greatest_bounded_index_bound + S (fom_index_pfp_normalized_exists_gcd_gcd_greatest_bounded) = G) -> exists fom_value_pfp_normalized_exists_gcd_gcd_greatest_bounded. ((((exists fom_beta_height_pfp_normalized_exists_gcd_gcd_greatest_bounded_entry. fom_beta_height_pfp_normalized_exists_gcd_gcd_greatest_bounded_entry + S (fom_value_pfp_normalized_exists_gcd_gcd_greatest_bounded) = S ((S (fom_index_pfp_normalized_exists_gcd_gcd_greatest_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_gcd_greatest_bounded_entry. gb = fom_beta_quotient_pfp_normalized_exists_gcd_gcd_greatest_bounded_entry * S ((S (fom_index_pfp_normalized_exists_gcd_gcd_greatest_bounded)) * gc) + (fom_value_pfp_normalized_exists_gcd_gcd_greatest_bounded))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_gcd_greatest_bounded_value_bound. fom_gap_pfp_normalized_exists_gcd_gcd_greatest_bounded_value_bound + S (fom_value_pfp_normalized_exists_gcd_gcd_greatest_bounded) = p))) /\\ ((exists pfgd_qb_normalized_exists_gcd_gcd_greatest pfgd_qc_normalized_exists_gcd_gcd_greatest pfgd_Q_normalized_exists_gcd_gcd_greatest pfgd_pb_normalized_exists_gcd_gcd_greatest pfgd_pc_normalized_exists_gcd_gcd_greatest pfgd_P_normalized_exists_gcd_gcd_greatest. ((((forall fom_index_pfp_normalized_exists_gcd_gcd_greatest_productleft. (exists fom_gap_pfp_normalized_exists_gcd_gcd_greatest_productleft_index_bound. fom_gap_pfp_normalized_exists_gcd_gcd_greatest_productleft_index_bound + S (fom_index_pfp_normalized_exists_gcd_gcd_greatest_productleft) = pfgd_Q_normalized_exists_gcd_gcd_greatest) -> exists fom_value_pfp_normalized_exists_gcd_gcd_greatest_productleft. ((((exists fom_beta_height_pfp_normalized_exists_gcd_gcd_greatest_productleft_entry. fom_beta_height_pfp_normalized_exists_gcd_gcd_greatest_productleft_entry + S (fom_value_pfp_normalized_exists_gcd_gcd_greatest_productleft) = S ((S (fom_index_pfp_normalized_exists_gcd_gcd_greatest_productleft)) * pfgd_qc_normalized_exists_gcd_gcd_greatest)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_gcd_greatest_productleft_entry. pfgd_qb_normalized_exists_gcd_gcd_greatest = fom_beta_quotient_pfp_normalized_exists_gcd_gcd_greatest_productleft_entry * S ((S (fom_index_pfp_normalized_exists_gcd_gcd_greatest_productleft)) * pfgd_qc_normalized_exists_gcd_gcd_greatest) + (fom_value_pfp_normalized_exists_gcd_gcd_greatest_productleft))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_gcd_greatest_productleft_value_bound. fom_gap_pfp_normalized_exists_gcd_gcd_greatest_productleft_value_bound + S (fom_value_pfp_normalized_exists_gcd_gcd_greatest_productleft) = p))) /\\ (((forall fom_index_pfp_normalized_exists_gcd_gcd_greatest_productright. (exists fom_gap_pfp_normalized_exists_gcd_gcd_greatest_productright_index_bound. fom_gap_pfp_normalized_exists_gcd_gcd_greatest_productright_index_bound + S (fom_index_pfp_normalized_exists_gcd_gcd_greatest_productright) = pfgg_D_normalized_exists_gcd_gcd) -> exists fom_value_pfp_normalized_exists_gcd_gcd_greatest_productright. ((((exists fom_beta_height_pfp_normalized_exists_gcd_gcd_greatest_productright_entry. fom_beta_height_pfp_normalized_exists_gcd_gcd_greatest_productright_entry + S (fom_value_pfp_normalized_exists_gcd_gcd_greatest_productright) = S ((S (fom_index_pfp_normalized_exists_gcd_gcd_greatest_productright)) * pfgg_dc_normalized_exists_gcd_gcd)) /\\ exists fom_beta_quotient_pfp_normalized_exists_gcd_gcd_greatest_productright_entry. pfgg_db_normalized_exists_gcd_gcd = fom_beta_quotient_pfp_normalized_exists_gcd_gcd_greatest_productright_entry * S ((S (fom_index_pfp_normalized_exists_gcd_gcd_greatest_productright)) * pfgg_dc_normalized_exists_gcd_gcd) + (fom_value_pfp_normalized_exists_gcd_gcd_greatest_productright))) /\\ (exists fom_gap_pfp_normalized_exists_gcd_gcd_greatest_productright_value_bound. fom_gap_pfp_normalized_exists_gcd_gcd_greatest_productright_value_bound + S (fom_value_pfp_normalized_exists_gcd_gcd_greatest_productright) = p))) /\\ (((((((pfgd_Q_normalized_exists_gcd_gcd_greatest)=0 \\/ (pfgg_D_normalized_exists_gcd_gcd)=0) /\\ (((pfgd_P_normalized_exists_gcd_gcd_greatest)=0)))) \\/ (((~((pfgd_Q_normalized_exists_gcd_gcd_greatest)=0)) /\\ (((~((pfgg_D_normalized_exists_gcd_gcd)=0)) /\\ (((pfgd_Q_normalized_exists_gcd_gcd_greatest)+(pfgg_D_normalized_exists_gcd_gcd)=S (pfgd_P_normalized_exists_gcd_gcd_greatest)))))))) /\\ ((forall pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficients. (exists pfa_gap_normalized_exists_gcd_gcd_greatest_productcoefficientsbound. pfa_gap_normalized_exists_gcd_gcd_greatest_productcoefficientsbound + S (pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficients) = (pfgd_P_normalized_exists_gcd_gcd_greatest)) -> exists pfc_value_normalized_exists_gcd_gcd_greatest_productcoefficients. ((((exists ff_h_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientsentry. ff_h_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientsentry + S (pfc_value_normalized_exists_gcd_gcd_greatest_productcoefficients) = S ((S (pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficients)) * pfgd_pc_normalized_exists_gcd_gcd_greatest)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientsentry. pfgd_pb_normalized_exists_gcd_gcd_greatest = ff_q_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientsentry * S ((S (pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficients)) * pfgd_pc_normalized_exists_gcd_gcd_greatest) + (pfc_value_normalized_exists_gcd_gcd_greatest_productcoefficients))) /\\ ((exists pfc_terms_code_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficient pfc_terms_scale_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficient pfc_natural_sum_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficient. ((forall pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonal. (exists pfa_gap_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalbound. pfa_gap_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalbound + S (pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonal) = (S (pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficients))) -> exists pfc_value_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalentry + S (pfc_value_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficient = ff_q_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficient) + (pfc_value_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm pfc_left_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm pfc_right_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm. (((pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonal)+pfc_complement_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm=(pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficients)) /\\ ((((((exists pfa_gap_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonal) = (pfgd_Q_normalized_exists_gcd_gcd_greatest)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonal)) * pfgd_qc_normalized_exists_gcd_gcd_greatest)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermleftentry. pfgd_qb_normalized_exists_gcd_gcd_greatest = ff_q_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonal)) * pfgd_qc_normalized_exists_gcd_gcd_greatest) + (pfc_left_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermleftoutside+(pfgd_Q_normalized_exists_gcd_gcd_greatest)=(pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm) = (pfgg_D_normalized_exists_gcd_gcd)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm)) * pfgg_dc_normalized_exists_gcd_gcd)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermrightentry. pfgg_db_normalized_exists_gcd_gcd = ff_q_pfp_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm)) * pfgg_dc_normalized_exists_gcd_gcd) + (pfc_right_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonaltermrightoutside+(pfgg_D_normalized_exists_gcd_gcd)=(pfc_complement_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonal)=pfc_left_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm*pfc_right_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum fs_v_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_start. fs_h_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_start. fs_u_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficient) = S ((S (S (pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficients))) * fs_v_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficients))) * fs_v_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum) + (pfc_natural_sum_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps = S (pfc_index_normalized_exists_gcd_gcd_greatest_productcoefficients)) -> exists fs_a_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps fs_r_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps fs_s_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficient = fs_q_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficient) + (fs_a_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum) + (fs_r_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum = fs_q_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum) + (fs_s_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientresiduebound. pfa_gap_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientresiduebound + S (pfc_value_normalized_exists_gcd_gcd_greatest_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientresiduecongruence pfa_offset_right_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficient) + (p) * pfa_offset_left_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientresiduecongruence = (pfc_value_normalized_exists_gcd_gcd_greatest_productcoefficients) + (p) * pfa_offset_right_normalized_exists_gcd_gcd_greatest_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normalized_exists_gcd_gcd_greatest_equivalent pfrep_left_normalized_exists_gcd_gcd_greatest_equivalent pfrep_right_normalized_exists_gcd_gcd_greatest_equivalent. ((exists pfrep_position_normalized_exists_gcd_gcd_greatest_equivalentfirst. ((pfrep_position_normalized_exists_gcd_gcd_greatest_equivalentfirst+S (pfrep_power_normalized_exists_gcd_gcd_greatest_equivalent)=(pfgd_P_normalized_exists_gcd_gcd_greatest)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_greatest_equivalentfirstentry. ff_h_pfp_normalized_exists_gcd_gcd_greatest_equivalentfirstentry + S (pfrep_left_normalized_exists_gcd_gcd_greatest_equivalent) = S ((S (pfrep_position_normalized_exists_gcd_gcd_greatest_equivalentfirst)) * pfgd_pc_normalized_exists_gcd_gcd_greatest)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_greatest_equivalentfirstentry. pfgd_pb_normalized_exists_gcd_gcd_greatest = ff_q_pfp_normalized_exists_gcd_gcd_greatest_equivalentfirstentry * S ((S (pfrep_position_normalized_exists_gcd_gcd_greatest_equivalentfirst)) * pfgd_pc_normalized_exists_gcd_gcd_greatest) + (pfrep_left_normalized_exists_gcd_gcd_greatest_equivalent)))))) \\/ (((exists pfrep_gap_normalized_exists_gcd_gcd_greatest_equivalentfirstoutside. pfrep_gap_normalized_exists_gcd_gcd_greatest_equivalentfirstoutside+(pfgd_P_normalized_exists_gcd_gcd_greatest)=(pfrep_power_normalized_exists_gcd_gcd_greatest_equivalent)) /\\ (((pfrep_left_normalized_exists_gcd_gcd_greatest_equivalent)=0))))) -> ((exists pfrep_position_normalized_exists_gcd_gcd_greatest_equivalentsecond. ((pfrep_position_normalized_exists_gcd_gcd_greatest_equivalentsecond+S (pfrep_power_normalized_exists_gcd_gcd_greatest_equivalent)=(G)) /\\ ((((exists ff_h_pfp_normalized_exists_gcd_gcd_greatest_equivalentsecondentry. ff_h_pfp_normalized_exists_gcd_gcd_greatest_equivalentsecondentry + S (pfrep_right_normalized_exists_gcd_gcd_greatest_equivalent) = S ((S (pfrep_position_normalized_exists_gcd_gcd_greatest_equivalentsecond)) * gc)) /\\ exists ff_q_pfp_normalized_exists_gcd_gcd_greatest_equivalentsecondentry. gb = ff_q_pfp_normalized_exists_gcd_gcd_greatest_equivalentsecondentry * S ((S (pfrep_position_normalized_exists_gcd_gcd_greatest_equivalentsecond)) * gc) + (pfrep_right_normalized_exists_gcd_gcd_greatest_equivalent)))))) \\/ (((exists pfrep_gap_normalized_exists_gcd_gcd_greatest_equivalentsecondoutside. pfrep_gap_normalized_exists_gcd_gcd_greatest_equivalentsecondoutside+(G)=(pfrep_power_normalized_exists_gcd_gcd_greatest_equivalent)) /\\ (((pfrep_right_normalized_exists_gcd_gcd_greatest_equivalent)=0))))) -> pfrep_left_normalized_exists_gcd_gcd_greatest_equivalent=pfrep_right_normalized_exists_gcd_gcd_greatest_equivalent)))))))))))))) /\\ ((exists pfgb_pb_normalized_exists_bezout pfgb_pc_normalized_exists_bezout pfgb_P_normalized_exists_bezout pfgb_qb_normalized_exists_bezout pfgb_qc_normalized_exists_bezout pfgb_Q_normalized_exists_bezout. ((((forall fom_index_pfp_normalized_exists_bezout_leftleft. (exists fom_gap_pfp_normalized_exists_bezout_leftleft_index_bound. fom_gap_pfp_normalized_exists_bezout_leftleft_index_bound + S (fom_index_pfp_normalized_exists_bezout_leftleft) = U) -> exists fom_value_pfp_normalized_exists_bezout_leftleft. ((((exists fom_beta_height_pfp_normalized_exists_bezout_leftleft_entry. fom_beta_height_pfp_normalized_exists_bezout_leftleft_entry + S (fom_value_pfp_normalized_exists_bezout_leftleft) = S ((S (fom_index_pfp_normalized_exists_bezout_leftleft)) * uc)) /\\ exists fom_beta_quotient_pfp_normalized_exists_bezout_leftleft_entry. ub = fom_beta_quotient_pfp_normalized_exists_bezout_leftleft_entry * S ((S (fom_index_pfp_normalized_exists_bezout_leftleft)) * uc) + (fom_value_pfp_normalized_exists_bezout_leftleft))) /\\ (exists fom_gap_pfp_normalized_exists_bezout_leftleft_value_bound. fom_gap_pfp_normalized_exists_bezout_leftleft_value_bound + S (fom_value_pfp_normalized_exists_bezout_leftleft) = p))) /\\ (((forall fom_index_pfp_normalized_exists_bezout_leftright. (exists fom_gap_pfp_normalized_exists_bezout_leftright_index_bound. fom_gap_pfp_normalized_exists_bezout_leftright_index_bound + S (fom_index_pfp_normalized_exists_bezout_leftright) = L) -> exists fom_value_pfp_normalized_exists_bezout_leftright. ((((exists fom_beta_height_pfp_normalized_exists_bezout_leftright_entry. fom_beta_height_pfp_normalized_exists_bezout_leftright_entry + S (fom_value_pfp_normalized_exists_bezout_leftright) = S ((S (fom_index_pfp_normalized_exists_bezout_leftright)) * ac)) /\\ exists fom_beta_quotient_pfp_normalized_exists_bezout_leftright_entry. ab = fom_beta_quotient_pfp_normalized_exists_bezout_leftright_entry * S ((S (fom_index_pfp_normalized_exists_bezout_leftright)) * ac) + (fom_value_pfp_normalized_exists_bezout_leftright))) /\\ (exists fom_gap_pfp_normalized_exists_bezout_leftright_value_bound. fom_gap_pfp_normalized_exists_bezout_leftright_value_bound + S (fom_value_pfp_normalized_exists_bezout_leftright) = p))) /\\ (((((((U)=0 \\/ (L)=0) /\\ (((pfgb_P_normalized_exists_bezout)=0)))) \\/ (((~((U)=0)) /\\ (((~((L)=0)) /\\ (((U)+(L)=S (pfgb_P_normalized_exists_bezout)))))))) /\\ ((forall pfc_index_normalized_exists_bezout_leftcoefficients. (exists pfa_gap_normalized_exists_bezout_leftcoefficientsbound. pfa_gap_normalized_exists_bezout_leftcoefficientsbound + S (pfc_index_normalized_exists_bezout_leftcoefficients) = (pfgb_P_normalized_exists_bezout)) -> exists pfc_value_normalized_exists_bezout_leftcoefficients. ((((exists ff_h_pfp_normalized_exists_bezout_leftcoefficientsentry. ff_h_pfp_normalized_exists_bezout_leftcoefficientsentry + S (pfc_value_normalized_exists_bezout_leftcoefficients) = S ((S (pfc_index_normalized_exists_bezout_leftcoefficients)) * pfgb_pc_normalized_exists_bezout)) /\\ exists ff_q_pfp_normalized_exists_bezout_leftcoefficientsentry. pfgb_pb_normalized_exists_bezout = ff_q_pfp_normalized_exists_bezout_leftcoefficientsentry * S ((S (pfc_index_normalized_exists_bezout_leftcoefficients)) * pfgb_pc_normalized_exists_bezout) + (pfc_value_normalized_exists_bezout_leftcoefficients))) /\\ ((exists pfc_terms_code_normalized_exists_bezout_leftcoefficientscoefficient pfc_terms_scale_normalized_exists_bezout_leftcoefficientscoefficient pfc_natural_sum_normalized_exists_bezout_leftcoefficientscoefficient. ((forall pfc_index_normalized_exists_bezout_leftcoefficientscoefficientdiagonal. (exists pfa_gap_normalized_exists_bezout_leftcoefficientscoefficientdiagonalbound. pfa_gap_normalized_exists_bezout_leftcoefficientscoefficientdiagonalbound + S (pfc_index_normalized_exists_bezout_leftcoefficientscoefficientdiagonal) = (S (pfc_index_normalized_exists_bezout_leftcoefficients))) -> exists pfc_value_normalized_exists_bezout_leftcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalized_exists_bezout_leftcoefficientscoefficientdiagonalentry. ff_h_pfp_normalized_exists_bezout_leftcoefficientscoefficientdiagonalentry + S (pfc_value_normalized_exists_bezout_leftcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalized_exists_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_bezout_leftcoefficientscoefficient)) /\\ exists ff_q_pfp_normalized_exists_bezout_leftcoefficientscoefficientdiagonalentry. pfc_terms_code_normalized_exists_bezout_leftcoefficientscoefficient = ff_q_pfp_normalized_exists_bezout_leftcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalized_exists_bezout_leftcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_bezout_leftcoefficientscoefficient) + (pfc_value_normalized_exists_bezout_leftcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm pfc_left_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm pfc_right_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm. (((pfc_index_normalized_exists_bezout_leftcoefficientscoefficientdiagonal)+pfc_complement_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm=(pfc_index_normalized_exists_bezout_leftcoefficients)) /\\ ((((((exists pfa_gap_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalized_exists_bezout_leftcoefficientscoefficientdiagonal) = (U)) /\\ ((((exists ff_h_pfp_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalized_exists_bezout_leftcoefficientscoefficientdiagonal)) * uc)) /\\ exists ff_q_pfp_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermleftentry. ub = ff_q_pfp_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalized_exists_bezout_leftcoefficientscoefficientdiagonal)) * uc) + (pfc_left_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermleftoutside+(U)=(pfc_index_normalized_exists_bezout_leftcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm) = (L)) /\\ ((((exists ff_h_pfp_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm)) * ac)) /\\ exists ff_q_pfp_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermrightentry. ab = ff_q_pfp_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm)) * ac) + (pfc_right_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalized_exists_bezout_leftcoefficientscoefficientdiagonaltermrightoutside+(L)=(pfc_complement_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalized_exists_bezout_leftcoefficientscoefficientdiagonal)=pfc_left_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm*pfc_right_normalized_exists_bezout_leftcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum fs_v_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum. ((((exists fs_h_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_start. fs_h_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_start. fs_u_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum = fs_q_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalized_exists_bezout_leftcoefficientscoefficient) = S ((S (S (pfc_index_normalized_exists_bezout_leftcoefficients))) * fs_v_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum = fs_q_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalized_exists_bezout_leftcoefficients))) * fs_v_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum) + (pfc_natural_sum_normalized_exists_bezout_leftcoefficientscoefficient))) /\\ forall fs_i_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps = S (pfc_index_normalized_exists_bezout_leftcoefficients)) -> exists fs_a_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps fs_r_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps fs_s_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_bezout_leftcoefficientscoefficient)) /\\ exists fs_q_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalized_exists_bezout_leftcoefficientscoefficient = fs_q_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_bezout_leftcoefficientscoefficient) + (fs_a_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum = fs_q_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum) + (fs_r_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum = fs_q_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum) + (fs_s_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps = fs_r_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps + fs_a_pfc_normalized_exists_bezout_leftcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalized_exists_bezout_leftcoefficientscoefficientresiduebound. pfa_gap_normalized_exists_bezout_leftcoefficientscoefficientresiduebound + S (pfc_value_normalized_exists_bezout_leftcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalized_exists_bezout_leftcoefficientscoefficientresiduecongruence pfa_offset_right_normalized_exists_bezout_leftcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalized_exists_bezout_leftcoefficientscoefficient) + (p) * pfa_offset_left_normalized_exists_bezout_leftcoefficientscoefficientresiduecongruence = (pfc_value_normalized_exists_bezout_leftcoefficients) + (p) * pfa_offset_right_normalized_exists_bezout_leftcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((((forall fom_index_pfp_normalized_exists_bezout_rightleft. (exists fom_gap_pfp_normalized_exists_bezout_rightleft_index_bound. fom_gap_pfp_normalized_exists_bezout_rightleft_index_bound + S (fom_index_pfp_normalized_exists_bezout_rightleft) = V) -> exists fom_value_pfp_normalized_exists_bezout_rightleft. ((((exists fom_beta_height_pfp_normalized_exists_bezout_rightleft_entry. fom_beta_height_pfp_normalized_exists_bezout_rightleft_entry + S (fom_value_pfp_normalized_exists_bezout_rightleft) = S ((S (fom_index_pfp_normalized_exists_bezout_rightleft)) * vc)) /\\ exists fom_beta_quotient_pfp_normalized_exists_bezout_rightleft_entry. vb = fom_beta_quotient_pfp_normalized_exists_bezout_rightleft_entry * S ((S (fom_index_pfp_normalized_exists_bezout_rightleft)) * vc) + (fom_value_pfp_normalized_exists_bezout_rightleft))) /\\ (exists fom_gap_pfp_normalized_exists_bezout_rightleft_value_bound. fom_gap_pfp_normalized_exists_bezout_rightleft_value_bound + S (fom_value_pfp_normalized_exists_bezout_rightleft) = p))) /\\ (((forall fom_index_pfp_normalized_exists_bezout_rightright. (exists fom_gap_pfp_normalized_exists_bezout_rightright_index_bound. fom_gap_pfp_normalized_exists_bezout_rightright_index_bound + S (fom_index_pfp_normalized_exists_bezout_rightright) = M) -> exists fom_value_pfp_normalized_exists_bezout_rightright. ((((exists fom_beta_height_pfp_normalized_exists_bezout_rightright_entry. fom_beta_height_pfp_normalized_exists_bezout_rightright_entry + S (fom_value_pfp_normalized_exists_bezout_rightright) = S ((S (fom_index_pfp_normalized_exists_bezout_rightright)) * bc)) /\\ exists fom_beta_quotient_pfp_normalized_exists_bezout_rightright_entry. bb = fom_beta_quotient_pfp_normalized_exists_bezout_rightright_entry * S ((S (fom_index_pfp_normalized_exists_bezout_rightright)) * bc) + (fom_value_pfp_normalized_exists_bezout_rightright))) /\\ (exists fom_gap_pfp_normalized_exists_bezout_rightright_value_bound. fom_gap_pfp_normalized_exists_bezout_rightright_value_bound + S (fom_value_pfp_normalized_exists_bezout_rightright) = p))) /\\ (((((((V)=0 \\/ (M)=0) /\\ (((pfgb_Q_normalized_exists_bezout)=0)))) \\/ (((~((V)=0)) /\\ (((~((M)=0)) /\\ (((V)+(M)=S (pfgb_Q_normalized_exists_bezout)))))))) /\\ ((forall pfc_index_normalized_exists_bezout_rightcoefficients. (exists pfa_gap_normalized_exists_bezout_rightcoefficientsbound. pfa_gap_normalized_exists_bezout_rightcoefficientsbound + S (pfc_index_normalized_exists_bezout_rightcoefficients) = (pfgb_Q_normalized_exists_bezout)) -> exists pfc_value_normalized_exists_bezout_rightcoefficients. ((((exists ff_h_pfp_normalized_exists_bezout_rightcoefficientsentry. ff_h_pfp_normalized_exists_bezout_rightcoefficientsentry + S (pfc_value_normalized_exists_bezout_rightcoefficients) = S ((S (pfc_index_normalized_exists_bezout_rightcoefficients)) * pfgb_qc_normalized_exists_bezout)) /\\ exists ff_q_pfp_normalized_exists_bezout_rightcoefficientsentry. pfgb_qb_normalized_exists_bezout = ff_q_pfp_normalized_exists_bezout_rightcoefficientsentry * S ((S (pfc_index_normalized_exists_bezout_rightcoefficients)) * pfgb_qc_normalized_exists_bezout) + (pfc_value_normalized_exists_bezout_rightcoefficients))) /\\ ((exists pfc_terms_code_normalized_exists_bezout_rightcoefficientscoefficient pfc_terms_scale_normalized_exists_bezout_rightcoefficientscoefficient pfc_natural_sum_normalized_exists_bezout_rightcoefficientscoefficient. ((forall pfc_index_normalized_exists_bezout_rightcoefficientscoefficientdiagonal. (exists pfa_gap_normalized_exists_bezout_rightcoefficientscoefficientdiagonalbound. pfa_gap_normalized_exists_bezout_rightcoefficientscoefficientdiagonalbound + S (pfc_index_normalized_exists_bezout_rightcoefficientscoefficientdiagonal) = (S (pfc_index_normalized_exists_bezout_rightcoefficients))) -> exists pfc_value_normalized_exists_bezout_rightcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normalized_exists_bezout_rightcoefficientscoefficientdiagonalentry. ff_h_pfp_normalized_exists_bezout_rightcoefficientscoefficientdiagonalentry + S (pfc_value_normalized_exists_bezout_rightcoefficientscoefficientdiagonal) = S ((S (pfc_index_normalized_exists_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_bezout_rightcoefficientscoefficient)) /\\ exists ff_q_pfp_normalized_exists_bezout_rightcoefficientscoefficientdiagonalentry. pfc_terms_code_normalized_exists_bezout_rightcoefficientscoefficient = ff_q_pfp_normalized_exists_bezout_rightcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normalized_exists_bezout_rightcoefficientscoefficientdiagonal)) * pfc_terms_scale_normalized_exists_bezout_rightcoefficientscoefficient) + (pfc_value_normalized_exists_bezout_rightcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm pfc_left_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm pfc_right_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm. (((pfc_index_normalized_exists_bezout_rightcoefficientscoefficientdiagonal)+pfc_complement_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm=(pfc_index_normalized_exists_bezout_rightcoefficients)) /\\ ((((((exists pfa_gap_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normalized_exists_bezout_rightcoefficientscoefficientdiagonal) = (V)) /\\ ((((exists ff_h_pfp_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normalized_exists_bezout_rightcoefficientscoefficientdiagonal)) * vc)) /\\ exists ff_q_pfp_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermleftentry. vb = ff_q_pfp_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normalized_exists_bezout_rightcoefficientscoefficientdiagonal)) * vc) + (pfc_left_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermleftoutside+(V)=(pfc_index_normalized_exists_bezout_rightcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm) = (M)) /\\ ((((exists ff_h_pfp_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm)) * bc)) /\\ exists ff_q_pfp_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermrightentry. bb = ff_q_pfp_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm)) * bc) + (pfc_right_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normalized_exists_bezout_rightcoefficientscoefficientdiagonaltermrightoutside+(M)=(pfc_complement_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normalized_exists_bezout_rightcoefficientscoefficientdiagonal)=pfc_left_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm*pfc_right_normalized_exists_bezout_rightcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum fs_v_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum. ((((exists fs_h_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_start. fs_h_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_start. fs_u_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum = fs_q_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_terminal. fs_h_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normalized_exists_bezout_rightcoefficientscoefficient) = S ((S (S (pfc_index_normalized_exists_bezout_rightcoefficients))) * fs_v_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_terminal. fs_u_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum = fs_q_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normalized_exists_bezout_rightcoefficients))) * fs_v_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum) + (pfc_natural_sum_normalized_exists_bezout_rightcoefficientscoefficient))) /\\ forall fs_i_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps = S (pfc_index_normalized_exists_bezout_rightcoefficients)) -> exists fs_a_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps fs_r_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps fs_s_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_bezout_rightcoefficientscoefficient)) /\\ exists fs_q_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normalized_exists_bezout_rightcoefficientscoefficient = fs_q_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normalized_exists_bezout_rightcoefficientscoefficient) + (fs_a_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum = fs_q_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum) + (fs_r_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum = fs_q_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum) + (fs_s_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps = fs_r_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps + fs_a_pfc_normalized_exists_bezout_rightcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normalized_exists_bezout_rightcoefficientscoefficientresiduebound. pfa_gap_normalized_exists_bezout_rightcoefficientscoefficientresiduebound + S (pfc_value_normalized_exists_bezout_rightcoefficients) = (p)) /\\ ((exists pfa_offset_left_normalized_exists_bezout_rightcoefficientscoefficientresiduecongruence pfa_offset_right_normalized_exists_bezout_rightcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normalized_exists_bezout_rightcoefficientscoefficient) + (p) * pfa_offset_left_normalized_exists_bezout_rightcoefficientscoefficientresiduecongruence = (pfc_value_normalized_exists_bezout_rightcoefficients) + (p) * pfa_offset_right_normalized_exists_bezout_rightcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((((forall fom_index_pfp_normalized_exists_bezout_sum_left_bounded. (exists fom_gap_pfp_normalized_exists_bezout_sum_left_bounded_index_bound. fom_gap_pfp_normalized_exists_bezout_sum_left_bounded_index_bound + S (fom_index_pfp_normalized_exists_bezout_sum_left_bounded) = pfgb_P_normalized_exists_bezout) -> exists fom_value_pfp_normalized_exists_bezout_sum_left_bounded. ((((exists fom_beta_height_pfp_normalized_exists_bezout_sum_left_bounded_entry. fom_beta_height_pfp_normalized_exists_bezout_sum_left_bounded_entry + S (fom_value_pfp_normalized_exists_bezout_sum_left_bounded) = S ((S (fom_index_pfp_normalized_exists_bezout_sum_left_bounded)) * pfgb_pc_normalized_exists_bezout)) /\\ exists fom_beta_quotient_pfp_normalized_exists_bezout_sum_left_bounded_entry. pfgb_pb_normalized_exists_bezout = fom_beta_quotient_pfp_normalized_exists_bezout_sum_left_bounded_entry * S ((S (fom_index_pfp_normalized_exists_bezout_sum_left_bounded)) * pfgb_pc_normalized_exists_bezout) + (fom_value_pfp_normalized_exists_bezout_sum_left_bounded))) /\\ (exists fom_gap_pfp_normalized_exists_bezout_sum_left_bounded_value_bound. fom_gap_pfp_normalized_exists_bezout_sum_left_bounded_value_bound + S (fom_value_pfp_normalized_exists_bezout_sum_left_bounded) = p))) /\\ (((forall fom_index_pfp_normalized_exists_bezout_sum_right_bounded. (exists fom_gap_pfp_normalized_exists_bezout_sum_right_bounded_index_bound. fom_gap_pfp_normalized_exists_bezout_sum_right_bounded_index_bound + S (fom_index_pfp_normalized_exists_bezout_sum_right_bounded) = pfgb_Q_normalized_exists_bezout) -> exists fom_value_pfp_normalized_exists_bezout_sum_right_bounded. ((((exists fom_beta_height_pfp_normalized_exists_bezout_sum_right_bounded_entry. fom_beta_height_pfp_normalized_exists_bezout_sum_right_bounded_entry + S (fom_value_pfp_normalized_exists_bezout_sum_right_bounded) = S ((S (fom_index_pfp_normalized_exists_bezout_sum_right_bounded)) * pfgb_qc_normalized_exists_bezout)) /\\ exists fom_beta_quotient_pfp_normalized_exists_bezout_sum_right_bounded_entry. pfgb_qb_normalized_exists_bezout = fom_beta_quotient_pfp_normalized_exists_bezout_sum_right_bounded_entry * S ((S (fom_index_pfp_normalized_exists_bezout_sum_right_bounded)) * pfgb_qc_normalized_exists_bezout) + (fom_value_pfp_normalized_exists_bezout_sum_right_bounded))) /\\ (exists fom_gap_pfp_normalized_exists_bezout_sum_right_bounded_value_bound. fom_gap_pfp_normalized_exists_bezout_sum_right_bounded_value_bound + S (fom_value_pfp_normalized_exists_bezout_sum_right_bounded) = p))) /\\ (((forall fom_index_pfp_normalized_exists_bezout_sum_result_bounded. (exists fom_gap_pfp_normalized_exists_bezout_sum_result_bounded_index_bound. fom_gap_pfp_normalized_exists_bezout_sum_result_bounded_index_bound + S (fom_index_pfp_normalized_exists_bezout_sum_result_bounded) = G) -> exists fom_value_pfp_normalized_exists_bezout_sum_result_bounded. ((((exists fom_beta_height_pfp_normalized_exists_bezout_sum_result_bounded_entry. fom_beta_height_pfp_normalized_exists_bezout_sum_result_bounded_entry + S (fom_value_pfp_normalized_exists_bezout_sum_result_bounded) = S ((S (fom_index_pfp_normalized_exists_bezout_sum_result_bounded)) * gc)) /\\ exists fom_beta_quotient_pfp_normalized_exists_bezout_sum_result_bounded_entry. gb = fom_beta_quotient_pfp_normalized_exists_bezout_sum_result_bounded_entry * S ((S (fom_index_pfp_normalized_exists_bezout_sum_result_bounded)) * gc) + (fom_value_pfp_normalized_exists_bezout_sum_result_bounded))) /\\ (exists fom_gap_pfp_normalized_exists_bezout_sum_result_bounded_value_bound. fom_gap_pfp_normalized_exists_bezout_sum_result_bounded_value_bound + S (fom_value_pfp_normalized_exists_bezout_sum_result_bounded) = p))) /\\ ((exists pfga_ub_normalized_exists_bezout_sum pfga_uc_normalized_exists_bezout_sum pfga_vb_normalized_exists_bezout_sum pfga_vc_normalized_exists_bezout_sum pfga_tb_normalized_exists_bezout_sum pfga_tc_normalized_exists_bezout_sum pfga_K_normalized_exists_bezout_sum. ((((forall pfrep_power_normalized_exists_bezout_sum_left pfrep_left_normalized_exists_bezout_sum_left pfrep_right_normalized_exists_bezout_sum_left. ((exists pfrep_position_normalized_exists_bezout_sum_leftfirst. ((pfrep_position_normalized_exists_bezout_sum_leftfirst+S (pfrep_power_normalized_exists_bezout_sum_left)=(pfgb_P_normalized_exists_bezout)) /\\ ((((exists ff_h_pfp_normalized_exists_bezout_sum_leftfirstentry. ff_h_pfp_normalized_exists_bezout_sum_leftfirstentry + S (pfrep_left_normalized_exists_bezout_sum_left) = S ((S (pfrep_position_normalized_exists_bezout_sum_leftfirst)) * pfgb_pc_normalized_exists_bezout)) /\\ exists ff_q_pfp_normalized_exists_bezout_sum_leftfirstentry. pfgb_pb_normalized_exists_bezout = ff_q_pfp_normalized_exists_bezout_sum_leftfirstentry * S ((S (pfrep_position_normalized_exists_bezout_sum_leftfirst)) * pfgb_pc_normalized_exists_bezout) + (pfrep_left_normalized_exists_bezout_sum_left)))))) \\/ (((exists pfrep_gap_normalized_exists_bezout_sum_leftfirstoutside. pfrep_gap_normalized_exists_bezout_sum_leftfirstoutside+(pfgb_P_normalized_exists_bezout)=(pfrep_power_normalized_exists_bezout_sum_left)) /\\ (((pfrep_left_normalized_exists_bezout_sum_left)=0))))) -> ((exists pfrep_position_normalized_exists_bezout_sum_leftsecond. ((pfrep_position_normalized_exists_bezout_sum_leftsecond+S (pfrep_power_normalized_exists_bezout_sum_left)=(pfga_K_normalized_exists_bezout_sum)) /\\ ((((exists ff_h_pfp_normalized_exists_bezout_sum_leftsecondentry. ff_h_pfp_normalized_exists_bezout_sum_leftsecondentry + S (pfrep_right_normalized_exists_bezout_sum_left) = S ((S (pfrep_position_normalized_exists_bezout_sum_leftsecond)) * pfga_uc_normalized_exists_bezout_sum)) /\\ exists ff_q_pfp_normalized_exists_bezout_sum_leftsecondentry. pfga_ub_normalized_exists_bezout_sum = ff_q_pfp_normalized_exists_bezout_sum_leftsecondentry * S ((S (pfrep_position_normalized_exists_bezout_sum_leftsecond)) * pfga_uc_normalized_exists_bezout_sum) + (pfrep_right_normalized_exists_bezout_sum_left)))))) \\/ (((exists pfrep_gap_normalized_exists_bezout_sum_leftsecondoutside. pfrep_gap_normalized_exists_bezout_sum_leftsecondoutside+(pfga_K_normalized_exists_bezout_sum)=(pfrep_power_normalized_exists_bezout_sum_left)) /\\ (((pfrep_right_normalized_exists_bezout_sum_left)=0))))) -> pfrep_left_normalized_exists_bezout_sum_left=pfrep_right_normalized_exists_bezout_sum_left) /\\ ((forall pfrep_power_normalized_exists_bezout_sum_right pfrep_left_normalized_exists_bezout_sum_right pfrep_right_normalized_exists_bezout_sum_right. ((exists pfrep_position_normalized_exists_bezout_sum_rightfirst. ((pfrep_position_normalized_exists_bezout_sum_rightfirst+S (pfrep_power_normalized_exists_bezout_sum_right)=(pfgb_Q_normalized_exists_bezout)) /\\ ((((exists ff_h_pfp_normalized_exists_bezout_sum_rightfirstentry. ff_h_pfp_normalized_exists_bezout_sum_rightfirstentry + S (pfrep_left_normalized_exists_bezout_sum_right) = S ((S (pfrep_position_normalized_exists_bezout_sum_rightfirst)) * pfgb_qc_normalized_exists_bezout)) /\\ exists ff_q_pfp_normalized_exists_bezout_sum_rightfirstentry. pfgb_qb_normalized_exists_bezout = ff_q_pfp_normalized_exists_bezout_sum_rightfirstentry * S ((S (pfrep_position_normalized_exists_bezout_sum_rightfirst)) * pfgb_qc_normalized_exists_bezout) + (pfrep_left_normalized_exists_bezout_sum_right)))))) \\/ (((exists pfrep_gap_normalized_exists_bezout_sum_rightfirstoutside. pfrep_gap_normalized_exists_bezout_sum_rightfirstoutside+(pfgb_Q_normalized_exists_bezout)=(pfrep_power_normalized_exists_bezout_sum_right)) /\\ (((pfrep_left_normalized_exists_bezout_sum_right)=0))))) -> ((exists pfrep_position_normalized_exists_bezout_sum_rightsecond. ((pfrep_position_normalized_exists_bezout_sum_rightsecond+S (pfrep_power_normalized_exists_bezout_sum_right)=(pfga_K_normalized_exists_bezout_sum)) /\\ ((((exists ff_h_pfp_normalized_exists_bezout_sum_rightsecondentry. ff_h_pfp_normalized_exists_bezout_sum_rightsecondentry + S (pfrep_right_normalized_exists_bezout_sum_right) = S ((S (pfrep_position_normalized_exists_bezout_sum_rightsecond)) * pfga_vc_normalized_exists_bezout_sum)) /\\ exists ff_q_pfp_normalized_exists_bezout_sum_rightsecondentry. pfga_vb_normalized_exists_bezout_sum = ff_q_pfp_normalized_exists_bezout_sum_rightsecondentry * S ((S (pfrep_position_normalized_exists_bezout_sum_rightsecond)) * pfga_vc_normalized_exists_bezout_sum) + (pfrep_right_normalized_exists_bezout_sum_right)))))) \\/ (((exists pfrep_gap_normalized_exists_bezout_sum_rightsecondoutside. pfrep_gap_normalized_exists_bezout_sum_rightsecondoutside+(pfga_K_normalized_exists_bezout_sum)=(pfrep_power_normalized_exists_bezout_sum_right)) /\\ (((pfrep_right_normalized_exists_bezout_sum_right)=0))))) -> pfrep_left_normalized_exists_bezout_sum_right=pfrep_right_normalized_exists_bezout_sum_right)))) /\\ (((forall pfp_index_normalized_exists_bezout_sum_add. (exists pfa_gap_normalized_exists_bezout_sum_addindex. pfa_gap_normalized_exists_bezout_sum_addindex + S (pfp_index_normalized_exists_bezout_sum_add) = (pfga_K_normalized_exists_bezout_sum)) -> exists pfp_left_normalized_exists_bezout_sum_add pfp_right_normalized_exists_bezout_sum_add pfp_value_normalized_exists_bezout_sum_add. ((((exists ff_h_pfp_normalized_exists_bezout_sum_addleft. ff_h_pfp_normalized_exists_bezout_sum_addleft + S (pfp_left_normalized_exists_bezout_sum_add) = S ((S (pfp_index_normalized_exists_bezout_sum_add)) * pfga_uc_normalized_exists_bezout_sum)) /\\ exists ff_q_pfp_normalized_exists_bezout_sum_addleft. pfga_ub_normalized_exists_bezout_sum = ff_q_pfp_normalized_exists_bezout_sum_addleft * S ((S (pfp_index_normalized_exists_bezout_sum_add)) * pfga_uc_normalized_exists_bezout_sum) + (pfp_left_normalized_exists_bezout_sum_add))) /\\ (((((exists ff_h_pfp_normalized_exists_bezout_sum_addright. ff_h_pfp_normalized_exists_bezout_sum_addright + S (pfp_right_normalized_exists_bezout_sum_add) = S ((S (pfp_index_normalized_exists_bezout_sum_add)) * pfga_vc_normalized_exists_bezout_sum)) /\\ exists ff_q_pfp_normalized_exists_bezout_sum_addright. pfga_vb_normalized_exists_bezout_sum = ff_q_pfp_normalized_exists_bezout_sum_addright * S ((S (pfp_index_normalized_exists_bezout_sum_add)) * pfga_vc_normalized_exists_bezout_sum) + (pfp_right_normalized_exists_bezout_sum_add))) /\\ (((((exists ff_h_pfp_normalized_exists_bezout_sum_addtarget. ff_h_pfp_normalized_exists_bezout_sum_addtarget + S (pfp_value_normalized_exists_bezout_sum_add) = S ((S (pfp_index_normalized_exists_bezout_sum_add)) * pfga_tc_normalized_exists_bezout_sum)) /\\ exists ff_q_pfp_normalized_exists_bezout_sum_addtarget. pfga_tb_normalized_exists_bezout_sum = ff_q_pfp_normalized_exists_bezout_sum_addtarget * S ((S (pfp_index_normalized_exists_bezout_sum_add)) * pfga_tc_normalized_exists_bezout_sum) + (pfp_value_normalized_exists_bezout_sum_add))) /\\ ((((exists pfa_gap_normalized_exists_bezout_sum_addoperationleft. pfa_gap_normalized_exists_bezout_sum_addoperationleft + S (pfp_left_normalized_exists_bezout_sum_add) = (p)) /\\ (((exists pfa_gap_normalized_exists_bezout_sum_addoperationright. pfa_gap_normalized_exists_bezout_sum_addoperationright + S (pfp_right_normalized_exists_bezout_sum_add) = (p)) /\\ ((((exists pfa_gap_normalized_exists_bezout_sum_addoperationresultbound. pfa_gap_normalized_exists_bezout_sum_addoperationresultbound + S (pfp_value_normalized_exists_bezout_sum_add) = (p)) /\\ ((exists pfa_offset_left_normalized_exists_bezout_sum_addoperationresultcongruence pfa_offset_right_normalized_exists_bezout_sum_addoperationresultcongruence. ((pfp_left_normalized_exists_bezout_sum_add) + (pfp_right_normalized_exists_bezout_sum_add)) + (p) * pfa_offset_left_normalized_exists_bezout_sum_addoperationresultcongruence = (pfp_value_normalized_exists_bezout_sum_add) + (p) * pfa_offset_right_normalized_exists_bezout_sum_addoperationresultcongruence)))))))))))))))) /\\ ((forall pfrep_power_normalized_exists_bezout_sum_result pfrep_left_normalized_exists_bezout_sum_result pfrep_right_normalized_exists_bezout_sum_result. ((exists pfrep_position_normalized_exists_bezout_sum_resultfirst. ((pfrep_position_normalized_exists_bezout_sum_resultfirst+S (pfrep_power_normalized_exists_bezout_sum_result)=(pfga_K_normalized_exists_bezout_sum)) /\\ ((((exists ff_h_pfp_normalized_exists_bezout_sum_resultfirstentry. ff_h_pfp_normalized_exists_bezout_sum_resultfirstentry + S (pfrep_left_normalized_exists_bezout_sum_result) = S ((S (pfrep_position_normalized_exists_bezout_sum_resultfirst)) * pfga_tc_normalized_exists_bezout_sum)) /\\ exists ff_q_pfp_normalized_exists_bezout_sum_resultfirstentry. pfga_tb_normalized_exists_bezout_sum = ff_q_pfp_normalized_exists_bezout_sum_resultfirstentry * S ((S (pfrep_position_normalized_exists_bezout_sum_resultfirst)) * pfga_tc_normalized_exists_bezout_sum) + (pfrep_left_normalized_exists_bezout_sum_result)))))) \\/ (((exists pfrep_gap_normalized_exists_bezout_sum_resultfirstoutside. pfrep_gap_normalized_exists_bezout_sum_resultfirstoutside+(pfga_K_normalized_exists_bezout_sum)=(pfrep_power_normalized_exists_bezout_sum_result)) /\\ (((pfrep_left_normalized_exists_bezout_sum_result)=0))))) -> ((exists pfrep_position_normalized_exists_bezout_sum_resultsecond. ((pfrep_position_normalized_exists_bezout_sum_resultsecond+S (pfrep_power_normalized_exists_bezout_sum_result)=(G)) /\\ ((((exists ff_h_pfp_normalized_exists_bezout_sum_resultsecondentry. ff_h_pfp_normalized_exists_bezout_sum_resultsecondentry + S (pfrep_right_normalized_exists_bezout_sum_result) = S ((S (pfrep_position_normalized_exists_bezout_sum_resultsecond)) * gc)) /\\ exists ff_q_pfp_normalized_exists_bezout_sum_resultsecondentry. gb = ff_q_pfp_normalized_exists_bezout_sum_resultsecondentry * S ((S (pfrep_position_normalized_exists_bezout_sum_resultsecond)) * gc) + (pfrep_right_normalized_exists_bezout_sum_result)))))) \\/ (((exists pfrep_gap_normalized_exists_bezout_sum_resultsecondoutside. pfrep_gap_normalized_exists_bezout_sum_resultsecondoutside+(G)=(pfrep_power_normalized_exists_bezout_sum_result)) /\\ (((pfrep_right_normalized_exists_bezout_sum_result)=0))))) -> pfrep_left_normalized_exists_bezout_sum_result=pfrep_right_normalized_exists_bezout_sum_result)))))))))))))))))))))", "statement_sha256": "d97cbfa3dc334fa5bcf7b9bd92bde2e117b29595864a9cddb093ffe842832463", "summary": "Every pair of canonical polynomials over a prime field has an actual zero-or-monic greatest common right divisor and actual Bezout coefficients. 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"source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_uniqueness_candidate.py", "sha256": "916c24ad6c59609612e97daee6e49347a9522cdb28b44f6f09c6c5760bff0b5b" }, "statement": "forall ab ac d bb bc M a. (((exists ff_h_pfp_length_head. ff_h_pfp_length_head + S (a) = S ((S (0)) * ac)) /\\ exists ff_q_pfp_length_head. ab = ff_q_pfp_length_head * S ((S (0)) * ac) + (a))) -> (~(a=0)) -> (forall pfrep_power_length_equivalent pfrep_left_length_equivalent pfrep_right_length_equivalent. ((exists pfrep_position_length_equivalentfirst. ((pfrep_position_length_equivalentfirst+S (pfrep_power_length_equivalent)=(S d)) /\\ ((((exists ff_h_pfp_length_equivalentfirstentry. ff_h_pfp_length_equivalentfirstentry + S (pfrep_left_length_equivalent) = S ((S (pfrep_position_length_equivalentfirst)) * ac)) /\\ exists ff_q_pfp_length_equivalentfirstentry. ab = ff_q_pfp_length_equivalentfirstentry * S ((S (pfrep_position_length_equivalentfirst)) * ac) + (pfrep_left_length_equivalent)))))) \\/ (((exists pfrep_gap_length_equivalentfirstoutside. pfrep_gap_length_equivalentfirstoutside+(S d)=(pfrep_power_length_equivalent)) /\\ (((pfrep_left_length_equivalent)=0))))) -> ((exists pfrep_position_length_equivalentsecond. ((pfrep_position_length_equivalentsecond+S (pfrep_power_length_equivalent)=(M)) /\\ ((((exists ff_h_pfp_length_equivalentsecondentry. ff_h_pfp_length_equivalentsecondentry + S (pfrep_right_length_equivalent) = S ((S (pfrep_position_length_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_length_equivalentsecondentry. bb = ff_q_pfp_length_equivalentsecondentry * S ((S (pfrep_position_length_equivalentsecond)) * bc) + (pfrep_right_length_equivalent)))))) \\/ (((exists pfrep_gap_length_equivalentsecondoutside. pfrep_gap_length_equivalentsecondoutside+(M)=(pfrep_power_length_equivalent)) /\\ (((pfrep_right_length_equivalent)=0))))) -> pfrep_left_length_equivalent=pfrep_right_length_equivalent) -> (exists pfc_gap_length_result. pfc_gap_length_result+(S d)=(M))", "statement_sha256": "28aac541e72d83aaab41b9ac1e757742f48b28507189f487b35a4433c95b2d8d", "summary": "A nonzero coefficient at the leading power cannot be matched by an outside-prefix zero. This length bound needs neither primality nor canonical coefficients.", "summary_sha256": "85ff452f2d3c45c438d791d70babcd517bd2f6e6e0b342055edb53de85c7ff0f" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "7f9dda9b7e6347a844df7d60237aabaa073bf3e0e5bee22a392ac6100b706fbe", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 472, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "916c24ad6c59609612e97daee6e49347a9522cdb28b44f6f09c6c5760bff0b5b", "test_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85" }, "body_checked": true, "body_receipt": { "command_count": 65, "dependency_count": 4, "dne_command_count": 0, "name": "prime_field_polynomial_equivalent_represented_degrees_equal", "proof_depth": 32, "proof_edges": 96, "proof_nodes": 97, "proof_objects": 97, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "prime_field_polynomial_nonzero_leading_equivalent_length_bound", "le_antisymm", "prime_field_polynomial_equivalent_symmetric", "succ_injective" ], "dependencies_sha256": "6213073b926af0568b06038fa12278bfaf774ce019d69fd011c51f36527e0a91", "empty_context_closure": { "body_proof_depth": 32, "body_proof_nodes": 97, "bundle_campaign": "polynomial-gcd-bezout", "bundle_dependency_edge_count": 1578, "bundle_node_count": 493, "bundle_node_id": 472, "bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "bundle_root_id": 492, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "720b099c222375155e17ebd54d2545ea237245be578d7ac8f3a468908d307549", "status": "checked" }, "enrollment_index": 4201, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "916c24ad6c59609612e97daee6e49347a9522cdb28b44f6f09c6c5760bff0b5b", "kind": "alpha_v34_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_uniqueness_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85", "kind": "alpha_v34_frontier_executable_audit", "path": "peano-lab/py/tests/test_campaign_research_v34_closure.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "kind": "alpha_v34_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-field-polynomial-gcd-bezout-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "kind": "alpha_v34_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=472]" }, { "document_sha256": "81943a545e7b9f667ad758c24de14bb47280e8cf3fbb3ca714efec55fb096408", "kind": "alpha_v34_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "0dd76edb9f2f4e4df0cb6c3caacefa049e65e3fdf218719e6b67ab44c4f39410", "membership": "alpha_only", "name": "prime_field_polynomial_equivalent_represented_degrees_equal", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro ab", "intro ac", "intro L", "intro d", "intro bb", "intro bc", "intro M", "intro e", "intro ha", "intro hb", "intro he", "cases ha", "cases ha_right", "cases hb", "cases hb_right", "have hsame : forall pfrep_power_degree_lengths pfrep_left_degree_lengths pfrep_right_degree_lengths. ((exists pfrep_position_degree_lengthsfirst. ((pfrep_position_degree_lengthsfirst+S (pfrep_power_degree_lengths)=(S d)) /\\ ((((exists ff_h_pfp_degree_lengthsfirstentry. ff_h_pfp_degree_lengthsfirstentry + S (pfrep_left_degree_lengths) = S ((S (pfrep_position_degree_lengthsfirst)) * ac)) /\\ exists ff_q_pfp_degree_lengthsfirstentry. ab = ff_q_pfp_degree_lengthsfirstentry * S ((S (pfrep_position_degree_lengthsfirst)) * ac) + (pfrep_left_degree_lengths)))))) \\/ (((exists pfrep_gap_degree_lengthsfirstoutside. pfrep_gap_degree_lengthsfirstoutside+(S d)=(pfrep_power_degree_lengths)) /\\ (((pfrep_left_degree_lengths)=0))))) -> ((exists pfrep_position_degree_lengthssecond. ((pfrep_position_degree_lengthssecond+S (pfrep_power_degree_lengths)=(S e)) /\\ ((((exists ff_h_pfp_degree_lengthssecondentry. ff_h_pfp_degree_lengthssecondentry + S (pfrep_right_degree_lengths) = S ((S (pfrep_position_degree_lengthssecond)) * bc)) /\\ exists ff_q_pfp_degree_lengthssecondentry. bb = ff_q_pfp_degree_lengthssecondentry * S ((S (pfrep_position_degree_lengthssecond)) * bc) + (pfrep_right_degree_lengths)))))) \\/ (((exists pfrep_gap_degree_lengthssecondoutside. pfrep_gap_degree_lengthssecondoutside+(S e)=(pfrep_power_degree_lengths)) /\\ (((pfrep_right_degree_lengths)=0))))) -> pfrep_left_degree_lengths=pfrep_right_degree_lengths", "have hcopy : forall pfrep_power_degree_equivalent pfrep_left_degree_equivalent pfrep_right_degree_equivalent. ((exists pfrep_position_degree_equivalentfirst. ((pfrep_position_degree_equivalentfirst+S (pfrep_power_degree_equivalent)=(L)) /\\ ((((exists ff_h_pfp_degree_equivalentfirstentry. ff_h_pfp_degree_equivalentfirstentry + S (pfrep_left_degree_equivalent) = S ((S (pfrep_position_degree_equivalentfirst)) * ac)) /\\ exists ff_q_pfp_degree_equivalentfirstentry. ab = ff_q_pfp_degree_equivalentfirstentry * S ((S (pfrep_position_degree_equivalentfirst)) * ac) + (pfrep_left_degree_equivalent)))))) \\/ (((exists pfrep_gap_degree_equivalentfirstoutside. pfrep_gap_degree_equivalentfirstoutside+(L)=(pfrep_power_degree_equivalent)) /\\ (((pfrep_left_degree_equivalent)=0))))) -> ((exists pfrep_position_degree_equivalentsecond. ((pfrep_position_degree_equivalentsecond+S (pfrep_power_degree_equivalent)=(M)) /\\ ((((exists ff_h_pfp_degree_equivalentsecondentry. ff_h_pfp_degree_equivalentsecondentry + S (pfrep_right_degree_equivalent) = S ((S (pfrep_position_degree_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_degree_equivalentsecondentry. bb = ff_q_pfp_degree_equivalentsecondentry * S ((S (pfrep_position_degree_equivalentsecond)) * bc) + (pfrep_right_degree_equivalent)))))) \\/ (((exists pfrep_gap_degree_equivalentsecondoutside. pfrep_gap_degree_equivalentsecondoutside+(M)=(pfrep_power_degree_equivalent)) /\\ (((pfrep_right_degree_equivalent)=0))))) -> pfrep_left_degree_equivalent=pfrep_right_degree_equivalent", "exact he", "rewrite ha_left at hcopy", "rewrite ha_left at hcopy", "rewrite hb_left at hcopy", "rewrite hb_left at hcopy", "exact hcopy", "cases ha_right_right", "cases ha_right_right_witness", "cases hb_right_right", "cases hb_right_right_witness", "have hlength : S d=S e", "specialize le_antisymm (S d)", "specialize le_antisymm (S e)", "apply le_antisymm", "specialize prime_field_polynomial_nonzero_leading_equivalent_length_bound (ab)", "specialize prime_field_polynomial_nonzero_leading_equivalent_length_bound (ac)", "specialize prime_field_polynomial_nonzero_leading_equivalent_length_bound (d)", "specialize prime_field_polynomial_nonzero_leading_equivalent_length_bound (bb)", "specialize prime_field_polynomial_nonzero_leading_equivalent_length_bound (bc)", "specialize prime_field_polynomial_nonzero_leading_equivalent_length_bound (S e)", "specialize prime_field_polynomial_nonzero_leading_equivalent_length_bound (x)", "apply prime_field_polynomial_nonzero_leading_equivalent_length_bound", "exact ha_right_right_witness_left", "exact ha_right_right_witness_right", "exact hsame", "specialize prime_field_polynomial_nonzero_leading_equivalent_length_bound (bb)", "specialize prime_field_polynomial_nonzero_leading_equivalent_length_bound (bc)", "specialize prime_field_polynomial_nonzero_leading_equivalent_length_bound (e)", "specialize prime_field_polynomial_nonzero_leading_equivalent_length_bound (ab)", "specialize prime_field_polynomial_nonzero_leading_equivalent_length_bound (ac)", "specialize prime_field_polynomial_nonzero_leading_equivalent_length_bound (S d)", "specialize prime_field_polynomial_nonzero_leading_equivalent_length_bound (x1)", "apply prime_field_polynomial_nonzero_leading_equivalent_length_bound", "exact hb_right_right_witness_left", "exact hb_right_right_witness_right", "specialize prime_field_polynomial_equivalent_symmetric (ab)", "specialize prime_field_polynomial_equivalent_symmetric (ac)", "specialize prime_field_polynomial_equivalent_symmetric (S d)", "specialize prime_field_polynomial_equivalent_symmetric (bb)", "specialize prime_field_polynomial_equivalent_symmetric (bc)", "specialize prime_field_polynomial_equivalent_symmetric (S e)", "apply prime_field_polynomial_equivalent_symmetric", "exact hsame", "specialize succ_injective (d)", "specialize succ_injective (e)", "apply succ_injective", "exact hlength" ], "script_sha256": "8cf7c983c2bcbae43d0c0e4e79af391db94c4579f1081c6c21fcf39916a5870c", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_uniqueness_candidate.py", "sha256": "916c24ad6c59609612e97daee6e49347a9522cdb28b44f6f09c6c5760bff0b5b" }, "statement": "forall p ab ac L d bb bc M e. ((((L)=S (d)) /\\ (((forall fom_index_pfp_degree_leftcoefficients. (exists fom_gap_pfp_degree_leftcoefficients_index_bound. fom_gap_pfp_degree_leftcoefficients_index_bound + S (fom_index_pfp_degree_leftcoefficients) = L) -> exists fom_value_pfp_degree_leftcoefficients. ((((exists fom_beta_height_pfp_degree_leftcoefficients_entry. fom_beta_height_pfp_degree_leftcoefficients_entry + S (fom_value_pfp_degree_leftcoefficients) = S ((S (fom_index_pfp_degree_leftcoefficients)) * ac)) /\\ exists fom_beta_quotient_pfp_degree_leftcoefficients_entry. ab = fom_beta_quotient_pfp_degree_leftcoefficients_entry * S ((S (fom_index_pfp_degree_leftcoefficients)) * ac) + (fom_value_pfp_degree_leftcoefficients))) /\\ (exists fom_gap_pfp_degree_leftcoefficients_value_bound. fom_gap_pfp_degree_leftcoefficients_value_bound + S (fom_value_pfp_degree_leftcoefficients) = p))) /\\ ((exists pfd_leading_degree_left. ((((exists ff_h_pfp_degree_leftentry. ff_h_pfp_degree_leftentry + S (pfd_leading_degree_left) = S ((S (0)) * ac)) /\\ exists ff_q_pfp_degree_leftentry. ab = ff_q_pfp_degree_leftentry * S ((S (0)) * ac) + (pfd_leading_degree_left))) /\\ ((~(pfd_leading_degree_left=0)))))))))) -> ((((M)=S (e)) /\\ (((forall fom_index_pfp_degree_rightcoefficients. (exists fom_gap_pfp_degree_rightcoefficients_index_bound. fom_gap_pfp_degree_rightcoefficients_index_bound + S (fom_index_pfp_degree_rightcoefficients) = M) -> exists fom_value_pfp_degree_rightcoefficients. ((((exists fom_beta_height_pfp_degree_rightcoefficients_entry. fom_beta_height_pfp_degree_rightcoefficients_entry + S (fom_value_pfp_degree_rightcoefficients) = S ((S (fom_index_pfp_degree_rightcoefficients)) * bc)) /\\ exists fom_beta_quotient_pfp_degree_rightcoefficients_entry. bb = fom_beta_quotient_pfp_degree_rightcoefficients_entry * S ((S (fom_index_pfp_degree_rightcoefficients)) * bc) + (fom_value_pfp_degree_rightcoefficients))) /\\ (exists fom_gap_pfp_degree_rightcoefficients_value_bound. fom_gap_pfp_degree_rightcoefficients_value_bound + S (fom_value_pfp_degree_rightcoefficients) = p))) /\\ ((exists pfd_leading_degree_right. ((((exists ff_h_pfp_degree_rightentry. ff_h_pfp_degree_rightentry + S (pfd_leading_degree_right) = S ((S (0)) * bc)) /\\ exists ff_q_pfp_degree_rightentry. bb = ff_q_pfp_degree_rightentry * S ((S (0)) * bc) + (pfd_leading_degree_right))) /\\ ((~(pfd_leading_degree_right=0)))))))))) -> (forall pfrep_power_degree_equivalent pfrep_left_degree_equivalent pfrep_right_degree_equivalent. ((exists pfrep_position_degree_equivalentfirst. ((pfrep_position_degree_equivalentfirst+S (pfrep_power_degree_equivalent)=(L)) /\\ ((((exists ff_h_pfp_degree_equivalentfirstentry. ff_h_pfp_degree_equivalentfirstentry + S (pfrep_left_degree_equivalent) = S ((S (pfrep_position_degree_equivalentfirst)) * ac)) /\\ exists ff_q_pfp_degree_equivalentfirstentry. ab = ff_q_pfp_degree_equivalentfirstentry * S ((S (pfrep_position_degree_equivalentfirst)) * ac) + (pfrep_left_degree_equivalent)))))) \\/ (((exists pfrep_gap_degree_equivalentfirstoutside. pfrep_gap_degree_equivalentfirstoutside+(L)=(pfrep_power_degree_equivalent)) /\\ (((pfrep_left_degree_equivalent)=0))))) -> ((exists pfrep_position_degree_equivalentsecond. ((pfrep_position_degree_equivalentsecond+S (pfrep_power_degree_equivalent)=(M)) /\\ ((((exists ff_h_pfp_degree_equivalentsecondentry. ff_h_pfp_degree_equivalentsecondentry + S (pfrep_right_degree_equivalent) = S ((S (pfrep_position_degree_equivalentsecond)) * bc)) /\\ exists ff_q_pfp_degree_equivalentsecondentry. bb = ff_q_pfp_degree_equivalentsecondentry * S ((S (pfrep_position_degree_equivalentsecond)) * bc) + (pfrep_right_degree_equivalent)))))) \\/ (((exists pfrep_gap_degree_equivalentsecondoutside. pfrep_gap_degree_equivalentsecondoutside+(M)=(pfrep_power_degree_equivalent)) /\\ (((pfrep_right_degree_equivalent)=0))))) -> pfrep_left_degree_equivalent=pfrep_right_degree_equivalent) -> (d=e)", "statement_sha256": "720b099c222375155e17ebd54d2545ea237245be578d7ac8f3a468908d307549", "summary": "Formal equivalence preserves genuine represented degree across independently encoded and independently length-annotated nonzero-leading prefixes.", "summary_sha256": 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((exists pfrep_position_nonempty_symmetricfirst. ((pfrep_position_nonempty_symmetricfirst+S (pfrep_power_nonempty_symmetric)=(L)) /\\ ((((exists ff_h_pfp_nonempty_symmetricfirstentry. ff_h_pfp_nonempty_symmetricfirstentry + S (pfrep_left_nonempty_symmetric) = S ((S (pfrep_position_nonempty_symmetricfirst)) * ac)) /\\ exists ff_q_pfp_nonempty_symmetricfirstentry. ab = ff_q_pfp_nonempty_symmetricfirstentry * S ((S (pfrep_position_nonempty_symmetricfirst)) * ac) + (pfrep_left_nonempty_symmetric)))))) \\/ (((exists pfrep_gap_nonempty_symmetricfirstoutside. pfrep_gap_nonempty_symmetricfirstoutside+(L)=(pfrep_power_nonempty_symmetric)) /\\ (((pfrep_left_nonempty_symmetric)=0))))) -> ((exists pfrep_position_nonempty_symmetricsecond. ((pfrep_position_nonempty_symmetricsecond+S (pfrep_power_nonempty_symmetric)=(P)) /\\ ((((exists ff_h_pfp_nonempty_symmetricsecondentry. ff_h_pfp_nonempty_symmetricsecondentry + S (pfrep_right_nonempty_symmetric) = S ((S (pfrep_position_nonempty_symmetricsecond)) * pc)) /\\ exists ff_q_pfp_nonempty_symmetricsecondentry. pb = ff_q_pfp_nonempty_symmetricsecondentry * S ((S (pfrep_position_nonempty_symmetricsecond)) * pc) + (pfrep_right_nonempty_symmetric)))))) \\/ (((exists pfrep_gap_nonempty_symmetricsecondoutside. pfrep_gap_nonempty_symmetricsecondoutside+(P)=(pfrep_power_nonempty_symmetric)) /\\ (((pfrep_right_nonempty_symmetric)=0))))) -> pfrep_left_nonempty_symmetric=pfrep_right_nonempty_symmetric", "specialize prime_field_polynomial_equivalent_symmetric (pb)", "specialize prime_field_polynomial_equivalent_symmetric (pc)", "specialize prime_field_polynomial_equivalent_symmetric (P)", "specialize prime_field_polynomial_equivalent_symmetric (ab)", "specialize prime_field_polynomial_equivalent_symmetric (ac)", "specialize prime_field_polynomial_equivalent_symmetric (L)", "apply prime_field_polynomial_equivalent_symmetric", "exact he", "rewrite ha_left at hsame", "rewrite ha_left at hsame", "exact hsame", "cases hlength", "cases hlength_left", "have hzero : S a=0", "specialize le_zero (S a)", "apply le_zero", "rewrite <- hlength_left_right", "exact hbound", "specialize succ_ne_zero (a)", "apply succ_ne_zero", "exact hzero", "cases hlength_right", "cases hlength_right_right", "apply hlength_right_left", "exact hz" ], "script_sha256": "0d706caf338a39cc1a86c63d1044e1d03c01bda78a63740694990bd5d5df2d77", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_uniqueness_candidate.py", "sha256": "916c24ad6c59609612e97daee6e49347a9522cdb28b44f6f09c6c5760bff0b5b" }, "statement": "forall p qb qc Q db dc D pb pc P ab ac L a. ((((L)=S (a)) /\\ (((forall fom_index_pfp_nonempty_targetcoefficients. (exists fom_gap_pfp_nonempty_targetcoefficients_index_bound. fom_gap_pfp_nonempty_targetcoefficients_index_bound + S (fom_index_pfp_nonempty_targetcoefficients) = L) -> exists fom_value_pfp_nonempty_targetcoefficients. ((((exists fom_beta_height_pfp_nonempty_targetcoefficients_entry. fom_beta_height_pfp_nonempty_targetcoefficients_entry + S (fom_value_pfp_nonempty_targetcoefficients) = S ((S (fom_index_pfp_nonempty_targetcoefficients)) * ac)) /\\ exists fom_beta_quotient_pfp_nonempty_targetcoefficients_entry. ab = fom_beta_quotient_pfp_nonempty_targetcoefficients_entry * S ((S (fom_index_pfp_nonempty_targetcoefficients)) * ac) + (fom_value_pfp_nonempty_targetcoefficients))) /\\ (exists fom_gap_pfp_nonempty_targetcoefficients_value_bound. fom_gap_pfp_nonempty_targetcoefficients_value_bound + S (fom_value_pfp_nonempty_targetcoefficients) = p))) /\\ ((exists pfd_leading_nonempty_target. ((((exists ff_h_pfp_nonempty_targetentry. ff_h_pfp_nonempty_targetentry + S (pfd_leading_nonempty_target) = S ((S (0)) * ac)) /\\ exists ff_q_pfp_nonempty_targetentry. ab = ff_q_pfp_nonempty_targetentry * S ((S (0)) * ac) + (pfd_leading_nonempty_target))) /\\ ((~(pfd_leading_nonempty_target=0)))))))))) -> (((forall fom_index_pfp_nonempty_productleft. (exists fom_gap_pfp_nonempty_productleft_index_bound. fom_gap_pfp_nonempty_productleft_index_bound + S (fom_index_pfp_nonempty_productleft) = Q) -> exists fom_value_pfp_nonempty_productleft. ((((exists fom_beta_height_pfp_nonempty_productleft_entry. fom_beta_height_pfp_nonempty_productleft_entry + S (fom_value_pfp_nonempty_productleft) = S ((S (fom_index_pfp_nonempty_productleft)) * qc)) /\\ exists fom_beta_quotient_pfp_nonempty_productleft_entry. qb = fom_beta_quotient_pfp_nonempty_productleft_entry * S ((S (fom_index_pfp_nonempty_productleft)) * qc) + (fom_value_pfp_nonempty_productleft))) /\\ (exists fom_gap_pfp_nonempty_productleft_value_bound. fom_gap_pfp_nonempty_productleft_value_bound + S (fom_value_pfp_nonempty_productleft) = p))) /\\ (((forall fom_index_pfp_nonempty_productright. (exists fom_gap_pfp_nonempty_productright_index_bound. fom_gap_pfp_nonempty_productright_index_bound + S (fom_index_pfp_nonempty_productright) = D) -> exists fom_value_pfp_nonempty_productright. ((((exists fom_beta_height_pfp_nonempty_productright_entry. fom_beta_height_pfp_nonempty_productright_entry + S (fom_value_pfp_nonempty_productright) = S ((S (fom_index_pfp_nonempty_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_nonempty_productright_entry. db = fom_beta_quotient_pfp_nonempty_productright_entry * S ((S (fom_index_pfp_nonempty_productright)) * dc) + (fom_value_pfp_nonempty_productright))) /\\ (exists fom_gap_pfp_nonempty_productright_value_bound. fom_gap_pfp_nonempty_productright_value_bound + S (fom_value_pfp_nonempty_productright) = p))) /\\ (((((((Q)=0 \\/ (D)=0) /\\ (((P)=0)))) \\/ (((~((Q)=0)) /\\ (((~((D)=0)) /\\ (((Q)+(D)=S (P)))))))) /\\ ((forall pfc_index_nonempty_productcoefficients. (exists pfa_gap_nonempty_productcoefficientsbound. pfa_gap_nonempty_productcoefficientsbound + S (pfc_index_nonempty_productcoefficients) = (P)) -> exists pfc_value_nonempty_productcoefficients. ((((exists ff_h_pfp_nonempty_productcoefficientsentry. ff_h_pfp_nonempty_productcoefficientsentry + S (pfc_value_nonempty_productcoefficients) = S ((S (pfc_index_nonempty_productcoefficients)) * pc)) /\\ exists ff_q_pfp_nonempty_productcoefficientsentry. pb = ff_q_pfp_nonempty_productcoefficientsentry * S ((S (pfc_index_nonempty_productcoefficients)) * pc) + (pfc_value_nonempty_productcoefficients))) /\\ ((exists pfc_terms_code_nonempty_productcoefficientscoefficient pfc_terms_scale_nonempty_productcoefficientscoefficient pfc_natural_sum_nonempty_productcoefficientscoefficient. ((forall pfc_index_nonempty_productcoefficientscoefficientdiagonal. (exists pfa_gap_nonempty_productcoefficientscoefficientdiagonalbound. pfa_gap_nonempty_productcoefficientscoefficientdiagonalbound + S (pfc_index_nonempty_productcoefficientscoefficientdiagonal) = (S (pfc_index_nonempty_productcoefficients))) -> exists pfc_value_nonempty_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_nonempty_productcoefficientscoefficientdiagonalentry. ff_h_pfp_nonempty_productcoefficientscoefficientdiagonalentry + S (pfc_value_nonempty_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_nonempty_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_nonempty_productcoefficientscoefficient)) /\\ exists ff_q_pfp_nonempty_productcoefficientscoefficientdiagonalentry. pfc_terms_code_nonempty_productcoefficientscoefficient = ff_q_pfp_nonempty_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_nonempty_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_nonempty_productcoefficientscoefficient) + (pfc_value_nonempty_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_nonempty_productcoefficientscoefficientdiagonalterm pfc_left_nonempty_productcoefficientscoefficientdiagonalterm pfc_right_nonempty_productcoefficientscoefficientdiagonalterm. (((pfc_index_nonempty_productcoefficientscoefficientdiagonal)+pfc_complement_nonempty_productcoefficientscoefficientdiagonalterm=(pfc_index_nonempty_productcoefficients)) /\\ ((((((exists pfa_gap_nonempty_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_nonempty_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_nonempty_productcoefficientscoefficientdiagonal) = (Q)) /\\ ((((exists ff_h_pfp_nonempty_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_nonempty_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_nonempty_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_nonempty_productcoefficientscoefficientdiagonal)) * qc)) /\\ exists ff_q_pfp_nonempty_productcoefficientscoefficientdiagonaltermleftentry. qb = ff_q_pfp_nonempty_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_nonempty_productcoefficientscoefficientdiagonal)) * qc) + (pfc_left_nonempty_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_nonempty_productcoefficientscoefficientdiagonaltermleftoutside+(Q)=(pfc_index_nonempty_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_nonempty_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_nonempty_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_nonempty_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_nonempty_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_nonempty_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_nonempty_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_nonempty_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_nonempty_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_nonempty_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_nonempty_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_nonempty_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_nonempty_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_nonempty_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_nonempty_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_nonempty_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_nonempty_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_nonempty_productcoefficientscoefficientdiagonal)=pfc_left_nonempty_productcoefficientscoefficientdiagonalterm*pfc_right_nonempty_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_nonempty_productcoefficientscoefficientsum fs_v_pfc_nonempty_productcoefficientscoefficientsum. ((((exists fs_h_pfc_nonempty_productcoefficientscoefficientsum_body_start. fs_h_pfc_nonempty_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_nonempty_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_productcoefficientscoefficientsum_body_start. fs_u_pfc_nonempty_productcoefficientscoefficientsum = fs_q_pfc_nonempty_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_nonempty_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_nonempty_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_nonempty_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_nonempty_productcoefficientscoefficient) = S ((S (S (pfc_index_nonempty_productcoefficients))) * fs_v_pfc_nonempty_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_nonempty_productcoefficientscoefficientsum = fs_q_pfc_nonempty_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_nonempty_productcoefficients))) * fs_v_pfc_nonempty_productcoefficientscoefficientsum) + (pfc_natural_sum_nonempty_productcoefficientscoefficient))) /\\ forall fs_i_pfc_nonempty_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_nonempty_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_nonempty_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_nonempty_productcoefficientscoefficientsum_body_steps = S (pfc_index_nonempty_productcoefficients)) -> exists fs_a_pfc_nonempty_productcoefficientscoefficientsum_body_steps fs_r_pfc_nonempty_productcoefficientscoefficientsum_body_steps fs_s_pfc_nonempty_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_nonempty_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_nonempty_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_nonempty_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_nonempty_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_nonempty_productcoefficientscoefficient)) /\\ exists fs_q_pfc_nonempty_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_nonempty_productcoefficientscoefficient = fs_q_pfc_nonempty_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_nonempty_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_nonempty_productcoefficientscoefficient) + (fs_a_pfc_nonempty_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_nonempty_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_nonempty_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_nonempty_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_nonempty_productcoefficientscoefficientsum = fs_q_pfc_nonempty_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_nonempty_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_productcoefficientscoefficientsum) + (fs_r_pfc_nonempty_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_nonempty_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_nonempty_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_nonempty_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_nonempty_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_nonempty_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_nonempty_productcoefficientscoefficientsum = fs_q_pfc_nonempty_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_nonempty_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_nonempty_productcoefficientscoefficientsum) + (fs_s_pfc_nonempty_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_nonempty_productcoefficientscoefficientsum_body_steps = fs_r_pfc_nonempty_productcoefficientscoefficientsum_body_steps + fs_a_pfc_nonempty_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_nonempty_productcoefficientscoefficientresiduebound. pfa_gap_nonempty_productcoefficientscoefficientresiduebound + S (pfc_value_nonempty_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_nonempty_productcoefficientscoefficientresiduecongruence pfa_offset_right_nonempty_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_nonempty_productcoefficientscoefficient) + (p) * pfa_offset_left_nonempty_productcoefficientscoefficientresiduecongruence = (pfc_value_nonempty_productcoefficients) + (p) * pfa_offset_right_nonempty_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_nonempty_target_equivalent pfrep_left_nonempty_target_equivalent pfrep_right_nonempty_target_equivalent. ((exists pfrep_position_nonempty_target_equivalentfirst. ((pfrep_position_nonempty_target_equivalentfirst+S (pfrep_power_nonempty_target_equivalent)=(P)) /\\ ((((exists ff_h_pfp_nonempty_target_equivalentfirstentry. ff_h_pfp_nonempty_target_equivalentfirstentry + S (pfrep_left_nonempty_target_equivalent) = S ((S (pfrep_position_nonempty_target_equivalentfirst)) * pc)) /\\ exists ff_q_pfp_nonempty_target_equivalentfirstentry. pb = ff_q_pfp_nonempty_target_equivalentfirstentry * S ((S (pfrep_position_nonempty_target_equivalentfirst)) * pc) + (pfrep_left_nonempty_target_equivalent)))))) \\/ (((exists pfrep_gap_nonempty_target_equivalentfirstoutside. pfrep_gap_nonempty_target_equivalentfirstoutside+(P)=(pfrep_power_nonempty_target_equivalent)) /\\ (((pfrep_left_nonempty_target_equivalent)=0))))) -> ((exists pfrep_position_nonempty_target_equivalentsecond. ((pfrep_position_nonempty_target_equivalentsecond+S (pfrep_power_nonempty_target_equivalent)=(L)) /\\ ((((exists ff_h_pfp_nonempty_target_equivalentsecondentry. ff_h_pfp_nonempty_target_equivalentsecondentry + S (pfrep_right_nonempty_target_equivalent) = S ((S (pfrep_position_nonempty_target_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_nonempty_target_equivalentsecondentry. ab = ff_q_pfp_nonempty_target_equivalentsecondentry * S ((S (pfrep_position_nonempty_target_equivalentsecond)) * ac) + (pfrep_right_nonempty_target_equivalent)))))) \\/ (((exists pfrep_gap_nonempty_target_equivalentsecondoutside. pfrep_gap_nonempty_target_equivalentsecondoutside+(L)=(pfrep_power_nonempty_target_equivalent)) /\\ (((pfrep_right_nonempty_target_equivalent)=0))))) -> pfrep_left_nonempty_target_equivalent=pfrep_right_nonempty_target_equivalent) -> (~(Q=0))", "statement_sha256": "a9e37f903eda17c613ed5c54b9889d5e78eccef9c3d1ef4085557a90f9ea8927", "summary": "An actual product formally equal to a nonzero-leading representation cannot have an empty left factor. 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[ "intro p", "intro db", "intro dc", "intro D", "intro d", "intro ab", "intro ac", "intro L", "intro a", "intro hp", "intro hd", "intro ha", "intro hrd", "have hpn : ~(p=0)", "intro hpzero", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hpzero", "have hdbound : forall fom_index_pfp_factor_divisor_bound. (exists fom_gap_pfp_factor_divisor_bound_index_bound. fom_gap_pfp_factor_divisor_bound_index_bound + S (fom_index_pfp_factor_divisor_bound) = D) -> exists fom_value_pfp_factor_divisor_bound. ((((exists fom_beta_height_pfp_factor_divisor_bound_entry. fom_beta_height_pfp_factor_divisor_bound_entry + S (fom_value_pfp_factor_divisor_bound) = S ((S (fom_index_pfp_factor_divisor_bound)) * dc)) /\\ exists fom_beta_quotient_pfp_factor_divisor_bound_entry. db = fom_beta_quotient_pfp_factor_divisor_bound_entry * S ((S (fom_index_pfp_factor_divisor_bound)) * dc) + (fom_value_pfp_factor_divisor_bound))) /\\ (exists fom_gap_pfp_factor_divisor_bound_value_bound. fom_gap_pfp_factor_divisor_bound_value_bound + S (fom_value_pfp_factor_divisor_bound) = p))", "cases hd", "cases hd_right", "exact hd_right_left", "cases hrd", "cases hrd_right", "cases hrd_right_witness", "cases hrd_right_witness_witness", "cases hrd_right_witness_witness_witness", "cases hrd_right_witness_witness_witness_witness", "cases hrd_right_witness_witness_witness_witness_witness", "cases hrd_right_witness_witness_witness_witness_witness_witness", "have hqbound : forall fom_index_pfp_factor_quotient_bound. (exists fom_gap_pfp_factor_quotient_bound_index_bound. fom_gap_pfp_factor_quotient_bound_index_bound + S (fom_index_pfp_factor_quotient_bound) = x2) -> exists fom_value_pfp_factor_quotient_bound. ((((exists fom_beta_height_pfp_factor_quotient_bound_entry. fom_beta_height_pfp_factor_quotient_bound_entry + S (fom_value_pfp_factor_quotient_bound) = S ((S (fom_index_pfp_factor_quotient_bound)) * x1)) /\\ exists fom_beta_quotient_pfp_factor_quotient_bound_entry. x = fom_beta_quotient_pfp_factor_quotient_bound_entry * S ((S (fom_index_pfp_factor_quotient_bound)) * x1) + (fom_value_pfp_factor_quotient_bound))) /\\ (exists fom_gap_pfp_factor_quotient_bound_value_bound. fom_gap_pfp_factor_quotient_bound_value_bound + S (fom_value_pfp_factor_quotient_bound) = p))", "cases hrd_right_witness_witness_witness_witness_witness_witness_left", "cases hrd_right_witness_witness_witness_witness_witness_witness_left_right", "cases hrd_right_witness_witness_witness_witness_witness_witness_left_right_right", "exact hrd_right_witness_witness_witness_witness_witness_witness_left_left", "have ht : exists t tb tc T. (((x2)=(t)+(T)) /\\ (((forall fom_index_pfp_factor_triminput. (exists fom_gap_pfp_factor_triminput_index_bound. fom_gap_pfp_factor_triminput_index_bound + S (fom_index_pfp_factor_triminput) = x2) -> exists fom_value_pfp_factor_triminput. ((((exists fom_beta_height_pfp_factor_triminput_entry. fom_beta_height_pfp_factor_triminput_entry + S (fom_value_pfp_factor_triminput) = S ((S (fom_index_pfp_factor_triminput)) * x1)) /\\ exists fom_beta_quotient_pfp_factor_triminput_entry. x = fom_beta_quotient_pfp_factor_triminput_entry * S ((S (fom_index_pfp_factor_triminput)) * x1) + (fom_value_pfp_factor_triminput))) /\\ (exists fom_gap_pfp_factor_triminput_value_bound. fom_gap_pfp_factor_triminput_value_bound + S (fom_value_pfp_factor_triminput) = p))) /\\ (((forall pfp_repeat_index_factor_trimremoved. (exists pfa_gap_factor_trimremovedindex. pfa_gap_factor_trimremovedindex + S (pfp_repeat_index_factor_trimremoved) = (t)) -> (((exists ff_h_pfp_factor_trimremovedentry. ff_h_pfp_factor_trimremovedentry + S (0) = S ((S (pfp_repeat_index_factor_trimremoved)) * x1)) /\\ exists ff_q_pfp_factor_trimremovedentry. x = ff_q_pfp_factor_trimremovedentry * S ((S (pfp_repeat_index_factor_trimremoved)) * x1) + (0)))) /\\ (((forall pftrim_index_factor_trimsuffix pftrim_value_factor_trimsuffix. (exists pfa_gap_factor_trimsuffixbound. pfa_gap_factor_trimsuffixbound + S (pftrim_index_factor_trimsuffix) = (T)) -> (((exists ff_h_pfp_factor_trimsuffixsource. ff_h_pfp_factor_trimsuffixsource + S (pftrim_value_factor_trimsuffix) = S ((S ((t)+pftrim_index_factor_trimsuffix)) * x1)) /\\ exists ff_q_pfp_factor_trimsuffixsource. x = ff_q_pfp_factor_trimsuffixsource * S ((S ((t)+pftrim_index_factor_trimsuffix)) * x1) + (pftrim_value_factor_trimsuffix))) -> (((exists ff_h_pfp_factor_trimsuffixoutput. ff_h_pfp_factor_trimsuffixoutput + S (pftrim_value_factor_trimsuffix) = S ((S (pftrim_index_factor_trimsuffix)) * tc)) /\\ exists ff_q_pfp_factor_trimsuffixoutput. tb = ff_q_pfp_factor_trimsuffixoutput * S ((S (pftrim_index_factor_trimsuffix)) * tc) + (pftrim_value_factor_trimsuffix)))) /\\ (((T)=0 \\/ (exists pftrim_leading_factor_trimnormal. ((((exists ff_h_pfp_factor_trimnormalentry. ff_h_pfp_factor_trimnormalentry + S (pftrim_leading_factor_trimnormal) = S ((S (0)) * tc)) /\\ exists ff_q_pfp_factor_trimnormalentry. tb = ff_q_pfp_factor_trimnormalentry * S ((S (0)) * tc) + (pftrim_leading_factor_trimnormal))) /\\ ((~(pftrim_leading_factor_trimnormal=0))))))))))))))", "specialize prime_field_polynomial_trim_exists (p)", "specialize prime_field_polynomial_trim_exists (x)", "specialize prime_field_polynomial_trim_exists (x1)", "specialize prime_field_polynomial_trim_exists (x2)", "apply prime_field_polynomial_trim_exists", "exact hqbound", "cases ht", "cases ht_witness", "cases ht_witness_witness", "cases ht_witness_witness_witness", "have htbound : forall fom_index_pfp_factor_trim_bound. (exists fom_gap_pfp_factor_trim_bound_index_bound. fom_gap_pfp_factor_trim_bound_index_bound + S (fom_index_pfp_factor_trim_bound) = x9) -> exists fom_value_pfp_factor_trim_bound. ((((exists fom_beta_height_pfp_factor_trim_bound_entry. fom_beta_height_pfp_factor_trim_bound_entry + S (fom_value_pfp_factor_trim_bound) = S ((S (fom_index_pfp_factor_trim_bound)) * x8)) /\\ exists fom_beta_quotient_pfp_factor_trim_bound_entry. x7 = fom_beta_quotient_pfp_factor_trim_bound_entry * S ((S (fom_index_pfp_factor_trim_bound)) * x8) + (fom_value_pfp_factor_trim_bound))) /\\ (exists fom_gap_pfp_factor_trim_bound_value_bound. fom_gap_pfp_factor_trim_bound_value_bound + S (fom_value_pfp_factor_trim_bound) = p))", "specialize prime_field_polynomial_trim_output_coefficients (p)", "specialize prime_field_polynomial_trim_output_coefficients (x)", "specialize prime_field_polynomial_trim_output_coefficients (x1)", "specialize prime_field_polynomial_trim_output_coefficients (x2)", "specialize prime_field_polynomial_trim_output_coefficients (x6)", "specialize prime_field_polynomial_trim_output_coefficients (x7)", "specialize prime_field_polynomial_trim_output_coefficients (x8)", "specialize prime_field_polynomial_trim_output_coefficients (x9)", "apply prime_field_polynomial_trim_output_coefficients", "exact ht_witness_witness_witness_witness", "have hqe : forall pfrep_power_factor_trim_equivalent pfrep_left_factor_trim_equivalent pfrep_right_factor_trim_equivalent. ((exists pfrep_position_factor_trim_equivalentfirst. ((pfrep_position_factor_trim_equivalentfirst+S (pfrep_power_factor_trim_equivalent)=(x2)) /\\ ((((exists ff_h_pfp_factor_trim_equivalentfirstentry. ff_h_pfp_factor_trim_equivalentfirstentry + S (pfrep_left_factor_trim_equivalent) = S ((S (pfrep_position_factor_trim_equivalentfirst)) * x1)) /\\ exists ff_q_pfp_factor_trim_equivalentfirstentry. x = ff_q_pfp_factor_trim_equivalentfirstentry * S ((S (pfrep_position_factor_trim_equivalentfirst)) * x1) + (pfrep_left_factor_trim_equivalent)))))) \\/ (((exists pfrep_gap_factor_trim_equivalentfirstoutside. pfrep_gap_factor_trim_equivalentfirstoutside+(x2)=(pfrep_power_factor_trim_equivalent)) /\\ (((pfrep_left_factor_trim_equivalent)=0))))) -> ((exists pfrep_position_factor_trim_equivalentsecond. ((pfrep_position_factor_trim_equivalentsecond+S (pfrep_power_factor_trim_equivalent)=(x9)) /\\ ((((exists ff_h_pfp_factor_trim_equivalentsecondentry. ff_h_pfp_factor_trim_equivalentsecondentry + S (pfrep_right_factor_trim_equivalent) = S ((S (pfrep_position_factor_trim_equivalentsecond)) * x8)) /\\ exists ff_q_pfp_factor_trim_equivalentsecondentry. x7 = ff_q_pfp_factor_trim_equivalentsecondentry * S ((S (pfrep_position_factor_trim_equivalentsecond)) * x8) + (pfrep_right_factor_trim_equivalent)))))) \\/ (((exists pfrep_gap_factor_trim_equivalentsecondoutside. pfrep_gap_factor_trim_equivalentsecondoutside+(x9)=(pfrep_power_factor_trim_equivalent)) /\\ (((pfrep_right_factor_trim_equivalent)=0))))) -> pfrep_left_factor_trim_equivalent=pfrep_right_factor_trim_equivalent", "specialize prime_field_polynomial_trim_equivalent (p)", "specialize prime_field_polynomial_trim_equivalent (x)", "specialize prime_field_polynomial_trim_equivalent (x1)", "specialize prime_field_polynomial_trim_equivalent (x2)", "specialize prime_field_polynomial_trim_equivalent (x6)", "specialize prime_field_polynomial_trim_equivalent (x7)", "specialize prime_field_polynomial_trim_equivalent (x8)", "specialize prime_field_polynomial_trim_equivalent (x9)", "apply prime_field_polynomial_trim_equivalent", "exact ht_witness_witness_witness_witness", "have hplen : exists N. ((((x9)=0 \\/ (D)=0) /\\ (((N)=0)))) \\/ (((~((x9)=0)) /\\ (((~((D)=0)) /\\ (((x9)+(D)=S (N)))))))", "specialize polynomial_product_length_exists (x9)", "specialize polynomial_product_length_exists (D)", "apply polynomial_product_length_exists", "cases hplen", "have hpnew : exists vb vc. ((forall fom_index_pfp_factor_new_productleft. (exists fom_gap_pfp_factor_new_productleft_index_bound. fom_gap_pfp_factor_new_productleft_index_bound + S (fom_index_pfp_factor_new_productleft) = x9) -> exists fom_value_pfp_factor_new_productleft. ((((exists fom_beta_height_pfp_factor_new_productleft_entry. fom_beta_height_pfp_factor_new_productleft_entry + S (fom_value_pfp_factor_new_productleft) = S ((S (fom_index_pfp_factor_new_productleft)) * x8)) /\\ exists fom_beta_quotient_pfp_factor_new_productleft_entry. x7 = fom_beta_quotient_pfp_factor_new_productleft_entry * S ((S (fom_index_pfp_factor_new_productleft)) * x8) + (fom_value_pfp_factor_new_productleft))) /\\ (exists fom_gap_pfp_factor_new_productleft_value_bound. fom_gap_pfp_factor_new_productleft_value_bound + S (fom_value_pfp_factor_new_productleft) = p))) /\\ (((forall fom_index_pfp_factor_new_productright. (exists fom_gap_pfp_factor_new_productright_index_bound. fom_gap_pfp_factor_new_productright_index_bound + S (fom_index_pfp_factor_new_productright) = D) -> exists fom_value_pfp_factor_new_productright. ((((exists fom_beta_height_pfp_factor_new_productright_entry. fom_beta_height_pfp_factor_new_productright_entry + S (fom_value_pfp_factor_new_productright) = S ((S (fom_index_pfp_factor_new_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_factor_new_productright_entry. db = fom_beta_quotient_pfp_factor_new_productright_entry * S ((S (fom_index_pfp_factor_new_productright)) * dc) + (fom_value_pfp_factor_new_productright))) /\\ (exists fom_gap_pfp_factor_new_productright_value_bound. fom_gap_pfp_factor_new_productright_value_bound + S (fom_value_pfp_factor_new_productright) = p))) /\\ (((((((x9)=0 \\/ (D)=0) /\\ (((x10)=0)))) \\/ (((~((x9)=0)) /\\ (((~((D)=0)) /\\ (((x9)+(D)=S (x10)))))))) /\\ ((forall pfc_index_factor_new_productcoefficients. (exists pfa_gap_factor_new_productcoefficientsbound. pfa_gap_factor_new_productcoefficientsbound + S (pfc_index_factor_new_productcoefficients) = (x10)) -> exists pfc_value_factor_new_productcoefficients. ((((exists ff_h_pfp_factor_new_productcoefficientsentry. ff_h_pfp_factor_new_productcoefficientsentry + S (pfc_value_factor_new_productcoefficients) = S ((S (pfc_index_factor_new_productcoefficients)) * vc)) /\\ exists ff_q_pfp_factor_new_productcoefficientsentry. vb = ff_q_pfp_factor_new_productcoefficientsentry * S ((S (pfc_index_factor_new_productcoefficients)) * vc) + (pfc_value_factor_new_productcoefficients))) /\\ ((exists pfc_terms_code_factor_new_productcoefficientscoefficient pfc_terms_scale_factor_new_productcoefficientscoefficient pfc_natural_sum_factor_new_productcoefficientscoefficient. ((forall pfc_index_factor_new_productcoefficientscoefficientdiagonal. (exists pfa_gap_factor_new_productcoefficientscoefficientdiagonalbound. pfa_gap_factor_new_productcoefficientscoefficientdiagonalbound + S (pfc_index_factor_new_productcoefficientscoefficientdiagonal) = (S (pfc_index_factor_new_productcoefficients))) -> exists pfc_value_factor_new_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_factor_new_productcoefficientscoefficientdiagonalentry. ff_h_pfp_factor_new_productcoefficientscoefficientdiagonalentry + S (pfc_value_factor_new_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_factor_new_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_factor_new_productcoefficientscoefficient)) /\\ exists ff_q_pfp_factor_new_productcoefficientscoefficientdiagonalentry. pfc_terms_code_factor_new_productcoefficientscoefficient = ff_q_pfp_factor_new_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_factor_new_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_factor_new_productcoefficientscoefficient) + (pfc_value_factor_new_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_factor_new_productcoefficientscoefficientdiagonalterm pfc_left_factor_new_productcoefficientscoefficientdiagonalterm pfc_right_factor_new_productcoefficientscoefficientdiagonalterm. (((pfc_index_factor_new_productcoefficientscoefficientdiagonal)+pfc_complement_factor_new_productcoefficientscoefficientdiagonalterm=(pfc_index_factor_new_productcoefficients)) /\\ ((((((exists pfa_gap_factor_new_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_factor_new_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_factor_new_productcoefficientscoefficientdiagonal) = (x9)) /\\ ((((exists ff_h_pfp_factor_new_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_factor_new_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_factor_new_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_factor_new_productcoefficientscoefficientdiagonal)) * x8)) /\\ exists ff_q_pfp_factor_new_productcoefficientscoefficientdiagonaltermleftentry. x7 = ff_q_pfp_factor_new_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_factor_new_productcoefficientscoefficientdiagonal)) * x8) + (pfc_left_factor_new_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_factor_new_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_factor_new_productcoefficientscoefficientdiagonaltermleftoutside+(x9)=(pfc_index_factor_new_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_factor_new_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_factor_new_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_factor_new_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_factor_new_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_factor_new_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_factor_new_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_factor_new_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_factor_new_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_factor_new_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_factor_new_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_factor_new_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_factor_new_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_factor_new_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_factor_new_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_factor_new_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_factor_new_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_factor_new_productcoefficientscoefficientdiagonal)=pfc_left_factor_new_productcoefficientscoefficientdiagonalterm*pfc_right_factor_new_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_factor_new_productcoefficientscoefficientsum fs_v_pfc_factor_new_productcoefficientscoefficientsum. ((((exists fs_h_pfc_factor_new_productcoefficientscoefficientsum_body_start. fs_h_pfc_factor_new_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_factor_new_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_new_productcoefficientscoefficientsum_body_start. fs_u_pfc_factor_new_productcoefficientscoefficientsum = fs_q_pfc_factor_new_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_factor_new_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_factor_new_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_factor_new_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_factor_new_productcoefficientscoefficient) = S ((S (S (pfc_index_factor_new_productcoefficients))) * fs_v_pfc_factor_new_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_new_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_factor_new_productcoefficientscoefficientsum = fs_q_pfc_factor_new_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_factor_new_productcoefficients))) * fs_v_pfc_factor_new_productcoefficientscoefficientsum) + (pfc_natural_sum_factor_new_productcoefficientscoefficient))) /\\ forall fs_i_pfc_factor_new_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_factor_new_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_factor_new_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_factor_new_productcoefficientscoefficientsum_body_steps = S (pfc_index_factor_new_productcoefficients)) -> exists fs_a_pfc_factor_new_productcoefficientscoefficientsum_body_steps fs_r_pfc_factor_new_productcoefficientscoefficientsum_body_steps fs_s_pfc_factor_new_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_factor_new_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_factor_new_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_factor_new_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_factor_new_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_factor_new_productcoefficientscoefficient)) /\\ exists fs_q_pfc_factor_new_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_factor_new_productcoefficientscoefficient = fs_q_pfc_factor_new_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_factor_new_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_factor_new_productcoefficientscoefficient) + (fs_a_pfc_factor_new_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_factor_new_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_factor_new_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_factor_new_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_factor_new_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_new_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_new_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_factor_new_productcoefficientscoefficientsum = fs_q_pfc_factor_new_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_factor_new_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_new_productcoefficientscoefficientsum) + (fs_r_pfc_factor_new_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_factor_new_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_factor_new_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_factor_new_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_factor_new_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_new_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_new_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_factor_new_productcoefficientscoefficientsum = fs_q_pfc_factor_new_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_factor_new_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_new_productcoefficientscoefficientsum) + (fs_s_pfc_factor_new_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_factor_new_productcoefficientscoefficientsum_body_steps = fs_r_pfc_factor_new_productcoefficientscoefficientsum_body_steps + fs_a_pfc_factor_new_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_factor_new_productcoefficientscoefficientresiduebound. pfa_gap_factor_new_productcoefficientscoefficientresiduebound + S (pfc_value_factor_new_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_factor_new_productcoefficientscoefficientresiduecongruence pfa_offset_right_factor_new_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_factor_new_productcoefficientscoefficient) + (p) * pfa_offset_left_factor_new_productcoefficientscoefficientresiduecongruence = (pfc_value_factor_new_productcoefficients) + (p) * pfa_offset_right_factor_new_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "specialize prime_field_polynomial_convolution_at_length_exists (p)", "specialize prime_field_polynomial_convolution_at_length_exists (x7)", "specialize prime_field_polynomial_convolution_at_length_exists (x8)", "specialize prime_field_polynomial_convolution_at_length_exists (x9)", "specialize prime_field_polynomial_convolution_at_length_exists (db)", "specialize prime_field_polynomial_convolution_at_length_exists (dc)", "specialize prime_field_polynomial_convolution_at_length_exists (D)", "specialize prime_field_polynomial_convolution_at_length_exists (x10)", "apply prime_field_polynomial_convolution_at_length_exists", "exact hpn", "exact htbound", "exact hdbound", "exact hplen_witness", "cases hpnew", "cases hpnew_witness", "have hequiv : forall pfrep_power_factor_actual_equivalent pfrep_left_factor_actual_equivalent pfrep_right_factor_actual_equivalent. ((exists pfrep_position_factor_actual_equivalentfirst. ((pfrep_position_factor_actual_equivalentfirst+S (pfrep_power_factor_actual_equivalent)=(x10)) /\\ ((((exists ff_h_pfp_factor_actual_equivalentfirstentry. ff_h_pfp_factor_actual_equivalentfirstentry + S (pfrep_left_factor_actual_equivalent) = S ((S (pfrep_position_factor_actual_equivalentfirst)) * x12)) /\\ exists ff_q_pfp_factor_actual_equivalentfirstentry. x11 = ff_q_pfp_factor_actual_equivalentfirstentry * S ((S (pfrep_position_factor_actual_equivalentfirst)) * x12) + (pfrep_left_factor_actual_equivalent)))))) \\/ (((exists pfrep_gap_factor_actual_equivalentfirstoutside. pfrep_gap_factor_actual_equivalentfirstoutside+(x10)=(pfrep_power_factor_actual_equivalent)) /\\ (((pfrep_left_factor_actual_equivalent)=0))))) -> ((exists pfrep_position_factor_actual_equivalentsecond. ((pfrep_position_factor_actual_equivalentsecond+S (pfrep_power_factor_actual_equivalent)=(L)) /\\ ((((exists ff_h_pfp_factor_actual_equivalentsecondentry. ff_h_pfp_factor_actual_equivalentsecondentry + S (pfrep_right_factor_actual_equivalent) = S ((S (pfrep_position_factor_actual_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_factor_actual_equivalentsecondentry. ab = ff_q_pfp_factor_actual_equivalentsecondentry * S ((S (pfrep_position_factor_actual_equivalentsecond)) * ac) + (pfrep_right_factor_actual_equivalent)))))) \\/ (((exists pfrep_gap_factor_actual_equivalentsecondoutside. pfrep_gap_factor_actual_equivalentsecondoutside+(L)=(pfrep_power_factor_actual_equivalent)) /\\ (((pfrep_right_factor_actual_equivalent)=0))))) -> pfrep_left_factor_actual_equivalent=pfrep_right_factor_actual_equivalent", "specialize prime_field_polynomial_equivalent_transitive (x11)", "specialize prime_field_polynomial_equivalent_transitive (x12)", "specialize prime_field_polynomial_equivalent_transitive (x10)", "specialize prime_field_polynomial_equivalent_transitive (x3)", "specialize prime_field_polynomial_equivalent_transitive (x4)", "specialize prime_field_polynomial_equivalent_transitive (x5)", "specialize prime_field_polynomial_equivalent_transitive (ab)", "specialize prime_field_polynomial_equivalent_transitive (ac)", "specialize prime_field_polynomial_equivalent_transitive (L)", "apply prime_field_polynomial_equivalent_transitive", "specialize prime_field_polynomial_equivalent_symmetric (x3)", "specialize prime_field_polynomial_equivalent_symmetric (x4)", "specialize prime_field_polynomial_equivalent_symmetric (x5)", "specialize prime_field_polynomial_equivalent_symmetric (x11)", "specialize prime_field_polynomial_equivalent_symmetric (x12)", "specialize prime_field_polynomial_equivalent_symmetric (x10)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (p)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x1)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x2)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (db)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (dc)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (D)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x3)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x4)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x5)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x7)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x8)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x9)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x11)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x12)", "specialize prime_field_polynomial_convolution_equivalent_congruent_left (x10)", "apply prime_field_polynomial_convolution_equivalent_congruent_left", "exact hpn", "exact hqe", "exact hrd_right_witness_witness_witness_witness_witness_witness_left", "exact hpnew_witness_witness", "exact hrd_right_witness_witness_witness_witness_witness_witness_right", "have hn : ~(x9=0)", "intro htzero", "specialize prime_field_polynomial_product_equivalent_nonzero_left_nonempty (p)", "specialize prime_field_polynomial_product_equivalent_nonzero_left_nonempty (x7)", "specialize prime_field_polynomial_product_equivalent_nonzero_left_nonempty (x8)", "specialize prime_field_polynomial_product_equivalent_nonzero_left_nonempty (x9)", "specialize prime_field_polynomial_product_equivalent_nonzero_left_nonempty (db)", "specialize prime_field_polynomial_product_equivalent_nonzero_left_nonempty (dc)", "specialize prime_field_polynomial_product_equivalent_nonzero_left_nonempty (D)", "specialize prime_field_polynomial_product_equivalent_nonzero_left_nonempty (x11)", "specialize prime_field_polynomial_product_equivalent_nonzero_left_nonempty (x12)", "specialize prime_field_polynomial_product_equivalent_nonzero_left_nonempty (x10)", "specialize prime_field_polynomial_product_equivalent_nonzero_left_nonempty (ab)", "specialize prime_field_polynomial_product_equivalent_nonzero_left_nonempty (ac)", "specialize prime_field_polynomial_product_equivalent_nonzero_left_nonempty (L)", "specialize prime_field_polynomial_product_equivalent_nonzero_left_nonempty (a)", "apply prime_field_polynomial_product_equivalent_nonzero_left_nonempty", "exact ha", "exact hpnew_witness_witness", "exact hequiv", "exact htzero", "have hqd : exists e. (((x9)=S (e)) /\\ (((forall fom_index_pfp_factor_trim_degreecoefficients. (exists fom_gap_pfp_factor_trim_degreecoefficients_index_bound. fom_gap_pfp_factor_trim_degreecoefficients_index_bound + S (fom_index_pfp_factor_trim_degreecoefficients) = x9) -> exists fom_value_pfp_factor_trim_degreecoefficients. ((((exists fom_beta_height_pfp_factor_trim_degreecoefficients_entry. fom_beta_height_pfp_factor_trim_degreecoefficients_entry + S (fom_value_pfp_factor_trim_degreecoefficients) = S ((S (fom_index_pfp_factor_trim_degreecoefficients)) * x8)) /\\ exists fom_beta_quotient_pfp_factor_trim_degreecoefficients_entry. x7 = fom_beta_quotient_pfp_factor_trim_degreecoefficients_entry * S ((S (fom_index_pfp_factor_trim_degreecoefficients)) * x8) + (fom_value_pfp_factor_trim_degreecoefficients))) /\\ (exists fom_gap_pfp_factor_trim_degreecoefficients_value_bound. fom_gap_pfp_factor_trim_degreecoefficients_value_bound + S (fom_value_pfp_factor_trim_degreecoefficients) = p))) /\\ ((exists pfd_leading_factor_trim_degree. ((((exists ff_h_pfp_factor_trim_degreeentry. ff_h_pfp_factor_trim_degreeentry + S (pfd_leading_factor_trim_degree) = S ((S (0)) * x8)) /\\ exists ff_q_pfp_factor_trim_degreeentry. x7 = ff_q_pfp_factor_trim_degreeentry * S ((S (0)) * x8) + (pfd_leading_factor_trim_degree))) /\\ ((~(pfd_leading_factor_trim_degree=0)))))))))", "specialize prime_field_polynomial_trim_nonempty_degree_exists (p)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (x)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (x1)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (x2)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (x6)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (x7)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (x8)", "specialize prime_field_polynomial_trim_nonempty_degree_exists (x9)", "apply prime_field_polynomial_trim_nonempty_degree_exists", "exact ht_witness_witness_witness_witness", "exact hn", "cases hqd", "have hpd : (((x10)=S (x13+d)) /\\ (((forall fom_index_pfp_factor_product_degreecoefficients. (exists fom_gap_pfp_factor_product_degreecoefficients_index_bound. fom_gap_pfp_factor_product_degreecoefficients_index_bound + S (fom_index_pfp_factor_product_degreecoefficients) = x10) -> exists fom_value_pfp_factor_product_degreecoefficients. ((((exists fom_beta_height_pfp_factor_product_degreecoefficients_entry. fom_beta_height_pfp_factor_product_degreecoefficients_entry + S (fom_value_pfp_factor_product_degreecoefficients) = S ((S (fom_index_pfp_factor_product_degreecoefficients)) * x12)) /\\ exists fom_beta_quotient_pfp_factor_product_degreecoefficients_entry. x11 = fom_beta_quotient_pfp_factor_product_degreecoefficients_entry * S ((S (fom_index_pfp_factor_product_degreecoefficients)) * x12) + (fom_value_pfp_factor_product_degreecoefficients))) /\\ (exists fom_gap_pfp_factor_product_degreecoefficients_value_bound. fom_gap_pfp_factor_product_degreecoefficients_value_bound + S (fom_value_pfp_factor_product_degreecoefficients) = p))) /\\ ((exists pfd_leading_factor_product_degree. ((((exists ff_h_pfp_factor_product_degreeentry. ff_h_pfp_factor_product_degreeentry + S (pfd_leading_factor_product_degree) = S ((S (0)) * x12)) /\\ exists ff_q_pfp_factor_product_degreeentry. x11 = ff_q_pfp_factor_product_degreeentry * S ((S (0)) * x12) + (pfd_leading_factor_product_degree))) /\\ ((~(pfd_leading_factor_product_degree=0)))))))))", "specialize prime_field_polynomial_convolution_represented_degree (p)", "specialize prime_field_polynomial_convolution_represented_degree (x7)", "specialize prime_field_polynomial_convolution_represented_degree (x8)", "specialize prime_field_polynomial_convolution_represented_degree (x9)", "specialize prime_field_polynomial_convolution_represented_degree (x13)", "specialize prime_field_polynomial_convolution_represented_degree (db)", "specialize prime_field_polynomial_convolution_represented_degree (dc)", "specialize prime_field_polynomial_convolution_represented_degree (D)", "specialize prime_field_polynomial_convolution_represented_degree (d)", "specialize prime_field_polynomial_convolution_represented_degree (x11)", "specialize prime_field_polynomial_convolution_represented_degree (x12)", "specialize prime_field_polynomial_convolution_represented_degree (x10)", "apply prime_field_polynomial_convolution_represented_degree", "exact hp", "exact hqd_witness", "exact hd", "exact hpnew_witness_witness", "have hsum : x13+d=a", "specialize prime_field_polynomial_equivalent_represented_degrees_equal (p)", "specialize prime_field_polynomial_equivalent_represented_degrees_equal (x11)", "specialize prime_field_polynomial_equivalent_represented_degrees_equal (x12)", "specialize prime_field_polynomial_equivalent_represented_degrees_equal (x10)", "specialize prime_field_polynomial_equivalent_represented_degrees_equal (x13+d)", "specialize prime_field_polynomial_equivalent_represented_degrees_equal (ab)", "specialize prime_field_polynomial_equivalent_represented_degrees_equal (ac)", "specialize prime_field_polynomial_equivalent_represented_degrees_equal (L)", "specialize prime_field_polynomial_equivalent_represented_degrees_equal (a)", "apply prime_field_polynomial_equivalent_represented_degrees_equal", "exact hpd", "exact ha", "exact hequiv", "have hqlen : x9=S x13", "cases hqd_witness", "exact hqd_witness_left", "have hplen2 : x10=S a", "cases hpd", "rewrite hpd_left", "rewrite hsum", "refl", "exists x7", "exists x8", "exists x13", "exists x11", "exists x12", "split", "have hqdnew : (((x9)=S (x13)) /\\ (((forall fom_index_pfp_factor_pack_degreecoefficients. (exists fom_gap_pfp_factor_pack_degreecoefficients_index_bound. fom_gap_pfp_factor_pack_degreecoefficients_index_bound + S (fom_index_pfp_factor_pack_degreecoefficients) = x9) -> exists fom_value_pfp_factor_pack_degreecoefficients. ((((exists fom_beta_height_pfp_factor_pack_degreecoefficients_entry. fom_beta_height_pfp_factor_pack_degreecoefficients_entry + S (fom_value_pfp_factor_pack_degreecoefficients) = S ((S (fom_index_pfp_factor_pack_degreecoefficients)) * x8)) /\\ exists fom_beta_quotient_pfp_factor_pack_degreecoefficients_entry. x7 = fom_beta_quotient_pfp_factor_pack_degreecoefficients_entry * S ((S (fom_index_pfp_factor_pack_degreecoefficients)) * x8) + (fom_value_pfp_factor_pack_degreecoefficients))) /\\ (exists fom_gap_pfp_factor_pack_degreecoefficients_value_bound. fom_gap_pfp_factor_pack_degreecoefficients_value_bound + S (fom_value_pfp_factor_pack_degreecoefficients) = p))) /\\ ((exists pfd_leading_factor_pack_degree. ((((exists ff_h_pfp_factor_pack_degreeentry. ff_h_pfp_factor_pack_degreeentry + S (pfd_leading_factor_pack_degree) = S ((S (0)) * x8)) /\\ exists ff_q_pfp_factor_pack_degreeentry. x7 = ff_q_pfp_factor_pack_degreeentry * S ((S (0)) * x8) + (pfd_leading_factor_pack_degree))) /\\ ((~(pfd_leading_factor_pack_degree=0)))))))))", "exact hqd_witness", "rewrite hqlen at hqdnew", "rewrite hqlen at hqdnew", "exact hqdnew", "split", "have hcp : ((forall fom_index_pfp_factor_actual_newleft. (exists fom_gap_pfp_factor_actual_newleft_index_bound. fom_gap_pfp_factor_actual_newleft_index_bound + S (fom_index_pfp_factor_actual_newleft) = x9) -> exists fom_value_pfp_factor_actual_newleft. ((((exists fom_beta_height_pfp_factor_actual_newleft_entry. fom_beta_height_pfp_factor_actual_newleft_entry + S (fom_value_pfp_factor_actual_newleft) = S ((S (fom_index_pfp_factor_actual_newleft)) * x8)) /\\ exists fom_beta_quotient_pfp_factor_actual_newleft_entry. x7 = fom_beta_quotient_pfp_factor_actual_newleft_entry * S ((S (fom_index_pfp_factor_actual_newleft)) * x8) + (fom_value_pfp_factor_actual_newleft))) /\\ (exists fom_gap_pfp_factor_actual_newleft_value_bound. fom_gap_pfp_factor_actual_newleft_value_bound + S (fom_value_pfp_factor_actual_newleft) = p))) /\\ (((forall fom_index_pfp_factor_actual_newright. (exists fom_gap_pfp_factor_actual_newright_index_bound. fom_gap_pfp_factor_actual_newright_index_bound + S (fom_index_pfp_factor_actual_newright) = D) -> exists fom_value_pfp_factor_actual_newright. ((((exists fom_beta_height_pfp_factor_actual_newright_entry. fom_beta_height_pfp_factor_actual_newright_entry + S (fom_value_pfp_factor_actual_newright) = S ((S (fom_index_pfp_factor_actual_newright)) * dc)) /\\ exists fom_beta_quotient_pfp_factor_actual_newright_entry. db = fom_beta_quotient_pfp_factor_actual_newright_entry * S ((S (fom_index_pfp_factor_actual_newright)) * dc) + (fom_value_pfp_factor_actual_newright))) /\\ (exists fom_gap_pfp_factor_actual_newright_value_bound. fom_gap_pfp_factor_actual_newright_value_bound + S (fom_value_pfp_factor_actual_newright) = p))) /\\ (((((((x9)=0 \\/ (D)=0) /\\ (((x10)=0)))) \\/ (((~((x9)=0)) /\\ (((~((D)=0)) /\\ (((x9)+(D)=S (x10)))))))) /\\ ((forall pfc_index_factor_actual_newcoefficients. (exists pfa_gap_factor_actual_newcoefficientsbound. pfa_gap_factor_actual_newcoefficientsbound + S (pfc_index_factor_actual_newcoefficients) = (x10)) -> exists pfc_value_factor_actual_newcoefficients. ((((exists ff_h_pfp_factor_actual_newcoefficientsentry. ff_h_pfp_factor_actual_newcoefficientsentry + S (pfc_value_factor_actual_newcoefficients) = S ((S (pfc_index_factor_actual_newcoefficients)) * x12)) /\\ exists ff_q_pfp_factor_actual_newcoefficientsentry. x11 = ff_q_pfp_factor_actual_newcoefficientsentry * S ((S (pfc_index_factor_actual_newcoefficients)) * x12) + (pfc_value_factor_actual_newcoefficients))) /\\ ((exists pfc_terms_code_factor_actual_newcoefficientscoefficient pfc_terms_scale_factor_actual_newcoefficientscoefficient pfc_natural_sum_factor_actual_newcoefficientscoefficient. ((forall pfc_index_factor_actual_newcoefficientscoefficientdiagonal. (exists pfa_gap_factor_actual_newcoefficientscoefficientdiagonalbound. pfa_gap_factor_actual_newcoefficientscoefficientdiagonalbound + S (pfc_index_factor_actual_newcoefficientscoefficientdiagonal) = (S (pfc_index_factor_actual_newcoefficients))) -> exists pfc_value_factor_actual_newcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_factor_actual_newcoefficientscoefficientdiagonalentry. ff_h_pfp_factor_actual_newcoefficientscoefficientdiagonalentry + S (pfc_value_factor_actual_newcoefficientscoefficientdiagonal) = S ((S (pfc_index_factor_actual_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_factor_actual_newcoefficientscoefficient)) /\\ exists ff_q_pfp_factor_actual_newcoefficientscoefficientdiagonalentry. pfc_terms_code_factor_actual_newcoefficientscoefficient = ff_q_pfp_factor_actual_newcoefficientscoefficientdiagonalentry * S ((S (pfc_index_factor_actual_newcoefficientscoefficientdiagonal)) * pfc_terms_scale_factor_actual_newcoefficientscoefficient) + (pfc_value_factor_actual_newcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_factor_actual_newcoefficientscoefficientdiagonalterm pfc_left_factor_actual_newcoefficientscoefficientdiagonalterm pfc_right_factor_actual_newcoefficientscoefficientdiagonalterm. (((pfc_index_factor_actual_newcoefficientscoefficientdiagonal)+pfc_complement_factor_actual_newcoefficientscoefficientdiagonalterm=(pfc_index_factor_actual_newcoefficients)) /\\ ((((((exists pfa_gap_factor_actual_newcoefficientscoefficientdiagonaltermleftinside. pfa_gap_factor_actual_newcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_factor_actual_newcoefficientscoefficientdiagonal) = (x9)) /\\ ((((exists ff_h_pfp_factor_actual_newcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_factor_actual_newcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_factor_actual_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_factor_actual_newcoefficientscoefficientdiagonal)) * x8)) /\\ exists ff_q_pfp_factor_actual_newcoefficientscoefficientdiagonaltermleftentry. x7 = ff_q_pfp_factor_actual_newcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_factor_actual_newcoefficientscoefficientdiagonal)) * x8) + (pfc_left_factor_actual_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_factor_actual_newcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_factor_actual_newcoefficientscoefficientdiagonaltermleftoutside+(x9)=(pfc_index_factor_actual_newcoefficientscoefficientdiagonal)) /\\ (((pfc_left_factor_actual_newcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_factor_actual_newcoefficientscoefficientdiagonaltermrightinside. pfa_gap_factor_actual_newcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_factor_actual_newcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_factor_actual_newcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_factor_actual_newcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_factor_actual_newcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_factor_actual_newcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_factor_actual_newcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_factor_actual_newcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_factor_actual_newcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_factor_actual_newcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_factor_actual_newcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_factor_actual_newcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_factor_actual_newcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_factor_actual_newcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_factor_actual_newcoefficientscoefficientdiagonal)=pfc_left_factor_actual_newcoefficientscoefficientdiagonalterm*pfc_right_factor_actual_newcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_factor_actual_newcoefficientscoefficientsum fs_v_pfc_factor_actual_newcoefficientscoefficientsum. ((((exists fs_h_pfc_factor_actual_newcoefficientscoefficientsum_body_start. fs_h_pfc_factor_actual_newcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_factor_actual_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_actual_newcoefficientscoefficientsum_body_start. fs_u_pfc_factor_actual_newcoefficientscoefficientsum = fs_q_pfc_factor_actual_newcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_factor_actual_newcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_factor_actual_newcoefficientscoefficientsum_body_terminal. fs_h_pfc_factor_actual_newcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_factor_actual_newcoefficientscoefficient) = S ((S (S (pfc_index_factor_actual_newcoefficients))) * fs_v_pfc_factor_actual_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_actual_newcoefficientscoefficientsum_body_terminal. fs_u_pfc_factor_actual_newcoefficientscoefficientsum = fs_q_pfc_factor_actual_newcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_factor_actual_newcoefficients))) * fs_v_pfc_factor_actual_newcoefficientscoefficientsum) + (pfc_natural_sum_factor_actual_newcoefficientscoefficient))) /\\ forall fs_i_pfc_factor_actual_newcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_factor_actual_newcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_factor_actual_newcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_factor_actual_newcoefficientscoefficientsum_body_steps = S (pfc_index_factor_actual_newcoefficients)) -> exists fs_a_pfc_factor_actual_newcoefficientscoefficientsum_body_steps fs_r_pfc_factor_actual_newcoefficientscoefficientsum_body_steps fs_s_pfc_factor_actual_newcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_factor_actual_newcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_factor_actual_newcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_factor_actual_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_factor_actual_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_factor_actual_newcoefficientscoefficient)) /\\ exists fs_q_pfc_factor_actual_newcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_factor_actual_newcoefficientscoefficient = fs_q_pfc_factor_actual_newcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_factor_actual_newcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_factor_actual_newcoefficientscoefficient) + (fs_a_pfc_factor_actual_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_factor_actual_newcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_factor_actual_newcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_factor_actual_newcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_factor_actual_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_actual_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_actual_newcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_factor_actual_newcoefficientscoefficientsum = fs_q_pfc_factor_actual_newcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_factor_actual_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_actual_newcoefficientscoefficientsum) + (fs_r_pfc_factor_actual_newcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_factor_actual_newcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_factor_actual_newcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_factor_actual_newcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_factor_actual_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_actual_newcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_actual_newcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_factor_actual_newcoefficientscoefficientsum = fs_q_pfc_factor_actual_newcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_factor_actual_newcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_actual_newcoefficientscoefficientsum) + (fs_s_pfc_factor_actual_newcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_factor_actual_newcoefficientscoefficientsum_body_steps = fs_r_pfc_factor_actual_newcoefficientscoefficientsum_body_steps + fs_a_pfc_factor_actual_newcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_factor_actual_newcoefficientscoefficientresiduebound. pfa_gap_factor_actual_newcoefficientscoefficientresiduebound + S (pfc_value_factor_actual_newcoefficients) = (p)) /\\ ((exists pfa_offset_left_factor_actual_newcoefficientscoefficientresiduecongruence pfa_offset_right_factor_actual_newcoefficientscoefficientresiduecongruence. (pfc_natural_sum_factor_actual_newcoefficientscoefficient) + (p) * pfa_offset_left_factor_actual_newcoefficientscoefficientresiduecongruence = (pfc_value_factor_actual_newcoefficients) + (p) * pfa_offset_right_factor_actual_newcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hpnew_witness_witness", "rewrite hqlen at hcp", "rewrite hqlen at hcp", "rewrite hqlen at hcp", "rewrite hqlen at hcp", "rewrite hqlen at hcp", "rewrite hqlen at hcp", "rewrite hplen2 at hcp", "rewrite hplen2 at hcp", "rewrite hplen2 at hcp", "exact hcp", "split", "have hep : forall pfrep_power_factor_actual_equivalent pfrep_left_factor_actual_equivalent pfrep_right_factor_actual_equivalent. ((exists pfrep_position_factor_actual_equivalentfirst. ((pfrep_position_factor_actual_equivalentfirst+S (pfrep_power_factor_actual_equivalent)=(x10)) /\\ ((((exists ff_h_pfp_factor_actual_equivalentfirstentry. ff_h_pfp_factor_actual_equivalentfirstentry + S (pfrep_left_factor_actual_equivalent) = S ((S (pfrep_position_factor_actual_equivalentfirst)) * x12)) /\\ exists ff_q_pfp_factor_actual_equivalentfirstentry. x11 = ff_q_pfp_factor_actual_equivalentfirstentry * S ((S (pfrep_position_factor_actual_equivalentfirst)) * x12) + (pfrep_left_factor_actual_equivalent)))))) \\/ (((exists pfrep_gap_factor_actual_equivalentfirstoutside. pfrep_gap_factor_actual_equivalentfirstoutside+(x10)=(pfrep_power_factor_actual_equivalent)) /\\ (((pfrep_left_factor_actual_equivalent)=0))))) -> ((exists pfrep_position_factor_actual_equivalentsecond. ((pfrep_position_factor_actual_equivalentsecond+S (pfrep_power_factor_actual_equivalent)=(L)) /\\ ((((exists ff_h_pfp_factor_actual_equivalentsecondentry. ff_h_pfp_factor_actual_equivalentsecondentry + S (pfrep_right_factor_actual_equivalent) = S ((S (pfrep_position_factor_actual_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_factor_actual_equivalentsecondentry. ab = ff_q_pfp_factor_actual_equivalentsecondentry * S ((S (pfrep_position_factor_actual_equivalentsecond)) * ac) + (pfrep_right_factor_actual_equivalent)))))) \\/ (((exists pfrep_gap_factor_actual_equivalentsecondoutside. pfrep_gap_factor_actual_equivalentsecondoutside+(L)=(pfrep_power_factor_actual_equivalent)) /\\ (((pfrep_right_factor_actual_equivalent)=0))))) -> pfrep_left_factor_actual_equivalent=pfrep_right_factor_actual_equivalent", "exact hequiv", "rewrite hplen2 at hep", "rewrite hplen2 at hep", "exact hep", "exact hsum" ], "script_sha256": "2055437e4d8f82a97e2f1df3e78d1890776d687dd7342593f866d51972b39ee0", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_uniqueness_candidate.py", "sha256": "916c24ad6c59609612e97daee6e49347a9522cdb28b44f6f09c6c5760bff0b5b" }, "statement": "forall p db dc D d ab ac L a. (~((p) = 1) /\\ forall pfa_factor_left_factor_prime pfa_factor_right_factor_prime. (p) = pfa_factor_left_factor_prime * pfa_factor_right_factor_prime -> pfa_factor_left_factor_prime = 1 \\/ pfa_factor_right_factor_prime = 1) -> ((((D)=S (d)) /\\ (((forall fom_index_pfp_factor_divisorcoefficients. (exists fom_gap_pfp_factor_divisorcoefficients_index_bound. fom_gap_pfp_factor_divisorcoefficients_index_bound + S (fom_index_pfp_factor_divisorcoefficients) = D) -> exists fom_value_pfp_factor_divisorcoefficients. ((((exists fom_beta_height_pfp_factor_divisorcoefficients_entry. fom_beta_height_pfp_factor_divisorcoefficients_entry + S (fom_value_pfp_factor_divisorcoefficients) = S ((S (fom_index_pfp_factor_divisorcoefficients)) * dc)) /\\ exists fom_beta_quotient_pfp_factor_divisorcoefficients_entry. db = fom_beta_quotient_pfp_factor_divisorcoefficients_entry * S ((S (fom_index_pfp_factor_divisorcoefficients)) * dc) + (fom_value_pfp_factor_divisorcoefficients))) /\\ (exists fom_gap_pfp_factor_divisorcoefficients_value_bound. fom_gap_pfp_factor_divisorcoefficients_value_bound + S (fom_value_pfp_factor_divisorcoefficients) = p))) /\\ ((exists pfd_leading_factor_divisor. ((((exists ff_h_pfp_factor_divisorentry. ff_h_pfp_factor_divisorentry + S (pfd_leading_factor_divisor) = S ((S (0)) * dc)) /\\ exists ff_q_pfp_factor_divisorentry. db = ff_q_pfp_factor_divisorentry * S ((S (0)) * dc) + (pfd_leading_factor_divisor))) /\\ ((~(pfd_leading_factor_divisor=0)))))))))) -> ((((L)=S (a)) /\\ (((forall fom_index_pfp_factor_targetcoefficients. (exists fom_gap_pfp_factor_targetcoefficients_index_bound. fom_gap_pfp_factor_targetcoefficients_index_bound + S (fom_index_pfp_factor_targetcoefficients) = L) -> exists fom_value_pfp_factor_targetcoefficients. ((((exists fom_beta_height_pfp_factor_targetcoefficients_entry. fom_beta_height_pfp_factor_targetcoefficients_entry + S (fom_value_pfp_factor_targetcoefficients) = S ((S (fom_index_pfp_factor_targetcoefficients)) * ac)) /\\ exists fom_beta_quotient_pfp_factor_targetcoefficients_entry. ab = fom_beta_quotient_pfp_factor_targetcoefficients_entry * S ((S (fom_index_pfp_factor_targetcoefficients)) * ac) + (fom_value_pfp_factor_targetcoefficients))) /\\ (exists fom_gap_pfp_factor_targetcoefficients_value_bound. fom_gap_pfp_factor_targetcoefficients_value_bound + S (fom_value_pfp_factor_targetcoefficients) = p))) /\\ ((exists pfd_leading_factor_target. ((((exists ff_h_pfp_factor_targetentry. ff_h_pfp_factor_targetentry + S (pfd_leading_factor_target) = S ((S (0)) * ac)) /\\ exists ff_q_pfp_factor_targetentry. ab = ff_q_pfp_factor_targetentry * S ((S (0)) * ac) + (pfd_leading_factor_target))) /\\ ((~(pfd_leading_factor_target=0)))))))))) -> (((forall fom_index_pfp_factor_divisibility_canonical. (exists fom_gap_pfp_factor_divisibility_canonical_index_bound. fom_gap_pfp_factor_divisibility_canonical_index_bound + S (fom_index_pfp_factor_divisibility_canonical) = L) -> exists fom_value_pfp_factor_divisibility_canonical. ((((exists fom_beta_height_pfp_factor_divisibility_canonical_entry. fom_beta_height_pfp_factor_divisibility_canonical_entry + S (fom_value_pfp_factor_divisibility_canonical) = S ((S (fom_index_pfp_factor_divisibility_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_factor_divisibility_canonical_entry. ab = fom_beta_quotient_pfp_factor_divisibility_canonical_entry * S ((S (fom_index_pfp_factor_divisibility_canonical)) * ac) + (fom_value_pfp_factor_divisibility_canonical))) /\\ (exists fom_gap_pfp_factor_divisibility_canonical_value_bound. fom_gap_pfp_factor_divisibility_canonical_value_bound + S (fom_value_pfp_factor_divisibility_canonical) = p))) /\\ ((exists pfgu_qb_factor_divisibility pfgu_qc_factor_divisibility pfgu_Q_factor_divisibility pfgu_pb_factor_divisibility pfgu_pc_factor_divisibility pfgu_P_factor_divisibility. ((((forall fom_index_pfp_factor_divisibility_productleft. (exists fom_gap_pfp_factor_divisibility_productleft_index_bound. fom_gap_pfp_factor_divisibility_productleft_index_bound + S (fom_index_pfp_factor_divisibility_productleft) = pfgu_Q_factor_divisibility) -> exists fom_value_pfp_factor_divisibility_productleft. ((((exists fom_beta_height_pfp_factor_divisibility_productleft_entry. fom_beta_height_pfp_factor_divisibility_productleft_entry + S (fom_value_pfp_factor_divisibility_productleft) = S ((S (fom_index_pfp_factor_divisibility_productleft)) * pfgu_qc_factor_divisibility)) /\\ exists fom_beta_quotient_pfp_factor_divisibility_productleft_entry. pfgu_qb_factor_divisibility = fom_beta_quotient_pfp_factor_divisibility_productleft_entry * S ((S (fom_index_pfp_factor_divisibility_productleft)) * pfgu_qc_factor_divisibility) + (fom_value_pfp_factor_divisibility_productleft))) /\\ (exists fom_gap_pfp_factor_divisibility_productleft_value_bound. fom_gap_pfp_factor_divisibility_productleft_value_bound + S (fom_value_pfp_factor_divisibility_productleft) = p))) /\\ (((forall fom_index_pfp_factor_divisibility_productright. (exists fom_gap_pfp_factor_divisibility_productright_index_bound. fom_gap_pfp_factor_divisibility_productright_index_bound + S (fom_index_pfp_factor_divisibility_productright) = D) -> exists fom_value_pfp_factor_divisibility_productright. ((((exists fom_beta_height_pfp_factor_divisibility_productright_entry. fom_beta_height_pfp_factor_divisibility_productright_entry + S (fom_value_pfp_factor_divisibility_productright) = S ((S (fom_index_pfp_factor_divisibility_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_factor_divisibility_productright_entry. db = fom_beta_quotient_pfp_factor_divisibility_productright_entry * S ((S (fom_index_pfp_factor_divisibility_productright)) * dc) + (fom_value_pfp_factor_divisibility_productright))) /\\ (exists fom_gap_pfp_factor_divisibility_productright_value_bound. fom_gap_pfp_factor_divisibility_productright_value_bound + S (fom_value_pfp_factor_divisibility_productright) = p))) /\\ (((((((pfgu_Q_factor_divisibility)=0 \\/ (D)=0) /\\ (((pfgu_P_factor_divisibility)=0)))) \\/ (((~((pfgu_Q_factor_divisibility)=0)) /\\ (((~((D)=0)) /\\ (((pfgu_Q_factor_divisibility)+(D)=S (pfgu_P_factor_divisibility)))))))) /\\ ((forall pfc_index_factor_divisibility_productcoefficients. (exists pfa_gap_factor_divisibility_productcoefficientsbound. pfa_gap_factor_divisibility_productcoefficientsbound + S (pfc_index_factor_divisibility_productcoefficients) = (pfgu_P_factor_divisibility)) -> exists pfc_value_factor_divisibility_productcoefficients. ((((exists ff_h_pfp_factor_divisibility_productcoefficientsentry. ff_h_pfp_factor_divisibility_productcoefficientsentry + S (pfc_value_factor_divisibility_productcoefficients) = S ((S (pfc_index_factor_divisibility_productcoefficients)) * pfgu_pc_factor_divisibility)) /\\ exists ff_q_pfp_factor_divisibility_productcoefficientsentry. pfgu_pb_factor_divisibility = ff_q_pfp_factor_divisibility_productcoefficientsentry * S ((S (pfc_index_factor_divisibility_productcoefficients)) * pfgu_pc_factor_divisibility) + (pfc_value_factor_divisibility_productcoefficients))) /\\ ((exists pfc_terms_code_factor_divisibility_productcoefficientscoefficient pfc_terms_scale_factor_divisibility_productcoefficientscoefficient pfc_natural_sum_factor_divisibility_productcoefficientscoefficient. ((forall pfc_index_factor_divisibility_productcoefficientscoefficientdiagonal. (exists pfa_gap_factor_divisibility_productcoefficientscoefficientdiagonalbound. pfa_gap_factor_divisibility_productcoefficientscoefficientdiagonalbound + S (pfc_index_factor_divisibility_productcoefficientscoefficientdiagonal) = (S (pfc_index_factor_divisibility_productcoefficients))) -> exists pfc_value_factor_divisibility_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_factor_divisibility_productcoefficientscoefficientdiagonalentry. ff_h_pfp_factor_divisibility_productcoefficientscoefficientdiagonalentry + S (pfc_value_factor_divisibility_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_factor_divisibility_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_factor_divisibility_productcoefficientscoefficient)) /\\ exists ff_q_pfp_factor_divisibility_productcoefficientscoefficientdiagonalentry. pfc_terms_code_factor_divisibility_productcoefficientscoefficient = ff_q_pfp_factor_divisibility_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_factor_divisibility_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_factor_divisibility_productcoefficientscoefficient) + (pfc_value_factor_divisibility_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_factor_divisibility_productcoefficientscoefficientdiagonalterm pfc_left_factor_divisibility_productcoefficientscoefficientdiagonalterm pfc_right_factor_divisibility_productcoefficientscoefficientdiagonalterm. (((pfc_index_factor_divisibility_productcoefficientscoefficientdiagonal)+pfc_complement_factor_divisibility_productcoefficientscoefficientdiagonalterm=(pfc_index_factor_divisibility_productcoefficients)) /\\ ((((((exists pfa_gap_factor_divisibility_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_factor_divisibility_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_factor_divisibility_productcoefficientscoefficientdiagonal) = (pfgu_Q_factor_divisibility)) /\\ ((((exists ff_h_pfp_factor_divisibility_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_factor_divisibility_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_factor_divisibility_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_factor_divisibility_productcoefficientscoefficientdiagonal)) * pfgu_qc_factor_divisibility)) /\\ exists ff_q_pfp_factor_divisibility_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_factor_divisibility = ff_q_pfp_factor_divisibility_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_factor_divisibility_productcoefficientscoefficientdiagonal)) * pfgu_qc_factor_divisibility) + (pfc_left_factor_divisibility_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_factor_divisibility_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_factor_divisibility_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_factor_divisibility)=(pfc_index_factor_divisibility_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_factor_divisibility_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_factor_divisibility_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_factor_divisibility_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_factor_divisibility_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_factor_divisibility_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_factor_divisibility_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_factor_divisibility_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_factor_divisibility_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_factor_divisibility_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_factor_divisibility_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_factor_divisibility_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_factor_divisibility_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_factor_divisibility_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_factor_divisibility_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_factor_divisibility_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_factor_divisibility_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_factor_divisibility_productcoefficientscoefficientdiagonal)=pfc_left_factor_divisibility_productcoefficientscoefficientdiagonalterm*pfc_right_factor_divisibility_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_factor_divisibility_productcoefficientscoefficientsum fs_v_pfc_factor_divisibility_productcoefficientscoefficientsum. ((((exists fs_h_pfc_factor_divisibility_productcoefficientscoefficientsum_body_start. fs_h_pfc_factor_divisibility_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_factor_divisibility_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_divisibility_productcoefficientscoefficientsum_body_start. fs_u_pfc_factor_divisibility_productcoefficientscoefficientsum = fs_q_pfc_factor_divisibility_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_factor_divisibility_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_factor_divisibility_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_factor_divisibility_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_factor_divisibility_productcoefficientscoefficient) = S ((S (S (pfc_index_factor_divisibility_productcoefficients))) * fs_v_pfc_factor_divisibility_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_divisibility_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_factor_divisibility_productcoefficientscoefficientsum = fs_q_pfc_factor_divisibility_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_factor_divisibility_productcoefficients))) * fs_v_pfc_factor_divisibility_productcoefficientscoefficientsum) + (pfc_natural_sum_factor_divisibility_productcoefficientscoefficient))) /\\ forall fs_i_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps = S (pfc_index_factor_divisibility_productcoefficients)) -> exists fs_a_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps fs_r_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps fs_s_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_factor_divisibility_productcoefficientscoefficient)) /\\ exists fs_q_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_factor_divisibility_productcoefficientscoefficient = fs_q_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_factor_divisibility_productcoefficientscoefficient) + (fs_a_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_divisibility_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_factor_divisibility_productcoefficientscoefficientsum = fs_q_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_divisibility_productcoefficientscoefficientsum) + (fs_r_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_divisibility_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_factor_divisibility_productcoefficientscoefficientsum = fs_q_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_divisibility_productcoefficientscoefficientsum) + (fs_s_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps = fs_r_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps + fs_a_pfc_factor_divisibility_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_factor_divisibility_productcoefficientscoefficientresiduebound. pfa_gap_factor_divisibility_productcoefficientscoefficientresiduebound + S (pfc_value_factor_divisibility_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_factor_divisibility_productcoefficientscoefficientresiduecongruence pfa_offset_right_factor_divisibility_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_factor_divisibility_productcoefficientscoefficient) + (p) * pfa_offset_left_factor_divisibility_productcoefficientscoefficientresiduecongruence = (pfc_value_factor_divisibility_productcoefficients) + (p) * pfa_offset_right_factor_divisibility_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_factor_divisibility_target pfrep_left_factor_divisibility_target pfrep_right_factor_divisibility_target. ((exists pfrep_position_factor_divisibility_targetfirst. ((pfrep_position_factor_divisibility_targetfirst+S (pfrep_power_factor_divisibility_target)=(pfgu_P_factor_divisibility)) /\\ ((((exists ff_h_pfp_factor_divisibility_targetfirstentry. ff_h_pfp_factor_divisibility_targetfirstentry + S (pfrep_left_factor_divisibility_target) = S ((S (pfrep_position_factor_divisibility_targetfirst)) * pfgu_pc_factor_divisibility)) /\\ exists ff_q_pfp_factor_divisibility_targetfirstentry. pfgu_pb_factor_divisibility = ff_q_pfp_factor_divisibility_targetfirstentry * S ((S (pfrep_position_factor_divisibility_targetfirst)) * pfgu_pc_factor_divisibility) + (pfrep_left_factor_divisibility_target)))))) \\/ (((exists pfrep_gap_factor_divisibility_targetfirstoutside. pfrep_gap_factor_divisibility_targetfirstoutside+(pfgu_P_factor_divisibility)=(pfrep_power_factor_divisibility_target)) /\\ (((pfrep_left_factor_divisibility_target)=0))))) -> ((exists pfrep_position_factor_divisibility_targetsecond. ((pfrep_position_factor_divisibility_targetsecond+S (pfrep_power_factor_divisibility_target)=(L)) /\\ ((((exists ff_h_pfp_factor_divisibility_targetsecondentry. ff_h_pfp_factor_divisibility_targetsecondentry + S (pfrep_right_factor_divisibility_target) = S ((S (pfrep_position_factor_divisibility_targetsecond)) * ac)) /\\ exists ff_q_pfp_factor_divisibility_targetsecondentry. ab = ff_q_pfp_factor_divisibility_targetsecondentry * S ((S (pfrep_position_factor_divisibility_targetsecond)) * ac) + (pfrep_right_factor_divisibility_target)))))) \\/ (((exists pfrep_gap_factor_divisibility_targetsecondoutside. pfrep_gap_factor_divisibility_targetsecondoutside+(L)=(pfrep_power_factor_divisibility_target)) /\\ (((pfrep_right_factor_divisibility_target)=0))))) -> pfrep_left_factor_divisibility_target=pfrep_right_factor_divisibility_target))))))) -> (exists pfgu_qb_factor_result pfgu_qc_factor_result pfgu_e_factor_result pfgu_pb_factor_result pfgu_pc_factor_result. (((((S (pfgu_e_factor_result))=S (pfgu_e_factor_result)) /\\ (((forall fom_index_pfp_factor_result_quotientcoefficients. (exists fom_gap_pfp_factor_result_quotientcoefficients_index_bound. fom_gap_pfp_factor_result_quotientcoefficients_index_bound + S (fom_index_pfp_factor_result_quotientcoefficients) = S (pfgu_e_factor_result)) -> exists fom_value_pfp_factor_result_quotientcoefficients. ((((exists fom_beta_height_pfp_factor_result_quotientcoefficients_entry. fom_beta_height_pfp_factor_result_quotientcoefficients_entry + S (fom_value_pfp_factor_result_quotientcoefficients) = S ((S (fom_index_pfp_factor_result_quotientcoefficients)) * pfgu_qc_factor_result)) /\\ exists fom_beta_quotient_pfp_factor_result_quotientcoefficients_entry. pfgu_qb_factor_result = fom_beta_quotient_pfp_factor_result_quotientcoefficients_entry * S ((S (fom_index_pfp_factor_result_quotientcoefficients)) * pfgu_qc_factor_result) + (fom_value_pfp_factor_result_quotientcoefficients))) /\\ (exists fom_gap_pfp_factor_result_quotientcoefficients_value_bound. fom_gap_pfp_factor_result_quotientcoefficients_value_bound + S (fom_value_pfp_factor_result_quotientcoefficients) = p))) /\\ ((exists pfd_leading_factor_result_quotient. ((((exists ff_h_pfp_factor_result_quotiententry. ff_h_pfp_factor_result_quotiententry + S (pfd_leading_factor_result_quotient) = S ((S (0)) * pfgu_qc_factor_result)) /\\ exists ff_q_pfp_factor_result_quotiententry. pfgu_qb_factor_result = ff_q_pfp_factor_result_quotiententry * S ((S (0)) * pfgu_qc_factor_result) + (pfd_leading_factor_result_quotient))) /\\ ((~(pfd_leading_factor_result_quotient=0)))))))))) /\\ (((((forall fom_index_pfp_factor_result_productleft. (exists fom_gap_pfp_factor_result_productleft_index_bound. fom_gap_pfp_factor_result_productleft_index_bound + S (fom_index_pfp_factor_result_productleft) = S (pfgu_e_factor_result)) -> exists fom_value_pfp_factor_result_productleft. ((((exists fom_beta_height_pfp_factor_result_productleft_entry. fom_beta_height_pfp_factor_result_productleft_entry + S (fom_value_pfp_factor_result_productleft) = S ((S (fom_index_pfp_factor_result_productleft)) * pfgu_qc_factor_result)) /\\ exists fom_beta_quotient_pfp_factor_result_productleft_entry. pfgu_qb_factor_result = fom_beta_quotient_pfp_factor_result_productleft_entry * S ((S (fom_index_pfp_factor_result_productleft)) * pfgu_qc_factor_result) + (fom_value_pfp_factor_result_productleft))) /\\ (exists fom_gap_pfp_factor_result_productleft_value_bound. fom_gap_pfp_factor_result_productleft_value_bound + S (fom_value_pfp_factor_result_productleft) = p))) /\\ (((forall fom_index_pfp_factor_result_productright. (exists fom_gap_pfp_factor_result_productright_index_bound. fom_gap_pfp_factor_result_productright_index_bound + S (fom_index_pfp_factor_result_productright) = D) -> exists fom_value_pfp_factor_result_productright. ((((exists fom_beta_height_pfp_factor_result_productright_entry. fom_beta_height_pfp_factor_result_productright_entry + S (fom_value_pfp_factor_result_productright) = S ((S (fom_index_pfp_factor_result_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_factor_result_productright_entry. db = fom_beta_quotient_pfp_factor_result_productright_entry * S ((S (fom_index_pfp_factor_result_productright)) * dc) + (fom_value_pfp_factor_result_productright))) /\\ (exists fom_gap_pfp_factor_result_productright_value_bound. fom_gap_pfp_factor_result_productright_value_bound + S (fom_value_pfp_factor_result_productright) = p))) /\\ (((((((S (pfgu_e_factor_result))=0 \\/ (D)=0) /\\ (((S (a))=0)))) \\/ (((~((S (pfgu_e_factor_result))=0)) /\\ (((~((D)=0)) /\\ (((S (pfgu_e_factor_result))+(D)=S (S (a))))))))) /\\ ((forall pfc_index_factor_result_productcoefficients. (exists pfa_gap_factor_result_productcoefficientsbound. pfa_gap_factor_result_productcoefficientsbound + S (pfc_index_factor_result_productcoefficients) = (S (a))) -> exists pfc_value_factor_result_productcoefficients. ((((exists ff_h_pfp_factor_result_productcoefficientsentry. ff_h_pfp_factor_result_productcoefficientsentry + S (pfc_value_factor_result_productcoefficients) = S ((S (pfc_index_factor_result_productcoefficients)) * pfgu_pc_factor_result)) /\\ exists ff_q_pfp_factor_result_productcoefficientsentry. pfgu_pb_factor_result = ff_q_pfp_factor_result_productcoefficientsentry * S ((S (pfc_index_factor_result_productcoefficients)) * pfgu_pc_factor_result) + (pfc_value_factor_result_productcoefficients))) /\\ ((exists pfc_terms_code_factor_result_productcoefficientscoefficient pfc_terms_scale_factor_result_productcoefficientscoefficient pfc_natural_sum_factor_result_productcoefficientscoefficient. ((forall pfc_index_factor_result_productcoefficientscoefficientdiagonal. (exists pfa_gap_factor_result_productcoefficientscoefficientdiagonalbound. pfa_gap_factor_result_productcoefficientscoefficientdiagonalbound + S (pfc_index_factor_result_productcoefficientscoefficientdiagonal) = (S (pfc_index_factor_result_productcoefficients))) -> exists pfc_value_factor_result_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_factor_result_productcoefficientscoefficientdiagonalentry. ff_h_pfp_factor_result_productcoefficientscoefficientdiagonalentry + S (pfc_value_factor_result_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_factor_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_factor_result_productcoefficientscoefficient)) /\\ exists ff_q_pfp_factor_result_productcoefficientscoefficientdiagonalentry. pfc_terms_code_factor_result_productcoefficientscoefficient = ff_q_pfp_factor_result_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_factor_result_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_factor_result_productcoefficientscoefficient) + (pfc_value_factor_result_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_factor_result_productcoefficientscoefficientdiagonalterm pfc_left_factor_result_productcoefficientscoefficientdiagonalterm pfc_right_factor_result_productcoefficientscoefficientdiagonalterm. (((pfc_index_factor_result_productcoefficientscoefficientdiagonal)+pfc_complement_factor_result_productcoefficientscoefficientdiagonalterm=(pfc_index_factor_result_productcoefficients)) /\\ ((((((exists pfa_gap_factor_result_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_factor_result_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_factor_result_productcoefficientscoefficientdiagonal) = (S (pfgu_e_factor_result))) /\\ ((((exists ff_h_pfp_factor_result_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_factor_result_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_factor_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_factor_result_productcoefficientscoefficientdiagonal)) * pfgu_qc_factor_result)) /\\ exists ff_q_pfp_factor_result_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_factor_result = ff_q_pfp_factor_result_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_factor_result_productcoefficientscoefficientdiagonal)) * pfgu_qc_factor_result) + (pfc_left_factor_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_factor_result_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_factor_result_productcoefficientscoefficientdiagonaltermleftoutside+(S (pfgu_e_factor_result))=(pfc_index_factor_result_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_factor_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_factor_result_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_factor_result_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_factor_result_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_factor_result_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_factor_result_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_factor_result_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_factor_result_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_factor_result_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_factor_result_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_factor_result_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_factor_result_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_factor_result_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_factor_result_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_factor_result_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_factor_result_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_factor_result_productcoefficientscoefficientdiagonal)=pfc_left_factor_result_productcoefficientscoefficientdiagonalterm*pfc_right_factor_result_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_factor_result_productcoefficientscoefficientsum fs_v_pfc_factor_result_productcoefficientscoefficientsum. ((((exists fs_h_pfc_factor_result_productcoefficientscoefficientsum_body_start. fs_h_pfc_factor_result_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_factor_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_result_productcoefficientscoefficientsum_body_start. fs_u_pfc_factor_result_productcoefficientscoefficientsum = fs_q_pfc_factor_result_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_factor_result_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_factor_result_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_factor_result_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_factor_result_productcoefficientscoefficient) = S ((S (S (pfc_index_factor_result_productcoefficients))) * fs_v_pfc_factor_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_result_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_factor_result_productcoefficientscoefficientsum = fs_q_pfc_factor_result_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_factor_result_productcoefficients))) * fs_v_pfc_factor_result_productcoefficientscoefficientsum) + (pfc_natural_sum_factor_result_productcoefficientscoefficient))) /\\ forall fs_i_pfc_factor_result_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_factor_result_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_factor_result_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_factor_result_productcoefficientscoefficientsum_body_steps = S (pfc_index_factor_result_productcoefficients)) -> exists fs_a_pfc_factor_result_productcoefficientscoefficientsum_body_steps fs_r_pfc_factor_result_productcoefficientscoefficientsum_body_steps fs_s_pfc_factor_result_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_factor_result_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_factor_result_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_factor_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_factor_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_factor_result_productcoefficientscoefficient)) /\\ exists fs_q_pfc_factor_result_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_factor_result_productcoefficientscoefficient = fs_q_pfc_factor_result_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_factor_result_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_factor_result_productcoefficientscoefficient) + (fs_a_pfc_factor_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_factor_result_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_factor_result_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_factor_result_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_factor_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_result_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_factor_result_productcoefficientscoefficientsum = fs_q_pfc_factor_result_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_factor_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_result_productcoefficientscoefficientsum) + (fs_r_pfc_factor_result_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_factor_result_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_factor_result_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_factor_result_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_factor_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_result_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_factor_result_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_factor_result_productcoefficientscoefficientsum = fs_q_pfc_factor_result_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_factor_result_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_factor_result_productcoefficientscoefficientsum) + (fs_s_pfc_factor_result_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_factor_result_productcoefficientscoefficientsum_body_steps = fs_r_pfc_factor_result_productcoefficientscoefficientsum_body_steps + fs_a_pfc_factor_result_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_factor_result_productcoefficientscoefficientresiduebound. pfa_gap_factor_result_productcoefficientscoefficientresiduebound + S (pfc_value_factor_result_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_factor_result_productcoefficientscoefficientresiduecongruence pfa_offset_right_factor_result_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_factor_result_productcoefficientscoefficient) + (p) * pfa_offset_left_factor_result_productcoefficientscoefficientresiduecongruence = (pfc_value_factor_result_productcoefficients) + (p) * pfa_offset_right_factor_result_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((forall pfrep_power_factor_result_equivalent pfrep_left_factor_result_equivalent pfrep_right_factor_result_equivalent. ((exists pfrep_position_factor_result_equivalentfirst. ((pfrep_position_factor_result_equivalentfirst+S (pfrep_power_factor_result_equivalent)=(S (a))) /\\ ((((exists ff_h_pfp_factor_result_equivalentfirstentry. ff_h_pfp_factor_result_equivalentfirstentry + S (pfrep_left_factor_result_equivalent) = S ((S (pfrep_position_factor_result_equivalentfirst)) * pfgu_pc_factor_result)) /\\ exists ff_q_pfp_factor_result_equivalentfirstentry. pfgu_pb_factor_result = ff_q_pfp_factor_result_equivalentfirstentry * S ((S (pfrep_position_factor_result_equivalentfirst)) * pfgu_pc_factor_result) + (pfrep_left_factor_result_equivalent)))))) \\/ (((exists pfrep_gap_factor_result_equivalentfirstoutside. pfrep_gap_factor_result_equivalentfirstoutside+(S (a))=(pfrep_power_factor_result_equivalent)) /\\ (((pfrep_left_factor_result_equivalent)=0))))) -> ((exists pfrep_position_factor_result_equivalentsecond. ((pfrep_position_factor_result_equivalentsecond+S (pfrep_power_factor_result_equivalent)=(L)) /\\ ((((exists ff_h_pfp_factor_result_equivalentsecondentry. ff_h_pfp_factor_result_equivalentsecondentry + S (pfrep_right_factor_result_equivalent) = S ((S (pfrep_position_factor_result_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_factor_result_equivalentsecondentry. ab = ff_q_pfp_factor_result_equivalentsecondentry * S ((S (pfrep_position_factor_result_equivalentsecond)) * ac) + (pfrep_right_factor_result_equivalent)))))) \\/ (((exists pfrep_gap_factor_result_equivalentsecondoutside. pfrep_gap_factor_result_equivalentsecondoutside+(L)=(pfrep_power_factor_result_equivalent)) /\\ (((pfrep_right_factor_result_equivalent)=0))))) -> pfrep_left_factor_result_equivalent=pfrep_right_factor_result_equivalent) /\\ (((pfgu_e_factor_result)+(d)=(a)))))))))", "statement_sha256": "0a9b0f4462f89c0987eb0cb2db57edaf595dc38af34776a76d0c26becf03ee75", "summary": "Trim the actual quotient, construct an independent proper-length product, and transport formal coefficients. 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(((((S (pfgu_e_bound_factor))=S (pfgu_e_bound_factor)) /\\ (((forall fom_index_pfp_bound_factor_quotientcoefficients. (exists fom_gap_pfp_bound_factor_quotientcoefficients_index_bound. fom_gap_pfp_bound_factor_quotientcoefficients_index_bound + S (fom_index_pfp_bound_factor_quotientcoefficients) = S (pfgu_e_bound_factor)) -> exists fom_value_pfp_bound_factor_quotientcoefficients. ((((exists fom_beta_height_pfp_bound_factor_quotientcoefficients_entry. fom_beta_height_pfp_bound_factor_quotientcoefficients_entry + S (fom_value_pfp_bound_factor_quotientcoefficients) = S ((S (fom_index_pfp_bound_factor_quotientcoefficients)) * pfgu_qc_bound_factor)) /\\ exists fom_beta_quotient_pfp_bound_factor_quotientcoefficients_entry. pfgu_qb_bound_factor = fom_beta_quotient_pfp_bound_factor_quotientcoefficients_entry * S ((S (fom_index_pfp_bound_factor_quotientcoefficients)) * pfgu_qc_bound_factor) + (fom_value_pfp_bound_factor_quotientcoefficients))) /\\ (exists fom_gap_pfp_bound_factor_quotientcoefficients_value_bound. fom_gap_pfp_bound_factor_quotientcoefficients_value_bound + S (fom_value_pfp_bound_factor_quotientcoefficients) = p))) /\\ ((exists pfd_leading_bound_factor_quotient. ((((exists ff_h_pfp_bound_factor_quotiententry. ff_h_pfp_bound_factor_quotiententry + S (pfd_leading_bound_factor_quotient) = S ((S (0)) * pfgu_qc_bound_factor)) /\\ exists ff_q_pfp_bound_factor_quotiententry. pfgu_qb_bound_factor = ff_q_pfp_bound_factor_quotiententry * S ((S (0)) * pfgu_qc_bound_factor) + (pfd_leading_bound_factor_quotient))) /\\ ((~(pfd_leading_bound_factor_quotient=0)))))))))) /\\ (((((forall fom_index_pfp_bound_factor_productleft. (exists fom_gap_pfp_bound_factor_productleft_index_bound. fom_gap_pfp_bound_factor_productleft_index_bound + S (fom_index_pfp_bound_factor_productleft) = S (pfgu_e_bound_factor)) -> exists fom_value_pfp_bound_factor_productleft. ((((exists fom_beta_height_pfp_bound_factor_productleft_entry. fom_beta_height_pfp_bound_factor_productleft_entry + S (fom_value_pfp_bound_factor_productleft) = S ((S (fom_index_pfp_bound_factor_productleft)) * pfgu_qc_bound_factor)) /\\ exists fom_beta_quotient_pfp_bound_factor_productleft_entry. pfgu_qb_bound_factor = fom_beta_quotient_pfp_bound_factor_productleft_entry * S ((S (fom_index_pfp_bound_factor_productleft)) * pfgu_qc_bound_factor) + (fom_value_pfp_bound_factor_productleft))) /\\ (exists fom_gap_pfp_bound_factor_productleft_value_bound. fom_gap_pfp_bound_factor_productleft_value_bound + S (fom_value_pfp_bound_factor_productleft) = p))) /\\ (((forall fom_index_pfp_bound_factor_productright. (exists fom_gap_pfp_bound_factor_productright_index_bound. fom_gap_pfp_bound_factor_productright_index_bound + S (fom_index_pfp_bound_factor_productright) = D) -> exists fom_value_pfp_bound_factor_productright. ((((exists fom_beta_height_pfp_bound_factor_productright_entry. fom_beta_height_pfp_bound_factor_productright_entry + S (fom_value_pfp_bound_factor_productright) = S ((S (fom_index_pfp_bound_factor_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_bound_factor_productright_entry. db = fom_beta_quotient_pfp_bound_factor_productright_entry * S ((S (fom_index_pfp_bound_factor_productright)) * dc) + (fom_value_pfp_bound_factor_productright))) /\\ (exists fom_gap_pfp_bound_factor_productright_value_bound. fom_gap_pfp_bound_factor_productright_value_bound + S (fom_value_pfp_bound_factor_productright) = p))) /\\ (((((((S (pfgu_e_bound_factor))=0 \\/ (D)=0) /\\ (((S (a))=0)))) \\/ (((~((S (pfgu_e_bound_factor))=0)) /\\ (((~((D)=0)) /\\ (((S (pfgu_e_bound_factor))+(D)=S (S (a))))))))) /\\ ((forall pfc_index_bound_factor_productcoefficients. (exists pfa_gap_bound_factor_productcoefficientsbound. pfa_gap_bound_factor_productcoefficientsbound + S (pfc_index_bound_factor_productcoefficients) = (S (a))) -> exists pfc_value_bound_factor_productcoefficients. ((((exists ff_h_pfp_bound_factor_productcoefficientsentry. ff_h_pfp_bound_factor_productcoefficientsentry + S (pfc_value_bound_factor_productcoefficients) = S ((S (pfc_index_bound_factor_productcoefficients)) * pfgu_pc_bound_factor)) /\\ exists ff_q_pfp_bound_factor_productcoefficientsentry. pfgu_pb_bound_factor = ff_q_pfp_bound_factor_productcoefficientsentry * S ((S (pfc_index_bound_factor_productcoefficients)) * pfgu_pc_bound_factor) + (pfc_value_bound_factor_productcoefficients))) /\\ ((exists pfc_terms_code_bound_factor_productcoefficientscoefficient pfc_terms_scale_bound_factor_productcoefficientscoefficient pfc_natural_sum_bound_factor_productcoefficientscoefficient. ((forall pfc_index_bound_factor_productcoefficientscoefficientdiagonal. (exists pfa_gap_bound_factor_productcoefficientscoefficientdiagonalbound. pfa_gap_bound_factor_productcoefficientscoefficientdiagonalbound + S (pfc_index_bound_factor_productcoefficientscoefficientdiagonal) = (S (pfc_index_bound_factor_productcoefficients))) -> exists pfc_value_bound_factor_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_bound_factor_productcoefficientscoefficientdiagonalentry. ff_h_pfp_bound_factor_productcoefficientscoefficientdiagonalentry + S (pfc_value_bound_factor_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_bound_factor_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bound_factor_productcoefficientscoefficient)) /\\ exists ff_q_pfp_bound_factor_productcoefficientscoefficientdiagonalentry. pfc_terms_code_bound_factor_productcoefficientscoefficient = ff_q_pfp_bound_factor_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_bound_factor_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bound_factor_productcoefficientscoefficient) + (pfc_value_bound_factor_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_bound_factor_productcoefficientscoefficientdiagonalterm pfc_left_bound_factor_productcoefficientscoefficientdiagonalterm pfc_right_bound_factor_productcoefficientscoefficientdiagonalterm. (((pfc_index_bound_factor_productcoefficientscoefficientdiagonal)+pfc_complement_bound_factor_productcoefficientscoefficientdiagonalterm=(pfc_index_bound_factor_productcoefficients)) /\\ ((((((exists pfa_gap_bound_factor_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_bound_factor_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_bound_factor_productcoefficientscoefficientdiagonal) = (S (pfgu_e_bound_factor))) /\\ ((((exists ff_h_pfp_bound_factor_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_bound_factor_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_bound_factor_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_bound_factor_productcoefficientscoefficientdiagonal)) * pfgu_qc_bound_factor)) /\\ exists ff_q_pfp_bound_factor_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_bound_factor = ff_q_pfp_bound_factor_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_bound_factor_productcoefficientscoefficientdiagonal)) * pfgu_qc_bound_factor) + (pfc_left_bound_factor_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bound_factor_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_bound_factor_productcoefficientscoefficientdiagonaltermleftoutside+(S (pfgu_e_bound_factor))=(pfc_index_bound_factor_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_bound_factor_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_bound_factor_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_bound_factor_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_bound_factor_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_bound_factor_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_bound_factor_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_bound_factor_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_bound_factor_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_bound_factor_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_bound_factor_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_bound_factor_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_bound_factor_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bound_factor_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_bound_factor_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_bound_factor_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_bound_factor_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_bound_factor_productcoefficientscoefficientdiagonal)=pfc_left_bound_factor_productcoefficientscoefficientdiagonalterm*pfc_right_bound_factor_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_bound_factor_productcoefficientscoefficientsum fs_v_pfc_bound_factor_productcoefficientscoefficientsum. ((((exists fs_h_pfc_bound_factor_productcoefficientscoefficientsum_body_start. fs_h_pfc_bound_factor_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_bound_factor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bound_factor_productcoefficientscoefficientsum_body_start. fs_u_pfc_bound_factor_productcoefficientscoefficientsum = fs_q_pfc_bound_factor_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_bound_factor_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_bound_factor_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_bound_factor_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_bound_factor_productcoefficientscoefficient) = S ((S (S (pfc_index_bound_factor_productcoefficients))) * fs_v_pfc_bound_factor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bound_factor_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_bound_factor_productcoefficientscoefficientsum = fs_q_pfc_bound_factor_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_bound_factor_productcoefficients))) * fs_v_pfc_bound_factor_productcoefficientscoefficientsum) + (pfc_natural_sum_bound_factor_productcoefficientscoefficient))) /\\ forall fs_i_pfc_bound_factor_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_bound_factor_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_bound_factor_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_bound_factor_productcoefficientscoefficientsum_body_steps = S (pfc_index_bound_factor_productcoefficients)) -> exists fs_a_pfc_bound_factor_productcoefficientscoefficientsum_body_steps fs_r_pfc_bound_factor_productcoefficientscoefficientsum_body_steps fs_s_pfc_bound_factor_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_bound_factor_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_bound_factor_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_bound_factor_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bound_factor_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bound_factor_productcoefficientscoefficient)) /\\ exists fs_q_pfc_bound_factor_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_bound_factor_productcoefficientscoefficient = fs_q_pfc_bound_factor_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_bound_factor_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bound_factor_productcoefficientscoefficient) + (fs_a_pfc_bound_factor_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bound_factor_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_bound_factor_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_bound_factor_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bound_factor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bound_factor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bound_factor_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_bound_factor_productcoefficientscoefficientsum = fs_q_pfc_bound_factor_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_bound_factor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bound_factor_productcoefficientscoefficientsum) + (fs_r_pfc_bound_factor_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bound_factor_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_bound_factor_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_bound_factor_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_bound_factor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bound_factor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bound_factor_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_bound_factor_productcoefficientscoefficientsum = fs_q_pfc_bound_factor_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_bound_factor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bound_factor_productcoefficientscoefficientsum) + (fs_s_pfc_bound_factor_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_bound_factor_productcoefficientscoefficientsum_body_steps = fs_r_pfc_bound_factor_productcoefficientscoefficientsum_body_steps + fs_a_pfc_bound_factor_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_bound_factor_productcoefficientscoefficientresiduebound. pfa_gap_bound_factor_productcoefficientscoefficientresiduebound + S (pfc_value_bound_factor_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_bound_factor_productcoefficientscoefficientresiduecongruence pfa_offset_right_bound_factor_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_bound_factor_productcoefficientscoefficient) + (p) * pfa_offset_left_bound_factor_productcoefficientscoefficientresiduecongruence = (pfc_value_bound_factor_productcoefficients) + (p) * pfa_offset_right_bound_factor_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((forall pfrep_power_bound_factor_equivalent pfrep_left_bound_factor_equivalent pfrep_right_bound_factor_equivalent. ((exists pfrep_position_bound_factor_equivalentfirst. ((pfrep_position_bound_factor_equivalentfirst+S (pfrep_power_bound_factor_equivalent)=(S (a))) /\\ ((((exists ff_h_pfp_bound_factor_equivalentfirstentry. ff_h_pfp_bound_factor_equivalentfirstentry + S (pfrep_left_bound_factor_equivalent) = S ((S (pfrep_position_bound_factor_equivalentfirst)) * pfgu_pc_bound_factor)) /\\ exists ff_q_pfp_bound_factor_equivalentfirstentry. pfgu_pb_bound_factor = ff_q_pfp_bound_factor_equivalentfirstentry * S ((S (pfrep_position_bound_factor_equivalentfirst)) * pfgu_pc_bound_factor) + (pfrep_left_bound_factor_equivalent)))))) \\/ (((exists pfrep_gap_bound_factor_equivalentfirstoutside. pfrep_gap_bound_factor_equivalentfirstoutside+(S (a))=(pfrep_power_bound_factor_equivalent)) /\\ (((pfrep_left_bound_factor_equivalent)=0))))) -> ((exists pfrep_position_bound_factor_equivalentsecond. ((pfrep_position_bound_factor_equivalentsecond+S (pfrep_power_bound_factor_equivalent)=(L)) /\\ ((((exists ff_h_pfp_bound_factor_equivalentsecondentry. ff_h_pfp_bound_factor_equivalentsecondentry + S (pfrep_right_bound_factor_equivalent) = S ((S (pfrep_position_bound_factor_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_bound_factor_equivalentsecondentry. ab = ff_q_pfp_bound_factor_equivalentsecondentry * S ((S (pfrep_position_bound_factor_equivalentsecond)) * ac) + (pfrep_right_bound_factor_equivalent)))))) \\/ (((exists pfrep_gap_bound_factor_equivalentsecondoutside. pfrep_gap_bound_factor_equivalentsecondoutside+(L)=(pfrep_power_bound_factor_equivalent)) /\\ (((pfrep_right_bound_factor_equivalent)=0))))) -> pfrep_left_bound_factor_equivalent=pfrep_right_bound_factor_equivalent) /\\ (((pfgu_e_bound_factor)+(d)=(a))))))))", "specialize prime_field_polynomial_right_divides_represented_factorization (p)", "specialize prime_field_polynomial_right_divides_represented_factorization (db)", "specialize prime_field_polynomial_right_divides_represented_factorization (dc)", "specialize prime_field_polynomial_right_divides_represented_factorization (D)", "specialize prime_field_polynomial_right_divides_represented_factorization (d)", "specialize prime_field_polynomial_right_divides_represented_factorization (ab)", "specialize prime_field_polynomial_right_divides_represented_factorization (ac)", "specialize prime_field_polynomial_right_divides_represented_factorization (L)", "specialize prime_field_polynomial_right_divides_represented_factorization (a)", "apply prime_field_polynomial_right_divides_represented_factorization", "exact hp", "exact hd", "exact ha", "exact hrd", "cases hf", "cases hf_witness", "cases hf_witness_witness", "cases hf_witness_witness_witness", "cases hf_witness_witness_witness_witness", "cases hf_witness_witness_witness_witness_witness", "cases hf_witness_witness_witness_witness_witness_right", "cases hf_witness_witness_witness_witness_witness_right_right", "exists x2", "exact hf_witness_witness_witness_witness_witness_right_right_right" ], "script_sha256": "a37cdeeeb0b780832d4d3c35d05f91407b1852a1d139dc9363cddfa15d80c909", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_uniqueness_candidate.py", "sha256": "916c24ad6c59609612e97daee6e49347a9522cdb28b44f6f09c6c5760bff0b5b" }, "statement": "forall p db dc D d ab ac L a. (~((p) = 1) /\\ forall pfa_factor_left_bound_prime pfa_factor_right_bound_prime. (p) = pfa_factor_left_bound_prime * pfa_factor_right_bound_prime -> pfa_factor_left_bound_prime = 1 \\/ pfa_factor_right_bound_prime = 1) -> ((((D)=S (d)) /\\ (((forall fom_index_pfp_bound_divisorcoefficients. (exists fom_gap_pfp_bound_divisorcoefficients_index_bound. fom_gap_pfp_bound_divisorcoefficients_index_bound + S (fom_index_pfp_bound_divisorcoefficients) = D) -> exists fom_value_pfp_bound_divisorcoefficients. ((((exists fom_beta_height_pfp_bound_divisorcoefficients_entry. fom_beta_height_pfp_bound_divisorcoefficients_entry + S (fom_value_pfp_bound_divisorcoefficients) = S ((S (fom_index_pfp_bound_divisorcoefficients)) * dc)) /\\ exists fom_beta_quotient_pfp_bound_divisorcoefficients_entry. db = fom_beta_quotient_pfp_bound_divisorcoefficients_entry * S ((S (fom_index_pfp_bound_divisorcoefficients)) * dc) + (fom_value_pfp_bound_divisorcoefficients))) /\\ (exists fom_gap_pfp_bound_divisorcoefficients_value_bound. fom_gap_pfp_bound_divisorcoefficients_value_bound + S (fom_value_pfp_bound_divisorcoefficients) = p))) /\\ ((exists pfd_leading_bound_divisor. ((((exists ff_h_pfp_bound_divisorentry. ff_h_pfp_bound_divisorentry + S (pfd_leading_bound_divisor) = S ((S (0)) * dc)) /\\ exists ff_q_pfp_bound_divisorentry. db = ff_q_pfp_bound_divisorentry * S ((S (0)) * dc) + (pfd_leading_bound_divisor))) /\\ ((~(pfd_leading_bound_divisor=0)))))))))) -> ((((L)=S (a)) /\\ (((forall fom_index_pfp_bound_targetcoefficients. (exists fom_gap_pfp_bound_targetcoefficients_index_bound. fom_gap_pfp_bound_targetcoefficients_index_bound + S (fom_index_pfp_bound_targetcoefficients) = L) -> exists fom_value_pfp_bound_targetcoefficients. ((((exists fom_beta_height_pfp_bound_targetcoefficients_entry. fom_beta_height_pfp_bound_targetcoefficients_entry + S (fom_value_pfp_bound_targetcoefficients) = S ((S (fom_index_pfp_bound_targetcoefficients)) * ac)) /\\ exists fom_beta_quotient_pfp_bound_targetcoefficients_entry. ab = fom_beta_quotient_pfp_bound_targetcoefficients_entry * S ((S (fom_index_pfp_bound_targetcoefficients)) * ac) + (fom_value_pfp_bound_targetcoefficients))) /\\ (exists fom_gap_pfp_bound_targetcoefficients_value_bound. fom_gap_pfp_bound_targetcoefficients_value_bound + S (fom_value_pfp_bound_targetcoefficients) = p))) /\\ ((exists pfd_leading_bound_target. ((((exists ff_h_pfp_bound_targetentry. ff_h_pfp_bound_targetentry + S (pfd_leading_bound_target) = S ((S (0)) * ac)) /\\ exists ff_q_pfp_bound_targetentry. ab = ff_q_pfp_bound_targetentry * S ((S (0)) * ac) + (pfd_leading_bound_target))) /\\ ((~(pfd_leading_bound_target=0)))))))))) -> (((forall fom_index_pfp_bound_divisibility_canonical. (exists fom_gap_pfp_bound_divisibility_canonical_index_bound. fom_gap_pfp_bound_divisibility_canonical_index_bound + S (fom_index_pfp_bound_divisibility_canonical) = L) -> exists fom_value_pfp_bound_divisibility_canonical. ((((exists fom_beta_height_pfp_bound_divisibility_canonical_entry. fom_beta_height_pfp_bound_divisibility_canonical_entry + S (fom_value_pfp_bound_divisibility_canonical) = S ((S (fom_index_pfp_bound_divisibility_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_bound_divisibility_canonical_entry. ab = fom_beta_quotient_pfp_bound_divisibility_canonical_entry * S ((S (fom_index_pfp_bound_divisibility_canonical)) * ac) + (fom_value_pfp_bound_divisibility_canonical))) /\\ (exists fom_gap_pfp_bound_divisibility_canonical_value_bound. fom_gap_pfp_bound_divisibility_canonical_value_bound + S (fom_value_pfp_bound_divisibility_canonical) = p))) /\\ ((exists pfgu_qb_bound_divisibility pfgu_qc_bound_divisibility pfgu_Q_bound_divisibility pfgu_pb_bound_divisibility pfgu_pc_bound_divisibility pfgu_P_bound_divisibility. ((((forall fom_index_pfp_bound_divisibility_productleft. (exists fom_gap_pfp_bound_divisibility_productleft_index_bound. fom_gap_pfp_bound_divisibility_productleft_index_bound + S (fom_index_pfp_bound_divisibility_productleft) = pfgu_Q_bound_divisibility) -> exists fom_value_pfp_bound_divisibility_productleft. ((((exists fom_beta_height_pfp_bound_divisibility_productleft_entry. fom_beta_height_pfp_bound_divisibility_productleft_entry + S (fom_value_pfp_bound_divisibility_productleft) = S ((S (fom_index_pfp_bound_divisibility_productleft)) * pfgu_qc_bound_divisibility)) /\\ exists fom_beta_quotient_pfp_bound_divisibility_productleft_entry. pfgu_qb_bound_divisibility = fom_beta_quotient_pfp_bound_divisibility_productleft_entry * S ((S (fom_index_pfp_bound_divisibility_productleft)) * pfgu_qc_bound_divisibility) + (fom_value_pfp_bound_divisibility_productleft))) /\\ (exists fom_gap_pfp_bound_divisibility_productleft_value_bound. fom_gap_pfp_bound_divisibility_productleft_value_bound + S (fom_value_pfp_bound_divisibility_productleft) = p))) /\\ (((forall fom_index_pfp_bound_divisibility_productright. (exists fom_gap_pfp_bound_divisibility_productright_index_bound. fom_gap_pfp_bound_divisibility_productright_index_bound + S (fom_index_pfp_bound_divisibility_productright) = D) -> exists fom_value_pfp_bound_divisibility_productright. ((((exists fom_beta_height_pfp_bound_divisibility_productright_entry. fom_beta_height_pfp_bound_divisibility_productright_entry + S (fom_value_pfp_bound_divisibility_productright) = S ((S (fom_index_pfp_bound_divisibility_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_bound_divisibility_productright_entry. db = fom_beta_quotient_pfp_bound_divisibility_productright_entry * S ((S (fom_index_pfp_bound_divisibility_productright)) * dc) + (fom_value_pfp_bound_divisibility_productright))) /\\ (exists fom_gap_pfp_bound_divisibility_productright_value_bound. fom_gap_pfp_bound_divisibility_productright_value_bound + S (fom_value_pfp_bound_divisibility_productright) = p))) /\\ (((((((pfgu_Q_bound_divisibility)=0 \\/ (D)=0) /\\ (((pfgu_P_bound_divisibility)=0)))) \\/ (((~((pfgu_Q_bound_divisibility)=0)) /\\ (((~((D)=0)) /\\ (((pfgu_Q_bound_divisibility)+(D)=S (pfgu_P_bound_divisibility)))))))) /\\ ((forall pfc_index_bound_divisibility_productcoefficients. (exists pfa_gap_bound_divisibility_productcoefficientsbound. pfa_gap_bound_divisibility_productcoefficientsbound + S (pfc_index_bound_divisibility_productcoefficients) = (pfgu_P_bound_divisibility)) -> exists pfc_value_bound_divisibility_productcoefficients. ((((exists ff_h_pfp_bound_divisibility_productcoefficientsentry. ff_h_pfp_bound_divisibility_productcoefficientsentry + S (pfc_value_bound_divisibility_productcoefficients) = S ((S (pfc_index_bound_divisibility_productcoefficients)) * pfgu_pc_bound_divisibility)) /\\ exists ff_q_pfp_bound_divisibility_productcoefficientsentry. pfgu_pb_bound_divisibility = ff_q_pfp_bound_divisibility_productcoefficientsentry * S ((S (pfc_index_bound_divisibility_productcoefficients)) * pfgu_pc_bound_divisibility) + (pfc_value_bound_divisibility_productcoefficients))) /\\ ((exists pfc_terms_code_bound_divisibility_productcoefficientscoefficient pfc_terms_scale_bound_divisibility_productcoefficientscoefficient pfc_natural_sum_bound_divisibility_productcoefficientscoefficient. ((forall pfc_index_bound_divisibility_productcoefficientscoefficientdiagonal. (exists pfa_gap_bound_divisibility_productcoefficientscoefficientdiagonalbound. pfa_gap_bound_divisibility_productcoefficientscoefficientdiagonalbound + S (pfc_index_bound_divisibility_productcoefficientscoefficientdiagonal) = (S (pfc_index_bound_divisibility_productcoefficients))) -> exists pfc_value_bound_divisibility_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_bound_divisibility_productcoefficientscoefficientdiagonalentry. ff_h_pfp_bound_divisibility_productcoefficientscoefficientdiagonalentry + S (pfc_value_bound_divisibility_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_bound_divisibility_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bound_divisibility_productcoefficientscoefficient)) /\\ exists ff_q_pfp_bound_divisibility_productcoefficientscoefficientdiagonalentry. pfc_terms_code_bound_divisibility_productcoefficientscoefficient = ff_q_pfp_bound_divisibility_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_bound_divisibility_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_bound_divisibility_productcoefficientscoefficient) + (pfc_value_bound_divisibility_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_bound_divisibility_productcoefficientscoefficientdiagonalterm pfc_left_bound_divisibility_productcoefficientscoefficientdiagonalterm pfc_right_bound_divisibility_productcoefficientscoefficientdiagonalterm. (((pfc_index_bound_divisibility_productcoefficientscoefficientdiagonal)+pfc_complement_bound_divisibility_productcoefficientscoefficientdiagonalterm=(pfc_index_bound_divisibility_productcoefficients)) /\\ ((((((exists pfa_gap_bound_divisibility_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_bound_divisibility_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_bound_divisibility_productcoefficientscoefficientdiagonal) = (pfgu_Q_bound_divisibility)) /\\ ((((exists ff_h_pfp_bound_divisibility_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_bound_divisibility_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_bound_divisibility_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_bound_divisibility_productcoefficientscoefficientdiagonal)) * pfgu_qc_bound_divisibility)) /\\ exists ff_q_pfp_bound_divisibility_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_bound_divisibility = ff_q_pfp_bound_divisibility_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_bound_divisibility_productcoefficientscoefficientdiagonal)) * pfgu_qc_bound_divisibility) + (pfc_left_bound_divisibility_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bound_divisibility_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_bound_divisibility_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_bound_divisibility)=(pfc_index_bound_divisibility_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_bound_divisibility_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_bound_divisibility_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_bound_divisibility_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_bound_divisibility_productcoefficientscoefficientdiagonalterm) = (D)) /\\ ((((exists ff_h_pfp_bound_divisibility_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_bound_divisibility_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_bound_divisibility_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_bound_divisibility_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_bound_divisibility_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_bound_divisibility_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_bound_divisibility_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_bound_divisibility_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_bound_divisibility_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_bound_divisibility_productcoefficientscoefficientdiagonaltermrightoutside+(D)=(pfc_complement_bound_divisibility_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_bound_divisibility_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_bound_divisibility_productcoefficientscoefficientdiagonal)=pfc_left_bound_divisibility_productcoefficientscoefficientdiagonalterm*pfc_right_bound_divisibility_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_bound_divisibility_productcoefficientscoefficientsum fs_v_pfc_bound_divisibility_productcoefficientscoefficientsum. ((((exists fs_h_pfc_bound_divisibility_productcoefficientscoefficientsum_body_start. fs_h_pfc_bound_divisibility_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_bound_divisibility_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bound_divisibility_productcoefficientscoefficientsum_body_start. fs_u_pfc_bound_divisibility_productcoefficientscoefficientsum = fs_q_pfc_bound_divisibility_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_bound_divisibility_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_bound_divisibility_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_bound_divisibility_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_bound_divisibility_productcoefficientscoefficient) = S ((S (S (pfc_index_bound_divisibility_productcoefficients))) * fs_v_pfc_bound_divisibility_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bound_divisibility_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_bound_divisibility_productcoefficientscoefficientsum = fs_q_pfc_bound_divisibility_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_bound_divisibility_productcoefficients))) * fs_v_pfc_bound_divisibility_productcoefficientscoefficientsum) + (pfc_natural_sum_bound_divisibility_productcoefficientscoefficient))) /\\ forall fs_i_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps = S (pfc_index_bound_divisibility_productcoefficients)) -> exists fs_a_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps fs_r_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps fs_s_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bound_divisibility_productcoefficientscoefficient)) /\\ exists fs_q_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_bound_divisibility_productcoefficientscoefficient = fs_q_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_bound_divisibility_productcoefficientscoefficient) + (fs_a_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bound_divisibility_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_bound_divisibility_productcoefficientscoefficientsum = fs_q_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bound_divisibility_productcoefficientscoefficientsum) + (fs_r_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bound_divisibility_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_bound_divisibility_productcoefficientscoefficientsum = fs_q_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_bound_divisibility_productcoefficientscoefficientsum) + (fs_s_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps = fs_r_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps + fs_a_pfc_bound_divisibility_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_bound_divisibility_productcoefficientscoefficientresiduebound. pfa_gap_bound_divisibility_productcoefficientscoefficientresiduebound + S (pfc_value_bound_divisibility_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_bound_divisibility_productcoefficientscoefficientresiduecongruence pfa_offset_right_bound_divisibility_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_bound_divisibility_productcoefficientscoefficient) + (p) * pfa_offset_left_bound_divisibility_productcoefficientscoefficientresiduecongruence = (pfc_value_bound_divisibility_productcoefficients) + (p) * pfa_offset_right_bound_divisibility_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_bound_divisibility_target pfrep_left_bound_divisibility_target pfrep_right_bound_divisibility_target. ((exists pfrep_position_bound_divisibility_targetfirst. ((pfrep_position_bound_divisibility_targetfirst+S (pfrep_power_bound_divisibility_target)=(pfgu_P_bound_divisibility)) /\\ ((((exists ff_h_pfp_bound_divisibility_targetfirstentry. ff_h_pfp_bound_divisibility_targetfirstentry + S (pfrep_left_bound_divisibility_target) = S ((S (pfrep_position_bound_divisibility_targetfirst)) * pfgu_pc_bound_divisibility)) /\\ exists ff_q_pfp_bound_divisibility_targetfirstentry. pfgu_pb_bound_divisibility = ff_q_pfp_bound_divisibility_targetfirstentry * S ((S (pfrep_position_bound_divisibility_targetfirst)) * pfgu_pc_bound_divisibility) + (pfrep_left_bound_divisibility_target)))))) \\/ (((exists pfrep_gap_bound_divisibility_targetfirstoutside. pfrep_gap_bound_divisibility_targetfirstoutside+(pfgu_P_bound_divisibility)=(pfrep_power_bound_divisibility_target)) /\\ (((pfrep_left_bound_divisibility_target)=0))))) -> ((exists pfrep_position_bound_divisibility_targetsecond. ((pfrep_position_bound_divisibility_targetsecond+S (pfrep_power_bound_divisibility_target)=(L)) /\\ ((((exists ff_h_pfp_bound_divisibility_targetsecondentry. ff_h_pfp_bound_divisibility_targetsecondentry + S (pfrep_right_bound_divisibility_target) = S ((S (pfrep_position_bound_divisibility_targetsecond)) * ac)) /\\ exists ff_q_pfp_bound_divisibility_targetsecondentry. ab = ff_q_pfp_bound_divisibility_targetsecondentry * S ((S (pfrep_position_bound_divisibility_targetsecond)) * ac) + (pfrep_right_bound_divisibility_target)))))) \\/ (((exists pfrep_gap_bound_divisibility_targetsecondoutside. pfrep_gap_bound_divisibility_targetsecondoutside+(L)=(pfrep_power_bound_divisibility_target)) /\\ (((pfrep_right_bound_divisibility_target)=0))))) -> pfrep_left_bound_divisibility_target=pfrep_right_bound_divisibility_target))))))) -> (exists pfc_gap_bound_result. pfc_gap_bound_result+(d)=(a))", "statement_sha256": "02b6e351058fcf3c7fe3e46e9b2b0f98897aa5b1c4298193b08d3922f844b6c2", "summary": "A nonzero represented right divisor has degree at most that of its nonzero represented multiple, using the actual retained quotient degree as the natural witness.", "summary_sha256": "6ed804d97a9ed2cfb430c01c23983c33452f456c4375a67d822a5385d5615f97" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "ec6290be77af036d9fc83182a86bcc6280e47201099745cadba25fd8859d3658", "bundle_campaign": "polynomial-gcd-bezout", "bundle_node_id": 485, "bundle_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "campaign": "polynomial-gcd-bezout", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "source_sha256": "916c24ad6c59609612e97daee6e49347a9522cdb28b44f6f09c6c5760bff0b5b", "test_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85" }, "body_checked": true, "body_receipt": { 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"bundle_path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "bundle_root_id": 492, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "a5b195a244527915f7f21ff6cee5081e2ee60923489cbe7fd2a6c4034ff7cee0", "status": "checked" }, "enrollment_index": 4205, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "916c24ad6c59609612e97daee6e49347a9522cdb28b44f6f09c6c5760bff0b5b", "kind": "alpha_v34_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_uniqueness_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "6bfcca310f97b5cdfd5de2cfb6667bc54ca16511bb8c757a0654a184b0548c85", "kind": "alpha_v34_frontier_executable_audit", "path": "peano-lab/py/tests/test_campaign_research_v34_closure.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "03e63a853943fd916c7f8270d3a5fa760d7289c508672b7ab4a469dff6dc84be", "kind": "alpha_v34_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-field-polynomial-gcd-bezout-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "3fe18ad2899cff7db5fbe19df8570ef70b1bfb902171d5212e9b036dda660a46", "kind": "alpha_v34_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/prime-field-polynomial-gcd-bezout-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=485]" }, { "document_sha256": "81943a545e7b9f667ad758c24de14bb47280e8cf3fbb3ca714efec55fb096408", "kind": "alpha_v34_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=polynomial-gcd-bezout]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "b98fbce9eb82b3f10b6b99f80dfb6cdae900d2f93f90dc9828eb9e0e514d7994", "membership": "alpha_only", "name": "prime_field_polynomial_monic_singleton_multiple_equivalent", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro kb", "intro kc", "intro db", "intro dc", "intro ab", "intro ac", "intro d", "intro pb", "intro pc", "intro hp", "intro hd", "intro ha", "intro hc", "intro he", "cases hd", "cases hd_right", "cases ha", "cases ha_right", "have hk : exists k. ((exists ff_h_pfp_singleton_head. ff_h_pfp_singleton_head + S (k) = S ((S (0)) * kc)) /\\ exists ff_q_pfp_singleton_head. kb = ff_q_pfp_singleton_head * S ((S (0)) * kc) + (k))", "specialize beta_at_exists (kb)", "specialize beta_at_exists (kc)", "specialize beta_at_exists (0)", "apply beta_at_exists", "cases hk", "have hpa : ((exists ff_h_pfp_singleton_product_head. ff_h_pfp_singleton_product_head + S (1) = S ((S (0)) * pc)) /\\ exists ff_q_pfp_singleton_product_head. pb = ff_q_pfp_singleton_product_head * S ((S (0)) * pc) + (1))", "have hprefix : forall mdr_i_pfp_singleton_prefix mdr_a_pfp_singleton_prefix. (exists mdr_gap_pfp_singleton_prefixb. mdr_gap_pfp_singleton_prefixb + S (mdr_i_pfp_singleton_prefix) = (S d)) -> (((exists ff_h_mdr_pfp_singleton_prefixo. ff_h_mdr_pfp_singleton_prefixo + S (mdr_a_pfp_singleton_prefix) = S ((S (mdr_i_pfp_singleton_prefix)) * ac)) /\\ exists ff_q_mdr_pfp_singleton_prefixo. ab = ff_q_mdr_pfp_singleton_prefixo * S ((S (mdr_i_pfp_singleton_prefix)) * ac) + (mdr_a_pfp_singleton_prefix))) -> (((exists ff_h_mdr_pfp_singleton_prefixn. ff_h_mdr_pfp_singleton_prefixn + S (mdr_a_pfp_singleton_prefix) = S ((S (mdr_i_pfp_singleton_prefix)) * pc)) /\\ exists ff_q_mdr_pfp_singleton_prefixn. pb = ff_q_mdr_pfp_singleton_prefixn * S ((S (mdr_i_pfp_singleton_prefix)) * pc) + (mdr_a_pfp_singleton_prefix)))", "specialize prime_field_polynomial_equivalent_implies_equal_same_length (ab)", "specialize prime_field_polynomial_equivalent_implies_equal_same_length (ac)", "specialize prime_field_polynomial_equivalent_implies_equal_same_length (pb)", "specialize prime_field_polynomial_equivalent_implies_equal_same_length (pc)", "specialize prime_field_polynomial_equivalent_implies_equal_same_length (S d)", "apply prime_field_polynomial_equivalent_implies_equal_same_length", "specialize prime_field_polynomial_equivalent_symmetric (pb)", "specialize prime_field_polynomial_equivalent_symmetric (pc)", "specialize prime_field_polynomial_equivalent_symmetric (S d)", "specialize prime_field_polynomial_equivalent_symmetric (ab)", "specialize prime_field_polynomial_equivalent_symmetric (ac)", "specialize prime_field_polynomial_equivalent_symmetric (S d)", "apply prime_field_polynomial_equivalent_symmetric", "exact he", "specialize hprefix (0)", "specialize hprefix (1)", "apply hprefix", "exists d", "simp", "exact ha_right_right", "have hm : ((exists pfa_gap_singleton_leading_multiplyleft. pfa_gap_singleton_leading_multiplyleft + S (x) = (p)) /\\ (((exists pfa_gap_singleton_leading_multiplyright. pfa_gap_singleton_leading_multiplyright + S (1) = (p)) /\\ ((((exists pfa_gap_singleton_leading_multiplyresultbound. pfa_gap_singleton_leading_multiplyresultbound + S (1) = (p)) /\\ ((exists pfa_offset_left_singleton_leading_multiplyresultcongruence pfa_offset_right_singleton_leading_multiplyresultcongruence. ((x) * (1)) + (p) * pfa_offset_left_singleton_leading_multiplyresultcongruence = (1) + (p) * pfa_offset_right_singleton_leading_multiplyresultcongruence))))))))", "specialize prime_field_polynomial_convolution_leading_coefficient (p)", "specialize prime_field_polynomial_convolution_leading_coefficient (kb)", "specialize prime_field_polynomial_convolution_leading_coefficient (kc)", "specialize prime_field_polynomial_convolution_leading_coefficient (0)", "specialize prime_field_polynomial_convolution_leading_coefficient (db)", "specialize prime_field_polynomial_convolution_leading_coefficient (dc)", "specialize prime_field_polynomial_convolution_leading_coefficient (d)", "specialize prime_field_polynomial_convolution_leading_coefficient (pb)", "specialize prime_field_polynomial_convolution_leading_coefficient (pc)", "specialize prime_field_polynomial_convolution_leading_coefficient (S d)", "specialize prime_field_polynomial_convolution_leading_coefficient (x)", "specialize prime_field_polynomial_convolution_leading_coefficient (1)", "specialize prime_field_polynomial_convolution_leading_coefficient (1)", "apply prime_field_polynomial_convolution_leading_coefficient", "exact hc", "exact hk_witness", "exact hd_right_right", "exact hpa", "have hone : x=1", "specialize prime_field_multiply_functional (p)", "specialize prime_field_multiply_functional (x)", "specialize prime_field_multiply_functional (1)", "specialize prime_field_multiply_functional (x)", "specialize prime_field_multiply_functional (1)", "apply prime_field_multiply_functional", "specialize prime_field_multiply_one_right (p)", "specialize prime_field_multiply_one_right (x)", "apply prime_field_multiply_one_right", "exact hp", "cases hm", "exact hm_left", "exact hm", "have hkone : ((exists ff_h_pfp_singleton_unit_head. ff_h_pfp_singleton_unit_head + S (1) = S ((S (0)) * kc)) /\\ exists ff_q_pfp_singleton_unit_head. kb = ff_q_pfp_singleton_unit_head * S ((S (0)) * kc) + (1))", "have hcopy : ((exists ff_h_pfp_singleton_copy_head. ff_h_pfp_singleton_copy_head + S (x) = S ((S (0)) * kc)) /\\ exists ff_q_pfp_singleton_copy_head. kb = ff_q_pfp_singleton_copy_head * S ((S (0)) * kc) + (x))", "exact hk_witness", "rewrite hone at hcopy", "rewrite hone at hcopy", "exact hcopy", "specialize prime_field_polynomial_equivalent_transitive (db)", "specialize prime_field_polynomial_equivalent_transitive (dc)", "specialize prime_field_polynomial_equivalent_transitive (S d)", "specialize prime_field_polynomial_equivalent_transitive (pb)", "specialize prime_field_polynomial_equivalent_transitive (pc)", "specialize prime_field_polynomial_equivalent_transitive (S d)", "specialize prime_field_polynomial_equivalent_transitive (ab)", "specialize prime_field_polynomial_equivalent_transitive (ac)", "specialize prime_field_polynomial_equivalent_transitive (S d)", "apply prime_field_polynomial_equivalent_transitive", "specialize prime_field_polynomial_equivalent_symmetric (pb)", "specialize prime_field_polynomial_equivalent_symmetric (pc)", "specialize prime_field_polynomial_equivalent_symmetric (S d)", "specialize prime_field_polynomial_equivalent_symmetric (db)", "specialize prime_field_polynomial_equivalent_symmetric (dc)", "specialize prime_field_polynomial_equivalent_symmetric (S d)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_convolution_left_unit_equivalent (p)", "specialize prime_field_polynomial_convolution_left_unit_equivalent (kb)", "specialize prime_field_polynomial_convolution_left_unit_equivalent (kc)", "specialize prime_field_polynomial_convolution_left_unit_equivalent (db)", "specialize prime_field_polynomial_convolution_left_unit_equivalent (dc)", "specialize prime_field_polynomial_convolution_left_unit_equivalent (S d)", "specialize prime_field_polynomial_convolution_left_unit_equivalent (pb)", "specialize prime_field_polynomial_convolution_left_unit_equivalent (pc)", "apply prime_field_polynomial_convolution_left_unit_equivalent", "exact hkone", "exact hc", "exact he" ], "script_sha256": "62756204497acb1806f71e9069fd9c755d5d6d35068d163640ee622778eabdcb", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_uniqueness_candidate.py", "sha256": "916c24ad6c59609612e97daee6e49347a9522cdb28b44f6f09c6c5760bff0b5b" }, "statement": "forall p kb kc db dc ab ac d pb pc. (~((p) = 1) /\\ forall pfa_factor_left_singleton_prime pfa_factor_right_singleton_prime. (p) = pfa_factor_left_singleton_prime * pfa_factor_right_singleton_prime -> pfa_factor_left_singleton_prime = 1 \\/ pfa_factor_right_singleton_prime = 1) -> (((~((S d) = 0)) /\\ (((forall fom_index_pfp_singleton_divisorcoefficients. (exists fom_gap_pfp_singleton_divisorcoefficients_index_bound. fom_gap_pfp_singleton_divisorcoefficients_index_bound + S (fom_index_pfp_singleton_divisorcoefficients) = S d) -> exists fom_value_pfp_singleton_divisorcoefficients. ((((exists fom_beta_height_pfp_singleton_divisorcoefficients_entry. fom_beta_height_pfp_singleton_divisorcoefficients_entry + S (fom_value_pfp_singleton_divisorcoefficients) = S ((S (fom_index_pfp_singleton_divisorcoefficients)) * dc)) /\\ exists fom_beta_quotient_pfp_singleton_divisorcoefficients_entry. db = fom_beta_quotient_pfp_singleton_divisorcoefficients_entry * S ((S (fom_index_pfp_singleton_divisorcoefficients)) * dc) + (fom_value_pfp_singleton_divisorcoefficients))) /\\ (exists fom_gap_pfp_singleton_divisorcoefficients_value_bound. fom_gap_pfp_singleton_divisorcoefficients_value_bound + S (fom_value_pfp_singleton_divisorcoefficients) = p))) /\\ ((((exists ff_h_pfp_singleton_divisorleading. ff_h_pfp_singleton_divisorleading + S (1) = S ((S (0)) * dc)) /\\ exists ff_q_pfp_singleton_divisorleading. db = ff_q_pfp_singleton_divisorleading * S ((S (0)) * dc) + (1)))))))) -> (((~((S d) = 0)) /\\ (((forall fom_index_pfp_singleton_targetcoefficients. (exists fom_gap_pfp_singleton_targetcoefficients_index_bound. fom_gap_pfp_singleton_targetcoefficients_index_bound + S (fom_index_pfp_singleton_targetcoefficients) = S d) -> exists fom_value_pfp_singleton_targetcoefficients. ((((exists fom_beta_height_pfp_singleton_targetcoefficients_entry. fom_beta_height_pfp_singleton_targetcoefficients_entry + S (fom_value_pfp_singleton_targetcoefficients) = S ((S (fom_index_pfp_singleton_targetcoefficients)) * ac)) /\\ exists fom_beta_quotient_pfp_singleton_targetcoefficients_entry. ab = fom_beta_quotient_pfp_singleton_targetcoefficients_entry * S ((S (fom_index_pfp_singleton_targetcoefficients)) * ac) + (fom_value_pfp_singleton_targetcoefficients))) /\\ (exists fom_gap_pfp_singleton_targetcoefficients_value_bound. fom_gap_pfp_singleton_targetcoefficients_value_bound + S (fom_value_pfp_singleton_targetcoefficients) = p))) /\\ ((((exists ff_h_pfp_singleton_targetleading. ff_h_pfp_singleton_targetleading + S (1) = S ((S (0)) * ac)) /\\ exists ff_q_pfp_singleton_targetleading. ab = ff_q_pfp_singleton_targetleading * S ((S (0)) * ac) + (1)))))))) -> (((forall fom_index_pfp_singleton_productleft. (exists fom_gap_pfp_singleton_productleft_index_bound. fom_gap_pfp_singleton_productleft_index_bound + S (fom_index_pfp_singleton_productleft) = 1) -> exists fom_value_pfp_singleton_productleft. ((((exists fom_beta_height_pfp_singleton_productleft_entry. fom_beta_height_pfp_singleton_productleft_entry + S (fom_value_pfp_singleton_productleft) = S ((S (fom_index_pfp_singleton_productleft)) * kc)) /\\ exists fom_beta_quotient_pfp_singleton_productleft_entry. kb = fom_beta_quotient_pfp_singleton_productleft_entry * S ((S (fom_index_pfp_singleton_productleft)) * kc) + (fom_value_pfp_singleton_productleft))) /\\ (exists fom_gap_pfp_singleton_productleft_value_bound. fom_gap_pfp_singleton_productleft_value_bound + S (fom_value_pfp_singleton_productleft) = p))) /\\ (((forall fom_index_pfp_singleton_productright. (exists fom_gap_pfp_singleton_productright_index_bound. fom_gap_pfp_singleton_productright_index_bound + S (fom_index_pfp_singleton_productright) = S d) -> exists fom_value_pfp_singleton_productright. ((((exists fom_beta_height_pfp_singleton_productright_entry. fom_beta_height_pfp_singleton_productright_entry + S (fom_value_pfp_singleton_productright) = S ((S (fom_index_pfp_singleton_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_singleton_productright_entry. db = fom_beta_quotient_pfp_singleton_productright_entry * S ((S (fom_index_pfp_singleton_productright)) * dc) + (fom_value_pfp_singleton_productright))) /\\ (exists fom_gap_pfp_singleton_productright_value_bound. fom_gap_pfp_singleton_productright_value_bound + S (fom_value_pfp_singleton_productright) = p))) /\\ (((((((1)=0 \\/ (S d)=0) /\\ (((S d)=0)))) \\/ (((~((1)=0)) /\\ (((~((S d)=0)) /\\ (((1)+(S d)=S (S d)))))))) /\\ ((forall pfc_index_singleton_productcoefficients. (exists pfa_gap_singleton_productcoefficientsbound. pfa_gap_singleton_productcoefficientsbound + S (pfc_index_singleton_productcoefficients) = (S d)) -> exists pfc_value_singleton_productcoefficients. ((((exists ff_h_pfp_singleton_productcoefficientsentry. ff_h_pfp_singleton_productcoefficientsentry + S (pfc_value_singleton_productcoefficients) = S ((S (pfc_index_singleton_productcoefficients)) * pc)) /\\ exists ff_q_pfp_singleton_productcoefficientsentry. pb = ff_q_pfp_singleton_productcoefficientsentry * S ((S (pfc_index_singleton_productcoefficients)) * pc) + (pfc_value_singleton_productcoefficients))) /\\ ((exists pfc_terms_code_singleton_productcoefficientscoefficient pfc_terms_scale_singleton_productcoefficientscoefficient pfc_natural_sum_singleton_productcoefficientscoefficient. ((forall pfc_index_singleton_productcoefficientscoefficientdiagonal. (exists pfa_gap_singleton_productcoefficientscoefficientdiagonalbound. pfa_gap_singleton_productcoefficientscoefficientdiagonalbound + S (pfc_index_singleton_productcoefficientscoefficientdiagonal) = (S (pfc_index_singleton_productcoefficients))) -> exists pfc_value_singleton_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_singleton_productcoefficientscoefficientdiagonalentry. ff_h_pfp_singleton_productcoefficientscoefficientdiagonalentry + S (pfc_value_singleton_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_singleton_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_singleton_productcoefficientscoefficient)) /\\ exists ff_q_pfp_singleton_productcoefficientscoefficientdiagonalentry. pfc_terms_code_singleton_productcoefficientscoefficient = ff_q_pfp_singleton_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_singleton_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_singleton_productcoefficientscoefficient) + (pfc_value_singleton_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_singleton_productcoefficientscoefficientdiagonalterm pfc_left_singleton_productcoefficientscoefficientdiagonalterm pfc_right_singleton_productcoefficientscoefficientdiagonalterm. (((pfc_index_singleton_productcoefficientscoefficientdiagonal)+pfc_complement_singleton_productcoefficientscoefficientdiagonalterm=(pfc_index_singleton_productcoefficients)) /\\ ((((((exists pfa_gap_singleton_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_singleton_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_singleton_productcoefficientscoefficientdiagonal) = (1)) /\\ ((((exists ff_h_pfp_singleton_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_singleton_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_singleton_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_singleton_productcoefficientscoefficientdiagonal)) * kc)) /\\ exists ff_q_pfp_singleton_productcoefficientscoefficientdiagonaltermleftentry. kb = ff_q_pfp_singleton_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_singleton_productcoefficientscoefficientdiagonal)) * kc) + (pfc_left_singleton_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_singleton_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_singleton_productcoefficientscoefficientdiagonaltermleftoutside+(1)=(pfc_index_singleton_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_singleton_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_singleton_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_singleton_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_singleton_productcoefficientscoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_singleton_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_singleton_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_singleton_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_singleton_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_singleton_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_singleton_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_singleton_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_singleton_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_singleton_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_singleton_productcoefficientscoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_singleton_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_singleton_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_singleton_productcoefficientscoefficientdiagonal)=pfc_left_singleton_productcoefficientscoefficientdiagonalterm*pfc_right_singleton_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_singleton_productcoefficientscoefficientsum fs_v_pfc_singleton_productcoefficientscoefficientsum. ((((exists fs_h_pfc_singleton_productcoefficientscoefficientsum_body_start. fs_h_pfc_singleton_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_singleton_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_singleton_productcoefficientscoefficientsum_body_start. fs_u_pfc_singleton_productcoefficientscoefficientsum = fs_q_pfc_singleton_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_singleton_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_singleton_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_singleton_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_singleton_productcoefficientscoefficient) = S ((S (S (pfc_index_singleton_productcoefficients))) * fs_v_pfc_singleton_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_singleton_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_singleton_productcoefficientscoefficientsum = fs_q_pfc_singleton_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_singleton_productcoefficients))) * fs_v_pfc_singleton_productcoefficientscoefficientsum) + (pfc_natural_sum_singleton_productcoefficientscoefficient))) /\\ forall fs_i_pfc_singleton_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_singleton_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_singleton_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_singleton_productcoefficientscoefficientsum_body_steps = S (pfc_index_singleton_productcoefficients)) -> exists fs_a_pfc_singleton_productcoefficientscoefficientsum_body_steps fs_r_pfc_singleton_productcoefficientscoefficientsum_body_steps fs_s_pfc_singleton_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_singleton_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_singleton_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_singleton_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_singleton_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_singleton_productcoefficientscoefficient)) /\\ exists fs_q_pfc_singleton_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_singleton_productcoefficientscoefficient = fs_q_pfc_singleton_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_singleton_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_singleton_productcoefficientscoefficient) + (fs_a_pfc_singleton_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_singleton_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_singleton_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_singleton_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_singleton_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_singleton_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_singleton_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_singleton_productcoefficientscoefficientsum = fs_q_pfc_singleton_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_singleton_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_singleton_productcoefficientscoefficientsum) + (fs_r_pfc_singleton_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_singleton_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_singleton_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_singleton_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_singleton_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_singleton_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_singleton_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_singleton_productcoefficientscoefficientsum = fs_q_pfc_singleton_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_singleton_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_singleton_productcoefficientscoefficientsum) + (fs_s_pfc_singleton_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_singleton_productcoefficientscoefficientsum_body_steps = fs_r_pfc_singleton_productcoefficientscoefficientsum_body_steps + fs_a_pfc_singleton_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_singleton_productcoefficientscoefficientresiduebound. pfa_gap_singleton_productcoefficientscoefficientresiduebound + S (pfc_value_singleton_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_singleton_productcoefficientscoefficientresiduecongruence pfa_offset_right_singleton_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_singleton_productcoefficientscoefficient) + (p) * pfa_offset_left_singleton_productcoefficientscoefficientresiduecongruence = (pfc_value_singleton_productcoefficients) + (p) * pfa_offset_right_singleton_productcoefficientscoefficientresiduecongruence))))))))))))))))))) -> (forall pfrep_power_singleton_target_equivalent pfrep_left_singleton_target_equivalent pfrep_right_singleton_target_equivalent. ((exists pfrep_position_singleton_target_equivalentfirst. ((pfrep_position_singleton_target_equivalentfirst+S (pfrep_power_singleton_target_equivalent)=(S d)) /\\ ((((exists ff_h_pfp_singleton_target_equivalentfirstentry. ff_h_pfp_singleton_target_equivalentfirstentry + S (pfrep_left_singleton_target_equivalent) = S ((S (pfrep_position_singleton_target_equivalentfirst)) * pc)) /\\ exists ff_q_pfp_singleton_target_equivalentfirstentry. pb = ff_q_pfp_singleton_target_equivalentfirstentry * S ((S (pfrep_position_singleton_target_equivalentfirst)) * pc) + (pfrep_left_singleton_target_equivalent)))))) \\/ (((exists pfrep_gap_singleton_target_equivalentfirstoutside. pfrep_gap_singleton_target_equivalentfirstoutside+(S d)=(pfrep_power_singleton_target_equivalent)) /\\ (((pfrep_left_singleton_target_equivalent)=0))))) -> ((exists pfrep_position_singleton_target_equivalentsecond. ((pfrep_position_singleton_target_equivalentsecond+S (pfrep_power_singleton_target_equivalent)=(S d)) /\\ ((((exists ff_h_pfp_singleton_target_equivalentsecondentry. ff_h_pfp_singleton_target_equivalentsecondentry + S (pfrep_right_singleton_target_equivalent) = S ((S (pfrep_position_singleton_target_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_singleton_target_equivalentsecondentry. ab = ff_q_pfp_singleton_target_equivalentsecondentry * S ((S (pfrep_position_singleton_target_equivalentsecond)) * ac) + (pfrep_right_singleton_target_equivalent)))))) \\/ (((exists pfrep_gap_singleton_target_equivalentsecondoutside. pfrep_gap_singleton_target_equivalentsecondoutside+(S d)=(pfrep_power_singleton_target_equivalent)) /\\ (((pfrep_right_singleton_target_equivalent)=0))))) -> pfrep_left_singleton_target_equivalent=pfrep_right_singleton_target_equivalent) -> (forall pfrep_power_singleton_result pfrep_left_singleton_result pfrep_right_singleton_result. ((exists pfrep_position_singleton_resultfirst. ((pfrep_position_singleton_resultfirst+S (pfrep_power_singleton_result)=(S d)) /\\ ((((exists ff_h_pfp_singleton_resultfirstentry. ff_h_pfp_singleton_resultfirstentry + S (pfrep_left_singleton_result) = S ((S (pfrep_position_singleton_resultfirst)) * dc)) /\\ exists ff_q_pfp_singleton_resultfirstentry. db = ff_q_pfp_singleton_resultfirstentry * S ((S (pfrep_position_singleton_resultfirst)) * dc) + (pfrep_left_singleton_result)))))) \\/ (((exists pfrep_gap_singleton_resultfirstoutside. pfrep_gap_singleton_resultfirstoutside+(S d)=(pfrep_power_singleton_result)) /\\ (((pfrep_left_singleton_result)=0))))) -> ((exists pfrep_position_singleton_resultsecond. ((pfrep_position_singleton_resultsecond+S (pfrep_power_singleton_result)=(S d)) /\\ ((((exists ff_h_pfp_singleton_resultsecondentry. ff_h_pfp_singleton_resultsecondentry + S (pfrep_right_singleton_result) = S ((S (pfrep_position_singleton_resultsecond)) * ac)) /\\ exists ff_q_pfp_singleton_resultsecondentry. ab = ff_q_pfp_singleton_resultsecondentry * S ((S (pfrep_position_singleton_resultsecond)) * ac) + (pfrep_right_singleton_result)))))) \\/ (((exists pfrep_gap_singleton_resultsecondoutside. pfrep_gap_singleton_resultsecondoutside+(S d)=(pfrep_power_singleton_result)) /\\ (((pfrep_right_singleton_result)=0))))) -> pfrep_left_singleton_result=pfrep_right_singleton_result)", "statement_sha256": "a5b195a244527915f7f21ff6cee5081e2ee60923489cbe7fd2a6c4034ff7cee0", "summary": "Both monic heads force an actual left singleton quotient to have coefficient one, by the ordered leading product k*1. 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(exists fom_gap_pfp_monic_degree_divisorcoefficients_index_bound. fom_gap_pfp_monic_degree_divisorcoefficients_index_bound + S (fom_index_pfp_monic_degree_divisorcoefficients) = S d) -> exists fom_value_pfp_monic_degree_divisorcoefficients. ((((exists fom_beta_height_pfp_monic_degree_divisorcoefficients_entry. fom_beta_height_pfp_monic_degree_divisorcoefficients_entry + S (fom_value_pfp_monic_degree_divisorcoefficients) = S ((S (fom_index_pfp_monic_degree_divisorcoefficients)) * dc)) /\\ exists fom_beta_quotient_pfp_monic_degree_divisorcoefficients_entry. db = fom_beta_quotient_pfp_monic_degree_divisorcoefficients_entry * S ((S (fom_index_pfp_monic_degree_divisorcoefficients)) * dc) + (fom_value_pfp_monic_degree_divisorcoefficients))) /\\ (exists fom_gap_pfp_monic_degree_divisorcoefficients_value_bound. fom_gap_pfp_monic_degree_divisorcoefficients_value_bound + S (fom_value_pfp_monic_degree_divisorcoefficients) = p))) /\\ ((exists pfd_leading_monic_degree_divisor. ((((exists ff_h_pfp_monic_degree_divisorentry. ff_h_pfp_monic_degree_divisorentry + S (pfd_leading_monic_degree_divisor) = S ((S (0)) * dc)) /\\ exists ff_q_pfp_monic_degree_divisorentry. db = ff_q_pfp_monic_degree_divisorentry * S ((S (0)) * dc) + (pfd_leading_monic_degree_divisor))) /\\ ((~(pfd_leading_monic_degree_divisor=0)))))))))", "specialize prime_field_polynomial_monic_represented_degree (p)", "specialize prime_field_polynomial_monic_represented_degree (db)", "specialize prime_field_polynomial_monic_represented_degree (dc)", "specialize prime_field_polynomial_monic_represented_degree (S d)", "specialize prime_field_polynomial_monic_represented_degree (d)", "apply prime_field_polynomial_monic_represented_degree", "exact hd", "refl", "have had : (((S d)=S (d)) /\\ (((forall fom_index_pfp_monic_degree_targetcoefficients. (exists fom_gap_pfp_monic_degree_targetcoefficients_index_bound. fom_gap_pfp_monic_degree_targetcoefficients_index_bound + S (fom_index_pfp_monic_degree_targetcoefficients) = S d) -> exists fom_value_pfp_monic_degree_targetcoefficients. ((((exists fom_beta_height_pfp_monic_degree_targetcoefficients_entry. fom_beta_height_pfp_monic_degree_targetcoefficients_entry + S (fom_value_pfp_monic_degree_targetcoefficients) = S ((S (fom_index_pfp_monic_degree_targetcoefficients)) * ac)) /\\ exists fom_beta_quotient_pfp_monic_degree_targetcoefficients_entry. ab = fom_beta_quotient_pfp_monic_degree_targetcoefficients_entry * S ((S (fom_index_pfp_monic_degree_targetcoefficients)) * ac) + (fom_value_pfp_monic_degree_targetcoefficients))) /\\ (exists fom_gap_pfp_monic_degree_targetcoefficients_value_bound. fom_gap_pfp_monic_degree_targetcoefficients_value_bound + S (fom_value_pfp_monic_degree_targetcoefficients) = p))) /\\ ((exists pfd_leading_monic_degree_target. ((((exists ff_h_pfp_monic_degree_targetentry. ff_h_pfp_monic_degree_targetentry + S (pfd_leading_monic_degree_target) = S ((S (0)) * ac)) /\\ exists ff_q_pfp_monic_degree_targetentry. ab = ff_q_pfp_monic_degree_targetentry * S ((S (0)) * ac) + (pfd_leading_monic_degree_target))) /\\ ((~(pfd_leading_monic_degree_target=0)))))))))", "specialize prime_field_polynomial_monic_represented_degree (p)", "specialize prime_field_polynomial_monic_represented_degree (ab)", "specialize prime_field_polynomial_monic_represented_degree (ac)", "specialize prime_field_polynomial_monic_represented_degree (S d)", "specialize prime_field_polynomial_monic_represented_degree (d)", "apply prime_field_polynomial_monic_represented_degree", "exact ha", "refl", "have hf : exists pfgu_qb_monic_degree_factor pfgu_qc_monic_degree_factor pfgu_e_monic_degree_factor pfgu_pb_monic_degree_factor pfgu_pc_monic_degree_factor. (((((S (pfgu_e_monic_degree_factor))=S (pfgu_e_monic_degree_factor)) /\\ (((forall fom_index_pfp_monic_degree_factor_quotientcoefficients. (exists fom_gap_pfp_monic_degree_factor_quotientcoefficients_index_bound. fom_gap_pfp_monic_degree_factor_quotientcoefficients_index_bound + S (fom_index_pfp_monic_degree_factor_quotientcoefficients) = S (pfgu_e_monic_degree_factor)) -> exists fom_value_pfp_monic_degree_factor_quotientcoefficients. ((((exists fom_beta_height_pfp_monic_degree_factor_quotientcoefficients_entry. fom_beta_height_pfp_monic_degree_factor_quotientcoefficients_entry + S (fom_value_pfp_monic_degree_factor_quotientcoefficients) = S ((S (fom_index_pfp_monic_degree_factor_quotientcoefficients)) * pfgu_qc_monic_degree_factor)) /\\ exists fom_beta_quotient_pfp_monic_degree_factor_quotientcoefficients_entry. pfgu_qb_monic_degree_factor = fom_beta_quotient_pfp_monic_degree_factor_quotientcoefficients_entry * S ((S (fom_index_pfp_monic_degree_factor_quotientcoefficients)) * pfgu_qc_monic_degree_factor) + (fom_value_pfp_monic_degree_factor_quotientcoefficients))) /\\ (exists fom_gap_pfp_monic_degree_factor_quotientcoefficients_value_bound. fom_gap_pfp_monic_degree_factor_quotientcoefficients_value_bound + S (fom_value_pfp_monic_degree_factor_quotientcoefficients) = p))) /\\ ((exists pfd_leading_monic_degree_factor_quotient. ((((exists ff_h_pfp_monic_degree_factor_quotiententry. ff_h_pfp_monic_degree_factor_quotiententry + S (pfd_leading_monic_degree_factor_quotient) = S ((S (0)) * pfgu_qc_monic_degree_factor)) /\\ exists ff_q_pfp_monic_degree_factor_quotiententry. pfgu_qb_monic_degree_factor = ff_q_pfp_monic_degree_factor_quotiententry * S ((S (0)) * pfgu_qc_monic_degree_factor) + (pfd_leading_monic_degree_factor_quotient))) /\\ ((~(pfd_leading_monic_degree_factor_quotient=0)))))))))) /\\ (((((forall fom_index_pfp_monic_degree_factor_productleft. (exists fom_gap_pfp_monic_degree_factor_productleft_index_bound. fom_gap_pfp_monic_degree_factor_productleft_index_bound + S (fom_index_pfp_monic_degree_factor_productleft) = S (pfgu_e_monic_degree_factor)) -> exists fom_value_pfp_monic_degree_factor_productleft. ((((exists fom_beta_height_pfp_monic_degree_factor_productleft_entry. fom_beta_height_pfp_monic_degree_factor_productleft_entry + S (fom_value_pfp_monic_degree_factor_productleft) = S ((S (fom_index_pfp_monic_degree_factor_productleft)) * pfgu_qc_monic_degree_factor)) /\\ exists fom_beta_quotient_pfp_monic_degree_factor_productleft_entry. pfgu_qb_monic_degree_factor = fom_beta_quotient_pfp_monic_degree_factor_productleft_entry * S ((S (fom_index_pfp_monic_degree_factor_productleft)) * pfgu_qc_monic_degree_factor) + (fom_value_pfp_monic_degree_factor_productleft))) /\\ (exists fom_gap_pfp_monic_degree_factor_productleft_value_bound. fom_gap_pfp_monic_degree_factor_productleft_value_bound + S (fom_value_pfp_monic_degree_factor_productleft) = p))) /\\ (((forall fom_index_pfp_monic_degree_factor_productright. (exists fom_gap_pfp_monic_degree_factor_productright_index_bound. fom_gap_pfp_monic_degree_factor_productright_index_bound + S (fom_index_pfp_monic_degree_factor_productright) = S d) -> exists fom_value_pfp_monic_degree_factor_productright. ((((exists fom_beta_height_pfp_monic_degree_factor_productright_entry. fom_beta_height_pfp_monic_degree_factor_productright_entry + S (fom_value_pfp_monic_degree_factor_productright) = S ((S (fom_index_pfp_monic_degree_factor_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_monic_degree_factor_productright_entry. db = fom_beta_quotient_pfp_monic_degree_factor_productright_entry * S ((S (fom_index_pfp_monic_degree_factor_productright)) * dc) + (fom_value_pfp_monic_degree_factor_productright))) /\\ (exists fom_gap_pfp_monic_degree_factor_productright_value_bound. fom_gap_pfp_monic_degree_factor_productright_value_bound + S (fom_value_pfp_monic_degree_factor_productright) = p))) /\\ (((((((S (pfgu_e_monic_degree_factor))=0 \\/ (S d)=0) /\\ (((S (d))=0)))) \\/ (((~((S (pfgu_e_monic_degree_factor))=0)) /\\ (((~((S d)=0)) /\\ (((S (pfgu_e_monic_degree_factor))+(S d)=S (S (d))))))))) /\\ ((forall pfc_index_monic_degree_factor_productcoefficients. (exists pfa_gap_monic_degree_factor_productcoefficientsbound. pfa_gap_monic_degree_factor_productcoefficientsbound + S (pfc_index_monic_degree_factor_productcoefficients) = (S (d))) -> exists pfc_value_monic_degree_factor_productcoefficients. ((((exists ff_h_pfp_monic_degree_factor_productcoefficientsentry. ff_h_pfp_monic_degree_factor_productcoefficientsentry + S (pfc_value_monic_degree_factor_productcoefficients) = S ((S (pfc_index_monic_degree_factor_productcoefficients)) * pfgu_pc_monic_degree_factor)) /\\ exists ff_q_pfp_monic_degree_factor_productcoefficientsentry. pfgu_pb_monic_degree_factor = ff_q_pfp_monic_degree_factor_productcoefficientsentry * S ((S (pfc_index_monic_degree_factor_productcoefficients)) * pfgu_pc_monic_degree_factor) + (pfc_value_monic_degree_factor_productcoefficients))) /\\ ((exists pfc_terms_code_monic_degree_factor_productcoefficientscoefficient pfc_terms_scale_monic_degree_factor_productcoefficientscoefficient pfc_natural_sum_monic_degree_factor_productcoefficientscoefficient. ((forall pfc_index_monic_degree_factor_productcoefficientscoefficientdiagonal. (exists pfa_gap_monic_degree_factor_productcoefficientscoefficientdiagonalbound. pfa_gap_monic_degree_factor_productcoefficientscoefficientdiagonalbound + S (pfc_index_monic_degree_factor_productcoefficientscoefficientdiagonal) = (S (pfc_index_monic_degree_factor_productcoefficients))) -> exists pfc_value_monic_degree_factor_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_monic_degree_factor_productcoefficientscoefficientdiagonalentry. ff_h_pfp_monic_degree_factor_productcoefficientscoefficientdiagonalentry + S (pfc_value_monic_degree_factor_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_monic_degree_factor_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_monic_degree_factor_productcoefficientscoefficient)) /\\ exists ff_q_pfp_monic_degree_factor_productcoefficientscoefficientdiagonalentry. pfc_terms_code_monic_degree_factor_productcoefficientscoefficient = ff_q_pfp_monic_degree_factor_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_monic_degree_factor_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_monic_degree_factor_productcoefficientscoefficient) + (pfc_value_monic_degree_factor_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_monic_degree_factor_productcoefficientscoefficientdiagonalterm pfc_left_monic_degree_factor_productcoefficientscoefficientdiagonalterm pfc_right_monic_degree_factor_productcoefficientscoefficientdiagonalterm. (((pfc_index_monic_degree_factor_productcoefficientscoefficientdiagonal)+pfc_complement_monic_degree_factor_productcoefficientscoefficientdiagonalterm=(pfc_index_monic_degree_factor_productcoefficients)) /\\ ((((((exists pfa_gap_monic_degree_factor_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_monic_degree_factor_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_monic_degree_factor_productcoefficientscoefficientdiagonal) = (S (pfgu_e_monic_degree_factor))) /\\ ((((exists ff_h_pfp_monic_degree_factor_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_monic_degree_factor_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_monic_degree_factor_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_monic_degree_factor_productcoefficientscoefficientdiagonal)) * pfgu_qc_monic_degree_factor)) /\\ exists ff_q_pfp_monic_degree_factor_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_monic_degree_factor = ff_q_pfp_monic_degree_factor_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_monic_degree_factor_productcoefficientscoefficientdiagonal)) * pfgu_qc_monic_degree_factor) + (pfc_left_monic_degree_factor_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_monic_degree_factor_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_monic_degree_factor_productcoefficientscoefficientdiagonaltermleftoutside+(S (pfgu_e_monic_degree_factor))=(pfc_index_monic_degree_factor_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_monic_degree_factor_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_monic_degree_factor_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_monic_degree_factor_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_monic_degree_factor_productcoefficientscoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_monic_degree_factor_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_monic_degree_factor_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_monic_degree_factor_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_monic_degree_factor_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_monic_degree_factor_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_monic_degree_factor_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_monic_degree_factor_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_monic_degree_factor_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_monic_degree_factor_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_monic_degree_factor_productcoefficientscoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_monic_degree_factor_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_monic_degree_factor_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_monic_degree_factor_productcoefficientscoefficientdiagonal)=pfc_left_monic_degree_factor_productcoefficientscoefficientdiagonalterm*pfc_right_monic_degree_factor_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_monic_degree_factor_productcoefficientscoefficientsum fs_v_pfc_monic_degree_factor_productcoefficientscoefficientsum. ((((exists fs_h_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_start. fs_h_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_monic_degree_factor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_start. fs_u_pfc_monic_degree_factor_productcoefficientscoefficientsum = fs_q_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_monic_degree_factor_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_monic_degree_factor_productcoefficientscoefficient) = S ((S (S (pfc_index_monic_degree_factor_productcoefficients))) * fs_v_pfc_monic_degree_factor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_monic_degree_factor_productcoefficientscoefficientsum = fs_q_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_monic_degree_factor_productcoefficients))) * fs_v_pfc_monic_degree_factor_productcoefficientscoefficientsum) + (pfc_natural_sum_monic_degree_factor_productcoefficientscoefficient))) /\\ forall fs_i_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps = S (pfc_index_monic_degree_factor_productcoefficients)) -> exists fs_a_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps fs_r_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps fs_s_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_monic_degree_factor_productcoefficientscoefficient)) /\\ exists fs_q_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_monic_degree_factor_productcoefficientscoefficient = fs_q_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_monic_degree_factor_productcoefficientscoefficient) + (fs_a_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_monic_degree_factor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_monic_degree_factor_productcoefficientscoefficientsum = fs_q_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_monic_degree_factor_productcoefficientscoefficientsum) + (fs_r_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_monic_degree_factor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_monic_degree_factor_productcoefficientscoefficientsum = fs_q_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_monic_degree_factor_productcoefficientscoefficientsum) + (fs_s_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps = fs_r_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps + fs_a_pfc_monic_degree_factor_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_monic_degree_factor_productcoefficientscoefficientresiduebound. pfa_gap_monic_degree_factor_productcoefficientscoefficientresiduebound + S (pfc_value_monic_degree_factor_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_monic_degree_factor_productcoefficientscoefficientresiduecongruence pfa_offset_right_monic_degree_factor_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_monic_degree_factor_productcoefficientscoefficient) + (p) * pfa_offset_left_monic_degree_factor_productcoefficientscoefficientresiduecongruence = (pfc_value_monic_degree_factor_productcoefficients) + (p) * pfa_offset_right_monic_degree_factor_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ (((forall pfrep_power_monic_degree_factor_equivalent pfrep_left_monic_degree_factor_equivalent pfrep_right_monic_degree_factor_equivalent. ((exists pfrep_position_monic_degree_factor_equivalentfirst. ((pfrep_position_monic_degree_factor_equivalentfirst+S (pfrep_power_monic_degree_factor_equivalent)=(S (d))) /\\ ((((exists ff_h_pfp_monic_degree_factor_equivalentfirstentry. ff_h_pfp_monic_degree_factor_equivalentfirstentry + S (pfrep_left_monic_degree_factor_equivalent) = S ((S (pfrep_position_monic_degree_factor_equivalentfirst)) * pfgu_pc_monic_degree_factor)) /\\ exists ff_q_pfp_monic_degree_factor_equivalentfirstentry. pfgu_pb_monic_degree_factor = ff_q_pfp_monic_degree_factor_equivalentfirstentry * S ((S (pfrep_position_monic_degree_factor_equivalentfirst)) * pfgu_pc_monic_degree_factor) + (pfrep_left_monic_degree_factor_equivalent)))))) \\/ (((exists pfrep_gap_monic_degree_factor_equivalentfirstoutside. pfrep_gap_monic_degree_factor_equivalentfirstoutside+(S (d))=(pfrep_power_monic_degree_factor_equivalent)) /\\ (((pfrep_left_monic_degree_factor_equivalent)=0))))) -> ((exists pfrep_position_monic_degree_factor_equivalentsecond. ((pfrep_position_monic_degree_factor_equivalentsecond+S (pfrep_power_monic_degree_factor_equivalent)=(S d)) /\\ ((((exists ff_h_pfp_monic_degree_factor_equivalentsecondentry. ff_h_pfp_monic_degree_factor_equivalentsecondentry + S (pfrep_right_monic_degree_factor_equivalent) = S ((S (pfrep_position_monic_degree_factor_equivalentsecond)) * ac)) /\\ exists ff_q_pfp_monic_degree_factor_equivalentsecondentry. ab = ff_q_pfp_monic_degree_factor_equivalentsecondentry * S ((S (pfrep_position_monic_degree_factor_equivalentsecond)) * ac) + (pfrep_right_monic_degree_factor_equivalent)))))) \\/ (((exists pfrep_gap_monic_degree_factor_equivalentsecondoutside. pfrep_gap_monic_degree_factor_equivalentsecondoutside+(S d)=(pfrep_power_monic_degree_factor_equivalent)) /\\ (((pfrep_right_monic_degree_factor_equivalent)=0))))) -> pfrep_left_monic_degree_factor_equivalent=pfrep_right_monic_degree_factor_equivalent) /\\ (((pfgu_e_monic_degree_factor)+(d)=(d))))))))", "specialize prime_field_polynomial_right_divides_represented_factorization (p)", "specialize prime_field_polynomial_right_divides_represented_factorization (db)", "specialize prime_field_polynomial_right_divides_represented_factorization (dc)", "specialize prime_field_polynomial_right_divides_represented_factorization (S d)", "specialize prime_field_polynomial_right_divides_represented_factorization (d)", "specialize prime_field_polynomial_right_divides_represented_factorization (ab)", "specialize prime_field_polynomial_right_divides_represented_factorization (ac)", "specialize prime_field_polynomial_right_divides_represented_factorization (S d)", "specialize prime_field_polynomial_right_divides_represented_factorization (d)", "apply prime_field_polynomial_right_divides_represented_factorization", "exact hp", "exact hdd", "exact had", "exact hrd", "cases hf", "cases hf_witness", "cases hf_witness_witness", "cases hf_witness_witness_witness", "cases hf_witness_witness_witness_witness", "cases hf_witness_witness_witness_witness_witness", "cases hf_witness_witness_witness_witness_witness_right", "cases hf_witness_witness_witness_witness_witness_right_right", "have hezero : x2=0", "specialize add_right_cancel (x2)", "specialize add_right_cancel (0)", "specialize add_right_cancel (d)", "apply add_right_cancel", "trans d", "exact hf_witness_witness_witness_witness_witness_right_right_right", "symm", "apply zero_add", "specialize prime_field_polynomial_monic_singleton_multiple_equivalent (p)", "specialize prime_field_polynomial_monic_singleton_multiple_equivalent (x)", "specialize prime_field_polynomial_monic_singleton_multiple_equivalent (x1)", "specialize prime_field_polynomial_monic_singleton_multiple_equivalent (db)", "specialize prime_field_polynomial_monic_singleton_multiple_equivalent (dc)", "specialize prime_field_polynomial_monic_singleton_multiple_equivalent (ab)", "specialize prime_field_polynomial_monic_singleton_multiple_equivalent (ac)", "specialize prime_field_polynomial_monic_singleton_multiple_equivalent (d)", "specialize prime_field_polynomial_monic_singleton_multiple_equivalent (x3)", "specialize prime_field_polynomial_monic_singleton_multiple_equivalent (x4)", "apply prime_field_polynomial_monic_singleton_multiple_equivalent", "exact hp", "exact hd", "exact ha", "have hproduct : ((forall fom_index_pfp_monic_degree_productleft. (exists fom_gap_pfp_monic_degree_productleft_index_bound. fom_gap_pfp_monic_degree_productleft_index_bound + S (fom_index_pfp_monic_degree_productleft) = S x2) -> exists fom_value_pfp_monic_degree_productleft. ((((exists fom_beta_height_pfp_monic_degree_productleft_entry. fom_beta_height_pfp_monic_degree_productleft_entry + S (fom_value_pfp_monic_degree_productleft) = S ((S (fom_index_pfp_monic_degree_productleft)) * x1)) /\\ exists fom_beta_quotient_pfp_monic_degree_productleft_entry. x = fom_beta_quotient_pfp_monic_degree_productleft_entry * S ((S (fom_index_pfp_monic_degree_productleft)) * x1) + (fom_value_pfp_monic_degree_productleft))) /\\ (exists fom_gap_pfp_monic_degree_productleft_value_bound. fom_gap_pfp_monic_degree_productleft_value_bound + S (fom_value_pfp_monic_degree_productleft) = p))) /\\ (((forall fom_index_pfp_monic_degree_productright. (exists fom_gap_pfp_monic_degree_productright_index_bound. fom_gap_pfp_monic_degree_productright_index_bound + S (fom_index_pfp_monic_degree_productright) = S d) -> exists fom_value_pfp_monic_degree_productright. ((((exists fom_beta_height_pfp_monic_degree_productright_entry. fom_beta_height_pfp_monic_degree_productright_entry + S (fom_value_pfp_monic_degree_productright) = S ((S (fom_index_pfp_monic_degree_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_monic_degree_productright_entry. db = fom_beta_quotient_pfp_monic_degree_productright_entry * S ((S (fom_index_pfp_monic_degree_productright)) * dc) + (fom_value_pfp_monic_degree_productright))) /\\ (exists fom_gap_pfp_monic_degree_productright_value_bound. fom_gap_pfp_monic_degree_productright_value_bound + S (fom_value_pfp_monic_degree_productright) = p))) /\\ (((((((S x2)=0 \\/ (S d)=0) /\\ (((S d)=0)))) \\/ (((~((S x2)=0)) /\\ (((~((S d)=0)) /\\ (((S x2)+(S d)=S (S d)))))))) /\\ ((forall pfc_index_monic_degree_productcoefficients. (exists pfa_gap_monic_degree_productcoefficientsbound. pfa_gap_monic_degree_productcoefficientsbound + S (pfc_index_monic_degree_productcoefficients) = (S d)) -> exists pfc_value_monic_degree_productcoefficients. ((((exists ff_h_pfp_monic_degree_productcoefficientsentry. ff_h_pfp_monic_degree_productcoefficientsentry + S (pfc_value_monic_degree_productcoefficients) = S ((S (pfc_index_monic_degree_productcoefficients)) * x4)) /\\ exists ff_q_pfp_monic_degree_productcoefficientsentry. x3 = ff_q_pfp_monic_degree_productcoefficientsentry * S ((S (pfc_index_monic_degree_productcoefficients)) * x4) + (pfc_value_monic_degree_productcoefficients))) /\\ ((exists pfc_terms_code_monic_degree_productcoefficientscoefficient pfc_terms_scale_monic_degree_productcoefficientscoefficient pfc_natural_sum_monic_degree_productcoefficientscoefficient. ((forall pfc_index_monic_degree_productcoefficientscoefficientdiagonal. (exists pfa_gap_monic_degree_productcoefficientscoefficientdiagonalbound. pfa_gap_monic_degree_productcoefficientscoefficientdiagonalbound + S (pfc_index_monic_degree_productcoefficientscoefficientdiagonal) = (S (pfc_index_monic_degree_productcoefficients))) -> exists pfc_value_monic_degree_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_monic_degree_productcoefficientscoefficientdiagonalentry. ff_h_pfp_monic_degree_productcoefficientscoefficientdiagonalentry + S (pfc_value_monic_degree_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_monic_degree_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_monic_degree_productcoefficientscoefficient)) /\\ exists ff_q_pfp_monic_degree_productcoefficientscoefficientdiagonalentry. pfc_terms_code_monic_degree_productcoefficientscoefficient = ff_q_pfp_monic_degree_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_monic_degree_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_monic_degree_productcoefficientscoefficient) + (pfc_value_monic_degree_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_monic_degree_productcoefficientscoefficientdiagonalterm pfc_left_monic_degree_productcoefficientscoefficientdiagonalterm pfc_right_monic_degree_productcoefficientscoefficientdiagonalterm. (((pfc_index_monic_degree_productcoefficientscoefficientdiagonal)+pfc_complement_monic_degree_productcoefficientscoefficientdiagonalterm=(pfc_index_monic_degree_productcoefficients)) /\\ ((((((exists pfa_gap_monic_degree_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_monic_degree_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_monic_degree_productcoefficientscoefficientdiagonal) = (S x2)) /\\ ((((exists ff_h_pfp_monic_degree_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_monic_degree_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_monic_degree_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_monic_degree_productcoefficientscoefficientdiagonal)) * x1)) /\\ exists ff_q_pfp_monic_degree_productcoefficientscoefficientdiagonaltermleftentry. x = ff_q_pfp_monic_degree_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_monic_degree_productcoefficientscoefficientdiagonal)) * x1) + (pfc_left_monic_degree_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_monic_degree_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_monic_degree_productcoefficientscoefficientdiagonaltermleftoutside+(S x2)=(pfc_index_monic_degree_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_monic_degree_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_monic_degree_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_monic_degree_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_monic_degree_productcoefficientscoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_monic_degree_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_monic_degree_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_monic_degree_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_monic_degree_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_monic_degree_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_monic_degree_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_monic_degree_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_monic_degree_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_monic_degree_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_monic_degree_productcoefficientscoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_monic_degree_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_monic_degree_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_monic_degree_productcoefficientscoefficientdiagonal)=pfc_left_monic_degree_productcoefficientscoefficientdiagonalterm*pfc_right_monic_degree_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_monic_degree_productcoefficientscoefficientsum fs_v_pfc_monic_degree_productcoefficientscoefficientsum. ((((exists fs_h_pfc_monic_degree_productcoefficientscoefficientsum_body_start. fs_h_pfc_monic_degree_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_monic_degree_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_monic_degree_productcoefficientscoefficientsum_body_start. fs_u_pfc_monic_degree_productcoefficientscoefficientsum = fs_q_pfc_monic_degree_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_monic_degree_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_monic_degree_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_monic_degree_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_monic_degree_productcoefficientscoefficient) = S ((S (S (pfc_index_monic_degree_productcoefficients))) * fs_v_pfc_monic_degree_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_monic_degree_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_monic_degree_productcoefficientscoefficientsum = fs_q_pfc_monic_degree_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_monic_degree_productcoefficients))) * fs_v_pfc_monic_degree_productcoefficientscoefficientsum) + (pfc_natural_sum_monic_degree_productcoefficientscoefficient))) /\\ forall fs_i_pfc_monic_degree_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_monic_degree_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_monic_degree_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_monic_degree_productcoefficientscoefficientsum_body_steps = S (pfc_index_monic_degree_productcoefficients)) -> exists fs_a_pfc_monic_degree_productcoefficientscoefficientsum_body_steps fs_r_pfc_monic_degree_productcoefficientscoefficientsum_body_steps fs_s_pfc_monic_degree_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_monic_degree_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_monic_degree_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_monic_degree_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_monic_degree_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_monic_degree_productcoefficientscoefficient)) /\\ exists fs_q_pfc_monic_degree_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_monic_degree_productcoefficientscoefficient = fs_q_pfc_monic_degree_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_monic_degree_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_monic_degree_productcoefficientscoefficient) + (fs_a_pfc_monic_degree_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_monic_degree_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_monic_degree_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_monic_degree_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_monic_degree_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_monic_degree_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_monic_degree_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_monic_degree_productcoefficientscoefficientsum = fs_q_pfc_monic_degree_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_monic_degree_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_monic_degree_productcoefficientscoefficientsum) + (fs_r_pfc_monic_degree_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_monic_degree_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_monic_degree_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_monic_degree_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_monic_degree_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_monic_degree_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_monic_degree_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_monic_degree_productcoefficientscoefficientsum = fs_q_pfc_monic_degree_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_monic_degree_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_monic_degree_productcoefficientscoefficientsum) + (fs_s_pfc_monic_degree_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_monic_degree_productcoefficientscoefficientsum_body_steps = fs_r_pfc_monic_degree_productcoefficientscoefficientsum_body_steps + fs_a_pfc_monic_degree_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_monic_degree_productcoefficientscoefficientresiduebound. pfa_gap_monic_degree_productcoefficientscoefficientresiduebound + S (pfc_value_monic_degree_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_monic_degree_productcoefficientscoefficientresiduecongruence pfa_offset_right_monic_degree_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_monic_degree_productcoefficientscoefficient) + (p) * pfa_offset_left_monic_degree_productcoefficientscoefficientresiduecongruence = (pfc_value_monic_degree_productcoefficients) + (p) * pfa_offset_right_monic_degree_productcoefficientscoefficientresiduecongruence))))))))))))))))))", "exact hf_witness_witness_witness_witness_witness_right_left", "rewrite hezero at hproduct", "rewrite hezero at hproduct", "rewrite hezero at hproduct", "rewrite hezero at hproduct", "rewrite hezero at hproduct", "rewrite hezero at hproduct", "exact hproduct", "exact hf_witness_witness_witness_witness_witness_right_right_left" ], "script_sha256": "40e66c589f215b972338c736ade7af1ad28145e0b8e5fa8ce44be7c7810e7244", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_uniqueness_candidate.py", "sha256": "916c24ad6c59609612e97daee6e49347a9522cdb28b44f6f09c6c5760bff0b5b" }, "statement": "forall p db dc ab ac d. (~((p) = 1) /\\ forall pfa_factor_left_monic_degree_prime pfa_factor_right_monic_degree_prime. (p) = pfa_factor_left_monic_degree_prime * pfa_factor_right_monic_degree_prime -> pfa_factor_left_monic_degree_prime = 1 \\/ pfa_factor_right_monic_degree_prime = 1) -> (((~((S d) = 0)) /\\ (((forall fom_index_pfp_monic_degree_Dcoefficients. (exists fom_gap_pfp_monic_degree_Dcoefficients_index_bound. fom_gap_pfp_monic_degree_Dcoefficients_index_bound + S (fom_index_pfp_monic_degree_Dcoefficients) = S d) -> exists fom_value_pfp_monic_degree_Dcoefficients. ((((exists fom_beta_height_pfp_monic_degree_Dcoefficients_entry. fom_beta_height_pfp_monic_degree_Dcoefficients_entry + S (fom_value_pfp_monic_degree_Dcoefficients) = S ((S (fom_index_pfp_monic_degree_Dcoefficients)) * dc)) /\\ exists fom_beta_quotient_pfp_monic_degree_Dcoefficients_entry. db = fom_beta_quotient_pfp_monic_degree_Dcoefficients_entry * S ((S (fom_index_pfp_monic_degree_Dcoefficients)) * dc) + (fom_value_pfp_monic_degree_Dcoefficients))) /\\ (exists fom_gap_pfp_monic_degree_Dcoefficients_value_bound. fom_gap_pfp_monic_degree_Dcoefficients_value_bound + S (fom_value_pfp_monic_degree_Dcoefficients) = p))) /\\ ((((exists ff_h_pfp_monic_degree_Dleading. ff_h_pfp_monic_degree_Dleading + S (1) = S ((S (0)) * dc)) /\\ exists ff_q_pfp_monic_degree_Dleading. db = ff_q_pfp_monic_degree_Dleading * S ((S (0)) * dc) + (1)))))))) -> (((~((S d) = 0)) /\\ (((forall fom_index_pfp_monic_degree_Acoefficients. (exists fom_gap_pfp_monic_degree_Acoefficients_index_bound. fom_gap_pfp_monic_degree_Acoefficients_index_bound + S (fom_index_pfp_monic_degree_Acoefficients) = S d) -> exists fom_value_pfp_monic_degree_Acoefficients. ((((exists fom_beta_height_pfp_monic_degree_Acoefficients_entry. fom_beta_height_pfp_monic_degree_Acoefficients_entry + S (fom_value_pfp_monic_degree_Acoefficients) = S ((S (fom_index_pfp_monic_degree_Acoefficients)) * ac)) /\\ exists fom_beta_quotient_pfp_monic_degree_Acoefficients_entry. ab = fom_beta_quotient_pfp_monic_degree_Acoefficients_entry * S ((S (fom_index_pfp_monic_degree_Acoefficients)) * ac) + (fom_value_pfp_monic_degree_Acoefficients))) /\\ (exists fom_gap_pfp_monic_degree_Acoefficients_value_bound. fom_gap_pfp_monic_degree_Acoefficients_value_bound + S (fom_value_pfp_monic_degree_Acoefficients) = p))) /\\ ((((exists ff_h_pfp_monic_degree_Aleading. ff_h_pfp_monic_degree_Aleading + S (1) = S ((S (0)) * ac)) /\\ exists ff_q_pfp_monic_degree_Aleading. ab = ff_q_pfp_monic_degree_Aleading * S ((S (0)) * ac) + (1)))))))) -> (((forall fom_index_pfp_monic_degree_RD_canonical. (exists fom_gap_pfp_monic_degree_RD_canonical_index_bound. fom_gap_pfp_monic_degree_RD_canonical_index_bound + S (fom_index_pfp_monic_degree_RD_canonical) = S d) -> exists fom_value_pfp_monic_degree_RD_canonical. ((((exists fom_beta_height_pfp_monic_degree_RD_canonical_entry. fom_beta_height_pfp_monic_degree_RD_canonical_entry + S (fom_value_pfp_monic_degree_RD_canonical) = S ((S (fom_index_pfp_monic_degree_RD_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_monic_degree_RD_canonical_entry. ab = fom_beta_quotient_pfp_monic_degree_RD_canonical_entry * S ((S (fom_index_pfp_monic_degree_RD_canonical)) * ac) + (fom_value_pfp_monic_degree_RD_canonical))) /\\ (exists fom_gap_pfp_monic_degree_RD_canonical_value_bound. fom_gap_pfp_monic_degree_RD_canonical_value_bound + S (fom_value_pfp_monic_degree_RD_canonical) = p))) /\\ ((exists pfgu_qb_monic_degree_RD pfgu_qc_monic_degree_RD pfgu_Q_monic_degree_RD pfgu_pb_monic_degree_RD pfgu_pc_monic_degree_RD pfgu_P_monic_degree_RD. ((((forall fom_index_pfp_monic_degree_RD_productleft. (exists fom_gap_pfp_monic_degree_RD_productleft_index_bound. fom_gap_pfp_monic_degree_RD_productleft_index_bound + S (fom_index_pfp_monic_degree_RD_productleft) = pfgu_Q_monic_degree_RD) -> exists fom_value_pfp_monic_degree_RD_productleft. ((((exists fom_beta_height_pfp_monic_degree_RD_productleft_entry. fom_beta_height_pfp_monic_degree_RD_productleft_entry + S (fom_value_pfp_monic_degree_RD_productleft) = S ((S (fom_index_pfp_monic_degree_RD_productleft)) * pfgu_qc_monic_degree_RD)) /\\ exists fom_beta_quotient_pfp_monic_degree_RD_productleft_entry. pfgu_qb_monic_degree_RD = fom_beta_quotient_pfp_monic_degree_RD_productleft_entry * S ((S (fom_index_pfp_monic_degree_RD_productleft)) * pfgu_qc_monic_degree_RD) + (fom_value_pfp_monic_degree_RD_productleft))) /\\ (exists fom_gap_pfp_monic_degree_RD_productleft_value_bound. fom_gap_pfp_monic_degree_RD_productleft_value_bound + S (fom_value_pfp_monic_degree_RD_productleft) = p))) /\\ (((forall fom_index_pfp_monic_degree_RD_productright. (exists fom_gap_pfp_monic_degree_RD_productright_index_bound. fom_gap_pfp_monic_degree_RD_productright_index_bound + S (fom_index_pfp_monic_degree_RD_productright) = S d) -> exists fom_value_pfp_monic_degree_RD_productright. ((((exists fom_beta_height_pfp_monic_degree_RD_productright_entry. fom_beta_height_pfp_monic_degree_RD_productright_entry + S (fom_value_pfp_monic_degree_RD_productright) = S ((S (fom_index_pfp_monic_degree_RD_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_monic_degree_RD_productright_entry. db = fom_beta_quotient_pfp_monic_degree_RD_productright_entry * S ((S (fom_index_pfp_monic_degree_RD_productright)) * dc) + (fom_value_pfp_monic_degree_RD_productright))) /\\ (exists fom_gap_pfp_monic_degree_RD_productright_value_bound. fom_gap_pfp_monic_degree_RD_productright_value_bound + S (fom_value_pfp_monic_degree_RD_productright) = p))) /\\ (((((((pfgu_Q_monic_degree_RD)=0 \\/ (S d)=0) /\\ (((pfgu_P_monic_degree_RD)=0)))) \\/ (((~((pfgu_Q_monic_degree_RD)=0)) /\\ (((~((S d)=0)) /\\ (((pfgu_Q_monic_degree_RD)+(S d)=S (pfgu_P_monic_degree_RD)))))))) /\\ ((forall pfc_index_monic_degree_RD_productcoefficients. (exists pfa_gap_monic_degree_RD_productcoefficientsbound. pfa_gap_monic_degree_RD_productcoefficientsbound + S (pfc_index_monic_degree_RD_productcoefficients) = (pfgu_P_monic_degree_RD)) -> exists pfc_value_monic_degree_RD_productcoefficients. ((((exists ff_h_pfp_monic_degree_RD_productcoefficientsentry. ff_h_pfp_monic_degree_RD_productcoefficientsentry + S (pfc_value_monic_degree_RD_productcoefficients) = S ((S (pfc_index_monic_degree_RD_productcoefficients)) * pfgu_pc_monic_degree_RD)) /\\ exists ff_q_pfp_monic_degree_RD_productcoefficientsentry. pfgu_pb_monic_degree_RD = ff_q_pfp_monic_degree_RD_productcoefficientsentry * S ((S (pfc_index_monic_degree_RD_productcoefficients)) * pfgu_pc_monic_degree_RD) + (pfc_value_monic_degree_RD_productcoefficients))) /\\ ((exists pfc_terms_code_monic_degree_RD_productcoefficientscoefficient pfc_terms_scale_monic_degree_RD_productcoefficientscoefficient pfc_natural_sum_monic_degree_RD_productcoefficientscoefficient. ((forall pfc_index_monic_degree_RD_productcoefficientscoefficientdiagonal. (exists pfa_gap_monic_degree_RD_productcoefficientscoefficientdiagonalbound. pfa_gap_monic_degree_RD_productcoefficientscoefficientdiagonalbound + S (pfc_index_monic_degree_RD_productcoefficientscoefficientdiagonal) = (S (pfc_index_monic_degree_RD_productcoefficients))) -> exists pfc_value_monic_degree_RD_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_monic_degree_RD_productcoefficientscoefficientdiagonalentry. ff_h_pfp_monic_degree_RD_productcoefficientscoefficientdiagonalentry + S (pfc_value_monic_degree_RD_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_monic_degree_RD_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_monic_degree_RD_productcoefficientscoefficient)) /\\ exists ff_q_pfp_monic_degree_RD_productcoefficientscoefficientdiagonalentry. pfc_terms_code_monic_degree_RD_productcoefficientscoefficient = ff_q_pfp_monic_degree_RD_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_monic_degree_RD_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_monic_degree_RD_productcoefficientscoefficient) + (pfc_value_monic_degree_RD_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_monic_degree_RD_productcoefficientscoefficientdiagonalterm pfc_left_monic_degree_RD_productcoefficientscoefficientdiagonalterm pfc_right_monic_degree_RD_productcoefficientscoefficientdiagonalterm. (((pfc_index_monic_degree_RD_productcoefficientscoefficientdiagonal)+pfc_complement_monic_degree_RD_productcoefficientscoefficientdiagonalterm=(pfc_index_monic_degree_RD_productcoefficients)) /\\ ((((((exists pfa_gap_monic_degree_RD_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_monic_degree_RD_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_monic_degree_RD_productcoefficientscoefficientdiagonal) = (pfgu_Q_monic_degree_RD)) /\\ ((((exists ff_h_pfp_monic_degree_RD_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_monic_degree_RD_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_monic_degree_RD_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_monic_degree_RD_productcoefficientscoefficientdiagonal)) * pfgu_qc_monic_degree_RD)) /\\ exists ff_q_pfp_monic_degree_RD_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_monic_degree_RD = ff_q_pfp_monic_degree_RD_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_monic_degree_RD_productcoefficientscoefficientdiagonal)) * pfgu_qc_monic_degree_RD) + (pfc_left_monic_degree_RD_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_monic_degree_RD_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_monic_degree_RD_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_monic_degree_RD)=(pfc_index_monic_degree_RD_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_monic_degree_RD_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_monic_degree_RD_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_monic_degree_RD_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_monic_degree_RD_productcoefficientscoefficientdiagonalterm) = (S d)) /\\ ((((exists ff_h_pfp_monic_degree_RD_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_monic_degree_RD_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_monic_degree_RD_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_monic_degree_RD_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_monic_degree_RD_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_monic_degree_RD_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_monic_degree_RD_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_monic_degree_RD_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_monic_degree_RD_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_monic_degree_RD_productcoefficientscoefficientdiagonaltermrightoutside+(S d)=(pfc_complement_monic_degree_RD_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_monic_degree_RD_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_monic_degree_RD_productcoefficientscoefficientdiagonal)=pfc_left_monic_degree_RD_productcoefficientscoefficientdiagonalterm*pfc_right_monic_degree_RD_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_monic_degree_RD_productcoefficientscoefficientsum fs_v_pfc_monic_degree_RD_productcoefficientscoefficientsum. ((((exists fs_h_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_start. fs_h_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_monic_degree_RD_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_start. fs_u_pfc_monic_degree_RD_productcoefficientscoefficientsum = fs_q_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_monic_degree_RD_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_monic_degree_RD_productcoefficientscoefficient) = S ((S (S (pfc_index_monic_degree_RD_productcoefficients))) * fs_v_pfc_monic_degree_RD_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_monic_degree_RD_productcoefficientscoefficientsum = fs_q_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_monic_degree_RD_productcoefficients))) * fs_v_pfc_monic_degree_RD_productcoefficientscoefficientsum) + (pfc_natural_sum_monic_degree_RD_productcoefficientscoefficient))) /\\ forall fs_i_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps = S (pfc_index_monic_degree_RD_productcoefficients)) -> exists fs_a_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps fs_r_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps fs_s_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_monic_degree_RD_productcoefficientscoefficient)) /\\ exists fs_q_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_monic_degree_RD_productcoefficientscoefficient = fs_q_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_monic_degree_RD_productcoefficientscoefficient) + (fs_a_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_monic_degree_RD_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_monic_degree_RD_productcoefficientscoefficientsum = fs_q_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_monic_degree_RD_productcoefficientscoefficientsum) + (fs_r_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_monic_degree_RD_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_monic_degree_RD_productcoefficientscoefficientsum = fs_q_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_monic_degree_RD_productcoefficientscoefficientsum) + (fs_s_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps = fs_r_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps + fs_a_pfc_monic_degree_RD_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_monic_degree_RD_productcoefficientscoefficientresiduebound. pfa_gap_monic_degree_RD_productcoefficientscoefficientresiduebound + S (pfc_value_monic_degree_RD_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_monic_degree_RD_productcoefficientscoefficientresiduecongruence pfa_offset_right_monic_degree_RD_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_monic_degree_RD_productcoefficientscoefficient) + (p) * pfa_offset_left_monic_degree_RD_productcoefficientscoefficientresiduecongruence = (pfc_value_monic_degree_RD_productcoefficients) + (p) * pfa_offset_right_monic_degree_RD_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_monic_degree_RD_target pfrep_left_monic_degree_RD_target pfrep_right_monic_degree_RD_target. ((exists pfrep_position_monic_degree_RD_targetfirst. ((pfrep_position_monic_degree_RD_targetfirst+S (pfrep_power_monic_degree_RD_target)=(pfgu_P_monic_degree_RD)) /\\ ((((exists ff_h_pfp_monic_degree_RD_targetfirstentry. ff_h_pfp_monic_degree_RD_targetfirstentry + S (pfrep_left_monic_degree_RD_target) = S ((S (pfrep_position_monic_degree_RD_targetfirst)) * pfgu_pc_monic_degree_RD)) /\\ exists ff_q_pfp_monic_degree_RD_targetfirstentry. pfgu_pb_monic_degree_RD = ff_q_pfp_monic_degree_RD_targetfirstentry * S ((S (pfrep_position_monic_degree_RD_targetfirst)) * pfgu_pc_monic_degree_RD) + (pfrep_left_monic_degree_RD_target)))))) \\/ (((exists pfrep_gap_monic_degree_RD_targetfirstoutside. pfrep_gap_monic_degree_RD_targetfirstoutside+(pfgu_P_monic_degree_RD)=(pfrep_power_monic_degree_RD_target)) /\\ (((pfrep_left_monic_degree_RD_target)=0))))) -> ((exists pfrep_position_monic_degree_RD_targetsecond. ((pfrep_position_monic_degree_RD_targetsecond+S (pfrep_power_monic_degree_RD_target)=(S d)) /\\ ((((exists ff_h_pfp_monic_degree_RD_targetsecondentry. ff_h_pfp_monic_degree_RD_targetsecondentry + S (pfrep_right_monic_degree_RD_target) = S ((S (pfrep_position_monic_degree_RD_targetsecond)) * ac)) /\\ exists ff_q_pfp_monic_degree_RD_targetsecondentry. ab = ff_q_pfp_monic_degree_RD_targetsecondentry * S ((S (pfrep_position_monic_degree_RD_targetsecond)) * ac) + (pfrep_right_monic_degree_RD_target)))))) \\/ (((exists pfrep_gap_monic_degree_RD_targetsecondoutside. pfrep_gap_monic_degree_RD_targetsecondoutside+(S d)=(pfrep_power_monic_degree_RD_target)) /\\ (((pfrep_right_monic_degree_RD_target)=0))))) -> pfrep_left_monic_degree_RD_target=pfrep_right_monic_degree_RD_target))))))) -> (forall pfrep_power_monic_degree_result pfrep_left_monic_degree_result pfrep_right_monic_degree_result. ((exists pfrep_position_monic_degree_resultfirst. ((pfrep_position_monic_degree_resultfirst+S (pfrep_power_monic_degree_result)=(S d)) /\\ ((((exists ff_h_pfp_monic_degree_resultfirstentry. ff_h_pfp_monic_degree_resultfirstentry + S (pfrep_left_monic_degree_result) = S ((S (pfrep_position_monic_degree_resultfirst)) * dc)) /\\ exists ff_q_pfp_monic_degree_resultfirstentry. db = ff_q_pfp_monic_degree_resultfirstentry * S ((S (pfrep_position_monic_degree_resultfirst)) * dc) + (pfrep_left_monic_degree_result)))))) \\/ (((exists pfrep_gap_monic_degree_resultfirstoutside. pfrep_gap_monic_degree_resultfirstoutside+(S d)=(pfrep_power_monic_degree_result)) /\\ (((pfrep_left_monic_degree_result)=0))))) -> ((exists pfrep_position_monic_degree_resultsecond. ((pfrep_position_monic_degree_resultsecond+S (pfrep_power_monic_degree_result)=(S d)) /\\ ((((exists ff_h_pfp_monic_degree_resultsecondentry. ff_h_pfp_monic_degree_resultsecondentry + S (pfrep_right_monic_degree_result) = S ((S (pfrep_position_monic_degree_resultsecond)) * ac)) /\\ exists ff_q_pfp_monic_degree_resultsecondentry. ab = ff_q_pfp_monic_degree_resultsecondentry * S ((S (pfrep_position_monic_degree_resultsecond)) * ac) + (pfrep_right_monic_degree_result)))))) \\/ (((exists pfrep_gap_monic_degree_resultsecondoutside. pfrep_gap_monic_degree_resultsecondoutside+(S d)=(pfrep_power_monic_degree_result)) /\\ (((pfrep_right_monic_degree_result)=0))))) -> pfrep_left_monic_degree_result=pfrep_right_monic_degree_result)", "statement_sha256": "bd0247a143301cfbe9b9b289378dd16170e408793fdfcc977bbe9f3dac519134", "summary": "Equal-degree monic right divisibility has a genuinely constructed degree-zero quotient. 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G=S d", "specialize nonzero_is_succ (G)", "apply nonzero_is_succ", "exact hgn", "cases hgl", "have hhl : exists e. H=S e", "specialize nonzero_is_succ (H)", "apply nonzero_is_succ", "exact hhn", "cases hhl", "have hgd : (((G)=S (x)) /\\ (((forall fom_index_pfp_associates_degree_Gcoefficients. (exists fom_gap_pfp_associates_degree_Gcoefficients_index_bound. fom_gap_pfp_associates_degree_Gcoefficients_index_bound + S (fom_index_pfp_associates_degree_Gcoefficients) = G) -> exists fom_value_pfp_associates_degree_Gcoefficients. ((((exists fom_beta_height_pfp_associates_degree_Gcoefficients_entry. fom_beta_height_pfp_associates_degree_Gcoefficients_entry + S (fom_value_pfp_associates_degree_Gcoefficients) = S ((S (fom_index_pfp_associates_degree_Gcoefficients)) * gc)) /\\ exists fom_beta_quotient_pfp_associates_degree_Gcoefficients_entry. gb = fom_beta_quotient_pfp_associates_degree_Gcoefficients_entry * S ((S (fom_index_pfp_associates_degree_Gcoefficients)) * gc) + (fom_value_pfp_associates_degree_Gcoefficients))) /\\ (exists fom_gap_pfp_associates_degree_Gcoefficients_value_bound. fom_gap_pfp_associates_degree_Gcoefficients_value_bound + S (fom_value_pfp_associates_degree_Gcoefficients) = p))) /\\ ((exists pfd_leading_associates_degree_G. ((((exists ff_h_pfp_associates_degree_Gentry. ff_h_pfp_associates_degree_Gentry + S (pfd_leading_associates_degree_G) = S ((S (0)) * gc)) /\\ exists ff_q_pfp_associates_degree_Gentry. gb = ff_q_pfp_associates_degree_Gentry * S ((S (0)) * gc) + (pfd_leading_associates_degree_G))) /\\ ((~(pfd_leading_associates_degree_G=0)))))))))", "specialize prime_field_polynomial_monic_represented_degree (p)", "specialize prime_field_polynomial_monic_represented_degree (gb)", "specialize prime_field_polynomial_monic_represented_degree (gc)", "specialize prime_field_polynomial_monic_represented_degree (G)", "specialize prime_field_polynomial_monic_represented_degree (x)", "apply prime_field_polynomial_monic_represented_degree", "exact hg", "exact hgl_witness", "have hhd : (((H)=S (x1)) /\\ (((forall fom_index_pfp_associates_degree_Hcoefficients. (exists fom_gap_pfp_associates_degree_Hcoefficients_index_bound. fom_gap_pfp_associates_degree_Hcoefficients_index_bound + S (fom_index_pfp_associates_degree_Hcoefficients) = H) -> exists fom_value_pfp_associates_degree_Hcoefficients. ((((exists fom_beta_height_pfp_associates_degree_Hcoefficients_entry. fom_beta_height_pfp_associates_degree_Hcoefficients_entry + S (fom_value_pfp_associates_degree_Hcoefficients) = S ((S (fom_index_pfp_associates_degree_Hcoefficients)) * hc)) /\\ exists fom_beta_quotient_pfp_associates_degree_Hcoefficients_entry. hb = fom_beta_quotient_pfp_associates_degree_Hcoefficients_entry * S ((S (fom_index_pfp_associates_degree_Hcoefficients)) * hc) + (fom_value_pfp_associates_degree_Hcoefficients))) /\\ (exists fom_gap_pfp_associates_degree_Hcoefficients_value_bound. fom_gap_pfp_associates_degree_Hcoefficients_value_bound + S (fom_value_pfp_associates_degree_Hcoefficients) = p))) /\\ ((exists pfd_leading_associates_degree_H. ((((exists ff_h_pfp_associates_degree_Hentry. ff_h_pfp_associates_degree_Hentry + S (pfd_leading_associates_degree_H) = S ((S (0)) * hc)) /\\ exists ff_q_pfp_associates_degree_Hentry. hb = ff_q_pfp_associates_degree_Hentry * S ((S (0)) * hc) + (pfd_leading_associates_degree_H))) /\\ ((~(pfd_leading_associates_degree_H=0)))))))))", "specialize prime_field_polynomial_monic_represented_degree (p)", "specialize prime_field_polynomial_monic_represented_degree (hb)", "specialize prime_field_polynomial_monic_represented_degree (hc)", "specialize prime_field_polynomial_monic_represented_degree (H)", "specialize prime_field_polynomial_monic_represented_degree (x1)", "apply prime_field_polynomial_monic_represented_degree", "exact hh", "exact hhl_witness", "have hde : x=x1", "specialize le_antisymm (x)", "specialize le_antisymm (x1)", "apply le_antisymm", "specialize prime_field_polynomial_right_divides_represented_degree_bound (p)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (gb)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (gc)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (G)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (x)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (hb)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (hc)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (H)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (x1)", "apply prime_field_polynomial_right_divides_represented_degree_bound", "exact hp", "exact hgd", "exact hhd", "exact hgh", "specialize prime_field_polynomial_right_divides_represented_degree_bound (p)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (hb)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (hc)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (H)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (x1)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (gb)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (gc)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (G)", "specialize prime_field_polynomial_right_divides_represented_degree_bound (x)", "apply prime_field_polynomial_right_divides_represented_degree_bound", "exact hp", "exact hhd", "exact hgd", "exact hhg", "have hsame : H=S x", "rewrite hhl_witness", "rewrite hde", "refl", "have hge : ((~((S x) = 0)) /\\ (((forall fom_index_pfp_associates_monic_Gcoefficients. (exists fom_gap_pfp_associates_monic_Gcoefficients_index_bound. fom_gap_pfp_associates_monic_Gcoefficients_index_bound + S (fom_index_pfp_associates_monic_Gcoefficients) = S x) -> exists fom_value_pfp_associates_monic_Gcoefficients. ((((exists fom_beta_height_pfp_associates_monic_Gcoefficients_entry. fom_beta_height_pfp_associates_monic_Gcoefficients_entry + S (fom_value_pfp_associates_monic_Gcoefficients) = S ((S (fom_index_pfp_associates_monic_Gcoefficients)) * gc)) /\\ exists fom_beta_quotient_pfp_associates_monic_Gcoefficients_entry. gb = fom_beta_quotient_pfp_associates_monic_Gcoefficients_entry * S ((S (fom_index_pfp_associates_monic_Gcoefficients)) * gc) + (fom_value_pfp_associates_monic_Gcoefficients))) /\\ (exists fom_gap_pfp_associates_monic_Gcoefficients_value_bound. fom_gap_pfp_associates_monic_Gcoefficients_value_bound + S (fom_value_pfp_associates_monic_Gcoefficients) = p))) /\\ ((((exists ff_h_pfp_associates_monic_Gleading. ff_h_pfp_associates_monic_Gleading + S (1) = S ((S (0)) * gc)) /\\ exists ff_q_pfp_associates_monic_Gleading. gb = ff_q_pfp_associates_monic_Gleading * S ((S (0)) * gc) + (1)))))))", "have hcopy : ((~((G) = 0)) /\\ (((forall fom_index_pfp_associates_monic_Gcopycoefficients. (exists fom_gap_pfp_associates_monic_Gcopycoefficients_index_bound. fom_gap_pfp_associates_monic_Gcopycoefficients_index_bound + S (fom_index_pfp_associates_monic_Gcopycoefficients) = G) -> exists fom_value_pfp_associates_monic_Gcopycoefficients. ((((exists fom_beta_height_pfp_associates_monic_Gcopycoefficients_entry. fom_beta_height_pfp_associates_monic_Gcopycoefficients_entry + S (fom_value_pfp_associates_monic_Gcopycoefficients) = S ((S (fom_index_pfp_associates_monic_Gcopycoefficients)) * gc)) /\\ exists fom_beta_quotient_pfp_associates_monic_Gcopycoefficients_entry. gb = fom_beta_quotient_pfp_associates_monic_Gcopycoefficients_entry * S ((S (fom_index_pfp_associates_monic_Gcopycoefficients)) * gc) + (fom_value_pfp_associates_monic_Gcopycoefficients))) /\\ (exists fom_gap_pfp_associates_monic_Gcopycoefficients_value_bound. fom_gap_pfp_associates_monic_Gcopycoefficients_value_bound + S (fom_value_pfp_associates_monic_Gcopycoefficients) = p))) /\\ ((((exists ff_h_pfp_associates_monic_Gcopyleading. ff_h_pfp_associates_monic_Gcopyleading + S (1) = S ((S (0)) * gc)) /\\ exists ff_q_pfp_associates_monic_Gcopyleading. gb = ff_q_pfp_associates_monic_Gcopyleading * S ((S (0)) * gc) + (1)))))))", "exact hg", "rewrite hgl_witness at hcopy", "rewrite hgl_witness at hcopy", "exact hcopy", "have hhe : ((~((S x) = 0)) /\\ (((forall fom_index_pfp_associates_monic_Hcoefficients. (exists fom_gap_pfp_associates_monic_Hcoefficients_index_bound. fom_gap_pfp_associates_monic_Hcoefficients_index_bound + S (fom_index_pfp_associates_monic_Hcoefficients) = S x) -> exists fom_value_pfp_associates_monic_Hcoefficients. ((((exists fom_beta_height_pfp_associates_monic_Hcoefficients_entry. fom_beta_height_pfp_associates_monic_Hcoefficients_entry + S (fom_value_pfp_associates_monic_Hcoefficients) = S ((S (fom_index_pfp_associates_monic_Hcoefficients)) * hc)) /\\ exists fom_beta_quotient_pfp_associates_monic_Hcoefficients_entry. hb = fom_beta_quotient_pfp_associates_monic_Hcoefficients_entry * S ((S (fom_index_pfp_associates_monic_Hcoefficients)) * hc) + (fom_value_pfp_associates_monic_Hcoefficients))) /\\ (exists fom_gap_pfp_associates_monic_Hcoefficients_value_bound. fom_gap_pfp_associates_monic_Hcoefficients_value_bound + S (fom_value_pfp_associates_monic_Hcoefficients) = p))) /\\ ((((exists ff_h_pfp_associates_monic_Hleading. ff_h_pfp_associates_monic_Hleading + S (1) = S ((S (0)) * hc)) /\\ exists ff_q_pfp_associates_monic_Hleading. hb = ff_q_pfp_associates_monic_Hleading * S ((S (0)) * hc) + (1)))))))", "have hcopy : ((~((H) = 0)) /\\ (((forall fom_index_pfp_associates_monic_Hcopycoefficients. (exists fom_gap_pfp_associates_monic_Hcopycoefficients_index_bound. fom_gap_pfp_associates_monic_Hcopycoefficients_index_bound + S (fom_index_pfp_associates_monic_Hcopycoefficients) = H) -> exists fom_value_pfp_associates_monic_Hcopycoefficients. ((((exists fom_beta_height_pfp_associates_monic_Hcopycoefficients_entry. fom_beta_height_pfp_associates_monic_Hcopycoefficients_entry + S (fom_value_pfp_associates_monic_Hcopycoefficients) = S ((S (fom_index_pfp_associates_monic_Hcopycoefficients)) * hc)) /\\ exists fom_beta_quotient_pfp_associates_monic_Hcopycoefficients_entry. hb = fom_beta_quotient_pfp_associates_monic_Hcopycoefficients_entry * S ((S (fom_index_pfp_associates_monic_Hcopycoefficients)) * hc) + (fom_value_pfp_associates_monic_Hcopycoefficients))) /\\ (exists fom_gap_pfp_associates_monic_Hcopycoefficients_value_bound. fom_gap_pfp_associates_monic_Hcopycoefficients_value_bound + S (fom_value_pfp_associates_monic_Hcopycoefficients) = p))) /\\ ((((exists ff_h_pfp_associates_monic_Hcopyleading. ff_h_pfp_associates_monic_Hcopyleading + S (1) = S ((S (0)) * hc)) /\\ exists ff_q_pfp_associates_monic_Hcopyleading. hb = ff_q_pfp_associates_monic_Hcopyleading * S ((S (0)) * hc) + (1)))))))", "exact hh", "rewrite hsame at hcopy", "rewrite hsame at hcopy", "exact hcopy", "have hr : ((forall fom_index_pfp_associates_RD_canonical. (exists fom_gap_pfp_associates_RD_canonical_index_bound. fom_gap_pfp_associates_RD_canonical_index_bound + S (fom_index_pfp_associates_RD_canonical) = S x) -> exists fom_value_pfp_associates_RD_canonical. ((((exists fom_beta_height_pfp_associates_RD_canonical_entry. fom_beta_height_pfp_associates_RD_canonical_entry + S (fom_value_pfp_associates_RD_canonical) = S ((S (fom_index_pfp_associates_RD_canonical)) * hc)) /\\ exists fom_beta_quotient_pfp_associates_RD_canonical_entry. hb = fom_beta_quotient_pfp_associates_RD_canonical_entry * S ((S (fom_index_pfp_associates_RD_canonical)) * hc) + (fom_value_pfp_associates_RD_canonical))) /\\ (exists fom_gap_pfp_associates_RD_canonical_value_bound. fom_gap_pfp_associates_RD_canonical_value_bound + S (fom_value_pfp_associates_RD_canonical) = p))) /\\ ((exists pfgu_qb_associates_RD pfgu_qc_associates_RD pfgu_Q_associates_RD pfgu_pb_associates_RD pfgu_pc_associates_RD pfgu_P_associates_RD. ((((forall fom_index_pfp_associates_RD_productleft. (exists fom_gap_pfp_associates_RD_productleft_index_bound. fom_gap_pfp_associates_RD_productleft_index_bound + S (fom_index_pfp_associates_RD_productleft) = pfgu_Q_associates_RD) -> exists fom_value_pfp_associates_RD_productleft. ((((exists fom_beta_height_pfp_associates_RD_productleft_entry. fom_beta_height_pfp_associates_RD_productleft_entry + S (fom_value_pfp_associates_RD_productleft) = S ((S (fom_index_pfp_associates_RD_productleft)) * pfgu_qc_associates_RD)) /\\ exists fom_beta_quotient_pfp_associates_RD_productleft_entry. pfgu_qb_associates_RD = fom_beta_quotient_pfp_associates_RD_productleft_entry * S ((S (fom_index_pfp_associates_RD_productleft)) * pfgu_qc_associates_RD) + (fom_value_pfp_associates_RD_productleft))) /\\ (exists fom_gap_pfp_associates_RD_productleft_value_bound. fom_gap_pfp_associates_RD_productleft_value_bound + S (fom_value_pfp_associates_RD_productleft) = p))) /\\ (((forall fom_index_pfp_associates_RD_productright. (exists fom_gap_pfp_associates_RD_productright_index_bound. fom_gap_pfp_associates_RD_productright_index_bound + S (fom_index_pfp_associates_RD_productright) = S x) -> exists fom_value_pfp_associates_RD_productright. ((((exists fom_beta_height_pfp_associates_RD_productright_entry. fom_beta_height_pfp_associates_RD_productright_entry + S (fom_value_pfp_associates_RD_productright) = S ((S (fom_index_pfp_associates_RD_productright)) * gc)) /\\ exists fom_beta_quotient_pfp_associates_RD_productright_entry. gb = fom_beta_quotient_pfp_associates_RD_productright_entry * S ((S (fom_index_pfp_associates_RD_productright)) * gc) + (fom_value_pfp_associates_RD_productright))) /\\ (exists fom_gap_pfp_associates_RD_productright_value_bound. fom_gap_pfp_associates_RD_productright_value_bound + S (fom_value_pfp_associates_RD_productright) = p))) /\\ (((((((pfgu_Q_associates_RD)=0 \\/ (S x)=0) /\\ (((pfgu_P_associates_RD)=0)))) \\/ (((~((pfgu_Q_associates_RD)=0)) /\\ (((~((S x)=0)) /\\ (((pfgu_Q_associates_RD)+(S x)=S (pfgu_P_associates_RD)))))))) /\\ ((forall pfc_index_associates_RD_productcoefficients. (exists pfa_gap_associates_RD_productcoefficientsbound. pfa_gap_associates_RD_productcoefficientsbound + S (pfc_index_associates_RD_productcoefficients) = (pfgu_P_associates_RD)) -> exists pfc_value_associates_RD_productcoefficients. ((((exists ff_h_pfp_associates_RD_productcoefficientsentry. ff_h_pfp_associates_RD_productcoefficientsentry + S (pfc_value_associates_RD_productcoefficients) = S ((S (pfc_index_associates_RD_productcoefficients)) * pfgu_pc_associates_RD)) /\\ exists ff_q_pfp_associates_RD_productcoefficientsentry. pfgu_pb_associates_RD = ff_q_pfp_associates_RD_productcoefficientsentry * S ((S (pfc_index_associates_RD_productcoefficients)) * pfgu_pc_associates_RD) + (pfc_value_associates_RD_productcoefficients))) /\\ ((exists pfc_terms_code_associates_RD_productcoefficientscoefficient pfc_terms_scale_associates_RD_productcoefficientscoefficient pfc_natural_sum_associates_RD_productcoefficientscoefficient. ((forall pfc_index_associates_RD_productcoefficientscoefficientdiagonal. (exists pfa_gap_associates_RD_productcoefficientscoefficientdiagonalbound. pfa_gap_associates_RD_productcoefficientscoefficientdiagonalbound + S (pfc_index_associates_RD_productcoefficientscoefficientdiagonal) = (S (pfc_index_associates_RD_productcoefficients))) -> exists pfc_value_associates_RD_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associates_RD_productcoefficientscoefficientdiagonalentry. ff_h_pfp_associates_RD_productcoefficientscoefficientdiagonalentry + S (pfc_value_associates_RD_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_associates_RD_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_associates_RD_productcoefficientscoefficient)) /\\ exists ff_q_pfp_associates_RD_productcoefficientscoefficientdiagonalentry. pfc_terms_code_associates_RD_productcoefficientscoefficient = ff_q_pfp_associates_RD_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_associates_RD_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_associates_RD_productcoefficientscoefficient) + (pfc_value_associates_RD_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associates_RD_productcoefficientscoefficientdiagonalterm pfc_left_associates_RD_productcoefficientscoefficientdiagonalterm pfc_right_associates_RD_productcoefficientscoefficientdiagonalterm. (((pfc_index_associates_RD_productcoefficientscoefficientdiagonal)+pfc_complement_associates_RD_productcoefficientscoefficientdiagonalterm=(pfc_index_associates_RD_productcoefficients)) /\\ ((((((exists pfa_gap_associates_RD_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_associates_RD_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associates_RD_productcoefficientscoefficientdiagonal) = (pfgu_Q_associates_RD)) /\\ ((((exists ff_h_pfp_associates_RD_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associates_RD_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associates_RD_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associates_RD_productcoefficientscoefficientdiagonal)) * pfgu_qc_associates_RD)) /\\ exists ff_q_pfp_associates_RD_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_associates_RD = ff_q_pfp_associates_RD_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associates_RD_productcoefficientscoefficientdiagonal)) * pfgu_qc_associates_RD) + (pfc_left_associates_RD_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associates_RD_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associates_RD_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_associates_RD)=(pfc_index_associates_RD_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_associates_RD_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associates_RD_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_associates_RD_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associates_RD_productcoefficientscoefficientdiagonalterm) = (S x)) /\\ ((((exists ff_h_pfp_associates_RD_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associates_RD_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associates_RD_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associates_RD_productcoefficientscoefficientdiagonalterm)) * gc)) /\\ exists ff_q_pfp_associates_RD_productcoefficientscoefficientdiagonaltermrightentry. gb = ff_q_pfp_associates_RD_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associates_RD_productcoefficientscoefficientdiagonalterm)) * gc) + (pfc_right_associates_RD_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associates_RD_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associates_RD_productcoefficientscoefficientdiagonaltermrightoutside+(S x)=(pfc_complement_associates_RD_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associates_RD_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associates_RD_productcoefficientscoefficientdiagonal)=pfc_left_associates_RD_productcoefficientscoefficientdiagonalterm*pfc_right_associates_RD_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associates_RD_productcoefficientscoefficientsum fs_v_pfc_associates_RD_productcoefficientscoefficientsum. ((((exists fs_h_pfc_associates_RD_productcoefficientscoefficientsum_body_start. fs_h_pfc_associates_RD_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associates_RD_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_RD_productcoefficientscoefficientsum_body_start. fs_u_pfc_associates_RD_productcoefficientscoefficientsum = fs_q_pfc_associates_RD_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associates_RD_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associates_RD_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_associates_RD_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associates_RD_productcoefficientscoefficient) = S ((S (S (pfc_index_associates_RD_productcoefficients))) * fs_v_pfc_associates_RD_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_RD_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_associates_RD_productcoefficientscoefficientsum = fs_q_pfc_associates_RD_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associates_RD_productcoefficients))) * fs_v_pfc_associates_RD_productcoefficientscoefficientsum) + (pfc_natural_sum_associates_RD_productcoefficientscoefficient))) /\\ forall fs_i_pfc_associates_RD_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associates_RD_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associates_RD_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associates_RD_productcoefficientscoefficientsum_body_steps = S (pfc_index_associates_RD_productcoefficients)) -> exists fs_a_pfc_associates_RD_productcoefficientscoefficientsum_body_steps fs_r_pfc_associates_RD_productcoefficientscoefficientsum_body_steps fs_s_pfc_associates_RD_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associates_RD_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associates_RD_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associates_RD_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associates_RD_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associates_RD_productcoefficientscoefficient)) /\\ exists fs_q_pfc_associates_RD_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associates_RD_productcoefficientscoefficient = fs_q_pfc_associates_RD_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associates_RD_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associates_RD_productcoefficientscoefficient) + (fs_a_pfc_associates_RD_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associates_RD_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associates_RD_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associates_RD_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associates_RD_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_RD_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_RD_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associates_RD_productcoefficientscoefficientsum = fs_q_pfc_associates_RD_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associates_RD_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_RD_productcoefficientscoefficientsum) + (fs_r_pfc_associates_RD_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associates_RD_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associates_RD_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associates_RD_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associates_RD_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_RD_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_RD_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associates_RD_productcoefficientscoefficientsum = fs_q_pfc_associates_RD_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associates_RD_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_RD_productcoefficientscoefficientsum) + (fs_s_pfc_associates_RD_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associates_RD_productcoefficientscoefficientsum_body_steps = fs_r_pfc_associates_RD_productcoefficientscoefficientsum_body_steps + fs_a_pfc_associates_RD_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associates_RD_productcoefficientscoefficientresiduebound. pfa_gap_associates_RD_productcoefficientscoefficientresiduebound + S (pfc_value_associates_RD_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_associates_RD_productcoefficientscoefficientresiduecongruence pfa_offset_right_associates_RD_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_associates_RD_productcoefficientscoefficient) + (p) * pfa_offset_left_associates_RD_productcoefficientscoefficientresiduecongruence = (pfc_value_associates_RD_productcoefficients) + (p) * pfa_offset_right_associates_RD_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_associates_RD_target pfrep_left_associates_RD_target pfrep_right_associates_RD_target. ((exists pfrep_position_associates_RD_targetfirst. ((pfrep_position_associates_RD_targetfirst+S (pfrep_power_associates_RD_target)=(pfgu_P_associates_RD)) /\\ ((((exists ff_h_pfp_associates_RD_targetfirstentry. ff_h_pfp_associates_RD_targetfirstentry + S (pfrep_left_associates_RD_target) = S ((S (pfrep_position_associates_RD_targetfirst)) * pfgu_pc_associates_RD)) /\\ exists ff_q_pfp_associates_RD_targetfirstentry. pfgu_pb_associates_RD = ff_q_pfp_associates_RD_targetfirstentry * S ((S (pfrep_position_associates_RD_targetfirst)) * pfgu_pc_associates_RD) + (pfrep_left_associates_RD_target)))))) \\/ (((exists pfrep_gap_associates_RD_targetfirstoutside. pfrep_gap_associates_RD_targetfirstoutside+(pfgu_P_associates_RD)=(pfrep_power_associates_RD_target)) /\\ (((pfrep_left_associates_RD_target)=0))))) -> ((exists pfrep_position_associates_RD_targetsecond. ((pfrep_position_associates_RD_targetsecond+S (pfrep_power_associates_RD_target)=(S x)) /\\ ((((exists ff_h_pfp_associates_RD_targetsecondentry. ff_h_pfp_associates_RD_targetsecondentry + S (pfrep_right_associates_RD_target) = S ((S (pfrep_position_associates_RD_targetsecond)) * hc)) /\\ exists ff_q_pfp_associates_RD_targetsecondentry. hb = ff_q_pfp_associates_RD_targetsecondentry * S ((S (pfrep_position_associates_RD_targetsecond)) * hc) + (pfrep_right_associates_RD_target)))))) \\/ (((exists pfrep_gap_associates_RD_targetsecondoutside. pfrep_gap_associates_RD_targetsecondoutside+(S x)=(pfrep_power_associates_RD_target)) /\\ (((pfrep_right_associates_RD_target)=0))))) -> pfrep_left_associates_RD_target=pfrep_right_associates_RD_target))))))", "have hcopy : ((forall fom_index_pfp_associates_RDcopy_canonical. (exists fom_gap_pfp_associates_RDcopy_canonical_index_bound. fom_gap_pfp_associates_RDcopy_canonical_index_bound + S (fom_index_pfp_associates_RDcopy_canonical) = H) -> exists fom_value_pfp_associates_RDcopy_canonical. ((((exists fom_beta_height_pfp_associates_RDcopy_canonical_entry. fom_beta_height_pfp_associates_RDcopy_canonical_entry + S (fom_value_pfp_associates_RDcopy_canonical) = S ((S (fom_index_pfp_associates_RDcopy_canonical)) * hc)) /\\ exists fom_beta_quotient_pfp_associates_RDcopy_canonical_entry. hb = fom_beta_quotient_pfp_associates_RDcopy_canonical_entry * S ((S (fom_index_pfp_associates_RDcopy_canonical)) * hc) + (fom_value_pfp_associates_RDcopy_canonical))) /\\ (exists fom_gap_pfp_associates_RDcopy_canonical_value_bound. fom_gap_pfp_associates_RDcopy_canonical_value_bound + S (fom_value_pfp_associates_RDcopy_canonical) = p))) /\\ ((exists pfgu_qb_associates_RDcopy pfgu_qc_associates_RDcopy pfgu_Q_associates_RDcopy pfgu_pb_associates_RDcopy pfgu_pc_associates_RDcopy pfgu_P_associates_RDcopy. ((((forall fom_index_pfp_associates_RDcopy_productleft. (exists fom_gap_pfp_associates_RDcopy_productleft_index_bound. fom_gap_pfp_associates_RDcopy_productleft_index_bound + S (fom_index_pfp_associates_RDcopy_productleft) = pfgu_Q_associates_RDcopy) -> exists fom_value_pfp_associates_RDcopy_productleft. ((((exists fom_beta_height_pfp_associates_RDcopy_productleft_entry. fom_beta_height_pfp_associates_RDcopy_productleft_entry + S (fom_value_pfp_associates_RDcopy_productleft) = S ((S (fom_index_pfp_associates_RDcopy_productleft)) * pfgu_qc_associates_RDcopy)) /\\ exists fom_beta_quotient_pfp_associates_RDcopy_productleft_entry. pfgu_qb_associates_RDcopy = fom_beta_quotient_pfp_associates_RDcopy_productleft_entry * S ((S (fom_index_pfp_associates_RDcopy_productleft)) * pfgu_qc_associates_RDcopy) + (fom_value_pfp_associates_RDcopy_productleft))) /\\ (exists fom_gap_pfp_associates_RDcopy_productleft_value_bound. fom_gap_pfp_associates_RDcopy_productleft_value_bound + S (fom_value_pfp_associates_RDcopy_productleft) = p))) /\\ (((forall fom_index_pfp_associates_RDcopy_productright. (exists fom_gap_pfp_associates_RDcopy_productright_index_bound. fom_gap_pfp_associates_RDcopy_productright_index_bound + S (fom_index_pfp_associates_RDcopy_productright) = G) -> exists fom_value_pfp_associates_RDcopy_productright. ((((exists fom_beta_height_pfp_associates_RDcopy_productright_entry. fom_beta_height_pfp_associates_RDcopy_productright_entry + S (fom_value_pfp_associates_RDcopy_productright) = S ((S (fom_index_pfp_associates_RDcopy_productright)) * gc)) /\\ exists fom_beta_quotient_pfp_associates_RDcopy_productright_entry. gb = fom_beta_quotient_pfp_associates_RDcopy_productright_entry * S ((S (fom_index_pfp_associates_RDcopy_productright)) * gc) + (fom_value_pfp_associates_RDcopy_productright))) /\\ (exists fom_gap_pfp_associates_RDcopy_productright_value_bound. fom_gap_pfp_associates_RDcopy_productright_value_bound + S (fom_value_pfp_associates_RDcopy_productright) = p))) /\\ (((((((pfgu_Q_associates_RDcopy)=0 \\/ (G)=0) /\\ (((pfgu_P_associates_RDcopy)=0)))) \\/ (((~((pfgu_Q_associates_RDcopy)=0)) /\\ (((~((G)=0)) /\\ (((pfgu_Q_associates_RDcopy)+(G)=S (pfgu_P_associates_RDcopy)))))))) /\\ ((forall pfc_index_associates_RDcopy_productcoefficients. (exists pfa_gap_associates_RDcopy_productcoefficientsbound. pfa_gap_associates_RDcopy_productcoefficientsbound + S (pfc_index_associates_RDcopy_productcoefficients) = (pfgu_P_associates_RDcopy)) -> exists pfc_value_associates_RDcopy_productcoefficients. ((((exists ff_h_pfp_associates_RDcopy_productcoefficientsentry. ff_h_pfp_associates_RDcopy_productcoefficientsentry + S (pfc_value_associates_RDcopy_productcoefficients) = S ((S (pfc_index_associates_RDcopy_productcoefficients)) * pfgu_pc_associates_RDcopy)) /\\ exists ff_q_pfp_associates_RDcopy_productcoefficientsentry. pfgu_pb_associates_RDcopy = ff_q_pfp_associates_RDcopy_productcoefficientsentry * S ((S (pfc_index_associates_RDcopy_productcoefficients)) * pfgu_pc_associates_RDcopy) + (pfc_value_associates_RDcopy_productcoefficients))) /\\ ((exists pfc_terms_code_associates_RDcopy_productcoefficientscoefficient pfc_terms_scale_associates_RDcopy_productcoefficientscoefficient pfc_natural_sum_associates_RDcopy_productcoefficientscoefficient. ((forall pfc_index_associates_RDcopy_productcoefficientscoefficientdiagonal. (exists pfa_gap_associates_RDcopy_productcoefficientscoefficientdiagonalbound. pfa_gap_associates_RDcopy_productcoefficientscoefficientdiagonalbound + S (pfc_index_associates_RDcopy_productcoefficientscoefficientdiagonal) = (S (pfc_index_associates_RDcopy_productcoefficients))) -> exists pfc_value_associates_RDcopy_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associates_RDcopy_productcoefficientscoefficientdiagonalentry. ff_h_pfp_associates_RDcopy_productcoefficientscoefficientdiagonalentry + S (pfc_value_associates_RDcopy_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_associates_RDcopy_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_associates_RDcopy_productcoefficientscoefficient)) /\\ exists ff_q_pfp_associates_RDcopy_productcoefficientscoefficientdiagonalentry. pfc_terms_code_associates_RDcopy_productcoefficientscoefficient = ff_q_pfp_associates_RDcopy_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_associates_RDcopy_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_associates_RDcopy_productcoefficientscoefficient) + (pfc_value_associates_RDcopy_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associates_RDcopy_productcoefficientscoefficientdiagonalterm pfc_left_associates_RDcopy_productcoefficientscoefficientdiagonalterm pfc_right_associates_RDcopy_productcoefficientscoefficientdiagonalterm. (((pfc_index_associates_RDcopy_productcoefficientscoefficientdiagonal)+pfc_complement_associates_RDcopy_productcoefficientscoefficientdiagonalterm=(pfc_index_associates_RDcopy_productcoefficients)) /\\ ((((((exists pfa_gap_associates_RDcopy_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_associates_RDcopy_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associates_RDcopy_productcoefficientscoefficientdiagonal) = (pfgu_Q_associates_RDcopy)) /\\ ((((exists ff_h_pfp_associates_RDcopy_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associates_RDcopy_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associates_RDcopy_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associates_RDcopy_productcoefficientscoefficientdiagonal)) * pfgu_qc_associates_RDcopy)) /\\ exists ff_q_pfp_associates_RDcopy_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_associates_RDcopy = ff_q_pfp_associates_RDcopy_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associates_RDcopy_productcoefficientscoefficientdiagonal)) * pfgu_qc_associates_RDcopy) + (pfc_left_associates_RDcopy_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associates_RDcopy_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associates_RDcopy_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_associates_RDcopy)=(pfc_index_associates_RDcopy_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_associates_RDcopy_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associates_RDcopy_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_associates_RDcopy_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associates_RDcopy_productcoefficientscoefficientdiagonalterm) = (G)) /\\ ((((exists ff_h_pfp_associates_RDcopy_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associates_RDcopy_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associates_RDcopy_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associates_RDcopy_productcoefficientscoefficientdiagonalterm)) * gc)) /\\ exists ff_q_pfp_associates_RDcopy_productcoefficientscoefficientdiagonaltermrightentry. gb = ff_q_pfp_associates_RDcopy_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associates_RDcopy_productcoefficientscoefficientdiagonalterm)) * gc) + (pfc_right_associates_RDcopy_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associates_RDcopy_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associates_RDcopy_productcoefficientscoefficientdiagonaltermrightoutside+(G)=(pfc_complement_associates_RDcopy_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associates_RDcopy_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associates_RDcopy_productcoefficientscoefficientdiagonal)=pfc_left_associates_RDcopy_productcoefficientscoefficientdiagonalterm*pfc_right_associates_RDcopy_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associates_RDcopy_productcoefficientscoefficientsum fs_v_pfc_associates_RDcopy_productcoefficientscoefficientsum. ((((exists fs_h_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_start. fs_h_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associates_RDcopy_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_start. fs_u_pfc_associates_RDcopy_productcoefficientscoefficientsum = fs_q_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associates_RDcopy_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associates_RDcopy_productcoefficientscoefficient) = S ((S (S (pfc_index_associates_RDcopy_productcoefficients))) * fs_v_pfc_associates_RDcopy_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_associates_RDcopy_productcoefficientscoefficientsum = fs_q_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associates_RDcopy_productcoefficients))) * fs_v_pfc_associates_RDcopy_productcoefficientscoefficientsum) + (pfc_natural_sum_associates_RDcopy_productcoefficientscoefficient))) /\\ forall fs_i_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps = S (pfc_index_associates_RDcopy_productcoefficients)) -> exists fs_a_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps fs_r_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps fs_s_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associates_RDcopy_productcoefficientscoefficient)) /\\ exists fs_q_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associates_RDcopy_productcoefficientscoefficient = fs_q_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associates_RDcopy_productcoefficientscoefficient) + (fs_a_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_RDcopy_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associates_RDcopy_productcoefficientscoefficientsum = fs_q_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_RDcopy_productcoefficientscoefficientsum) + (fs_r_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_RDcopy_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associates_RDcopy_productcoefficientscoefficientsum = fs_q_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_RDcopy_productcoefficientscoefficientsum) + (fs_s_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps = fs_r_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps + fs_a_pfc_associates_RDcopy_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associates_RDcopy_productcoefficientscoefficientresiduebound. pfa_gap_associates_RDcopy_productcoefficientscoefficientresiduebound + S (pfc_value_associates_RDcopy_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_associates_RDcopy_productcoefficientscoefficientresiduecongruence pfa_offset_right_associates_RDcopy_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_associates_RDcopy_productcoefficientscoefficient) + (p) * pfa_offset_left_associates_RDcopy_productcoefficientscoefficientresiduecongruence = (pfc_value_associates_RDcopy_productcoefficients) + (p) * pfa_offset_right_associates_RDcopy_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_associates_RDcopy_target pfrep_left_associates_RDcopy_target pfrep_right_associates_RDcopy_target. ((exists pfrep_position_associates_RDcopy_targetfirst. ((pfrep_position_associates_RDcopy_targetfirst+S (pfrep_power_associates_RDcopy_target)=(pfgu_P_associates_RDcopy)) /\\ ((((exists ff_h_pfp_associates_RDcopy_targetfirstentry. ff_h_pfp_associates_RDcopy_targetfirstentry + S (pfrep_left_associates_RDcopy_target) = S ((S (pfrep_position_associates_RDcopy_targetfirst)) * pfgu_pc_associates_RDcopy)) /\\ exists ff_q_pfp_associates_RDcopy_targetfirstentry. pfgu_pb_associates_RDcopy = ff_q_pfp_associates_RDcopy_targetfirstentry * S ((S (pfrep_position_associates_RDcopy_targetfirst)) * pfgu_pc_associates_RDcopy) + (pfrep_left_associates_RDcopy_target)))))) \\/ (((exists pfrep_gap_associates_RDcopy_targetfirstoutside. pfrep_gap_associates_RDcopy_targetfirstoutside+(pfgu_P_associates_RDcopy)=(pfrep_power_associates_RDcopy_target)) /\\ (((pfrep_left_associates_RDcopy_target)=0))))) -> ((exists pfrep_position_associates_RDcopy_targetsecond. ((pfrep_position_associates_RDcopy_targetsecond+S (pfrep_power_associates_RDcopy_target)=(H)) /\\ ((((exists ff_h_pfp_associates_RDcopy_targetsecondentry. ff_h_pfp_associates_RDcopy_targetsecondentry + S (pfrep_right_associates_RDcopy_target) = S ((S (pfrep_position_associates_RDcopy_targetsecond)) * hc)) /\\ exists ff_q_pfp_associates_RDcopy_targetsecondentry. hb = ff_q_pfp_associates_RDcopy_targetsecondentry * S ((S (pfrep_position_associates_RDcopy_targetsecond)) * hc) + (pfrep_right_associates_RDcopy_target)))))) \\/ (((exists pfrep_gap_associates_RDcopy_targetsecondoutside. pfrep_gap_associates_RDcopy_targetsecondoutside+(H)=(pfrep_power_associates_RDcopy_target)) /\\ (((pfrep_right_associates_RDcopy_target)=0))))) -> pfrep_left_associates_RDcopy_target=pfrep_right_associates_RDcopy_target))))))", "exact hgh", "rewrite hgl_witness at hcopy", "rewrite hgl_witness at hcopy", "rewrite hgl_witness at hcopy", "rewrite hgl_witness at hcopy", "rewrite hgl_witness at hcopy", "rewrite hgl_witness at hcopy", "rewrite hsame at hcopy", "rewrite hsame at hcopy", "rewrite hsame at hcopy", "exact hcopy", "rewrite hgl_witness", "rewrite hgl_witness", "rewrite hsame", "rewrite hsame", "specialize prime_field_polynomial_monic_equal_degree_right_divides_equivalent (p)", "specialize prime_field_polynomial_monic_equal_degree_right_divides_equivalent (gb)", "specialize prime_field_polynomial_monic_equal_degree_right_divides_equivalent (gc)", "specialize prime_field_polynomial_monic_equal_degree_right_divides_equivalent (hb)", "specialize prime_field_polynomial_monic_equal_degree_right_divides_equivalent (hc)", "specialize prime_field_polynomial_monic_equal_degree_right_divides_equivalent (x)", "apply prime_field_polynomial_monic_equal_degree_right_divides_equivalent", "exact hp", "exact hge", "exact hhe", "exact hr" ], "script_sha256": "b91a9deae6653e7faec18adfb803ec6a5f25ad79834cf032819b34ed8a7798d0", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_uniqueness_candidate.py", "sha256": "916c24ad6c59609612e97daee6e49347a9522cdb28b44f6f09c6c5760bff0b5b" }, "statement": "forall p gb gc G hb hc H. (~((p) = 1) /\\ forall pfa_factor_left_associates_prime pfa_factor_right_associates_prime. (p) = pfa_factor_left_associates_prime * pfa_factor_right_associates_prime -> pfa_factor_left_associates_prime = 1 \\/ pfa_factor_right_associates_prime = 1) -> (((~((G) = 0)) /\\ (((forall fom_index_pfp_associates_Gcoefficients. (exists fom_gap_pfp_associates_Gcoefficients_index_bound. fom_gap_pfp_associates_Gcoefficients_index_bound + S (fom_index_pfp_associates_Gcoefficients) = G) -> exists fom_value_pfp_associates_Gcoefficients. ((((exists fom_beta_height_pfp_associates_Gcoefficients_entry. fom_beta_height_pfp_associates_Gcoefficients_entry + S (fom_value_pfp_associates_Gcoefficients) = S ((S (fom_index_pfp_associates_Gcoefficients)) * gc)) /\\ exists fom_beta_quotient_pfp_associates_Gcoefficients_entry. gb = fom_beta_quotient_pfp_associates_Gcoefficients_entry * S ((S (fom_index_pfp_associates_Gcoefficients)) * gc) + (fom_value_pfp_associates_Gcoefficients))) /\\ (exists fom_gap_pfp_associates_Gcoefficients_value_bound. fom_gap_pfp_associates_Gcoefficients_value_bound + S (fom_value_pfp_associates_Gcoefficients) = p))) /\\ ((((exists ff_h_pfp_associates_Gleading. ff_h_pfp_associates_Gleading + S (1) = S ((S (0)) * gc)) /\\ exists ff_q_pfp_associates_Gleading. gb = ff_q_pfp_associates_Gleading * S ((S (0)) * gc) + (1)))))))) -> (((~((H) = 0)) /\\ (((forall fom_index_pfp_associates_Hcoefficients. (exists fom_gap_pfp_associates_Hcoefficients_index_bound. fom_gap_pfp_associates_Hcoefficients_index_bound + S (fom_index_pfp_associates_Hcoefficients) = H) -> exists fom_value_pfp_associates_Hcoefficients. ((((exists fom_beta_height_pfp_associates_Hcoefficients_entry. fom_beta_height_pfp_associates_Hcoefficients_entry + S (fom_value_pfp_associates_Hcoefficients) = S ((S (fom_index_pfp_associates_Hcoefficients)) * hc)) /\\ exists fom_beta_quotient_pfp_associates_Hcoefficients_entry. hb = fom_beta_quotient_pfp_associates_Hcoefficients_entry * S ((S (fom_index_pfp_associates_Hcoefficients)) * hc) + (fom_value_pfp_associates_Hcoefficients))) /\\ (exists fom_gap_pfp_associates_Hcoefficients_value_bound. fom_gap_pfp_associates_Hcoefficients_value_bound + S (fom_value_pfp_associates_Hcoefficients) = p))) /\\ ((((exists ff_h_pfp_associates_Hleading. ff_h_pfp_associates_Hleading + S (1) = S ((S (0)) * hc)) /\\ exists ff_q_pfp_associates_Hleading. hb = ff_q_pfp_associates_Hleading * S ((S (0)) * hc) + (1)))))))) -> (((forall fom_index_pfp_associates_GH_canonical. (exists fom_gap_pfp_associates_GH_canonical_index_bound. fom_gap_pfp_associates_GH_canonical_index_bound + S (fom_index_pfp_associates_GH_canonical) = H) -> exists fom_value_pfp_associates_GH_canonical. ((((exists fom_beta_height_pfp_associates_GH_canonical_entry. fom_beta_height_pfp_associates_GH_canonical_entry + S (fom_value_pfp_associates_GH_canonical) = S ((S (fom_index_pfp_associates_GH_canonical)) * hc)) /\\ exists fom_beta_quotient_pfp_associates_GH_canonical_entry. hb = fom_beta_quotient_pfp_associates_GH_canonical_entry * S ((S (fom_index_pfp_associates_GH_canonical)) * hc) + (fom_value_pfp_associates_GH_canonical))) /\\ (exists fom_gap_pfp_associates_GH_canonical_value_bound. fom_gap_pfp_associates_GH_canonical_value_bound + S (fom_value_pfp_associates_GH_canonical) = p))) /\\ ((exists pfgu_qb_associates_GH pfgu_qc_associates_GH pfgu_Q_associates_GH pfgu_pb_associates_GH pfgu_pc_associates_GH pfgu_P_associates_GH. ((((forall fom_index_pfp_associates_GH_productleft. (exists fom_gap_pfp_associates_GH_productleft_index_bound. fom_gap_pfp_associates_GH_productleft_index_bound + S (fom_index_pfp_associates_GH_productleft) = pfgu_Q_associates_GH) -> exists fom_value_pfp_associates_GH_productleft. ((((exists fom_beta_height_pfp_associates_GH_productleft_entry. fom_beta_height_pfp_associates_GH_productleft_entry + S (fom_value_pfp_associates_GH_productleft) = S ((S (fom_index_pfp_associates_GH_productleft)) * pfgu_qc_associates_GH)) /\\ exists fom_beta_quotient_pfp_associates_GH_productleft_entry. pfgu_qb_associates_GH = fom_beta_quotient_pfp_associates_GH_productleft_entry * S ((S (fom_index_pfp_associates_GH_productleft)) * pfgu_qc_associates_GH) + (fom_value_pfp_associates_GH_productleft))) /\\ (exists fom_gap_pfp_associates_GH_productleft_value_bound. fom_gap_pfp_associates_GH_productleft_value_bound + S (fom_value_pfp_associates_GH_productleft) = p))) /\\ (((forall fom_index_pfp_associates_GH_productright. (exists fom_gap_pfp_associates_GH_productright_index_bound. fom_gap_pfp_associates_GH_productright_index_bound + S (fom_index_pfp_associates_GH_productright) = G) -> exists fom_value_pfp_associates_GH_productright. ((((exists fom_beta_height_pfp_associates_GH_productright_entry. fom_beta_height_pfp_associates_GH_productright_entry + S (fom_value_pfp_associates_GH_productright) = S ((S (fom_index_pfp_associates_GH_productright)) * gc)) /\\ exists fom_beta_quotient_pfp_associates_GH_productright_entry. gb = fom_beta_quotient_pfp_associates_GH_productright_entry * S ((S (fom_index_pfp_associates_GH_productright)) * gc) + (fom_value_pfp_associates_GH_productright))) /\\ (exists fom_gap_pfp_associates_GH_productright_value_bound. fom_gap_pfp_associates_GH_productright_value_bound + S (fom_value_pfp_associates_GH_productright) = p))) /\\ (((((((pfgu_Q_associates_GH)=0 \\/ (G)=0) /\\ (((pfgu_P_associates_GH)=0)))) \\/ (((~((pfgu_Q_associates_GH)=0)) /\\ (((~((G)=0)) /\\ (((pfgu_Q_associates_GH)+(G)=S (pfgu_P_associates_GH)))))))) /\\ ((forall pfc_index_associates_GH_productcoefficients. (exists pfa_gap_associates_GH_productcoefficientsbound. pfa_gap_associates_GH_productcoefficientsbound + S (pfc_index_associates_GH_productcoefficients) = (pfgu_P_associates_GH)) -> exists pfc_value_associates_GH_productcoefficients. ((((exists ff_h_pfp_associates_GH_productcoefficientsentry. ff_h_pfp_associates_GH_productcoefficientsentry + S (pfc_value_associates_GH_productcoefficients) = S ((S (pfc_index_associates_GH_productcoefficients)) * pfgu_pc_associates_GH)) /\\ exists ff_q_pfp_associates_GH_productcoefficientsentry. pfgu_pb_associates_GH = ff_q_pfp_associates_GH_productcoefficientsentry * S ((S (pfc_index_associates_GH_productcoefficients)) * pfgu_pc_associates_GH) + (pfc_value_associates_GH_productcoefficients))) /\\ ((exists pfc_terms_code_associates_GH_productcoefficientscoefficient pfc_terms_scale_associates_GH_productcoefficientscoefficient pfc_natural_sum_associates_GH_productcoefficientscoefficient. ((forall pfc_index_associates_GH_productcoefficientscoefficientdiagonal. (exists pfa_gap_associates_GH_productcoefficientscoefficientdiagonalbound. pfa_gap_associates_GH_productcoefficientscoefficientdiagonalbound + S (pfc_index_associates_GH_productcoefficientscoefficientdiagonal) = (S (pfc_index_associates_GH_productcoefficients))) -> exists pfc_value_associates_GH_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associates_GH_productcoefficientscoefficientdiagonalentry. ff_h_pfp_associates_GH_productcoefficientscoefficientdiagonalentry + S (pfc_value_associates_GH_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_associates_GH_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_associates_GH_productcoefficientscoefficient)) /\\ exists ff_q_pfp_associates_GH_productcoefficientscoefficientdiagonalentry. pfc_terms_code_associates_GH_productcoefficientscoefficient = ff_q_pfp_associates_GH_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_associates_GH_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_associates_GH_productcoefficientscoefficient) + (pfc_value_associates_GH_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associates_GH_productcoefficientscoefficientdiagonalterm pfc_left_associates_GH_productcoefficientscoefficientdiagonalterm pfc_right_associates_GH_productcoefficientscoefficientdiagonalterm. (((pfc_index_associates_GH_productcoefficientscoefficientdiagonal)+pfc_complement_associates_GH_productcoefficientscoefficientdiagonalterm=(pfc_index_associates_GH_productcoefficients)) /\\ ((((((exists pfa_gap_associates_GH_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_associates_GH_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associates_GH_productcoefficientscoefficientdiagonal) = (pfgu_Q_associates_GH)) /\\ ((((exists ff_h_pfp_associates_GH_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associates_GH_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associates_GH_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associates_GH_productcoefficientscoefficientdiagonal)) * pfgu_qc_associates_GH)) /\\ exists ff_q_pfp_associates_GH_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_associates_GH = ff_q_pfp_associates_GH_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associates_GH_productcoefficientscoefficientdiagonal)) * pfgu_qc_associates_GH) + (pfc_left_associates_GH_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associates_GH_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associates_GH_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_associates_GH)=(pfc_index_associates_GH_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_associates_GH_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associates_GH_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_associates_GH_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associates_GH_productcoefficientscoefficientdiagonalterm) = (G)) /\\ ((((exists ff_h_pfp_associates_GH_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associates_GH_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associates_GH_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associates_GH_productcoefficientscoefficientdiagonalterm)) * gc)) /\\ exists ff_q_pfp_associates_GH_productcoefficientscoefficientdiagonaltermrightentry. gb = ff_q_pfp_associates_GH_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associates_GH_productcoefficientscoefficientdiagonalterm)) * gc) + (pfc_right_associates_GH_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associates_GH_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associates_GH_productcoefficientscoefficientdiagonaltermrightoutside+(G)=(pfc_complement_associates_GH_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associates_GH_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associates_GH_productcoefficientscoefficientdiagonal)=pfc_left_associates_GH_productcoefficientscoefficientdiagonalterm*pfc_right_associates_GH_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associates_GH_productcoefficientscoefficientsum fs_v_pfc_associates_GH_productcoefficientscoefficientsum. ((((exists fs_h_pfc_associates_GH_productcoefficientscoefficientsum_body_start. fs_h_pfc_associates_GH_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associates_GH_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_GH_productcoefficientscoefficientsum_body_start. fs_u_pfc_associates_GH_productcoefficientscoefficientsum = fs_q_pfc_associates_GH_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associates_GH_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associates_GH_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_associates_GH_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associates_GH_productcoefficientscoefficient) = S ((S (S (pfc_index_associates_GH_productcoefficients))) * fs_v_pfc_associates_GH_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_GH_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_associates_GH_productcoefficientscoefficientsum = fs_q_pfc_associates_GH_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associates_GH_productcoefficients))) * fs_v_pfc_associates_GH_productcoefficientscoefficientsum) + (pfc_natural_sum_associates_GH_productcoefficientscoefficient))) /\\ forall fs_i_pfc_associates_GH_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associates_GH_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associates_GH_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associates_GH_productcoefficientscoefficientsum_body_steps = S (pfc_index_associates_GH_productcoefficients)) -> exists fs_a_pfc_associates_GH_productcoefficientscoefficientsum_body_steps fs_r_pfc_associates_GH_productcoefficientscoefficientsum_body_steps fs_s_pfc_associates_GH_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associates_GH_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associates_GH_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associates_GH_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associates_GH_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associates_GH_productcoefficientscoefficient)) /\\ exists fs_q_pfc_associates_GH_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associates_GH_productcoefficientscoefficient = fs_q_pfc_associates_GH_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associates_GH_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associates_GH_productcoefficientscoefficient) + (fs_a_pfc_associates_GH_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associates_GH_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associates_GH_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associates_GH_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associates_GH_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_GH_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_GH_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associates_GH_productcoefficientscoefficientsum = fs_q_pfc_associates_GH_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associates_GH_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_GH_productcoefficientscoefficientsum) + (fs_r_pfc_associates_GH_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associates_GH_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associates_GH_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associates_GH_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associates_GH_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_GH_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_GH_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associates_GH_productcoefficientscoefficientsum = fs_q_pfc_associates_GH_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associates_GH_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_GH_productcoefficientscoefficientsum) + (fs_s_pfc_associates_GH_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associates_GH_productcoefficientscoefficientsum_body_steps = fs_r_pfc_associates_GH_productcoefficientscoefficientsum_body_steps + fs_a_pfc_associates_GH_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associates_GH_productcoefficientscoefficientresiduebound. pfa_gap_associates_GH_productcoefficientscoefficientresiduebound + S (pfc_value_associates_GH_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_associates_GH_productcoefficientscoefficientresiduecongruence pfa_offset_right_associates_GH_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_associates_GH_productcoefficientscoefficient) + (p) * pfa_offset_left_associates_GH_productcoefficientscoefficientresiduecongruence = (pfc_value_associates_GH_productcoefficients) + (p) * pfa_offset_right_associates_GH_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_associates_GH_target pfrep_left_associates_GH_target pfrep_right_associates_GH_target. ((exists pfrep_position_associates_GH_targetfirst. ((pfrep_position_associates_GH_targetfirst+S (pfrep_power_associates_GH_target)=(pfgu_P_associates_GH)) /\\ ((((exists ff_h_pfp_associates_GH_targetfirstentry. ff_h_pfp_associates_GH_targetfirstentry + S (pfrep_left_associates_GH_target) = S ((S (pfrep_position_associates_GH_targetfirst)) * pfgu_pc_associates_GH)) /\\ exists ff_q_pfp_associates_GH_targetfirstentry. pfgu_pb_associates_GH = ff_q_pfp_associates_GH_targetfirstentry * S ((S (pfrep_position_associates_GH_targetfirst)) * pfgu_pc_associates_GH) + (pfrep_left_associates_GH_target)))))) \\/ (((exists pfrep_gap_associates_GH_targetfirstoutside. pfrep_gap_associates_GH_targetfirstoutside+(pfgu_P_associates_GH)=(pfrep_power_associates_GH_target)) /\\ (((pfrep_left_associates_GH_target)=0))))) -> ((exists pfrep_position_associates_GH_targetsecond. ((pfrep_position_associates_GH_targetsecond+S (pfrep_power_associates_GH_target)=(H)) /\\ ((((exists ff_h_pfp_associates_GH_targetsecondentry. ff_h_pfp_associates_GH_targetsecondentry + S (pfrep_right_associates_GH_target) = S ((S (pfrep_position_associates_GH_targetsecond)) * hc)) /\\ exists ff_q_pfp_associates_GH_targetsecondentry. hb = ff_q_pfp_associates_GH_targetsecondentry * S ((S (pfrep_position_associates_GH_targetsecond)) * hc) + (pfrep_right_associates_GH_target)))))) \\/ (((exists pfrep_gap_associates_GH_targetsecondoutside. pfrep_gap_associates_GH_targetsecondoutside+(H)=(pfrep_power_associates_GH_target)) /\\ (((pfrep_right_associates_GH_target)=0))))) -> pfrep_left_associates_GH_target=pfrep_right_associates_GH_target))))))) -> (((forall fom_index_pfp_associates_HG_canonical. (exists fom_gap_pfp_associates_HG_canonical_index_bound. fom_gap_pfp_associates_HG_canonical_index_bound + S (fom_index_pfp_associates_HG_canonical) = G) -> exists fom_value_pfp_associates_HG_canonical. ((((exists fom_beta_height_pfp_associates_HG_canonical_entry. fom_beta_height_pfp_associates_HG_canonical_entry + S (fom_value_pfp_associates_HG_canonical) = S ((S (fom_index_pfp_associates_HG_canonical)) * gc)) /\\ exists fom_beta_quotient_pfp_associates_HG_canonical_entry. gb = fom_beta_quotient_pfp_associates_HG_canonical_entry * S ((S (fom_index_pfp_associates_HG_canonical)) * gc) + (fom_value_pfp_associates_HG_canonical))) /\\ (exists fom_gap_pfp_associates_HG_canonical_value_bound. fom_gap_pfp_associates_HG_canonical_value_bound + S (fom_value_pfp_associates_HG_canonical) = p))) /\\ ((exists pfgu_qb_associates_HG pfgu_qc_associates_HG pfgu_Q_associates_HG pfgu_pb_associates_HG pfgu_pc_associates_HG pfgu_P_associates_HG. ((((forall fom_index_pfp_associates_HG_productleft. (exists fom_gap_pfp_associates_HG_productleft_index_bound. fom_gap_pfp_associates_HG_productleft_index_bound + S (fom_index_pfp_associates_HG_productleft) = pfgu_Q_associates_HG) -> exists fom_value_pfp_associates_HG_productleft. ((((exists fom_beta_height_pfp_associates_HG_productleft_entry. fom_beta_height_pfp_associates_HG_productleft_entry + S (fom_value_pfp_associates_HG_productleft) = S ((S (fom_index_pfp_associates_HG_productleft)) * pfgu_qc_associates_HG)) /\\ exists fom_beta_quotient_pfp_associates_HG_productleft_entry. pfgu_qb_associates_HG = fom_beta_quotient_pfp_associates_HG_productleft_entry * S ((S (fom_index_pfp_associates_HG_productleft)) * pfgu_qc_associates_HG) + (fom_value_pfp_associates_HG_productleft))) /\\ (exists fom_gap_pfp_associates_HG_productleft_value_bound. fom_gap_pfp_associates_HG_productleft_value_bound + S (fom_value_pfp_associates_HG_productleft) = p))) /\\ (((forall fom_index_pfp_associates_HG_productright. (exists fom_gap_pfp_associates_HG_productright_index_bound. fom_gap_pfp_associates_HG_productright_index_bound + S (fom_index_pfp_associates_HG_productright) = H) -> exists fom_value_pfp_associates_HG_productright. ((((exists fom_beta_height_pfp_associates_HG_productright_entry. fom_beta_height_pfp_associates_HG_productright_entry + S (fom_value_pfp_associates_HG_productright) = S ((S (fom_index_pfp_associates_HG_productright)) * hc)) /\\ exists fom_beta_quotient_pfp_associates_HG_productright_entry. hb = fom_beta_quotient_pfp_associates_HG_productright_entry * S ((S (fom_index_pfp_associates_HG_productright)) * hc) + (fom_value_pfp_associates_HG_productright))) /\\ (exists fom_gap_pfp_associates_HG_productright_value_bound. fom_gap_pfp_associates_HG_productright_value_bound + S (fom_value_pfp_associates_HG_productright) = p))) /\\ (((((((pfgu_Q_associates_HG)=0 \\/ (H)=0) /\\ (((pfgu_P_associates_HG)=0)))) \\/ (((~((pfgu_Q_associates_HG)=0)) /\\ (((~((H)=0)) /\\ (((pfgu_Q_associates_HG)+(H)=S (pfgu_P_associates_HG)))))))) /\\ ((forall pfc_index_associates_HG_productcoefficients. (exists pfa_gap_associates_HG_productcoefficientsbound. pfa_gap_associates_HG_productcoefficientsbound + S (pfc_index_associates_HG_productcoefficients) = (pfgu_P_associates_HG)) -> exists pfc_value_associates_HG_productcoefficients. ((((exists ff_h_pfp_associates_HG_productcoefficientsentry. ff_h_pfp_associates_HG_productcoefficientsentry + S (pfc_value_associates_HG_productcoefficients) = S ((S (pfc_index_associates_HG_productcoefficients)) * pfgu_pc_associates_HG)) /\\ exists ff_q_pfp_associates_HG_productcoefficientsentry. pfgu_pb_associates_HG = ff_q_pfp_associates_HG_productcoefficientsentry * S ((S (pfc_index_associates_HG_productcoefficients)) * pfgu_pc_associates_HG) + (pfc_value_associates_HG_productcoefficients))) /\\ ((exists pfc_terms_code_associates_HG_productcoefficientscoefficient pfc_terms_scale_associates_HG_productcoefficientscoefficient pfc_natural_sum_associates_HG_productcoefficientscoefficient. ((forall pfc_index_associates_HG_productcoefficientscoefficientdiagonal. (exists pfa_gap_associates_HG_productcoefficientscoefficientdiagonalbound. pfa_gap_associates_HG_productcoefficientscoefficientdiagonalbound + S (pfc_index_associates_HG_productcoefficientscoefficientdiagonal) = (S (pfc_index_associates_HG_productcoefficients))) -> exists pfc_value_associates_HG_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_associates_HG_productcoefficientscoefficientdiagonalentry. ff_h_pfp_associates_HG_productcoefficientscoefficientdiagonalentry + S (pfc_value_associates_HG_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_associates_HG_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_associates_HG_productcoefficientscoefficient)) /\\ exists ff_q_pfp_associates_HG_productcoefficientscoefficientdiagonalentry. pfc_terms_code_associates_HG_productcoefficientscoefficient = ff_q_pfp_associates_HG_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_associates_HG_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_associates_HG_productcoefficientscoefficient) + (pfc_value_associates_HG_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_associates_HG_productcoefficientscoefficientdiagonalterm pfc_left_associates_HG_productcoefficientscoefficientdiagonalterm pfc_right_associates_HG_productcoefficientscoefficientdiagonalterm. (((pfc_index_associates_HG_productcoefficientscoefficientdiagonal)+pfc_complement_associates_HG_productcoefficientscoefficientdiagonalterm=(pfc_index_associates_HG_productcoefficients)) /\\ ((((((exists pfa_gap_associates_HG_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_associates_HG_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_associates_HG_productcoefficientscoefficientdiagonal) = (pfgu_Q_associates_HG)) /\\ ((((exists ff_h_pfp_associates_HG_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_associates_HG_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_associates_HG_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_associates_HG_productcoefficientscoefficientdiagonal)) * pfgu_qc_associates_HG)) /\\ exists ff_q_pfp_associates_HG_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_associates_HG = ff_q_pfp_associates_HG_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_associates_HG_productcoefficientscoefficientdiagonal)) * pfgu_qc_associates_HG) + (pfc_left_associates_HG_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associates_HG_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_associates_HG_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_associates_HG)=(pfc_index_associates_HG_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_associates_HG_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_associates_HG_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_associates_HG_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_associates_HG_productcoefficientscoefficientdiagonalterm) = (H)) /\\ ((((exists ff_h_pfp_associates_HG_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_associates_HG_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_associates_HG_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_associates_HG_productcoefficientscoefficientdiagonalterm)) * hc)) /\\ exists ff_q_pfp_associates_HG_productcoefficientscoefficientdiagonaltermrightentry. hb = ff_q_pfp_associates_HG_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_associates_HG_productcoefficientscoefficientdiagonalterm)) * hc) + (pfc_right_associates_HG_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_associates_HG_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_associates_HG_productcoefficientscoefficientdiagonaltermrightoutside+(H)=(pfc_complement_associates_HG_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_associates_HG_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_associates_HG_productcoefficientscoefficientdiagonal)=pfc_left_associates_HG_productcoefficientscoefficientdiagonalterm*pfc_right_associates_HG_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_associates_HG_productcoefficientscoefficientsum fs_v_pfc_associates_HG_productcoefficientscoefficientsum. ((((exists fs_h_pfc_associates_HG_productcoefficientscoefficientsum_body_start. fs_h_pfc_associates_HG_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_associates_HG_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_HG_productcoefficientscoefficientsum_body_start. fs_u_pfc_associates_HG_productcoefficientscoefficientsum = fs_q_pfc_associates_HG_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_associates_HG_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_associates_HG_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_associates_HG_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_associates_HG_productcoefficientscoefficient) = S ((S (S (pfc_index_associates_HG_productcoefficients))) * fs_v_pfc_associates_HG_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_HG_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_associates_HG_productcoefficientscoefficientsum = fs_q_pfc_associates_HG_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_associates_HG_productcoefficients))) * fs_v_pfc_associates_HG_productcoefficientscoefficientsum) + (pfc_natural_sum_associates_HG_productcoefficientscoefficient))) /\\ forall fs_i_pfc_associates_HG_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_associates_HG_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_associates_HG_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_associates_HG_productcoefficientscoefficientsum_body_steps = S (pfc_index_associates_HG_productcoefficients)) -> exists fs_a_pfc_associates_HG_productcoefficientscoefficientsum_body_steps fs_r_pfc_associates_HG_productcoefficientscoefficientsum_body_steps fs_s_pfc_associates_HG_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_associates_HG_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_associates_HG_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_associates_HG_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associates_HG_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associates_HG_productcoefficientscoefficient)) /\\ exists fs_q_pfc_associates_HG_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_associates_HG_productcoefficientscoefficient = fs_q_pfc_associates_HG_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_associates_HG_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_associates_HG_productcoefficientscoefficient) + (fs_a_pfc_associates_HG_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associates_HG_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_associates_HG_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_associates_HG_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_associates_HG_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_HG_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_HG_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_associates_HG_productcoefficientscoefficientsum = fs_q_pfc_associates_HG_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_associates_HG_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_HG_productcoefficientscoefficientsum) + (fs_r_pfc_associates_HG_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_associates_HG_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_associates_HG_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_associates_HG_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_associates_HG_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_HG_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_associates_HG_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_associates_HG_productcoefficientscoefficientsum = fs_q_pfc_associates_HG_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_associates_HG_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_associates_HG_productcoefficientscoefficientsum) + (fs_s_pfc_associates_HG_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_associates_HG_productcoefficientscoefficientsum_body_steps = fs_r_pfc_associates_HG_productcoefficientscoefficientsum_body_steps + fs_a_pfc_associates_HG_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_associates_HG_productcoefficientscoefficientresiduebound. pfa_gap_associates_HG_productcoefficientscoefficientresiduebound + S (pfc_value_associates_HG_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_associates_HG_productcoefficientscoefficientresiduecongruence pfa_offset_right_associates_HG_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_associates_HG_productcoefficientscoefficient) + (p) * pfa_offset_left_associates_HG_productcoefficientscoefficientresiduecongruence = (pfc_value_associates_HG_productcoefficients) + (p) * pfa_offset_right_associates_HG_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_associates_HG_target pfrep_left_associates_HG_target pfrep_right_associates_HG_target. ((exists pfrep_position_associates_HG_targetfirst. ((pfrep_position_associates_HG_targetfirst+S (pfrep_power_associates_HG_target)=(pfgu_P_associates_HG)) /\\ ((((exists ff_h_pfp_associates_HG_targetfirstentry. ff_h_pfp_associates_HG_targetfirstentry + S (pfrep_left_associates_HG_target) = S ((S (pfrep_position_associates_HG_targetfirst)) * pfgu_pc_associates_HG)) /\\ exists ff_q_pfp_associates_HG_targetfirstentry. pfgu_pb_associates_HG = ff_q_pfp_associates_HG_targetfirstentry * S ((S (pfrep_position_associates_HG_targetfirst)) * pfgu_pc_associates_HG) + (pfrep_left_associates_HG_target)))))) \\/ (((exists pfrep_gap_associates_HG_targetfirstoutside. pfrep_gap_associates_HG_targetfirstoutside+(pfgu_P_associates_HG)=(pfrep_power_associates_HG_target)) /\\ (((pfrep_left_associates_HG_target)=0))))) -> ((exists pfrep_position_associates_HG_targetsecond. ((pfrep_position_associates_HG_targetsecond+S (pfrep_power_associates_HG_target)=(G)) /\\ ((((exists ff_h_pfp_associates_HG_targetsecondentry. ff_h_pfp_associates_HG_targetsecondentry + S (pfrep_right_associates_HG_target) = S ((S (pfrep_position_associates_HG_targetsecond)) * gc)) /\\ exists ff_q_pfp_associates_HG_targetsecondentry. gb = ff_q_pfp_associates_HG_targetsecondentry * S ((S (pfrep_position_associates_HG_targetsecond)) * gc) + (pfrep_right_associates_HG_target)))))) \\/ (((exists pfrep_gap_associates_HG_targetsecondoutside. pfrep_gap_associates_HG_targetsecondoutside+(G)=(pfrep_power_associates_HG_target)) /\\ (((pfrep_right_associates_HG_target)=0))))) -> pfrep_left_associates_HG_target=pfrep_right_associates_HG_target))))))) -> (forall pfrep_power_associates_result pfrep_left_associates_result pfrep_right_associates_result. ((exists pfrep_position_associates_resultfirst. ((pfrep_position_associates_resultfirst+S (pfrep_power_associates_result)=(G)) /\\ ((((exists ff_h_pfp_associates_resultfirstentry. ff_h_pfp_associates_resultfirstentry + S (pfrep_left_associates_result) = S ((S (pfrep_position_associates_resultfirst)) * gc)) /\\ exists ff_q_pfp_associates_resultfirstentry. gb = ff_q_pfp_associates_resultfirstentry * S ((S (pfrep_position_associates_resultfirst)) * gc) + (pfrep_left_associates_result)))))) \\/ (((exists pfrep_gap_associates_resultfirstoutside. pfrep_gap_associates_resultfirstoutside+(G)=(pfrep_power_associates_result)) /\\ (((pfrep_left_associates_result)=0))))) -> ((exists pfrep_position_associates_resultsecond. ((pfrep_position_associates_resultsecond+S (pfrep_power_associates_result)=(H)) /\\ ((((exists ff_h_pfp_associates_resultsecondentry. ff_h_pfp_associates_resultsecondentry + S (pfrep_right_associates_result) = S ((S (pfrep_position_associates_resultsecond)) * hc)) /\\ exists ff_q_pfp_associates_resultsecondentry. hb = ff_q_pfp_associates_resultsecondentry * S ((S (pfrep_position_associates_resultsecond)) * hc) + (pfrep_right_associates_result)))))) \\/ (((exists pfrep_gap_associates_resultsecondoutside. pfrep_gap_associates_resultsecondoutside+(H)=(pfrep_power_associates_result)) /\\ (((pfrep_right_associates_result)=0))))) -> pfrep_left_associates_result=pfrep_right_associates_result)", "statement_sha256": "b75140b759eeada9689d42997559c207b51954b445ddc0c3f8d9962c5a8892f0", "summary": "Mutual right divisibility forces equal represented degrees. 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(0)=0) /\\ (((x5)=0)))) \\/ (((~((x2)=0)) /\\ (((~((0)=0)) /\\ (((x2)+(0)=S (x5)))))))", "cases hrd_right_witness_witness_witness_witness_witness_witness_left", "cases hrd_right_witness_witness_witness_witness_witness_witness_left_right", "cases hrd_right_witness_witness_witness_witness_witness_witness_left_right_right", "exact hrd_right_witness_witness_witness_witness_witness_witness_left_right_right_left", "cases hlength", "cases hlength_left", "specialize prime_field_polynomial_equivalent_transitive (ab)", "specialize prime_field_polynomial_equivalent_transitive (ac)", "specialize prime_field_polynomial_equivalent_transitive (L)", "specialize prime_field_polynomial_equivalent_transitive (x3)", "specialize prime_field_polynomial_equivalent_transitive (x4)", "specialize prime_field_polynomial_equivalent_transitive (0)", "specialize prime_field_polynomial_equivalent_transitive (db)", "specialize prime_field_polynomial_equivalent_transitive (dc)", "specialize prime_field_polynomial_equivalent_transitive (0)", "apply prime_field_polynomial_equivalent_transitive", "specialize prime_field_polynomial_equivalent_symmetric (x3)", "specialize prime_field_polynomial_equivalent_symmetric (x4)", "specialize prime_field_polynomial_equivalent_symmetric (0)", "specialize prime_field_polynomial_equivalent_symmetric (ab)", "specialize prime_field_polynomial_equivalent_symmetric (ac)", "specialize prime_field_polynomial_equivalent_symmetric (L)", "apply prime_field_polynomial_equivalent_symmetric", "have hcopy : forall pfrep_power_empty_copy pfrep_left_empty_copy pfrep_right_empty_copy. ((exists pfrep_position_empty_copyfirst. ((pfrep_position_empty_copyfirst+S (pfrep_power_empty_copy)=(x5)) /\\ ((((exists ff_h_pfp_empty_copyfirstentry. ff_h_pfp_empty_copyfirstentry + S (pfrep_left_empty_copy) = S ((S (pfrep_position_empty_copyfirst)) * x4)) /\\ exists ff_q_pfp_empty_copyfirstentry. x3 = ff_q_pfp_empty_copyfirstentry * S ((S (pfrep_position_empty_copyfirst)) * x4) + (pfrep_left_empty_copy)))))) \\/ (((exists pfrep_gap_empty_copyfirstoutside. pfrep_gap_empty_copyfirstoutside+(x5)=(pfrep_power_empty_copy)) /\\ (((pfrep_left_empty_copy)=0))))) -> ((exists pfrep_position_empty_copysecond. ((pfrep_position_empty_copysecond+S (pfrep_power_empty_copy)=(L)) /\\ ((((exists ff_h_pfp_empty_copysecondentry. ff_h_pfp_empty_copysecondentry + S (pfrep_right_empty_copy) = S ((S (pfrep_position_empty_copysecond)) * ac)) /\\ exists ff_q_pfp_empty_copysecondentry. ab = ff_q_pfp_empty_copysecondentry * S ((S (pfrep_position_empty_copysecond)) * ac) + (pfrep_right_empty_copy)))))) \\/ (((exists pfrep_gap_empty_copysecondoutside. pfrep_gap_empty_copysecondoutside+(L)=(pfrep_power_empty_copy)) /\\ (((pfrep_right_empty_copy)=0))))) -> pfrep_left_empty_copy=pfrep_right_empty_copy", "exact hrd_right_witness_witness_witness_witness_witness_witness_right", "rewrite hlength_left_right at hcopy", "rewrite hlength_left_right at hcopy", "exact hcopy", "specialize prime_field_polynomial_equal_implies_equivalent (x3)", "specialize prime_field_polynomial_equal_implies_equivalent (x4)", "specialize prime_field_polynomial_equal_implies_equivalent (db)", "specialize prime_field_polynomial_equal_implies_equivalent (dc)", "specialize prime_field_polynomial_equal_implies_equivalent (0)", "apply prime_field_polynomial_equal_implies_equivalent", "intro i", "intro a", "intro hi", "intro ha", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hi", "specialize zero_le (i)", "apply zero_le", "cases hlength_right", "cases hlength_right_right", "exfalso", "apply hlength_right_right_left", "refl" ], "script_sha256": "d99f405effb7868e4f18d82023c51277923a89b745e0ddcd3aa6c2e5a82a98fb", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_uniqueness_candidate.py", "sha256": "916c24ad6c59609612e97daee6e49347a9522cdb28b44f6f09c6c5760bff0b5b" }, "statement": "forall p db dc ab ac L. (((forall fom_index_pfp_empty_divisor_canonical. (exists fom_gap_pfp_empty_divisor_canonical_index_bound. fom_gap_pfp_empty_divisor_canonical_index_bound + S (fom_index_pfp_empty_divisor_canonical) = L) -> exists fom_value_pfp_empty_divisor_canonical. ((((exists fom_beta_height_pfp_empty_divisor_canonical_entry. fom_beta_height_pfp_empty_divisor_canonical_entry + S (fom_value_pfp_empty_divisor_canonical) = S ((S (fom_index_pfp_empty_divisor_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_empty_divisor_canonical_entry. ab = fom_beta_quotient_pfp_empty_divisor_canonical_entry * S ((S (fom_index_pfp_empty_divisor_canonical)) * ac) + (fom_value_pfp_empty_divisor_canonical))) /\\ (exists fom_gap_pfp_empty_divisor_canonical_value_bound. fom_gap_pfp_empty_divisor_canonical_value_bound + S (fom_value_pfp_empty_divisor_canonical) = p))) /\\ ((exists pfgu_qb_empty_divisor pfgu_qc_empty_divisor pfgu_Q_empty_divisor pfgu_pb_empty_divisor pfgu_pc_empty_divisor pfgu_P_empty_divisor. ((((forall fom_index_pfp_empty_divisor_productleft. (exists fom_gap_pfp_empty_divisor_productleft_index_bound. fom_gap_pfp_empty_divisor_productleft_index_bound + S (fom_index_pfp_empty_divisor_productleft) = pfgu_Q_empty_divisor) -> exists fom_value_pfp_empty_divisor_productleft. ((((exists fom_beta_height_pfp_empty_divisor_productleft_entry. fom_beta_height_pfp_empty_divisor_productleft_entry + S (fom_value_pfp_empty_divisor_productleft) = S ((S (fom_index_pfp_empty_divisor_productleft)) * pfgu_qc_empty_divisor)) /\\ exists fom_beta_quotient_pfp_empty_divisor_productleft_entry. pfgu_qb_empty_divisor = fom_beta_quotient_pfp_empty_divisor_productleft_entry * S ((S (fom_index_pfp_empty_divisor_productleft)) * pfgu_qc_empty_divisor) + (fom_value_pfp_empty_divisor_productleft))) /\\ (exists fom_gap_pfp_empty_divisor_productleft_value_bound. fom_gap_pfp_empty_divisor_productleft_value_bound + S (fom_value_pfp_empty_divisor_productleft) = p))) /\\ (((forall fom_index_pfp_empty_divisor_productright. (exists fom_gap_pfp_empty_divisor_productright_index_bound. fom_gap_pfp_empty_divisor_productright_index_bound + S (fom_index_pfp_empty_divisor_productright) = 0) -> exists fom_value_pfp_empty_divisor_productright. ((((exists fom_beta_height_pfp_empty_divisor_productright_entry. fom_beta_height_pfp_empty_divisor_productright_entry + S (fom_value_pfp_empty_divisor_productright) = S ((S (fom_index_pfp_empty_divisor_productright)) * dc)) /\\ exists fom_beta_quotient_pfp_empty_divisor_productright_entry. db = fom_beta_quotient_pfp_empty_divisor_productright_entry * S ((S (fom_index_pfp_empty_divisor_productright)) * dc) + (fom_value_pfp_empty_divisor_productright))) /\\ (exists fom_gap_pfp_empty_divisor_productright_value_bound. fom_gap_pfp_empty_divisor_productright_value_bound + S (fom_value_pfp_empty_divisor_productright) = p))) /\\ (((((((pfgu_Q_empty_divisor)=0 \\/ (0)=0) /\\ (((pfgu_P_empty_divisor)=0)))) \\/ (((~((pfgu_Q_empty_divisor)=0)) /\\ (((~((0)=0)) /\\ (((pfgu_Q_empty_divisor)+(0)=S (pfgu_P_empty_divisor)))))))) /\\ ((forall pfc_index_empty_divisor_productcoefficients. (exists pfa_gap_empty_divisor_productcoefficientsbound. pfa_gap_empty_divisor_productcoefficientsbound + S (pfc_index_empty_divisor_productcoefficients) = (pfgu_P_empty_divisor)) -> exists pfc_value_empty_divisor_productcoefficients. ((((exists ff_h_pfp_empty_divisor_productcoefficientsentry. ff_h_pfp_empty_divisor_productcoefficientsentry + S (pfc_value_empty_divisor_productcoefficients) = S ((S (pfc_index_empty_divisor_productcoefficients)) * pfgu_pc_empty_divisor)) /\\ exists ff_q_pfp_empty_divisor_productcoefficientsentry. pfgu_pb_empty_divisor = ff_q_pfp_empty_divisor_productcoefficientsentry * S ((S (pfc_index_empty_divisor_productcoefficients)) * pfgu_pc_empty_divisor) + (pfc_value_empty_divisor_productcoefficients))) /\\ ((exists pfc_terms_code_empty_divisor_productcoefficientscoefficient pfc_terms_scale_empty_divisor_productcoefficientscoefficient pfc_natural_sum_empty_divisor_productcoefficientscoefficient. ((forall pfc_index_empty_divisor_productcoefficientscoefficientdiagonal. (exists pfa_gap_empty_divisor_productcoefficientscoefficientdiagonalbound. pfa_gap_empty_divisor_productcoefficientscoefficientdiagonalbound + S (pfc_index_empty_divisor_productcoefficientscoefficientdiagonal) = (S (pfc_index_empty_divisor_productcoefficients))) -> exists pfc_value_empty_divisor_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_empty_divisor_productcoefficientscoefficientdiagonalentry. ff_h_pfp_empty_divisor_productcoefficientscoefficientdiagonalentry + S (pfc_value_empty_divisor_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_empty_divisor_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_empty_divisor_productcoefficientscoefficient)) /\\ exists ff_q_pfp_empty_divisor_productcoefficientscoefficientdiagonalentry. pfc_terms_code_empty_divisor_productcoefficientscoefficient = ff_q_pfp_empty_divisor_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_empty_divisor_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_empty_divisor_productcoefficientscoefficient) + (pfc_value_empty_divisor_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_empty_divisor_productcoefficientscoefficientdiagonalterm pfc_left_empty_divisor_productcoefficientscoefficientdiagonalterm pfc_right_empty_divisor_productcoefficientscoefficientdiagonalterm. (((pfc_index_empty_divisor_productcoefficientscoefficientdiagonal)+pfc_complement_empty_divisor_productcoefficientscoefficientdiagonalterm=(pfc_index_empty_divisor_productcoefficients)) /\\ ((((((exists pfa_gap_empty_divisor_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_empty_divisor_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_empty_divisor_productcoefficientscoefficientdiagonal) = (pfgu_Q_empty_divisor)) /\\ ((((exists ff_h_pfp_empty_divisor_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_empty_divisor_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_empty_divisor_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_empty_divisor_productcoefficientscoefficientdiagonal)) * pfgu_qc_empty_divisor)) /\\ exists ff_q_pfp_empty_divisor_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_empty_divisor = ff_q_pfp_empty_divisor_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_empty_divisor_productcoefficientscoefficientdiagonal)) * pfgu_qc_empty_divisor) + (pfc_left_empty_divisor_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_empty_divisor_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_empty_divisor_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_empty_divisor)=(pfc_index_empty_divisor_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_empty_divisor_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_empty_divisor_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_empty_divisor_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_empty_divisor_productcoefficientscoefficientdiagonalterm) = (0)) /\\ ((((exists ff_h_pfp_empty_divisor_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_empty_divisor_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_empty_divisor_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_empty_divisor_productcoefficientscoefficientdiagonalterm)) * dc)) /\\ exists ff_q_pfp_empty_divisor_productcoefficientscoefficientdiagonaltermrightentry. db = ff_q_pfp_empty_divisor_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_empty_divisor_productcoefficientscoefficientdiagonalterm)) * dc) + (pfc_right_empty_divisor_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_empty_divisor_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_empty_divisor_productcoefficientscoefficientdiagonaltermrightoutside+(0)=(pfc_complement_empty_divisor_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_empty_divisor_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_empty_divisor_productcoefficientscoefficientdiagonal)=pfc_left_empty_divisor_productcoefficientscoefficientdiagonalterm*pfc_right_empty_divisor_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_empty_divisor_productcoefficientscoefficientsum fs_v_pfc_empty_divisor_productcoefficientscoefficientsum. ((((exists fs_h_pfc_empty_divisor_productcoefficientscoefficientsum_body_start. fs_h_pfc_empty_divisor_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_empty_divisor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_empty_divisor_productcoefficientscoefficientsum_body_start. fs_u_pfc_empty_divisor_productcoefficientscoefficientsum = fs_q_pfc_empty_divisor_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_empty_divisor_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_empty_divisor_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_empty_divisor_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_empty_divisor_productcoefficientscoefficient) = S ((S (S (pfc_index_empty_divisor_productcoefficients))) * fs_v_pfc_empty_divisor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_empty_divisor_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_empty_divisor_productcoefficientscoefficientsum = fs_q_pfc_empty_divisor_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_empty_divisor_productcoefficients))) * fs_v_pfc_empty_divisor_productcoefficientscoefficientsum) + (pfc_natural_sum_empty_divisor_productcoefficientscoefficient))) /\\ forall fs_i_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps = S (pfc_index_empty_divisor_productcoefficients)) -> exists fs_a_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps fs_r_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps fs_s_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_empty_divisor_productcoefficientscoefficient)) /\\ exists fs_q_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_empty_divisor_productcoefficientscoefficient = fs_q_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_empty_divisor_productcoefficientscoefficient) + (fs_a_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_empty_divisor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_empty_divisor_productcoefficientscoefficientsum = fs_q_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_empty_divisor_productcoefficientscoefficientsum) + (fs_r_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_empty_divisor_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_empty_divisor_productcoefficientscoefficientsum = fs_q_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_empty_divisor_productcoefficientscoefficientsum) + (fs_s_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps = fs_r_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps + fs_a_pfc_empty_divisor_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_empty_divisor_productcoefficientscoefficientresiduebound. pfa_gap_empty_divisor_productcoefficientscoefficientresiduebound + S (pfc_value_empty_divisor_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_empty_divisor_productcoefficientscoefficientresiduecongruence pfa_offset_right_empty_divisor_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_empty_divisor_productcoefficientscoefficient) + (p) * pfa_offset_left_empty_divisor_productcoefficientscoefficientresiduecongruence = (pfc_value_empty_divisor_productcoefficients) + (p) * pfa_offset_right_empty_divisor_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_empty_divisor_target pfrep_left_empty_divisor_target pfrep_right_empty_divisor_target. ((exists pfrep_position_empty_divisor_targetfirst. ((pfrep_position_empty_divisor_targetfirst+S (pfrep_power_empty_divisor_target)=(pfgu_P_empty_divisor)) /\\ ((((exists ff_h_pfp_empty_divisor_targetfirstentry. ff_h_pfp_empty_divisor_targetfirstentry + S (pfrep_left_empty_divisor_target) = S ((S (pfrep_position_empty_divisor_targetfirst)) * pfgu_pc_empty_divisor)) /\\ exists ff_q_pfp_empty_divisor_targetfirstentry. pfgu_pb_empty_divisor = ff_q_pfp_empty_divisor_targetfirstentry * S ((S (pfrep_position_empty_divisor_targetfirst)) * pfgu_pc_empty_divisor) + (pfrep_left_empty_divisor_target)))))) \\/ (((exists pfrep_gap_empty_divisor_targetfirstoutside. pfrep_gap_empty_divisor_targetfirstoutside+(pfgu_P_empty_divisor)=(pfrep_power_empty_divisor_target)) /\\ (((pfrep_left_empty_divisor_target)=0))))) -> ((exists pfrep_position_empty_divisor_targetsecond. ((pfrep_position_empty_divisor_targetsecond+S (pfrep_power_empty_divisor_target)=(L)) /\\ ((((exists ff_h_pfp_empty_divisor_targetsecondentry. ff_h_pfp_empty_divisor_targetsecondentry + S (pfrep_right_empty_divisor_target) = S ((S (pfrep_position_empty_divisor_targetsecond)) * ac)) /\\ exists ff_q_pfp_empty_divisor_targetsecondentry. ab = ff_q_pfp_empty_divisor_targetsecondentry * S ((S (pfrep_position_empty_divisor_targetsecond)) * ac) + (pfrep_right_empty_divisor_target)))))) \\/ (((exists pfrep_gap_empty_divisor_targetsecondoutside. pfrep_gap_empty_divisor_targetsecondoutside+(L)=(pfrep_power_empty_divisor_target)) /\\ (((pfrep_right_empty_divisor_target)=0))))) -> pfrep_left_empty_divisor_target=pfrep_right_empty_divisor_target))))))) -> (forall pfrep_power_empty_divisor_result pfrep_left_empty_divisor_result pfrep_right_empty_divisor_result. ((exists pfrep_position_empty_divisor_resultfirst. ((pfrep_position_empty_divisor_resultfirst+S (pfrep_power_empty_divisor_result)=(L)) /\\ ((((exists ff_h_pfp_empty_divisor_resultfirstentry. ff_h_pfp_empty_divisor_resultfirstentry + S (pfrep_left_empty_divisor_result) = S ((S (pfrep_position_empty_divisor_resultfirst)) * ac)) /\\ exists ff_q_pfp_empty_divisor_resultfirstentry. ab = ff_q_pfp_empty_divisor_resultfirstentry * S ((S (pfrep_position_empty_divisor_resultfirst)) * ac) + (pfrep_left_empty_divisor_result)))))) \\/ (((exists pfrep_gap_empty_divisor_resultfirstoutside. pfrep_gap_empty_divisor_resultfirstoutside+(L)=(pfrep_power_empty_divisor_result)) /\\ (((pfrep_left_empty_divisor_result)=0))))) -> ((exists pfrep_position_empty_divisor_resultsecond. ((pfrep_position_empty_divisor_resultsecond+S (pfrep_power_empty_divisor_result)=(0)) /\\ ((((exists ff_h_pfp_empty_divisor_resultsecondentry. ff_h_pfp_empty_divisor_resultsecondentry + S (pfrep_right_empty_divisor_result) = S ((S (pfrep_position_empty_divisor_resultsecond)) * dc)) /\\ exists ff_q_pfp_empty_divisor_resultsecondentry. db = ff_q_pfp_empty_divisor_resultsecondentry * S ((S (pfrep_position_empty_divisor_resultsecond)) * dc) + (pfrep_right_empty_divisor_result)))))) \\/ (((exists pfrep_gap_empty_divisor_resultsecondoutside. pfrep_gap_empty_divisor_resultsecondoutside+(0)=(pfrep_power_empty_divisor_result)) /\\ (((pfrep_right_empty_divisor_result)=0))))) -> pfrep_left_empty_divisor_result=pfrep_right_empty_divisor_result)", "statement_sha256": "aa3334bf9dfb6a31df12ced2b5f9b73b5b3def10a1b7b88aea709787185dc307", "summary": "An empty right divisor has only formally zero multiples, at any target representation length. 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"6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "polynomial-gcd-bezout", "logical_spec_sha256": "9bb53b122aeae78633e4f7f5020dabac4058658e72de562d48d7998c6ccf7f5f", "membership": "alpha_only", "name": "prime_field_polynomial_normal_right_associates_equivalent", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro gb", "intro gc", "intro G", "intro hb", "intro hc", "intro H", "intro hp", "intro hg", "intro hh", "intro hgh", "intro hhg", "cases hg", "rewrite hg_left", "rewrite hg_left", "specialize prime_field_polynomial_equivalent_symmetric (hb)", "specialize prime_field_polynomial_equivalent_symmetric (hc)", "specialize prime_field_polynomial_equivalent_symmetric (H)", "specialize prime_field_polynomial_equivalent_symmetric (gb)", "specialize prime_field_polynomial_equivalent_symmetric (gc)", "specialize prime_field_polynomial_equivalent_symmetric (0)", "apply prime_field_polynomial_equivalent_symmetric", "specialize prime_field_polynomial_empty_right_divisor_implies_equivalent_zero (p)", "specialize prime_field_polynomial_empty_right_divisor_implies_equivalent_zero (gb)", "specialize prime_field_polynomial_empty_right_divisor_implies_equivalent_zero (gc)", "specialize prime_field_polynomial_empty_right_divisor_implies_equivalent_zero (hb)", "specialize prime_field_polynomial_empty_right_divisor_implies_equivalent_zero (hc)", "specialize prime_field_polynomial_empty_right_divisor_implies_equivalent_zero (H)", "apply prime_field_polynomial_empty_right_divisor_implies_equivalent_zero", "have hcopy : ((forall fom_index_pfp_normal_empty_G_canonical. (exists fom_gap_pfp_normal_empty_G_canonical_index_bound. fom_gap_pfp_normal_empty_G_canonical_index_bound + S (fom_index_pfp_normal_empty_G_canonical) = H) -> exists fom_value_pfp_normal_empty_G_canonical. ((((exists fom_beta_height_pfp_normal_empty_G_canonical_entry. fom_beta_height_pfp_normal_empty_G_canonical_entry + S (fom_value_pfp_normal_empty_G_canonical) = S ((S (fom_index_pfp_normal_empty_G_canonical)) * hc)) /\\ exists fom_beta_quotient_pfp_normal_empty_G_canonical_entry. hb = fom_beta_quotient_pfp_normal_empty_G_canonical_entry * S ((S (fom_index_pfp_normal_empty_G_canonical)) * hc) + (fom_value_pfp_normal_empty_G_canonical))) /\\ (exists fom_gap_pfp_normal_empty_G_canonical_value_bound. fom_gap_pfp_normal_empty_G_canonical_value_bound + S (fom_value_pfp_normal_empty_G_canonical) = p))) /\\ ((exists pfgu_qb_normal_empty_G pfgu_qc_normal_empty_G pfgu_Q_normal_empty_G pfgu_pb_normal_empty_G pfgu_pc_normal_empty_G pfgu_P_normal_empty_G. ((((forall fom_index_pfp_normal_empty_G_productleft. (exists fom_gap_pfp_normal_empty_G_productleft_index_bound. fom_gap_pfp_normal_empty_G_productleft_index_bound + S (fom_index_pfp_normal_empty_G_productleft) = pfgu_Q_normal_empty_G) -> exists fom_value_pfp_normal_empty_G_productleft. ((((exists fom_beta_height_pfp_normal_empty_G_productleft_entry. fom_beta_height_pfp_normal_empty_G_productleft_entry + S (fom_value_pfp_normal_empty_G_productleft) = S ((S (fom_index_pfp_normal_empty_G_productleft)) * pfgu_qc_normal_empty_G)) /\\ exists fom_beta_quotient_pfp_normal_empty_G_productleft_entry. pfgu_qb_normal_empty_G = fom_beta_quotient_pfp_normal_empty_G_productleft_entry * S ((S (fom_index_pfp_normal_empty_G_productleft)) * pfgu_qc_normal_empty_G) + (fom_value_pfp_normal_empty_G_productleft))) /\\ (exists fom_gap_pfp_normal_empty_G_productleft_value_bound. fom_gap_pfp_normal_empty_G_productleft_value_bound + S (fom_value_pfp_normal_empty_G_productleft) = p))) /\\ (((forall fom_index_pfp_normal_empty_G_productright. (exists fom_gap_pfp_normal_empty_G_productright_index_bound. fom_gap_pfp_normal_empty_G_productright_index_bound + S (fom_index_pfp_normal_empty_G_productright) = G) -> exists fom_value_pfp_normal_empty_G_productright. ((((exists fom_beta_height_pfp_normal_empty_G_productright_entry. fom_beta_height_pfp_normal_empty_G_productright_entry + S (fom_value_pfp_normal_empty_G_productright) = S ((S (fom_index_pfp_normal_empty_G_productright)) * gc)) /\\ exists fom_beta_quotient_pfp_normal_empty_G_productright_entry. gb = fom_beta_quotient_pfp_normal_empty_G_productright_entry * S ((S (fom_index_pfp_normal_empty_G_productright)) * gc) + (fom_value_pfp_normal_empty_G_productright))) /\\ (exists fom_gap_pfp_normal_empty_G_productright_value_bound. fom_gap_pfp_normal_empty_G_productright_value_bound + S (fom_value_pfp_normal_empty_G_productright) = p))) /\\ (((((((pfgu_Q_normal_empty_G)=0 \\/ (G)=0) /\\ (((pfgu_P_normal_empty_G)=0)))) \\/ (((~((pfgu_Q_normal_empty_G)=0)) /\\ (((~((G)=0)) /\\ (((pfgu_Q_normal_empty_G)+(G)=S (pfgu_P_normal_empty_G)))))))) /\\ ((forall pfc_index_normal_empty_G_productcoefficients. (exists pfa_gap_normal_empty_G_productcoefficientsbound. pfa_gap_normal_empty_G_productcoefficientsbound + S (pfc_index_normal_empty_G_productcoefficients) = (pfgu_P_normal_empty_G)) -> exists pfc_value_normal_empty_G_productcoefficients. ((((exists ff_h_pfp_normal_empty_G_productcoefficientsentry. ff_h_pfp_normal_empty_G_productcoefficientsentry + S (pfc_value_normal_empty_G_productcoefficients) = S ((S (pfc_index_normal_empty_G_productcoefficients)) * pfgu_pc_normal_empty_G)) /\\ exists ff_q_pfp_normal_empty_G_productcoefficientsentry. pfgu_pb_normal_empty_G = ff_q_pfp_normal_empty_G_productcoefficientsentry * S ((S (pfc_index_normal_empty_G_productcoefficients)) * pfgu_pc_normal_empty_G) + (pfc_value_normal_empty_G_productcoefficients))) /\\ ((exists pfc_terms_code_normal_empty_G_productcoefficientscoefficient pfc_terms_scale_normal_empty_G_productcoefficientscoefficient pfc_natural_sum_normal_empty_G_productcoefficientscoefficient. ((forall pfc_index_normal_empty_G_productcoefficientscoefficientdiagonal. (exists pfa_gap_normal_empty_G_productcoefficientscoefficientdiagonalbound. pfa_gap_normal_empty_G_productcoefficientscoefficientdiagonalbound + S (pfc_index_normal_empty_G_productcoefficientscoefficientdiagonal) = (S (pfc_index_normal_empty_G_productcoefficients))) -> exists pfc_value_normal_empty_G_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normal_empty_G_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normal_empty_G_productcoefficientscoefficientdiagonalentry + S (pfc_value_normal_empty_G_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normal_empty_G_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normal_empty_G_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normal_empty_G_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normal_empty_G_productcoefficientscoefficient = ff_q_pfp_normal_empty_G_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normal_empty_G_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normal_empty_G_productcoefficientscoefficient) + (pfc_value_normal_empty_G_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normal_empty_G_productcoefficientscoefficientdiagonalterm pfc_left_normal_empty_G_productcoefficientscoefficientdiagonalterm pfc_right_normal_empty_G_productcoefficientscoefficientdiagonalterm. (((pfc_index_normal_empty_G_productcoefficientscoefficientdiagonal)+pfc_complement_normal_empty_G_productcoefficientscoefficientdiagonalterm=(pfc_index_normal_empty_G_productcoefficients)) /\\ ((((((exists pfa_gap_normal_empty_G_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normal_empty_G_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normal_empty_G_productcoefficientscoefficientdiagonal) = (pfgu_Q_normal_empty_G)) /\\ ((((exists ff_h_pfp_normal_empty_G_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normal_empty_G_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normal_empty_G_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normal_empty_G_productcoefficientscoefficientdiagonal)) * pfgu_qc_normal_empty_G)) /\\ exists ff_q_pfp_normal_empty_G_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_normal_empty_G = ff_q_pfp_normal_empty_G_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normal_empty_G_productcoefficientscoefficientdiagonal)) * pfgu_qc_normal_empty_G) + (pfc_left_normal_empty_G_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normal_empty_G_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normal_empty_G_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_normal_empty_G)=(pfc_index_normal_empty_G_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normal_empty_G_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normal_empty_G_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normal_empty_G_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normal_empty_G_productcoefficientscoefficientdiagonalterm) = (G)) /\\ ((((exists ff_h_pfp_normal_empty_G_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normal_empty_G_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normal_empty_G_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normal_empty_G_productcoefficientscoefficientdiagonalterm)) * gc)) /\\ exists ff_q_pfp_normal_empty_G_productcoefficientscoefficientdiagonaltermrightentry. gb = ff_q_pfp_normal_empty_G_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normal_empty_G_productcoefficientscoefficientdiagonalterm)) * gc) + (pfc_right_normal_empty_G_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normal_empty_G_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normal_empty_G_productcoefficientscoefficientdiagonaltermrightoutside+(G)=(pfc_complement_normal_empty_G_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normal_empty_G_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normal_empty_G_productcoefficientscoefficientdiagonal)=pfc_left_normal_empty_G_productcoefficientscoefficientdiagonalterm*pfc_right_normal_empty_G_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normal_empty_G_productcoefficientscoefficientsum fs_v_pfc_normal_empty_G_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normal_empty_G_productcoefficientscoefficientsum_body_start. fs_h_pfc_normal_empty_G_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normal_empty_G_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_empty_G_productcoefficientscoefficientsum_body_start. fs_u_pfc_normal_empty_G_productcoefficientscoefficientsum = fs_q_pfc_normal_empty_G_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normal_empty_G_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normal_empty_G_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normal_empty_G_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normal_empty_G_productcoefficientscoefficient) = S ((S (S (pfc_index_normal_empty_G_productcoefficients))) * fs_v_pfc_normal_empty_G_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_empty_G_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normal_empty_G_productcoefficientscoefficientsum = fs_q_pfc_normal_empty_G_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normal_empty_G_productcoefficients))) * fs_v_pfc_normal_empty_G_productcoefficientscoefficientsum) + (pfc_natural_sum_normal_empty_G_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps = S (pfc_index_normal_empty_G_productcoefficients)) -> exists fs_a_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps fs_r_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps fs_s_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normal_empty_G_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normal_empty_G_productcoefficientscoefficient = fs_q_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normal_empty_G_productcoefficientscoefficient) + (fs_a_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_empty_G_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normal_empty_G_productcoefficientscoefficientsum = fs_q_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_empty_G_productcoefficientscoefficientsum) + (fs_r_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_empty_G_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normal_empty_G_productcoefficientscoefficientsum = fs_q_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_empty_G_productcoefficientscoefficientsum) + (fs_s_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normal_empty_G_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normal_empty_G_productcoefficientscoefficientresiduebound. pfa_gap_normal_empty_G_productcoefficientscoefficientresiduebound + S (pfc_value_normal_empty_G_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normal_empty_G_productcoefficientscoefficientresiduecongruence pfa_offset_right_normal_empty_G_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normal_empty_G_productcoefficientscoefficient) + (p) * pfa_offset_left_normal_empty_G_productcoefficientscoefficientresiduecongruence = (pfc_value_normal_empty_G_productcoefficients) + (p) * pfa_offset_right_normal_empty_G_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normal_empty_G_target pfrep_left_normal_empty_G_target pfrep_right_normal_empty_G_target. ((exists pfrep_position_normal_empty_G_targetfirst. ((pfrep_position_normal_empty_G_targetfirst+S (pfrep_power_normal_empty_G_target)=(pfgu_P_normal_empty_G)) /\\ ((((exists ff_h_pfp_normal_empty_G_targetfirstentry. ff_h_pfp_normal_empty_G_targetfirstentry + S (pfrep_left_normal_empty_G_target) = S ((S (pfrep_position_normal_empty_G_targetfirst)) * pfgu_pc_normal_empty_G)) /\\ exists ff_q_pfp_normal_empty_G_targetfirstentry. pfgu_pb_normal_empty_G = ff_q_pfp_normal_empty_G_targetfirstentry * S ((S (pfrep_position_normal_empty_G_targetfirst)) * pfgu_pc_normal_empty_G) + (pfrep_left_normal_empty_G_target)))))) \\/ (((exists pfrep_gap_normal_empty_G_targetfirstoutside. pfrep_gap_normal_empty_G_targetfirstoutside+(pfgu_P_normal_empty_G)=(pfrep_power_normal_empty_G_target)) /\\ (((pfrep_left_normal_empty_G_target)=0))))) -> ((exists pfrep_position_normal_empty_G_targetsecond. ((pfrep_position_normal_empty_G_targetsecond+S (pfrep_power_normal_empty_G_target)=(H)) /\\ ((((exists ff_h_pfp_normal_empty_G_targetsecondentry. ff_h_pfp_normal_empty_G_targetsecondentry + S (pfrep_right_normal_empty_G_target) = S ((S (pfrep_position_normal_empty_G_targetsecond)) * hc)) /\\ exists ff_q_pfp_normal_empty_G_targetsecondentry. hb = ff_q_pfp_normal_empty_G_targetsecondentry * S ((S (pfrep_position_normal_empty_G_targetsecond)) * hc) + (pfrep_right_normal_empty_G_target)))))) \\/ (((exists pfrep_gap_normal_empty_G_targetsecondoutside. pfrep_gap_normal_empty_G_targetsecondoutside+(H)=(pfrep_power_normal_empty_G_target)) /\\ (((pfrep_right_normal_empty_G_target)=0))))) -> pfrep_left_normal_empty_G_target=pfrep_right_normal_empty_G_target))))))", "exact hgh", "rewrite hg_left at hcopy", "rewrite hg_left at hcopy", "rewrite hg_left at hcopy", "rewrite hg_left at hcopy", "rewrite hg_left at hcopy", "rewrite hg_left at hcopy", "exact hcopy", "cases hh", "rewrite hh_left", "rewrite hh_left", "specialize prime_field_polynomial_empty_right_divisor_implies_equivalent_zero (p)", "specialize prime_field_polynomial_empty_right_divisor_implies_equivalent_zero (hb)", "specialize prime_field_polynomial_empty_right_divisor_implies_equivalent_zero (hc)", "specialize prime_field_polynomial_empty_right_divisor_implies_equivalent_zero (gb)", "specialize prime_field_polynomial_empty_right_divisor_implies_equivalent_zero (gc)", "specialize prime_field_polynomial_empty_right_divisor_implies_equivalent_zero (G)", "apply prime_field_polynomial_empty_right_divisor_implies_equivalent_zero", "have hcopy : ((forall fom_index_pfp_normal_empty_H_canonical. (exists fom_gap_pfp_normal_empty_H_canonical_index_bound. fom_gap_pfp_normal_empty_H_canonical_index_bound + S (fom_index_pfp_normal_empty_H_canonical) = G) -> exists fom_value_pfp_normal_empty_H_canonical. ((((exists fom_beta_height_pfp_normal_empty_H_canonical_entry. fom_beta_height_pfp_normal_empty_H_canonical_entry + S (fom_value_pfp_normal_empty_H_canonical) = S ((S (fom_index_pfp_normal_empty_H_canonical)) * gc)) /\\ exists fom_beta_quotient_pfp_normal_empty_H_canonical_entry. gb = fom_beta_quotient_pfp_normal_empty_H_canonical_entry * S ((S (fom_index_pfp_normal_empty_H_canonical)) * gc) + (fom_value_pfp_normal_empty_H_canonical))) /\\ (exists fom_gap_pfp_normal_empty_H_canonical_value_bound. fom_gap_pfp_normal_empty_H_canonical_value_bound + S (fom_value_pfp_normal_empty_H_canonical) = p))) /\\ ((exists pfgu_qb_normal_empty_H pfgu_qc_normal_empty_H pfgu_Q_normal_empty_H pfgu_pb_normal_empty_H pfgu_pc_normal_empty_H pfgu_P_normal_empty_H. ((((forall fom_index_pfp_normal_empty_H_productleft. (exists fom_gap_pfp_normal_empty_H_productleft_index_bound. fom_gap_pfp_normal_empty_H_productleft_index_bound + S (fom_index_pfp_normal_empty_H_productleft) = pfgu_Q_normal_empty_H) -> exists fom_value_pfp_normal_empty_H_productleft. ((((exists fom_beta_height_pfp_normal_empty_H_productleft_entry. fom_beta_height_pfp_normal_empty_H_productleft_entry + S (fom_value_pfp_normal_empty_H_productleft) = S ((S (fom_index_pfp_normal_empty_H_productleft)) * pfgu_qc_normal_empty_H)) /\\ exists fom_beta_quotient_pfp_normal_empty_H_productleft_entry. pfgu_qb_normal_empty_H = fom_beta_quotient_pfp_normal_empty_H_productleft_entry * S ((S (fom_index_pfp_normal_empty_H_productleft)) * pfgu_qc_normal_empty_H) + (fom_value_pfp_normal_empty_H_productleft))) /\\ (exists fom_gap_pfp_normal_empty_H_productleft_value_bound. fom_gap_pfp_normal_empty_H_productleft_value_bound + S (fom_value_pfp_normal_empty_H_productleft) = p))) /\\ (((forall fom_index_pfp_normal_empty_H_productright. (exists fom_gap_pfp_normal_empty_H_productright_index_bound. fom_gap_pfp_normal_empty_H_productright_index_bound + S (fom_index_pfp_normal_empty_H_productright) = H) -> exists fom_value_pfp_normal_empty_H_productright. ((((exists fom_beta_height_pfp_normal_empty_H_productright_entry. fom_beta_height_pfp_normal_empty_H_productright_entry + S (fom_value_pfp_normal_empty_H_productright) = S ((S (fom_index_pfp_normal_empty_H_productright)) * hc)) /\\ exists fom_beta_quotient_pfp_normal_empty_H_productright_entry. hb = fom_beta_quotient_pfp_normal_empty_H_productright_entry * S ((S (fom_index_pfp_normal_empty_H_productright)) * hc) + (fom_value_pfp_normal_empty_H_productright))) /\\ (exists fom_gap_pfp_normal_empty_H_productright_value_bound. fom_gap_pfp_normal_empty_H_productright_value_bound + S (fom_value_pfp_normal_empty_H_productright) = p))) /\\ (((((((pfgu_Q_normal_empty_H)=0 \\/ (H)=0) /\\ (((pfgu_P_normal_empty_H)=0)))) \\/ (((~((pfgu_Q_normal_empty_H)=0)) /\\ (((~((H)=0)) /\\ (((pfgu_Q_normal_empty_H)+(H)=S (pfgu_P_normal_empty_H)))))))) /\\ ((forall pfc_index_normal_empty_H_productcoefficients. (exists pfa_gap_normal_empty_H_productcoefficientsbound. pfa_gap_normal_empty_H_productcoefficientsbound + S (pfc_index_normal_empty_H_productcoefficients) = (pfgu_P_normal_empty_H)) -> exists pfc_value_normal_empty_H_productcoefficients. ((((exists ff_h_pfp_normal_empty_H_productcoefficientsentry. ff_h_pfp_normal_empty_H_productcoefficientsentry + S (pfc_value_normal_empty_H_productcoefficients) = S ((S (pfc_index_normal_empty_H_productcoefficients)) * pfgu_pc_normal_empty_H)) /\\ exists ff_q_pfp_normal_empty_H_productcoefficientsentry. pfgu_pb_normal_empty_H = ff_q_pfp_normal_empty_H_productcoefficientsentry * S ((S (pfc_index_normal_empty_H_productcoefficients)) * pfgu_pc_normal_empty_H) + (pfc_value_normal_empty_H_productcoefficients))) /\\ ((exists pfc_terms_code_normal_empty_H_productcoefficientscoefficient pfc_terms_scale_normal_empty_H_productcoefficientscoefficient pfc_natural_sum_normal_empty_H_productcoefficientscoefficient. ((forall pfc_index_normal_empty_H_productcoefficientscoefficientdiagonal. (exists pfa_gap_normal_empty_H_productcoefficientscoefficientdiagonalbound. pfa_gap_normal_empty_H_productcoefficientscoefficientdiagonalbound + S (pfc_index_normal_empty_H_productcoefficientscoefficientdiagonal) = (S (pfc_index_normal_empty_H_productcoefficients))) -> exists pfc_value_normal_empty_H_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normal_empty_H_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normal_empty_H_productcoefficientscoefficientdiagonalentry + S (pfc_value_normal_empty_H_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normal_empty_H_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normal_empty_H_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normal_empty_H_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normal_empty_H_productcoefficientscoefficient = ff_q_pfp_normal_empty_H_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normal_empty_H_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normal_empty_H_productcoefficientscoefficient) + (pfc_value_normal_empty_H_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normal_empty_H_productcoefficientscoefficientdiagonalterm pfc_left_normal_empty_H_productcoefficientscoefficientdiagonalterm pfc_right_normal_empty_H_productcoefficientscoefficientdiagonalterm. (((pfc_index_normal_empty_H_productcoefficientscoefficientdiagonal)+pfc_complement_normal_empty_H_productcoefficientscoefficientdiagonalterm=(pfc_index_normal_empty_H_productcoefficients)) /\\ ((((((exists pfa_gap_normal_empty_H_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normal_empty_H_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normal_empty_H_productcoefficientscoefficientdiagonal) = (pfgu_Q_normal_empty_H)) /\\ ((((exists ff_h_pfp_normal_empty_H_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normal_empty_H_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normal_empty_H_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normal_empty_H_productcoefficientscoefficientdiagonal)) * pfgu_qc_normal_empty_H)) /\\ exists ff_q_pfp_normal_empty_H_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_normal_empty_H = ff_q_pfp_normal_empty_H_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normal_empty_H_productcoefficientscoefficientdiagonal)) * pfgu_qc_normal_empty_H) + (pfc_left_normal_empty_H_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normal_empty_H_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normal_empty_H_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_normal_empty_H)=(pfc_index_normal_empty_H_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normal_empty_H_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normal_empty_H_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normal_empty_H_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normal_empty_H_productcoefficientscoefficientdiagonalterm) = (H)) /\\ ((((exists ff_h_pfp_normal_empty_H_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normal_empty_H_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normal_empty_H_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normal_empty_H_productcoefficientscoefficientdiagonalterm)) * hc)) /\\ exists ff_q_pfp_normal_empty_H_productcoefficientscoefficientdiagonaltermrightentry. hb = ff_q_pfp_normal_empty_H_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normal_empty_H_productcoefficientscoefficientdiagonalterm)) * hc) + (pfc_right_normal_empty_H_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normal_empty_H_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normal_empty_H_productcoefficientscoefficientdiagonaltermrightoutside+(H)=(pfc_complement_normal_empty_H_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normal_empty_H_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normal_empty_H_productcoefficientscoefficientdiagonal)=pfc_left_normal_empty_H_productcoefficientscoefficientdiagonalterm*pfc_right_normal_empty_H_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normal_empty_H_productcoefficientscoefficientsum fs_v_pfc_normal_empty_H_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normal_empty_H_productcoefficientscoefficientsum_body_start. fs_h_pfc_normal_empty_H_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normal_empty_H_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_empty_H_productcoefficientscoefficientsum_body_start. fs_u_pfc_normal_empty_H_productcoefficientscoefficientsum = fs_q_pfc_normal_empty_H_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normal_empty_H_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normal_empty_H_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normal_empty_H_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normal_empty_H_productcoefficientscoefficient) = S ((S (S (pfc_index_normal_empty_H_productcoefficients))) * fs_v_pfc_normal_empty_H_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_empty_H_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normal_empty_H_productcoefficientscoefficientsum = fs_q_pfc_normal_empty_H_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normal_empty_H_productcoefficients))) * fs_v_pfc_normal_empty_H_productcoefficientscoefficientsum) + (pfc_natural_sum_normal_empty_H_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps = S (pfc_index_normal_empty_H_productcoefficients)) -> exists fs_a_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps fs_r_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps fs_s_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normal_empty_H_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normal_empty_H_productcoefficientscoefficient = fs_q_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normal_empty_H_productcoefficientscoefficient) + (fs_a_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_empty_H_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normal_empty_H_productcoefficientscoefficientsum = fs_q_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_empty_H_productcoefficientscoefficientsum) + (fs_r_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_empty_H_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normal_empty_H_productcoefficientscoefficientsum = fs_q_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_empty_H_productcoefficientscoefficientsum) + (fs_s_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normal_empty_H_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normal_empty_H_productcoefficientscoefficientresiduebound. pfa_gap_normal_empty_H_productcoefficientscoefficientresiduebound + S (pfc_value_normal_empty_H_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normal_empty_H_productcoefficientscoefficientresiduecongruence pfa_offset_right_normal_empty_H_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normal_empty_H_productcoefficientscoefficient) + (p) * pfa_offset_left_normal_empty_H_productcoefficientscoefficientresiduecongruence = (pfc_value_normal_empty_H_productcoefficients) + (p) * pfa_offset_right_normal_empty_H_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normal_empty_H_target pfrep_left_normal_empty_H_target pfrep_right_normal_empty_H_target. ((exists pfrep_position_normal_empty_H_targetfirst. ((pfrep_position_normal_empty_H_targetfirst+S (pfrep_power_normal_empty_H_target)=(pfgu_P_normal_empty_H)) /\\ ((((exists ff_h_pfp_normal_empty_H_targetfirstentry. ff_h_pfp_normal_empty_H_targetfirstentry + S (pfrep_left_normal_empty_H_target) = S ((S (pfrep_position_normal_empty_H_targetfirst)) * pfgu_pc_normal_empty_H)) /\\ exists ff_q_pfp_normal_empty_H_targetfirstentry. pfgu_pb_normal_empty_H = ff_q_pfp_normal_empty_H_targetfirstentry * S ((S (pfrep_position_normal_empty_H_targetfirst)) * pfgu_pc_normal_empty_H) + (pfrep_left_normal_empty_H_target)))))) \\/ (((exists pfrep_gap_normal_empty_H_targetfirstoutside. pfrep_gap_normal_empty_H_targetfirstoutside+(pfgu_P_normal_empty_H)=(pfrep_power_normal_empty_H_target)) /\\ (((pfrep_left_normal_empty_H_target)=0))))) -> ((exists pfrep_position_normal_empty_H_targetsecond. ((pfrep_position_normal_empty_H_targetsecond+S (pfrep_power_normal_empty_H_target)=(G)) /\\ ((((exists ff_h_pfp_normal_empty_H_targetsecondentry. ff_h_pfp_normal_empty_H_targetsecondentry + S (pfrep_right_normal_empty_H_target) = S ((S (pfrep_position_normal_empty_H_targetsecond)) * gc)) /\\ exists ff_q_pfp_normal_empty_H_targetsecondentry. gb = ff_q_pfp_normal_empty_H_targetsecondentry * S ((S (pfrep_position_normal_empty_H_targetsecond)) * gc) + (pfrep_right_normal_empty_H_target)))))) \\/ (((exists pfrep_gap_normal_empty_H_targetsecondoutside. pfrep_gap_normal_empty_H_targetsecondoutside+(G)=(pfrep_power_normal_empty_H_target)) /\\ (((pfrep_right_normal_empty_H_target)=0))))) -> pfrep_left_normal_empty_H_target=pfrep_right_normal_empty_H_target))))))", "exact hhg", "rewrite hh_left at hcopy", "rewrite hh_left at hcopy", "rewrite hh_left at hcopy", "rewrite hh_left at hcopy", "rewrite hh_left at hcopy", "rewrite hh_left at hcopy", "exact hcopy", "specialize prime_field_polynomial_monic_right_associates_equivalent (p)", "specialize prime_field_polynomial_monic_right_associates_equivalent (gb)", "specialize prime_field_polynomial_monic_right_associates_equivalent (gc)", "specialize prime_field_polynomial_monic_right_associates_equivalent (G)", "specialize prime_field_polynomial_monic_right_associates_equivalent (hb)", "specialize prime_field_polynomial_monic_right_associates_equivalent (hc)", "specialize prime_field_polynomial_monic_right_associates_equivalent (H)", "apply prime_field_polynomial_monic_right_associates_equivalent", "exact hp", "exact hg_right", "exact hh_right", "exact hgh", "exact hhg" ], "script_sha256": "0e8166b7021e181aa74518aaf85c447a686b2f30f12ccfa948a47fc7d697b56d", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_field_polynomial_gcd_uniqueness_candidate.py", "sha256": "916c24ad6c59609612e97daee6e49347a9522cdb28b44f6f09c6c5760bff0b5b" }, "statement": "forall p gb gc G hb hc H. (~((p) = 1) /\\ forall pfa_factor_left_normal_prime pfa_factor_right_normal_prime. (p) = pfa_factor_left_normal_prime * pfa_factor_right_normal_prime -> pfa_factor_left_normal_prime = 1 \\/ pfa_factor_right_normal_prime = 1) -> ((G)=0 \\/ (((~((G) = 0)) /\\ (((forall fom_index_pfp_normal_G_moniccoefficients. (exists fom_gap_pfp_normal_G_moniccoefficients_index_bound. fom_gap_pfp_normal_G_moniccoefficients_index_bound + S (fom_index_pfp_normal_G_moniccoefficients) = G) -> exists fom_value_pfp_normal_G_moniccoefficients. ((((exists fom_beta_height_pfp_normal_G_moniccoefficients_entry. fom_beta_height_pfp_normal_G_moniccoefficients_entry + S (fom_value_pfp_normal_G_moniccoefficients) = S ((S (fom_index_pfp_normal_G_moniccoefficients)) * gc)) /\\ exists fom_beta_quotient_pfp_normal_G_moniccoefficients_entry. gb = fom_beta_quotient_pfp_normal_G_moniccoefficients_entry * S ((S (fom_index_pfp_normal_G_moniccoefficients)) * gc) + (fom_value_pfp_normal_G_moniccoefficients))) /\\ (exists fom_gap_pfp_normal_G_moniccoefficients_value_bound. fom_gap_pfp_normal_G_moniccoefficients_value_bound + S (fom_value_pfp_normal_G_moniccoefficients) = p))) /\\ ((((exists ff_h_pfp_normal_G_monicleading. ff_h_pfp_normal_G_monicleading + S (1) = S ((S (0)) * gc)) /\\ exists ff_q_pfp_normal_G_monicleading. gb = ff_q_pfp_normal_G_monicleading * S ((S (0)) * gc) + (1))))))))) -> ((H)=0 \\/ (((~((H) = 0)) /\\ (((forall fom_index_pfp_normal_H_moniccoefficients. (exists fom_gap_pfp_normal_H_moniccoefficients_index_bound. fom_gap_pfp_normal_H_moniccoefficients_index_bound + S (fom_index_pfp_normal_H_moniccoefficients) = H) -> exists fom_value_pfp_normal_H_moniccoefficients. ((((exists fom_beta_height_pfp_normal_H_moniccoefficients_entry. fom_beta_height_pfp_normal_H_moniccoefficients_entry + S (fom_value_pfp_normal_H_moniccoefficients) = S ((S (fom_index_pfp_normal_H_moniccoefficients)) * hc)) /\\ exists fom_beta_quotient_pfp_normal_H_moniccoefficients_entry. hb = fom_beta_quotient_pfp_normal_H_moniccoefficients_entry * S ((S (fom_index_pfp_normal_H_moniccoefficients)) * hc) + (fom_value_pfp_normal_H_moniccoefficients))) /\\ (exists fom_gap_pfp_normal_H_moniccoefficients_value_bound. fom_gap_pfp_normal_H_moniccoefficients_value_bound + S (fom_value_pfp_normal_H_moniccoefficients) = p))) /\\ ((((exists ff_h_pfp_normal_H_monicleading. ff_h_pfp_normal_H_monicleading + S (1) = S ((S (0)) * hc)) /\\ exists ff_q_pfp_normal_H_monicleading. hb = ff_q_pfp_normal_H_monicleading * S ((S (0)) * hc) + (1))))))))) -> (((forall fom_index_pfp_normal_GH_canonical. (exists fom_gap_pfp_normal_GH_canonical_index_bound. fom_gap_pfp_normal_GH_canonical_index_bound + S (fom_index_pfp_normal_GH_canonical) = H) -> exists fom_value_pfp_normal_GH_canonical. ((((exists fom_beta_height_pfp_normal_GH_canonical_entry. fom_beta_height_pfp_normal_GH_canonical_entry + S (fom_value_pfp_normal_GH_canonical) = S ((S (fom_index_pfp_normal_GH_canonical)) * hc)) /\\ exists fom_beta_quotient_pfp_normal_GH_canonical_entry. hb = fom_beta_quotient_pfp_normal_GH_canonical_entry * S ((S (fom_index_pfp_normal_GH_canonical)) * hc) + (fom_value_pfp_normal_GH_canonical))) /\\ (exists fom_gap_pfp_normal_GH_canonical_value_bound. fom_gap_pfp_normal_GH_canonical_value_bound + S (fom_value_pfp_normal_GH_canonical) = p))) /\\ ((exists pfgu_qb_normal_GH pfgu_qc_normal_GH pfgu_Q_normal_GH pfgu_pb_normal_GH pfgu_pc_normal_GH pfgu_P_normal_GH. ((((forall fom_index_pfp_normal_GH_productleft. (exists fom_gap_pfp_normal_GH_productleft_index_bound. fom_gap_pfp_normal_GH_productleft_index_bound + S (fom_index_pfp_normal_GH_productleft) = pfgu_Q_normal_GH) -> exists fom_value_pfp_normal_GH_productleft. ((((exists fom_beta_height_pfp_normal_GH_productleft_entry. fom_beta_height_pfp_normal_GH_productleft_entry + S (fom_value_pfp_normal_GH_productleft) = S ((S (fom_index_pfp_normal_GH_productleft)) * pfgu_qc_normal_GH)) /\\ exists fom_beta_quotient_pfp_normal_GH_productleft_entry. pfgu_qb_normal_GH = fom_beta_quotient_pfp_normal_GH_productleft_entry * S ((S (fom_index_pfp_normal_GH_productleft)) * pfgu_qc_normal_GH) + (fom_value_pfp_normal_GH_productleft))) /\\ (exists fom_gap_pfp_normal_GH_productleft_value_bound. fom_gap_pfp_normal_GH_productleft_value_bound + S (fom_value_pfp_normal_GH_productleft) = p))) /\\ (((forall fom_index_pfp_normal_GH_productright. (exists fom_gap_pfp_normal_GH_productright_index_bound. fom_gap_pfp_normal_GH_productright_index_bound + S (fom_index_pfp_normal_GH_productright) = G) -> exists fom_value_pfp_normal_GH_productright. ((((exists fom_beta_height_pfp_normal_GH_productright_entry. fom_beta_height_pfp_normal_GH_productright_entry + S (fom_value_pfp_normal_GH_productright) = S ((S (fom_index_pfp_normal_GH_productright)) * gc)) /\\ exists fom_beta_quotient_pfp_normal_GH_productright_entry. gb = fom_beta_quotient_pfp_normal_GH_productright_entry * S ((S (fom_index_pfp_normal_GH_productright)) * gc) + (fom_value_pfp_normal_GH_productright))) /\\ (exists fom_gap_pfp_normal_GH_productright_value_bound. fom_gap_pfp_normal_GH_productright_value_bound + S (fom_value_pfp_normal_GH_productright) = p))) /\\ (((((((pfgu_Q_normal_GH)=0 \\/ (G)=0) /\\ (((pfgu_P_normal_GH)=0)))) \\/ (((~((pfgu_Q_normal_GH)=0)) /\\ (((~((G)=0)) /\\ (((pfgu_Q_normal_GH)+(G)=S (pfgu_P_normal_GH)))))))) /\\ ((forall pfc_index_normal_GH_productcoefficients. (exists pfa_gap_normal_GH_productcoefficientsbound. pfa_gap_normal_GH_productcoefficientsbound + S (pfc_index_normal_GH_productcoefficients) = (pfgu_P_normal_GH)) -> exists pfc_value_normal_GH_productcoefficients. ((((exists ff_h_pfp_normal_GH_productcoefficientsentry. ff_h_pfp_normal_GH_productcoefficientsentry + S (pfc_value_normal_GH_productcoefficients) = S ((S (pfc_index_normal_GH_productcoefficients)) * pfgu_pc_normal_GH)) /\\ exists ff_q_pfp_normal_GH_productcoefficientsentry. pfgu_pb_normal_GH = ff_q_pfp_normal_GH_productcoefficientsentry * S ((S (pfc_index_normal_GH_productcoefficients)) * pfgu_pc_normal_GH) + (pfc_value_normal_GH_productcoefficients))) /\\ ((exists pfc_terms_code_normal_GH_productcoefficientscoefficient pfc_terms_scale_normal_GH_productcoefficientscoefficient pfc_natural_sum_normal_GH_productcoefficientscoefficient. ((forall pfc_index_normal_GH_productcoefficientscoefficientdiagonal. (exists pfa_gap_normal_GH_productcoefficientscoefficientdiagonalbound. pfa_gap_normal_GH_productcoefficientscoefficientdiagonalbound + S (pfc_index_normal_GH_productcoefficientscoefficientdiagonal) = (S (pfc_index_normal_GH_productcoefficients))) -> exists pfc_value_normal_GH_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normal_GH_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normal_GH_productcoefficientscoefficientdiagonalentry + S (pfc_value_normal_GH_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normal_GH_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normal_GH_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normal_GH_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normal_GH_productcoefficientscoefficient = ff_q_pfp_normal_GH_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normal_GH_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normal_GH_productcoefficientscoefficient) + (pfc_value_normal_GH_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normal_GH_productcoefficientscoefficientdiagonalterm pfc_left_normal_GH_productcoefficientscoefficientdiagonalterm pfc_right_normal_GH_productcoefficientscoefficientdiagonalterm. (((pfc_index_normal_GH_productcoefficientscoefficientdiagonal)+pfc_complement_normal_GH_productcoefficientscoefficientdiagonalterm=(pfc_index_normal_GH_productcoefficients)) /\\ ((((((exists pfa_gap_normal_GH_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normal_GH_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normal_GH_productcoefficientscoefficientdiagonal) = (pfgu_Q_normal_GH)) /\\ ((((exists ff_h_pfp_normal_GH_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normal_GH_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normal_GH_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normal_GH_productcoefficientscoefficientdiagonal)) * pfgu_qc_normal_GH)) /\\ exists ff_q_pfp_normal_GH_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_normal_GH = ff_q_pfp_normal_GH_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normal_GH_productcoefficientscoefficientdiagonal)) * pfgu_qc_normal_GH) + (pfc_left_normal_GH_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normal_GH_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normal_GH_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_normal_GH)=(pfc_index_normal_GH_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normal_GH_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normal_GH_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normal_GH_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normal_GH_productcoefficientscoefficientdiagonalterm) = (G)) /\\ ((((exists ff_h_pfp_normal_GH_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normal_GH_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normal_GH_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normal_GH_productcoefficientscoefficientdiagonalterm)) * gc)) /\\ exists ff_q_pfp_normal_GH_productcoefficientscoefficientdiagonaltermrightentry. gb = ff_q_pfp_normal_GH_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normal_GH_productcoefficientscoefficientdiagonalterm)) * gc) + (pfc_right_normal_GH_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normal_GH_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normal_GH_productcoefficientscoefficientdiagonaltermrightoutside+(G)=(pfc_complement_normal_GH_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normal_GH_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normal_GH_productcoefficientscoefficientdiagonal)=pfc_left_normal_GH_productcoefficientscoefficientdiagonalterm*pfc_right_normal_GH_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normal_GH_productcoefficientscoefficientsum fs_v_pfc_normal_GH_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normal_GH_productcoefficientscoefficientsum_body_start. fs_h_pfc_normal_GH_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normal_GH_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_GH_productcoefficientscoefficientsum_body_start. fs_u_pfc_normal_GH_productcoefficientscoefficientsum = fs_q_pfc_normal_GH_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normal_GH_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normal_GH_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normal_GH_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normal_GH_productcoefficientscoefficient) = S ((S (S (pfc_index_normal_GH_productcoefficients))) * fs_v_pfc_normal_GH_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_GH_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normal_GH_productcoefficientscoefficientsum = fs_q_pfc_normal_GH_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normal_GH_productcoefficients))) * fs_v_pfc_normal_GH_productcoefficientscoefficientsum) + (pfc_natural_sum_normal_GH_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normal_GH_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normal_GH_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normal_GH_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normal_GH_productcoefficientscoefficientsum_body_steps = S (pfc_index_normal_GH_productcoefficients)) -> exists fs_a_pfc_normal_GH_productcoefficientscoefficientsum_body_steps fs_r_pfc_normal_GH_productcoefficientscoefficientsum_body_steps fs_s_pfc_normal_GH_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normal_GH_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normal_GH_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normal_GH_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normal_GH_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normal_GH_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normal_GH_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normal_GH_productcoefficientscoefficient = fs_q_pfc_normal_GH_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normal_GH_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normal_GH_productcoefficientscoefficient) + (fs_a_pfc_normal_GH_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normal_GH_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normal_GH_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normal_GH_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normal_GH_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_GH_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_GH_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normal_GH_productcoefficientscoefficientsum = fs_q_pfc_normal_GH_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normal_GH_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_GH_productcoefficientscoefficientsum) + (fs_r_pfc_normal_GH_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normal_GH_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normal_GH_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normal_GH_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normal_GH_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_GH_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_GH_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normal_GH_productcoefficientscoefficientsum = fs_q_pfc_normal_GH_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normal_GH_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_GH_productcoefficientscoefficientsum) + (fs_s_pfc_normal_GH_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normal_GH_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normal_GH_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normal_GH_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normal_GH_productcoefficientscoefficientresiduebound. pfa_gap_normal_GH_productcoefficientscoefficientresiduebound + S (pfc_value_normal_GH_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normal_GH_productcoefficientscoefficientresiduecongruence pfa_offset_right_normal_GH_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normal_GH_productcoefficientscoefficient) + (p) * pfa_offset_left_normal_GH_productcoefficientscoefficientresiduecongruence = (pfc_value_normal_GH_productcoefficients) + (p) * pfa_offset_right_normal_GH_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normal_GH_target pfrep_left_normal_GH_target pfrep_right_normal_GH_target. ((exists pfrep_position_normal_GH_targetfirst. ((pfrep_position_normal_GH_targetfirst+S (pfrep_power_normal_GH_target)=(pfgu_P_normal_GH)) /\\ ((((exists ff_h_pfp_normal_GH_targetfirstentry. ff_h_pfp_normal_GH_targetfirstentry + S (pfrep_left_normal_GH_target) = S ((S (pfrep_position_normal_GH_targetfirst)) * pfgu_pc_normal_GH)) /\\ exists ff_q_pfp_normal_GH_targetfirstentry. pfgu_pb_normal_GH = ff_q_pfp_normal_GH_targetfirstentry * S ((S (pfrep_position_normal_GH_targetfirst)) * pfgu_pc_normal_GH) + (pfrep_left_normal_GH_target)))))) \\/ (((exists pfrep_gap_normal_GH_targetfirstoutside. pfrep_gap_normal_GH_targetfirstoutside+(pfgu_P_normal_GH)=(pfrep_power_normal_GH_target)) /\\ (((pfrep_left_normal_GH_target)=0))))) -> ((exists pfrep_position_normal_GH_targetsecond. ((pfrep_position_normal_GH_targetsecond+S (pfrep_power_normal_GH_target)=(H)) /\\ ((((exists ff_h_pfp_normal_GH_targetsecondentry. ff_h_pfp_normal_GH_targetsecondentry + S (pfrep_right_normal_GH_target) = S ((S (pfrep_position_normal_GH_targetsecond)) * hc)) /\\ exists ff_q_pfp_normal_GH_targetsecondentry. hb = ff_q_pfp_normal_GH_targetsecondentry * S ((S (pfrep_position_normal_GH_targetsecond)) * hc) + (pfrep_right_normal_GH_target)))))) \\/ (((exists pfrep_gap_normal_GH_targetsecondoutside. pfrep_gap_normal_GH_targetsecondoutside+(H)=(pfrep_power_normal_GH_target)) /\\ (((pfrep_right_normal_GH_target)=0))))) -> pfrep_left_normal_GH_target=pfrep_right_normal_GH_target))))))) -> (((forall fom_index_pfp_normal_HG_canonical. (exists fom_gap_pfp_normal_HG_canonical_index_bound. fom_gap_pfp_normal_HG_canonical_index_bound + S (fom_index_pfp_normal_HG_canonical) = G) -> exists fom_value_pfp_normal_HG_canonical. ((((exists fom_beta_height_pfp_normal_HG_canonical_entry. fom_beta_height_pfp_normal_HG_canonical_entry + S (fom_value_pfp_normal_HG_canonical) = S ((S (fom_index_pfp_normal_HG_canonical)) * gc)) /\\ exists fom_beta_quotient_pfp_normal_HG_canonical_entry. gb = fom_beta_quotient_pfp_normal_HG_canonical_entry * S ((S (fom_index_pfp_normal_HG_canonical)) * gc) + (fom_value_pfp_normal_HG_canonical))) /\\ (exists fom_gap_pfp_normal_HG_canonical_value_bound. fom_gap_pfp_normal_HG_canonical_value_bound + S (fom_value_pfp_normal_HG_canonical) = p))) /\\ ((exists pfgu_qb_normal_HG pfgu_qc_normal_HG pfgu_Q_normal_HG pfgu_pb_normal_HG pfgu_pc_normal_HG pfgu_P_normal_HG. ((((forall fom_index_pfp_normal_HG_productleft. (exists fom_gap_pfp_normal_HG_productleft_index_bound. fom_gap_pfp_normal_HG_productleft_index_bound + S (fom_index_pfp_normal_HG_productleft) = pfgu_Q_normal_HG) -> exists fom_value_pfp_normal_HG_productleft. ((((exists fom_beta_height_pfp_normal_HG_productleft_entry. fom_beta_height_pfp_normal_HG_productleft_entry + S (fom_value_pfp_normal_HG_productleft) = S ((S (fom_index_pfp_normal_HG_productleft)) * pfgu_qc_normal_HG)) /\\ exists fom_beta_quotient_pfp_normal_HG_productleft_entry. pfgu_qb_normal_HG = fom_beta_quotient_pfp_normal_HG_productleft_entry * S ((S (fom_index_pfp_normal_HG_productleft)) * pfgu_qc_normal_HG) + (fom_value_pfp_normal_HG_productleft))) /\\ (exists fom_gap_pfp_normal_HG_productleft_value_bound. fom_gap_pfp_normal_HG_productleft_value_bound + S (fom_value_pfp_normal_HG_productleft) = p))) /\\ (((forall fom_index_pfp_normal_HG_productright. (exists fom_gap_pfp_normal_HG_productright_index_bound. fom_gap_pfp_normal_HG_productright_index_bound + S (fom_index_pfp_normal_HG_productright) = H) -> exists fom_value_pfp_normal_HG_productright. ((((exists fom_beta_height_pfp_normal_HG_productright_entry. fom_beta_height_pfp_normal_HG_productright_entry + S (fom_value_pfp_normal_HG_productright) = S ((S (fom_index_pfp_normal_HG_productright)) * hc)) /\\ exists fom_beta_quotient_pfp_normal_HG_productright_entry. hb = fom_beta_quotient_pfp_normal_HG_productright_entry * S ((S (fom_index_pfp_normal_HG_productright)) * hc) + (fom_value_pfp_normal_HG_productright))) /\\ (exists fom_gap_pfp_normal_HG_productright_value_bound. fom_gap_pfp_normal_HG_productright_value_bound + S (fom_value_pfp_normal_HG_productright) = p))) /\\ (((((((pfgu_Q_normal_HG)=0 \\/ (H)=0) /\\ (((pfgu_P_normal_HG)=0)))) \\/ (((~((pfgu_Q_normal_HG)=0)) /\\ (((~((H)=0)) /\\ (((pfgu_Q_normal_HG)+(H)=S (pfgu_P_normal_HG)))))))) /\\ ((forall pfc_index_normal_HG_productcoefficients. (exists pfa_gap_normal_HG_productcoefficientsbound. pfa_gap_normal_HG_productcoefficientsbound + S (pfc_index_normal_HG_productcoefficients) = (pfgu_P_normal_HG)) -> exists pfc_value_normal_HG_productcoefficients. ((((exists ff_h_pfp_normal_HG_productcoefficientsentry. ff_h_pfp_normal_HG_productcoefficientsentry + S (pfc_value_normal_HG_productcoefficients) = S ((S (pfc_index_normal_HG_productcoefficients)) * pfgu_pc_normal_HG)) /\\ exists ff_q_pfp_normal_HG_productcoefficientsentry. pfgu_pb_normal_HG = ff_q_pfp_normal_HG_productcoefficientsentry * S ((S (pfc_index_normal_HG_productcoefficients)) * pfgu_pc_normal_HG) + (pfc_value_normal_HG_productcoefficients))) /\\ ((exists pfc_terms_code_normal_HG_productcoefficientscoefficient pfc_terms_scale_normal_HG_productcoefficientscoefficient pfc_natural_sum_normal_HG_productcoefficientscoefficient. ((forall pfc_index_normal_HG_productcoefficientscoefficientdiagonal. (exists pfa_gap_normal_HG_productcoefficientscoefficientdiagonalbound. pfa_gap_normal_HG_productcoefficientscoefficientdiagonalbound + S (pfc_index_normal_HG_productcoefficientscoefficientdiagonal) = (S (pfc_index_normal_HG_productcoefficients))) -> exists pfc_value_normal_HG_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_normal_HG_productcoefficientscoefficientdiagonalentry. ff_h_pfp_normal_HG_productcoefficientscoefficientdiagonalentry + S (pfc_value_normal_HG_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_normal_HG_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normal_HG_productcoefficientscoefficient)) /\\ exists ff_q_pfp_normal_HG_productcoefficientscoefficientdiagonalentry. pfc_terms_code_normal_HG_productcoefficientscoefficient = ff_q_pfp_normal_HG_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_normal_HG_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_normal_HG_productcoefficientscoefficient) + (pfc_value_normal_HG_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_normal_HG_productcoefficientscoefficientdiagonalterm pfc_left_normal_HG_productcoefficientscoefficientdiagonalterm pfc_right_normal_HG_productcoefficientscoefficientdiagonalterm. (((pfc_index_normal_HG_productcoefficientscoefficientdiagonal)+pfc_complement_normal_HG_productcoefficientscoefficientdiagonalterm=(pfc_index_normal_HG_productcoefficients)) /\\ ((((((exists pfa_gap_normal_HG_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_normal_HG_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_normal_HG_productcoefficientscoefficientdiagonal) = (pfgu_Q_normal_HG)) /\\ ((((exists ff_h_pfp_normal_HG_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_normal_HG_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_normal_HG_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_normal_HG_productcoefficientscoefficientdiagonal)) * pfgu_qc_normal_HG)) /\\ exists ff_q_pfp_normal_HG_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_normal_HG = ff_q_pfp_normal_HG_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_normal_HG_productcoefficientscoefficientdiagonal)) * pfgu_qc_normal_HG) + (pfc_left_normal_HG_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normal_HG_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_normal_HG_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_normal_HG)=(pfc_index_normal_HG_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_normal_HG_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_normal_HG_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_normal_HG_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_normal_HG_productcoefficientscoefficientdiagonalterm) = (H)) /\\ ((((exists ff_h_pfp_normal_HG_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_normal_HG_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_normal_HG_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_normal_HG_productcoefficientscoefficientdiagonalterm)) * hc)) /\\ exists ff_q_pfp_normal_HG_productcoefficientscoefficientdiagonaltermrightentry. hb = ff_q_pfp_normal_HG_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_normal_HG_productcoefficientscoefficientdiagonalterm)) * hc) + (pfc_right_normal_HG_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_normal_HG_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_normal_HG_productcoefficientscoefficientdiagonaltermrightoutside+(H)=(pfc_complement_normal_HG_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_normal_HG_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_normal_HG_productcoefficientscoefficientdiagonal)=pfc_left_normal_HG_productcoefficientscoefficientdiagonalterm*pfc_right_normal_HG_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_normal_HG_productcoefficientscoefficientsum fs_v_pfc_normal_HG_productcoefficientscoefficientsum. ((((exists fs_h_pfc_normal_HG_productcoefficientscoefficientsum_body_start. fs_h_pfc_normal_HG_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_normal_HG_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_HG_productcoefficientscoefficientsum_body_start. fs_u_pfc_normal_HG_productcoefficientscoefficientsum = fs_q_pfc_normal_HG_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_normal_HG_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_normal_HG_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_normal_HG_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_normal_HG_productcoefficientscoefficient) = S ((S (S (pfc_index_normal_HG_productcoefficients))) * fs_v_pfc_normal_HG_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_HG_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_normal_HG_productcoefficientscoefficientsum = fs_q_pfc_normal_HG_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_normal_HG_productcoefficients))) * fs_v_pfc_normal_HG_productcoefficientscoefficientsum) + (pfc_natural_sum_normal_HG_productcoefficientscoefficient))) /\\ forall fs_i_pfc_normal_HG_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_normal_HG_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_normal_HG_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_normal_HG_productcoefficientscoefficientsum_body_steps = S (pfc_index_normal_HG_productcoefficients)) -> exists fs_a_pfc_normal_HG_productcoefficientscoefficientsum_body_steps fs_r_pfc_normal_HG_productcoefficientscoefficientsum_body_steps fs_s_pfc_normal_HG_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_normal_HG_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_normal_HG_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_normal_HG_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normal_HG_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normal_HG_productcoefficientscoefficient)) /\\ exists fs_q_pfc_normal_HG_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_normal_HG_productcoefficientscoefficient = fs_q_pfc_normal_HG_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_normal_HG_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_normal_HG_productcoefficientscoefficient) + (fs_a_pfc_normal_HG_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normal_HG_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_normal_HG_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_normal_HG_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_normal_HG_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_HG_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_HG_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_normal_HG_productcoefficientscoefficientsum = fs_q_pfc_normal_HG_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_normal_HG_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_HG_productcoefficientscoefficientsum) + (fs_r_pfc_normal_HG_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_normal_HG_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_normal_HG_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_normal_HG_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_normal_HG_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_HG_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_normal_HG_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_normal_HG_productcoefficientscoefficientsum = fs_q_pfc_normal_HG_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_normal_HG_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_normal_HG_productcoefficientscoefficientsum) + (fs_s_pfc_normal_HG_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_normal_HG_productcoefficientscoefficientsum_body_steps = fs_r_pfc_normal_HG_productcoefficientscoefficientsum_body_steps + fs_a_pfc_normal_HG_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_normal_HG_productcoefficientscoefficientresiduebound. pfa_gap_normal_HG_productcoefficientscoefficientresiduebound + S (pfc_value_normal_HG_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_normal_HG_productcoefficientscoefficientresiduecongruence pfa_offset_right_normal_HG_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_normal_HG_productcoefficientscoefficient) + (p) * pfa_offset_left_normal_HG_productcoefficientscoefficientresiduecongruence = (pfc_value_normal_HG_productcoefficients) + (p) * pfa_offset_right_normal_HG_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_normal_HG_target pfrep_left_normal_HG_target pfrep_right_normal_HG_target. ((exists pfrep_position_normal_HG_targetfirst. ((pfrep_position_normal_HG_targetfirst+S (pfrep_power_normal_HG_target)=(pfgu_P_normal_HG)) /\\ ((((exists ff_h_pfp_normal_HG_targetfirstentry. ff_h_pfp_normal_HG_targetfirstentry + S (pfrep_left_normal_HG_target) = S ((S (pfrep_position_normal_HG_targetfirst)) * pfgu_pc_normal_HG)) /\\ exists ff_q_pfp_normal_HG_targetfirstentry. pfgu_pb_normal_HG = ff_q_pfp_normal_HG_targetfirstentry * S ((S (pfrep_position_normal_HG_targetfirst)) * pfgu_pc_normal_HG) + (pfrep_left_normal_HG_target)))))) \\/ (((exists pfrep_gap_normal_HG_targetfirstoutside. pfrep_gap_normal_HG_targetfirstoutside+(pfgu_P_normal_HG)=(pfrep_power_normal_HG_target)) /\\ (((pfrep_left_normal_HG_target)=0))))) -> ((exists pfrep_position_normal_HG_targetsecond. ((pfrep_position_normal_HG_targetsecond+S (pfrep_power_normal_HG_target)=(G)) /\\ ((((exists ff_h_pfp_normal_HG_targetsecondentry. ff_h_pfp_normal_HG_targetsecondentry + S (pfrep_right_normal_HG_target) = S ((S (pfrep_position_normal_HG_targetsecond)) * gc)) /\\ exists ff_q_pfp_normal_HG_targetsecondentry. gb = ff_q_pfp_normal_HG_targetsecondentry * S ((S (pfrep_position_normal_HG_targetsecond)) * gc) + (pfrep_right_normal_HG_target)))))) \\/ (((exists pfrep_gap_normal_HG_targetsecondoutside. pfrep_gap_normal_HG_targetsecondoutside+(G)=(pfrep_power_normal_HG_target)) /\\ (((pfrep_right_normal_HG_target)=0))))) -> pfrep_left_normal_HG_target=pfrep_right_normal_HG_target))))))) -> (forall pfrep_power_normal_result pfrep_left_normal_result pfrep_right_normal_result. ((exists pfrep_position_normal_resultfirst. ((pfrep_position_normal_resultfirst+S (pfrep_power_normal_result)=(G)) /\\ ((((exists ff_h_pfp_normal_resultfirstentry. ff_h_pfp_normal_resultfirstentry + S (pfrep_left_normal_result) = S ((S (pfrep_position_normal_resultfirst)) * gc)) /\\ exists ff_q_pfp_normal_resultfirstentry. gb = ff_q_pfp_normal_resultfirstentry * S ((S (pfrep_position_normal_resultfirst)) * gc) + (pfrep_left_normal_result)))))) \\/ (((exists pfrep_gap_normal_resultfirstoutside. pfrep_gap_normal_resultfirstoutside+(G)=(pfrep_power_normal_result)) /\\ (((pfrep_left_normal_result)=0))))) -> ((exists pfrep_position_normal_resultsecond. ((pfrep_position_normal_resultsecond+S (pfrep_power_normal_result)=(H)) /\\ ((((exists ff_h_pfp_normal_resultsecondentry. ff_h_pfp_normal_resultsecondentry + S (pfrep_right_normal_result) = S ((S (pfrep_position_normal_resultsecond)) * hc)) /\\ exists ff_q_pfp_normal_resultsecondentry. hb = ff_q_pfp_normal_resultsecondentry * S ((S (pfrep_position_normal_resultsecond)) * hc) + (pfrep_right_normal_result)))))) \\/ (((exists pfrep_gap_normal_resultsecondoutside. pfrep_gap_normal_resultsecondoutside+(H)=(pfrep_power_normal_result)) /\\ (((pfrep_right_normal_result)=0))))) -> pfrep_left_normal_result=pfrep_right_normal_result)", "statement_sha256": "3f1accbdc5d052befcd3269832021e993b13e2cc05c96cc07a425520f9e3f9de", "summary": "Two zero-or-monic mutual right associates are formally equivalent, including both-zero and one-empty branches. 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(~((p) = 1) /\\ forall pfa_factor_left_gcd_unique_prime pfa_factor_right_gcd_unique_prime. (p) = pfa_factor_left_gcd_unique_prime * pfa_factor_right_gcd_unique_prime -> pfa_factor_left_gcd_unique_prime = 1 \\/ pfa_factor_right_gcd_unique_prime = 1) -> ((((G)=0 \\/ (((~((G) = 0)) /\\ (((forall fom_index_pfp_gcd_unique_G_normal_moniccoefficients. (exists fom_gap_pfp_gcd_unique_G_normal_moniccoefficients_index_bound. fom_gap_pfp_gcd_unique_G_normal_moniccoefficients_index_bound + S (fom_index_pfp_gcd_unique_G_normal_moniccoefficients) = G) -> exists fom_value_pfp_gcd_unique_G_normal_moniccoefficients. ((((exists fom_beta_height_pfp_gcd_unique_G_normal_moniccoefficients_entry. fom_beta_height_pfp_gcd_unique_G_normal_moniccoefficients_entry + S (fom_value_pfp_gcd_unique_G_normal_moniccoefficients) = S ((S (fom_index_pfp_gcd_unique_G_normal_moniccoefficients)) * gc)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_normal_moniccoefficients_entry. gb = fom_beta_quotient_pfp_gcd_unique_G_normal_moniccoefficients_entry * S ((S (fom_index_pfp_gcd_unique_G_normal_moniccoefficients)) * gc) + (fom_value_pfp_gcd_unique_G_normal_moniccoefficients))) /\\ (exists fom_gap_pfp_gcd_unique_G_normal_moniccoefficients_value_bound. fom_gap_pfp_gcd_unique_G_normal_moniccoefficients_value_bound + S (fom_value_pfp_gcd_unique_G_normal_moniccoefficients) = p))) /\\ ((((exists ff_h_pfp_gcd_unique_G_normal_monicleading. ff_h_pfp_gcd_unique_G_normal_monicleading + S (1) = S ((S (0)) * gc)) /\\ exists ff_q_pfp_gcd_unique_G_normal_monicleading. gb = ff_q_pfp_gcd_unique_G_normal_monicleading * S ((S (0)) * gc) + (1))))))))) /\\ ((((((((forall fom_index_pfp_gcd_unique_G_gcd_common_left_canonical. (exists fom_gap_pfp_gcd_unique_G_gcd_common_left_canonical_index_bound. fom_gap_pfp_gcd_unique_G_gcd_common_left_canonical_index_bound + S (fom_index_pfp_gcd_unique_G_gcd_common_left_canonical) = L) -> exists fom_value_pfp_gcd_unique_G_gcd_common_left_canonical. ((((exists fom_beta_height_pfp_gcd_unique_G_gcd_common_left_canonical_entry. fom_beta_height_pfp_gcd_unique_G_gcd_common_left_canonical_entry + S (fom_value_pfp_gcd_unique_G_gcd_common_left_canonical) = S ((S (fom_index_pfp_gcd_unique_G_gcd_common_left_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_gcd_common_left_canonical_entry. ab = fom_beta_quotient_pfp_gcd_unique_G_gcd_common_left_canonical_entry * S ((S (fom_index_pfp_gcd_unique_G_gcd_common_left_canonical)) * ac) + (fom_value_pfp_gcd_unique_G_gcd_common_left_canonical))) /\\ (exists fom_gap_pfp_gcd_unique_G_gcd_common_left_canonical_value_bound. fom_gap_pfp_gcd_unique_G_gcd_common_left_canonical_value_bound + S (fom_value_pfp_gcd_unique_G_gcd_common_left_canonical) = p))) /\\ ((exists pfgu_qb_gcd_unique_G_gcd_common_left pfgu_qc_gcd_unique_G_gcd_common_left pfgu_Q_gcd_unique_G_gcd_common_left pfgu_pb_gcd_unique_G_gcd_common_left pfgu_pc_gcd_unique_G_gcd_common_left pfgu_P_gcd_unique_G_gcd_common_left. ((((forall fom_index_pfp_gcd_unique_G_gcd_common_left_productleft. (exists fom_gap_pfp_gcd_unique_G_gcd_common_left_productleft_index_bound. fom_gap_pfp_gcd_unique_G_gcd_common_left_productleft_index_bound + S (fom_index_pfp_gcd_unique_G_gcd_common_left_productleft) = pfgu_Q_gcd_unique_G_gcd_common_left) -> exists fom_value_pfp_gcd_unique_G_gcd_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_unique_G_gcd_common_left_productleft_entry. fom_beta_height_pfp_gcd_unique_G_gcd_common_left_productleft_entry + S (fom_value_pfp_gcd_unique_G_gcd_common_left_productleft) = S ((S (fom_index_pfp_gcd_unique_G_gcd_common_left_productleft)) * pfgu_qc_gcd_unique_G_gcd_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_gcd_common_left_productleft_entry. pfgu_qb_gcd_unique_G_gcd_common_left = fom_beta_quotient_pfp_gcd_unique_G_gcd_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_unique_G_gcd_common_left_productleft)) * pfgu_qc_gcd_unique_G_gcd_common_left) + (fom_value_pfp_gcd_unique_G_gcd_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_unique_G_gcd_common_left_productleft_value_bound. fom_gap_pfp_gcd_unique_G_gcd_common_left_productleft_value_bound + S (fom_value_pfp_gcd_unique_G_gcd_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_unique_G_gcd_common_left_productright. (exists fom_gap_pfp_gcd_unique_G_gcd_common_left_productright_index_bound. fom_gap_pfp_gcd_unique_G_gcd_common_left_productright_index_bound + S (fom_index_pfp_gcd_unique_G_gcd_common_left_productright) = G) -> exists fom_value_pfp_gcd_unique_G_gcd_common_left_productright. ((((exists fom_beta_height_pfp_gcd_unique_G_gcd_common_left_productright_entry. fom_beta_height_pfp_gcd_unique_G_gcd_common_left_productright_entry + S (fom_value_pfp_gcd_unique_G_gcd_common_left_productright) = S ((S (fom_index_pfp_gcd_unique_G_gcd_common_left_productright)) * gc)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_gcd_common_left_productright_entry. gb = fom_beta_quotient_pfp_gcd_unique_G_gcd_common_left_productright_entry * S ((S (fom_index_pfp_gcd_unique_G_gcd_common_left_productright)) * gc) + (fom_value_pfp_gcd_unique_G_gcd_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_unique_G_gcd_common_left_productright_value_bound. fom_gap_pfp_gcd_unique_G_gcd_common_left_productright_value_bound + S (fom_value_pfp_gcd_unique_G_gcd_common_left_productright) = p))) /\\ (((((((pfgu_Q_gcd_unique_G_gcd_common_left)=0 \\/ (G)=0) /\\ (((pfgu_P_gcd_unique_G_gcd_common_left)=0)))) \\/ (((~((pfgu_Q_gcd_unique_G_gcd_common_left)=0)) /\\ (((~((G)=0)) /\\ (((pfgu_Q_gcd_unique_G_gcd_common_left)+(G)=S (pfgu_P_gcd_unique_G_gcd_common_left)))))))) /\\ ((forall pfc_index_gcd_unique_G_gcd_common_left_productcoefficients. (exists pfa_gap_gcd_unique_G_gcd_common_left_productcoefficientsbound. pfa_gap_gcd_unique_G_gcd_common_left_productcoefficientsbound + S (pfc_index_gcd_unique_G_gcd_common_left_productcoefficients) = (pfgu_P_gcd_unique_G_gcd_common_left)) -> exists pfc_value_gcd_unique_G_gcd_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_unique_G_gcd_common_left_productcoefficientsentry. ff_h_pfp_gcd_unique_G_gcd_common_left_productcoefficientsentry + S (pfc_value_gcd_unique_G_gcd_common_left_productcoefficients) = S ((S (pfc_index_gcd_unique_G_gcd_common_left_productcoefficients)) * pfgu_pc_gcd_unique_G_gcd_common_left)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_common_left_productcoefficientsentry. pfgu_pb_gcd_unique_G_gcd_common_left = ff_q_pfp_gcd_unique_G_gcd_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_unique_G_gcd_common_left_productcoefficients)) * pfgu_pc_gcd_unique_G_gcd_common_left) + (pfc_value_gcd_unique_G_gcd_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_unique_G_gcd_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_unique_G_gcd_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_unique_G_gcd_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_unique_G_gcd_common_left_productcoefficients))) -> exists pfc_value_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_G_gcd_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_unique_G_gcd_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_G_gcd_common_left_productcoefficientscoefficient) + (pfc_value_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_unique_G_gcd_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonal) = (pfgu_Q_gcd_unique_G_gcd_common_left)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_G_gcd_common_left)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_gcd_unique_G_gcd_common_left = ff_q_pfp_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_G_gcd_common_left) + (pfc_left_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_gcd_unique_G_gcd_common_left)=(pfc_index_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm) = (G)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm)) * gc)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermrightentry. gb = ff_q_pfp_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm)) * gc) + (pfc_right_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(G)=(pfc_complement_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_unique_G_gcd_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_unique_G_gcd_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_unique_G_gcd_common_left_productcoefficients))) * fs_v_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_unique_G_gcd_common_left_productcoefficients))) * fs_v_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_unique_G_gcd_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_unique_G_gcd_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_G_gcd_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_unique_G_gcd_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_G_gcd_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_unique_G_gcd_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_unique_G_gcd_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_unique_G_gcd_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_unique_G_gcd_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_unique_G_gcd_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_unique_G_gcd_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_unique_G_gcd_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_unique_G_gcd_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_unique_G_gcd_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_unique_G_gcd_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_unique_G_gcd_common_left_target pfrep_left_gcd_unique_G_gcd_common_left_target pfrep_right_gcd_unique_G_gcd_common_left_target. ((exists pfrep_position_gcd_unique_G_gcd_common_left_targetfirst. ((pfrep_position_gcd_unique_G_gcd_common_left_targetfirst+S (pfrep_power_gcd_unique_G_gcd_common_left_target)=(pfgu_P_gcd_unique_G_gcd_common_left)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_common_left_targetfirstentry. ff_h_pfp_gcd_unique_G_gcd_common_left_targetfirstentry + S (pfrep_left_gcd_unique_G_gcd_common_left_target) = S ((S (pfrep_position_gcd_unique_G_gcd_common_left_targetfirst)) * pfgu_pc_gcd_unique_G_gcd_common_left)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_common_left_targetfirstentry. pfgu_pb_gcd_unique_G_gcd_common_left = ff_q_pfp_gcd_unique_G_gcd_common_left_targetfirstentry * S ((S (pfrep_position_gcd_unique_G_gcd_common_left_targetfirst)) * pfgu_pc_gcd_unique_G_gcd_common_left) + (pfrep_left_gcd_unique_G_gcd_common_left_target)))))) \\/ (((exists pfrep_gap_gcd_unique_G_gcd_common_left_targetfirstoutside. pfrep_gap_gcd_unique_G_gcd_common_left_targetfirstoutside+(pfgu_P_gcd_unique_G_gcd_common_left)=(pfrep_power_gcd_unique_G_gcd_common_left_target)) /\\ (((pfrep_left_gcd_unique_G_gcd_common_left_target)=0))))) -> ((exists pfrep_position_gcd_unique_G_gcd_common_left_targetsecond. ((pfrep_position_gcd_unique_G_gcd_common_left_targetsecond+S (pfrep_power_gcd_unique_G_gcd_common_left_target)=(L)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_common_left_targetsecondentry. ff_h_pfp_gcd_unique_G_gcd_common_left_targetsecondentry + S (pfrep_right_gcd_unique_G_gcd_common_left_target) = S ((S (pfrep_position_gcd_unique_G_gcd_common_left_targetsecond)) * ac)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_common_left_targetsecondentry. ab = ff_q_pfp_gcd_unique_G_gcd_common_left_targetsecondentry * S ((S (pfrep_position_gcd_unique_G_gcd_common_left_targetsecond)) * ac) + (pfrep_right_gcd_unique_G_gcd_common_left_target)))))) \\/ (((exists pfrep_gap_gcd_unique_G_gcd_common_left_targetsecondoutside. pfrep_gap_gcd_unique_G_gcd_common_left_targetsecondoutside+(L)=(pfrep_power_gcd_unique_G_gcd_common_left_target)) /\\ (((pfrep_right_gcd_unique_G_gcd_common_left_target)=0))))) -> pfrep_left_gcd_unique_G_gcd_common_left_target=pfrep_right_gcd_unique_G_gcd_common_left_target))))))) /\\ ((((forall fom_index_pfp_gcd_unique_G_gcd_common_right_canonical. (exists fom_gap_pfp_gcd_unique_G_gcd_common_right_canonical_index_bound. fom_gap_pfp_gcd_unique_G_gcd_common_right_canonical_index_bound + S (fom_index_pfp_gcd_unique_G_gcd_common_right_canonical) = M) -> exists fom_value_pfp_gcd_unique_G_gcd_common_right_canonical. ((((exists fom_beta_height_pfp_gcd_unique_G_gcd_common_right_canonical_entry. fom_beta_height_pfp_gcd_unique_G_gcd_common_right_canonical_entry + S (fom_value_pfp_gcd_unique_G_gcd_common_right_canonical) = S ((S (fom_index_pfp_gcd_unique_G_gcd_common_right_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_gcd_common_right_canonical_entry. bb = fom_beta_quotient_pfp_gcd_unique_G_gcd_common_right_canonical_entry * S ((S (fom_index_pfp_gcd_unique_G_gcd_common_right_canonical)) * bc) + (fom_value_pfp_gcd_unique_G_gcd_common_right_canonical))) /\\ (exists fom_gap_pfp_gcd_unique_G_gcd_common_right_canonical_value_bound. fom_gap_pfp_gcd_unique_G_gcd_common_right_canonical_value_bound + S (fom_value_pfp_gcd_unique_G_gcd_common_right_canonical) = p))) /\\ ((exists pfgu_qb_gcd_unique_G_gcd_common_right pfgu_qc_gcd_unique_G_gcd_common_right pfgu_Q_gcd_unique_G_gcd_common_right pfgu_pb_gcd_unique_G_gcd_common_right pfgu_pc_gcd_unique_G_gcd_common_right pfgu_P_gcd_unique_G_gcd_common_right. ((((forall fom_index_pfp_gcd_unique_G_gcd_common_right_productleft. (exists fom_gap_pfp_gcd_unique_G_gcd_common_right_productleft_index_bound. fom_gap_pfp_gcd_unique_G_gcd_common_right_productleft_index_bound + S (fom_index_pfp_gcd_unique_G_gcd_common_right_productleft) = pfgu_Q_gcd_unique_G_gcd_common_right) -> exists fom_value_pfp_gcd_unique_G_gcd_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_unique_G_gcd_common_right_productleft_entry. fom_beta_height_pfp_gcd_unique_G_gcd_common_right_productleft_entry + S (fom_value_pfp_gcd_unique_G_gcd_common_right_productleft) = S ((S (fom_index_pfp_gcd_unique_G_gcd_common_right_productleft)) * pfgu_qc_gcd_unique_G_gcd_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_gcd_common_right_productleft_entry. pfgu_qb_gcd_unique_G_gcd_common_right = fom_beta_quotient_pfp_gcd_unique_G_gcd_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_unique_G_gcd_common_right_productleft)) * pfgu_qc_gcd_unique_G_gcd_common_right) + (fom_value_pfp_gcd_unique_G_gcd_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_unique_G_gcd_common_right_productleft_value_bound. fom_gap_pfp_gcd_unique_G_gcd_common_right_productleft_value_bound + S (fom_value_pfp_gcd_unique_G_gcd_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_unique_G_gcd_common_right_productright. (exists fom_gap_pfp_gcd_unique_G_gcd_common_right_productright_index_bound. fom_gap_pfp_gcd_unique_G_gcd_common_right_productright_index_bound + S (fom_index_pfp_gcd_unique_G_gcd_common_right_productright) = G) -> exists fom_value_pfp_gcd_unique_G_gcd_common_right_productright. ((((exists fom_beta_height_pfp_gcd_unique_G_gcd_common_right_productright_entry. fom_beta_height_pfp_gcd_unique_G_gcd_common_right_productright_entry + S (fom_value_pfp_gcd_unique_G_gcd_common_right_productright) = S ((S (fom_index_pfp_gcd_unique_G_gcd_common_right_productright)) * gc)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_gcd_common_right_productright_entry. gb = fom_beta_quotient_pfp_gcd_unique_G_gcd_common_right_productright_entry * S ((S (fom_index_pfp_gcd_unique_G_gcd_common_right_productright)) * gc) + (fom_value_pfp_gcd_unique_G_gcd_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_unique_G_gcd_common_right_productright_value_bound. fom_gap_pfp_gcd_unique_G_gcd_common_right_productright_value_bound + S (fom_value_pfp_gcd_unique_G_gcd_common_right_productright) = p))) /\\ (((((((pfgu_Q_gcd_unique_G_gcd_common_right)=0 \\/ (G)=0) /\\ (((pfgu_P_gcd_unique_G_gcd_common_right)=0)))) \\/ (((~((pfgu_Q_gcd_unique_G_gcd_common_right)=0)) /\\ (((~((G)=0)) /\\ (((pfgu_Q_gcd_unique_G_gcd_common_right)+(G)=S (pfgu_P_gcd_unique_G_gcd_common_right)))))))) /\\ ((forall pfc_index_gcd_unique_G_gcd_common_right_productcoefficients. (exists pfa_gap_gcd_unique_G_gcd_common_right_productcoefficientsbound. pfa_gap_gcd_unique_G_gcd_common_right_productcoefficientsbound + S (pfc_index_gcd_unique_G_gcd_common_right_productcoefficients) = (pfgu_P_gcd_unique_G_gcd_common_right)) -> exists pfc_value_gcd_unique_G_gcd_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_unique_G_gcd_common_right_productcoefficientsentry. ff_h_pfp_gcd_unique_G_gcd_common_right_productcoefficientsentry + S (pfc_value_gcd_unique_G_gcd_common_right_productcoefficients) = S ((S (pfc_index_gcd_unique_G_gcd_common_right_productcoefficients)) * pfgu_pc_gcd_unique_G_gcd_common_right)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_common_right_productcoefficientsentry. pfgu_pb_gcd_unique_G_gcd_common_right = ff_q_pfp_gcd_unique_G_gcd_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_unique_G_gcd_common_right_productcoefficients)) * pfgu_pc_gcd_unique_G_gcd_common_right) + (pfc_value_gcd_unique_G_gcd_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_unique_G_gcd_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_unique_G_gcd_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_unique_G_gcd_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_unique_G_gcd_common_right_productcoefficients))) -> exists pfc_value_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_G_gcd_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_unique_G_gcd_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_G_gcd_common_right_productcoefficientscoefficient) + (pfc_value_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_unique_G_gcd_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonal) = (pfgu_Q_gcd_unique_G_gcd_common_right)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_G_gcd_common_right)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_gcd_unique_G_gcd_common_right = ff_q_pfp_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_G_gcd_common_right) + (pfc_left_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_gcd_unique_G_gcd_common_right)=(pfc_index_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm) = (G)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm)) * gc)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermrightentry. gb = ff_q_pfp_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm)) * gc) + (pfc_right_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(G)=(pfc_complement_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_unique_G_gcd_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_unique_G_gcd_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_unique_G_gcd_common_right_productcoefficients))) * fs_v_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_unique_G_gcd_common_right_productcoefficients))) * fs_v_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_unique_G_gcd_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_unique_G_gcd_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_G_gcd_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_unique_G_gcd_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_G_gcd_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_unique_G_gcd_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_unique_G_gcd_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_unique_G_gcd_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_unique_G_gcd_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_unique_G_gcd_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_unique_G_gcd_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_unique_G_gcd_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_unique_G_gcd_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_unique_G_gcd_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_unique_G_gcd_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_unique_G_gcd_common_right_target pfrep_left_gcd_unique_G_gcd_common_right_target pfrep_right_gcd_unique_G_gcd_common_right_target. ((exists pfrep_position_gcd_unique_G_gcd_common_right_targetfirst. ((pfrep_position_gcd_unique_G_gcd_common_right_targetfirst+S (pfrep_power_gcd_unique_G_gcd_common_right_target)=(pfgu_P_gcd_unique_G_gcd_common_right)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_common_right_targetfirstentry. ff_h_pfp_gcd_unique_G_gcd_common_right_targetfirstentry + S (pfrep_left_gcd_unique_G_gcd_common_right_target) = S ((S (pfrep_position_gcd_unique_G_gcd_common_right_targetfirst)) * pfgu_pc_gcd_unique_G_gcd_common_right)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_common_right_targetfirstentry. pfgu_pb_gcd_unique_G_gcd_common_right = ff_q_pfp_gcd_unique_G_gcd_common_right_targetfirstentry * S ((S (pfrep_position_gcd_unique_G_gcd_common_right_targetfirst)) * pfgu_pc_gcd_unique_G_gcd_common_right) + (pfrep_left_gcd_unique_G_gcd_common_right_target)))))) \\/ (((exists pfrep_gap_gcd_unique_G_gcd_common_right_targetfirstoutside. pfrep_gap_gcd_unique_G_gcd_common_right_targetfirstoutside+(pfgu_P_gcd_unique_G_gcd_common_right)=(pfrep_power_gcd_unique_G_gcd_common_right_target)) /\\ (((pfrep_left_gcd_unique_G_gcd_common_right_target)=0))))) -> ((exists pfrep_position_gcd_unique_G_gcd_common_right_targetsecond. ((pfrep_position_gcd_unique_G_gcd_common_right_targetsecond+S (pfrep_power_gcd_unique_G_gcd_common_right_target)=(M)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_common_right_targetsecondentry. ff_h_pfp_gcd_unique_G_gcd_common_right_targetsecondentry + S (pfrep_right_gcd_unique_G_gcd_common_right_target) = S ((S (pfrep_position_gcd_unique_G_gcd_common_right_targetsecond)) * bc)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_common_right_targetsecondentry. bb = ff_q_pfp_gcd_unique_G_gcd_common_right_targetsecondentry * S ((S (pfrep_position_gcd_unique_G_gcd_common_right_targetsecond)) * bc) + (pfrep_right_gcd_unique_G_gcd_common_right_target)))))) \\/ (((exists pfrep_gap_gcd_unique_G_gcd_common_right_targetsecondoutside. pfrep_gap_gcd_unique_G_gcd_common_right_targetsecondoutside+(M)=(pfrep_power_gcd_unique_G_gcd_common_right_target)) /\\ (((pfrep_right_gcd_unique_G_gcd_common_right_target)=0))))) -> pfrep_left_gcd_unique_G_gcd_common_right_target=pfrep_right_gcd_unique_G_gcd_common_right_target)))))))))) /\\ ((forall pfgu_db_gcd_unique_G_gcd pfgu_dc_gcd_unique_G_gcd pfgu_D_gcd_unique_G_gcd. (((((forall fom_index_pfp_gcd_unique_G_gcd_divisor_left_canonical. (exists fom_gap_pfp_gcd_unique_G_gcd_divisor_left_canonical_index_bound. fom_gap_pfp_gcd_unique_G_gcd_divisor_left_canonical_index_bound + S (fom_index_pfp_gcd_unique_G_gcd_divisor_left_canonical) = L) -> exists fom_value_pfp_gcd_unique_G_gcd_divisor_left_canonical. ((((exists fom_beta_height_pfp_gcd_unique_G_gcd_divisor_left_canonical_entry. fom_beta_height_pfp_gcd_unique_G_gcd_divisor_left_canonical_entry + S (fom_value_pfp_gcd_unique_G_gcd_divisor_left_canonical) = S ((S (fom_index_pfp_gcd_unique_G_gcd_divisor_left_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_gcd_divisor_left_canonical_entry. ab = fom_beta_quotient_pfp_gcd_unique_G_gcd_divisor_left_canonical_entry * S ((S (fom_index_pfp_gcd_unique_G_gcd_divisor_left_canonical)) * ac) + (fom_value_pfp_gcd_unique_G_gcd_divisor_left_canonical))) /\\ (exists fom_gap_pfp_gcd_unique_G_gcd_divisor_left_canonical_value_bound. fom_gap_pfp_gcd_unique_G_gcd_divisor_left_canonical_value_bound + S (fom_value_pfp_gcd_unique_G_gcd_divisor_left_canonical) = p))) /\\ ((exists pfgu_qb_gcd_unique_G_gcd_divisor_left pfgu_qc_gcd_unique_G_gcd_divisor_left pfgu_Q_gcd_unique_G_gcd_divisor_left pfgu_pb_gcd_unique_G_gcd_divisor_left pfgu_pc_gcd_unique_G_gcd_divisor_left pfgu_P_gcd_unique_G_gcd_divisor_left. ((((forall fom_index_pfp_gcd_unique_G_gcd_divisor_left_productleft. (exists fom_gap_pfp_gcd_unique_G_gcd_divisor_left_productleft_index_bound. fom_gap_pfp_gcd_unique_G_gcd_divisor_left_productleft_index_bound + S (fom_index_pfp_gcd_unique_G_gcd_divisor_left_productleft) = pfgu_Q_gcd_unique_G_gcd_divisor_left) -> exists fom_value_pfp_gcd_unique_G_gcd_divisor_left_productleft. ((((exists fom_beta_height_pfp_gcd_unique_G_gcd_divisor_left_productleft_entry. fom_beta_height_pfp_gcd_unique_G_gcd_divisor_left_productleft_entry + S (fom_value_pfp_gcd_unique_G_gcd_divisor_left_productleft) = S ((S (fom_index_pfp_gcd_unique_G_gcd_divisor_left_productleft)) * pfgu_qc_gcd_unique_G_gcd_divisor_left)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_gcd_divisor_left_productleft_entry. pfgu_qb_gcd_unique_G_gcd_divisor_left = fom_beta_quotient_pfp_gcd_unique_G_gcd_divisor_left_productleft_entry * S ((S (fom_index_pfp_gcd_unique_G_gcd_divisor_left_productleft)) * pfgu_qc_gcd_unique_G_gcd_divisor_left) + (fom_value_pfp_gcd_unique_G_gcd_divisor_left_productleft))) /\\ (exists fom_gap_pfp_gcd_unique_G_gcd_divisor_left_productleft_value_bound. fom_gap_pfp_gcd_unique_G_gcd_divisor_left_productleft_value_bound + S (fom_value_pfp_gcd_unique_G_gcd_divisor_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_unique_G_gcd_divisor_left_productright. (exists fom_gap_pfp_gcd_unique_G_gcd_divisor_left_productright_index_bound. fom_gap_pfp_gcd_unique_G_gcd_divisor_left_productright_index_bound + S (fom_index_pfp_gcd_unique_G_gcd_divisor_left_productright) = pfgu_D_gcd_unique_G_gcd) -> exists fom_value_pfp_gcd_unique_G_gcd_divisor_left_productright. ((((exists fom_beta_height_pfp_gcd_unique_G_gcd_divisor_left_productright_entry. fom_beta_height_pfp_gcd_unique_G_gcd_divisor_left_productright_entry + S (fom_value_pfp_gcd_unique_G_gcd_divisor_left_productright) = S ((S (fom_index_pfp_gcd_unique_G_gcd_divisor_left_productright)) * pfgu_dc_gcd_unique_G_gcd)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_gcd_divisor_left_productright_entry. pfgu_db_gcd_unique_G_gcd = fom_beta_quotient_pfp_gcd_unique_G_gcd_divisor_left_productright_entry * S ((S (fom_index_pfp_gcd_unique_G_gcd_divisor_left_productright)) * pfgu_dc_gcd_unique_G_gcd) + (fom_value_pfp_gcd_unique_G_gcd_divisor_left_productright))) /\\ (exists fom_gap_pfp_gcd_unique_G_gcd_divisor_left_productright_value_bound. fom_gap_pfp_gcd_unique_G_gcd_divisor_left_productright_value_bound + S (fom_value_pfp_gcd_unique_G_gcd_divisor_left_productright) = p))) /\\ (((((((pfgu_Q_gcd_unique_G_gcd_divisor_left)=0 \\/ (pfgu_D_gcd_unique_G_gcd)=0) /\\ (((pfgu_P_gcd_unique_G_gcd_divisor_left)=0)))) \\/ (((~((pfgu_Q_gcd_unique_G_gcd_divisor_left)=0)) /\\ (((~((pfgu_D_gcd_unique_G_gcd)=0)) /\\ (((pfgu_Q_gcd_unique_G_gcd_divisor_left)+(pfgu_D_gcd_unique_G_gcd)=S (pfgu_P_gcd_unique_G_gcd_divisor_left)))))))) /\\ ((forall pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficients. (exists pfa_gap_gcd_unique_G_gcd_divisor_left_productcoefficientsbound. pfa_gap_gcd_unique_G_gcd_divisor_left_productcoefficientsbound + S (pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficients) = (pfgu_P_gcd_unique_G_gcd_divisor_left)) -> exists pfc_value_gcd_unique_G_gcd_divisor_left_productcoefficients. ((((exists ff_h_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientsentry. ff_h_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientsentry + S (pfc_value_gcd_unique_G_gcd_divisor_left_productcoefficients) = S ((S (pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficients)) * pfgu_pc_gcd_unique_G_gcd_divisor_left)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientsentry. pfgu_pb_gcd_unique_G_gcd_divisor_left = ff_q_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientsentry * S ((S (pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficients)) * pfgu_pc_gcd_unique_G_gcd_divisor_left) + (pfc_value_gcd_unique_G_gcd_divisor_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficient pfc_terms_scale_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficient pfc_natural_sum_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficient. ((forall pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficients))) -> exists pfc_value_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficient = ff_q_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficient) + (pfc_value_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonal) = (pfgu_Q_gcd_unique_G_gcd_divisor_left)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_G_gcd_divisor_left)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_gcd_unique_G_gcd_divisor_left = ff_q_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_G_gcd_divisor_left) + (pfc_left_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_gcd_unique_G_gcd_divisor_left)=(pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm) = (pfgu_D_gcd_unique_G_gcd)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)) * pfgu_dc_gcd_unique_G_gcd)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightentry. pfgu_db_gcd_unique_G_gcd = ff_q_pfp_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)) * pfgu_dc_gcd_unique_G_gcd) + (pfc_right_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightoutside+(pfgu_D_gcd_unique_G_gcd)=(pfc_complement_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum fs_v_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficients))) * fs_v_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficients))) * fs_v_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_unique_G_gcd_divisor_left_productcoefficients)) -> exists fs_a_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficient = fs_q_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_unique_G_gcd_divisor_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_unique_G_gcd_divisor_left_productcoefficients) + (p) * pfa_offset_right_gcd_unique_G_gcd_divisor_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_unique_G_gcd_divisor_left_target pfrep_left_gcd_unique_G_gcd_divisor_left_target pfrep_right_gcd_unique_G_gcd_divisor_left_target. ((exists pfrep_position_gcd_unique_G_gcd_divisor_left_targetfirst. ((pfrep_position_gcd_unique_G_gcd_divisor_left_targetfirst+S (pfrep_power_gcd_unique_G_gcd_divisor_left_target)=(pfgu_P_gcd_unique_G_gcd_divisor_left)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_divisor_left_targetfirstentry. ff_h_pfp_gcd_unique_G_gcd_divisor_left_targetfirstentry + S (pfrep_left_gcd_unique_G_gcd_divisor_left_target) = S ((S (pfrep_position_gcd_unique_G_gcd_divisor_left_targetfirst)) * pfgu_pc_gcd_unique_G_gcd_divisor_left)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_divisor_left_targetfirstentry. pfgu_pb_gcd_unique_G_gcd_divisor_left = ff_q_pfp_gcd_unique_G_gcd_divisor_left_targetfirstentry * S ((S (pfrep_position_gcd_unique_G_gcd_divisor_left_targetfirst)) * pfgu_pc_gcd_unique_G_gcd_divisor_left) + (pfrep_left_gcd_unique_G_gcd_divisor_left_target)))))) \\/ (((exists pfrep_gap_gcd_unique_G_gcd_divisor_left_targetfirstoutside. pfrep_gap_gcd_unique_G_gcd_divisor_left_targetfirstoutside+(pfgu_P_gcd_unique_G_gcd_divisor_left)=(pfrep_power_gcd_unique_G_gcd_divisor_left_target)) /\\ (((pfrep_left_gcd_unique_G_gcd_divisor_left_target)=0))))) -> ((exists pfrep_position_gcd_unique_G_gcd_divisor_left_targetsecond. ((pfrep_position_gcd_unique_G_gcd_divisor_left_targetsecond+S (pfrep_power_gcd_unique_G_gcd_divisor_left_target)=(L)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_divisor_left_targetsecondentry. ff_h_pfp_gcd_unique_G_gcd_divisor_left_targetsecondentry + S (pfrep_right_gcd_unique_G_gcd_divisor_left_target) = S ((S (pfrep_position_gcd_unique_G_gcd_divisor_left_targetsecond)) * ac)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_divisor_left_targetsecondentry. ab = ff_q_pfp_gcd_unique_G_gcd_divisor_left_targetsecondentry * S ((S (pfrep_position_gcd_unique_G_gcd_divisor_left_targetsecond)) * ac) + (pfrep_right_gcd_unique_G_gcd_divisor_left_target)))))) \\/ (((exists pfrep_gap_gcd_unique_G_gcd_divisor_left_targetsecondoutside. pfrep_gap_gcd_unique_G_gcd_divisor_left_targetsecondoutside+(L)=(pfrep_power_gcd_unique_G_gcd_divisor_left_target)) /\\ (((pfrep_right_gcd_unique_G_gcd_divisor_left_target)=0))))) -> pfrep_left_gcd_unique_G_gcd_divisor_left_target=pfrep_right_gcd_unique_G_gcd_divisor_left_target))))))) /\\ ((((forall fom_index_pfp_gcd_unique_G_gcd_divisor_right_canonical. (exists fom_gap_pfp_gcd_unique_G_gcd_divisor_right_canonical_index_bound. fom_gap_pfp_gcd_unique_G_gcd_divisor_right_canonical_index_bound + S (fom_index_pfp_gcd_unique_G_gcd_divisor_right_canonical) = M) -> exists fom_value_pfp_gcd_unique_G_gcd_divisor_right_canonical. ((((exists fom_beta_height_pfp_gcd_unique_G_gcd_divisor_right_canonical_entry. fom_beta_height_pfp_gcd_unique_G_gcd_divisor_right_canonical_entry + S (fom_value_pfp_gcd_unique_G_gcd_divisor_right_canonical) = S ((S (fom_index_pfp_gcd_unique_G_gcd_divisor_right_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_gcd_divisor_right_canonical_entry. bb = fom_beta_quotient_pfp_gcd_unique_G_gcd_divisor_right_canonical_entry * S ((S (fom_index_pfp_gcd_unique_G_gcd_divisor_right_canonical)) * bc) + (fom_value_pfp_gcd_unique_G_gcd_divisor_right_canonical))) /\\ (exists fom_gap_pfp_gcd_unique_G_gcd_divisor_right_canonical_value_bound. fom_gap_pfp_gcd_unique_G_gcd_divisor_right_canonical_value_bound + S (fom_value_pfp_gcd_unique_G_gcd_divisor_right_canonical) = p))) /\\ ((exists pfgu_qb_gcd_unique_G_gcd_divisor_right pfgu_qc_gcd_unique_G_gcd_divisor_right pfgu_Q_gcd_unique_G_gcd_divisor_right pfgu_pb_gcd_unique_G_gcd_divisor_right pfgu_pc_gcd_unique_G_gcd_divisor_right pfgu_P_gcd_unique_G_gcd_divisor_right. ((((forall fom_index_pfp_gcd_unique_G_gcd_divisor_right_productleft. (exists fom_gap_pfp_gcd_unique_G_gcd_divisor_right_productleft_index_bound. fom_gap_pfp_gcd_unique_G_gcd_divisor_right_productleft_index_bound + S (fom_index_pfp_gcd_unique_G_gcd_divisor_right_productleft) = pfgu_Q_gcd_unique_G_gcd_divisor_right) -> exists fom_value_pfp_gcd_unique_G_gcd_divisor_right_productleft. ((((exists fom_beta_height_pfp_gcd_unique_G_gcd_divisor_right_productleft_entry. fom_beta_height_pfp_gcd_unique_G_gcd_divisor_right_productleft_entry + S (fom_value_pfp_gcd_unique_G_gcd_divisor_right_productleft) = S ((S (fom_index_pfp_gcd_unique_G_gcd_divisor_right_productleft)) * pfgu_qc_gcd_unique_G_gcd_divisor_right)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_gcd_divisor_right_productleft_entry. pfgu_qb_gcd_unique_G_gcd_divisor_right = fom_beta_quotient_pfp_gcd_unique_G_gcd_divisor_right_productleft_entry * S ((S (fom_index_pfp_gcd_unique_G_gcd_divisor_right_productleft)) * pfgu_qc_gcd_unique_G_gcd_divisor_right) + (fom_value_pfp_gcd_unique_G_gcd_divisor_right_productleft))) /\\ (exists fom_gap_pfp_gcd_unique_G_gcd_divisor_right_productleft_value_bound. fom_gap_pfp_gcd_unique_G_gcd_divisor_right_productleft_value_bound + S (fom_value_pfp_gcd_unique_G_gcd_divisor_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_unique_G_gcd_divisor_right_productright. (exists fom_gap_pfp_gcd_unique_G_gcd_divisor_right_productright_index_bound. fom_gap_pfp_gcd_unique_G_gcd_divisor_right_productright_index_bound + S (fom_index_pfp_gcd_unique_G_gcd_divisor_right_productright) = pfgu_D_gcd_unique_G_gcd) -> exists fom_value_pfp_gcd_unique_G_gcd_divisor_right_productright. ((((exists fom_beta_height_pfp_gcd_unique_G_gcd_divisor_right_productright_entry. fom_beta_height_pfp_gcd_unique_G_gcd_divisor_right_productright_entry + S (fom_value_pfp_gcd_unique_G_gcd_divisor_right_productright) = S ((S (fom_index_pfp_gcd_unique_G_gcd_divisor_right_productright)) * pfgu_dc_gcd_unique_G_gcd)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_gcd_divisor_right_productright_entry. pfgu_db_gcd_unique_G_gcd = fom_beta_quotient_pfp_gcd_unique_G_gcd_divisor_right_productright_entry * S ((S (fom_index_pfp_gcd_unique_G_gcd_divisor_right_productright)) * pfgu_dc_gcd_unique_G_gcd) + (fom_value_pfp_gcd_unique_G_gcd_divisor_right_productright))) /\\ (exists fom_gap_pfp_gcd_unique_G_gcd_divisor_right_productright_value_bound. fom_gap_pfp_gcd_unique_G_gcd_divisor_right_productright_value_bound + S (fom_value_pfp_gcd_unique_G_gcd_divisor_right_productright) = p))) /\\ (((((((pfgu_Q_gcd_unique_G_gcd_divisor_right)=0 \\/ (pfgu_D_gcd_unique_G_gcd)=0) /\\ (((pfgu_P_gcd_unique_G_gcd_divisor_right)=0)))) \\/ (((~((pfgu_Q_gcd_unique_G_gcd_divisor_right)=0)) /\\ (((~((pfgu_D_gcd_unique_G_gcd)=0)) /\\ (((pfgu_Q_gcd_unique_G_gcd_divisor_right)+(pfgu_D_gcd_unique_G_gcd)=S (pfgu_P_gcd_unique_G_gcd_divisor_right)))))))) /\\ ((forall pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficients. (exists pfa_gap_gcd_unique_G_gcd_divisor_right_productcoefficientsbound. pfa_gap_gcd_unique_G_gcd_divisor_right_productcoefficientsbound + S (pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficients) = (pfgu_P_gcd_unique_G_gcd_divisor_right)) -> exists pfc_value_gcd_unique_G_gcd_divisor_right_productcoefficients. ((((exists ff_h_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientsentry. ff_h_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientsentry + S (pfc_value_gcd_unique_G_gcd_divisor_right_productcoefficients) = S ((S (pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficients)) * pfgu_pc_gcd_unique_G_gcd_divisor_right)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientsentry. pfgu_pb_gcd_unique_G_gcd_divisor_right = ff_q_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientsentry * S ((S (pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficients)) * pfgu_pc_gcd_unique_G_gcd_divisor_right) + (pfc_value_gcd_unique_G_gcd_divisor_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficient pfc_terms_scale_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficient pfc_natural_sum_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficient. ((forall pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficients))) -> exists pfc_value_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficient = ff_q_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficient) + (pfc_value_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonal) = (pfgu_Q_gcd_unique_G_gcd_divisor_right)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_G_gcd_divisor_right)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_gcd_unique_G_gcd_divisor_right = ff_q_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_G_gcd_divisor_right) + (pfc_left_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_gcd_unique_G_gcd_divisor_right)=(pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm) = (pfgu_D_gcd_unique_G_gcd)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)) * pfgu_dc_gcd_unique_G_gcd)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightentry. pfgu_db_gcd_unique_G_gcd = ff_q_pfp_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)) * pfgu_dc_gcd_unique_G_gcd) + (pfc_right_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightoutside+(pfgu_D_gcd_unique_G_gcd)=(pfc_complement_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum fs_v_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficients))) * fs_v_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficients))) * fs_v_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_unique_G_gcd_divisor_right_productcoefficients)) -> exists fs_a_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficient = fs_q_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_unique_G_gcd_divisor_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_unique_G_gcd_divisor_right_productcoefficients) + (p) * pfa_offset_right_gcd_unique_G_gcd_divisor_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_unique_G_gcd_divisor_right_target pfrep_left_gcd_unique_G_gcd_divisor_right_target pfrep_right_gcd_unique_G_gcd_divisor_right_target. ((exists pfrep_position_gcd_unique_G_gcd_divisor_right_targetfirst. ((pfrep_position_gcd_unique_G_gcd_divisor_right_targetfirst+S (pfrep_power_gcd_unique_G_gcd_divisor_right_target)=(pfgu_P_gcd_unique_G_gcd_divisor_right)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_divisor_right_targetfirstentry. ff_h_pfp_gcd_unique_G_gcd_divisor_right_targetfirstentry + S (pfrep_left_gcd_unique_G_gcd_divisor_right_target) = S ((S (pfrep_position_gcd_unique_G_gcd_divisor_right_targetfirst)) * pfgu_pc_gcd_unique_G_gcd_divisor_right)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_divisor_right_targetfirstentry. pfgu_pb_gcd_unique_G_gcd_divisor_right = ff_q_pfp_gcd_unique_G_gcd_divisor_right_targetfirstentry * S ((S (pfrep_position_gcd_unique_G_gcd_divisor_right_targetfirst)) * pfgu_pc_gcd_unique_G_gcd_divisor_right) + (pfrep_left_gcd_unique_G_gcd_divisor_right_target)))))) \\/ (((exists pfrep_gap_gcd_unique_G_gcd_divisor_right_targetfirstoutside. pfrep_gap_gcd_unique_G_gcd_divisor_right_targetfirstoutside+(pfgu_P_gcd_unique_G_gcd_divisor_right)=(pfrep_power_gcd_unique_G_gcd_divisor_right_target)) /\\ (((pfrep_left_gcd_unique_G_gcd_divisor_right_target)=0))))) -> ((exists pfrep_position_gcd_unique_G_gcd_divisor_right_targetsecond. ((pfrep_position_gcd_unique_G_gcd_divisor_right_targetsecond+S (pfrep_power_gcd_unique_G_gcd_divisor_right_target)=(M)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_divisor_right_targetsecondentry. ff_h_pfp_gcd_unique_G_gcd_divisor_right_targetsecondentry + S (pfrep_right_gcd_unique_G_gcd_divisor_right_target) = S ((S (pfrep_position_gcd_unique_G_gcd_divisor_right_targetsecond)) * bc)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_divisor_right_targetsecondentry. bb = ff_q_pfp_gcd_unique_G_gcd_divisor_right_targetsecondentry * S ((S (pfrep_position_gcd_unique_G_gcd_divisor_right_targetsecond)) * bc) + (pfrep_right_gcd_unique_G_gcd_divisor_right_target)))))) \\/ (((exists pfrep_gap_gcd_unique_G_gcd_divisor_right_targetsecondoutside. pfrep_gap_gcd_unique_G_gcd_divisor_right_targetsecondoutside+(M)=(pfrep_power_gcd_unique_G_gcd_divisor_right_target)) /\\ (((pfrep_right_gcd_unique_G_gcd_divisor_right_target)=0))))) -> pfrep_left_gcd_unique_G_gcd_divisor_right_target=pfrep_right_gcd_unique_G_gcd_divisor_right_target)))))))))) -> (((forall fom_index_pfp_gcd_unique_G_gcd_greatest_canonical. (exists fom_gap_pfp_gcd_unique_G_gcd_greatest_canonical_index_bound. fom_gap_pfp_gcd_unique_G_gcd_greatest_canonical_index_bound + S (fom_index_pfp_gcd_unique_G_gcd_greatest_canonical) = G) -> exists fom_value_pfp_gcd_unique_G_gcd_greatest_canonical. ((((exists fom_beta_height_pfp_gcd_unique_G_gcd_greatest_canonical_entry. fom_beta_height_pfp_gcd_unique_G_gcd_greatest_canonical_entry + S (fom_value_pfp_gcd_unique_G_gcd_greatest_canonical) = S ((S (fom_index_pfp_gcd_unique_G_gcd_greatest_canonical)) * gc)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_gcd_greatest_canonical_entry. gb = fom_beta_quotient_pfp_gcd_unique_G_gcd_greatest_canonical_entry * S ((S (fom_index_pfp_gcd_unique_G_gcd_greatest_canonical)) * gc) + (fom_value_pfp_gcd_unique_G_gcd_greatest_canonical))) /\\ (exists fom_gap_pfp_gcd_unique_G_gcd_greatest_canonical_value_bound. fom_gap_pfp_gcd_unique_G_gcd_greatest_canonical_value_bound + S (fom_value_pfp_gcd_unique_G_gcd_greatest_canonical) = p))) /\\ ((exists pfgu_qb_gcd_unique_G_gcd_greatest pfgu_qc_gcd_unique_G_gcd_greatest pfgu_Q_gcd_unique_G_gcd_greatest pfgu_pb_gcd_unique_G_gcd_greatest pfgu_pc_gcd_unique_G_gcd_greatest pfgu_P_gcd_unique_G_gcd_greatest. ((((forall fom_index_pfp_gcd_unique_G_gcd_greatest_productleft. (exists fom_gap_pfp_gcd_unique_G_gcd_greatest_productleft_index_bound. fom_gap_pfp_gcd_unique_G_gcd_greatest_productleft_index_bound + S (fom_index_pfp_gcd_unique_G_gcd_greatest_productleft) = pfgu_Q_gcd_unique_G_gcd_greatest) -> exists fom_value_pfp_gcd_unique_G_gcd_greatest_productleft. ((((exists fom_beta_height_pfp_gcd_unique_G_gcd_greatest_productleft_entry. fom_beta_height_pfp_gcd_unique_G_gcd_greatest_productleft_entry + S (fom_value_pfp_gcd_unique_G_gcd_greatest_productleft) = S ((S (fom_index_pfp_gcd_unique_G_gcd_greatest_productleft)) * pfgu_qc_gcd_unique_G_gcd_greatest)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_gcd_greatest_productleft_entry. pfgu_qb_gcd_unique_G_gcd_greatest = fom_beta_quotient_pfp_gcd_unique_G_gcd_greatest_productleft_entry * S ((S (fom_index_pfp_gcd_unique_G_gcd_greatest_productleft)) * pfgu_qc_gcd_unique_G_gcd_greatest) + (fom_value_pfp_gcd_unique_G_gcd_greatest_productleft))) /\\ (exists fom_gap_pfp_gcd_unique_G_gcd_greatest_productleft_value_bound. fom_gap_pfp_gcd_unique_G_gcd_greatest_productleft_value_bound + S (fom_value_pfp_gcd_unique_G_gcd_greatest_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_unique_G_gcd_greatest_productright. (exists fom_gap_pfp_gcd_unique_G_gcd_greatest_productright_index_bound. fom_gap_pfp_gcd_unique_G_gcd_greatest_productright_index_bound + S (fom_index_pfp_gcd_unique_G_gcd_greatest_productright) = pfgu_D_gcd_unique_G_gcd) -> exists fom_value_pfp_gcd_unique_G_gcd_greatest_productright. ((((exists fom_beta_height_pfp_gcd_unique_G_gcd_greatest_productright_entry. fom_beta_height_pfp_gcd_unique_G_gcd_greatest_productright_entry + S (fom_value_pfp_gcd_unique_G_gcd_greatest_productright) = S ((S (fom_index_pfp_gcd_unique_G_gcd_greatest_productright)) * pfgu_dc_gcd_unique_G_gcd)) /\\ exists fom_beta_quotient_pfp_gcd_unique_G_gcd_greatest_productright_entry. pfgu_db_gcd_unique_G_gcd = fom_beta_quotient_pfp_gcd_unique_G_gcd_greatest_productright_entry * S ((S (fom_index_pfp_gcd_unique_G_gcd_greatest_productright)) * pfgu_dc_gcd_unique_G_gcd) + (fom_value_pfp_gcd_unique_G_gcd_greatest_productright))) /\\ (exists fom_gap_pfp_gcd_unique_G_gcd_greatest_productright_value_bound. fom_gap_pfp_gcd_unique_G_gcd_greatest_productright_value_bound + S (fom_value_pfp_gcd_unique_G_gcd_greatest_productright) = p))) /\\ (((((((pfgu_Q_gcd_unique_G_gcd_greatest)=0 \\/ (pfgu_D_gcd_unique_G_gcd)=0) /\\ (((pfgu_P_gcd_unique_G_gcd_greatest)=0)))) \\/ (((~((pfgu_Q_gcd_unique_G_gcd_greatest)=0)) /\\ (((~((pfgu_D_gcd_unique_G_gcd)=0)) /\\ (((pfgu_Q_gcd_unique_G_gcd_greatest)+(pfgu_D_gcd_unique_G_gcd)=S (pfgu_P_gcd_unique_G_gcd_greatest)))))))) /\\ ((forall pfc_index_gcd_unique_G_gcd_greatest_productcoefficients. (exists pfa_gap_gcd_unique_G_gcd_greatest_productcoefficientsbound. pfa_gap_gcd_unique_G_gcd_greatest_productcoefficientsbound + S (pfc_index_gcd_unique_G_gcd_greatest_productcoefficients) = (pfgu_P_gcd_unique_G_gcd_greatest)) -> exists pfc_value_gcd_unique_G_gcd_greatest_productcoefficients. ((((exists ff_h_pfp_gcd_unique_G_gcd_greatest_productcoefficientsentry. ff_h_pfp_gcd_unique_G_gcd_greatest_productcoefficientsentry + S (pfc_value_gcd_unique_G_gcd_greatest_productcoefficients) = S ((S (pfc_index_gcd_unique_G_gcd_greatest_productcoefficients)) * pfgu_pc_gcd_unique_G_gcd_greatest)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_greatest_productcoefficientsentry. pfgu_pb_gcd_unique_G_gcd_greatest = ff_q_pfp_gcd_unique_G_gcd_greatest_productcoefficientsentry * S ((S (pfc_index_gcd_unique_G_gcd_greatest_productcoefficients)) * pfgu_pc_gcd_unique_G_gcd_greatest) + (pfc_value_gcd_unique_G_gcd_greatest_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_unique_G_gcd_greatest_productcoefficientscoefficient pfc_terms_scale_gcd_unique_G_gcd_greatest_productcoefficientscoefficient pfc_natural_sum_gcd_unique_G_gcd_greatest_productcoefficientscoefficient. ((forall pfc_index_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_unique_G_gcd_greatest_productcoefficients))) -> exists pfc_value_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_G_gcd_greatest_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_unique_G_gcd_greatest_productcoefficientscoefficient = ff_q_pfp_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_G_gcd_greatest_productcoefficientscoefficient) + (pfc_value_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm pfc_left_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm pfc_right_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_unique_G_gcd_greatest_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonal) = (pfgu_Q_gcd_unique_G_gcd_greatest)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_G_gcd_greatest)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_gcd_unique_G_gcd_greatest = ff_q_pfp_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_G_gcd_greatest) + (pfc_left_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_gcd_unique_G_gcd_greatest)=(pfc_index_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm) = (pfgu_D_gcd_unique_G_gcd)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm)) * pfgu_dc_gcd_unique_G_gcd)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermrightentry. pfgu_db_gcd_unique_G_gcd = ff_q_pfp_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm)) * pfgu_dc_gcd_unique_G_gcd) + (pfc_right_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonaltermrightoutside+(pfgu_D_gcd_unique_G_gcd)=(pfc_complement_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonal)=pfc_left_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_unique_G_gcd_greatest_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum fs_v_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_unique_G_gcd_greatest_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_unique_G_gcd_greatest_productcoefficients))) * fs_v_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_unique_G_gcd_greatest_productcoefficients))) * fs_v_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_unique_G_gcd_greatest_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_unique_G_gcd_greatest_productcoefficients)) -> exists fs_a_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_G_gcd_greatest_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_unique_G_gcd_greatest_productcoefficientscoefficient = fs_q_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_G_gcd_greatest_productcoefficientscoefficient) + (fs_a_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_unique_G_gcd_greatest_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_unique_G_gcd_greatest_productcoefficientscoefficientresiduebound. pfa_gap_gcd_unique_G_gcd_greatest_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_unique_G_gcd_greatest_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_unique_G_gcd_greatest_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_unique_G_gcd_greatest_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_unique_G_gcd_greatest_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_unique_G_gcd_greatest_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_unique_G_gcd_greatest_productcoefficients) + (p) * pfa_offset_right_gcd_unique_G_gcd_greatest_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_unique_G_gcd_greatest_target pfrep_left_gcd_unique_G_gcd_greatest_target pfrep_right_gcd_unique_G_gcd_greatest_target. ((exists pfrep_position_gcd_unique_G_gcd_greatest_targetfirst. ((pfrep_position_gcd_unique_G_gcd_greatest_targetfirst+S (pfrep_power_gcd_unique_G_gcd_greatest_target)=(pfgu_P_gcd_unique_G_gcd_greatest)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_greatest_targetfirstentry. ff_h_pfp_gcd_unique_G_gcd_greatest_targetfirstentry + S (pfrep_left_gcd_unique_G_gcd_greatest_target) = S ((S (pfrep_position_gcd_unique_G_gcd_greatest_targetfirst)) * pfgu_pc_gcd_unique_G_gcd_greatest)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_greatest_targetfirstentry. pfgu_pb_gcd_unique_G_gcd_greatest = ff_q_pfp_gcd_unique_G_gcd_greatest_targetfirstentry * S ((S (pfrep_position_gcd_unique_G_gcd_greatest_targetfirst)) * pfgu_pc_gcd_unique_G_gcd_greatest) + (pfrep_left_gcd_unique_G_gcd_greatest_target)))))) \\/ (((exists pfrep_gap_gcd_unique_G_gcd_greatest_targetfirstoutside. pfrep_gap_gcd_unique_G_gcd_greatest_targetfirstoutside+(pfgu_P_gcd_unique_G_gcd_greatest)=(pfrep_power_gcd_unique_G_gcd_greatest_target)) /\\ (((pfrep_left_gcd_unique_G_gcd_greatest_target)=0))))) -> ((exists pfrep_position_gcd_unique_G_gcd_greatest_targetsecond. ((pfrep_position_gcd_unique_G_gcd_greatest_targetsecond+S (pfrep_power_gcd_unique_G_gcd_greatest_target)=(G)) /\\ ((((exists ff_h_pfp_gcd_unique_G_gcd_greatest_targetsecondentry. ff_h_pfp_gcd_unique_G_gcd_greatest_targetsecondentry + S (pfrep_right_gcd_unique_G_gcd_greatest_target) = S ((S (pfrep_position_gcd_unique_G_gcd_greatest_targetsecond)) * gc)) /\\ exists ff_q_pfp_gcd_unique_G_gcd_greatest_targetsecondentry. gb = ff_q_pfp_gcd_unique_G_gcd_greatest_targetsecondentry * S ((S (pfrep_position_gcd_unique_G_gcd_greatest_targetsecond)) * gc) + (pfrep_right_gcd_unique_G_gcd_greatest_target)))))) \\/ (((exists pfrep_gap_gcd_unique_G_gcd_greatest_targetsecondoutside. pfrep_gap_gcd_unique_G_gcd_greatest_targetsecondoutside+(G)=(pfrep_power_gcd_unique_G_gcd_greatest_target)) /\\ (((pfrep_right_gcd_unique_G_gcd_greatest_target)=0))))) -> pfrep_left_gcd_unique_G_gcd_greatest_target=pfrep_right_gcd_unique_G_gcd_greatest_target)))))))))))))) -> ((((H)=0 \\/ (((~((H) = 0)) /\\ (((forall fom_index_pfp_gcd_unique_H_normal_moniccoefficients. (exists fom_gap_pfp_gcd_unique_H_normal_moniccoefficients_index_bound. fom_gap_pfp_gcd_unique_H_normal_moniccoefficients_index_bound + S (fom_index_pfp_gcd_unique_H_normal_moniccoefficients) = H) -> exists fom_value_pfp_gcd_unique_H_normal_moniccoefficients. ((((exists fom_beta_height_pfp_gcd_unique_H_normal_moniccoefficients_entry. fom_beta_height_pfp_gcd_unique_H_normal_moniccoefficients_entry + S (fom_value_pfp_gcd_unique_H_normal_moniccoefficients) = S ((S (fom_index_pfp_gcd_unique_H_normal_moniccoefficients)) * hc)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_normal_moniccoefficients_entry. hb = fom_beta_quotient_pfp_gcd_unique_H_normal_moniccoefficients_entry * S ((S (fom_index_pfp_gcd_unique_H_normal_moniccoefficients)) * hc) + (fom_value_pfp_gcd_unique_H_normal_moniccoefficients))) /\\ (exists fom_gap_pfp_gcd_unique_H_normal_moniccoefficients_value_bound. fom_gap_pfp_gcd_unique_H_normal_moniccoefficients_value_bound + S (fom_value_pfp_gcd_unique_H_normal_moniccoefficients) = p))) /\\ ((((exists ff_h_pfp_gcd_unique_H_normal_monicleading. ff_h_pfp_gcd_unique_H_normal_monicleading + S (1) = S ((S (0)) * hc)) /\\ exists ff_q_pfp_gcd_unique_H_normal_monicleading. hb = ff_q_pfp_gcd_unique_H_normal_monicleading * S ((S (0)) * hc) + (1))))))))) /\\ ((((((((forall fom_index_pfp_gcd_unique_H_gcd_common_left_canonical. (exists fom_gap_pfp_gcd_unique_H_gcd_common_left_canonical_index_bound. fom_gap_pfp_gcd_unique_H_gcd_common_left_canonical_index_bound + S (fom_index_pfp_gcd_unique_H_gcd_common_left_canonical) = L) -> exists fom_value_pfp_gcd_unique_H_gcd_common_left_canonical. ((((exists fom_beta_height_pfp_gcd_unique_H_gcd_common_left_canonical_entry. fom_beta_height_pfp_gcd_unique_H_gcd_common_left_canonical_entry + S (fom_value_pfp_gcd_unique_H_gcd_common_left_canonical) = S ((S (fom_index_pfp_gcd_unique_H_gcd_common_left_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_gcd_common_left_canonical_entry. ab = fom_beta_quotient_pfp_gcd_unique_H_gcd_common_left_canonical_entry * S ((S (fom_index_pfp_gcd_unique_H_gcd_common_left_canonical)) * ac) + (fom_value_pfp_gcd_unique_H_gcd_common_left_canonical))) /\\ (exists fom_gap_pfp_gcd_unique_H_gcd_common_left_canonical_value_bound. fom_gap_pfp_gcd_unique_H_gcd_common_left_canonical_value_bound + S (fom_value_pfp_gcd_unique_H_gcd_common_left_canonical) = p))) /\\ ((exists pfgu_qb_gcd_unique_H_gcd_common_left pfgu_qc_gcd_unique_H_gcd_common_left pfgu_Q_gcd_unique_H_gcd_common_left pfgu_pb_gcd_unique_H_gcd_common_left pfgu_pc_gcd_unique_H_gcd_common_left pfgu_P_gcd_unique_H_gcd_common_left. ((((forall fom_index_pfp_gcd_unique_H_gcd_common_left_productleft. (exists fom_gap_pfp_gcd_unique_H_gcd_common_left_productleft_index_bound. fom_gap_pfp_gcd_unique_H_gcd_common_left_productleft_index_bound + S (fom_index_pfp_gcd_unique_H_gcd_common_left_productleft) = pfgu_Q_gcd_unique_H_gcd_common_left) -> exists fom_value_pfp_gcd_unique_H_gcd_common_left_productleft. ((((exists fom_beta_height_pfp_gcd_unique_H_gcd_common_left_productleft_entry. fom_beta_height_pfp_gcd_unique_H_gcd_common_left_productleft_entry + S (fom_value_pfp_gcd_unique_H_gcd_common_left_productleft) = S ((S (fom_index_pfp_gcd_unique_H_gcd_common_left_productleft)) * pfgu_qc_gcd_unique_H_gcd_common_left)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_gcd_common_left_productleft_entry. pfgu_qb_gcd_unique_H_gcd_common_left = fom_beta_quotient_pfp_gcd_unique_H_gcd_common_left_productleft_entry * S ((S (fom_index_pfp_gcd_unique_H_gcd_common_left_productleft)) * pfgu_qc_gcd_unique_H_gcd_common_left) + (fom_value_pfp_gcd_unique_H_gcd_common_left_productleft))) /\\ (exists fom_gap_pfp_gcd_unique_H_gcd_common_left_productleft_value_bound. fom_gap_pfp_gcd_unique_H_gcd_common_left_productleft_value_bound + S (fom_value_pfp_gcd_unique_H_gcd_common_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_unique_H_gcd_common_left_productright. (exists fom_gap_pfp_gcd_unique_H_gcd_common_left_productright_index_bound. fom_gap_pfp_gcd_unique_H_gcd_common_left_productright_index_bound + S (fom_index_pfp_gcd_unique_H_gcd_common_left_productright) = H) -> exists fom_value_pfp_gcd_unique_H_gcd_common_left_productright. ((((exists fom_beta_height_pfp_gcd_unique_H_gcd_common_left_productright_entry. fom_beta_height_pfp_gcd_unique_H_gcd_common_left_productright_entry + S (fom_value_pfp_gcd_unique_H_gcd_common_left_productright) = S ((S (fom_index_pfp_gcd_unique_H_gcd_common_left_productright)) * hc)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_gcd_common_left_productright_entry. hb = fom_beta_quotient_pfp_gcd_unique_H_gcd_common_left_productright_entry * S ((S (fom_index_pfp_gcd_unique_H_gcd_common_left_productright)) * hc) + (fom_value_pfp_gcd_unique_H_gcd_common_left_productright))) /\\ (exists fom_gap_pfp_gcd_unique_H_gcd_common_left_productright_value_bound. fom_gap_pfp_gcd_unique_H_gcd_common_left_productright_value_bound + S (fom_value_pfp_gcd_unique_H_gcd_common_left_productright) = p))) /\\ (((((((pfgu_Q_gcd_unique_H_gcd_common_left)=0 \\/ (H)=0) /\\ (((pfgu_P_gcd_unique_H_gcd_common_left)=0)))) \\/ (((~((pfgu_Q_gcd_unique_H_gcd_common_left)=0)) /\\ (((~((H)=0)) /\\ (((pfgu_Q_gcd_unique_H_gcd_common_left)+(H)=S (pfgu_P_gcd_unique_H_gcd_common_left)))))))) /\\ ((forall pfc_index_gcd_unique_H_gcd_common_left_productcoefficients. (exists pfa_gap_gcd_unique_H_gcd_common_left_productcoefficientsbound. pfa_gap_gcd_unique_H_gcd_common_left_productcoefficientsbound + S (pfc_index_gcd_unique_H_gcd_common_left_productcoefficients) = (pfgu_P_gcd_unique_H_gcd_common_left)) -> exists pfc_value_gcd_unique_H_gcd_common_left_productcoefficients. ((((exists ff_h_pfp_gcd_unique_H_gcd_common_left_productcoefficientsentry. ff_h_pfp_gcd_unique_H_gcd_common_left_productcoefficientsentry + S (pfc_value_gcd_unique_H_gcd_common_left_productcoefficients) = S ((S (pfc_index_gcd_unique_H_gcd_common_left_productcoefficients)) * pfgu_pc_gcd_unique_H_gcd_common_left)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_common_left_productcoefficientsentry. pfgu_pb_gcd_unique_H_gcd_common_left = ff_q_pfp_gcd_unique_H_gcd_common_left_productcoefficientsentry * S ((S (pfc_index_gcd_unique_H_gcd_common_left_productcoefficients)) * pfgu_pc_gcd_unique_H_gcd_common_left) + (pfc_value_gcd_unique_H_gcd_common_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_unique_H_gcd_common_left_productcoefficientscoefficient pfc_terms_scale_gcd_unique_H_gcd_common_left_productcoefficientscoefficient pfc_natural_sum_gcd_unique_H_gcd_common_left_productcoefficientscoefficient. ((forall pfc_index_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_unique_H_gcd_common_left_productcoefficients))) -> exists pfc_value_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_H_gcd_common_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_unique_H_gcd_common_left_productcoefficientscoefficient = ff_q_pfp_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_H_gcd_common_left_productcoefficientscoefficient) + (pfc_value_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_unique_H_gcd_common_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonal) = (pfgu_Q_gcd_unique_H_gcd_common_left)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_H_gcd_common_left)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_gcd_unique_H_gcd_common_left = ff_q_pfp_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_H_gcd_common_left) + (pfc_left_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_gcd_unique_H_gcd_common_left)=(pfc_index_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm) = (H)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm)) * hc)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermrightentry. hb = ff_q_pfp_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm)) * hc) + (pfc_right_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonaltermrightoutside+(H)=(pfc_complement_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_unique_H_gcd_common_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum fs_v_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_unique_H_gcd_common_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_unique_H_gcd_common_left_productcoefficients))) * fs_v_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_unique_H_gcd_common_left_productcoefficients))) * fs_v_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_unique_H_gcd_common_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_unique_H_gcd_common_left_productcoefficients)) -> exists fs_a_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_H_gcd_common_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_unique_H_gcd_common_left_productcoefficientscoefficient = fs_q_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_H_gcd_common_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_unique_H_gcd_common_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_unique_H_gcd_common_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_unique_H_gcd_common_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_unique_H_gcd_common_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_unique_H_gcd_common_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_unique_H_gcd_common_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_unique_H_gcd_common_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_unique_H_gcd_common_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_unique_H_gcd_common_left_productcoefficients) + (p) * pfa_offset_right_gcd_unique_H_gcd_common_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_unique_H_gcd_common_left_target pfrep_left_gcd_unique_H_gcd_common_left_target pfrep_right_gcd_unique_H_gcd_common_left_target. ((exists pfrep_position_gcd_unique_H_gcd_common_left_targetfirst. ((pfrep_position_gcd_unique_H_gcd_common_left_targetfirst+S (pfrep_power_gcd_unique_H_gcd_common_left_target)=(pfgu_P_gcd_unique_H_gcd_common_left)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_common_left_targetfirstentry. ff_h_pfp_gcd_unique_H_gcd_common_left_targetfirstentry + S (pfrep_left_gcd_unique_H_gcd_common_left_target) = S ((S (pfrep_position_gcd_unique_H_gcd_common_left_targetfirst)) * pfgu_pc_gcd_unique_H_gcd_common_left)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_common_left_targetfirstentry. pfgu_pb_gcd_unique_H_gcd_common_left = ff_q_pfp_gcd_unique_H_gcd_common_left_targetfirstentry * S ((S (pfrep_position_gcd_unique_H_gcd_common_left_targetfirst)) * pfgu_pc_gcd_unique_H_gcd_common_left) + (pfrep_left_gcd_unique_H_gcd_common_left_target)))))) \\/ (((exists pfrep_gap_gcd_unique_H_gcd_common_left_targetfirstoutside. pfrep_gap_gcd_unique_H_gcd_common_left_targetfirstoutside+(pfgu_P_gcd_unique_H_gcd_common_left)=(pfrep_power_gcd_unique_H_gcd_common_left_target)) /\\ (((pfrep_left_gcd_unique_H_gcd_common_left_target)=0))))) -> ((exists pfrep_position_gcd_unique_H_gcd_common_left_targetsecond. ((pfrep_position_gcd_unique_H_gcd_common_left_targetsecond+S (pfrep_power_gcd_unique_H_gcd_common_left_target)=(L)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_common_left_targetsecondentry. ff_h_pfp_gcd_unique_H_gcd_common_left_targetsecondentry + S (pfrep_right_gcd_unique_H_gcd_common_left_target) = S ((S (pfrep_position_gcd_unique_H_gcd_common_left_targetsecond)) * ac)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_common_left_targetsecondentry. ab = ff_q_pfp_gcd_unique_H_gcd_common_left_targetsecondentry * S ((S (pfrep_position_gcd_unique_H_gcd_common_left_targetsecond)) * ac) + (pfrep_right_gcd_unique_H_gcd_common_left_target)))))) \\/ (((exists pfrep_gap_gcd_unique_H_gcd_common_left_targetsecondoutside. pfrep_gap_gcd_unique_H_gcd_common_left_targetsecondoutside+(L)=(pfrep_power_gcd_unique_H_gcd_common_left_target)) /\\ (((pfrep_right_gcd_unique_H_gcd_common_left_target)=0))))) -> pfrep_left_gcd_unique_H_gcd_common_left_target=pfrep_right_gcd_unique_H_gcd_common_left_target))))))) /\\ ((((forall fom_index_pfp_gcd_unique_H_gcd_common_right_canonical. (exists fom_gap_pfp_gcd_unique_H_gcd_common_right_canonical_index_bound. fom_gap_pfp_gcd_unique_H_gcd_common_right_canonical_index_bound + S (fom_index_pfp_gcd_unique_H_gcd_common_right_canonical) = M) -> exists fom_value_pfp_gcd_unique_H_gcd_common_right_canonical. ((((exists fom_beta_height_pfp_gcd_unique_H_gcd_common_right_canonical_entry. fom_beta_height_pfp_gcd_unique_H_gcd_common_right_canonical_entry + S (fom_value_pfp_gcd_unique_H_gcd_common_right_canonical) = S ((S (fom_index_pfp_gcd_unique_H_gcd_common_right_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_gcd_common_right_canonical_entry. bb = fom_beta_quotient_pfp_gcd_unique_H_gcd_common_right_canonical_entry * S ((S (fom_index_pfp_gcd_unique_H_gcd_common_right_canonical)) * bc) + (fom_value_pfp_gcd_unique_H_gcd_common_right_canonical))) /\\ (exists fom_gap_pfp_gcd_unique_H_gcd_common_right_canonical_value_bound. fom_gap_pfp_gcd_unique_H_gcd_common_right_canonical_value_bound + S (fom_value_pfp_gcd_unique_H_gcd_common_right_canonical) = p))) /\\ ((exists pfgu_qb_gcd_unique_H_gcd_common_right pfgu_qc_gcd_unique_H_gcd_common_right pfgu_Q_gcd_unique_H_gcd_common_right pfgu_pb_gcd_unique_H_gcd_common_right pfgu_pc_gcd_unique_H_gcd_common_right pfgu_P_gcd_unique_H_gcd_common_right. ((((forall fom_index_pfp_gcd_unique_H_gcd_common_right_productleft. (exists fom_gap_pfp_gcd_unique_H_gcd_common_right_productleft_index_bound. fom_gap_pfp_gcd_unique_H_gcd_common_right_productleft_index_bound + S (fom_index_pfp_gcd_unique_H_gcd_common_right_productleft) = pfgu_Q_gcd_unique_H_gcd_common_right) -> exists fom_value_pfp_gcd_unique_H_gcd_common_right_productleft. ((((exists fom_beta_height_pfp_gcd_unique_H_gcd_common_right_productleft_entry. fom_beta_height_pfp_gcd_unique_H_gcd_common_right_productleft_entry + S (fom_value_pfp_gcd_unique_H_gcd_common_right_productleft) = S ((S (fom_index_pfp_gcd_unique_H_gcd_common_right_productleft)) * pfgu_qc_gcd_unique_H_gcd_common_right)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_gcd_common_right_productleft_entry. pfgu_qb_gcd_unique_H_gcd_common_right = fom_beta_quotient_pfp_gcd_unique_H_gcd_common_right_productleft_entry * S ((S (fom_index_pfp_gcd_unique_H_gcd_common_right_productleft)) * pfgu_qc_gcd_unique_H_gcd_common_right) + (fom_value_pfp_gcd_unique_H_gcd_common_right_productleft))) /\\ (exists fom_gap_pfp_gcd_unique_H_gcd_common_right_productleft_value_bound. fom_gap_pfp_gcd_unique_H_gcd_common_right_productleft_value_bound + S (fom_value_pfp_gcd_unique_H_gcd_common_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_unique_H_gcd_common_right_productright. (exists fom_gap_pfp_gcd_unique_H_gcd_common_right_productright_index_bound. fom_gap_pfp_gcd_unique_H_gcd_common_right_productright_index_bound + S (fom_index_pfp_gcd_unique_H_gcd_common_right_productright) = H) -> exists fom_value_pfp_gcd_unique_H_gcd_common_right_productright. ((((exists fom_beta_height_pfp_gcd_unique_H_gcd_common_right_productright_entry. fom_beta_height_pfp_gcd_unique_H_gcd_common_right_productright_entry + S (fom_value_pfp_gcd_unique_H_gcd_common_right_productright) = S ((S (fom_index_pfp_gcd_unique_H_gcd_common_right_productright)) * hc)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_gcd_common_right_productright_entry. hb = fom_beta_quotient_pfp_gcd_unique_H_gcd_common_right_productright_entry * S ((S (fom_index_pfp_gcd_unique_H_gcd_common_right_productright)) * hc) + (fom_value_pfp_gcd_unique_H_gcd_common_right_productright))) /\\ (exists fom_gap_pfp_gcd_unique_H_gcd_common_right_productright_value_bound. fom_gap_pfp_gcd_unique_H_gcd_common_right_productright_value_bound + S (fom_value_pfp_gcd_unique_H_gcd_common_right_productright) = p))) /\\ (((((((pfgu_Q_gcd_unique_H_gcd_common_right)=0 \\/ (H)=0) /\\ (((pfgu_P_gcd_unique_H_gcd_common_right)=0)))) \\/ (((~((pfgu_Q_gcd_unique_H_gcd_common_right)=0)) /\\ (((~((H)=0)) /\\ (((pfgu_Q_gcd_unique_H_gcd_common_right)+(H)=S (pfgu_P_gcd_unique_H_gcd_common_right)))))))) /\\ ((forall pfc_index_gcd_unique_H_gcd_common_right_productcoefficients. (exists pfa_gap_gcd_unique_H_gcd_common_right_productcoefficientsbound. pfa_gap_gcd_unique_H_gcd_common_right_productcoefficientsbound + S (pfc_index_gcd_unique_H_gcd_common_right_productcoefficients) = (pfgu_P_gcd_unique_H_gcd_common_right)) -> exists pfc_value_gcd_unique_H_gcd_common_right_productcoefficients. ((((exists ff_h_pfp_gcd_unique_H_gcd_common_right_productcoefficientsentry. ff_h_pfp_gcd_unique_H_gcd_common_right_productcoefficientsentry + S (pfc_value_gcd_unique_H_gcd_common_right_productcoefficients) = S ((S (pfc_index_gcd_unique_H_gcd_common_right_productcoefficients)) * pfgu_pc_gcd_unique_H_gcd_common_right)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_common_right_productcoefficientsentry. pfgu_pb_gcd_unique_H_gcd_common_right = ff_q_pfp_gcd_unique_H_gcd_common_right_productcoefficientsentry * S ((S (pfc_index_gcd_unique_H_gcd_common_right_productcoefficients)) * pfgu_pc_gcd_unique_H_gcd_common_right) + (pfc_value_gcd_unique_H_gcd_common_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_unique_H_gcd_common_right_productcoefficientscoefficient pfc_terms_scale_gcd_unique_H_gcd_common_right_productcoefficientscoefficient pfc_natural_sum_gcd_unique_H_gcd_common_right_productcoefficientscoefficient. ((forall pfc_index_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_unique_H_gcd_common_right_productcoefficients))) -> exists pfc_value_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_H_gcd_common_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_unique_H_gcd_common_right_productcoefficientscoefficient = ff_q_pfp_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_H_gcd_common_right_productcoefficientscoefficient) + (pfc_value_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_unique_H_gcd_common_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonal) = (pfgu_Q_gcd_unique_H_gcd_common_right)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_H_gcd_common_right)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_gcd_unique_H_gcd_common_right = ff_q_pfp_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_H_gcd_common_right) + (pfc_left_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_gcd_unique_H_gcd_common_right)=(pfc_index_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm) = (H)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm)) * hc)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermrightentry. hb = ff_q_pfp_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm)) * hc) + (pfc_right_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonaltermrightoutside+(H)=(pfc_complement_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_unique_H_gcd_common_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum fs_v_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_unique_H_gcd_common_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_unique_H_gcd_common_right_productcoefficients))) * fs_v_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_unique_H_gcd_common_right_productcoefficients))) * fs_v_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_unique_H_gcd_common_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_unique_H_gcd_common_right_productcoefficients)) -> exists fs_a_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_H_gcd_common_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_unique_H_gcd_common_right_productcoefficientscoefficient = fs_q_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_H_gcd_common_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_unique_H_gcd_common_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_unique_H_gcd_common_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_unique_H_gcd_common_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_unique_H_gcd_common_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_unique_H_gcd_common_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_unique_H_gcd_common_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_unique_H_gcd_common_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_unique_H_gcd_common_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_unique_H_gcd_common_right_productcoefficients) + (p) * pfa_offset_right_gcd_unique_H_gcd_common_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_unique_H_gcd_common_right_target pfrep_left_gcd_unique_H_gcd_common_right_target pfrep_right_gcd_unique_H_gcd_common_right_target. ((exists pfrep_position_gcd_unique_H_gcd_common_right_targetfirst. ((pfrep_position_gcd_unique_H_gcd_common_right_targetfirst+S (pfrep_power_gcd_unique_H_gcd_common_right_target)=(pfgu_P_gcd_unique_H_gcd_common_right)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_common_right_targetfirstentry. ff_h_pfp_gcd_unique_H_gcd_common_right_targetfirstentry + S (pfrep_left_gcd_unique_H_gcd_common_right_target) = S ((S (pfrep_position_gcd_unique_H_gcd_common_right_targetfirst)) * pfgu_pc_gcd_unique_H_gcd_common_right)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_common_right_targetfirstentry. pfgu_pb_gcd_unique_H_gcd_common_right = ff_q_pfp_gcd_unique_H_gcd_common_right_targetfirstentry * S ((S (pfrep_position_gcd_unique_H_gcd_common_right_targetfirst)) * pfgu_pc_gcd_unique_H_gcd_common_right) + (pfrep_left_gcd_unique_H_gcd_common_right_target)))))) \\/ (((exists pfrep_gap_gcd_unique_H_gcd_common_right_targetfirstoutside. pfrep_gap_gcd_unique_H_gcd_common_right_targetfirstoutside+(pfgu_P_gcd_unique_H_gcd_common_right)=(pfrep_power_gcd_unique_H_gcd_common_right_target)) /\\ (((pfrep_left_gcd_unique_H_gcd_common_right_target)=0))))) -> ((exists pfrep_position_gcd_unique_H_gcd_common_right_targetsecond. ((pfrep_position_gcd_unique_H_gcd_common_right_targetsecond+S (pfrep_power_gcd_unique_H_gcd_common_right_target)=(M)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_common_right_targetsecondentry. ff_h_pfp_gcd_unique_H_gcd_common_right_targetsecondentry + S (pfrep_right_gcd_unique_H_gcd_common_right_target) = S ((S (pfrep_position_gcd_unique_H_gcd_common_right_targetsecond)) * bc)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_common_right_targetsecondentry. bb = ff_q_pfp_gcd_unique_H_gcd_common_right_targetsecondentry * S ((S (pfrep_position_gcd_unique_H_gcd_common_right_targetsecond)) * bc) + (pfrep_right_gcd_unique_H_gcd_common_right_target)))))) \\/ (((exists pfrep_gap_gcd_unique_H_gcd_common_right_targetsecondoutside. pfrep_gap_gcd_unique_H_gcd_common_right_targetsecondoutside+(M)=(pfrep_power_gcd_unique_H_gcd_common_right_target)) /\\ (((pfrep_right_gcd_unique_H_gcd_common_right_target)=0))))) -> pfrep_left_gcd_unique_H_gcd_common_right_target=pfrep_right_gcd_unique_H_gcd_common_right_target)))))))))) /\\ ((forall pfgu_db_gcd_unique_H_gcd pfgu_dc_gcd_unique_H_gcd pfgu_D_gcd_unique_H_gcd. (((((forall fom_index_pfp_gcd_unique_H_gcd_divisor_left_canonical. (exists fom_gap_pfp_gcd_unique_H_gcd_divisor_left_canonical_index_bound. fom_gap_pfp_gcd_unique_H_gcd_divisor_left_canonical_index_bound + S (fom_index_pfp_gcd_unique_H_gcd_divisor_left_canonical) = L) -> exists fom_value_pfp_gcd_unique_H_gcd_divisor_left_canonical. ((((exists fom_beta_height_pfp_gcd_unique_H_gcd_divisor_left_canonical_entry. fom_beta_height_pfp_gcd_unique_H_gcd_divisor_left_canonical_entry + S (fom_value_pfp_gcd_unique_H_gcd_divisor_left_canonical) = S ((S (fom_index_pfp_gcd_unique_H_gcd_divisor_left_canonical)) * ac)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_gcd_divisor_left_canonical_entry. ab = fom_beta_quotient_pfp_gcd_unique_H_gcd_divisor_left_canonical_entry * S ((S (fom_index_pfp_gcd_unique_H_gcd_divisor_left_canonical)) * ac) + (fom_value_pfp_gcd_unique_H_gcd_divisor_left_canonical))) /\\ (exists fom_gap_pfp_gcd_unique_H_gcd_divisor_left_canonical_value_bound. fom_gap_pfp_gcd_unique_H_gcd_divisor_left_canonical_value_bound + S (fom_value_pfp_gcd_unique_H_gcd_divisor_left_canonical) = p))) /\\ ((exists pfgu_qb_gcd_unique_H_gcd_divisor_left pfgu_qc_gcd_unique_H_gcd_divisor_left pfgu_Q_gcd_unique_H_gcd_divisor_left pfgu_pb_gcd_unique_H_gcd_divisor_left pfgu_pc_gcd_unique_H_gcd_divisor_left pfgu_P_gcd_unique_H_gcd_divisor_left. ((((forall fom_index_pfp_gcd_unique_H_gcd_divisor_left_productleft. (exists fom_gap_pfp_gcd_unique_H_gcd_divisor_left_productleft_index_bound. fom_gap_pfp_gcd_unique_H_gcd_divisor_left_productleft_index_bound + S (fom_index_pfp_gcd_unique_H_gcd_divisor_left_productleft) = pfgu_Q_gcd_unique_H_gcd_divisor_left) -> exists fom_value_pfp_gcd_unique_H_gcd_divisor_left_productleft. ((((exists fom_beta_height_pfp_gcd_unique_H_gcd_divisor_left_productleft_entry. fom_beta_height_pfp_gcd_unique_H_gcd_divisor_left_productleft_entry + S (fom_value_pfp_gcd_unique_H_gcd_divisor_left_productleft) = S ((S (fom_index_pfp_gcd_unique_H_gcd_divisor_left_productleft)) * pfgu_qc_gcd_unique_H_gcd_divisor_left)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_gcd_divisor_left_productleft_entry. pfgu_qb_gcd_unique_H_gcd_divisor_left = fom_beta_quotient_pfp_gcd_unique_H_gcd_divisor_left_productleft_entry * S ((S (fom_index_pfp_gcd_unique_H_gcd_divisor_left_productleft)) * pfgu_qc_gcd_unique_H_gcd_divisor_left) + (fom_value_pfp_gcd_unique_H_gcd_divisor_left_productleft))) /\\ (exists fom_gap_pfp_gcd_unique_H_gcd_divisor_left_productleft_value_bound. fom_gap_pfp_gcd_unique_H_gcd_divisor_left_productleft_value_bound + S (fom_value_pfp_gcd_unique_H_gcd_divisor_left_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_unique_H_gcd_divisor_left_productright. (exists fom_gap_pfp_gcd_unique_H_gcd_divisor_left_productright_index_bound. fom_gap_pfp_gcd_unique_H_gcd_divisor_left_productright_index_bound + S (fom_index_pfp_gcd_unique_H_gcd_divisor_left_productright) = pfgu_D_gcd_unique_H_gcd) -> exists fom_value_pfp_gcd_unique_H_gcd_divisor_left_productright. ((((exists fom_beta_height_pfp_gcd_unique_H_gcd_divisor_left_productright_entry. fom_beta_height_pfp_gcd_unique_H_gcd_divisor_left_productright_entry + S (fom_value_pfp_gcd_unique_H_gcd_divisor_left_productright) = S ((S (fom_index_pfp_gcd_unique_H_gcd_divisor_left_productright)) * pfgu_dc_gcd_unique_H_gcd)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_gcd_divisor_left_productright_entry. pfgu_db_gcd_unique_H_gcd = fom_beta_quotient_pfp_gcd_unique_H_gcd_divisor_left_productright_entry * S ((S (fom_index_pfp_gcd_unique_H_gcd_divisor_left_productright)) * pfgu_dc_gcd_unique_H_gcd) + (fom_value_pfp_gcd_unique_H_gcd_divisor_left_productright))) /\\ (exists fom_gap_pfp_gcd_unique_H_gcd_divisor_left_productright_value_bound. fom_gap_pfp_gcd_unique_H_gcd_divisor_left_productright_value_bound + S (fom_value_pfp_gcd_unique_H_gcd_divisor_left_productright) = p))) /\\ (((((((pfgu_Q_gcd_unique_H_gcd_divisor_left)=0 \\/ (pfgu_D_gcd_unique_H_gcd)=0) /\\ (((pfgu_P_gcd_unique_H_gcd_divisor_left)=0)))) \\/ (((~((pfgu_Q_gcd_unique_H_gcd_divisor_left)=0)) /\\ (((~((pfgu_D_gcd_unique_H_gcd)=0)) /\\ (((pfgu_Q_gcd_unique_H_gcd_divisor_left)+(pfgu_D_gcd_unique_H_gcd)=S (pfgu_P_gcd_unique_H_gcd_divisor_left)))))))) /\\ ((forall pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficients. (exists pfa_gap_gcd_unique_H_gcd_divisor_left_productcoefficientsbound. pfa_gap_gcd_unique_H_gcd_divisor_left_productcoefficientsbound + S (pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficients) = (pfgu_P_gcd_unique_H_gcd_divisor_left)) -> exists pfc_value_gcd_unique_H_gcd_divisor_left_productcoefficients. ((((exists ff_h_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientsentry. ff_h_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientsentry + S (pfc_value_gcd_unique_H_gcd_divisor_left_productcoefficients) = S ((S (pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficients)) * pfgu_pc_gcd_unique_H_gcd_divisor_left)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientsentry. pfgu_pb_gcd_unique_H_gcd_divisor_left = ff_q_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientsentry * S ((S (pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficients)) * pfgu_pc_gcd_unique_H_gcd_divisor_left) + (pfc_value_gcd_unique_H_gcd_divisor_left_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficient pfc_terms_scale_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficient pfc_natural_sum_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficient. ((forall pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficients))) -> exists pfc_value_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficient = ff_q_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficient) + (pfc_value_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm pfc_left_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm pfc_right_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonal) = (pfgu_Q_gcd_unique_H_gcd_divisor_left)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_H_gcd_divisor_left)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_gcd_unique_H_gcd_divisor_left = ff_q_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_H_gcd_divisor_left) + (pfc_left_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_gcd_unique_H_gcd_divisor_left)=(pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm) = (pfgu_D_gcd_unique_H_gcd)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)) * pfgu_dc_gcd_unique_H_gcd)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightentry. pfgu_db_gcd_unique_H_gcd = ff_q_pfp_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)) * pfgu_dc_gcd_unique_H_gcd) + (pfc_right_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonaltermrightoutside+(pfgu_D_gcd_unique_H_gcd)=(pfc_complement_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonal)=pfc_left_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum fs_v_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficients))) * fs_v_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficients))) * fs_v_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_unique_H_gcd_divisor_left_productcoefficients)) -> exists fs_a_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficient = fs_q_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficient) + (fs_a_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientresiduebound. pfa_gap_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_unique_H_gcd_divisor_left_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_unique_H_gcd_divisor_left_productcoefficients) + (p) * pfa_offset_right_gcd_unique_H_gcd_divisor_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_unique_H_gcd_divisor_left_target pfrep_left_gcd_unique_H_gcd_divisor_left_target pfrep_right_gcd_unique_H_gcd_divisor_left_target. ((exists pfrep_position_gcd_unique_H_gcd_divisor_left_targetfirst. ((pfrep_position_gcd_unique_H_gcd_divisor_left_targetfirst+S (pfrep_power_gcd_unique_H_gcd_divisor_left_target)=(pfgu_P_gcd_unique_H_gcd_divisor_left)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_divisor_left_targetfirstentry. ff_h_pfp_gcd_unique_H_gcd_divisor_left_targetfirstentry + S (pfrep_left_gcd_unique_H_gcd_divisor_left_target) = S ((S (pfrep_position_gcd_unique_H_gcd_divisor_left_targetfirst)) * pfgu_pc_gcd_unique_H_gcd_divisor_left)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_divisor_left_targetfirstentry. pfgu_pb_gcd_unique_H_gcd_divisor_left = ff_q_pfp_gcd_unique_H_gcd_divisor_left_targetfirstentry * S ((S (pfrep_position_gcd_unique_H_gcd_divisor_left_targetfirst)) * pfgu_pc_gcd_unique_H_gcd_divisor_left) + (pfrep_left_gcd_unique_H_gcd_divisor_left_target)))))) \\/ (((exists pfrep_gap_gcd_unique_H_gcd_divisor_left_targetfirstoutside. pfrep_gap_gcd_unique_H_gcd_divisor_left_targetfirstoutside+(pfgu_P_gcd_unique_H_gcd_divisor_left)=(pfrep_power_gcd_unique_H_gcd_divisor_left_target)) /\\ (((pfrep_left_gcd_unique_H_gcd_divisor_left_target)=0))))) -> ((exists pfrep_position_gcd_unique_H_gcd_divisor_left_targetsecond. ((pfrep_position_gcd_unique_H_gcd_divisor_left_targetsecond+S (pfrep_power_gcd_unique_H_gcd_divisor_left_target)=(L)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_divisor_left_targetsecondentry. ff_h_pfp_gcd_unique_H_gcd_divisor_left_targetsecondentry + S (pfrep_right_gcd_unique_H_gcd_divisor_left_target) = S ((S (pfrep_position_gcd_unique_H_gcd_divisor_left_targetsecond)) * ac)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_divisor_left_targetsecondentry. ab = ff_q_pfp_gcd_unique_H_gcd_divisor_left_targetsecondentry * S ((S (pfrep_position_gcd_unique_H_gcd_divisor_left_targetsecond)) * ac) + (pfrep_right_gcd_unique_H_gcd_divisor_left_target)))))) \\/ (((exists pfrep_gap_gcd_unique_H_gcd_divisor_left_targetsecondoutside. pfrep_gap_gcd_unique_H_gcd_divisor_left_targetsecondoutside+(L)=(pfrep_power_gcd_unique_H_gcd_divisor_left_target)) /\\ (((pfrep_right_gcd_unique_H_gcd_divisor_left_target)=0))))) -> pfrep_left_gcd_unique_H_gcd_divisor_left_target=pfrep_right_gcd_unique_H_gcd_divisor_left_target))))))) /\\ ((((forall fom_index_pfp_gcd_unique_H_gcd_divisor_right_canonical. (exists fom_gap_pfp_gcd_unique_H_gcd_divisor_right_canonical_index_bound. fom_gap_pfp_gcd_unique_H_gcd_divisor_right_canonical_index_bound + S (fom_index_pfp_gcd_unique_H_gcd_divisor_right_canonical) = M) -> exists fom_value_pfp_gcd_unique_H_gcd_divisor_right_canonical. ((((exists fom_beta_height_pfp_gcd_unique_H_gcd_divisor_right_canonical_entry. fom_beta_height_pfp_gcd_unique_H_gcd_divisor_right_canonical_entry + S (fom_value_pfp_gcd_unique_H_gcd_divisor_right_canonical) = S ((S (fom_index_pfp_gcd_unique_H_gcd_divisor_right_canonical)) * bc)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_gcd_divisor_right_canonical_entry. bb = fom_beta_quotient_pfp_gcd_unique_H_gcd_divisor_right_canonical_entry * S ((S (fom_index_pfp_gcd_unique_H_gcd_divisor_right_canonical)) * bc) + (fom_value_pfp_gcd_unique_H_gcd_divisor_right_canonical))) /\\ (exists fom_gap_pfp_gcd_unique_H_gcd_divisor_right_canonical_value_bound. fom_gap_pfp_gcd_unique_H_gcd_divisor_right_canonical_value_bound + S (fom_value_pfp_gcd_unique_H_gcd_divisor_right_canonical) = p))) /\\ ((exists pfgu_qb_gcd_unique_H_gcd_divisor_right pfgu_qc_gcd_unique_H_gcd_divisor_right pfgu_Q_gcd_unique_H_gcd_divisor_right pfgu_pb_gcd_unique_H_gcd_divisor_right pfgu_pc_gcd_unique_H_gcd_divisor_right pfgu_P_gcd_unique_H_gcd_divisor_right. ((((forall fom_index_pfp_gcd_unique_H_gcd_divisor_right_productleft. (exists fom_gap_pfp_gcd_unique_H_gcd_divisor_right_productleft_index_bound. fom_gap_pfp_gcd_unique_H_gcd_divisor_right_productleft_index_bound + S (fom_index_pfp_gcd_unique_H_gcd_divisor_right_productleft) = pfgu_Q_gcd_unique_H_gcd_divisor_right) -> exists fom_value_pfp_gcd_unique_H_gcd_divisor_right_productleft. ((((exists fom_beta_height_pfp_gcd_unique_H_gcd_divisor_right_productleft_entry. fom_beta_height_pfp_gcd_unique_H_gcd_divisor_right_productleft_entry + S (fom_value_pfp_gcd_unique_H_gcd_divisor_right_productleft) = S ((S (fom_index_pfp_gcd_unique_H_gcd_divisor_right_productleft)) * pfgu_qc_gcd_unique_H_gcd_divisor_right)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_gcd_divisor_right_productleft_entry. pfgu_qb_gcd_unique_H_gcd_divisor_right = fom_beta_quotient_pfp_gcd_unique_H_gcd_divisor_right_productleft_entry * S ((S (fom_index_pfp_gcd_unique_H_gcd_divisor_right_productleft)) * pfgu_qc_gcd_unique_H_gcd_divisor_right) + (fom_value_pfp_gcd_unique_H_gcd_divisor_right_productleft))) /\\ (exists fom_gap_pfp_gcd_unique_H_gcd_divisor_right_productleft_value_bound. fom_gap_pfp_gcd_unique_H_gcd_divisor_right_productleft_value_bound + S (fom_value_pfp_gcd_unique_H_gcd_divisor_right_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_unique_H_gcd_divisor_right_productright. (exists fom_gap_pfp_gcd_unique_H_gcd_divisor_right_productright_index_bound. fom_gap_pfp_gcd_unique_H_gcd_divisor_right_productright_index_bound + S (fom_index_pfp_gcd_unique_H_gcd_divisor_right_productright) = pfgu_D_gcd_unique_H_gcd) -> exists fom_value_pfp_gcd_unique_H_gcd_divisor_right_productright. ((((exists fom_beta_height_pfp_gcd_unique_H_gcd_divisor_right_productright_entry. fom_beta_height_pfp_gcd_unique_H_gcd_divisor_right_productright_entry + S (fom_value_pfp_gcd_unique_H_gcd_divisor_right_productright) = S ((S (fom_index_pfp_gcd_unique_H_gcd_divisor_right_productright)) * pfgu_dc_gcd_unique_H_gcd)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_gcd_divisor_right_productright_entry. pfgu_db_gcd_unique_H_gcd = fom_beta_quotient_pfp_gcd_unique_H_gcd_divisor_right_productright_entry * S ((S (fom_index_pfp_gcd_unique_H_gcd_divisor_right_productright)) * pfgu_dc_gcd_unique_H_gcd) + (fom_value_pfp_gcd_unique_H_gcd_divisor_right_productright))) /\\ (exists fom_gap_pfp_gcd_unique_H_gcd_divisor_right_productright_value_bound. fom_gap_pfp_gcd_unique_H_gcd_divisor_right_productright_value_bound + S (fom_value_pfp_gcd_unique_H_gcd_divisor_right_productright) = p))) /\\ (((((((pfgu_Q_gcd_unique_H_gcd_divisor_right)=0 \\/ (pfgu_D_gcd_unique_H_gcd)=0) /\\ (((pfgu_P_gcd_unique_H_gcd_divisor_right)=0)))) \\/ (((~((pfgu_Q_gcd_unique_H_gcd_divisor_right)=0)) /\\ (((~((pfgu_D_gcd_unique_H_gcd)=0)) /\\ (((pfgu_Q_gcd_unique_H_gcd_divisor_right)+(pfgu_D_gcd_unique_H_gcd)=S (pfgu_P_gcd_unique_H_gcd_divisor_right)))))))) /\\ ((forall pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficients. (exists pfa_gap_gcd_unique_H_gcd_divisor_right_productcoefficientsbound. pfa_gap_gcd_unique_H_gcd_divisor_right_productcoefficientsbound + S (pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficients) = (pfgu_P_gcd_unique_H_gcd_divisor_right)) -> exists pfc_value_gcd_unique_H_gcd_divisor_right_productcoefficients. ((((exists ff_h_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientsentry. ff_h_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientsentry + S (pfc_value_gcd_unique_H_gcd_divisor_right_productcoefficients) = S ((S (pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficients)) * pfgu_pc_gcd_unique_H_gcd_divisor_right)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientsentry. pfgu_pb_gcd_unique_H_gcd_divisor_right = ff_q_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientsentry * S ((S (pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficients)) * pfgu_pc_gcd_unique_H_gcd_divisor_right) + (pfc_value_gcd_unique_H_gcd_divisor_right_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficient pfc_terms_scale_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficient pfc_natural_sum_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficient. ((forall pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficients))) -> exists pfc_value_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficient = ff_q_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficient) + (pfc_value_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm pfc_left_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm pfc_right_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonal) = (pfgu_Q_gcd_unique_H_gcd_divisor_right)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_H_gcd_divisor_right)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_gcd_unique_H_gcd_divisor_right = ff_q_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_H_gcd_divisor_right) + (pfc_left_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_gcd_unique_H_gcd_divisor_right)=(pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm) = (pfgu_D_gcd_unique_H_gcd)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)) * pfgu_dc_gcd_unique_H_gcd)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightentry. pfgu_db_gcd_unique_H_gcd = ff_q_pfp_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)) * pfgu_dc_gcd_unique_H_gcd) + (pfc_right_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonaltermrightoutside+(pfgu_D_gcd_unique_H_gcd)=(pfc_complement_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonal)=pfc_left_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum fs_v_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficients))) * fs_v_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficients))) * fs_v_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_unique_H_gcd_divisor_right_productcoefficients)) -> exists fs_a_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficient = fs_q_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficient) + (fs_a_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientresiduebound. pfa_gap_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_unique_H_gcd_divisor_right_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_unique_H_gcd_divisor_right_productcoefficients) + (p) * pfa_offset_right_gcd_unique_H_gcd_divisor_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_unique_H_gcd_divisor_right_target pfrep_left_gcd_unique_H_gcd_divisor_right_target pfrep_right_gcd_unique_H_gcd_divisor_right_target. ((exists pfrep_position_gcd_unique_H_gcd_divisor_right_targetfirst. ((pfrep_position_gcd_unique_H_gcd_divisor_right_targetfirst+S (pfrep_power_gcd_unique_H_gcd_divisor_right_target)=(pfgu_P_gcd_unique_H_gcd_divisor_right)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_divisor_right_targetfirstentry. ff_h_pfp_gcd_unique_H_gcd_divisor_right_targetfirstentry + S (pfrep_left_gcd_unique_H_gcd_divisor_right_target) = S ((S (pfrep_position_gcd_unique_H_gcd_divisor_right_targetfirst)) * pfgu_pc_gcd_unique_H_gcd_divisor_right)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_divisor_right_targetfirstentry. pfgu_pb_gcd_unique_H_gcd_divisor_right = ff_q_pfp_gcd_unique_H_gcd_divisor_right_targetfirstentry * S ((S (pfrep_position_gcd_unique_H_gcd_divisor_right_targetfirst)) * pfgu_pc_gcd_unique_H_gcd_divisor_right) + (pfrep_left_gcd_unique_H_gcd_divisor_right_target)))))) \\/ (((exists pfrep_gap_gcd_unique_H_gcd_divisor_right_targetfirstoutside. pfrep_gap_gcd_unique_H_gcd_divisor_right_targetfirstoutside+(pfgu_P_gcd_unique_H_gcd_divisor_right)=(pfrep_power_gcd_unique_H_gcd_divisor_right_target)) /\\ (((pfrep_left_gcd_unique_H_gcd_divisor_right_target)=0))))) -> ((exists pfrep_position_gcd_unique_H_gcd_divisor_right_targetsecond. ((pfrep_position_gcd_unique_H_gcd_divisor_right_targetsecond+S (pfrep_power_gcd_unique_H_gcd_divisor_right_target)=(M)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_divisor_right_targetsecondentry. ff_h_pfp_gcd_unique_H_gcd_divisor_right_targetsecondentry + S (pfrep_right_gcd_unique_H_gcd_divisor_right_target) = S ((S (pfrep_position_gcd_unique_H_gcd_divisor_right_targetsecond)) * bc)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_divisor_right_targetsecondentry. bb = ff_q_pfp_gcd_unique_H_gcd_divisor_right_targetsecondentry * S ((S (pfrep_position_gcd_unique_H_gcd_divisor_right_targetsecond)) * bc) + (pfrep_right_gcd_unique_H_gcd_divisor_right_target)))))) \\/ (((exists pfrep_gap_gcd_unique_H_gcd_divisor_right_targetsecondoutside. pfrep_gap_gcd_unique_H_gcd_divisor_right_targetsecondoutside+(M)=(pfrep_power_gcd_unique_H_gcd_divisor_right_target)) /\\ (((pfrep_right_gcd_unique_H_gcd_divisor_right_target)=0))))) -> pfrep_left_gcd_unique_H_gcd_divisor_right_target=pfrep_right_gcd_unique_H_gcd_divisor_right_target)))))))))) -> (((forall fom_index_pfp_gcd_unique_H_gcd_greatest_canonical. (exists fom_gap_pfp_gcd_unique_H_gcd_greatest_canonical_index_bound. fom_gap_pfp_gcd_unique_H_gcd_greatest_canonical_index_bound + S (fom_index_pfp_gcd_unique_H_gcd_greatest_canonical) = H) -> exists fom_value_pfp_gcd_unique_H_gcd_greatest_canonical. ((((exists fom_beta_height_pfp_gcd_unique_H_gcd_greatest_canonical_entry. fom_beta_height_pfp_gcd_unique_H_gcd_greatest_canonical_entry + S (fom_value_pfp_gcd_unique_H_gcd_greatest_canonical) = S ((S (fom_index_pfp_gcd_unique_H_gcd_greatest_canonical)) * hc)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_gcd_greatest_canonical_entry. hb = fom_beta_quotient_pfp_gcd_unique_H_gcd_greatest_canonical_entry * S ((S (fom_index_pfp_gcd_unique_H_gcd_greatest_canonical)) * hc) + (fom_value_pfp_gcd_unique_H_gcd_greatest_canonical))) /\\ (exists fom_gap_pfp_gcd_unique_H_gcd_greatest_canonical_value_bound. fom_gap_pfp_gcd_unique_H_gcd_greatest_canonical_value_bound + S (fom_value_pfp_gcd_unique_H_gcd_greatest_canonical) = p))) /\\ ((exists pfgu_qb_gcd_unique_H_gcd_greatest pfgu_qc_gcd_unique_H_gcd_greatest pfgu_Q_gcd_unique_H_gcd_greatest pfgu_pb_gcd_unique_H_gcd_greatest pfgu_pc_gcd_unique_H_gcd_greatest pfgu_P_gcd_unique_H_gcd_greatest. ((((forall fom_index_pfp_gcd_unique_H_gcd_greatest_productleft. (exists fom_gap_pfp_gcd_unique_H_gcd_greatest_productleft_index_bound. fom_gap_pfp_gcd_unique_H_gcd_greatest_productleft_index_bound + S (fom_index_pfp_gcd_unique_H_gcd_greatest_productleft) = pfgu_Q_gcd_unique_H_gcd_greatest) -> exists fom_value_pfp_gcd_unique_H_gcd_greatest_productleft. ((((exists fom_beta_height_pfp_gcd_unique_H_gcd_greatest_productleft_entry. fom_beta_height_pfp_gcd_unique_H_gcd_greatest_productleft_entry + S (fom_value_pfp_gcd_unique_H_gcd_greatest_productleft) = S ((S (fom_index_pfp_gcd_unique_H_gcd_greatest_productleft)) * pfgu_qc_gcd_unique_H_gcd_greatest)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_gcd_greatest_productleft_entry. pfgu_qb_gcd_unique_H_gcd_greatest = fom_beta_quotient_pfp_gcd_unique_H_gcd_greatest_productleft_entry * S ((S (fom_index_pfp_gcd_unique_H_gcd_greatest_productleft)) * pfgu_qc_gcd_unique_H_gcd_greatest) + (fom_value_pfp_gcd_unique_H_gcd_greatest_productleft))) /\\ (exists fom_gap_pfp_gcd_unique_H_gcd_greatest_productleft_value_bound. fom_gap_pfp_gcd_unique_H_gcd_greatest_productleft_value_bound + S (fom_value_pfp_gcd_unique_H_gcd_greatest_productleft) = p))) /\\ (((forall fom_index_pfp_gcd_unique_H_gcd_greatest_productright. (exists fom_gap_pfp_gcd_unique_H_gcd_greatest_productright_index_bound. fom_gap_pfp_gcd_unique_H_gcd_greatest_productright_index_bound + S (fom_index_pfp_gcd_unique_H_gcd_greatest_productright) = pfgu_D_gcd_unique_H_gcd) -> exists fom_value_pfp_gcd_unique_H_gcd_greatest_productright. ((((exists fom_beta_height_pfp_gcd_unique_H_gcd_greatest_productright_entry. fom_beta_height_pfp_gcd_unique_H_gcd_greatest_productright_entry + S (fom_value_pfp_gcd_unique_H_gcd_greatest_productright) = S ((S (fom_index_pfp_gcd_unique_H_gcd_greatest_productright)) * pfgu_dc_gcd_unique_H_gcd)) /\\ exists fom_beta_quotient_pfp_gcd_unique_H_gcd_greatest_productright_entry. pfgu_db_gcd_unique_H_gcd = fom_beta_quotient_pfp_gcd_unique_H_gcd_greatest_productright_entry * S ((S (fom_index_pfp_gcd_unique_H_gcd_greatest_productright)) * pfgu_dc_gcd_unique_H_gcd) + (fom_value_pfp_gcd_unique_H_gcd_greatest_productright))) /\\ (exists fom_gap_pfp_gcd_unique_H_gcd_greatest_productright_value_bound. fom_gap_pfp_gcd_unique_H_gcd_greatest_productright_value_bound + S (fom_value_pfp_gcd_unique_H_gcd_greatest_productright) = p))) /\\ (((((((pfgu_Q_gcd_unique_H_gcd_greatest)=0 \\/ (pfgu_D_gcd_unique_H_gcd)=0) /\\ (((pfgu_P_gcd_unique_H_gcd_greatest)=0)))) \\/ (((~((pfgu_Q_gcd_unique_H_gcd_greatest)=0)) /\\ (((~((pfgu_D_gcd_unique_H_gcd)=0)) /\\ (((pfgu_Q_gcd_unique_H_gcd_greatest)+(pfgu_D_gcd_unique_H_gcd)=S (pfgu_P_gcd_unique_H_gcd_greatest)))))))) /\\ ((forall pfc_index_gcd_unique_H_gcd_greatest_productcoefficients. (exists pfa_gap_gcd_unique_H_gcd_greatest_productcoefficientsbound. pfa_gap_gcd_unique_H_gcd_greatest_productcoefficientsbound + S (pfc_index_gcd_unique_H_gcd_greatest_productcoefficients) = (pfgu_P_gcd_unique_H_gcd_greatest)) -> exists pfc_value_gcd_unique_H_gcd_greatest_productcoefficients. ((((exists ff_h_pfp_gcd_unique_H_gcd_greatest_productcoefficientsentry. ff_h_pfp_gcd_unique_H_gcd_greatest_productcoefficientsentry + S (pfc_value_gcd_unique_H_gcd_greatest_productcoefficients) = S ((S (pfc_index_gcd_unique_H_gcd_greatest_productcoefficients)) * pfgu_pc_gcd_unique_H_gcd_greatest)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_greatest_productcoefficientsentry. pfgu_pb_gcd_unique_H_gcd_greatest = ff_q_pfp_gcd_unique_H_gcd_greatest_productcoefficientsentry * S ((S (pfc_index_gcd_unique_H_gcd_greatest_productcoefficients)) * pfgu_pc_gcd_unique_H_gcd_greatest) + (pfc_value_gcd_unique_H_gcd_greatest_productcoefficients))) /\\ ((exists pfc_terms_code_gcd_unique_H_gcd_greatest_productcoefficientscoefficient pfc_terms_scale_gcd_unique_H_gcd_greatest_productcoefficientscoefficient pfc_natural_sum_gcd_unique_H_gcd_greatest_productcoefficientscoefficient. ((forall pfc_index_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonal. (exists pfa_gap_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalbound. pfa_gap_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalbound + S (pfc_index_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonal) = (S (pfc_index_gcd_unique_H_gcd_greatest_productcoefficients))) -> exists pfc_value_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalentry. ff_h_pfp_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalentry + S (pfc_value_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_H_gcd_greatest_productcoefficientscoefficient)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalentry. pfc_terms_code_gcd_unique_H_gcd_greatest_productcoefficientscoefficient = ff_q_pfp_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_gcd_unique_H_gcd_greatest_productcoefficientscoefficient) + (pfc_value_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonal))) /\\ ((exists pfc_complement_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm pfc_left_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm pfc_right_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm. (((pfc_index_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonal)+pfc_complement_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm=(pfc_index_gcd_unique_H_gcd_greatest_productcoefficients)) /\\ ((((((exists pfa_gap_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonal) = (pfgu_Q_gcd_unique_H_gcd_greatest)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_H_gcd_greatest)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermleftentry. pfgu_qb_gcd_unique_H_gcd_greatest = ff_q_pfp_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonal)) * pfgu_qc_gcd_unique_H_gcd_greatest) + (pfc_left_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermleftoutside+(pfgu_Q_gcd_unique_H_gcd_greatest)=(pfc_index_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonal)) /\\ (((pfc_left_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm)=0))))) /\\ ((((((exists pfa_gap_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm) = (pfgu_D_gcd_unique_H_gcd)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm)) * pfgu_dc_gcd_unique_H_gcd)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermrightentry. pfgu_db_gcd_unique_H_gcd = ff_q_pfp_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm)) * pfgu_dc_gcd_unique_H_gcd) + (pfc_right_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm)))))) \\/ (((exists pfc_gap_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonaltermrightoutside+(pfgu_D_gcd_unique_H_gcd)=(pfc_complement_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm)) /\\ (((pfc_right_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm)=0))))) /\\ (((pfc_value_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonal)=pfc_left_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm*pfc_right_gcd_unique_H_gcd_greatest_productcoefficientscoefficientdiagonalterm))))))))))) /\\ (((exists fs_u_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum fs_v_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum. ((((exists fs_h_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_start. fs_h_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_start. fs_u_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum) + (0))) /\\ ((((exists fs_h_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_gcd_unique_H_gcd_greatest_productcoefficientscoefficient) = S ((S (S (pfc_index_gcd_unique_H_gcd_greatest_productcoefficients))) * fs_v_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_gcd_unique_H_gcd_greatest_productcoefficients))) * fs_v_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum) + (pfc_natural_sum_gcd_unique_H_gcd_greatest_productcoefficientscoefficient))) /\\ forall fs_i_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps = S (pfc_index_gcd_unique_H_gcd_greatest_productcoefficients)) -> exists fs_a_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps fs_r_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps fs_s_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_H_gcd_greatest_productcoefficientscoefficient)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_gcd_unique_H_gcd_greatest_productcoefficientscoefficient = fs_q_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_gcd_unique_H_gcd_greatest_productcoefficientscoefficient) + (fs_a_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum) + (fs_r_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps))) /\\ ((((exists fs_h_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum)) /\\ exists fs_q_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum = fs_q_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum) + (fs_s_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps))) /\\ fs_s_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps = fs_r_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps + fs_a_pfc_gcd_unique_H_gcd_greatest_productcoefficientscoefficientsum_body_steps)))))) /\\ ((((exists pfa_gap_gcd_unique_H_gcd_greatest_productcoefficientscoefficientresiduebound. pfa_gap_gcd_unique_H_gcd_greatest_productcoefficientscoefficientresiduebound + S (pfc_value_gcd_unique_H_gcd_greatest_productcoefficients) = (p)) /\\ ((exists pfa_offset_left_gcd_unique_H_gcd_greatest_productcoefficientscoefficientresiduecongruence pfa_offset_right_gcd_unique_H_gcd_greatest_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_gcd_unique_H_gcd_greatest_productcoefficientscoefficient) + (p) * pfa_offset_left_gcd_unique_H_gcd_greatest_productcoefficientscoefficientresiduecongruence = (pfc_value_gcd_unique_H_gcd_greatest_productcoefficients) + (p) * pfa_offset_right_gcd_unique_H_gcd_greatest_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\\ ((forall pfrep_power_gcd_unique_H_gcd_greatest_target pfrep_left_gcd_unique_H_gcd_greatest_target pfrep_right_gcd_unique_H_gcd_greatest_target. ((exists pfrep_position_gcd_unique_H_gcd_greatest_targetfirst. ((pfrep_position_gcd_unique_H_gcd_greatest_targetfirst+S (pfrep_power_gcd_unique_H_gcd_greatest_target)=(pfgu_P_gcd_unique_H_gcd_greatest)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_greatest_targetfirstentry. ff_h_pfp_gcd_unique_H_gcd_greatest_targetfirstentry + S (pfrep_left_gcd_unique_H_gcd_greatest_target) = S ((S (pfrep_position_gcd_unique_H_gcd_greatest_targetfirst)) * pfgu_pc_gcd_unique_H_gcd_greatest)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_greatest_targetfirstentry. pfgu_pb_gcd_unique_H_gcd_greatest = ff_q_pfp_gcd_unique_H_gcd_greatest_targetfirstentry * S ((S (pfrep_position_gcd_unique_H_gcd_greatest_targetfirst)) * pfgu_pc_gcd_unique_H_gcd_greatest) + (pfrep_left_gcd_unique_H_gcd_greatest_target)))))) \\/ (((exists pfrep_gap_gcd_unique_H_gcd_greatest_targetfirstoutside. pfrep_gap_gcd_unique_H_gcd_greatest_targetfirstoutside+(pfgu_P_gcd_unique_H_gcd_greatest)=(pfrep_power_gcd_unique_H_gcd_greatest_target)) /\\ (((pfrep_left_gcd_unique_H_gcd_greatest_target)=0))))) -> ((exists pfrep_position_gcd_unique_H_gcd_greatest_targetsecond. ((pfrep_position_gcd_unique_H_gcd_greatest_targetsecond+S (pfrep_power_gcd_unique_H_gcd_greatest_target)=(H)) /\\ ((((exists ff_h_pfp_gcd_unique_H_gcd_greatest_targetsecondentry. ff_h_pfp_gcd_unique_H_gcd_greatest_targetsecondentry + S (pfrep_right_gcd_unique_H_gcd_greatest_target) = S ((S (pfrep_position_gcd_unique_H_gcd_greatest_targetsecond)) * hc)) /\\ exists ff_q_pfp_gcd_unique_H_gcd_greatest_targetsecondentry. hb = ff_q_pfp_gcd_unique_H_gcd_greatest_targetsecondentry * S ((S (pfrep_position_gcd_unique_H_gcd_greatest_targetsecond)) * hc) + (pfrep_right_gcd_unique_H_gcd_greatest_target)))))) \\/ (((exists pfrep_gap_gcd_unique_H_gcd_greatest_targetsecondoutside. pfrep_gap_gcd_unique_H_gcd_greatest_targetsecondoutside+(H)=(pfrep_power_gcd_unique_H_gcd_greatest_target)) /\\ (((pfrep_right_gcd_unique_H_gcd_greatest_target)=0))))) -> pfrep_left_gcd_unique_H_gcd_greatest_target=pfrep_right_gcd_unique_H_gcd_greatest_target)))))))))))))) -> (forall pfrep_power_gcd_unique_result pfrep_left_gcd_unique_result pfrep_right_gcd_unique_result. ((exists pfrep_position_gcd_unique_resultfirst. ((pfrep_position_gcd_unique_resultfirst+S (pfrep_power_gcd_unique_result)=(G)) /\\ ((((exists ff_h_pfp_gcd_unique_resultfirstentry. ff_h_pfp_gcd_unique_resultfirstentry + S (pfrep_left_gcd_unique_result) = S ((S (pfrep_position_gcd_unique_resultfirst)) * gc)) /\\ exists ff_q_pfp_gcd_unique_resultfirstentry. gb = ff_q_pfp_gcd_unique_resultfirstentry * S ((S (pfrep_position_gcd_unique_resultfirst)) * gc) + (pfrep_left_gcd_unique_result)))))) \\/ (((exists pfrep_gap_gcd_unique_resultfirstoutside. pfrep_gap_gcd_unique_resultfirstoutside+(G)=(pfrep_power_gcd_unique_result)) /\\ (((pfrep_left_gcd_unique_result)=0))))) -> ((exists pfrep_position_gcd_unique_resultsecond. ((pfrep_position_gcd_unique_resultsecond+S (pfrep_power_gcd_unique_result)=(H)) /\\ ((((exists ff_h_pfp_gcd_unique_resultsecondentry. ff_h_pfp_gcd_unique_resultsecondentry + S (pfrep_right_gcd_unique_result) = S ((S (pfrep_position_gcd_unique_resultsecond)) * hc)) /\\ exists ff_q_pfp_gcd_unique_resultsecondentry. hb = ff_q_pfp_gcd_unique_resultsecondentry * S ((S (pfrep_position_gcd_unique_resultsecond)) * hc) + (pfrep_right_gcd_unique_result)))))) \\/ (((exists pfrep_gap_gcd_unique_resultsecondoutside. pfrep_gap_gcd_unique_resultsecondoutside+(H)=(pfrep_power_gcd_unique_result)) /\\ (((pfrep_right_gcd_unique_result)=0))))) -> pfrep_left_gcd_unique_result=pfrep_right_gcd_unique_result)", "statement_sha256": "302df17d7792e85eb95dc25ff3b82ef61c84f67da66a886c1ef383f1115ef7a7", "summary": "The grouped normal/common-divisor/greatestness graph yields mutual right associates, hence uniqueness only up to formal coefficient equivalence. This does not assert unique beta codes or Bezout coefficients.", "summary_sha256": "3b67ff92805ee961660c402a09a8b32ee959e10d3c0d871893a5bf26ef777c0d" } ]